0$ .","The model problem (REF )-() arises from many scientific applications such as fluid flow in porous media.","Mostly importantly, this model problem has served, and still serves, the scientific computing community as a testbed in the search and design of new and efficient computational algorithms for partial differential equations.","The classical Galerkin finite element method (see, e.g., [10], [28], [16]) is particularly a numerical technique originated from the study of elliptic problems closed related to (REF )-() or its variations.","In the last three decades, various finite element methods using discontinuous trial and test functions, including discontinuous Galerkin (DG) methods and weak Galerkin (WG) methods, have been developed for numerical solutions of partial differential equations.","These developments were often tested over testbed problems such as (REF )-() before they were generalized or applied to more complex problems in science and engineering.","The DG method, also known as the interior penalty method in different contexts, was originated in early 70s of the last century for a numerical study of model problems such as (REF )-(); see, e.g., [3], [14], [25], [38] for early incubations and [1], [13], [17], [27] for a detailed discussion and recent developments.","The weak Galerkin finite element method is a recently developed discretization framework for partial differential equations [36], [37], [24], [34].","With new concepts referred to as weak differential operators (e.g., weak gradient, weak curl, weak Laplacian etc.)","and weak continuity through the use of various stabilizers, the method allows the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent or free of locking [33].","For the convection-diffusion-reaction equation (REF )-(), the recent work in the context of weak Galerkin includes the algorithm developed and analyzed in [9], the one in [20] for singularly perturbed problems, and an earlier one in [39].","The WG finite element method has been rapidly developed and applied to several different types of problems, including second order elliptic problems, the Stokes and Navier-Stokes equations, the biharmonic and elasticity equations, div-curl systems and the Maxwell's equations, etc.","The latest development of the WG methods is the prime-dual formulation for problems that are either nonsymmetric or do not have variational forms friendly for numerical use.","Details on the new developments can be found in [30] for second order elliptic equations in nondivergence form, [31] for the Fokker-Planck equation, and [32] for elliptic Cauchy problems.","The typical WG method for the model problem (REF )-() seeks weak finite element approximations $u_h=\\lbrace u_0, u_b\\rbrace $ satisfying $u_b|_{\\partial \\Omega } = Q_b g$ and $S(u_h, v)+ (\\alpha \\nabla _w u_h, \\nabla _w v) + ({\\beta }\\cdot \\nabla _w u_h, v_0) + (c u_0, v_0) = (f,v_0)$ for all test functions $v=\\lbrace v_0, v_b\\rbrace $ satisfying $v_b|_{\\partial \\Omega }=0$ , where $Q_bg$ is an interpolation of the Dirichlet boundary data, $\\nabla _w$ is the discrete weak gradient operator, and $S(\\cdot ,\\cdot )$ is a properly selected stabilizer that gives weak continuities for the numerical solutions.","The numerical solution $u_h$ consists of two components: the approximation $u_0$ on each element and the approximation $u_b$ on the boundary of each element.","To reduce the computational complexity, some hybridized formulations have been introduced in [22], [29] for the method when applied to the diffusion equation and the biharmonic equation through the elimination of the degrees of freedom associated with the unknown function $u_0$ locally on each element.","In the superconvergence study for WG [18] on rectangular elements, this hybridized formulation was further simplified in the description of the numerical algorithm, yielding a simplified weak Galerkin (SWG) finite element scheme for the diffusion equation.","In our further investigation of the SWG to the convection-diffusion-reaction equation (REF ), we came to the conclusion that SWG represents a new discretization scheme that is different from the usual WG through a simple elimination of the unknown $u_0$ .","As a result, we believe that a systematic study of the SWG for the convection-diffusion-reaction problem (REF )-() should be conducted for its stability and convergence.","This paper is in response to this observation and shall provide a mathematical theory for the stability and the convergence of the simplified weak Galerkin finite element method for the model problem (REF )-().","We believe that the result of this paper can be extended to other types of modeling equations.","The paper is organized as follows: In Section , we shall describe the simplified weak Galerkin finite element method for (REF )-() on general polygonal partitions.","In Section , we shall present a computational formula for the element stiffness matrices and the element load vectors from SWG.","In Section , we provide a mathematical theory for the stability and well-posedness of the SWG scheme.","Sections and are devoted to a discussion of the error estimates in a discrete $H^1$ and the $L^2$ norm for the numerical solutions.","Finally, in Section , we present some numerical results to demonstrate the efficiency and accuracy of the SWG method.","Throughout the rest of the paper, we assume $d=2$ and shall use the standard notations for Sobolev spaces and norms [10], [16].","For any open set $D\\subset \\mathbb {R}^{2}$ , $\\Vert \\cdot \\Vert _{s,D}$ and $(\\cdot ,\\cdot )_{s,D}$ denote the norm and inner-product in the Sobolev space $H^s(D)$ consisting of square integrable partial derivatives up to order $s$ .","When $s=0$ or $D=\\Omega $ , we shall drop the corresponding subscripts in the norm and inner-product notation."],["Algorithm on Polymesh","Assume that the domain is of polygonal type and is partitioned into non-overlap polygons ${\\mathcal {T}}_h=\\lbrace T\\rbrace $ that are shape regular.","For each $T\\in {\\mathcal {T}}_h$ , denote by $h_T$ its diameter and by $N$ the number of edges.","For each edge $e_i, \\ i=1,\\ldots , N$ , denote by $M_i$ the midpoints and ${\\bf n}_i$ the outward normal direction of $e_i$ (see Fig.","REF for an illuatration).","The meshsize of ${\\mathcal {T}}_h$ is defined as $h=\\max _{T\\in {\\mathcal {T}}_h} h_T$ .","Let $v_b$ be a piecewise constant function defined on the boundary of $T$ , i.e., $v_b|_{e_i} =v_{b,i},$ with $v_{b,i}$ being a constant.","We define the weak gradient of $v_b$ on $T$ by: $\\nabla _w v_b:=\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\bf {n_i},$ where $|e_i|$ is the length of the edge $e_i$ and $|T|$ is the area of the element $T$ .","It is not hard to see that the weak gradient $\\nabla _w v_b$ satisfies the following equation: $(\\nabla _w v_b, \\mathbf {\\phi })_T=\\langle v_b, \\mathbf {\\phi }\\cdot {\\bf n}\\rangle _{\\partial T}$ for all constant vector $\\mathbf {\\phi }$ .","Here and in what follows of the paper, $\\langle \\cdot ,\\cdot \\rangle _{\\partial T}$ stands for the usual inner product in $L^2({\\partial T})$ .","Denote by $W(T)$ the space of piecewise constant functions on ${\\partial T}$ .","The global finite element space $W({\\mathcal {T}}_h)$ is constructed by patching together all the local elements $W(T)$ through single values on interior edges.","The subspace of $W_h({\\mathcal {T}}_h)$ consisting of functions with vanishing boundary value is denoted as $W_h^0({\\mathcal {T}}_h)$ .","We use the conventional notation of ${P}_j(T)$ for the space of polynomials of degree $j\\ge 0$ on $T$ .","For each $v_b\\in W(T)$ , we associate it with a linear extension in $T$ , denoted as ${\\mathfrak {s}}(v_b)\\in {P}_1 (T)$ , satisfying $\\sum _{i=1}^{N}({\\mathfrak {s}}(v_b)(M_i) -v_{b,i})\\phi (M_i)|e_i|=0,\\quad \\forall \\; \\phi \\in {P}_1(T).$ It is easy to see that ${\\mathfrak {s}}(u_b)$ is well defined by (REF ), and its computation is local and straightforward.","In fact, ${\\mathfrak {s}}(u_b)$ can be viewed as an extension of $u_b$ from $\\partial T$ to $T$ through a least-squares fitting.","Figure: An illustrative polygonal element.On each element $T \\in {\\mathcal {T}}_h $ , we introduce the following bilinear forms: $a_T(u_b,v_b)&:= & (\\alpha \\nabla _w u_b, \\nabla _w v_b)_T, \\\\b_T(u_b,v_b)&:= & ({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T,\\\\c_T(u_b,v_b)&:= & (c{\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ For simplicity, we set $\\mathcal {B}_T(u_b,v_b):=a_T(u_b,v_b) + b_T(u_b,v_b) + c_T(u_b,v_b)$ for $u_b, v_b \\in W(T)$ .","We further introduce the stabilizer $\\begin{split}S_T(u_b,v_b):= & h^{-1}\\sum _{i=1}^N ({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\= & h^{-1}\\langle Q_b{\\mathfrak {s}}(u_b)-u_{b},Q_b{\\mathfrak {s}}(v_b)-v_{b}\\rangle _{\\partial T},\\end{split}$ where $Q_b$ is the $L^2$ projection operator onto $W(T)$ ; namely $Q_b u$ is the average of $u$ on each edge.","In particular, $Q_b(g)$ is well-defined and takes the average of the Dirichlet data on each boundary edge.","SWG Algorithm 2.1 The simplified weak Galerkin (SWG) scheme for the elliptic equation (REF )-() seeks $u_b \\in W_h({\\mathcal {T}}_h)$ satisfying $u_b = Q_b(g)$ on $\\partial \\Omega $ and ${\\mathcal {A}}(u_b, v_b) =(f, {\\mathfrak {s}}(v_b))\\qquad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where ${\\mathcal {A}}(u_b,v_b):=\\kappa S(u_b,v_b)+ {\\mathcal {B}}(u_b, v_b)$ , $S(u_b,v_b) &=& \\sum _{T\\in {\\mathcal {T}}_h} S_T(u_b,v_b),\\\\{\\mathcal {B}}(u_b,v_b)&=& \\sum _{T\\in {\\mathcal {T}}_h} \\mathcal {B}_T(u_b, v_b)$ are bilinear forms in $W_h({\\mathcal {T}}_h)$ and $(f,{\\mathfrak {s}}(v_b)):=\\sum _{T\\in {\\mathcal {T}}_h} (f, {\\mathfrak {s}}(v_b))_T$ is a linear form in $W_h({\\mathcal {T}}_h)$ ."],["Element Stiffness Matrices","The simplified weak Galerkin finite element method (REF ) is user-friendly in computer implementation.","In this section, we present a formula for the computation of the element stiffness matrices and the element load vector on general polygonal elements.","Let $T\\in {\\mathcal {T}}_h$ be a polygonal element of $N$ sides.","Denote by $X_{u_b}$ the vector representation of $u_b$ given by $(u_{b,1},u_{b,2},\\ldots ,u_{b,N})^T$ .","Then, the element stiffness matrix and the element load vector for the SWG scheme (REF ) are given in a block matrix form as follows: $(\\kappa h^{-1}A^T + B + R + C)X_{u_b} \\cong F,$ where the block components in (REF ) are given by: (1) $A:=\\lbrace a_{i,j}\\rbrace _{i,j=1}^N = E- EM(M^TEM)^{-1}M^TE$ , (2) $B:=\\lbrace b_{i,j}\\rbrace _{i,j=1}^N$ , with $b_{i,j} = (\\alpha \\mathbf {n}_i,\\mathbf {n}_j)_{T}\\displaystyle \\frac{|e_i||e_j|}{|T|^2}$ , (3) $R:=\\lbrace r_{ij}\\rbrace _{i,j=1}^N$ , with $r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT$ , (4) $C:=\\lbrace c_{ij}\\rbrace _{i,j=1}^N$ , with $c_{ij}= \\int _T c\\zeta _j\\zeta _i dT$ , (5) $F:=\\lbrace f_{i}\\rbrace _{i=1}^N$ , with $f_{i} = \\int _{T}f(x,y) \\zeta _i(x,y) dT$ , (6) $D:=\\lbrace d_{j,i}\\rbrace _{3\\times N}=(M^TEM)^{-1}M^TE$ and $\\zeta _i = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T)$ , (7) $M$ and $E$ are given by $M=\\begin{bmatrix}1 & x_{1} - x_T & y_{1} - y_T\\\\1 & x_{2} - x_T & y_{2} - y_T\\\\\\vdots & \\vdots & \\vdots \\\\1 & x_{N} - x_T & y_{N} - y_T\\\\\\end{bmatrix}_{N\\times 3},\\;E=\\begin{bmatrix}|e_1| & & & \\\\& |e_2| & & \\\\& & \\ddots & \\\\& & & |e_N| \\\\\\end{bmatrix}_{N\\times N}.$ Here $M_T=(x_T,y_T)$ is any point on the plane (e.g., the center of $T$ as a specific case), $(x_i, y_i)$ is the midpoint of $e_i$ , $|e_i|$ is the length of edge $e_i$ , $\\mathbf {n}_i$ is the unit outward normal vector on $e_i$ , and $|T|$ is the area of the element $T$ .","From (REF ), the element stiffness matrix on $T\\in {\\mathcal {T}}_h$ consists of two sub-matrices corresponding to the following forms: $S_T(u_b,v_b)\\ \\mbox{and } {\\mathcal {B}}_T(u_b, v_b).$ The bilinear form ${\\mathcal {B}}_T(\\cdot ,\\cdot )$ is composed of three bilinear forms given by (REF ).","The rest of this section is devoted to a computation of the element stiffness matrices for each of the bilinear forms involved."],["The stiffness matrix for $S_T(\\cdot ,\\cdot )$","For the element stiffness matrix corresponding to $S_T(u_b,v_b)$ , the key is to compute ${\\mathfrak {s}}(u_b)$ and ${\\mathfrak {s}}(v_b)$ which can be accomplished through its definition (REF ); readers are referred to [21] for a detailed derivation.","Specifically, let $M_T=(x_T,y_T)$ be the center of T (or any point on the plane), the extension ${\\mathfrak {s}}(u_b)$ can be represented as follows: ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ From ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T)$ and (REF ), we have $\\begin{bmatrix}{\\mathfrak {s}}(u_{b})(M_1) \\\\{\\mathfrak {s}}(u_{b})(M_2) \\\\\\vdots \\\\{\\mathfrak {s}}(u_{b})(M_N) \\\\\\end{bmatrix}=M\\begin{bmatrix}\\gamma _0\\\\\\gamma _1\\\\\\gamma _2\\\\\\end{bmatrix}=M(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ Let $v_b\\in W(T)$ be the basis function corresponding to the edge $e_j$ of $T$ : $v_b=\\left\\lbrace \\begin{array}{lllll}1, \\qquad \\text{ on } e_j,\\\\0, \\qquad \\text{ otherwise}.\\\\\\end{array}\\right.$ Then the coefficient $(\\tilde{\\gamma }_0,\\tilde{\\gamma }_1,\\tilde{\\gamma }_2)^T$ for ${\\mathfrak {s}}(v_b)$ is given by $\\begin{bmatrix}\\tilde{\\gamma }_0 \\\\\\tilde{\\gamma }_1 \\\\\\tilde{\\gamma }_2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1}\\\\\\vdots \\\\v_{b,j}\\\\\\vdots \\\\v_{b,N}\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}0\\\\\\vdots \\\\1\\\\\\vdots \\\\0\\\\\\end{bmatrix}\\triangleq \\begin{bmatrix}d_{1,j}\\\\d_{2,j}\\\\d_{3,j}\\\\\\end{bmatrix}.$ It follows that $\\begin{split}S_T(u_b,v_b)=&h^{-1}\\sum _{i=1}^N({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\=&h^{-1}\\sum _{i=1}^N (u_{b,i}-{\\mathfrak {s}}(u_b)(M_i))v_{b,i}|e_i|\\\\=&h^{-1}\\left((I_N-M(M^TEM)^{-1}M^TE)\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}\\right)_j |e_j| \\\\=&h^{-1}\\sum _{i=1}^N a_{j,i}u_{b,i},\\end{split}$ where $I_N$ is the identity matrix of size $N\\times N$ ."],["The stiffness matrix for $a_T(\\cdot ,\\cdot )$","For a computation of the element stiffness matrix corresponding to the bilinear form $a_T(u_b,v_b)=(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T$ , we have from the weak gradient formula (REF ) that $(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T&=&(\\alpha \\displaystyle \\frac{1}{|T|}\\sum _{j=1}^N u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}\\mathbf {n}_i|e_i|)_{T}\\\\&=&\\sum _{i,j=1}^N(\\alpha \\displaystyle \\frac{1}{|T|} u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|} v_{b,i}\\mathbf {n}_i|e_i|)_{T} \\nonumber \\\\&=&\\sum _{i,j=1}^N \\frac{|e_j||e_i|}{|T|^2}(\\alpha \\mathbf {n}_j,\\mathbf {n}_i)_{T}u_{b,j}v_{b,i},\\nonumber \\\\&=&\\sum _{i,j=1}^N b_{i,j}u_{b,j}v_{b,i},\\nonumber $ which leads to the block matrix $B$ in the element stiffness matrix."],["The stiffness matrix for $b_T(\\cdot ,\\cdot )$","Recall that the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $b_T(u_b, v_b)=({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T.$ Note that the extension ${\\mathfrak {s}}(v_b)$ has the following representation: ${\\mathfrak {s}}(v_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1} \\\\v_{b,2} \\\\\\vdots \\\\v_{b,N} \\\\\\end{bmatrix}.$ Thus, with $D=(M^TEM)^{-1}M^TE$ , we have from the weak gradient formula (REF ) that $\\begin{split}& ({\\beta }\\cdot \\nabla _w u_b,{\\mathfrak {s}}(v_b))_T \\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N ({\\beta }\\cdot \\mathbf {n}_j, d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))_T |e_j| u_{b,j} v_{b,i}\\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N \\int _T {\\beta }\\cdot \\mathbf {n}_j (d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))dT |e_j| u_{b,j} v_{b,i}.\\end{split}$ For simplicity, we introduce the following functions: $\\zeta _i(x,y) = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T),\\qquad i=1,\\ldots , N.$ Then, the equation (REF ) indicates that the element stiffness matrix corresponding to the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $R=\\lbrace r_{ij}\\rbrace _{N\\times N},\\ \\ r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT.$"],["The stiffness matrix for $c_T(\\cdot ,\\cdot )$","Recall that the bilinear form $c_T(\\cdot ,\\cdot )$ is given by $c_T(u_b, v_b)=(c {\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ Thus, the element stiffness matrix corresponding to $c_T(\\cdot ,\\cdot )$ has the following formula: $C=\\lbrace c_{ij}\\rbrace _{N\\times N},\\quad c_{ij} = \\int _T c(x,y) \\zeta _j\\zeta _i dT,$ where $\\zeta _i$ is the function defined in (REF )."],["The element load vector","Finally, the element load vector can be obtained from $(f, {\\mathfrak {s}}(v_b))_T& =& \\int _{T}f {\\mathfrak {s}}(v_b)dT \\\\& =& \\int _{T}f(x,y) (d_{1,i} + d_{2,i}(x-x_T) + d_{3,i}(y-y_T)) dT\\\\& =& \\int _{T}f(x,y) \\zeta _i(x,y) dT$ for $i=1,\\ldots , N$ ."],["Stability and Well-Posedness","The SWG scheme (REF ) can be derived from the classical weak Galerkin finite element method [36], [24], [37] by eliminating the degrees of freedom associated with the interior of each element when ${\\beta }=0$ and $c=0$ .","But for the general case of ${\\beta }$ and $c$ , the SWG finite element method (REF ) is different from the weak Galerkin schemes in existing literature.","It is thus necessary to provide a mathematical theory for the stability and well-posedness of the numerical scheme (REF ).","Let ${\\mathcal {T}}_h$ be a shape-regular polygonal partition of the domain $\\Omega $ .","There exists a constant $C$ such that $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),\\\\\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}^2&\\le & C h \\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1} \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).$ Moreover, the following Poincaré-type estimate holds true: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right).$ From the formula (REF ) for the weak gradient, we have for any constant vector $\\mathbf {\\phi }$ that $\\begin{split}(\\nabla _w v_b, \\mathbf {\\phi })_T=&\\langle v_b, \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ (\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T,\\end{split}$ which gives $(\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T = (\\nabla _w v_b, \\mathbf {\\phi })_T - \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}.$ Hence, by letting $\\mathbf {\\phi } = \\nabla {\\mathfrak {s}}(v_b)$ we arrive at $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 \\le C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),$ which verifies (REF ).","Next, from the usual error estimate for the $L^2$ projection operator $Q_b$ and the estimate (REF ), we have $\\begin{split}\\Vert {\\mathfrak {s}}(v_b) - Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}} \\le & C h^2 \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2 \\\\\\le & C h \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert ^2_T \\\\\\le & C\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).\\end{split}$ It follows that $\\begin{split}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\le & \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}} + \\Vert {\\mathfrak {s}}(v_b)-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\\\\\le & C \\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{1/2},\\end{split}$ which verifies the estimate ().","To derive the inequality (REF ), we note the following discrete Poincaré inequality: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 \\le C\\sum _{T\\in {\\mathcal {T}}_h} \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2 + h_T^{-1}\\Vert {\\mathfrak {s}}(v_b) - v_b\\Vert _{\\partial T}^2\\right).$ Combining the above estimate with (REF ) and (REF ) gives rise to the desired inequality (REF ).","This completes the proof of the lemma.","On each element $T\\in {\\mathcal {T}}_h$ , the following identity holds true: $\\begin{split}b_T(v_b,v_b) =&\\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where $\\overline{{\\mathfrak {s}}(v_b){\\beta }}$ is the average of ${\\mathfrak {s}}(v_b){\\beta }$ on the element $T$ .","From the formula (REF ), we have $\\begin{split}b_T(v_b,v_b) =& ({\\beta }\\cdot \\nabla _w v_b, {\\mathfrak {s}}(v_b))_T \\\\=& (\\nabla _w v_b, {{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& (\\nabla _w v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& \\langle v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Note that $\\begin{split}\\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}&= (\\nabla {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\&= (\\nabla {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b){\\beta })_T\\\\&= \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T.\\end{split}$ Substituting the above identity into (REF ) yields $\\begin{split}b_T(v_b,v_b) =& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle v_b,{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ + \\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T,\\end{split}$ which leads to the identify (REF ).","In the finite element space $W_h({\\mathcal {T}}_h)$ , we introduce the following semi-norm: ${|\\!|\\!|} v_b{|\\!|\\!|}^2: = \\sum _{T\\in {\\mathcal {T}}_h} \\left( \\kappa S_T(v_b, v_b) + a_T(v_b, v_b) \\right)$ We claim that ${|\\!|\\!|}\\cdot {|\\!|\\!|}$ defines a norm in the closed subspace $W_h^0({\\mathcal {T}}_h)$ .","It suffices to show that $v_b \\equiv 0$ for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ satisfying ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ .","In fact, if ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ , then from (REF ) we have $\\kappa \\sum _{T}S_T(v_b,v_b)+\\sum _{T}(\\alpha \\nabla _w v_b,\\nabla _w v_b)_T =0.$ It follows that on each element $T\\in {\\mathcal {T}}_h$ $\\nabla _w v_b=0,\\quad (v_b - {\\mathfrak {s}}(v_b))(M_i)=0$ for $i=1,\\ldots , N$ .","Thus, $\\displaystyle \\nabla {\\mathfrak {s}}(v_b)=\\frac{1}{|T|}\\sum _{i=1}^N {\\mathfrak {s}}(v_b)(M_i)|e_i|\\mathbf {n}_i=\\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\mathbf {n}_i=\\nabla _w v_b=0,$ so that ${\\mathfrak {s}}(v_b)$ has constant value on each element $T\\in {\\mathcal {T}}_h$ .","By using (REF ) we see that $v_b={\\mathfrak {s}}(v_b)=const$ on each edge, which, together with the fact that $v_b=0$ on $\\partial \\Omega $ , leads to $v_b\\equiv 0$ in $\\Omega $ .","For the model problem (REF ), assume that ${\\beta }\\in W^{1,\\infty }(\\Omega )$ and the condition (REF ) is satisfied.","Then, the bilinear form $\\kappa S(\\cdot ,\\cdot )+{\\mathcal {B}}(\\cdot ,\\cdot )$ is bounded and coercive in the finite element space $W_h^0({\\mathcal {T}}_h)$ ; i.e., there exist constants $M$ and $\\Lambda >0$ such that $|\\kappa S(v_b,w_b)+{\\mathcal {B}}(v_b, w_b)| & \\le & M {|\\!|\\!|} v_b{|\\!|\\!|} {|\\!|\\!|} w_b{|\\!|\\!|} \\qquad \\forall v_b, w_b \\in W_h^0({\\mathcal {T}}_h),\\\\\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) & \\ge & \\Lambda {|\\!|\\!|} v_b{|\\!|\\!|}^2\\qquad \\forall v_b\\in W_h^0({\\mathcal {T}}_h),$ provided that the meshsize $h$ of ${\\mathcal {T}}_h$ is sufficiently small.","Recall that for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ we have $\\begin{split}{\\mathcal {B}}(v_b, w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} \\left(a_T(v_b,w_b)+ b_T(v_b, w_b) + c_T(v_b,w_b)\\right),\\\\S(v_b,w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} S_T(v_b,w_b).\\end{split}$ The boundedness estimate (REF ) is then straightforward from the usual Cauchy-Schwarz and the inequality (REF ).","We shall focus on the derivation of the coercivity inequality () in the rest of the proof.","In comparison with (REF ), the key to the coercivity inequality () is to derive an estimate of the following type: $\\sum _{T\\in {\\mathcal {T}}_h} \\left(b_T(v_b, v_b) + c_T(v_b,v_b)\\right)\\ge \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2,$ where $\\eta \\ge 0$ and $\\varepsilon (h)$ is a parameter satisfying $\\varepsilon (h) \\rightarrow 0$ as $h\\rightarrow 0$ .","If (REF ) indeed holds true, then we have from (REF ) that $\\begin{split}\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) \\ge & {|\\!|\\!|} v_b {|\\!|\\!|}^2 + \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2\\\\\\ge & (1- \\varepsilon (h)) {|\\!|\\!|} v_b{|\\!|\\!|}^2,\\end{split}$ which implies the coercivity () for sufficiently small $h$ .","It remains to derive the estimate (REF ).","To this end, we sum up the identify in Lemma to obtain $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} b_T(v_b,v_b)= & -\\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h} ( \\nabla \\cdot {\\beta }{\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where we have used the fact that $\\sum _{T\\in {\\mathcal {T}}_h} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}=0$ .","Thus, $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} (b_T(v_b,v_b) + c_T(v_b,v_b))= & \\sum _{T\\in {\\mathcal {T}}_h} ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Next, from () we have $\\begin{split}\\left|\\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| & \\le C \\left( h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b - Q_b {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right) \\\\& \\le C h \\sum _{T\\in {\\mathcal {T}}_h} \\left( a_T(v_b,v_b) + S_T(v_b,v_b)\\right)\\\\&\\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.\\end{split}$ As to the last term in (REF ), we have $\\begin{split}& \\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\\\\\le & \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\Vert \\overline{{\\mathfrak {s}}(v_b){\\beta }} - {{\\mathfrak {s}}(v_b){\\beta }}\\Vert _{\\partial T}\\\\\\le & C h^{\\frac{1}{2}} \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T\\right)\\\\\\le & C h \\left(\\sum _{T\\in {\\mathcal {T}}_h} h^{-1}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right)^{\\frac{1}{2}} \\left(\\sum _{T\\in {\\mathcal {T}}_h} \\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T^2 + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2\\right)\\right)^{\\frac{1}{2}}\\end{split}$ Combining the estimates (REF ), (), and (REF ) with (REF ) yields $\\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.$ Now by substituting (REF ) and (REF ) into (REF ) we obtain the inequality (REF ) with $\\eta = ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))\\ge 0$ and $\\varepsilon (h) = Ch$ .","This completes the proof of the lemma.","The following is a direct application of Lemma .","Under the assumptions of Lemma , there exists a small, but fixed number $h_0>0$ , such that the numerical scheme (REF ) has one and only one solution $u_b\\in W_h({\\mathcal {T}}_h)$ for sufficiently fine finite element partitions ${\\mathcal {T}}_h$ satisfying $h\\le h_0$ .","It suffices to show that the homogeneous problem has only the trivial solution.","To this end, let $u_b \\in W_h^0({\\mathcal {T}}_h)$ , be the solution of scheme (REF ) with homogeneous data $f=0$ and $g=0$ .","By taking $v_b = u_b$ in (REF ) we obtain $\\kappa S(u_b,u_b)+{\\mathcal {B}}(u_b,u_b)=0,$ which, from the coercivity inequality (), gives $\\Lambda {|\\!|\\!|} u_b{|\\!|\\!|}^2 \\le \\kappa S(u_b,u_b) + {\\mathcal {B}}(u_b,u_b)=0$ , and hence $u_b\\equiv 0$ for sufficiently small $h$ ."],["Error Estimates in $H^1$","Let $u$ be the exact solution of the model problem (REF )-() and $u_b \\in W_h^0({\\mathcal {T}}_h)$ be the numerical approximation arising from the SWG scheme (REF ).","Let $Q_b u$ be the $L^2$ projection of $u$ in the space $W_h^0({\\mathcal {T}}_h)$ .","The error function refers to the difference between the $L^2$ projection and the SWG approximation: $e_b := Q_b u - u_b,$ The goal of this section is to establish an estimate for the error function $e_b$ in a discrete Sobolev norm.","Let us first state an error equation which plays an important role in the convergence analysis of the SWG scheme.","Assume that the coefficient $\\alpha $ of the model problem (REF )-() has piecewise constant values with respect to the finite element partition ${\\mathcal {T}}_h$ .","Then the following equation holds true $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b,v_b)= \\ell _u(v_b) \\quad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where $\\ell _u(\\cdot )$ is a linear functional given by $\\begin{split}\\ell _u(v_b):=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}+\\kappa S(Q_bu,v_b)\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ where $Q_0(\\nabla u)$ is the $L^2$ projection of $\\nabla u$ in the space $[P_0({\\mathcal {T}}_h)]^2$ , and $\\mathbf {n}$ is the outward normal vector on ${\\partial T}$ .","We first consider the weak gradient of $Q_b u$ , for any constant vector $\\mathbf {\\phi }$ , we have $(\\nabla _w Q_b u,\\mathbf {\\phi })_{T}&=& \\langle Q_b u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}= \\langle u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}\\\\&=& (\\nabla u, \\mathbf {\\phi })_{T}= (Q_0(\\nabla u), \\mathbf {\\phi })_{T},$ which implies $\\nabla _w Q_b u \\equiv Q_0(\\nabla u)$ .","Thus, for any $v_b \\in W_h^0(T_h)$ , we have $\\begin{split}&(\\alpha \\nabla _w Q_b u,\\nabla _w v_b)= \\sum _{T}(\\alpha Q_0(\\nabla u),\\nabla _w v_b)_T \\\\= & \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}-\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} + \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(\\alpha \\nabla u,\\nabla {\\mathfrak {s}}(v_b))_{T} \\\\=& \\sum _{T}\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))_{T} + \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& (-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))+ \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}.\\end{split}$ Next, from $\\nabla _w (Q_bu) = Q_0(\\nabla u)$ , we have $\\begin{split}\\sum _T ({\\beta }\\cdot \\nabla _w (Q_bu), {\\mathfrak {s}}(v_b))_T = & \\sum _T ({\\beta }\\cdot (Q_0 \\nabla u), {\\mathfrak {s}}(v_b))_T \\\\= & \\sum _T ({\\beta }\\cdot \\nabla u, {\\mathfrak {s}}(v_b))_T + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T,\\end{split}$ and $\\sum _T (c{\\mathfrak {s}}(Q_bu), {\\mathfrak {s}}(v_b))_T = (cu, {\\mathfrak {s}}(v_b)) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).$ The sum of (REF ), (REF ), and (REF ) gives rise to $\\begin{split}{\\mathcal {B}}(Q_b u, v_b) = &\\ (f, {\\mathfrak {s}}(v_b)) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ which, combined with $ (f, {\\mathfrak {s}}(v_b) = \\kappa S(u_b, v_b) + {\\mathcal {B}}(u_b,v_b)$ , leads to $\\begin{split}{\\mathcal {B}}(Q_b u - u_b, v_b) = &\\ \\kappa S(u_b,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ and $\\begin{split}&\\kappa S(Q_bu-u_b,v_b)+{\\mathcal {B}}(Q_b u - u_b, v_b) \\\\= &\\ \\kappa S(Q_bu,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).\\end{split}$ This completes the proof of the lemma.","Remark 5.1 It should be pointed out that Lemma can be extended to the case when $\\alpha $ is in $L^\\infty (\\Omega )$ and piecewise smooth with respect to the finite element partition ${\\mathcal {T}}_h$ .","Detailed analysis can be established by following the approach presented in [35].","The following result is concerned with the error estimate for the SWG numerical solutions in a discrete $H^1$ norm.","Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).","Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and that (REF ) is satisfied.","Then, the following error estimate holds true $\\kappa S(e_b ,e_b) +(\\alpha \\nabla _w e_b, \\nabla _w e_b )\\le Ch^2\\Vert u\\Vert _2^2,$ provided that the meshsize $h$ is sufficiently small.","Consequently, we have $\\Vert \\nabla _w u_b- \\nabla u\\Vert _0 \\le Ch \\Vert u\\Vert _2, $ The proof is based on the error equation (REF ) through a thorough analysis for the linear functional $\\ell _u(\\cdot )$ given in (REF ).","For the first term on the righ-hand side of (REF ), from the usual Cauchy-Schwarz inequality we have $\\begin{split}&\\ |\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & \\ \\Vert \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n}\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}\\\\\\le & \\ \\Vert \\alpha \\Vert _{\\infty } \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}.\\end{split}$ Now using the estimate () in the above inequality and then summing over all the element $T\\in {\\mathcal {T}}_h$ we arrive at the following: $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h}|\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & C \\Vert \\alpha \\Vert _{\\infty } \\sum _{T\\in {\\mathcal {T}}_h} \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{\\partial T}\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T +\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{\\frac{1}{2}}\\\\\\le & C\\Vert \\alpha \\Vert _{\\infty }\\left(\\Vert \\nabla u -Q_0(\\nabla u)\\Vert ^2_{0} + h^2\\Vert \\nabla ^2 u \\Vert _{0}^2\\right)^{\\frac{1}{2}}\\left( \\Vert \\nabla _w v_b\\Vert ^2+ \\kappa S(v_b, v_b)\\right)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b {|\\!|\\!|}.\\end{split}$ As to the second term on the right hand side of (REF ), we have $\\begin{split}|S(Q_bu,v_b)| =&\\sum _{T}h^{-1}\\langle Q_bu-Q_b{\\mathfrak {s}}(Q_bu),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu -Q_b(Q_1 u),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=&\\sum _{T}h^{-1}\\langle u -Q_1 u,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\\\le &\\left(\\sum _{T} h^{-1} \\int _{\\partial T}|u-Q_1u|^2ds \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C\\left(h^{-2}\\Vert u-Q_1 u\\Vert ^2 + \\Vert u-Q_1 u\\Vert _1^2 \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ The third term on the right hand side of (REF ) can be bounded by using the usual error estimate for $L^2$ projections as follows: $\\begin{split}|((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })| = & |((Q_0-I) \\nabla u, (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta }))| \\\\\\le & \\Vert (Q_0-I) \\nabla u\\Vert \\ \\Vert (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta })\\Vert \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert + \\Vert {\\mathfrak {s}}(v_b)\\Vert \\right) \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert {|\\!|\\!|} v_b{|\\!|\\!|},\\end{split}$ where we have used the estimates (REF ) and (REF ) in the last line.","The last term on the right hand side of (REF ) can be estimated as follows: $\\begin{split}|(c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b))|\\le & \\Vert c\\Vert _\\infty \\Vert {\\mathfrak {s}}(Q_bu)-u\\Vert \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C (\\Vert {\\mathfrak {s}}(Q_bu)-Q_1 u\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu)-{\\mathfrak {s}}(Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu-Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C h^2 \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ Substituting the estimates (REF )-(REF ) into the error equation (REF ) yields $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b, v_b) \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|},$ which, together with the coercivity (), leads to $\\Lambda {|\\!|\\!|} e_b{|\\!|\\!|}^2 \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}.$ The last inequality implies the error estimate (REF ).","Finally, from the triangle inequality and the error estimate (REF ), we obtain $\\begin{split}\\Vert \\nabla _w u_b - \\nabla u\\Vert \\le & \\Vert \\nabla _w (u_b - Q_b u)\\Vert + \\Vert \\nabla _w (Q_b u) - \\nabla u\\Vert \\\\= & \\Vert \\nabla _w e_b\\Vert + \\Vert Q_0 (\\nabla u) - \\nabla u\\Vert \\\\\\le & C h \\Vert u\\Vert _2,\\end{split}$ which gives rise to (REF ).","This completes the proof of the theorem."],["Error Estimates in $L^2$","We use the usual duality argument to derive an error estimate in $L^2$ for the numerical solutions arising from (REF ).","The analysis to be presented is a modified version of those developed in [36], [24], [35].","Consider the following auxiliary problem that seeks $\\Phi \\in H_0^1(\\Omega )$ such that $-\\nabla \\cdot (\\alpha \\nabla \\Phi ) - \\nabla \\cdot ({\\beta }\\Phi ) + c\\Phi &=&\\chi \\quad {\\rm in}\\ \\Omega \\\\\\Phi &=&0\\quad {\\rm on}\\ \\partial \\Omega , $ where $\\chi \\in L^2(\\Omega )$ .","Assume that the solution of the problem (REF )-() exists and has the $H^2$ -regularity: $\\Vert \\Phi \\Vert _2 \\le C \\Vert \\chi \\Vert ,$ where $C$ is a constant depending only on the domain and the coefficients $\\alpha , {\\beta }$ , and $c$ .","Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).","Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and the conditions (REF ) and (REF ) are satisfied.","Then, the following $L^2$ error estimate holds true $\\Vert u - {\\mathfrak {s}}(u_b)\\Vert \\le Ch^2\\Vert u\\Vert _2,$ provided that the meshsize $h$ is sufficiently small.","On each element $T\\in {\\mathcal {T}}_h$ , we test (REF ) against the linear function ${\\mathfrak {s}}(e_b)$ to obtain $(\\chi , {\\mathfrak {s}}(e_b))_T &=& (\\alpha \\nabla \\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + ({\\beta }\\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (Q_0({\\beta }\\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla _w e_b)_T + (Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&& - \\langle \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}- \\langle Q_0({\\beta }\\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}$ By using $Q_0(\\nabla \\Phi ) = \\nabla _w (Q_b\\Phi )$ and $(Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T = ({\\beta }\\cdot \\nabla _w e_b, \\Phi )_T$ in the above equation, we have from summing over all $T\\in {\\mathcal {T}}_h$ that $\\begin{split}(\\chi , {\\mathfrak {s}}(e_b))=& (\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi ) \\\\& - \\sum _T \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}- \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}\\\\& -\\langle {\\beta }\\Phi \\cdot {\\bf n}- Q_0({\\beta }\\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}.\\end{split}$ The last two terms on the right-hand side of (REF ) can be bounded by $C h \\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}$ through the Cauchy-Schwarz inequality.","Thus, we have $\\begin{split}|(\\chi , {\\mathfrak {s}}(e_b))|\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi )| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}\\\\\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|},\\end{split}$ where have also used $\\Vert \\Phi -{\\mathfrak {s}}(Q_b\\Phi )\\Vert \\le C h^2\\Vert \\Phi \\Vert _2$ .","Now, recall that $(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))= {\\mathcal {B}}(e_b, Q_b\\Phi ),$ and from the error equation (REF ), we have $\\begin{split}{\\mathcal {B}}(e_b, Q_b\\Phi ) = & \\ell _u(Q_b\\Phi ) - \\kappa S(e_b, Q_b\\Phi )\\\\=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}+\\kappa S(u_b,Q_b\\Phi )\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi )),\\end{split}$ The last two terms on the right-hand side of (REF ) have the following estimate: $\\left|((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi ))\\right| \\le C h^2 \\Vert u\\Vert _2 \\Vert \\Phi \\Vert _1.$ The second term, $\\kappa S(u_b,Q_b\\Phi )$ , can be dealt with as follows: $\\begin{split}\\kappa S(u_b,Q_b\\Phi ) = & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), Q_b\\Phi - Q_b{\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\= & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\\\le & \\kappa h^{-1} \\sum _T \\Vert u_b-Q_b{\\mathfrak {s}}(u_b)\\Vert _{\\partial T}\\Vert \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\Vert _{\\partial T}\\\\\\le & C h ({|\\!|\\!|} e_b{|\\!|\\!|} + h\\Vert u\\Vert _2)\\Vert \\Phi \\Vert _2.\\end{split}$ As to the first term, we note from the definition of $Q_b$ and $\\Phi |_{\\partial \\Omega }=0$ that $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= 0.\\end{split}$ Thus, we have $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}\\\\= & \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-\\Phi \\rangle _{\\partial T}\\\\\\le & C h^2 \\Vert u\\Vert _2\\Vert \\Phi \\Vert _2.\\end{split}$ Substituting (REF ), (REF ), and (REF ) into (REF ) yields the following estimate: $|{\\mathcal {B}}(e_b, Q_b\\Phi )| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2,$ which, together with (REF ), leads to $|(\\chi , {\\mathfrak {s}}(e_b))| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2 \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\chi \\Vert ,$ where the regularity assumption (REF ) has been employed in the last inequality.","Next, from (REF ) and the $H^1$ error estimate (REF ) in Theorem , we have $|(\\chi , {\\mathfrak {s}}(e_b))| \\le Ch^2 \\Vert u\\Vert _2 \\Vert \\chi \\Vert ,$ which leads to $\\Vert {\\mathfrak {s}}(e_b\\Vert \\le C h^2 \\Vert u\\Vert _2.$ Finally, we arrive at $\\Vert u-{\\mathfrak {s}}(u_b)\\Vert \\le \\Vert u-{\\mathfrak {s}}(Q_bu)\\Vert + \\Vert {\\mathfrak {s}}(e_b)\\Vert \\le C h^2\\Vert u\\Vert _2,$ which completes the proof of the theorem."],["Numerical Experiments","The goal of this section is to numerically verify the error estimates developed in the previous sections for the numerical scheme (REF ).","The following metrics are employed to measure the magnitude of the error function: $&&\\text{Discrete $L^2$-norm: }\\\\&&\\Vert u_b-u \\Vert _{0}=h\\left(\\sum _{i=1}^{n+1}\\sum _{j=1}^{n}|u_{i-\\frac{1}{2},j}-u(x_{i-\\frac{1}{2}},y_j)|^2 + \\sum _{i=1}^{n}\\sum _{j=1}^{n+1}|u_{i,j-\\frac{1}{2}}-u(x_{i},y_{j-\\frac{1}{2}})|^2\\right)^{1/2},\\\\&&\\text{Discrete $H^1$-norm: }\\\\&&\\Vert u_b-u\\Vert _1= h \\left(\\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i+\\frac{1}{2},j}-u_{i-\\frac{1}{2},j}}{h} - \\frac{\\partial u}{\\partial x} (x_{i},y_{j})\\right|^2 \\right.","\\\\&&\\qquad \\qquad \\qquad \\left.","+ \\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i,j+\\frac{1}{2}}-u_{i,j-\\frac{1}{2}}}{h} - \\frac{\\partial u}{\\partial y} (x_{i},y_{j})\\right|^2 \\right)^{1/2} ,\\\\$ Our numerical experiments are conducted for the model problem (REF )-() on polygonal domains.","The following set of test cases are considered: $\\left\\lbrace \\begin{split}&u=xy, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u= 3x^2+2xy, \\\\&\\alpha =\\begin{bmatrix}2 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y) + x^2-y^2, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\2\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y), \\\\&\\alpha =\\begin{bmatrix}xy+1 &0\\\\0 & 3xy\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}x^3y+xy+1\\\\3x^2y+xy+2\\end{bmatrix},\\quad c=x^4y^2+xy+1;\\end{split}\\right.$ The right-hand side function $f$ and the Dirichlet boundary data $g$ are chosen to match the exact solution $u=u(x,y)$ for each test case.","Table: Error and convergence performance of the SWG scheme () with κ=4.0\\kappa =4.0 and uniform square partitions on the unit square domain Ω=(0,1) 2 \\Omega =(0,1)^2.Table REF shows the performance of the SWG scheme for each of the above test problems with the stabilizer parameter $\\kappa =4$ on uniform square partitions.","The results indicate that the numerical approximation is in the machine accuracy for the test problem (REF ) where the exact solution is a bilinear function.","For the other three test problems, the numerical solutions have the optimal rate of convergence $r=2$ in the discrete $L^2$ norm and a superconvergence of order ${\\mathcal {O}}(h^2)$ in the discrete $H^1$ norm.","The numerical results are consistent with the theoretical prediction in the discrete $L^2$ norm, but they outperform the theory in the discrete $H^1$ norm.","It should be pointed out that the superconvergence theory in [18] was developed for the diffusion equation only; but a slight modification of the analysis there will yield a superconvergence of order ${\\mathcal {O}}(h^2)$ for the SWG solutions of the full convection-diffusion equation (REF )-().","Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."],["On the influence of the stabilizer parameter","The goal of this subsection is to test the influence of the stabilizer parameter $\\kappa $ on the numerical solutions.","This part of the numerical experiment considers only the test cases (REF ) and (REF ) with the following six values of $\\kappa =0.01, \\ 0.1, \\ 1.0, \\ 4.0, \\ 6.0, \\ 20.0$ .","The case of $\\kappa =0$ is not a viable choice , as it was not covered in the convergence theory.","In fact, our computation does not suggest any convergence of the scheme when $\\kappa =0$ .","Tables REF -REF illustrate the numerical performance of the SWG scheme with different values of the stabilizer parameter $\\kappa $ .","Note that, for both test cases, the rate of convergence deteriorates as $\\kappa $ gets small (e.g.","$\\kappa =0.01$ ), particulary on coarse finite element partitions, but the rate of convergence begins to improve when the meshsize $h$ gets small.","Optimal rate of convergence and the supercovergence of order ${\\mathcal {O}}(h^2)$ are clearly shown in the tables when $\\kappa $ is away from 0 (e.g., $\\kappa \\ge 0.1$ ).","The stability and accuracy of the SWG scheme is insensitive to the value of $\\kappa $ as long as it stays away from 0.","Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."],["SWG with general polygonal partitions","The SWG scheme was applied to the test problem (REF ) with general polygonal partitions.","Table REF shows the error and convergence performance of the scheme on four types of polygonal partitions.","The stabilization parameter was set as $\\kappa =4$ in all these tests.","Optimal order of convergence in the discrete $L^2$ norm can be observed for each polygonal partition, but the superconvergence in the discrete $H^1$ norm was only seen for rectangular partitions.","The table shows a numerical rate of convergence of $r=1$ in the discrete $H^1$ norm for three other type of partitions.","The result is clearly in consistency with the error estimate developed in Section .","Fig.","REF illustrates the contour plots of the numerical solutions on different type of polygonal partitions.","It also shows the shape of the polygonal elements in our computation.","Table: Error and convergence performance of the SWG scheme () for the test problem () on general polygonal partitions for Ω=(0,1) 2 \\Omega =(0,1)^2, with κ=4\\kappa = 4.Figure: Comparison of numerical solutions obtained from SWG and the exact solution for the test problem () on various polygonal partitions of h=1/8h=1/8 and κ=1\\kappa =1."],["Numerical results on a non-convex domain","The SWG scheme with the stabilization parameter $\\kappa =4$ was applied to the test problem (REF ) on the L-shaped domain $\\Omega :=(-1,1)\\times (-1,1)/ (0,1)\\times (-1,0)$ partitioned into triangles or rectangles.","The corresponding numerical results are summarized in Table REF , which shows a convergence of order ${\\mathcal {O}}(h^2)$ in the $L^2$ norm for both the triangular and rectangular partitions.","A superconvergence of order ${\\mathcal {O}}(h^2)$ was observed in the discrete $H^1$ norm on rectangular partitions, while the optimal order of convergence with $r=1$ is confirmed numerically on triangular partitions.","It should be pointed out that the $H^2$ -regularity assumption (REF ) is not valid for non-convex polygonal domains so that the optimal order of error estimate (REF ) is not known theoretically on the L-shaped domain.","The numerical results therefore outperform the theory in the usual $L^2$ norm.","Table: Error and convergence performance of the SWG scheme () for test case () on Lshape domain, κ=4\\kappa = 4.Figure: Comparison of numerical solution obtained from SWG and the exact solution for test case () on L-shaped domain with h=1/8h=1/8."]],"string":"[\n [\n \"A Simplified Weak Galerkin Finite Element Method: Algorithm and Error\\n Estimates\"\n ],\n [\n \"Abstract In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations.\",\n \"The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method.\",\n \"A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions.\",\n \"Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.\"\n ],\n [\n \"Introduction\",\n \"This paper is concerned with the development of a simplified formulation for the weak Galerkin finite element method for second order elliptic equations.\",\n \"For simplicity, consider the model problem that seeks an unknown function $u=u(x)$ satisfying $-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) + {\\\\beta }\\\\cdot \\\\nabla u + cu&=&f\\\\quad {\\\\rm in}\\\\ \\\\Omega \\\\\\\\u&=&g\\\\quad {\\\\rm on}\\\\ \\\\partial \\\\Omega $ where $\\\\Omega $ is a bounded polytopal domain in $\\\\mathbb {R}^d \\\\;(d\\\\ge 2)$ with boundary $\\\\partial \\\\Omega $ , $\\\\alpha =\\\\alpha (x)$ is the diffusion coefficient, ${\\\\beta }={\\\\beta }(x)$ is the convection, and $c=c(x)$ is the reaction coefficient in relevant applications.\",\n \"We assume that $\\\\alpha $ is sufficient smooth, ${\\\\beta }\\\\in [W^{1,\\\\infty }(\\\\Omega )]^d$ , and $c$ is piecewise smooth with respect to a partition of the domain.\",\n \"For well-posedness of the problem (REF )-(), we assume $f=f(x)\\\\in L^2(\\\\Omega )$ , $g=g(x)\\\\in H^{\\\\frac{1}{2}}(\\\\partial \\\\Omega )$ , and $c-\\\\frac{1}{2}\\\\nabla \\\\cdot {\\\\beta }\\\\ge 0,\\\\qquad \\\\alpha (x) \\\\ge \\\\alpha _0 \\\\qquad \\\\forall x \\\\in \\\\Omega $ for a constant $\\\\alpha _0>0$ .\",\n \"The model problem (REF )-() arises from many scientific applications such as fluid flow in porous media.\",\n \"Mostly importantly, this model problem has served, and still serves, the scientific computing community as a testbed in the search and design of new and efficient computational algorithms for partial differential equations.\",\n \"The classical Galerkin finite element method (see, e.g., [10], [28], [16]) is particularly a numerical technique originated from the study of elliptic problems closed related to (REF )-() or its variations.\",\n \"In the last three decades, various finite element methods using discontinuous trial and test functions, including discontinuous Galerkin (DG) methods and weak Galerkin (WG) methods, have been developed for numerical solutions of partial differential equations.\",\n \"These developments were often tested over testbed problems such as (REF )-() before they were generalized or applied to more complex problems in science and engineering.\",\n \"The DG method, also known as the interior penalty method in different contexts, was originated in early 70s of the last century for a numerical study of model problems such as (REF )-(); see, e.g., [3], [14], [25], [38] for early incubations and [1], [13], [17], [27] for a detailed discussion and recent developments.\",\n \"The weak Galerkin finite element method is a recently developed discretization framework for partial differential equations [36], [37], [24], [34].\",\n \"With new concepts referred to as weak differential operators (e.g., weak gradient, weak curl, weak Laplacian etc.)\",\n \"and weak continuity through the use of various stabilizers, the method allows the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent or free of locking [33].\",\n \"For the convection-diffusion-reaction equation (REF )-(), the recent work in the context of weak Galerkin includes the algorithm developed and analyzed in [9], the one in [20] for singularly perturbed problems, and an earlier one in [39].\",\n \"The WG finite element method has been rapidly developed and applied to several different types of problems, including second order elliptic problems, the Stokes and Navier-Stokes equations, the biharmonic and elasticity equations, div-curl systems and the Maxwell's equations, etc.\",\n \"The latest development of the WG methods is the prime-dual formulation for problems that are either nonsymmetric or do not have variational forms friendly for numerical use.\",\n \"Details on the new developments can be found in [30] for second order elliptic equations in nondivergence form, [31] for the Fokker-Planck equation, and [32] for elliptic Cauchy problems.\",\n \"The typical WG method for the model problem (REF )-() seeks weak finite element approximations $u_h=\\\\lbrace u_0, u_b\\\\rbrace $ satisfying $u_b|_{\\\\partial \\\\Omega } = Q_b g$ and $S(u_h, v)+ (\\\\alpha \\\\nabla _w u_h, \\\\nabla _w v) + ({\\\\beta }\\\\cdot \\\\nabla _w u_h, v_0) + (c u_0, v_0) = (f,v_0)$ for all test functions $v=\\\\lbrace v_0, v_b\\\\rbrace $ satisfying $v_b|_{\\\\partial \\\\Omega }=0$ , where $Q_bg$ is an interpolation of the Dirichlet boundary data, $\\\\nabla _w$ is the discrete weak gradient operator, and $S(\\\\cdot ,\\\\cdot )$ is a properly selected stabilizer that gives weak continuities for the numerical solutions.\",\n \"The numerical solution $u_h$ consists of two components: the approximation $u_0$ on each element and the approximation $u_b$ on the boundary of each element.\",\n \"To reduce the computational complexity, some hybridized formulations have been introduced in [22], [29] for the method when applied to the diffusion equation and the biharmonic equation through the elimination of the degrees of freedom associated with the unknown function $u_0$ locally on each element.\",\n \"In the superconvergence study for WG [18] on rectangular elements, this hybridized formulation was further simplified in the description of the numerical algorithm, yielding a simplified weak Galerkin (SWG) finite element scheme for the diffusion equation.\",\n \"In our further investigation of the SWG to the convection-diffusion-reaction equation (REF ), we came to the conclusion that SWG represents a new discretization scheme that is different from the usual WG through a simple elimination of the unknown $u_0$ .\",\n \"As a result, we believe that a systematic study of the SWG for the convection-diffusion-reaction problem (REF )-() should be conducted for its stability and convergence.\",\n \"This paper is in response to this observation and shall provide a mathematical theory for the stability and the convergence of the simplified weak Galerkin finite element method for the model problem (REF )-().\",\n \"We believe that the result of this paper can be extended to other types of modeling equations.\",\n \"The paper is organized as follows: In Section , we shall describe the simplified weak Galerkin finite element method for (REF )-() on general polygonal partitions.\",\n \"In Section , we shall present a computational formula for the element stiffness matrices and the element load vectors from SWG.\",\n \"In Section , we provide a mathematical theory for the stability and well-posedness of the SWG scheme.\",\n \"Sections and are devoted to a discussion of the error estimates in a discrete $H^1$ and the $L^2$ norm for the numerical solutions.\",\n \"Finally, in Section , we present some numerical results to demonstrate the efficiency and accuracy of the SWG method.\",\n \"Throughout the rest of the paper, we assume $d=2$ and shall use the standard notations for Sobolev spaces and norms [10], [16].\",\n \"For any open set $D\\\\subset \\\\mathbb {R}^{2}$ , $\\\\Vert \\\\cdot \\\\Vert _{s,D}$ and $(\\\\cdot ,\\\\cdot )_{s,D}$ denote the norm and inner-product in the Sobolev space $H^s(D)$ consisting of square integrable partial derivatives up to order $s$ .\",\n \"When $s=0$ or $D=\\\\Omega $ , we shall drop the corresponding subscripts in the norm and inner-product notation.\"\n ],\n [\n \"Algorithm on Polymesh\",\n \"Assume that the domain is of polygonal type and is partitioned into non-overlap polygons ${\\\\mathcal {T}}_h=\\\\lbrace T\\\\rbrace $ that are shape regular.\",\n \"For each $T\\\\in {\\\\mathcal {T}}_h$ , denote by $h_T$ its diameter and by $N$ the number of edges.\",\n \"For each edge $e_i, \\\\ i=1,\\\\ldots , N$ , denote by $M_i$ the midpoints and ${\\\\bf n}_i$ the outward normal direction of $e_i$ (see Fig.\",\n \"REF for an illuatration).\",\n \"The meshsize of ${\\\\mathcal {T}}_h$ is defined as $h=\\\\max _{T\\\\in {\\\\mathcal {T}}_h} h_T$ .\",\n \"Let $v_b$ be a piecewise constant function defined on the boundary of $T$ , i.e., $v_b|_{e_i} =v_{b,i},$ with $v_{b,i}$ being a constant.\",\n \"We define the weak gradient of $v_b$ on $T$ by: $\\\\nabla _w v_b:=\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}|e_i|\\\\bf {n_i},$ where $|e_i|$ is the length of the edge $e_i$ and $|T|$ is the area of the element $T$ .\",\n \"It is not hard to see that the weak gradient $\\\\nabla _w v_b$ satisfies the following equation: $(\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T=\\\\langle v_b, \\\\mathbf {\\\\phi }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}$ for all constant vector $\\\\mathbf {\\\\phi }$ .\",\n \"Here and in what follows of the paper, $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle _{\\\\partial T}$ stands for the usual inner product in $L^2({\\\\partial T})$ .\",\n \"Denote by $W(T)$ the space of piecewise constant functions on ${\\\\partial T}$ .\",\n \"The global finite element space $W({\\\\mathcal {T}}_h)$ is constructed by patching together all the local elements $W(T)$ through single values on interior edges.\",\n \"The subspace of $W_h({\\\\mathcal {T}}_h)$ consisting of functions with vanishing boundary value is denoted as $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"We use the conventional notation of ${P}_j(T)$ for the space of polynomials of degree $j\\\\ge 0$ on $T$ .\",\n \"For each $v_b\\\\in W(T)$ , we associate it with a linear extension in $T$ , denoted as ${\\\\mathfrak {s}}(v_b)\\\\in {P}_1 (T)$ , satisfying $\\\\sum _{i=1}^{N}({\\\\mathfrak {s}}(v_b)(M_i) -v_{b,i})\\\\phi (M_i)|e_i|=0,\\\\quad \\\\forall \\\\; \\\\phi \\\\in {P}_1(T).$ It is easy to see that ${\\\\mathfrak {s}}(u_b)$ is well defined by (REF ), and its computation is local and straightforward.\",\n \"In fact, ${\\\\mathfrak {s}}(u_b)$ can be viewed as an extension of $u_b$ from $\\\\partial T$ to $T$ through a least-squares fitting.\",\n \"Figure: An illustrative polygonal element.On each element $T \\\\in {\\\\mathcal {T}}_h $ , we introduce the following bilinear forms: $a_T(u_b,v_b)&:= & (\\\\alpha \\\\nabla _w u_b, \\\\nabla _w v_b)_T, \\\\\\\\b_T(u_b,v_b)&:= & ({\\\\beta }\\\\cdot \\\\nabla _w u_b, {\\\\mathfrak {s}}(v_b))_T,\\\\\\\\c_T(u_b,v_b)&:= & (c{\\\\mathfrak {s}}(u_b), {\\\\mathfrak {s}}(v_b))_T.$ For simplicity, we set $\\\\mathcal {B}_T(u_b,v_b):=a_T(u_b,v_b) + b_T(u_b,v_b) + c_T(u_b,v_b)$ for $u_b, v_b \\\\in W(T)$ .\",\n \"We further introduce the stabilizer $\\\\begin{split}S_T(u_b,v_b):= & h^{-1}\\\\sum _{i=1}^N ({\\\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\\\\\= & h^{-1}\\\\langle Q_b{\\\\mathfrak {s}}(u_b)-u_{b},Q_b{\\\\mathfrak {s}}(v_b)-v_{b}\\\\rangle _{\\\\partial T},\\\\end{split}$ where $Q_b$ is the $L^2$ projection operator onto $W(T)$ ; namely $Q_b u$ is the average of $u$ on each edge.\",\n \"In particular, $Q_b(g)$ is well-defined and takes the average of the Dirichlet data on each boundary edge.\",\n \"SWG Algorithm 2.1 The simplified weak Galerkin (SWG) scheme for the elliptic equation (REF )-() seeks $u_b \\\\in W_h({\\\\mathcal {T}}_h)$ satisfying $u_b = Q_b(g)$ on $\\\\partial \\\\Omega $ and ${\\\\mathcal {A}}(u_b, v_b) =(f, {\\\\mathfrak {s}}(v_b))\\\\qquad \\\\forall v_b \\\\in W_h^0({\\\\mathcal {T}}_h),$ where ${\\\\mathcal {A}}(u_b,v_b):=\\\\kappa S(u_b,v_b)+ {\\\\mathcal {B}}(u_b, v_b)$ , $S(u_b,v_b) &=& \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} S_T(u_b,v_b),\\\\\\\\{\\\\mathcal {B}}(u_b,v_b)&=& \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\mathcal {B}_T(u_b, v_b)$ are bilinear forms in $W_h({\\\\mathcal {T}}_h)$ and $(f,{\\\\mathfrak {s}}(v_b)):=\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} (f, {\\\\mathfrak {s}}(v_b))_T$ is a linear form in $W_h({\\\\mathcal {T}}_h)$ .\"\n ],\n [\n \"Element Stiffness Matrices\",\n \"The simplified weak Galerkin finite element method (REF ) is user-friendly in computer implementation.\",\n \"In this section, we present a formula for the computation of the element stiffness matrices and the element load vector on general polygonal elements.\",\n \"Let $T\\\\in {\\\\mathcal {T}}_h$ be a polygonal element of $N$ sides.\",\n \"Denote by $X_{u_b}$ the vector representation of $u_b$ given by $(u_{b,1},u_{b,2},\\\\ldots ,u_{b,N})^T$ .\",\n \"Then, the element stiffness matrix and the element load vector for the SWG scheme (REF ) are given in a block matrix form as follows: $(\\\\kappa h^{-1}A^T + B + R + C)X_{u_b} \\\\cong F,$ where the block components in (REF ) are given by: (1) $A:=\\\\lbrace a_{i,j}\\\\rbrace _{i,j=1}^N = E- EM(M^TEM)^{-1}M^TE$ , (2) $B:=\\\\lbrace b_{i,j}\\\\rbrace _{i,j=1}^N$ , with $b_{i,j} = (\\\\alpha \\\\mathbf {n}_i,\\\\mathbf {n}_j)_{T}\\\\displaystyle \\\\frac{|e_i||e_j|}{|T|^2}$ , (3) $R:=\\\\lbrace r_{ij}\\\\rbrace _{i,j=1}^N$ , with $r_{ij}= \\\\frac{|e_j|}{|T|}\\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j \\\\zeta _i dT$ , (4) $C:=\\\\lbrace c_{ij}\\\\rbrace _{i,j=1}^N$ , with $c_{ij}= \\\\int _T c\\\\zeta _j\\\\zeta _i dT$ , (5) $F:=\\\\lbrace f_{i}\\\\rbrace _{i=1}^N$ , with $f_{i} = \\\\int _{T}f(x,y) \\\\zeta _i(x,y) dT$ , (6) $D:=\\\\lbrace d_{j,i}\\\\rbrace _{3\\\\times N}=(M^TEM)^{-1}M^TE$ and $\\\\zeta _i = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T)$ , (7) $M$ and $E$ are given by $M=\\\\begin{bmatrix}1 & x_{1} - x_T & y_{1} - y_T\\\\\\\\1 & x_{2} - x_T & y_{2} - y_T\\\\\\\\\\\\vdots & \\\\vdots & \\\\vdots \\\\\\\\1 & x_{N} - x_T & y_{N} - y_T\\\\\\\\\\\\end{bmatrix}_{N\\\\times 3},\\\\;E=\\\\begin{bmatrix}|e_1| & & & \\\\\\\\& |e_2| & & \\\\\\\\& & \\\\ddots & \\\\\\\\& & & |e_N| \\\\\\\\\\\\end{bmatrix}_{N\\\\times N}.$ Here $M_T=(x_T,y_T)$ is any point on the plane (e.g., the center of $T$ as a specific case), $(x_i, y_i)$ is the midpoint of $e_i$ , $|e_i|$ is the length of edge $e_i$ , $\\\\mathbf {n}_i$ is the unit outward normal vector on $e_i$ , and $|T|$ is the area of the element $T$ .\",\n \"From (REF ), the element stiffness matrix on $T\\\\in {\\\\mathcal {T}}_h$ consists of two sub-matrices corresponding to the following forms: $S_T(u_b,v_b)\\\\ \\\\mbox{and } {\\\\mathcal {B}}_T(u_b, v_b).$ The bilinear form ${\\\\mathcal {B}}_T(\\\\cdot ,\\\\cdot )$ is composed of three bilinear forms given by (REF ).\",\n \"The rest of this section is devoted to a computation of the element stiffness matrices for each of the bilinear forms involved.\"\n ],\n [\n \"The stiffness matrix for $S_T(\\\\cdot ,\\\\cdot )$\",\n \"For the element stiffness matrix corresponding to $S_T(u_b,v_b)$ , the key is to compute ${\\\\mathfrak {s}}(u_b)$ and ${\\\\mathfrak {s}}(v_b)$ which can be accomplished through its definition (REF ); readers are referred to [21] for a detailed derivation.\",\n \"Specifically, let $M_T=(x_T,y_T)$ be the center of T (or any point on the plane), the extension ${\\\\mathfrak {s}}(u_b)$ can be represented as follows: ${\\\\mathfrak {s}}(u_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T),$ where $\\\\begin{bmatrix}\\\\gamma _0 \\\\\\\\\\\\gamma _1 \\\\\\\\\\\\gamma _2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}.$ From ${\\\\mathfrak {s}}(u_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T)$ and (REF ), we have $\\\\begin{bmatrix}{\\\\mathfrak {s}}(u_{b})(M_1) \\\\\\\\{\\\\mathfrak {s}}(u_{b})(M_2) \\\\\\\\\\\\vdots \\\\\\\\{\\\\mathfrak {s}}(u_{b})(M_N) \\\\\\\\\\\\end{bmatrix}=M\\\\begin{bmatrix}\\\\gamma _0\\\\\\\\\\\\gamma _1\\\\\\\\\\\\gamma _2\\\\\\\\\\\\end{bmatrix}=M(M^TEM)^{-1}M^TE\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}.$ Let $v_b\\\\in W(T)$ be the basis function corresponding to the edge $e_j$ of $T$ : $v_b=\\\\left\\\\lbrace \\\\begin{array}{lllll}1, \\\\qquad \\\\text{ on } e_j,\\\\\\\\0, \\\\qquad \\\\text{ otherwise}.\\\\\\\\\\\\end{array}\\\\right.$ Then the coefficient $(\\\\tilde{\\\\gamma }_0,\\\\tilde{\\\\gamma }_1,\\\\tilde{\\\\gamma }_2)^T$ for ${\\\\mathfrak {s}}(v_b)$ is given by $\\\\begin{bmatrix}\\\\tilde{\\\\gamma }_0 \\\\\\\\\\\\tilde{\\\\gamma }_1 \\\\\\\\\\\\tilde{\\\\gamma }_2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}v_{b,1}\\\\\\\\\\\\vdots \\\\\\\\v_{b,j}\\\\\\\\\\\\vdots \\\\\\\\v_{b,N}\\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}0\\\\\\\\\\\\vdots \\\\\\\\1\\\\\\\\\\\\vdots \\\\\\\\0\\\\\\\\\\\\end{bmatrix}\\\\triangleq \\\\begin{bmatrix}d_{1,j}\\\\\\\\d_{2,j}\\\\\\\\d_{3,j}\\\\\\\\\\\\end{bmatrix}.$ It follows that $\\\\begin{split}S_T(u_b,v_b)=&h^{-1}\\\\sum _{i=1}^N({\\\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\\\\\=&h^{-1}\\\\sum _{i=1}^N (u_{b,i}-{\\\\mathfrak {s}}(u_b)(M_i))v_{b,i}|e_i|\\\\\\\\=&h^{-1}\\\\left((I_N-M(M^TEM)^{-1}M^TE)\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}\\\\right)_j |e_j| \\\\\\\\=&h^{-1}\\\\sum _{i=1}^N a_{j,i}u_{b,i},\\\\end{split}$ where $I_N$ is the identity matrix of size $N\\\\times N$ .\"\n ],\n [\n \"The stiffness matrix for $a_T(\\\\cdot ,\\\\cdot )$\",\n \"For a computation of the element stiffness matrix corresponding to the bilinear form $a_T(u_b,v_b)=(\\\\alpha \\\\nabla _w u_b,\\\\nabla _w v_b )_T$ , we have from the weak gradient formula (REF ) that $(\\\\alpha \\\\nabla _w u_b,\\\\nabla _w v_b )_T&=&(\\\\alpha \\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{j=1}^N u_{b,j}\\\\mathbf {n}_j|e_j|,\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}\\\\mathbf {n}_i|e_i|)_{T}\\\\\\\\&=&\\\\sum _{i,j=1}^N(\\\\alpha \\\\displaystyle \\\\frac{1}{|T|} u_{b,j}\\\\mathbf {n}_j|e_j|,\\\\displaystyle \\\\frac{1}{|T|} v_{b,i}\\\\mathbf {n}_i|e_i|)_{T} \\\\nonumber \\\\\\\\&=&\\\\sum _{i,j=1}^N \\\\frac{|e_j||e_i|}{|T|^2}(\\\\alpha \\\\mathbf {n}_j,\\\\mathbf {n}_i)_{T}u_{b,j}v_{b,i},\\\\nonumber \\\\\\\\&=&\\\\sum _{i,j=1}^N b_{i,j}u_{b,j}v_{b,i},\\\\nonumber $ which leads to the block matrix $B$ in the element stiffness matrix.\"\n ],\n [\n \"The stiffness matrix for $b_T(\\\\cdot ,\\\\cdot )$\",\n \"Recall that the bilinear form $b_T(\\\\cdot ,\\\\cdot )$ is given by $b_T(u_b, v_b)=({\\\\beta }\\\\cdot \\\\nabla _w u_b, {\\\\mathfrak {s}}(v_b))_T.$ Note that the extension ${\\\\mathfrak {s}}(v_b)$ has the following representation: ${\\\\mathfrak {s}}(v_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T),$ where $\\\\begin{bmatrix}\\\\gamma _0 \\\\\\\\\\\\gamma _1 \\\\\\\\\\\\gamma _2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}v_{b,1} \\\\\\\\v_{b,2} \\\\\\\\\\\\vdots \\\\\\\\v_{b,N} \\\\\\\\\\\\end{bmatrix}.$ Thus, with $D=(M^TEM)^{-1}M^TE$ , we have from the weak gradient formula (REF ) that $\\\\begin{split}& ({\\\\beta }\\\\cdot \\\\nabla _w u_b,{\\\\mathfrak {s}}(v_b))_T \\\\\\\\=&\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i,j=1}^N ({\\\\beta }\\\\cdot \\\\mathbf {n}_j, d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))_T |e_j| u_{b,j} v_{b,i}\\\\\\\\=&\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i,j=1}^N \\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j (d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))dT |e_j| u_{b,j} v_{b,i}.\\\\end{split}$ For simplicity, we introduce the following functions: $\\\\zeta _i(x,y) = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T),\\\\qquad i=1,\\\\ldots , N.$ Then, the equation (REF ) indicates that the element stiffness matrix corresponding to the bilinear form $b_T(\\\\cdot ,\\\\cdot )$ is given by $R=\\\\lbrace r_{ij}\\\\rbrace _{N\\\\times N},\\\\ \\\\ r_{ij}= \\\\frac{|e_j|}{|T|}\\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j \\\\zeta _i dT.$\"\n ],\n [\n \"The stiffness matrix for $c_T(\\\\cdot ,\\\\cdot )$\",\n \"Recall that the bilinear form $c_T(\\\\cdot ,\\\\cdot )$ is given by $c_T(u_b, v_b)=(c {\\\\mathfrak {s}}(u_b), {\\\\mathfrak {s}}(v_b))_T.$ Thus, the element stiffness matrix corresponding to $c_T(\\\\cdot ,\\\\cdot )$ has the following formula: $C=\\\\lbrace c_{ij}\\\\rbrace _{N\\\\times N},\\\\quad c_{ij} = \\\\int _T c(x,y) \\\\zeta _j\\\\zeta _i dT,$ where $\\\\zeta _i$ is the function defined in (REF ).\"\n ],\n [\n \"The element load vector\",\n \"Finally, the element load vector can be obtained from $(f, {\\\\mathfrak {s}}(v_b))_T& =& \\\\int _{T}f {\\\\mathfrak {s}}(v_b)dT \\\\\\\\& =& \\\\int _{T}f(x,y) (d_{1,i} + d_{2,i}(x-x_T) + d_{3,i}(y-y_T)) dT\\\\\\\\& =& \\\\int _{T}f(x,y) \\\\zeta _i(x,y) dT$ for $i=1,\\\\ldots , N$ .\"\n ],\n [\n \"Stability and Well-Posedness\",\n \"The SWG scheme (REF ) can be derived from the classical weak Galerkin finite element method [36], [24], [37] by eliminating the degrees of freedom associated with the interior of each element when ${\\\\beta }=0$ and $c=0$ .\",\n \"But for the general case of ${\\\\beta }$ and $c$ , the SWG finite element method (REF ) is different from the weak Galerkin schemes in existing literature.\",\n \"It is thus necessary to provide a mathematical theory for the stability and well-posedness of the numerical scheme (REF ).\",\n \"Let ${\\\\mathcal {T}}_h$ be a shape-regular polygonal partition of the domain $\\\\Omega $ .\",\n \"There exists a constant $C$ such that $\\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{T}^2 & \\\\le & C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right),\\\\\\\\\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}^2&\\\\le & C h \\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1} \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right).$ Moreover, the following Poincaré-type estimate holds true: $\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert ^2 & \\\\le & C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right).$ From the formula (REF ) for the weak gradient, we have for any constant vector $\\\\mathbf {\\\\phi }$ that $\\\\begin{split}(\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T=&\\\\langle v_b, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b- {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}+ \\\\langle {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b- Q_b{\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}+ (\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi })_T,\\\\end{split}$ which gives $(\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi })_T = (\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T - \\\\langle v_b- Q_b{\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}.$ Hence, by letting $\\\\mathbf {\\\\phi } = \\\\nabla {\\\\mathfrak {s}}(v_b)$ we arrive at $\\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{T}^2 \\\\le C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right),$ which verifies (REF ).\",\n \"Next, from the usual error estimate for the $L^2$ projection operator $Q_b$ and the estimate (REF ), we have $\\\\begin{split}\\\\Vert {\\\\mathfrak {s}}(v_b) - Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}} \\\\le & C h^2 \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2 \\\\\\\\\\\\le & C h \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert ^2_T \\\\\\\\\\\\le & C\\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right).\\\\end{split}$ It follows that $\\\\begin{split}\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}\\\\le & \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}} + \\\\Vert {\\\\mathfrak {s}}(v_b)-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}\\\\\\\\\\\\le & C \\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right)^{1/2},\\\\end{split}$ which verifies the estimate ().\",\n \"To derive the inequality (REF ), we note the following discrete Poincaré inequality: $\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert ^2 \\\\le C\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2 + h_T^{-1}\\\\Vert {\\\\mathfrak {s}}(v_b) - v_b\\\\Vert _{\\\\partial T}^2\\\\right).$ Combining the above estimate with (REF ) and (REF ) gives rise to the desired inequality (REF ).\",\n \"This completes the proof of the lemma.\",\n \"On each element $T\\\\in {\\\\mathcal {T}}_h$ , the following identity holds true: $\\\\begin{split}b_T(v_b,v_b) =&\\\\frac{1}{2} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T},\\\\end{split}$ where $\\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}$ is the average of ${\\\\mathfrak {s}}(v_b){\\\\beta }$ on the element $T$ .\",\n \"From the formula (REF ), we have $\\\\begin{split}b_T(v_b,v_b) =& ({\\\\beta }\\\\cdot \\\\nabla _w v_b, {\\\\mathfrak {s}}(v_b))_T \\\\\\\\=& (\\\\nabla _w v_b, {{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\=& (\\\\nabla _w v_b, \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\=& \\\\langle v_b, \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\langle {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}.\\\\end{split}$ Note that $\\\\begin{split}\\\\langle {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}&= (\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\&= (\\\\nabla {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b){\\\\beta })_T\\\\\\\\&= \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T.\\\\end{split}$ Substituting the above identity into (REF ) yields $\\\\begin{split}b_T(v_b,v_b) =& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ \\\\ - \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\langle v_b,{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ \\\\ - \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ - \\\\frac{1}{2} \\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ + \\\\frac{1}{2} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T,\\\\end{split}$ which leads to the identify (REF ).\",\n \"In the finite element space $W_h({\\\\mathcal {T}}_h)$ , we introduce the following semi-norm: ${|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2: = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( \\\\kappa S_T(v_b, v_b) + a_T(v_b, v_b) \\\\right)$ We claim that ${|\\\\!|\\\\!|}\\\\cdot {|\\\\!|\\\\!|}$ defines a norm in the closed subspace $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"It suffices to show that $v_b \\\\equiv 0$ for any $v_b\\\\in W_h^0({\\\\mathcal {T}}_h)$ satisfying ${|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|} =0$ .\",\n \"In fact, if ${|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|} =0$ , then from (REF ) we have $\\\\kappa \\\\sum _{T}S_T(v_b,v_b)+\\\\sum _{T}(\\\\alpha \\\\nabla _w v_b,\\\\nabla _w v_b)_T =0.$ It follows that on each element $T\\\\in {\\\\mathcal {T}}_h$ $\\\\nabla _w v_b=0,\\\\quad (v_b - {\\\\mathfrak {s}}(v_b))(M_i)=0$ for $i=1,\\\\ldots , N$ .\",\n \"Thus, $\\\\displaystyle \\\\nabla {\\\\mathfrak {s}}(v_b)=\\\\frac{1}{|T|}\\\\sum _{i=1}^N {\\\\mathfrak {s}}(v_b)(M_i)|e_i|\\\\mathbf {n}_i=\\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}|e_i|\\\\mathbf {n}_i=\\\\nabla _w v_b=0,$ so that ${\\\\mathfrak {s}}(v_b)$ has constant value on each element $T\\\\in {\\\\mathcal {T}}_h$ .\",\n \"By using (REF ) we see that $v_b={\\\\mathfrak {s}}(v_b)=const$ on each edge, which, together with the fact that $v_b=0$ on $\\\\partial \\\\Omega $ , leads to $v_b\\\\equiv 0$ in $\\\\Omega $ .\",\n \"For the model problem (REF ), assume that ${\\\\beta }\\\\in W^{1,\\\\infty }(\\\\Omega )$ and the condition (REF ) is satisfied.\",\n \"Then, the bilinear form $\\\\kappa S(\\\\cdot ,\\\\cdot )+{\\\\mathcal {B}}(\\\\cdot ,\\\\cdot )$ is bounded and coercive in the finite element space $W_h^0({\\\\mathcal {T}}_h)$ ; i.e., there exist constants $M$ and $\\\\Lambda >0$ such that $|\\\\kappa S(v_b,w_b)+{\\\\mathcal {B}}(v_b, w_b)| & \\\\le & M {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|} {|\\\\!|\\\\!|} w_b{|\\\\!|\\\\!|} \\\\qquad \\\\forall v_b, w_b \\\\in W_h^0({\\\\mathcal {T}}_h),\\\\\\\\\\\\kappa S(v_b,v_b)+{\\\\mathcal {B}}(v_b, v_b) & \\\\ge & \\\\Lambda {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2\\\\qquad \\\\forall v_b\\\\in W_h^0({\\\\mathcal {T}}_h),$ provided that the meshsize $h$ of ${\\\\mathcal {T}}_h$ is sufficiently small.\",\n \"Recall that for any $v_b\\\\in W_h^0({\\\\mathcal {T}}_h)$ we have $\\\\begin{split}{\\\\mathcal {B}}(v_b, w_b) & \\\\ = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(a_T(v_b,w_b)+ b_T(v_b, w_b) + c_T(v_b,w_b)\\\\right),\\\\\\\\S(v_b,w_b) & \\\\ = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} S_T(v_b,w_b).\\\\end{split}$ The boundedness estimate (REF ) is then straightforward from the usual Cauchy-Schwarz and the inequality (REF ).\",\n \"We shall focus on the derivation of the coercivity inequality () in the rest of the proof.\",\n \"In comparison with (REF ), the key to the coercivity inequality () is to derive an estimate of the following type: $\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(b_T(v_b, v_b) + c_T(v_b,v_b)\\\\right)\\\\ge \\\\eta - \\\\varepsilon (h) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2,$ where $\\\\eta \\\\ge 0$ and $\\\\varepsilon (h)$ is a parameter satisfying $\\\\varepsilon (h) \\\\rightarrow 0$ as $h\\\\rightarrow 0$ .\",\n \"If (REF ) indeed holds true, then we have from (REF ) that $\\\\begin{split}\\\\kappa S(v_b,v_b)+{\\\\mathcal {B}}(v_b, v_b) \\\\ge & {|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|}^2 + \\\\eta - \\\\varepsilon (h) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2\\\\\\\\\\\\ge & (1- \\\\varepsilon (h)) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2,\\\\end{split}$ which implies the coercivity () for sufficiently small $h$ .\",\n \"It remains to derive the estimate (REF ).\",\n \"To this end, we sum up the identify in Lemma to obtain $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} b_T(v_b,v_b)= & -\\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} ( \\\\nabla \\\\cdot {\\\\beta }{\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T},\\\\end{split}$ where we have used the fact that $\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}=0$ .\",\n \"Thus, $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} (b_T(v_b,v_b) + c_T(v_b,v_b))= & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} ( (c-\\\\frac{1}{2} \\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}.\\\\end{split}$ Next, from () we have $\\\\begin{split}\\\\left|\\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| & \\\\le C \\\\left( h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b - Q_b {\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2\\\\right) \\\\\\\\& \\\\le C h \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( a_T(v_b,v_b) + S_T(v_b,v_b)\\\\right)\\\\\\\\&\\\\le C h {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2.\\\\end{split}$ As to the last term in (REF ), we have $\\\\begin{split}& \\\\left| \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| \\\\\\\\\\\\le & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}\\\\Vert \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }} - {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\Vert _{\\\\partial T}\\\\\\\\\\\\le & C h^{\\\\frac{1}{2}} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}\\\\left(\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert _T + \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T\\\\right)\\\\\\\\\\\\le & C h \\\\left(\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} h^{-1}\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2\\\\right)^{\\\\frac{1}{2}} \\\\left(\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2 + \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\end{split}$ Combining the estimates (REF ), (), and (REF ) with (REF ) yields $\\\\left| \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| \\\\le C h {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2.$ Now by substituting (REF ) and (REF ) into (REF ) we obtain the inequality (REF ) with $\\\\eta = ( (c-\\\\frac{1}{2} \\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))\\\\ge 0$ and $\\\\varepsilon (h) = Ch$ .\",\n \"This completes the proof of the lemma.\",\n \"The following is a direct application of Lemma .\",\n \"Under the assumptions of Lemma , there exists a small, but fixed number $h_0>0$ , such that the numerical scheme (REF ) has one and only one solution $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ for sufficiently fine finite element partitions ${\\\\mathcal {T}}_h$ satisfying $h\\\\le h_0$ .\",\n \"It suffices to show that the homogeneous problem has only the trivial solution.\",\n \"To this end, let $u_b \\\\in W_h^0({\\\\mathcal {T}}_h)$ , be the solution of scheme (REF ) with homogeneous data $f=0$ and $g=0$ .\",\n \"By taking $v_b = u_b$ in (REF ) we obtain $\\\\kappa S(u_b,u_b)+{\\\\mathcal {B}}(u_b,u_b)=0,$ which, from the coercivity inequality (), gives $\\\\Lambda {|\\\\!|\\\\!|} u_b{|\\\\!|\\\\!|}^2 \\\\le \\\\kappa S(u_b,u_b) + {\\\\mathcal {B}}(u_b,u_b)=0$ , and hence $u_b\\\\equiv 0$ for sufficiently small $h$ .\"\n ],\n [\n \"Error Estimates in $H^1$\",\n \"Let $u$ be the exact solution of the model problem (REF )-() and $u_b \\\\in W_h^0({\\\\mathcal {T}}_h)$ be the numerical approximation arising from the SWG scheme (REF ).\",\n \"Let $Q_b u$ be the $L^2$ projection of $u$ in the space $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"The error function refers to the difference between the $L^2$ projection and the SWG approximation: $e_b := Q_b u - u_b,$ The goal of this section is to establish an estimate for the error function $e_b$ in a discrete Sobolev norm.\",\n \"Let us first state an error equation which plays an important role in the convergence analysis of the SWG scheme.\",\n \"Assume that the coefficient $\\\\alpha $ of the model problem (REF )-() has piecewise constant values with respect to the finite element partition ${\\\\mathcal {T}}_h$ .\",\n \"Then the following equation holds true $\\\\kappa S(e_b, v_b) + {\\\\mathcal {B}}(e_b,v_b)= \\\\ell _u(v_b) \\\\quad \\\\forall v_b \\\\in W_h^0({\\\\mathcal {T}}_h),$ where $\\\\ell _u(\\\\cdot )$ is a linear functional given by $\\\\begin{split}\\\\ell _u(v_b):=&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}+\\\\kappa S(Q_bu,v_b)\\\\\\\\& + ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ where $Q_0(\\\\nabla u)$ is the $L^2$ projection of $\\\\nabla u$ in the space $[P_0({\\\\mathcal {T}}_h)]^2$ , and $\\\\mathbf {n}$ is the outward normal vector on ${\\\\partial T}$ .\",\n \"We first consider the weak gradient of $Q_b u$ , for any constant vector $\\\\mathbf {\\\\phi }$ , we have $(\\\\nabla _w Q_b u,\\\\mathbf {\\\\phi })_{T}&=& \\\\langle Q_b u, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n} \\\\rangle _{\\\\partial T}= \\\\langle u, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n} \\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\nabla u, \\\\mathbf {\\\\phi })_{T}= (Q_0(\\\\nabla u), \\\\mathbf {\\\\phi })_{T},$ which implies $\\\\nabla _w Q_b u \\\\equiv Q_0(\\\\nabla u)$ .\",\n \"Thus, for any $v_b \\\\in W_h^0(T_h)$ , we have $\\\\begin{split}&(\\\\alpha \\\\nabla _w Q_b u,\\\\nabla _w v_b)= \\\\sum _{T}(\\\\alpha Q_0(\\\\nabla u),\\\\nabla _w v_b)_T \\\\\\\\= & \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b\\\\rangle _{\\\\partial T}-\\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} + \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b-{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} +(\\\\alpha \\\\nabla u,\\\\nabla {\\\\mathfrak {s}}(v_b))_{T} \\\\\\\\=& \\\\sum _{T}\\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b-{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} +(-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) ,{\\\\mathfrak {s}}(v_b))_{T} + \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=& (-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) ,{\\\\mathfrak {s}}(v_b))+ \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}.\\\\end{split}$ Next, from $\\\\nabla _w (Q_bu) = Q_0(\\\\nabla u)$ , we have $\\\\begin{split}\\\\sum _T ({\\\\beta }\\\\cdot \\\\nabla _w (Q_bu), {\\\\mathfrak {s}}(v_b))_T = & \\\\sum _T ({\\\\beta }\\\\cdot (Q_0 \\\\nabla u), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\= & \\\\sum _T ({\\\\beta }\\\\cdot \\\\nabla u, {\\\\mathfrak {s}}(v_b))_T + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T,\\\\end{split}$ and $\\\\sum _T (c{\\\\mathfrak {s}}(Q_bu), {\\\\mathfrak {s}}(v_b))_T = (cu, {\\\\mathfrak {s}}(v_b)) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)).$ The sum of (REF ), (REF ), and (REF ) gives rise to $\\\\begin{split}{\\\\mathcal {B}}(Q_b u, v_b) = &\\\\ (f, {\\\\mathfrak {s}}(v_b)) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ which, combined with $ (f, {\\\\mathfrak {s}}(v_b) = \\\\kappa S(u_b, v_b) + {\\\\mathcal {B}}(u_b,v_b)$ , leads to $\\\\begin{split}{\\\\mathcal {B}}(Q_b u - u_b, v_b) = &\\\\ \\\\kappa S(u_b,v_b) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ and $\\\\begin{split}&\\\\kappa S(Q_bu-u_b,v_b)+{\\\\mathcal {B}}(Q_b u - u_b, v_b) \\\\\\\\= &\\\\ \\\\kappa S(Q_bu,v_b) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)).\\\\end{split}$ This completes the proof of the lemma.\",\n \"Remark 5.1 It should be pointed out that Lemma can be extended to the case when $\\\\alpha $ is in $L^\\\\infty (\\\\Omega )$ and piecewise smooth with respect to the finite element partition ${\\\\mathcal {T}}_h$ .\",\n \"Detailed analysis can be established by following the approach presented in [35].\",\n \"The following result is concerned with the error estimate for the SWG numerical solutions in a discrete $H^1$ norm.\",\n \"Let $u\\\\in H^2(\\\\Omega )$ be the exact solution of (REF )-() and $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).\",\n \"Assume ${\\\\beta }\\\\in C^1(\\\\bar{\\\\Omega })$ and that (REF ) is satisfied.\",\n \"Then, the following error estimate holds true $\\\\kappa S(e_b ,e_b) +(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w e_b )\\\\le Ch^2\\\\Vert u\\\\Vert _2^2,$ provided that the meshsize $h$ is sufficiently small.\",\n \"Consequently, we have $\\\\Vert \\\\nabla _w u_b- \\\\nabla u\\\\Vert _0 \\\\le Ch \\\\Vert u\\\\Vert _2, $ The proof is based on the error equation (REF ) through a thorough analysis for the linear functional $\\\\ell _u(\\\\cdot )$ given in (REF ).\",\n \"For the first term on the righ-hand side of (REF ), from the usual Cauchy-Schwarz inequality we have $\\\\begin{split}&\\\\ |\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}| \\\\\\\\\\\\le & \\\\ \\\\Vert \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n}\\\\Vert _{0,{\\\\partial T}} \\\\Vert {\\\\mathfrak {s}}(v_b)-v_b\\\\Vert _{0,{\\\\partial T}}\\\\\\\\\\\\le & \\\\ \\\\Vert \\\\alpha \\\\Vert _{\\\\infty } \\\\Vert \\\\nabla u- Q_0(\\\\nabla u)\\\\Vert _{0,{\\\\partial T}} \\\\Vert {\\\\mathfrak {s}}(v_b)-v_b\\\\Vert _{0,{\\\\partial T}}.\\\\end{split}$ Now using the estimate () in the above inequality and then summing over all the element $T\\\\in {\\\\mathcal {T}}_h$ we arrive at the following: $\\\\begin{split}&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h}|\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}| \\\\\\\\\\\\le & C \\\\Vert \\\\alpha \\\\Vert _{\\\\infty } \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert \\\\nabla u- Q_0(\\\\nabla u)\\\\Vert _{\\\\partial T}\\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T +\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C\\\\Vert \\\\alpha \\\\Vert _{\\\\infty }\\\\left(\\\\Vert \\\\nabla u -Q_0(\\\\nabla u)\\\\Vert ^2_{0} + h^2\\\\Vert \\\\nabla ^2 u \\\\Vert _{0}^2\\\\right)^{\\\\frac{1}{2}}\\\\left( \\\\Vert \\\\nabla _w v_b\\\\Vert ^2+ \\\\kappa S(v_b, v_b)\\\\right)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|}.\\\\end{split}$ As to the second term on the right hand side of (REF ), we have $\\\\begin{split}|S(Q_bu,v_b)| =&\\\\sum _{T}h^{-1}\\\\langle Q_bu-Q_b{\\\\mathfrak {s}}(Q_bu),v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} \\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle Q_bu,v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} \\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle Q_bu -Q_b(Q_1 u),v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle u -Q_1 u,v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\\\\\le &\\\\left(\\\\sum _{T} h^{-1} \\\\int _{\\\\partial T}|u-Q_1u|^2ds \\\\right)^{\\\\frac{1}{2}}S(v_b, v_b)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C\\\\left(h^{-2}\\\\Vert u-Q_1 u\\\\Vert ^2 + \\\\Vert u-Q_1 u\\\\Vert _1^2 \\\\right)^{\\\\frac{1}{2}}S(v_b, v_b)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}.\\\\end{split}$ The third term on the right hand side of (REF ) can be bounded by using the usual error estimate for $L^2$ projections as follows: $\\\\begin{split}|((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })| = & |((Q_0-I) \\\\nabla u, (Q_0-I) ({\\\\mathfrak {s}}(v_b){\\\\beta }))| \\\\\\\\\\\\le & \\\\Vert (Q_0-I) \\\\nabla u\\\\Vert \\\\ \\\\Vert (Q_0-I) ({\\\\mathfrak {s}}(v_b){\\\\beta })\\\\Vert \\\\\\\\\\\\le & C h^2 \\\\Vert \\\\nabla ^2 u\\\\Vert \\\\left( \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert + \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\right) \\\\\\\\\\\\le & C h^2 \\\\Vert \\\\nabla ^2 u\\\\Vert {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|},\\\\end{split}$ where we have used the estimates (REF ) and (REF ) in the last line.\",\n \"The last term on the right hand side of (REF ) can be estimated as follows: $\\\\begin{split}|(c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b))|\\\\le & \\\\Vert c\\\\Vert _\\\\infty \\\\Vert {\\\\mathfrak {s}}(Q_bu)-u\\\\Vert \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C (\\\\Vert {\\\\mathfrak {s}}(Q_bu)-Q_1 u\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C ( \\\\Vert {\\\\mathfrak {s}}(Q_bu)-{\\\\mathfrak {s}}(Q_1 u)\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C ( \\\\Vert {\\\\mathfrak {s}}(Q_bu-Q_1 u)\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C h^2 \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}.\\\\end{split}$ Substituting the estimates (REF )-(REF ) into the error equation (REF ) yields $\\\\kappa S(e_b, v_b) + {\\\\mathcal {B}}(e_b, v_b) \\\\le C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|},$ which, together with the coercivity (), leads to $\\\\Lambda {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}^2 \\\\le C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}.$ The last inequality implies the error estimate (REF ).\",\n \"Finally, from the triangle inequality and the error estimate (REF ), we obtain $\\\\begin{split}\\\\Vert \\\\nabla _w u_b - \\\\nabla u\\\\Vert \\\\le & \\\\Vert \\\\nabla _w (u_b - Q_b u)\\\\Vert + \\\\Vert \\\\nabla _w (Q_b u) - \\\\nabla u\\\\Vert \\\\\\\\= & \\\\Vert \\\\nabla _w e_b\\\\Vert + \\\\Vert Q_0 (\\\\nabla u) - \\\\nabla u\\\\Vert \\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2,\\\\end{split}$ which gives rise to (REF ).\",\n \"This completes the proof of the theorem.\"\n ],\n [\n \"Error Estimates in $L^2$\",\n \"We use the usual duality argument to derive an error estimate in $L^2$ for the numerical solutions arising from (REF ).\",\n \"The analysis to be presented is a modified version of those developed in [36], [24], [35].\",\n \"Consider the following auxiliary problem that seeks $\\\\Phi \\\\in H_0^1(\\\\Omega )$ such that $-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla \\\\Phi ) - \\\\nabla \\\\cdot ({\\\\beta }\\\\Phi ) + c\\\\Phi &=&\\\\chi \\\\quad {\\\\rm in}\\\\ \\\\Omega \\\\\\\\\\\\Phi &=&0\\\\quad {\\\\rm on}\\\\ \\\\partial \\\\Omega , $ where $\\\\chi \\\\in L^2(\\\\Omega )$ .\",\n \"Assume that the solution of the problem (REF )-() exists and has the $H^2$ -regularity: $\\\\Vert \\\\Phi \\\\Vert _2 \\\\le C \\\\Vert \\\\chi \\\\Vert ,$ where $C$ is a constant depending only on the domain and the coefficients $\\\\alpha , {\\\\beta }$ , and $c$ .\",\n \"Let $u\\\\in H^2(\\\\Omega )$ be the exact solution of (REF )-() and $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).\",\n \"Assume ${\\\\beta }\\\\in C^1(\\\\bar{\\\\Omega })$ and the conditions (REF ) and (REF ) are satisfied.\",\n \"Then, the following $L^2$ error estimate holds true $\\\\Vert u - {\\\\mathfrak {s}}(u_b)\\\\Vert \\\\le Ch^2\\\\Vert u\\\\Vert _2,$ provided that the meshsize $h$ is sufficiently small.\",\n \"On each element $T\\\\in {\\\\mathcal {T}}_h$ , we test (REF ) against the linear function ${\\\\mathfrak {s}}(e_b)$ to obtain $(\\\\chi , {\\\\mathfrak {s}}(e_b))_T &=& (\\\\alpha \\\\nabla \\\\Phi , \\\\nabla {\\\\mathfrak {s}}(e_b))_T + ({\\\\beta }\\\\Phi , \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\alpha Q_0(\\\\nabla \\\\Phi ), \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (Q_0({\\\\beta }\\\\Phi ), \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\alpha Q_0(\\\\nabla \\\\Phi ), \\\\nabla _w e_b)_T + (Q_0({\\\\beta }\\\\Phi ), \\\\nabla _w e_b)_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&& - \\\\langle \\\\alpha Q_0(\\\\nabla \\\\Phi )\\\\cdot {\\\\bf n}, e_b-{\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}- \\\\langle Q_0({\\\\beta }\\\\Phi )\\\\cdot {\\\\bf n}, e_b-{\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}$ By using $Q_0(\\\\nabla \\\\Phi ) = \\\\nabla _w (Q_b\\\\Phi )$ and $(Q_0({\\\\beta }\\\\Phi ), \\\\nabla _w e_b)_T = ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi )_T$ in the above equation, we have from summing over all $T\\\\in {\\\\mathcal {T}}_h$ that $\\\\begin{split}(\\\\chi , {\\\\mathfrak {s}}(e_b))=& (\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi ) + (c{\\\\mathfrak {s}}(e_b), \\\\Phi ) \\\\\\\\& - \\\\sum _T \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}- \\\\alpha Q_0(\\\\nabla \\\\Phi )\\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)-e_b\\\\rangle _{\\\\partial T}\\\\\\\\& -\\\\langle {\\\\beta }\\\\Phi \\\\cdot {\\\\bf n}- Q_0({\\\\beta }\\\\Phi )\\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)-e_b\\\\rangle _{\\\\partial T}.\\\\end{split}$ The last two terms on the right-hand side of (REF ) can be bounded by $C h \\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}$ through the Cauchy-Schwarz inequality.\",\n \"Thus, we have $\\\\begin{split}|(\\\\chi , {\\\\mathfrak {s}}(e_b))|\\\\le & |(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi ) + (c{\\\\mathfrak {s}}(e_b), \\\\Phi )| \\\\\\\\& + Ch\\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}\\\\\\\\\\\\le & |(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, {\\\\mathfrak {s}}(Q_b\\\\Phi )) + (c{\\\\mathfrak {s}}(e_b), {\\\\mathfrak {s}}(Q_b\\\\Phi ))| \\\\\\\\& + Ch\\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|},\\\\end{split}$ where have also used $\\\\Vert \\\\Phi -{\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\Vert \\\\le C h^2\\\\Vert \\\\Phi \\\\Vert _2$ .\",\n \"Now, recall that $(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, {\\\\mathfrak {s}}(Q_b\\\\Phi )) + (c{\\\\mathfrak {s}}(e_b), {\\\\mathfrak {s}}(Q_b\\\\Phi ))= {\\\\mathcal {B}}(e_b, Q_b\\\\Phi ),$ and from the error equation (REF ), we have $\\\\begin{split}{\\\\mathcal {B}}(e_b, Q_b\\\\Phi ) = & \\\\ell _u(Q_b\\\\Phi ) - \\\\kappa S(e_b, Q_b\\\\Phi )\\\\\\\\=&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-Q_b\\\\Phi \\\\rangle _{\\\\partial T}+\\\\kappa S(u_b,Q_b\\\\Phi )\\\\\\\\& + ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(Q_b\\\\Phi ){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(Q_b\\\\Phi )),\\\\end{split}$ The last two terms on the right-hand side of (REF ) have the following estimate: $\\\\left|((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(Q_b\\\\Phi ){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(Q_b\\\\Phi ))\\\\right| \\\\le C h^2 \\\\Vert u\\\\Vert _2 \\\\Vert \\\\Phi \\\\Vert _1.$ The second term, $\\\\kappa S(u_b,Q_b\\\\Phi )$ , can be dealt with as follows: $\\\\begin{split}\\\\kappa S(u_b,Q_b\\\\Phi ) = & \\\\kappa h^{-1} \\\\sum _T \\\\langle u_b-Q_b{\\\\mathfrak {s}}(u_b), Q_b\\\\Phi - Q_b{\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\rangle _{\\\\partial T}\\\\\\\\= & \\\\kappa h^{-1} \\\\sum _T \\\\langle u_b-Q_b{\\\\mathfrak {s}}(u_b), \\\\Phi - {\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\rangle _{\\\\partial T}\\\\\\\\\\\\le & \\\\kappa h^{-1} \\\\sum _T \\\\Vert u_b-Q_b{\\\\mathfrak {s}}(u_b)\\\\Vert _{\\\\partial T}\\\\Vert \\\\Phi - {\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\Vert _{\\\\partial T}\\\\\\\\\\\\le & C h ({|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|} + h\\\\Vert u\\\\Vert _2)\\\\Vert \\\\Phi \\\\Vert _2.\\\\end{split}$ As to the first term, we note from the definition of $Q_b$ and $\\\\Phi |_{\\\\partial \\\\Omega }=0$ that $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},\\\\Phi -Q_b\\\\Phi \\\\rangle _{\\\\partial T}= \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}},\\\\Phi -Q_b\\\\Phi \\\\rangle _{\\\\partial T}= 0.\\\\end{split}$ Thus, we have $\\\\begin{split}&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-Q_b\\\\Phi \\\\rangle _{\\\\partial T}\\\\\\\\= & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-\\\\Phi \\\\rangle _{\\\\partial T}\\\\\\\\\\\\le & C h^2 \\\\Vert u\\\\Vert _2\\\\Vert \\\\Phi \\\\Vert _2.\\\\end{split}$ Substituting (REF ), (REF ), and (REF ) into (REF ) yields the following estimate: $|{\\\\mathcal {B}}(e_b, Q_b\\\\Phi )| \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\Phi \\\\Vert _2,$ which, together with (REF ), leads to $|(\\\\chi , {\\\\mathfrak {s}}(e_b))| \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\Phi \\\\Vert _2 \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\chi \\\\Vert ,$ where the regularity assumption (REF ) has been employed in the last inequality.\",\n \"Next, from (REF ) and the $H^1$ error estimate (REF ) in Theorem , we have $|(\\\\chi , {\\\\mathfrak {s}}(e_b))| \\\\le Ch^2 \\\\Vert u\\\\Vert _2 \\\\Vert \\\\chi \\\\Vert ,$ which leads to $\\\\Vert {\\\\mathfrak {s}}(e_b\\\\Vert \\\\le C h^2 \\\\Vert u\\\\Vert _2.$ Finally, we arrive at $\\\\Vert u-{\\\\mathfrak {s}}(u_b)\\\\Vert \\\\le \\\\Vert u-{\\\\mathfrak {s}}(Q_bu)\\\\Vert + \\\\Vert {\\\\mathfrak {s}}(e_b)\\\\Vert \\\\le C h^2\\\\Vert u\\\\Vert _2,$ which completes the proof of the theorem.\"\n ],\n [\n \"Numerical Experiments\",\n \"The goal of this section is to numerically verify the error estimates developed in the previous sections for the numerical scheme (REF ).\",\n \"The following metrics are employed to measure the magnitude of the error function: $&&\\\\text{Discrete $L^2$-norm: }\\\\\\\\&&\\\\Vert u_b-u \\\\Vert _{0}=h\\\\left(\\\\sum _{i=1}^{n+1}\\\\sum _{j=1}^{n}|u_{i-\\\\frac{1}{2},j}-u(x_{i-\\\\frac{1}{2}},y_j)|^2 + \\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n+1}|u_{i,j-\\\\frac{1}{2}}-u(x_{i},y_{j-\\\\frac{1}{2}})|^2\\\\right)^{1/2},\\\\\\\\&&\\\\text{Discrete $H^1$-norm: }\\\\\\\\&&\\\\Vert u_b-u\\\\Vert _1= h \\\\left(\\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n} \\\\left|\\\\frac{u_{i+\\\\frac{1}{2},j}-u_{i-\\\\frac{1}{2},j}}{h} - \\\\frac{\\\\partial u}{\\\\partial x} (x_{i},y_{j})\\\\right|^2 \\\\right.\",\n \"\\\\\\\\&&\\\\qquad \\\\qquad \\\\qquad \\\\left.\",\n \"+ \\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n} \\\\left|\\\\frac{u_{i,j+\\\\frac{1}{2}}-u_{i,j-\\\\frac{1}{2}}}{h} - \\\\frac{\\\\partial u}{\\\\partial y} (x_{i},y_{j})\\\\right|^2 \\\\right)^{1/2} ,\\\\\\\\$ Our numerical experiments are conducted for the model problem (REF )-() on polygonal domains.\",\n \"The following set of test cases are considered: $\\\\left\\\\lbrace \\\\begin{split}&u=xy, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}1 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\1\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u= 3x^2+2xy, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}2 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\1\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u=\\\\sin (\\\\pi x)\\\\sin (\\\\pi y) + x^2-y^2, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}1 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\2\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u=\\\\sin (\\\\pi x)\\\\sin (\\\\pi y), \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}xy+1 &0\\\\\\\\0 & 3xy\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}x^3y+xy+1\\\\\\\\3x^2y+xy+2\\\\end{bmatrix},\\\\quad c=x^4y^2+xy+1;\\\\end{split}\\\\right.$ The right-hand side function $f$ and the Dirichlet boundary data $g$ are chosen to match the exact solution $u=u(x,y)$ for each test case.\",\n \"Table: Error and convergence performance of the SWG scheme () with κ=4.0\\\\kappa =4.0 and uniform square partitions on the unit square domain Ω=(0,1) 2 \\\\Omega =(0,1)^2.Table REF shows the performance of the SWG scheme for each of the above test problems with the stabilizer parameter $\\\\kappa =4$ on uniform square partitions.\",\n \"The results indicate that the numerical approximation is in the machine accuracy for the test problem (REF ) where the exact solution is a bilinear function.\",\n \"For the other three test problems, the numerical solutions have the optimal rate of convergence $r=2$ in the discrete $L^2$ norm and a superconvergence of order ${\\\\mathcal {O}}(h^2)$ in the discrete $H^1$ norm.\",\n \"The numerical results are consistent with the theoretical prediction in the discrete $L^2$ norm, but they outperform the theory in the discrete $H^1$ norm.\",\n \"It should be pointed out that the superconvergence theory in [18] was developed for the diffusion equation only; but a slight modification of the analysis there will yield a superconvergence of order ${\\\\mathcal {O}}(h^2)$ for the SWG solutions of the full convection-diffusion equation (REF )-().\",\n \"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\\\kappa on uniform square partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2.\"\n ],\n [\n \"On the influence of the stabilizer parameter\",\n \"The goal of this subsection is to test the influence of the stabilizer parameter $\\\\kappa $ on the numerical solutions.\",\n \"This part of the numerical experiment considers only the test cases (REF ) and (REF ) with the following six values of $\\\\kappa =0.01, \\\\ 0.1, \\\\ 1.0, \\\\ 4.0, \\\\ 6.0, \\\\ 20.0$ .\",\n \"The case of $\\\\kappa =0$ is not a viable choice , as it was not covered in the convergence theory.\",\n \"In fact, our computation does not suggest any convergence of the scheme when $\\\\kappa =0$ .\",\n \"Tables REF -REF illustrate the numerical performance of the SWG scheme with different values of the stabilizer parameter $\\\\kappa $ .\",\n \"Note that, for both test cases, the rate of convergence deteriorates as $\\\\kappa $ gets small (e.g.\",\n \"$\\\\kappa =0.01$ ), particulary on coarse finite element partitions, but the rate of convergence begins to improve when the meshsize $h$ gets small.\",\n \"Optimal rate of convergence and the supercovergence of order ${\\\\mathcal {O}}(h^2)$ are clearly shown in the tables when $\\\\kappa $ is away from 0 (e.g., $\\\\kappa \\\\ge 0.1$ ).\",\n \"The stability and accuracy of the SWG scheme is insensitive to the value of $\\\\kappa $ as long as it stays away from 0.\",\n \"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\\\kappa on uniform square partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2.\"\n ],\n [\n \"SWG with general polygonal partitions\",\n \"The SWG scheme was applied to the test problem (REF ) with general polygonal partitions.\",\n \"Table REF shows the error and convergence performance of the scheme on four types of polygonal partitions.\",\n \"The stabilization parameter was set as $\\\\kappa =4$ in all these tests.\",\n \"Optimal order of convergence in the discrete $L^2$ norm can be observed for each polygonal partition, but the superconvergence in the discrete $H^1$ norm was only seen for rectangular partitions.\",\n \"The table shows a numerical rate of convergence of $r=1$ in the discrete $H^1$ norm for three other type of partitions.\",\n \"The result is clearly in consistency with the error estimate developed in Section .\",\n \"Fig.\",\n \"REF illustrates the contour plots of the numerical solutions on different type of polygonal partitions.\",\n \"It also shows the shape of the polygonal elements in our computation.\",\n \"Table: Error and convergence performance of the SWG scheme () for the test problem () on general polygonal partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2, with κ=4\\\\kappa = 4.Figure: Comparison of numerical solutions obtained from SWG and the exact solution for the test problem () on various polygonal partitions of h=1/8h=1/8 and κ=1\\\\kappa =1.\"\n ],\n [\n \"Numerical results on a non-convex domain\",\n \"The SWG scheme with the stabilization parameter $\\\\kappa =4$ was applied to the test problem (REF ) on the L-shaped domain $\\\\Omega :=(-1,1)\\\\times (-1,1)/ (0,1)\\\\times (-1,0)$ partitioned into triangles or rectangles.\",\n \"The corresponding numerical results are summarized in Table REF , which shows a convergence of order ${\\\\mathcal {O}}(h^2)$ in the $L^2$ norm for both the triangular and rectangular partitions.\",\n \"A superconvergence of order ${\\\\mathcal {O}}(h^2)$ was observed in the discrete $H^1$ norm on rectangular partitions, while the optimal order of convergence with $r=1$ is confirmed numerically on triangular partitions.\",\n \"It should be pointed out that the $H^2$ -regularity assumption (REF ) is not valid for non-convex polygonal domains so that the optimal order of error estimate (REF ) is not known theoretically on the L-shaped domain.\",\n \"The numerical results therefore outperform the theory in the usual $L^2$ norm.\",\n \"Table: Error and convergence performance of the SWG scheme () for test case () on Lshape domain, κ=4\\\\kappa = 4.Figure: Comparison of numerical solution obtained from SWG and the exact solution for test case () on L-shaped domain with h=1/8h=1/8.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08667"}}},{"rowIdx":194,"cells":{"text":{"kind":"list like","value":[["Symmetries and Mass Degeneracies in the Scalar Sector"],["Abstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars.","In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum.","In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars.","We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars.","Here, CP4 is a generalized CP symmetry with the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.","The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields.","The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists).","A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices."],["Introduction","After the initial discovery of the Higgs boson in 2012[1], [2], certain anomalies in the Higgs data (which have since disappeared) motivated the exploration of the possibility that the 125 GeV Higgs signal was comprised of two nearly mass-degenerate scalar states[3], [4], [5], [6], [7], [8], [9], [10], [11].","Although the present Higgs data is consistent with the Standard Model[12], [13], [14], one cannot yet rule out the presence of a mass degenerate scalar state at 125 GeV[15].","In this work, we consider the implications of a mass degeneracy among two (or more) scalar states of an extended Higgs sector.","Such a mass degeneracy can be either accidental or the result of a symmetry.","A trivial example of such a phenomenon arises in any doublet extended Higgs model.","All such models possess a mass degenerate state, namely the charged Higgs boson, $H^\\pm $ .","Indeed, $H^+$ and $H^-$ are mass-degenerate due to the U(1)$_{\\rm EM}$ gauge symmetry.","Moreover, the $H^+$ and $H^-$ are distinguishable by their electric charge, which can be experimentally probed using photons.","Suppose that this probe were unavailable (or equivalently, suppose one could turn off electromagnetism).","In this case, would it be possible for an experiment to reveal the existence of a mass-degenerate scalar?","In this very simple example, one could not physically distinguish (on an event by event basis) between the two degenerate states that comprise the charged Higgs scalar.","Nevertheless, there would in principle be observables that are sensitive to the number of mass-degenerate scalar states present.","For example, in the CP-conserving two Higgs doublet model, the decay rate for the decay of a heavy CP-even neutral scalar, $H\\rightarrow H^+ H^-$ (if kinematically allowed) is proportional to the number of degrees of freedom in the final state.","If we express the charged Higgs field as a linear combination of real scalar fields, $H^\\pm =(\\phi _1\\pm i\\phi _2)/\\sqrt{2}$ , then the decay rate for $H\\rightarrow H^+ H^-$ is the (incoherent) sum of the decay rates for $H\\rightarrow \\phi _1\\phi _1$ and $\\phi _2\\phi _2$ .","These two rates are identical, and the sum yields a multiplicity factor of 2.","This multiplicity factor provides the experimental signal for mass-degenerate scalars.Note that the decay rate for $Z\\rightarrow H^+ H^-$ is equal to the decay rate for $Z\\rightarrow \\phi _1\\phi _2$ .","In this case, the off-diagonal nature of the $Z\\phi _1\\phi _2$ coupling implies that no multiplicity factor is present.","Nevertheless, one can still infer the existence of mass-degenerate states, since the decays $Z\\rightarrow \\phi _1\\phi _1, \\phi _2\\phi _2$ are forbidden by Bose statistics.","Apart from the trivial mass degeneracy of $H^\\pm $ , we would like to explore in this paper the possibility of exactly mass-degenerate neutral scalars and/or mass-degenerate charged Higgs pairs in extended Higgs sectors.","In each case, the critical questions to ask are: (i) is the origin of the exact mass degeneracy natural?","and (ii) how can the mass degenerate scalars be distinguished experimentally?","Exact mass degeneracies are natural if they are a consequence of an unbroken symmetry.","In particular, accidental mass degeneracies require an artificial fine-tuning of the scalar potential parameters, and in this sense we shall call them unnatural (and in our view not especially interesting).","If mass-degenerate states are present, it is of interest to determine how to probe them experimentally.","In some cases, one can identify the presence of mass degenerate states on an event by event basis.","In other cases, the only signal of the mass degeneracy is a measurable multiplicity factor that can be determined when averaging over initial state degeneracies and summing over final state degeneracies.For example, a quark of a given flavor is a mass degenerate state due to its three possible colors.","Although the color of a quark cannot be identified experimentally, the presence of the color degree of freedom can be experimentally verified by the color multiplicity factor (most famously exhibited in the observed cross section for $e^+e^-$ annihilation into quark-antiquark pairs.)","Our focus in this paper is extended multi-doublet Higgs sectors with mass-degenerate scalar states.","It is particularly instructive to discuss mass degeneracy in the scalar sector starting from the so-called Higgs basis.","This corresponds to a subset of all possible scalar field parameterizations in which only one Higgs doublet, denoted by $H_1$ , acquires a non-zero positive vacuum expectation value (vev), while all the other scalar fields of the Higgs basis ($H_2$ , $H_3,\\ldots H_n$ ) have zero vev [16], [17], [18], [19], [20].","The neutral and charged Goldstone bosons reside entirely in $H_1$ , as this is the only doublet that possesses a non-zero vev, together with a neutral Higgs field that, in the absence of mixing with the neutral fields of the other Higgs doublets, behaves like the Standard Model (SM) Higgs boson.","In this sense, this subset of scalar bases may be viewed as Standard Model aligned bases.","That is, the Higgs basis is actually a family of basis choices, since one is always free to perform an arbitrary U($n-1$ ) transformation among $H_2$ , $H_3,\\ldots H_n$ while preserving the vev of $H_1$ [21].","There is no loss of generality in choosing any particular scalar basis as a starting point.","In order to identify the physical neutral scalars one must diagonalize the corresponding scalar squared-mass matrix.","As expected, the end result is independent of the initial choice of basis for the scalar fields.","In section , we study under what circumstances there is an exact mass degeneracy in the familiar two-Higgs-doublet model (2HDM) [22], [23].","In this model there are three physical neutral fields and one charged field, so we only consider potential mass degeneracies among the neutral fields.One can also examine mass degeneracies between the charged Higgs boson and one of the neutral Higgs bosons.","For example, in a custodial symmetric 2HDM, the CP-odd Higgs scalar is degenerate with the charged Higgs boson [24], [25], [26], [27].","However, custodial symmetry is not an exact symmetry of the full electroweak Lagrangian.","Thus, given a custodial symmetric 2HDM scalar potential, any potential mass degeneracies between neutral and charged scalars is at best approximate.","We do not consider such mass degeneracies further in this work.","We begin our 2HDM analysis by studying possible mass degeneracies among the neutral scalar states of the inert doublet model (IDM) [28], [29].","The scalar potential of this model possesses a discrete $\\mathbb {Z}_2$ symmetry that is unbroken by the vacuum.","In this case, the CP-even neutral component of $H_1$ in the Higgs basis is a mass eigenstate whose tree-level couplings are precisely those of the SM Higgs boson.","The real and imaginary parts of the neutral component of $H_2$ are odd under the discrete $\\mathbb {Z}_2$ symmetry, and have opposite signs under CP.","We will denote these two neutral states by $H$ and $A$ , although there is no way to identify which of these two states is CP-even and which is CP-odd.Equivalently, one can propose two different definitions of CP (called, say, CPa and CPb), such that $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.","Either definition can be consistently used to define the CP symmetry of the bosonic sector of the IDM.","It is possible that $h$ is degenerate in mass with either $H$ or $A$ , but such mass degeneracies are accidental in nature since neither case can arise due to a symmetry.","Moreover, these mass-degenerate states are physically distinguishable, since $h$ is even whereas $H$ and $A$ are odd under the $\\mathbb {Z}_2$ symmetry.","In contrast, an exact mass degeneracy of $H$ and $A$ can arise if the $\\mathbb {Z}_2$ symmetry of the scalar potential is promoted to a continuous U(1) symmetry.","In our terminology, this mass degeneracy of $H$ and $A$ is natural.","Nevertheless, the two mass-degenerate states can still be physically distinguished due to the coupling of these states to $W^\\pm H^\\mp $ .","One can now extend the above analysis to an arbitrary 2HDM.","One can show that with one exception, all 2HDM mass degeneracies are accidental.","The one exceptional case of a natural mass degeneracy is precisely the case of $m_H=m_A$ in the IDM.","This conclusion can also be obtained by considering all possible symmetries of the 2HDM scalar potential.","Among these symmetries, we can identify those that can potentially guarantee the mass degeneracy of scalar states.","By examining the consequences of these symmetries, we again confirm that the only possible neutral scalar mass degeneracy in the 2HDM arises in the IDM as previously noted.","In section we consider possible mass-degeneracies in the three Higgs doublet model.","Using the previous 2HDM analysis of mass degeneracies of the IDM, we construct a three Higgs doublet model (3HDM) generalization of the IDM, which we call the replicated inert doublet model (RIDM).","In this model, two of the three Higgs doublets are inert, and four mass-degenerate scalar pairs exist (two involving the charged scalar states from the inert doublets and two involving the neutral scalar states from the inert doublets).","We can explicitly identify the symmetries that are responsible for these mass degeneracies.","We then investigate the possibility of adding new terms to the scalar potential that partially break these symmetries while preserving the mass degeneracies.","In this way, we arrive at a model first proposed by Ivanov and Silva [30].","The Ivanov and Silva scalar potential possesses a discrete subgroup of the continuous symmetries that govern the RIDM, that maintains the mass degeneracies of the RIDM.","This discrete subgroup is the generalized CP symmetry, CP4, which has the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.","The CP4 symmetry is distinguished from the conventional CP symmetry (denoted henceforth by CP2), which has the property that $(\\rm {CP}2)^2$ is the identity operator.","Some properties of specialized 3HDMs have also been analyzed recently in Ref.[31].","One of the most notable properties of the Ivanov-Silva (IS) model is that one can write down the most general CP4-invariant scalar potential with three Higgs doublets, which has the feature that at least one of the coefficients of the quartic terms of the scalar potential must be complex (with a nonvanishing imaginary part).","Indeed, as demonstrated explicitly in Appendix A, one cannot redefine the scalar fields within the family of Higgs bases such that all the coefficients of the scalar potential are real.","In this case, we say that no real Higgs basis exists.","This means that CP2 is not a symmetry of the IS scalar potential and vacuum.","In section , we identify the existence of a physical observable that is present if no CP2 symmetry exists that commutes with the CP4 symmetry of the IS model.However, this leaves open the possibility of the existence of a CP2 symmetry that does not commute with the CP4 symmetry [32]; in this case, a real Higgs basis exists and a conventional CP symmetry can be defined.","As an example, we focus on $Z$ decay into four inert neutral scalars (with some details relegated to Appendix B).","Nevertheless, the CP4 invariance guarantees that all CP-violating observables involving the Higgs/gauge boson sector of the theory must be absent.","For example, we provide an instructive analysis in section that shows how the CP4 symmetry of the IS model with no real Higgs basis ensures the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZW^+ W^-$ vertices up to three-loop order.","Finally, we state our conclusions in section ."],["2HDM mass degeneracies","Consider the 2HDM, consisting of two hypercharge-one, doublet scalar fields, $\\Phi _1$ and $\\Phi _2$ .","The most general gauge-invariant renormalizable scalar potential is $\\mathcal {V}&=& m_{11}^2 \\Phi _1^\\dagger \\Phi _1+ m_{22}^2 \\Phi _2^\\dagger \\Phi _2 -[m_{12}^2\\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}]+\\tfrac{1}{2}\\lambda _1(\\Phi _1^\\dagger \\Phi _1)^2+\\tfrac{1}{2}\\lambda _2(\\Phi _2^\\dagger \\Phi _2)^2+\\lambda _3(\\Phi _1^\\dagger \\Phi _1)(\\Phi _2^\\dagger \\Phi _2)\\nonumber \\\\&&\\quad +\\lambda _4( \\Phi _1^\\dagger \\Phi _2)(\\Phi _2^\\dagger \\Phi _1)+\\left\\lbrace \\tfrac{1}{2}\\lambda _5 (\\Phi _1^\\dagger \\Phi _2)^2 +\\big [\\lambda _6 (\\Phi _1^\\dagger \\Phi _1) +\\lambda _7 (\\Phi _2^\\dagger \\Phi _2)\\big ] \\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}\\right\\rbrace \\,.$ We shall assume that the minimum of the scalar potential is electric charge conserving, in which case only the neutral scalar fields possess a nonzero vacuum expectation value (vev), $\\langle \\Phi _i^0 \\rangle =v_i/\\sqrt{2}$ , where the $v_i$ are potentially complex.","The Fermi constant, $G_F$ fixes the value of $v^2\\equiv |v_1|^2+|v_2|^2=(\\sqrt{2}G_F)^{-1}\\simeq (246~{\\rm GeV})^2\\,.$ Employing a new scalar field basis consisting of two orthonormal linear combinations of $\\Phi _1$ and $\\Phi _2$ does not modify the physical predictions of the model.","One convenient choice is the Higgs basis, in which the redefined doublet fields (denoted below by $H_1$ and $H_2$ ) have the property that $H_1$ has a non-zero vev whereas $H_2$ has a zero vev [16], [17], [18].","In particular, we define new Higgs doublet fields: $H_1=\\begin{pmatrix}H_1^+\\\\ H_1^0\\end{pmatrix}\\equiv \\frac{1}{v}(v_1^* \\Phi _1+v_2^*\\Phi _2)\\,,\\qquad \\quad H_2=\\begin{pmatrix} H_2^+\\\\ H_2^0\\end{pmatrix}\\equiv \\frac{1}{v}(-v_2 \\Phi _1+v_1\\Phi _2)\\,.$ It follows that $\\langle H_1^0 \\rangle =v$ and $\\langle H_2^0 \\rangle =0$ .","The Higgs basis is uniquely defined up to an overall rephasing, $H_2\\rightarrow e^{i\\chi } H_2$ (which does not alter the fact that $\\langle H_2^0 \\rangle =0$ ).","In the Higgs basis, the scalar potential of Eq.","(REF ) is denoted as [19], [20]: $\\mathcal {V}&=& Y_1 H_1^\\dagger H_1+ Y_2 H_2^\\dagger H_2 +[Y_3H_1^\\dagger H_2+{\\rm h.c.}]+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2)^2+Z_3(H_1^\\dagger H_1)(H_2^\\dagger H_2)\\nonumber \\\\&&\\quad +Z_4( H_1^\\dagger H_2)(H_2^\\dagger H_1)+\\left\\lbrace \\tfrac{1}{2}Z_5 (H_1^\\dagger H_2)^2 +\\big [Z_6 (H_1^\\dagger H_1) +Z_7 (H_2^\\dagger H_2)\\big ] H_1^\\dagger H_2+{\\rm h.c.}\\right\\rbrace \\,,$ where $Y_1$ , $Y_2$ and $Z_1,\\ldots ,Z_4$ are real parameters, whereas $Y_3$ , $Z_5$ , $Z_6$ and $Z_7$ are potentially complex parameters.","Imposing the scalar potential minimum conditions yields, $Y_1=-\\tfrac{1}{2}Z_1 v^2\\,,\\qquad \\quad Y_3=-\\tfrac{1}{2}Z_6 v^2\\,.$"],["Mass degeneracies of the inert doublet model (IDM)","We wish to study the consequences of a 2HDM in which two or three of the neutral Higgs scalars are degenerate in mass.","For simplicity, we shall first specialize to the inert 2HDM (the so-called IDM)[28], [29] in which there is an exact discrete $\\mathbb {Z}_2$ symmetry that is preserved by the vacuum, under which all particles of the SM and one of the two Higgs doublet fields (which contains the observed Higgs boson) are even and the second Higgs doublet field is odd under the multiplicative discrete symmetry.","In particular, the discrete symmetry of the IDM is manifest in the Higgs basis, where we identify $H_1$ as even and $H_2$ as odd under the $\\mathbb {Z}_2$ symmetry.","It then follows that $Y_3=Z_6=Z_7=0$ .","The IDM scalar potential is CP-conserving since one can eliminate the phase of $Z_5$ (the only remaining potentially complex scalar potential parameter) by appropriately rephasing the Higgs basis field $H_2$ .","The Higgs basis fields are $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h+iG^0\\bigr ] \\end{pmatrix}\\,,\\qquad \\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix}\\,,$ where $G^\\pm $ and $G^0$ are the Goldstone bosons that provide the longitudinal degrees of freedom of the massive $W^\\pm $ and $Z^0$ gauge bosons.","The physical mass spectrum of the IDM is given by, $&&m_h^2=Z_1 v^2\\,,\\qquad \\qquad \\qquad \\qquad \\quad m^2_{H^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\\\&&m_{A}^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4- Z_5)v^2\\,,\\qquad \\,\\, m_H^2=m_A^2+Z_5 v^2\\,.$ For completeness, we exhibit the Higgs couplings of the IDM in the unitary gauge below (where the Goldstone fields are set to zero).","First, the interactions of the Higgs bosons and the gauge bosons are governed by,The photon field $A_\\mu $ should not be confused with the scalar field $A$ .","${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h^2+H^2+A^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right]H^+H^-\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)H^-(H+iA) +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\[8pt]{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H -12g[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA) +h.c.]","+[ieA+igcW(12-sW2) Z]H+$\\leftrightarrow $$\\partial $ H-, where $s_W\\equiv \\sin \\theta _W$ , $c_W\\equiv \\cos \\theta _W$ .","The trilinear and quadrilinear Higgs self-interactions are governed by ${L}_{3h}&=&-\\tfrac{1}{2}v\\bigl [Z_1 h^3+(Z_3+Z_4)h(H^2+A^2)+Z_5h(H^2-A^2)\\bigr ]-vZ_3hH^+H^-\\,, \\\\[6pt]{L}_{4h}&=&-\\tfrac{1}{8}\\bigl [Z_1 h^4+Z_2(H^2+A^2)^2+2(Z_3+Z_4)h^2(H^2+A^2)+2Z_5 h^2(H^2-A^2)\\bigr ]\\nonumber \\\\[6pt]&&-\\tfrac{1}{2}H^+ H^-\\bigl [Z_2(H^2+A^2+H^+ H^-)+Z_3 h^2\\bigr ].","$ The tree-level couplings of $h$ to SM particles obtained above are precisely those of the SM Higgs boson, corresponding to the exact Higgs alignment limit [33], [34], [35], [36], [37], [38], [39], [40], [41] (as expected in light of $Z_6=0$ ).","Moreover, an examination of the above couplings implies that $h$ is CP-even (to be identified with the SM Higgs boson) and $H$ and $A$ have opposite CP-quantum numbers (one is odd and the other is even) based on the $ZAH$ coupling.Under the rephasing, $H_2\\rightarrow iH_2$ , we note that $Z_5\\rightarrow -Z_5$ , $H\\rightarrow -A$ and $A\\rightarrow H$ .","One can check that the masses of $H$ and $A$ and their couplings are invariant under this rephasing.","Note that the CP is not uniquely defined by the IDM interactions, since two candidate definitions of CP exist (called CPa and CPb in footnote REF ), where $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.","Either definition of CP can be used consistently in exploring the phenomenology of the IDM.","Finally, we note that under the $\\mathbb {Z}_2$ symmetry of the IDM, the quarks and leptons can be chosen to be even.","Consequently, the tree-level couplings of $h$ to fermion pairs are identical to those of the SM Higgs boson, whereas $H$ , $A$ and $H^\\pm $ do not couple to the SM fermions.","We now examine the possibility of mass degeneracies in the IDM.","First, consider the case of $m_h=m_H$ or $m_h=m_A$ .","In this case, it is possible to physically distinguish between $h$ and its mass-degenerate partner due to their opposite $\\mathbb {Z}_2$ quantum numbers.","For example, since all SM bosons and fermions are even under the $\\mathbb {Z}_2$ symmetry, it follows that the gluon-gluon (via a top quark loop), $WW$ and $ZZ$ fusion processes can only produce $h$ whereas Drell-Yan production (via virtual $s$ -channel $Z$ exchange) can only produce $H$ in association with $A$ .","Hence, despite the mass degeneracy, the two mass-degenerate scalars are physically distinguishable.","Note that the mass degeneracy of $h$ and its scalar partner is not radiatively stable.","For example, if $h$ and $H$ are mass degenerate states, then the one-loop contributions to the $hh$ two-point function (such as $ZZ$ and $WW$ intermediate states) differ from the corresponding contributions to the $HH$ two-point function (e.g.","the $AZ$ intermediate state).","Indeed, the tree-level condition for the mass degeneracy of $h$ and $H$ , $Z_1v^2=Y_2+\\tfrac{1}{2}Z_{345}v^2,$ where $Z_{345}\\equiv Z_3+Z_4+Z_5$ , is unnatural; i.e., Eq.","(REF ) is not the result of some symmetry.The scenario where $m_h=m_H=m_A$ is a special case of the $h$ –$H$ mass degeneracy.","In the triply mass-degenerate scenario, $h$ is also distinguished from $\\phi ^\\pm $ by its U(1) charge, which is zero.","For example, there is no coupling of $ZW^\\pm H^\\mp h$ in contrast to the $Z W^\\pm H^\\mp \\phi ^\\pm $ couplings exhibited in Eq.","(REF ).","Second, consider the case of $m_H=m_A$ , which corresponds to $Z_5=0$ .","In this case, the IDM scalar potential possesses a continuous U(1) symmetry, which is not spontaneously broken by the vacuum.This case has also been noted in Ref. [41].","It is this symmetry that is responsible for the mass degenerate states $H$ and $A$ .","One can now define eigenstates of U(1) charge, $\\phi ^\\pm =\\frac{1}{\\sqrt{2}}\\bigl [H\\pm iA\\bigr ]\\,.$ The relevant interaction terms of $\\phi ^{\\pm }$ are ${L}_{\\rm int}&=&\\left[\\tfrac{1}{2}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{4c_W^2}Z_\\mu Z^\\mu \\right]\\phi ^+\\phi ^- +\\frac{ig}{2c_W}Z^\\mu \\phi ^-\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ + -g2[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ + +h.c.]","+eg2 (AW+H-+ + AW-H+-) -g2sW22cW(ZW+H-+ + ZW-H+-) - v(Z3+Z4)h+–12[Z2(+-)2+(Z3+Z4)h2+-]-Z2 H+ H-+- .","Although $\\phi ^\\pm $ are mass degenerate states, they can be distinguished.","For example, Drell-Yan production via a virtual $s$ -channel $W^+$ exchange can produce $H^+$ in association with $\\phi ^-$ , whereas virtual $s$ -channel $W^-$ exchange can produce $H^-$ in association with $\\phi ^+$ .","Thus, the sign of the charged Higgs boson reveals the U(1)-charge of the produced neutral scalar.","The origin of this correlation lies in the fact that, by construction, $H^+$ and $\\phi ^+$ both reside in $H_2$ , whereas $H^-$ and $\\phi ^-$ both reside in $H_2^\\dagger $ ."],["2HDM mass degeneracies beyond the IDM","Although the IDM is a rather special case among all possible 2HDMs, the conclusions concerning mass degeneracies are robust.","Allowing for the most general 2HDM scalar potential, the Higgs sector is CP-violating if the relative phases among $Z_5$ , $Z_6$ and $Z_7$ cannot be removed by rephasing the Higgs basis field $H_2$ .","It is convenient to introduce three invariant quantities[42], [18], [20], whose imaginary parts are given by,Basis-invariant expressions for the $J_i$ are given in Ref.[20].","$\\operatorname{Im}J_1=\\operatorname{Im}(Z_6^* Z_7)\\,,\\qquad \\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)\\,,\\qquad \\operatorname{Im}J_3=\\operatorname{Im}\\bigl [Z_5^*(Z_6+Z_7)^2\\bigr ]\\,.$ The Higgs sector of the 2HDM is CP-violating unless $\\operatorname{Im}J_1=\\operatorname{Im}J_2=\\operatorname{Im}J_3=0$ .","The origin of the CP-violation can either be explicit or spontaneous [43], [44].To test for explicit CP-violation, one must employ invariant quantities that are independent of the scalar field vacuum expectation values [45], [46], [47].","Note that the neutral scalar squared-mass matrix does not involve the Higgs basis parameter $Z_7$ .","In particular, $Z_7$ only enters in the Higgs boson cubic and quartic self-couplings.","Hence, if $\\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)=0$ , then the neutral Higgs mass eigenstates behave like eigenstates of CP in their tree-level interactions with the gauge bosons and fermions (independently of the values of $\\operatorname{Im}J_1$ and $\\operatorname{Im}J_3$ ).","Moreover, the neutral scalar squared-mass matrix breaks up into a block diagonal form consisting of a $2\\times 2$ block (whose diagonalization yields the two CP-even neutral scalars) and a $1\\times 1$ block (which yields the CP-odd neutral scalar).","Consider the possibility of mass degeneracies among the neutral scalars of the most general 2HDM.","We now recall a remarkable tree-level relation of the CP-violating 2HDM[48], [42], [49], $ \\operatorname{Im}J_2&={\\rm Im}(Z_5^* Z_6^2)=\\frac{2 s_{13}c_{13}^2 s_{12}c_{12}}{v^6}(m_2^2-m_1^2)(m_3^2-m_1^2)(m_3^2-m_2^2)\\,,$ where the $m_i$ ($i=1,2,3$ ) are the masses of the three neutral Higgs bosons of the 2HDM, $s_{12}\\equiv \\sin \\theta _{12}$ , $c_{12}\\equiv \\cos \\theta _{12}$ , etc., and $\\theta _{12}$ and $\\theta _{13}$ are invariant mixing angles that are associated with the diagonalization of the neutral Higgs squared-mass matrix in the Higgs basis.Details on the definition of the mixing angles and their relations to the Higgs basis scalar potential parameters can be found in Ref.[49].","However, we will not need any of these details for the present argument.","In Ref.","[50], the three CP-odd invariants $\\operatorname{Im}J_i$ have been expressed in terms of the neutral scalar masses and the couplings of the neutral Higgs mass eigenstates to charged pairs, $e_i$ $(H_iW^+W^-)$ and $q_i$ $(H_iH^+H^-)$ .","In particular,In Ref.","[42], the invariant defined in Eq.","(REF ) is called $J_1$ .","$ \\operatorname{Im}J_2=\\frac{2e_1 e_2 e_3}{v^9}(m_1^2-m_2^2)(m_2^2-m_3^2)(m_3^2-m_1^2).$ If any two of the three neutral Higgs bosons are mass-degenerate, then either Eq.","(REF ) or (REF ) implies that $\\operatorname{Im}J_2=0$ , and the corresponding neutral scalar mass-eigenstates will behave as states of definite CP in their interactions with gauge bosons and fermions.","Nevertheless, if $\\operatorname{Im}J_1\\ne 0$ and/or $\\operatorname{Im}J_3\\ne 0$ (which would imply that $\\operatorname{Im}Z_7\\ne 0$ in the Higgs basis where $Z_5$ and $Z_6$ are simultaneously real), then CP-violating Higgs self-couplings must be present.","Moreover, radiative corrections will generate a non-zero $\\operatorname{Im}J_2$ and yield neutral Higgs states of indefinite CP.","That is, if CP-violation in the scalar sector is present, the tree-level relation $\\operatorname{Im}J_2=0$ can only be realized via an artificial fine-tuning of the parameters.","Nevertheless, one can consider the implications of a tree-level mass degeneracy among the neutral Higgs scalars of the 2HDM.","The above discussion illustrates the power of using scalar basis invariant conditions to analyze the CP properties of multi-Higgs models [45], [46], [51], [52], [53].","In light of Eq.","(REF ), if a tree-level mass degeneracy among the neutral Higgs scalars is present, then it is possible to rephase the Higgs basis field $H_2$ such that $Z_5$ and $Z_6$ are simultaneously real.","Thus, in the analysis that follows, we shall analyze the most general 2HDM scalar potential assuming that $Z_5$ and $Z_6$ are real parameters.","The squared-masses of the charged Higgs boson, $H^\\pm $ , and the CP-odd Higgs boson, $A$ are given by,In the notation employed in Eqs.","(REF )–(), $h$ and $H$ [$A$ ] refer to the neutral scalars that behave as CP-even [odd] mass eigenstates in their tree-level interactions with the gauge bosons and fermions.","Indeed, CP-violating interactions are present in Eqs.","(REF ) and () if $\\operatorname{Im}Z_7\\ne 0$ .","$m_{H^\\pm }^2=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad \\quad m_A^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4-Z_5)v^2\\,.$ The squared-masses of the CP-even Higgs bosons, $h$ and $H$ are the eigenvalues of the $2\\times 2$ matrix, $\\mathcal {M}_H^2=\\begin{pmatrix} Z_1 v^2 & \\quad Z_6 v^2 \\\\ Z_6 v^2 & \\quad m_A^2+Z_5 v^2\\end{pmatrix}.$ That is, $m^2_{H,h}=\\frac{1}{2}\\biggl \\lbrace m_A^2+(Z_1+Z_5)v^2\\pm \\sqrt{[m_A^2-(Z_1-Z_5)v^2]^2+4Z_6^2 v^4\\,}\\biggr \\rbrace \\,.$ Mass degenerate states arise if one of the following two quantities is zero, $Z_5(m_A^2-Z_1 v^2)+Z_6^2 v^2=0\\quad \\text{or}\\quad \\bigl [m_A^2-(Z_1-Z_5)v^2\\bigr ]^2+4Z_6^2 v^4=0\\,,$ where $m_A^2$ is given by Eq.","(REF ).","The case of $m_h=m_H$ arises when the second condition given in Eq.","(REF ) is satisfied.","It then follows that $m_A^2=(Z_1-Z_5)v^2$ and $Z_6=0$ , and the latter then yields the IDM mass spectrum.","As in the case of the IDM, the mass degeneracy of $h$ and $H$ requires a fine tuning of the parameters shown in Eq.","(REF ).","In principle, it is possible that $Z_7\\ne 0$ , but in this case, $Z_6=0$ is not a natural condition since the $\\mathbb {Z}_2$ symmetry of the IDM is not present.","Nevertheless, even if one accepts the two fine tuned conditions needed in this scenario, the arguments presented above Eq.","(REF ) still apply.","Namely, $Z_6=0$ corresponds to the exact alignment limit (at tree-level), in which case the tree-level interactions of the Higgs bosons and gauge bosons are still the same as those of the IDM [cf. Eqs.","(REF )–(REF )], whereas the tree-level trilinear and quadrilinear Higgs self-interactions given in Eqs.","(REF ) and () are modified by the addition of the following terms, $\\delta {L}_{3h}&=&-\\tfrac{1}{4} v \\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)\\,, \\\\[5pt]\\delta {L}_{4h}&=&-\\tfrac{1}{4}\\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)h\\,,$ after rephasing the Higgs basis field $H_2$ such that $Z_5$ is real.","The cases $m_h=m_A$ or $m_H=m_A$ arise when the first condition given in Eq.","(REF ) is satisfied.","This condition also requires a fine-tuning of the parameters.","Moreover, approximate Higgs alignment (as suggested by the LHC Higgs data) is not achieved unless $m_A^2\\gg Z_1 v^2$ or $|Z_6|\\ll 1$ .","Nevertheless, the physical distinction of the mass degenerate states is due to the CP quantum numbers of the neutral scalar states (which are preserved in the tree-level Higgs interactions with gauge bosons and with fermions).","One can therefore distinguish between the corresponding production mechanisms of the degenerate scalars that are mediated by gauge boson fusion or Drell-Yan production via $s$ -channel gauge boson exchange.","Finally, we consider the triply mass-degenerate case of $m_h=m_H=m_A$ .","In this case, both conditions given in Eq.","(REF ) must be satisfied, which yields $Z_5=Z_6=0$ and $m_A^2=Z_1 v^2$ .","This leaves $Z_7$ as the only potentially complex parameter of the scalar potential in the Higgs basis.","Thus, one is free to rephase the Higgs basis field $H_2$ such that $Z_7$ is real, and we conclude that the Higgs scalar potential and vacuum must be CP-conserving.","However, as long as $Z_7\\ne 0$ , the triply mass-degenerate case is unnatural, since the $\\mathbb {Z}_2$ symmetry of the IDM is not present."],["Natural 2HDM mass degeneracies: a symmetry based approach","In sections REF and REF , we derived the conditions that yield mass degeneracies among the neutral scalars of the 2HDM by brute force.","Namely, we obtained explicit expressions for the neutral scalar masses and then derived the corresponding relations among Higgs basis parameters for which mass degeneracies were present.","We then checked whether any of these relations were a consequence of a symmetry, and if yes we concluded that the corresponding mass degeneracy was natural.","In this section, we will obtain the same result by considering all possible symmetries of the 2HDM scalar potential.","Since the complete list of such symmetries is known [54], [55], [56], [57], [58], [59], we can be sure that our catalog of natural mass degeneracies of the 2HDM is complete.","We shall make use of the classification of symmetries presented in Ref.","[56], which identifies three possible Higgs family symmetries, $\\mathbb {Z}_2$ , U(1) and SO(3), and three classes of generalized CP-symmetries, denoted by GCP1, GCP2 and GCP3, respectively, as summarized in Table .In Ref.","[56] the three classes of generalized CP transformations are denoted by CP1, CP2 and CP3 respectively.","This nomenclature for the generalized CP-symmetries is awkward, in light of the notation that will be employed in section .","To avoid confusion, we have appended the letter G (for “general”) in denoting the three classes of generalized CP transformations of the 2HDM.","In the GCP transformation laws of Table , we have introduced the conjugation symbol ${}^{\\scriptscriptstyle {\\bigstar }}$ , which when applied to an SU(2) multiplet of scalar fields is defined by $\\Phi ^{\\scriptscriptstyle {\\bigstar }}\\equiv \\bigl [\\Phi ^\\dagger \\bigr ]^{\\scriptstyle T}$$,where the dagger refers both to hermitian conjugation of the quantum field operator when acting on the Hilbert space, and to complex conjugate transpose when acting on an SU(2) multiplet of fields.\\vspace{-0.72229pt}\\begin{table}[hb!","]\\begin{tabular}{|cccc|}\\hline \\rule {0pt}{}symmetry & \\hspace{36.135pt} transformation law & \\hspace{50.58878pt} \\phantom{transformation law} & \\\\[2pt]\\hline \\rule {0pt}{}\\mathbb {Z}_2 & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow -\\Phi _2 &\\\\U(1) & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow e^{2i\\theta }\\Phi _2 &\\\\SO(3) & \\Phi _a\\rightarrow U_{ab}\\Phi _b & U\\in {\\rm U}(2)/{\\rm U}(1)_{\\rm Y}& (\\text{for $a$, $b=1,2$} )\\\\GCP1 & \\Phi _1\\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP2 & \\Phi _1\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP3 & \\Phi _1 \\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} \\cos \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }} \\sin \\theta &\\Phi _2 \\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }}\\sin \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }}\\cos \\theta & \\text{(for $0<\\theta <\\tfrac{1}{2}\\pi $)} \\\\[2pt]\\hline \\hline \\rule {0pt}{}\\Pi _2& \\Phi _1\\rightarrow \\Phi _2 & \\Phi _2\\rightarrow \\Phi _1 & \\\\[2pt]\\hline \\end{tabular}\\caption {\\small Possible symmetries of the 2HDM scalar potential that are respected by the SU(2)\\times U(1)_{\\rm Y} gauge kinetic terms of the scalar fields.","The corresponding symmetry transformation laws are given in a basis where the symmetry is manifest.","Note that a scalar potential that is invariant under the mirror discretesymmetry, \\Pi _2, is also invariant under the \\mathbb {Z}_2 in another scalar field basis~\\cite {Davidson:2005cw}.","}\\end{table}$ Table: Impact of the symmetries defined in Table on the coefficients of the 2HDM scalar potential [cf. Eq.","()] in abasis where the symmetry is manifest.","A short dash indicates the absence of a constraint.","A scalar potential that is simultaneously invariant under ℤ 2 \\mathbb {Z}_2 and Π 2 \\Pi _2 isalso invariant under GCP2 in another scalar field basis , .","Likewise, a scalar potential that is simultaneously invariant under U(1) and Π 2 \\Pi _2 is also invariant under GCP3 in another scalar field basis .","The symbol ⊕\\oplus is being used above to indicate that two symmetries are enforced simultaneously within the same scalar field basis.We shall not consider the seven additional accidental symmetries of the 2HDM scalar potential identified in Refs.","[58], [59], that utilized mixed Higgs family and generalized CP transformations that leave the SU(2) gauge kinetic terms of the scalar fields invariant.","An example of such a symmetry is the well known custodial symmetry that is respected by the 2HDM scalar potential when $m_{H^\\pm }=m_A$ [24], [25], [26], [27].","However, this class of symmetries is violated by the U(1)$_{\\rm Y}$ gauge kinetic term of the scalar potential (as well as by the Yukawa couplings that are responsible for mass differences between up and down-type fermions).","Hence, any exact mass degeneracies arising from these seven accidental symmetries will be spoiled, in the absence of an artificial fine tuning of the Higgs scalar potential parameters.In cases of accidental symmetries, i.e.","symmetries of the scalar potential that are not respected by the full theory, the would-be mass degeneracies are only approximate, with calculable mass splittings.","The possibility of such approximate mass degeneracies, although technically natural, is not the subject of this paper.","Possible natural mass degeneracy of the 2HDM must be the consequence of one of the symmetries listed in Table .","Starting from a generic scalar potential given by Eq.","(REF ), if the scalar potential respects one of the symmetries listed in Table , then a scalar basis is picked out in which the symmetry is manifest.","In this basis, the coefficients of the scalar potential are constrained according to Table REF .It can be shown that for each of the symmetries listed in Table REF , a scalar field basis exists in which all scalar potential parameters and the neutral scalar field vacuum expectation values are simultaneously real, in which case CP (as defined by GCP1 in Table ) is conserved by the scalar sector Lagrangian and vacuum.","It is straightforward to check that the possible discrete symmetries of the 2HDM, namely $\\mathbb {Z}_2$ , GCP1, GCP2 (or equivalently, $\\mathbb {Z}_2\\oplus \\Pi _2$ ), do not yield scalar potentials that lead to scalar mass degeneracies.","Thus, we henceforth focus on U(1), SO(3) and GCP3 (and the related U(1)$\\oplus \\Pi _2$ symmetry).","Given a 2HDM scalar potential with a Peccei-Quinn [U(1)$_{\\rm PQ}$ ] symmetry [60] (or equivalently the U(1) transformation specified in Table In Ref.","[60], a U(1)$_{\\rm PQ}$ transformation of the 2HDM scalar fields is given by $\\Phi _1\\rightarrow e^{-i\\theta }\\Phi _1$ and $\\Phi _2\\rightarrow e^{i\\theta }\\Phi _2$ .","The U(1) transformation specified in Table corresponds to combining the U(1)$_{\\rm PQ}$ transformation with a hypercharge U(1)$_{\\rm Y}$ transformation, $\\Phi _i\\rightarrow e^{i\\theta }\\Phi _i$ (for $i=1,2$ ).)","that is spontaneously broken by the vacuum, the scalar sector will contain a massless CP-odd (Goldstone) scalar [61], [62].","In such cases, no mass degeneracy is present (without further constraints on the scalar potential parameters).","However, if the U(1) symmetry is manifestly realized in the Higgs basis, then the U(1) symmetry is unbroken by the vacuum, resulting in a mass degeneracy between the two neutral scalars residing in the Higgs basis field $H_2$ .","Indeed, this has already been shown in Section REF [see text above Eq.","(REF )], in the case of the mass degeneracy, $m_H=m_A$ , of the IDM with $Z_5=0$ .","In the case of a 2HDM scalar potential with an SO(3) symmetry, the form of the scalar potential is invariant with respect to all possible changes of the scalar basis.","Hence, it follows that the scalar potential parameters in the Higgs basis satisfy $Y_1=Y_2$ , $Z_1=Z_2=Z_3+Z_4$ and $Y_3=Z_5=Z_6=Z_7=0$ .","Using Eqs.","(REF ), (REF ) and (), it follows that $m_H=m_A=0$ .","The presence of two massless (Goldstone) scalars is a consequence of the spontaneous breaking of the SO(3) global symmetry by the vacuum.","This scalar mass degeneracy is a special case of the mass degeneracy in the case of the IDM with $Z_5=0$ , in which additional constraints among the Higgs basis parameters result in the pair of massless scalar states.","Finally, let us consider the case of a 2HDM scalar potential with a GCP3 symmetry.","Suppose that the GCP3 symmetry is manifestly realized in a basis where $\\langle \\Phi _1^0 \\rangle =\\frac{v_1}{\\sqrt{2}}\\,,\\qquad \\quad \\langle \\Phi _2^0 \\rangle =\\frac{v_2}{\\sqrt{2}}e^{i\\xi }\\,,$ where $v_1$ and $v_2$ are positive.","We define $\\tan \\beta \\equiv v_2/v_1$ (so that $0<\\beta <\\tfrac{1}{2}\\pi $ ).","Then, in light of the constraints on the GCP3 scalar potential parameters given in Table REF , the scalar potential minimum conditions yield (e.g., see eqs.","(3)–(5) of Ref.","[63]), $m_{11}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\sin ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{22}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\cos ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{12}^2\\sin \\xi &=& v^2\\lambda _5 \\sin \\beta \\cos \\beta \\sin \\xi \\cos \\xi \\,.$ We can assume that $\\lambda _5\\ne 0$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.","Setting $m_{11}^2=m_{22}^2$ and $m_{12}^2=0$ then yields two conditions, $\\sin ^2 \\xi \\cos 2\\beta =0\\,,\\qquad \\quad \\sin \\xi \\cos \\xi \\sin 2\\beta =0\\,.$ Hence, there are two classes of vacua, $\\sin \\xi =0$ and $\\beta $ arbitrary ($0<\\beta <\\tfrac{1}{2}\\pi $ ) , $\\cos \\xi =0$ and $\\cos 2\\beta =0$ .","We now can calculate the parameters of the GCP3 scalar potential in the Higgs basis (e.g., see eqs.","(11)–(20) of Ref.","[63]) in the two Cases A and B defined above, $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1, \\qquad \\quad Z_3+Z_4=\\lambda _1-\\lambda _5,\\quad Z_5=\\lambda _5$ , $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1-\\lambda _5, \\quad Z_3+Z_4=\\lambda _1+\\lambda _5,\\quad Z_5=0$ .","In particular, $Z_1=Z_2=Z_3+Z_4+Z_5$ in Case A, and $Z_1=Z_2\\ne Z_3+Z_4$ (and $Z_5=0$ ) in Case B.","We can now make use of Eqs.","(REF ) and () [along with Eq.","(REF ) to eliminate $Y_2$ by virtue of $Y_1=Y_2$ ] to compute the neutral scalar mass spectrum in the two cases,Note that the positivity of the squared masses are consistent with the conditions on the 2HDM scalar potential parameters first obtained in Ref. [54].","$m_h^2=Z_1 v^2,\\quad m_H^2=0,\\quad m_A^2=(Z_3+Z_4-Z_1)v^2,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ , $m_h^2=Z_1 v^2,\\quad m^2_H=m^2_A=\\tfrac{1}{2}(Z_3+Z_4-Z_1) v^2,\\qquad \\,\\,\\, m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ .","Note that Cases A and B correspond to degenerate vacua, since in both cases the value of the scalar potential (in the Higgs basis) at its minimum is $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .","In light of Table REF , we can identify case A as corresponding to realizing a GCP3 symmetry in the Higgs basis,This means that we can relax the restrictions of $\\beta \\ne 0, \\tfrac{1}{2}\\pi $ in defining Case A.","Note that for $\\beta =\\tfrac{1}{2}\\pi $ one can simply interchange the definitions of $\\Phi _1$ and $\\Phi _2$ to recover the Higgs basis result (corresponding to $\\beta =0$ ).","and case B corresponding to realizing a U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.This result is not surprising given that U(1)$\\oplus \\Pi _2$ is equivalent to GCP3 in another scalar field basis.","In particular, case A exhibits a massless Goldstone boson corresponding to the spontaneous breaking of GCP3 by the vacuum.","In contrast, in case B, the GCP3 symmetry possesses a continuous U(1) subgroup, which is unbroken by the vacuum, that protects the mass degeneracy, $m_H=m_A$ .","Indeed, this case is again a special case of the IDM with $Z_5=0$ , where the additional constraints, $Y_1=Y_2$ and $Z_1=Z_2$ are imposed.","As a check, it is instructive to evaluate the consequences of a 2HDM scalar potential with a U(1)$\\oplus \\Pi _2$ symmetry that is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.","In the following, we employ primed coefficients, $\\lambda _i^\\prime $ to distinguish this case from the one above where the GCP3 symmetry is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.","Using the results of Table REF , the scalar potential minimum conditions yield, $m_{11}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\cos ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\sin ^2\\beta \\bigr ]\\,,\\\\m_{22}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\sin ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\cos ^2\\beta \\bigr ]\\,,$ under the assumption that $0<\\beta <\\tfrac{1}{2}\\pi $ .","We can assume that $\\lambda ^\\prime _1\\ne \\lambda ^\\prime _3+\\lambda ^\\prime _4$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.","Setting $m_{11}^2=m_{22}^2$ then yields $\\cos 2\\beta =0$ , with $\\xi $ arbitrary.","Without loss of generality, one can set $\\xi =0$ , since the scalar potential is unchanged under a rephasing of $\\Phi _2$ .","As before, we can now compute the scalar potential parameters in the Higgs basis, $&&Y_1=Y_2,\\qquad Y_3=Z_6=Z_7=0,\\qquad Z_1=Z_2=\\tfrac{1}{2}(\\lambda ^\\prime _1+\\lambda ^\\prime _3+\\lambda ^\\prime _4),\\nonumber \\\\&& Z_3+Z_4=\\lambda ^\\prime _1,\\qquad Z_5=\\tfrac{1}{2}(\\lambda ^\\prime _1-\\lambda ^\\prime _3-\\lambda ^\\prime _4)\\,.","$ In particular, note that $Z_1=Z_2=Z_3+Z_4-Z_5$ .","Using Eqs.","(REF ), (REF ) and (), we obtain $m_h^2=Z_1 v^2\\,,\\quad m_H^2=(Z_3+Z_4-Z_1)v^2\\,,\\quad m_A^2=0\\,,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2\\,,$ which is the same mass spectrum as Case A of the GCP3-symmetric scalar potential with $H$ and $A$ interchanged.","This result can be understood by noting that Eq.","(REF ) takes the standard form of the GCP3-symmetric scalar potential in the Higgs basis after rephasing the Higgs basis field $H_2\\rightarrow iH_2$ , which interchanges $H$ and $A$ and transforms $Z_5\\rightarrow -Z_5$ .","Finally, the case of $\\beta =0$ or $\\beta =\\tfrac{1}{2}\\pi $ must be treated separately and corresponds to a manifest realization of the U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.","This vacuum is degenerate with the one considered above, since in both cases, $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .","Indeed, this latter case corresponds to Case B of the GCP3-symmetric scalar potential treated above, where the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .","In summary, massless scalar (Goldstone boson) states $A$ , $H$ or ($A$ , $H$ ) exist in the 2HDM with a scalar potential that exhibits, respectively, a U(1), GCP3 or SO(3) symmetry manifestly realized in a generic $\\Phi _1$ –$\\Phi _2$ basis, which agrees with the results of Table 2 of Ref. [59].","Nevertheless, in the special cases where U(1) or U(1)$\\oplus \\Pi _2$ are manifestly realized in the Higgs basis (the latter corresponding to the Case B solution of the GCP3-symmetric scalar potential), the corresponding U(1) subgroups of theses symmetries are not spontaneously broken by the vacuum, and the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .","In the case of the SO(3)-symmetric scalar potential, this mass degeneracy is realized by a pair of massless Goldstone boson states.","Thus, we conclude that mass-degenerate neutral scalars can arise naturally in the 2HDM only in the case of the IDM with $Z_5=0$ .","All other cases of mass-degenerate scalars require an artificial fine-tuning of the scalar potential parameters, in agreement with the analysis of section REF .","Furthermore, this conclusion is unaffected by the interactions of the scalars with the vector bosons.","Indeed, the Higgs boson–gauge boson interactions, ${L}_{\\rm int}$ , given by Eq.","(REF ) show that the global U(1) symmetry responsible for the mass degeneracy of $H$ and $A$ is an exact symmetry of ${L}_{\\rm int}$ .","Finally, as previously noted, the Higgs basis field $H_2$ of the IDM is odd whereas all other scalar, fermion and vector fields are even under the discrete $\\mathbb {Z}_2$ symmetry.","This can be achieved by employing Type-I Yukawa couplings [64] where fermions couple only to the Higgs basis field $H_1$ .","In this case, the global U(1) symmetry of the IDM scalar potential with $Z_5=0$ will also be respected by the Yukawa interactions.","However, a GCP3 [or equivalently U(1)$\\oplus \\Pi _2$ ] or SO(3) symmetry of the IDM scalar potential will be explicitly broken by the Yukawa interactions.","Hence, the U(1)-symmetric IDM is the only 2HDM for which an exact mass degeneracy of $H$ and $A$ can be preserved."],["3HDM mass degeneracies and the Ivanov Silva model","In extended Higgs sectors with more than two scalar doublets, it is now possible to have mass-degenerate charged Higgs pairs as well as mass-degenerate neutral scalars [65].","In this section, we explore new phenomena associated with mass degenerate scalars that arises for the first time in the three-Higgs doublet model (3HDM).","As a warmup exercise, we return to the IDM and add a second inert doublet and consider possible mass degeneracies among the scalar fields of the two inert doublets.","We then perturb the resulting model to obtain a version of the 3HDM that is equivalent to a model first introduced by Ivanov and Silva[30]."],["The replicated inert doublet model (RIDM)","The IDM introduced in section REF can be generalized by introducing additional inert scalar doublets.","In this section, we consider a 3HDM that consists of two inert hypercharge-one electroweak doublets, in which the inert doublets contain mass-degenerate scalar states.","The resulting models shall be called the replicated inert doublet model (RIDM).","As in the case of the IDM, we work in the Higgs basis in which the first Higgs doublet field $H_1$ contains the SM Higgs boson.","The RIDM consists of $H_1$ , with $\\langle H_1^0 \\rangle =v/\\sqrt{2},$ and two inert doublet fields $H_2$ and $H_3$ , with $\\langle H_2 \\rangle =\\langle H_3 \\rangle =0$ , and a scalar potential given by, $\\mathcal {V}_{\\rm RIDM}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\left(H_2^\\dagger H_2 +H_3^\\dagger H_3\\right)+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2+H_3^\\dagger H_3)^2\\nonumber \\\\&&+Z_3(H_1^\\dagger H_1)\\left(H_2^\\dagger H_2+H_3^\\dagger H_3\\right)+Z_4\\left[( H_1^\\dagger H_2)(H_2^\\dagger H_1)+( H_1^\\dagger H_3)(H_3^\\dagger H_1)\\right]\\nonumber \\\\&&+\\tfrac{1}{2}Z_5\\left\\lbrace (H_1^\\dagger H_2)^2 +(H_2^\\dagger H_1)^2+(H_1^\\dagger H_3)^2 +(H_3^\\dagger H_1)^2\\right\\rbrace \\,.$ Without loss of generality, we have chosen $Z_5$ real and non-negative in Eq.","(REF ), which is always possible by an appropriate rephasing of the scalar fields $H_2$ and $H_3$ .","Hence, if follows that the bosonic sector of the RIDM is CP-conserving.","The charged and neutral components of the Higgs basis doublet fields of the RIDM are also mass eigenstate fields, $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h_{\\rm SM}+iG^0\\bigr ] \\end{pmatrix},\\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix},\\quad H_3=\\begin{pmatrix} h^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [h+ia\\bigr ]\\end{pmatrix},$ with a minor change of notation from the IDM.","The corresponding squared masses of the neutral and charged scalars are given by, $m^2_{H^\\pm }&=&m^2_{h^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad m^2_H=m^2_h=Y_2+\\tfrac{1}{2}(Z_3+Z_4+Z_5)v^2\\,,\\nonumber \\\\m^2_A&=&m^2_a=Y_2+\\tfrac{1}{2}(Z_3+Z_4-Z_5)v^2\\,.$ By assumption, $Z_5\\ge 0$ , in which case $m_H=m_h\\ge m_A=m_a$ .In particular, note that if $Z_5=0$ then there is an enhanced mass degeneracy in which $m_H=m_h=m_A=m_a$ .","Thus, the RIDM possesses four mass-degenerate scalar pairs: $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ .","These mass degeneracies can be understood as a consequence of a continuous global Higgs flavor symmetry (where Higgs flavor corresponds to the multiplicity of Higgs doublets).","In order to explicitly exhibit the relevant symmetries, it is convenient to focus on the neutral scalar states of the doublet fields $H_2$ and $H_3$ , denoted henceforth by the complex fields, $H^0\\equiv \\frac{H+iA}{\\sqrt{2}}\\,,\\qquad \\quad h^0\\equiv \\frac{h+ia}{\\sqrt{2}}\\,,$ respectively.","Let us first focus on the kinetic energy terms and the terms in Eq.","(REF ) in the absence of the term proportional to $Z_5$ .","Then, one can check that the neutral complex scalar fields $H^0$ and $h^0$ appear only in the combination $H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0=\\tfrac{1}{2}(H^2+h^2+A^2+a^2)$ .","Thus, excluding $Z_5$ , the scalar Lagrangian possesses an O(4) global symmetry, that is responsible for four mass-degenerate neutral scalar states.","It is instructive to see how this symmetry arises when employing the complex basis $\\varphi _i=\\lbrace H^0,h^0\\rbrace $ (for $i=1,2$ ).","Noting that $\\varphi ^{\\dagger \\,i}\\varphi _i=H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0$ (the sum over the repeated index $i$ is implicit), it is clear that the scalar Lagrangian (in the absence of $Z_5$ ) is invariant under a U(2) global symmetry, $\\varphi _i\\rightarrow U_i{}^j\\varphi _j$ , with $U\\in {\\rm U(2)}$ .","However, the corresponding symmetry group is in fact larger than U(2).","Working in the complex basis, it is straightforward to verify that the quantity $\\varphi ^{\\dagger \\,i}\\varphi _i$ is invariant with respect to $\\varphi _i\\rightarrow U_i{}^j\\varphi _j+(V^*)^i{}_j\\varphi ^{\\dagger \\,j}\\,,$ where $U$ and $V$ are complex $2\\times 2$ matrices (which are not in general unitary), provided that the following two conditions are satisfied: $&&(i)~~~~(U^\\dagger U+V^\\dagger V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii)~~~V^T U~\\hbox{\\rm is an antisymmetric matrix}\\,.$ One can now check that Eq.","(REF ) corresponds to an O(4) symmetry transformation.","More explicitly, the $4\\times 4$ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}\\operatorname{Re}(U+V) &\\,\\, -\\operatorname{Im}(U+V) \\\\\\operatorname{Im}(U-V) & \\phantom{-}\\,\\, \\operatorname{Re}(U-V) \\end{array}\\right)$ is an orthogonal matrix if and only if $U$ and $V$ satisfy Eqs.","(REF ) and ().","Indeed, one can check that in light of Eqs.","(REF ) and (), the global symmetry specified by Eq.","(REF ) is governed by 6 continuous parameters as expected for an O(4) transformation.","Two special cases of Eq.","(REF ) are noteworthy.","First, if $V=0$ , then $U$ is unitary and we recover the U(2) global symmetry mentioned previously.","Second, if $U=0$ then Eq.","(REF ) corresponds to a generalized CP transformation [cf. Eq.","(REF )].The unified treatment of Higgs family transformations and generalized CP transformations has been advocated previously in Ref. [58].","A related discussion emphasizing the promotion of the U(2) basis transformation to an enlarged group of O(4) transformations appears in Ref. [66].","Both symmetries are present in the scalar Lagrangian if the $Z_5$ coupling is neglected, and either one would be sufficient to guarantee the mass degeneracy of $H$ , $h$ , $A$ and $a$ .","In the absence of the $Z_5$ coupling, the full O(4) global symmetry is respected by the pure scalar Lagrangian.","However, when we include the coupling of the scalar doublets to the gauge bosons, one must replace the ordinary derivative, $\\partial _\\mu $ , with the SU(2)$\\times $ U(1) gauge covariant derivative, $D_\\mu $ , in the scalar kinetic energy term.","The resulting coupling of the scalars to the vector bosons partially breaks the O(4) symmetry.","Employing the complex basis, it is easy to check that the symmetry transformation specified by Eq.","(REF ) is unbroken if and only if either $U=0$ or $V=0$ , namely the two special cases just highlighted above.This result is not surprising given that Eq.","(REF ) transforms the scalar field into a linear combination of two fields of opposite hypercharge unless either $U=0$ or $V=0$ .","That is, the kinetic energy term $(D^\\mu \\varphi )^{i\\,\\dagger } (D_\\mu \\varphi )_i$ is invariant under a U(2) symmetry (corresponding to $V=0$ ) and under the generalized CP symmetry (corresponding to $U=0$ ).","Mathematically, the unbroken global symmetry that remains is the semi-direct product U(2)$\\rtimes \\mathbb {Z}_2$ .This symmetry is a generalization of the U(1) symmetry (and the associated CP symmetry) of the IDM with $Z_5=0$ treated in section REF , and provides the motivation for our choice of the RIDM scalar potential given in Eq.","(REF ).","We now examine the consequence of including the term of Eq.","(REF ) proportional to $Z_5$ .","Focusing again on the neutral complex scalar fields $H^0$ and $h^0$ [cf. Eq.","(REF )], we see that a new combination of fields arises, $\\varphi _i\\varphi _i+{\\rm h.c.}=(H^0)^2+(H^{0\\,\\dagger })^2+(h^0)^2+(h^{0\\,\\dagger })^2$ .","This term is invariant with respect to Eq.","(REF ) provided that the conditions specified in Eqs.","(REF ) and () are replaced by 1.25 $&&(i^\\prime )~~~~(U^T U+V^T V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii^\\prime )~~~V^\\dagger U~\\hbox{\\rm is an antihermitian matrix}\\,.$ The conditions specified by Eqs.","(REF ) and () are compatible with those of Eqs.","(REF ) and () if $U$ and $V$ are real matrices.","Consequently, $\\mathcal {Q}$ specified in Eq.","(REF ) is now a block diagonal orthogonal $4\\times 4 $ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}U+V &\\,\\, 0 \\\\0 & \\phantom{-}\\,\\, U-V \\end{array}\\right)\\,,$ where $(U\\pm V)^T(U\\pm V)={1}_{2\\times 2}$ as a consequence of Eqs.","() and (REF ).","That is, the scalar Lagrangian is invariant under a global O(2)$\\times $ O(2) symmetry, which explains the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.","The breaking of the four-fold mass degeneracy to the two mass-degenerate pairs is due to the scalar potential term proportional to $Z_5$ , as is evident from Eq.","(REF ).","Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is O(2)$\\rtimes \\mathbb {Z}_2$ .","For completeness, we note that the degeneracy of the charged Higgs scalars $(H^\\pm ,h^\\pm )$ is governed by the full O(4) symmetry, which is broken down to U(2)$\\rtimes \\mathbb {Z}_2$ after promoting the derivatives of the scalar kinetic energy term to gauge covariant derivatives.","This is easily seen by noting that in the unitary gauge (in which the Goldstone fields do not explicitly appear), the physical charged scalar fields do not appear in the scalar potential term proportional to $Z_5$ .","Finally, if $Z_4=Z_5=0$ , we can make use of the vertical SU(2) global symmetry (which when gauged corresponds to the SU(2) electroweak gauge group) to conclude that all eight charged and neutral inert scalars are mass-degenerate.","Next, we examine all the bosonic couplings of the RIDM in the unitary gauge (where the Goldstone fields are set to zero).","The Higgs boson interactions with the gauge bosons and the Higgs boson self couplings of the RIDM are listed below.","${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h_{\\rm SM}\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h_{\\rm SM}^2+H^2+h^2+A^2+a^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](H^+H^- +h^+ h^-)\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [H^-(H+iA)+h^-(h+ia)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu ( A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H+a $\\leftrightarrow $$\\partial $ h) -12g{iW+[ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA)+h-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (h+ia)] +h.c.}","+[ieA+igcW(12-sW2) Z](H+$\\leftrightarrow $$\\partial $ H-+h+$\\leftrightarrow $$\\partial $ h-), In the RIDM, there is no experimental measurement that can physically distinguish the degenerate scalars, $(H^\\pm ,h^\\pm )$ , $(H,h)$ and $(A,a)$ .","However, a multiplicity factor will appear after summing over final mass-degenerate states, e.g., $Z\\rightarrow HA$ , $ha$ doubles the rate into a pair of neutral scalars."],["An alternative basis choice for the RIDM","So far, our discussion has employed the $\\lbrace H_1,H_2,H_3\\rbrace $ basis of doublet scalar fields.","This is one choice among a family of Higgs bases defined such that $\\langle H_1^0 \\rangle =v/\\sqrt{2}$ and $\\langle H_2^0 \\rangle =\\langle H_3^0 \\rangle =0$ .","Indeed, the Higgs basis is unique only up to an arbitrary U(2) transformation of the doublet fields $H_2$ and $H_3$ .","In the following, we shall denote the $\\lbrace H_1,H_2,H_3\\rbrace $ basis as the $H23$ -basis, since the scalar potential of Eq.","(REF ) provides a simple 3HDM extension of the inert 2HDM.","It will prove useful to consider another choice of scalar field basis that is related to the $H23$ -basis as follows,Further details are provided in Appendix REF .","$&&\\mathcal {R}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2+iH_3\\bigr )=\\begin{pmatrix} R^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P+iQ^\\dagger \\bigr )\\end{pmatrix}\\,,\\nonumber \\\\[15pt]&&\\mathcal {S}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2-iH_3\\bigr )=\\begin{pmatrix} S^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P^\\dagger +iQ\\bigr )\\end{pmatrix}\\,.$ This defines the $\\lbrace H_1,\\mathcal {R},\\mathcal {S}\\rbrace $ basis of doublet scalar field, henceforth denoted as the $RS$ -basis.","Note that since the real neutral fields $(H,h)$ and $(A,a)$ are mass-degenerate pairs, respectively, one can combine the mass-degenerate real fields into complex fields, $P\\equiv \\frac{H+ih}{\\sqrt{2}}\\,,\\qquad Q\\equiv \\frac{A-ia}{\\sqrt{2}}\\,,$ where $M_P\\ge M_Q$ (in our convention where $Z_5\\ge 0$ ).The relative minus sign in the definition of the imaginary parts of $P$ and $Q$ has been introduced for later convenience.","The corresponding conjugate fields are $P^\\dagger \\equiv \\frac{H-ih}{\\sqrt{2}}\\,,\\qquad Q^\\dagger \\equiv \\frac{A+ia}{\\sqrt{2}}\\,,$ Likewise, since $H^\\pm $ and $h^\\pm $ are mass-degenerate charged fields, one is free to define, $R&=\\frac{H^- - ih^-}{\\sqrt{2}}, &\\quad S&=\\frac{H^- + ih^-}{\\sqrt{2}}, \\\\R^\\dagger &=\\frac{H^+ + ih^+}{\\sqrt{2}}, &\\quad S^\\dagger &=\\frac{H^+ - ih^+}{\\sqrt{2}},$ where $R$ and $S$ are negatively charged mass-degenerate scalars and the corresponding conjugate fields, $R^\\dagger $ and $S^\\dagger $ , are positively charged mass-degenerate scalars.","In the $RS$ -basis, the scalar potential is given by $\\!\\!\\!\\!\\!\\!\\mathcal {V}_{\\rm RIDM-RS}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\bigl (\\mathcal {R}^\\dagger \\mathcal {R} +\\mathcal {S}^\\dagger \\mathcal {S}\\bigr )+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}{\\bar{Z}}_2(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S})^2+Z_3(H_1^\\dagger H_1)(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S}) \\nonumber \\\\&& +Z_4\\bigl [( H_1^\\dagger \\mathcal {R})(\\mathcal {R}^\\dagger H_1)+( H_1^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger H_1)\\bigr ]+ \\bar{Z}^\\prime _5\\bigl [(H_1^\\dagger \\mathcal {R})(H_1^\\dagger \\mathcal {S}) +(\\mathcal {R}^\\dagger H_1)(\\mathcal {S}^\\dagger H_1)\\bigr ]\\,,$ where $\\bar{Z}_2=Z_2$ and $\\bar{Z}_5^\\prime =Z_5$ .The reason for introducing the notation $\\bar{Z}_2$ and $\\bar{Z}^\\prime _5$ in Eq.","(REF ) is clarified in section REF .","One can then rewrite the RIDM couplings given in Eqs.","(REF )–() in terms of the neutral scalar fields $P$ and $Q$ and the charged scalar fields $R$ and $S$ (and the corresponding conjugated fields), $&&\\hspace{-14.45377pt}{L}_{VVHH}=\\left[\\tfrac{1}{4} g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right]\\bigl (h^2_{\\rm SM}+2|P|^2+2|Q|^2\\bigr ) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](R^\\dagger R+S^\\dagger S)\\nonumber \\\\&&+\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [R(P+iQ^\\dagger )+S(P^\\dagger +iQ)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace ,\\\\&&\\hspace{-14.45377pt}{L}_{VHH}=\\frac{g}{2c_W}\\,Z^\\mu ( Q\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ P+Q $\\leftrightarrow $$\\partial $ P) +[ieA+igcW(12-sW2) Z] (R$\\leftrightarrow $$\\partial $ R+S$\\leftrightarrow $$\\partial $ S) -12g{iW+[R $\\leftrightarrow $$\\partial $ (P+iQ) +S $\\leftrightarrow $$\\partial $ (P+iQ)] +h.c.}","L3h=-12vZ1 hSM3-v[(Z3+Z4)hSM(|P|2+|Q|2)+Z5hSM(|P|2-|Q|2)] -vZ3hSM(RR+SS) , line .","L4h=-18 Z1 hSM4 -12Z2(|P|2+|Q|2)(|P|2+|Q|2+2RR + 2SS)-12(Z3+Z4) hSM2(|P|2+|Q|2) -12Z5 hSM2 (|P|2-|Q|2)-12[Z2(RR+SS) +Z3 h2SM] (RR+SS) ."],["Mass degeneracies beyond the RIDM","In this section, we add additional terms to the RIDM scalar potential while preserving the mass degeneracies of the model.","Naively, one can add to the RIDM scalar potential any gauge invariant quartic term involving the doublet fields $H_2$ and $H_3$ without upsetting the mass degeneracies of Eq.","(REF ).","However, the resulting tree-level mass degeneracies will be unnatural unless they are a consequence of a symmetry.","The simplest possible modification of the RIDM is to remove the $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ term entirely from the scalar potential.","That is, we can define a RIDM$^\\prime $ scalar potential as, $\\mathcal {V}_{\\rm RIDM^{\\prime }}= \\mathcal {V}_{\\rm RIDM}-Z_2(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)\\,.$ Note that the term in $\\mathcal {V}_{\\rm RIDM^{\\prime }}$ that is proportional to $Z_2$ is now given by $\\tfrac{1}{2}\\bigl [(H_2^\\dagger H_2)^2+(H_3^\\dagger H_3)^2\\bigr ]$ .","Indeed, one can argue that Eq.","(REF ) provides the simplest 3HDM generalization of the IDM.","In the case of the RIDM$^\\prime $ , the tree-level mass degeneracies are no longer a consequence of a continuous symmetry, which is now explicitly broken by the presence of the explicit term in Eq.","(REF ) that is proportional to $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ .","Indeed, this term is invariant only under a discrete subgroup of O(2)$\\times $ O(2) [which is the symmetry group of the RIDM scalar Lagrangian as discussed in section REF ].","In the notation of Eqs.","(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=g\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=g\\,,$ where $g$ is a $2\\times 2$ matrix that acts on the Higgs basis fields $H_2$ and $H_3$ regarded as a two dimensional vector.","Then, the term $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ is invariant under the two discrete subgroups above if $g\\in D_4\\mathchoice{\\cong }{\\cong }{\\vbox {{ \\scriptstyle \\hfil # \\hfil \\crcr \\sim \\crcr = \\crcr }}}{\\cong }\\lbrace {1},-{1}, R,-R,S,-S,Z,-Z\\rbrace $ , where 1 is the $2\\times 2$ identity matrix and $R=\\begin{pmatrix} 1 &\\,\\,\\,\\phantom{-}0 \\\\ 0 &\\,\\,\\, \\phantom{-}1\\end{pmatrix}\\,,\\qquad \\quad S=\\begin{pmatrix} 1 &\\,\\,\\, \\phantom{-}0 \\\\ 0 &\\,\\,\\,-1\\end{pmatrix}\\,,\\qquad \\quad Z=\\begin{pmatrix} 0 &\\,\\,\\, -1 \\\\ 1 &\\,\\,\\, \\phantom{-}0\\end{pmatrix}\\,.$ We recognize $D_4$ as the dihedral group of order eight, which is the symmetry group of the square [67].","Both discrete subgroups [Eqs.","(REF ) and ()] are isomorphic to $D_4$ .","Following the discussion below Eq.","(REF ), we conclude that the RIDM$^\\prime $ scalar Lagrangian is invariant under a discrete $D_4\\times D_4$ symmetry, which is responsible for the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.","Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is $D_4\\rtimes \\mathbb {Z}_2$ .","A comprehensive treatment of natural scalar mass degeneracies in the 3HDM would require a complete classification of 3HDM scalar potential symmetries, along the lines of the 2HDM analysis given in section REF .A complete catalog of all possible finite symmetry groups of the 3HDM is known (as well as some additional partial results); however the complete classification of all possible symmetries of the 3HDM remains an open problem [68], [69].","In this paper, we shall ask a less ambitious question: can one break the discrete symmetry identified above further while still naturally maintaining the mass-degenerate states of the RIDM.","The answer turns out to be affirmative.","This investigation led us to a particular 3HDM originally introduced by Ivanov and Silva [30] for other reasons that will be reviewed below.","The Ivanov-Silva (IS) model was constructed to exhibit a number of curious properties [30], [66], which appear to rely on the existence of degenerate states in the scalar spectrum.","In particular, the IS scalar potential does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ , where the latter satisfies $({\\rm CP})^2={1}$ , but instead respects a generalized CP symmetry of the form $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ for some unitary matrix $X$ .","In particular, the generalized CP symmetry of the IS scalar potential, denoted by CP4, is of order 4, signifying that $({\\rm CP4})^4={1}$ and $({\\rm CP4})^2\\ne {1}$ .","Moreover, no Higgs basis of scalar fields exists in which all the parameters of the IS scalar potential are simultaneously real.","As noted in section REF , this property is in stark contrast with the 2HDM in which the existence of any generalized CP symmetry implies that the 2HDM scalar potential automatically respects the conventional CP symmetry, i.e.","a basis of scalar fields exists such that the corresponding 2HDM scalar potential parameters are real [56], [57].","In Appendix A, we demonstrate that starting from the IS scalar potential, one can perform a basis change in order to obtain a more convenient form of the scalar potential.","By making an appropriate U(2) transformation to define the Higgs basis fields, $H_2$ and $H_3$ , we find that the IS scalar potential takes on the following form in the $H23$ -basis, $\\mathcal {V}_{\\rm IS}= \\mathcal {V}_{\\rm RIDM}+Z_3^\\prime (H_2^\\dagger H_2)(H_3^\\dagger H_3)+Z_4^\\prime (H_2^\\dagger H_3)(H_3^\\dagger H_2) +\\bigl [Z_8 (H_2^\\dagger H_3)^2+Z_9(H_2^\\dagger H_3)(H_2^\\dagger H_2-H_3^\\dagger H_3)+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM}$ is given in Eq.","(REF ).","In general, $Z_8$ and $Z_9$ are complex parameters.As shown in Appendix REF , one can perform an SO(2) rotation to redefine the fields $H_2$ and $H_3$ to remove the complex phase from either $Z_8$ or $Z_9$ .","We shall continue to use Eq.","(REF ) to express the Higgs basis fields in terms of mass-eigenstate fields.","Since none of the extra terms in Eq.","(REF ) involve the Higgs basis field $H_1$ , the tree-level mass relations of Eq.","(REF ) are not modified.","We now argue that the mass-degeneracies of $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ are stable due to the presence of a symmetry.","The O(2)$\\times $ O(2) symmetry of the RIDM (prior to gauging the scalar kinetic energy terms) that is responsible for the mass degeneracies among the neutral Higgs mass eigenstates is broken by the new terms beyond $\\mathcal {V}_{\\rm RIDM}$ contained in Eq.","(REF ).","Indeed, after the extra terms are included, no unbroken continuous subgroup of O(2)$\\times $ O(2) remains.","In the notation of Eqs.","(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=Z\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=Z\\,,$ where $Z$ is given by Eq.","(REF ).","The $2\\times 2$ matrix $Z$ acts on the Higgs basis fields $H_2$ and $H_3$ .","Both discrete subgroups [Eqs.","(REF ) and ()] are isomorphic to $\\mathbb {Z}_4=\\bigl \\lbrace {1},-{1},Z,-Z\\bigr \\rbrace $ .","Note that $Z^2=-{1}$ , where 1 is the $2\\times 2$ identity matrix.In this case, gauging the scalar kinetic energy terms does not reduce the symmetry group further.","Consider first the discrete symmetry defined in Eq.","(REF ).","The fields $H_2$ and $H_3$ are odd under $-{1}$ , which simply identifies the two inert doublets.","The elements $Z$ (and $-Z$ ) act non-trivially on the inert doublets.","However, Eq.","(REF ) is invariant with respect to $\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\rightarrow \\begin{pmatrix} 0 & -1\\\\ 1 & \\phantom{-}0\\end{pmatrix}\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\,,$ if and only if $Z_8$ and $Z_9$ are both real.","In the model of IS where there is an unremovable complex phase in the scalar potential, only the subgroup $\\mathbb {Z}_2=\\lbrace {1},-{1}\\rbrace $ of $\\mathbb {Z}_4$ survives.","In particular, the residual symmetry in this case is not sufficient to explain the mass degeneracies of the IS model.","The discrete symmetry defined in Eq.","() is a generalized CP symmetry.","In particular, the IS scalar potential is invariant under $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}\\,, \\qquad \\quad \\text{where $X=\\begin{pmatrix} 1 &\\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1 \\\\ 0 & \\phantom{-}1 & \\phantom{-}0\\end{pmatrix}$,}$ This symmetry, which is also isomorphic to $\\mathbb {Z}_4$ , is the CP4 symmetry advertised above.","Moreover, this discrete symmetry is sufficient to explain the mass degeneracies of the IS model (in the case of an unremovable complex phase in the IS scalar potential).","It is instructive to consider the Higgs couplings of the IS model.","Only the quartic Higgs couplings of the RIDM are modified as follows, $\\delta {L}_{4h}&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime )\\bigl [(H^2+A^2)(h^2+a^2)+4H^+ H^- h^+ h^-\\bigr ]-\\tfrac{1}{2}Z_3^\\prime \\bigl [(H^2+A^2)h^+ h^-+(h^2+a^2)H^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{2}Z^{\\prime }_4\\bigl [\\bigl (Hh+Aa+i(Ha-hA)\\bigr )H^+ h^- + \\bigl (Hh+Aa-i(Ha-hA)\\bigr )h^+ H^-\\bigr ] \\nonumber \\\\&& -\\tfrac{1}{4} Z_8\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]^2-\\tfrac{1}{4} Z^*_8\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]^2 \\nonumber \\\\&& -\\tfrac{1}{4} Z_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{4} Z^*_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]\\,.$ It is convenient to re-express the neutral scalar fields appearing in Eq.","(REF ) in terms of the complex neutral fields $P$ and $Q$ and their conjugates introduced in Eqs.","(REF ) and (REF ), and the charged fields $R$ and $S$ and their conjugates defined in Eqs.","(REF ) and ().","Note that the fields $P$ , $Q$ and the corresponding conjugate fields $P^\\dagger $ and $Q^\\dagger $ are each eigenstates of CP4.This means that each of the four states, $P$ , $Q$ , $P^\\dagger $ and $Q^\\dagger $ , are CP4-self conjugate (they are their own antiparticles).","Moreover, $P$ and the corresponding conjugate state $P^\\dagger $ are mass-degenerate, but are otherwise unrelated fields (and similarly for $Q$ and $Q^\\dagger $ ).","In particular, under a CP4 transformation, $P\\rightarrow iP$ , $Q\\rightarrow iQ$ , $P^\\dagger \\rightarrow -iP^\\dagger $ , and $Q^\\dagger \\rightarrow -iQ^\\dagger $ .","Likewise, under a CP4 transformation, $R\\rightarrow -iS^\\dagger $ , $R^\\dagger \\rightarrow iS$ , $S\\rightarrow iR^\\dagger $ and $S^\\dagger \\rightarrow -iR$ .","Note that these transformation properties are consistent with the requirement that $({\\rm CP}4)^4={1}$ .","We can evaluate the four-scalar interaction Lagrangian directly in the $RS$ -basis.","We first must rewrite Eq.","(REF ) in the RS-basis, $\\mathcal {V}_{\\rm IS-RS}= \\mathcal {V}_{\\rm RIDM-RS}+\\bar{Z}_3^\\prime (\\mathcal {R}^\\dagger \\mathcal {R})(\\mathcal {S}^\\dagger \\mathcal {S})+\\bar{Z}_4^\\prime (\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger \\mathcal {R}) +\\bigl [\\bar{Z}_8 (\\mathcal {R}^\\dagger \\mathcal {S})^2+\\bar{Z}_9(\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {R}^\\dagger \\mathcal {R}-\\mathcal {S}^\\dagger \\mathcal {S})+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM-RS}$ is given by Eq.","(REF ).","The relations between the unbarred and barred parameters are derived in Appendix REF , $\\bar{Z}_2&=&Z_2+\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\quad \\,\\,\\,\\,\\bar{Z}_3^\\prime =-Z_4^\\prime +2\\operatorname{Re}Z_8\\,,\\\\\\bar{Z}_4^\\prime &=&\\tfrac{1}{2}(Z_4^\\prime -Z_3^\\prime +2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\qquad \\qquad \\bar{Z}_5^\\prime =Z_5\\,,\\\\\\bar{Z}_8&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime +2\\operatorname{Re}Z_8)+i\\operatorname{Re}Z_9\\,,\\qquad \\quad \\bar{Z}_9= \\operatorname{Im}Z_9+i\\operatorname{Im}Z_8\\,.","$ The quartic interactions given in Eq.","(REF ) are then modified by employing the new definition of $\\bar{Z_2}$ given in Eq.","(REF ) and adding the following terms, We now consider the possible effects of the Yukawa interactions.","It is remarkable that it is possible to construct a CP4-invariant Yukawa interaction Lagrangian where the fermions transform nontrivially under a CP4 transformation [66], [70], [71].","In such a model, the mass degeneracies identified above that are a consequence of the CP4 symmetry are of course maintained.","Alternatively, if the fermions couple exclusively to the Higgs basis field $H_1$ (as in the case of the IDM), then the Yukawa interactions are invariant with respect to the $\\mathbb {Z}_2$ discrete symmetry defined below Eq.","(REF ),Note that this $\\mathbb {Z}_2$ symmetry is isomorphic to $(\\rm {CP}4)^2$ , which remains an exact symmetry of the model.","under which the inert doublet fields, $H_2$ and $H_3$ , are odd and all other fields of the model ($H_1$ , gauge bosons and fermions) are even.","However, the CP4 symmetry is no longer a symmetry of the complete model.","That is, if we define the CP4 transformation to be the conventional CP transformation when acting on the fermions and gauge fields, then the CP4 symmetry of the model will be violated by the presence of the unremovable CP-violating phase in the CKM mixing matrix.","Nevertheless, it is not clear whether this violation is sufficient to remove the scalar mass degeneracies of the IS model that were protected by the (now accidental) CP4 symmetry of the scalar potential.","This is an open question that we hope to revisit in a future work.","Finally, it is instructive to note that the scalar mass degeneracies of the CP4-invariant 3HDM is just the simplest example of a larger class of multi-Higgs models with degenerate scalars that are a consequence of a generalized CP symmetry.","In Ref.","[72], Ivanov and Laletin demonstrate how to construct $N$ Higgs doublet models with a generalized CP symmetry of order $2k$ (denoted by CP$2k$ ) with positive integer $k$ .","Nontrivial cases arise only for $2k=2^p$ with integer $p\\ge 1$ .","The simplest nontrivial models of this type (CP8 and CP16) require at least $N=5$ Higgs doublets.","Such models necessarily have mass-degenerate neutral scalars and mass-degenerate charged Higgs pairs.","A further exploration of models of this type is beyond the scope of this work."],["An observable distinction between CP2 and CP4","The distinction between the IS scalar potential in the $H23$ -basis with $Z_8$ and $Z_9$ real or complex is physical.In making this assertion, we have implicitly assumed that $Z_5\\ne 0$ .","The case of $Z_5=0$ , which is special due to the enhanced mass degeneracy noted in footnote REF , will be treated at the end of this section.","To demonstrate this assertion, we focus on the neutral scalar self-interactions in $\\delta {L}_{4h}$ that are linear in the fields $P$ or $Q$ (or their complex conjugates), $\\delta {L}_{4h}\\ni \\tfrac{1}{2}\\operatorname{Im}Z_8\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2-Q^2-P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]+\\tfrac{1}{2}i\\operatorname{Im}Z_9\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2+Q^2+P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]\\,,$ where we have used Eq.","() to re-express $\\bar{Z}_9$ [which appears in Eq.","(REF )] in terms of the $H23$ -basis parameters, $\\operatorname{Im}Z_8$ and $\\operatorname{Im}Z_9$ .","Self-interaction terms of this type are absent if $Z_8$ and $Z_9$ are both real.","Hence, the presence of these terms signals a CP4-symmetric IS scalar potential that does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ .","Here we provide two specific examples.","First, Eq.","(REF ) shows the existence of a $ZPQ$ interaction, which would permit the decay $Z\\rightarrow PQ, P^*Q^*$ , if kinematically available.","Since $M_Q\\le M_P$ , let us further suppose that $M_Q<\\tfrac{1}{4} m_Z0$ , defining the vacuum state.","Its $Q$ function is shown on the left in the top row of Fig.","REF , and the corresponding quasiprobability distribution is the original Glauber-Sudarshan distribution.","Furthermore, the case $s=0$ interestingly describes an operator with the maximally possible singularities a Glauber-Sudarshan distribution can exhibit [41]; see also the center plot in the top row of Fig.","REF for its $Q$ function.","There, we observe that the spread (i.e., variance) in phase space is reduced in all directions when compared to the vacuum state, further highlighting that it is a genuine quasistate that is incompatible with the uncertainty relation which holds true for any physical state.","The underlying phase-space distribution $P_{F^{-1}\\Omega F}$ is the Wigner-Weyl distribution.","Finally, the quasistates for the Husimi-Kano distribution are obtained in the limit $s\\rightarrow -1$ , and its corresponding $Q$ function is described by a singular Dirac distribution.","Beyond real-valued $s$ parameters, we can additionally consider complex values as well [58].","Then, the eigenvalues (REF ) are complex too.","The example $s=i$ is shown on the left of the bottom row in Fig.","REF .","While the amplitude $|Q|$ is identical with the vacuum state, the quasistate characteristics are clearly visible through the nontrivial phases, $\\arg Q\\ne 0$ ."],["Non-$s$ -parametrized quasistates","Moreover, we can also consider the quasistates for cases with $p,q\\ne 0$ .","As we demand $|q|=|p|$ , we have two extremal scenarios, $q=p^\\ast $ and $q=-p^\\ast $ .","In addition to the $s$ -parametrized quasiprobabilities in Table REF , we also listed the Agarwal-Wolf distributions.","They correspond to the case $q=-p^\\ast =\\mp 1$ (also, $s=0$ ).","The complex-valued $Q$ function of the resulting quasistate $\\hat{\\Delta }_{F^{-1}\\Omega F}$ is shown in the bottom-right plot of Fig.","REF .","The amplitude $|Q|$ is again compatible with the uncertainty principle bounded by the vacuum state, similarly to the previously discussed scenario of a complex $s$ parameter.","Here, however, the functional behavior of the phase $\\arg Q$ does not possess a radial symmetry.","Rather, the quasistates to the Agarwal-Wolf distributions exhibit hyperbolic isophase contours.","Note that this dismisses a sometimes held believe that Agarwal-Wolf distributions are an example of a $s$ -parametrized quasiprobability.","Finally, we can ask ourselves what happens in the case $q=p^\\ast $ .","An example is shown in the top-right plot in Fig.","REF .","Interestingly, the resulting quasistates are squeezed.","Specifically for a squeezing parameter $\\zeta $ , we can choose $s=\\frac{1+\\tanh ^2|\\zeta |}{1-\\tanh ^2|\\zeta |}\\text{ and }q=\\frac{-2e^{i\\arg \\zeta }\\tanh |\\zeta |}{1-\\tanh ^2|\\zeta |}=p^\\ast .$ This choice implies $\\bar{n}=0$ and means that the quasistates are pure squeezed states, cf.","Eqs.","(REF ) and (REF ).","Consequently, the quasiprobability density $P_{F^{-1}\\Omega F}(\\alpha )$ now describes the decomposition of the density operator in terms of displaced squeezed states with a coherent amplitude $\\alpha $ and a squeezing defined by $\\zeta $ .","This further implies that we have a generalized Glauber-Sudarshan quasiprobability distribution which is, however, a phase-space representation that expands the state in terms of squeezed states.","Thus, the case $q=p^\\ast $ significantly extends the notion of $s$ -parametrized distributions to additionally include arbitrary squeezed states."],["Quantum correlations","In a recent work [19], we generalized the concept of quasiprobability representations to more general notions of quantum coherence [62], beyond the specific example of harmonic oscillators studied in the previous section.","Among the various types of quantumness, quantum correlations between multiple degrees of freedom play an outstanding role for applications in quantum technologies [1], [63].","For this reason, let us study the entanglement of quantum systems within the framework of quasistates.","A comprehensive introduction to the theory of entanglement (likewise, inseparability) can be found in Ref.","[63]."],["Entanglement quasiprobabilities","A pure separable state in a multipartite system is described by a tensor-product vector, $|(a,b,\\ldots )\\rangle =|a\\rangle \\otimes |b\\rangle \\otimes \\cdots ,$ where $a\\in \\mathcal {S}_A$ , $b\\in \\mathcal {S}_B$ , etc.","In addition, the inclusion of classical correlations in terms of statistical mixtures, given by a classical probability density $P$ , yields the definition of a mixed separable state [64], $\\hat{\\rho }{=}\\int _{S_A{\\times } S_B{\\times }\\cdots } da\\,db\\cdots P(a,b,\\ldots )|(a,b,\\ldots )\\rangle \\langle (a,b,\\ldots )|.$ A state is defined to be inseparable (i.e., entangled) if it cannot be written in the form of Eq.","(REF ).","However, when we allow for $P$ to be a quasiprobability, even entangled states can be expanded in terms of separable states using Eq.","(REF ) [21].","A construction approach for bipartite entanglement quasiprobabilities was introduced in Ref.","[22].","As this approach can be generalized from the bipartite scenario to the multipartite case [19], we restrict ourselves to bipartite systems in the following.","The optimal decomposition of entangled states in terms of separable ones warrants that the entanglement is uniquely identified through negativities in $P(a,b)$ [22], [19].","The separable states $|(a,b)\\rangle $ needed for the decomposition are obtained from the solution of the so-called separability eigenvalue equations (SEEs), $\\hat{\\rho }_a|b\\rangle =g|b\\rangle \\text{ and }\\hat{\\rho }_b|a\\rangle =g|a\\rangle ,$ with $\\hat{\\rho }_a=\\mathrm {tr}_A[\\hat{\\rho }(|a\\rangle \\langle a|\\otimes \\hat{1}_B)]$ and $\\hat{\\rho }_b=\\mathrm {tr}_B[\\hat{\\rho }(\\hat{1}_A\\otimes |b\\rangle \\langle b|)]$ .","The SEEs approach generalizes the eigenvalue problem to composite systems while respecting the tensor-product structure of the corresponding eigenvectors and was initially introduced to construct entanglement witnesses [65], [66].","The solutions $(a,b)\\in \\mathcal {S}$ allow us to construct the distribution $P$ by solving the linear problem $G\\vec{p}=\\vec{g},$ where $G=[|\\langle (a,b)|(a^{\\prime },b^{\\prime })\\rangle |^2]_{(a,b),(a^{\\prime },b^{\\prime })\\in \\mathcal {S}}$ is the Gram-Schmidt matrix of separability eigenvectors.","Furthermore, the solution vector $\\vec{p}=[P(a,b)]_{(a,b)\\in \\mathcal {S}}$ defines the quasiprobabilities, and $\\vec{g}=[g]_{(a,b)\\in \\mathcal {S}}$ is the vector of separability eigenvalues, cf.","Eq.","(REF ).","Let us point out that the here-used Gram-Schmidt matrix also relates to the reconstruction approach in Sec.","[Eq.","(REF )] and is a result of a underlying principle of convex decomposition, discussed in greater detail in Ref.","[19].","One can always find a set $\\mathcal {S}_0$ such that $\\mathcal {S}\\subset \\mathcal {S}_0\\times \\mathcal {S}_0$ ; we then set $P(a,b)=0$ for all $(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0\\setminus \\mathcal {S}$ .","Using the construction to find optimal entanglement quasiprobabilities via the SEEs, we can decompose any state, be it entangled or separable [22], [19], as $\\hat{\\rho }=\\sum _{(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(a,b)|(a,b)\\rangle \\langle (a,b)|.$ It is worth mentioning that the entanglement quasiprobability is always real valued and normalized."],["Quasistate representation","A universally applicable approach to diminish the negativities in quasiprobabilities for a single system is a convolution of the form $P_K(\\chi )=rP(\\chi )+\\frac{1-r}{|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}P(\\psi ),$ with $0< r\\le 1$ and $|\\mathcal {S}_0|$ being the cardinality of the set; see Appendix for details.","The kernel under study consequently leads to quasistates of the form $\\hat{\\Delta }_K(\\chi )=\\frac{1}{r}|\\chi \\rangle \\langle \\chi |-\\frac{1-r}{r|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}|\\psi \\rangle \\langle \\psi |.$ We emphasize that quasistates defined in this manner are both Hermitian, $\\hat{\\Delta }_K(\\chi )=\\hat{\\Delta }_K(\\chi )^\\dag $ , and normalized, $\\mathrm {tr}[\\hat{\\Delta }_K(\\chi )]=1$ .","Thus, the only distinction to true states is the fact that $\\hat{\\Delta }_K(\\chi )$ is, in general, not a positive semidefinite operator.","The convolution can be generalized to composite systems via product kernels.","Choosing, for example, the same $r$ for both systems, we can reformulate Eq.","(REF ) as $\\hat{\\rho }=\\sum _{(\\tilde{a},\\tilde{b})\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P_{K\\otimes K}(\\tilde{a},\\tilde{b})\\,\\hat{\\Delta }_K(\\tilde{a})\\otimes \\hat{\\Delta }_K(\\tilde{b}).$ In particular, we have $\\begin{aligned}P_{K\\otimes K}(\\tilde{a},\\tilde{b})=&r^2P(\\tilde{a},\\tilde{b})+\\frac{(1-r)^2}{|\\mathcal {S}_0|^2}\\\\&+\\frac{r(1-r)}{|\\mathcal {S}_0|}\\left[P(\\tilde{a})+P(\\tilde{b})\\right],\\end{aligned}$ using the marginal distributions $P(\\tilde{a})=\\sum _{b\\in \\mathcal {S}_0}P(\\tilde{a},b)$ and $P(\\tilde{b})=\\sum _{a\\in \\mathcal {S}_0}P(a,\\tilde{b})$ .","Eventually, we get a nonnegative distribution $P_{K\\otimes K}$ for a sufficiently large $r$ , which is also demonstrated later.","This means that an entangled state can be decomposed according to Eq.","(REF ) in terms of a classical joint probability distribution $P_{K\\otimes K}(a,b)\\ge 0$ and tensor-product quasistates $\\hat{\\Delta }_K(a)\\otimes \\hat{\\Delta }_K(b)\\ngeq 0$ [cf.","Eq.","(REF )].","Since we can chose $r=0$ for separable states, we find that quasistates are not required in this case.","Therefore, we are able to conclude that entanglement is unambiguously identified in terms of classical joint probability distributions if and only if the tensor-product operators of the decomposition have to be unphysical quasistates."],["Example","As a proof of concept, let us consider an entangled two-qubit quantum state parametrized as $\\hat{\\rho }=\\frac{1}{4}\\left[\\hat{1}\\otimes \\hat{1}+\\sum _{j\\in \\lbrace x,y,z\\rbrace }\\rho (j)\\,\\hat{\\sigma }_j\\otimes \\hat{\\sigma }_j\\right].$ This corresponds to a physical density operator if and only if the real coefficients satisfy $[\\rho (x),\\rho (y),\\rho (z)]\\in \\mathrm {conv}\\lbrace [-1,1,1],[1,-1,1],[1,1,-1],[-1,-1,-1]\\rbrace $ , where the extremal points represent Bell states.","In Ref.","[19], we solved the SEEs [Eq.","(REF )] for this family of states.","In particular, the separability eigenvectors are tensor products of eigenstates of Pauli operators ($\\hat{\\sigma }_x$ , $\\hat{\\sigma }_y$ , and $\\hat{\\sigma }_z$ ).","This implies that we have $\\mathcal {S}_0=\\lbrace x_{+},x_{-},y_{+},y_{-},z_{+},z_{-}\\rbrace ,$ where $\\hat{\\sigma }_w|w_s\\rangle =s|w_{s}\\rangle $ for $w_s\\in \\mathcal {S}_0$ .","The exact entanglement quasiprobability reads [19] $P(a_s,b_t)=\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right],$ with $a,b\\in \\lbrace x,y,z\\rbrace $ and $s,t\\in \\lbrace +1,-1\\rbrace $ , identifying the separability eigenvectors, $(a_s, b_t)\\in \\mathcal {S}_0\\times \\mathcal {S}_0$ .","Of particular importance is the parameter $q=1-|\\rho (x)|-|\\rho (y)|-|\\rho (z)|$ as it determines if the state is separable; namely, $\\hat{\\rho }$ in Eq.","(REF ) is separable if and only if $q\\ge 0$ [19].","Conversely, the negativity of the quasiprobability in the case of entanglement is given by this quantity as well; it reads $\\min _{(u, v)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(u,v)=q/12\\ge -1/6$ , and the lower bound is attained for Bell states.","Figure: (Color online)The top panel depicts the entanglement quasiprobabilities [Eq.","()] for a Bell state with ρ(x)=ρ(y)=ρ(z)=-1\\rho (x)=\\rho (y)=\\rho (z)=-1.Each patch in the AA-BB plane corresponds to a product state |a〉〈a|⊗|b〉〈b||a\\rangle \\langle a|\\otimes |b\\rangle \\langle b|, with a,b∈𝒮 0 a,b\\in \\mathcal {S}_0 according to Eq.","().The bottom panel shows the eigenvalues of those separable-state projectors.The negativities in the quasiprobabilities PP identify the entanglement.Now we can apply the kernel in Eq.","(REF ).","Inserting the distribution in Eq.","(REF ) yields $\\begin{aligned}&P_{K\\otimes K}(a_s,b_t)\\\\= & r^2\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right]+\\frac{(1-r)^2}{36}\\\\&+\\frac{r(1-r)}{6}\\left[\\frac{q}{3}+\\frac{|\\rho (a)|}{2}+\\frac{|\\rho (b)|}{2}\\right].\\end{aligned}$ After some straightforward algebra, we find that $P_{K\\otimes K}$ is always nonnegative for $00$ and $\\mathrm {Re}[z^2-4xy]>0$ .","We use the branch with a nonnegative real part for the square root of a complex number.","The above relation can be extended to integrals of normally ordered expressions because under this prescription, we have an algebra of commuting operators.","In the case that operator ordering becomes relevant, let us formulate additional relations.","From the observations that $\\hat{a}^mf(\\hat{n})|n\\rangle =f(\\hat{n}+m)\\hat{a}^m|n\\rangle $ and $f(\\hat{n})\\hat{a}^{\\dag m}|n\\rangle =\\hat{a}^{\\dag m}f(\\hat{n}+m)|n\\rangle $ hold true, we conclude that $e^{x\\hat{a}^2}\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}e^{\\omega ^2x\\hat{a}^2}\\text{ and }\\omega ^{\\hat{n}}e^{y\\hat{a}^{\\dag 2}}=e^{\\omega ^2y\\hat{a}^{\\dag 2}}\\omega ^{\\hat{n}}.$ In the same manner, we get $\\exp (u\\hat{a})\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}\\exp (\\omega u\\hat{a})$ and $\\omega ^{\\hat{n}}\\exp (v\\hat{a}^\\dag )=\\exp (\\omega v\\hat{a}^\\dag )\\omega ^{\\hat{n}}$ .","Furthermore, using the above integral, $|\\alpha \\rangle \\langle \\alpha |={:}\\exp (-[\\hat{a}-\\alpha ]^\\dag [\\hat{a}-\\alpha ]){:}$ , and $\\pi \\hat{1}=\\int _{\\mathbb {C}}d\\alpha \\,|\\alpha \\rangle \\langle \\alpha |$ , we obtain $\\begin{aligned}&e^{x\\hat{a}^2}e^{y\\hat{a}^{\\dag 2}}=\\exp [x\\hat{a}^2]\\hat{1}\\exp [y\\hat{a}^{\\dag 2}]\\\\=&\\frac{1}{\\sqrt{1-4xy}}{:}\\exp \\left(\\frac{4xy\\hat{a}^\\dag \\hat{a}+x\\hat{a}^2+y\\hat{a}^{\\dag 2}}{1-4uv}\\right){:}\\\\=&e^{v\\hat{a}^{\\dag 2}/(1-4uv)}\\left(\\frac{1}{1-4uv}\\right)^{\\hat{n}+1/2}e^{u\\hat{a}^2/(1-4uv)}.\\end{aligned}$ The same approach yields $\\exp [u\\hat{a}]\\exp [y\\hat{a}^{\\dag 2}]=\\exp [y(\\hat{a}^\\dag +u)^2]\\exp [u\\hat{a}]$ and $\\exp [x\\hat{a}^2]\\exp [v\\hat{a}^\\dag ]=\\exp [v\\hat{a}^\\dag ]\\exp [x(\\hat{a}+v)^2]$ .","Finally, let us also mention the well-known formula $\\exp (u\\hat{a})\\exp (v\\hat{a}^\\dag )=\\exp (uv)\\exp (v\\hat{a}^\\dag )\\exp (u\\hat{a})$ ."],["Spectral decomposition of Gaussian quasistates","In the main part of this contribution, we consider filter functions of a Gaussian form [cf.","Eq.","(REF )].","The exact analysis for the resulting quasistates is performed here.","For zero displacement, the integral in Eq.","(REF ) yields $\\begin{aligned}\\hat{\\Delta }=&\\int _{\\mathbb {C}}d\\beta \\,\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}e^{-\\beta \\hat{a}^\\dag }e^{\\beta ^\\ast \\hat{a}}\\\\=&\\frac{2\\,{:}\\exp \\left(\\frac{-2[1+s]\\hat{a}^\\dag \\hat{a} +q\\hat{a}^{\\dag 2}+p\\hat{a}^2}{(1+s)^2-pq}\\right){:}}{\\sqrt{(1+s)^2-pq}}.\\end{aligned}$ To characterize this quasistate, let us consider its decomposition in terms of exponential operators.","First, the squeezing operator $\\hat{S}=\\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)$ can be equivalently given as [58] $\\hat{S}=\\exp \\left(-\\frac{\\tau }{2}\\hat{a}^{\\dag 2}\\right)\\sqrt{1-|\\tau |^2}^{\\hat{n}+1/2}\\exp \\left(\\frac{\\tau ^\\ast }{2}\\hat{a}^2\\right),$ where $\\tau =e^{i\\arg \\zeta }\\tanh |\\zeta |$ .","Note that $|\\tau |<1$ , and we have infinite squeezing for $|\\tau |\\rightarrow 1$ .","In addition, we consider thermal-state-like operator $\\begin{aligned}\\hat{W}=&(1-\\omega )\\omega ^{\\hat{n}}=\\sum _{n\\in \\mathbb {N}}(1-\\omega )\\omega ^{n}|n\\rangle \\langle n|\\\\=&(1-\\omega ){:}e^{-(1-\\omega )\\hat{n}}{:}\\\\=&\\int _{\\mathbb {C}}d\\alpha \\,\\frac{1-\\omega }{\\pi \\omega }\\exp \\left(-\\frac{(1-\\omega )|\\alpha |^2}{\\omega }\\right)|\\alpha \\rangle \\langle \\alpha |.\\end{aligned}$ See Ref.","[68] for a derivation and additional considerations.","Also note that the vacuum state $\\hat{W}=|0\\rangle \\langle 0|$ is obtained in the limit $\\omega \\rightarrow 0$ .","Now, we defined an operator $\\hat{T}$ to be compared with the quasistate $\\hat{\\Delta }$ , $\\hat{T}=\\hat{S}\\hat{W}\\hat{S}^\\dag .$ Using the exchange relations in Eqs.","(REF ) and (REF ), we find after some algebra a normally ordered expression, $\\begin{aligned}\\hat{T}=&(1-\\omega )\\sqrt{\\frac{1-|\\tau |^2}{1-|\\tau |^2\\omega ^2}}\\\\&\\times \\exp \\left(-\\frac{\\tau }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^{\\dag 2}\\right)\\\\&\\times {:}\\exp \\left(-\\frac{[1-\\omega ][1+\\omega |\\tau |^2]}{1-|\\tau |^2\\omega ^2}\\hat{n}\\right){:}\\\\&\\times \\exp \\left(-\\frac{\\tau ^\\ast }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^2\\right).\\end{aligned}$ Finally, we can compare this expression with Eq.","(REF ), $\\hat{\\Delta }=\\hat{T}$ .","Namely, equating coefficient yields $\\omega =\\frac{r-1}{r+1},\\,|\\tau |^2=\\frac{s-r}{s+r},\\,\\text{and}\\,\\frac{\\tau }{\\tau ^\\ast }=\\frac{q}{p},$ where $r=\\sqrt{s^2-pq}$ .","This also implies $\\tau =-q/(s+r)$ and $\\tau ^\\ast =-p/(s+r)$ .","Conversely, we can deduce parameters $s$ , $p$ , and $q$ from given values of $\\omega $ and $\\tau $ , $\\begin{aligned}p=\\frac{-2(1+\\omega )\\tau ^\\ast }{(1-\\omega )(1-|\\tau |^2)},\\quad q=\\frac{-2(1+\\omega )\\tau }{(1-\\omega )(1-|\\tau |^2)},\\\\\\text{and}\\quad s+1=\\frac{2(1+\\omega |\\tau |^2)}{(1-\\omega )(1-|\\tau |^2)}.\\end{aligned}$"],["Uniform attenuation","In the main text, we consider a specific example of a kernel.","For studying its properties, let us consider the corresponding convolution $P_K(\\chi )=\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )K(\\psi ,\\chi )=a\\,P(\\chi )+b\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )$ for a set $\\mathcal {S}$ of states.","Note that $\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )=1$ is the normalization condition.","To ensure that the result is normalized as well, $1=\\sum _{\\chi \\in \\mathcal {S}}P_K(\\chi )=a+b|\\mathcal {S}|$ , we find $b=\\frac{1-a}{|\\mathcal {S}|},$ using the cardinality $|\\mathcal {S}|$ of the set of states.","Further, when $0\\le a\\le 1$ is satisfied, the non-negativity of $P_K$ is guaranteed for any nonnegative $P$ .","The considered convolution can be formulated via the $|\\mathcal {S}|\\times |\\mathcal {S}|$ map $K=a\\,\\mathrm {id}+b\\,nn^\\mathrm {T},$ using the identity $\\mathrm {id}$ and the $|\\mathcal {S}|$ -dimensional vector $n=[1,\\ldots ,1]^\\mathrm {T}$ .","Then the inverse takes a similar form, $K^{-1}=\\frac{1}{a}\\mathrm {id}-\\frac{b/a}{a+b|\\mathcal {S}|}nn^\\mathrm {T},$ where $a+b|\\mathcal {S}|=1$ for the given $b$ .","Using the normalization, the convolution kernel $K$ corresponds to uniform addition of noise to the probabilities.","On the level of the operators, we get $\\hat{\\Delta }_K(\\chi )=\\frac{1}{a}|\\chi \\rangle \\langle \\chi |-\\frac{1-a}{a|\\mathcal {S}|}\\sum _{\\psi \\in \\mathcal {S}}|\\psi \\rangle \\langle \\psi |.$ In the continuous case, we can generalize the above relations as $P_K(\\chi )=\\int _{\\mathcal {S}} d\\psi \\,P(\\psi )\\left[a\\delta (\\psi ,\\chi )+\\frac{1-a}{|\\mathcal {S}|}\\right],$ where the volume $|\\mathcal {S}|=\\int _{\\mathcal {S}}d\\psi $ is used instead.","Furthermore, the projection operators transform as $\\hat{\\Delta }_{K}(\\chi )=\\int _{\\mathcal {S}} d\\psi \\left[\\frac{1}{a}\\delta (\\psi ,\\chi )-\\frac{1-a}{a|\\mathcal {S}|}\\right]|\\psi \\rangle \\langle \\psi |.$ As a discrete-variable example, we study a qubit system with orthogonal basis $\\lbrace |0\\rangle ,|1\\rangle \\rbrace $ .","We can identify the Pauli matrices as $\\hat{\\sigma }_z=|1\\rangle \\langle 1|-|0\\rangle \\langle 0|$ , $\\hat{\\sigma }_x=|0\\rangle \\langle 1|+|1\\rangle \\langle 0|$ , and $\\hat{\\sigma }_y=i|0\\rangle \\langle 1|-i|1\\rangle \\langle 0|$ .","The Hermitian operator basis can be completed with the operator $\\hat{1}=|0\\rangle \\langle 0|+|1\\rangle \\langle 1|$ .","The eigenvectors $|j_{\\pm 1}\\rangle $ of the Pauli matrices $\\hat{\\sigma }_j$ to the eigenvalues $\\pm 1$ for $j\\in \\lbrace z,x,y\\rbrace $ form the set $\\mathcal {S}$ .","Then we can decompose any density operator as $\\hat{\\rho }=\\sum _{j\\in \\lbrace x,y,z\\rbrace ,s\\in \\lbrace +1,-1\\rbrace }P(j_s)|j_{s}\\rangle \\langle j_{s}|.$ Conversely, the attenuated distribution reads $P_{K}(j_s)=aP(j_s)+(1-a)/6$ , and the transformed operators take the form $\\hat{\\Delta }_K(j_s)=a^{-1}|j_s\\rangle \\langle j_s|-(1-a)(2a)^{-1}\\hat{1}$ ."]],"string":"[\n [\n \"Quasistates and quasiprobabilities\"\n ],\n [\n \"Abstract The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator.\",\n \"While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently.\",\n \"In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates.\",\n \"In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system.\",\n \"Therefore, our method presents a complementary approach to prominent quasiprobability representations.\",\n \"We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light.\",\n \"In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.\"\n ],\n [\n \"Introduction\",\n \"Quantum systems can exhibit properties which have no analog in the classical realm, opening possibilities for technological innovations beyond the limitations posed by classical physics [1].\",\n \"For assessing the quantumness of a state, the bisection of a system into a classically accessible domain and a genuinely quantum-mechanical sector has become one of the major challenges of current research.\",\n \"A successful approach to performing such a desired separation are quasiprobability distributions, such as the Wigner-Weyl function [2], [3].\",\n \"Within this framework, a state is identified to be nonclassical when the corresponding quasiprobability does not have the properties of a classical probability distribution.\",\n \"Beyond this nonclassicality aspect, another purpose of quasiprobabilities is the full characterization of a quantum state, for example, using the Glauber-Sudarshan representation to describe quantized harmonic oscillators [4], [5].\",\n \"As such, the description of a density operator in terms of quasiprobabilities and classical reference states, e.g., coherent states, is an equally important feature of quasiprobabilities.\",\n \"However, despite having such a vital impact on the state's representation, the density operator expansion is, by far, less frequently considered.\",\n \"This is at least partially a result of a missing framework which focuses on the decomposition of density operators.\",\n \"Thus, the main objective of this contributions is to devise such a missing methodology that naturally leads to the useful and complementary concept of quasistates.\",\n \"The notion of quasiprobabilities has become one of the main hallmarks for certifying quantum features since it allows us to study resources for practical tasks.\",\n \"For instance, negative quasiprobabilities have been identified as indicators for the state's usefulness in performing quantum computation protocols [6], [7].\",\n \"Moreover, negativities in quasiprobabilities can be related to a broad range of concepts of nonclassicality, such as contextuality [8] and quantum entanglement [9].\",\n \"In addition, quasiprobabilities can be helpful to determine the actual probabilities of measurement outcomes [10], [11].\",\n \"Furthermore, quasiprobabilities can be defined for many relevant physical systems, such as continuous-variable harmonic oscillators [4], [5] or discrete-variable angular-momentum-type degrees of freedom [12], [13].\",\n \"Typically, the underlying quasiprobabilities are obtained by generalizing a classical phase space to the quantum domain; see Ref.\",\n \"[14] for an early study and Ref.\",\n \"[15] for a recent generalization.\",\n \"For example, quasiprobabilities in finite-dimensional systems can be introduced in this manner, cf., e.g., Refs.\",\n \"[16], [17], [18], [19].\",\n \"With regard to the quantum state representation, this typically requires us to identify pure reference states which are compatible with the properties of classical physics [20].\",\n \"Based on such classical reference states, a general quasiprobability decomposition of a quantum state can be constructed [19].\",\n \"Examples relevant for quantum technologies include the formulation of entanglement quasiprobabilities [21], [22], which become negative for quantum correlated states.\",\n \"Once a quasiprobability representation is constructed, it can be further generalized.\",\n \"Again, pioneering applications can be found in quantum optics, where the Glauber-Sudarshan and Wigner-Weyl quasiprobabilities can be unified in terms of so-called $s$ -parametrized quasiprobabilities [23], [24], which also include the well-known Husimi-Kano distributions [25], [26].\",\n \"This unification is achieved by a convolution of the quasiprobability with a Gaussian function and can be further generalized [27], [28], [29], including non-Gaussian scenarios [30], [31].\",\n \"A successful application of the $s$ -parameter approach in a finite-dimensional system has been reported as well [32].\",\n \"In addition, joint non-Gaussian quasiprobabilities for multiple optical modes and points in time have been introduced to study quantum correlated light [33], [34].\",\n \"Although such formal aspects led to profound insights into quantum physics when compared to classical statistical theories (see, e.g., [35], [36]), the method of quasiprobabilities is also of experimental importance.\",\n \"The most frequent implementation is the identification of nonclassical light in terms of negative Wigner-Weyl functions; see Refs.\",\n \"[37], [38] for early and recent examples.\",\n \"While the Glauber-Sudarshan distribution can be highly singular or even ambiguous [39], [40], [41], the reconstruction of this function is feasible in certain experiments as well [42].\",\n \"Moreover, the non-Gaussian generalization of this distribution is always experimentally accessible, such as reported for squeezed light [43].\",\n \"Even imperfect detection schemes can be employed for the reconstruction of nonclassical phase-space distribution of light [44].\",\n \"Conversely, a phase-space representation of the detector can experimentally verify quantum properties of the detection device used [45].\",\n \"Another approach [46], [47] enables an experimental reconstruction of phase-space distributions and entirely circumvents a detector characterization by analyzing certain data patterns [48].\",\n \"Beyond quantum light, equally insightful is the characterization of matter systems using quasiprobabilities, such as demonstrated for motional states of ions [49] and large atomic ensembles [50].\",\n \"It is also worth mentioning that quantum correlations in composite hybrid systems are also accessible via generalized quasiprobabilities [51], [52].\",\n \"Therefore, quasiprobabilities present a highly successful and versatile approach to identifying nonclassical properties of quantum states in theory and experiment.\",\n \"Still, as outlined above, another important aspect is the quantum state decomposition.\",\n \"Yet, in contrast to the vast number of examples for the nonclassicality certification, studies of the decomposition properties are rarely done; a gap we aim to close in this article.\",\n \"Based on the decomposition of states in terms of quasiprobabilities, we introduce a generalized expansion of quantum states in terms of quasistates.\",\n \"This method is complementary to the previously known approaches in which a density operator is decomposed in terms of classical reference states and a negative quasiprobability density.\",\n \"In our general method, we can find a decomposition which results in a nonnegative (i.e., classical) probability density by overcoming the usage of physical reference states in the decomposition.\",\n \"This naturally establishes the concept of quasistates, which then become the relevant objects for certifying the different kinds of nonclassicality.\",\n \"The usefulness of this change of perspective from quasiprobabilities to quasistates is studied in detail.\",\n \"One practical application relates to the experimental reconstruction of density operators.\",\n \"Moreover, as quasiprobabilities are most frequently applied in optical systems, we also perform a detailed analysis of quasistates that correspond to prominent phase-space representations of light.\",\n \"Finally, we investigate how quantum entanglement can be uniquely characterized via classical joint probability distributions when employing quasistates.\",\n \"Therefore, our approach offers a toolbox for the characterization and decomposition of quantum states.\",\n \"This article is structured as follows.\",\n \"In Sec.\",\n \", we discuss the general methodological framework.\",\n \"An application to the quantum state reconstruction is developed in Sec.\",\n \".\",\n \"Phase-space based quasistates for quantized light are studied in Sec.\",\n \".\",\n \"In Sec.\",\n \", quantum correlations are analyzed within the proposed framework.\",\n \"Finally, concluding discussions are presented in Sec.\",\n \".\"\n ],\n [\n \"Conceptual framework\",\n \"To characterize the state of a quantum system, a representation of the density operator is required.\",\n \"In general, a state decomposition consists of a family of pure states, defining a set $\\\\mathcal {S}$ , and corresponding expansion coefficients.\",\n \"Then the density operator takes the form $\\\\hat{\\\\rho }=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\, P(\\\\psi ) |\\\\psi \\\\rangle \\\\langle \\\\psi |,$ where $d\\\\psi $ indicates an integral over the volume of $\\\\mathcal {S}$ and $P$ is a correspondingly defined density.\",\n \"In the case of a discrete decomposition, the integral may be replaced by a sum.\",\n \"Note that we deliberately choose not to specify the set $\\\\mathcal {S}$ to keep our treatment as broadly applicable as possible.\",\n \"Arguably the most well-known example of a representation (REF ) is the spectral decomposition.\",\n \"In this case, $\\\\mathcal {S}$ is the set of eigenstates and the probability mass function $P$ returns the corresponding eigenvalues.\",\n \"Other examples for a decomposition (REF ) are studied in the continuation of this work and have applications, for example, in quantum optics and quantum information.\",\n \"It is worth mentioning that a general method to construct quasiprobabilities $P$ for a given set $\\\\mathcal {S}$ has been recently devised [19].\",\n \"In that approach, it is ensured that $P$ is a classical probability distribution when the state is in the convex hull $\\\\mathrm {conv}\\\\lbrace |\\\\psi \\\\rangle \\\\langle \\\\psi |:\\\\psi \\\\in \\\\mathcal {S}\\\\rbrace $ , which is a nontrivial task as the decomposition (REF ) is typically not unique [19].\",\n \"For our purpose, we may generalize the above treatment.\",\n \"Specifically, a convolution kernel $K:\\\\mathcal {S}\\\\rightarrow \\\\mathcal {S}$ is introduced together with the kernel $K^{-1}$ for the corresponding deconvolution.\",\n \"Then we get an equivalent decomposition as $\\\\hat{\\\\rho }=\\\\int _{\\\\mathcal {S}} d\\\\chi \\\\, P_K(\\\\chi )\\\\hat{\\\\Delta }_{K}(\\\\chi ).$ Therein we have a modified density $P_K$ and a modified family of operators $\\\\hat{\\\\Delta }_{K}$ , $P_K(\\\\chi )=&\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\, P(\\\\psi )K(\\\\psi ,\\\\chi ),\\\\\\\\\\\\hat{\\\\Delta }_{K}(\\\\chi )=&\\\\int _{\\\\mathcal {S}} d\\\\psi ^{\\\\prime }\\\\, K^{-1}(\\\\chi ,\\\\psi ^{\\\\prime }) |\\\\psi ^{\\\\prime }\\\\rangle \\\\langle \\\\psi ^{\\\\prime }|.$ Using the properties that the operations used are inverse to one another, $\\\\delta (\\\\psi ,\\\\psi ^{\\\\prime })=\\\\int _{\\\\mathcal {S}} d\\\\chi K(\\\\psi ,\\\\chi )K^{-1}(\\\\chi ,\\\\psi ^{\\\\prime })$ with $\\\\delta $ being the Dirac distribution, one can directly see that Eqs.\",\n \"(REF ) and (REF ) describe the same density operator $\\\\hat{\\\\rho }$ .\",\n \"In general, the distribution $P_K$ in Eq.\",\n \"(REF ) is not a probability density, even if $P$ was.\",\n \"For such generalized distributions, the name quasiprobabilities was established.\",\n \"As discussed in the Introduction, such quasiprobability densities cover a wide range of applications, mainly for purpose of identifying quantum features.\",\n \"In close analogy to the notion of a quasiprobability, we are going to demonstrate that the operators $\\\\hat{\\\\Delta }_K$ in Eq.\",\n \"() are, in general, not physical density operators.\",\n \"Consequently, we refer to such operators as quasistates.\",\n \"A main focus of previous research was devoted to characterizing quasiprobabilities.\",\n \"Often, the idea of a decomposition of the quantum state [cf.\",\n \"Eqs.\",\n \"(REF ) and (REF )] in terms of such distributions was neglected.\",\n \"In particular, a general characterization of the distinctive features of quasistates and their applications beyond being a mathematical tool does not exist.\",\n \"For this reason, we are going to perform the missing comprehensive investigation of quasistates as defined in Eq.\",\n \"() and discuss useful applications of such operators.\"\n ],\n [\n \"State reconstruction\",\n \"One interesting application of quasistates are state reconstruction protocols as we show in this section.\",\n \"The following considerations are based on a recent work [11] in which the dual representation of measurement operators has been introduced.\",\n \"Here we demonstrate how this method relates to quasistates and can be used for the general reconstruction of density operators.\"\n ],\n [\n \"Dual representation\",\n \"Let us briefly revisit the findings in Ref.\",\n \"[11] with regards to our method.\",\n \"Suppose $\\\\lbrace \\\\hat{\\\\Pi }(j)\\\\rbrace _{j\\\\in \\\\mathcal {S}}$ is a positive operator-valued measure (POVM).\",\n \"As a complementary set of operators, the notion of contravariant operator-valued measured (COVM) was introduced.\",\n \"This defines a set $\\\\lbrace \\\\hat{\\\\Gamma }(j^{\\\\prime })\\\\rbrace _{j^{\\\\prime }\\\\in \\\\mathcal {S}}$ that satisfies the orthonormality relation $\\\\mathrm {tr}\\\\big [\\\\hat{\\\\Gamma }(j^{\\\\prime })\\\\hat{\\\\Pi }(j)\\\\big ]=\\\\delta (j^{\\\\prime },j).$ This means that the COVM represent dual basis operators to the measured POVM.\",\n \"In contrast to the POVM, COVM operators are not necessarily positive semidefinite.\",\n \"However, they are of particular interest, for example, when considering imperfections in the measurement process [11].\",\n \"The construction of COVM operators is based on a kernel defined via the elements $K(l,j)=\\\\mathrm {tr}\\\\big [\\\\hat{\\\\Pi }(l)\\\\hat{\\\\Pi }(j)\\\\big ].$ Specifically, it was shown that $\\\\hat{\\\\Gamma }(j^{\\\\prime })=\\\\sum _{l\\\\in \\\\mathcal {S}}K^{-1}(j^{\\\\prime },l)\\\\hat{\\\\Pi }(l),$ for the discrete case.\",\n \"In the following, let us explore the relation between the COVM and quasistates.\",\n \"For simplicity, we assume that the considered complex Hilbert space is finite dimensional, $\\\\dim _{\\\\mathbb {C}}\\\\mathcal {H}=d<\\\\infty $ .\",\n \"The set of Hermitian operators over $\\\\mathcal {H}$ is a real valued space with the dimension $\\\\dim _{\\\\mathbb {R}}\\\\mathrm {Herm}(\\\\mathcal {H})=d^2$ .\",\n \"Further, rather than restricting to a POVM for a single observable as done previously, we consider an informationally complete set $\\\\lbrace \\\\hat{\\\\Pi }(\\\\psi )\\\\rbrace _{\\\\psi \\\\in \\\\mathcal {S}}$ , with $\\\\hat{\\\\Pi }(\\\\psi )=|\\\\psi \\\\rangle \\\\langle \\\\psi |,$ which guarantees that a reconstruction of the full quantum state is possible and implies the cardinality $|\\\\mathcal {S}|=d^2$ .\",\n \"However, this also means that not all elements of $\\\\mathcal {S}$ can be pairwise orthogonal vectors.\",\n \"See Refs.\",\n \"[53], [54], [55] for introductions to informationally complete measurements.\",\n \"Because of the properties of the now studied POVM, we can expand the quantum state as given in Eq.\",\n \"(REF ), $\\\\hat{\\\\rho }=\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi ) \\\\hat{\\\\Pi }(\\\\psi ).$ It is important to mention that $P(\\\\psi )$ , in general, does not correspond to the probability of the measurement of $\\\\hat{\\\\Pi }(\\\\psi )$ , i.e., $\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Pi }(\\\\psi )]\\\\ne P(\\\\psi )$ .\",\n \"However, we can employ the same formalism as discussed above, cf.\",\n \"Eqs.\",\n \"(REF ) and (REF ), to expand the state as [11] $\\\\hat{\\\\rho }=\\\\sum _{\\\\chi \\\\in \\\\mathcal {S}}\\\\rho (\\\\chi ) \\\\hat{\\\\Gamma }(\\\\chi ).$ Now we can identify the coefficient with the measurement probability, $\\\\rho (\\\\psi ^{\\\\prime })=\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Pi }(\\\\psi ^{\\\\prime })]$ for all $\\\\psi ^{\\\\prime }\\\\in \\\\mathcal {S}$ , because of relation (REF ).\",\n \"In other words, the expansion in Eq.\",\n \"(REF ) can be used to directly reconstruct the density operator in terms of measurement outcomes and dual operators.\",\n \"Let us specifically relate the found reconstruction with the generalized decomposition (REF ).\",\n \"Using the definitions (REF ) and (REF ), we find that the kernel under study is given by the $d^2\\\\times d^2$ Gram-Schmidt matrix $K=\\\\left[|\\\\langle \\\\psi |\\\\chi \\\\rangle |^2\\\\right]_{\\\\psi ,\\\\chi \\\\in \\\\mathcal {S}}.$ Recall that information completeness implies that this matrix is a bijection, thus invertible.\",\n \"Further, as a result of Eqs.\",\n \"() and (REF ), the thought-after quasistates are identical to the COVM elements, $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\hat{\\\\Gamma }(\\\\chi ).$ Consequently, it also holds true that $P_K(\\\\chi )=\\\\rho (\\\\chi )$ ; see Eqs.\",\n \"(REF ) and (REF ).\",\n \"Therefore, quasistates allow for the direct reconstruction of the density operator under study.\",\n \"It is worth mentioning that imperfections in any measurement, meaning that $\\\\hat{\\\\Pi }(\\\\psi )$ is not a pure-state projector, can be treated similarly to the thorough discussion for a single observable carried out in Ref.\",\n \"[11].\",\n \"Furthermore, if the POVM does not allow for a full reconstruction or is over-complete (e.g., because $|\\\\mathcal {S}|\\\\ne d^2$ ), then the state in the spanned subspace can be reconstructed or a pseudo-inversion of $K$ can be performed [11].\"\n ],\n [\n \"Example\",\n \"To further discuss the above findings, we study the specific example of a qubit system ($d=2$ ), defined via the orthonormal Hilbert space basis $\\\\lbrace |0\\\\rangle ,|1\\\\rangle \\\\rbrace $ .\",\n \"The four rank-one projection operators $|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|$ in Eq.\",\n \"(REF ), also defining the set $\\\\mathcal {S}$ , can be chosen according to the states $\\\\nonumber |\\\\psi _1\\\\rangle =|1\\\\rangle ,|\\\\psi _2\\\\rangle =&\\\\frac{\\\\sqrt{2} |0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},|\\\\psi _3\\\\rangle =\\\\frac{\\\\sqrt{2} w|0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},\\\\\\\\\\\\text{and }|\\\\psi _4\\\\rangle =&\\\\frac{\\\\sqrt{2} w^2|0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},$ where $w=\\\\exp [2\\\\pi i/3]$ ; see the top-left panel in Fig.\",\n \"REF for their geometric Bloch-sphere representation.\",\n \"This allows us to construct $K$ via Eq.\",\n \"(REF ) and compute its inverse, $K=\\\\frac{1}{3}\\\\begin{bmatrix}3 & 1 & 1 & 1\\\\\\\\1 & 3 & 1 & 1\\\\\\\\1 & 1 & 3 & 1\\\\\\\\1 & 1 & 1 & 3\\\\end{bmatrix}\\\\!,\\\\,K^{-1}=\\\\frac{1}{4}\\\\begin{bmatrix}5 & -1 & -1 & -1\\\\\\\\-1 & 5 & -1 & -1\\\\\\\\-1 & -1 & 5 & -1\\\\\\\\-1 & -1 & -1 & 5\\\\end{bmatrix}\\\\!.$ This then yields the quasistates as $\\\\hat{\\\\Delta }_K(\\\\psi _1)=&|1\\\\rangle \\\\langle 1|-\\\\frac{1}{2}|0\\\\rangle \\\\langle 0|\\\\text{ and}\\\\\\\\\\\\nonumber \\\\hat{\\\\Delta }_K(\\\\psi _j)=&\\\\frac{|0\\\\rangle \\\\langle 0|+\\\\sqrt{2}w^{j-2}|0\\\\rangle \\\\langle 1|+\\\\sqrt{2}w^{\\\\ast (j-2)}|1\\\\rangle \\\\langle 0|}{2},$ for $j=2,3,4$ .\",\n \"As we can see from the top-right panel in Fig.\",\n \"REF , these quasistates do not resemble physical quantum states because of the negative eigenvalue.\",\n \"Figure: (Color online)The elements of 𝒮\\\\mathcal {S} given in Eq.\",\n \"() in a tetrahedron configuration (larger green bullets) are shown in the top-left Bloch-sphere representation.The smaller red bullet indicates the state ρ ^=|0〉〈0|\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0| to be reconstructed.The top-right panel shows the two eigenvalues of the resulting quasistates Δ ^ K (ψ j )\\\\hat{\\\\Delta }_K(\\\\psi _j), being identical for j=1,...,4j=1,\\\\ldots ,4.For the reconstruction of the state under study, the bottom panels depict the expansion coefficients P(ψ j )P(\\\\psi _j) and P K (ψ j )P_K(\\\\psi _j) to be used with |ψ j 〉〈ψ j ||\\\\psi _j\\\\rangle \\\\langle \\\\psi _j| and Δ ^ K (ψ j )\\\\hat{\\\\Delta }_K(\\\\psi _j) [Eqs.\",\n \"() and ()], respectively.Let us now consider the specific state $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ for the reconstruction.\",\n \"Then, the measured and, therefore, nonnegative expansion coefficients are $P_K(\\\\psi _j)=\\\\mathrm {tr}(\\\\hat{\\\\rho }|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|)=\\\\langle \\\\psi _j|\\\\hat{\\\\rho }|\\\\psi _j\\\\rangle ,$ using the projectors defined via the states in Eq.\",\n \"(REF ).\",\n \"In addition, the corresponding COVM elements, being the quasistates in Eq.\",\n \"(REF ), result in the expansion coefficients $P(\\\\psi _j)=\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Delta }_K(\\\\psi _j)]$ , which resembles a quasiprobability distribution.\",\n \"The bottom part in Fig.\",\n \"REF shows the values of $P_K$ and $P$ for the state $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ .\",\n \"Using the nonnegative $P_K$ (see Fig.\",\n \"REF ), we can now directly confirm that the density operator $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ is expanded as described in Eq.\",\n \"(REF ) while using the quasistates in Eq.\",\n \"(REF ), $\\\\hat{\\\\rho }=\\\\sum _{j=1,\\\\ldots ,4}P_K(\\\\psi _j)\\\\hat{\\\\Delta }_K(\\\\psi _j)$ .\",\n \"The possibly negative elements of $P$ also allow for the decomposition of the state, $\\\\hat{\\\\rho }=\\\\sum _{j=1,\\\\ldots ,4}P(\\\\psi _j)|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|$ .\",\n \"However, $P$ does not correspond to a directly measured quantity.\",\n \"Thus, quasistates provide a beneficial tool to reconstruct density operators using measured quantities.\",\n \"Let us also briefly comment on the extension of our approach to continuous-variable systems, $d=\\\\infty $ .\",\n \"In fact, already in Refs.\",\n \"[56], [57], a method was introduced to expand the state of a single-mode radiation field in terms of so-called pattern functions.\",\n \"Using the here developed framework, we can directly identify these pattern functions with the wave functions of the corresponding quasistates.\",\n \"Beyond this reconstruction approach, we study the expansion of nonclassical states of light in the following section.\"\n ],\n [\n \"Quantum Light\",\n \"We now demonstrate the application of quasistates for the description of nonclassical quantum states.\",\n \"In particular, we consider a harmonic oscillator, which can, for example, describe a quantized radiation mode.\",\n \"To describe the system's nonclassicality, the prominent Glauber-Sudarshan representation has been developed [4], [5], $\\\\hat{\\\\rho }=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,P(\\\\alpha )|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.$ Here this decomposition implies that we identify the set $\\\\mathcal {S}$ with the complex coherent amplitudes $\\\\alpha \\\\in \\\\mathbb {C}$ , defining the coherent states $|\\\\alpha \\\\rangle $ .\",\n \"For a thorough introduction to quantum optics in phase space, see Ref.\",\n \"[58].\",\n \"A state is referred to as nonclassical if $P$ does not exhibit the properties of a classical probability distribution.\",\n \"In the following, we begin our considerations of quasistates for nonclassical light in the Fourier picture and then proceed to prominent examples of quasiprobabilities and their dual representation via quasistates.\"\n ],\n [\n \"Fourier representation\",\n \"The characteristic function $\\\\Phi (\\\\beta )$ is the Fourier transform $F$ of a distribution, which is accessible via the kernel $F(\\\\alpha ,\\\\beta )=\\\\exp \\\\left(\\\\beta \\\\alpha ^\\\\ast -\\\\beta ^\\\\ast \\\\alpha \\\\right),$ i.e., in our notation, $P_{F}=\\\\Phi $ .\",\n \"The inverse Fourier transform, defined via $F^{-1}(\\\\beta ,\\\\alpha )=F(\\\\alpha ,\\\\beta )^\\\\ast /\\\\pi ^2$ , results in the corresponding quasistates [Eq.\",\n \"()], $\\\\hat{\\\\Delta }_{F}(\\\\beta )=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\frac{\\\\exp \\\\left(\\\\beta ^\\\\ast \\\\alpha -\\\\beta \\\\alpha ^\\\\ast \\\\right)}{\\\\pi ^2}|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.$ For the evaluation of the integral, it is useful to represent the coherent-state projectors in terms of normally ordered exponential of creation ($\\\\hat{a}^\\\\dag $ ) and annihilation ($\\\\hat{a}$ ) operators, which reads $|\\\\alpha \\\\rangle \\\\langle \\\\alpha |={:}\\\\exp (-[\\\\hat{a}-\\\\alpha ]^\\\\ast [\\\\hat{a}-\\\\alpha ]){:}$ [58].\",\n \"This representation yields $\\\\hat{\\\\Delta }_{F}(\\\\beta )=\\\\frac{\\\\exp (-|\\\\beta |^2)}{\\\\pi }{:}\\\\hat{D}(-\\\\beta ){:},$ where $\\\\hat{D}(\\\\gamma )=\\\\exp (\\\\gamma \\\\hat{a}^\\\\dag -\\\\gamma ^\\\\ast \\\\hat{a})=\\\\exp (-|\\\\gamma |^2/2){:}\\\\hat{D}(\\\\gamma ){:}$ is the unitary displacement operator.\",\n \"Therefore, the quasistates $\\\\hat{\\\\Delta }_{F}(\\\\beta )$ correspond (up to a rescaling) to unitary operators in the case that the convolution kernel is the Fourier transformation.\",\n \"Note that instead of the Fourier transform, the Laplace transform can be advantageous in certain scenarios as well [59].\",\n \"One application, which also connects to the previous section, is the state reconstruction via balanced homodyne detection.\",\n \"For this purpose, let us recall that the characteristic function of the Wigner-Weyl distribution takes the form [58] $\\\\Phi (\\\\beta )e^{-|\\\\beta |^2/2}=\\\\langle \\\\hat{D}(\\\\beta )\\\\rangle =\\\\langle \\\\exp \\\\left(i|\\\\beta |\\\\hat{x}[\\\\pi /2-\\\\arg \\\\beta ]\\\\right)\\\\rangle ,$ where we used the quadrature operator $\\\\hat{x}[\\\\varphi ]=e^{i\\\\varphi }\\\\hat{a}+e^{-i\\\\varphi }\\\\hat{a}^\\\\dag $ .\",\n \"This means for a balanced homodyne measurement, providing a data set $\\\\lbrace x_j[\\\\varphi _j]\\\\rbrace _{j=1,\\\\ldots ,N}$ of quadratures $x_j$ for the phase values $\\\\varphi _j$ , we can approximate $\\\\begin{aligned}\\\\hat{\\\\rho }=&\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\Phi (\\\\beta )\\\\hat{\\\\Delta }_{F}(\\\\beta )\\\\\\\\=&\\\\int _{-\\\\infty }^\\\\infty dr\\\\int _{-\\\\pi /2}^{\\\\pi /2} d\\\\varphi \\\\,\\\\frac{\\\\pi }{\\\\pi }e^{r^2/2}\\\\langle e^{ir\\\\hat{x}[\\\\varphi ]}\\\\rangle \\\\hat{\\\\Delta }_{F}(ire^{-i\\\\varphi })\\\\\\\\\\\\approx &\\\\int _{-\\\\infty }^\\\\infty dr\\\\, e^{r^2/2} \\\\frac{\\\\pi }{N}\\\\sum _{j=1}^N e^{irx_j[\\\\varphi _j]}\\\\hat{\\\\Delta }_{F}(ire^{-i\\\\varphi _j}),\\\\end{aligned}$ which corresponds to a sampling approach as a consequence of a quasistate representation [see specifically the first line of Eq.\",\n \"(REF ) and the use of $\\\\hat{\\\\Delta }_{F}$ ].\",\n \"In fact, further analyses show that this approach is indeed equivalent to established reconstruction methods bases on balanced homodyne detection and using pattern functions [56], [57], [60].\",\n \"In addition, it is worth emphasizing that the phases $\\\\varphi $ have to be uniformly distributed in the interval $[-\\\\pi /2,\\\\pi /2]$ to ensure an optimal sampling [61].\",\n \"Figure: (Color online)The QQ function [Eq.\",\n \"()] of the quasistate [Eq.\",\n \"()] for different parameters ss, pp, and qq.The top-left plot corresponds to a coherent (vacuum) state.The top-center plot shows the maximally singular quasistate .The top-right plot includes, in addition, squeezing.The bottom row depicts cases in which the QQ function becomes complex;the larger and corresponding smaller plots show the amplitude |Q||Q| and phase argQ\\\\arg Q, respectively.Note the difference in the phases when comparing the left and right plots.\"\n ],\n [\n \"Generalized optical quasistates\",\n \"Translational types of convolutions in the original space, e.g., $P_{\\\\kappa }(\\\\alpha ^{\\\\prime })=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,P(\\\\alpha )\\\\kappa (\\\\alpha ^{\\\\prime }-\\\\alpha )$ , take a product form in the Fourier domain.\",\n \"In this scenario, we can consider the representation $\\\\hat{\\\\rho }=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\left[\\\\Phi (\\\\beta )\\\\Omega (\\\\beta )\\\\right]\\\\left[\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}{:}\\\\hat{D}(-\\\\beta ){:}\\\\right],$ for a kernel $\\\\Omega $ .\",\n \"The inverse Fourier transform gives the distribution $P_{F^{-1}\\\\Omega F}$ as well as the quasistates $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(\\\\alpha )=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,e^{\\\\beta \\\\alpha ^\\\\ast -\\\\beta ^\\\\ast \\\\alpha }\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}{:}\\\\hat{D}(-\\\\beta ){:}.$ Such operators have been extensively studied in connection to operator orderings and their resulting quasiprobabilities [23], [24], [27], [28], [29].\",\n \"Here, however, let us analyze them based on their own merit as quasistates.\",\n \"Of particular interest are the Gaussian functions $\\\\Omega (\\\\beta )=\\\\exp \\\\left(-\\\\frac{1-s}{2}|\\\\beta |^2-\\\\frac{p}{4}\\\\beta ^2-\\\\frac{q}{4}\\\\beta ^\\\\ast \\\\right),$ where $|p|=|q|$ [27], [28], [29].\",\n \"In Table REF , specific examples and their connection to known quasiprobabilities are given.\",\n \"Evaluating the integrals as demonstrated in Appendix , we get the quasistates in normal ordering as $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)=\\\\frac{2\\\\,{:}\\\\exp \\\\left(\\\\frac{-2[1+s]\\\\hat{a}^\\\\dag \\\\hat{a} +q\\\\hat{a}^{\\\\dag 2}+p\\\\hat{a}^2}{(1+s)^2-pq}\\\\right){:}}{\\\\sqrt{(1+s)^2-pq}}.$ Note that because of the convolution property, we have $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(\\\\alpha )=\\\\hat{D}(\\\\alpha )\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)\\\\hat{D}(\\\\alpha )^\\\\dag $ for the displaced quasistates.\",\n \"In Fig.\",\n \"REF , several examples of the phase-space $Q$ -function representation, $Q(\\\\alpha )=\\\\frac{\\\\langle \\\\alpha |\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)|\\\\alpha \\\\rangle }{\\\\pi },$ are shown for different parameters that define the quasistates in Eq.\",\n \"(REF ).\",\n \"It is also worth mentioning that the $Q$ function is the Husimi-Kano distribution.\",\n \"Table: Parameters [cf.\",\n \"Eq.\",\n \"()] for prominent quasiprobability distributions frequently used in quantum optics.Interestingly, the considered class of quasistates, given in Eq.\",\n \"(REF ), can be related to squeezed versions of thermal-like quasistates, $\\\\begin{aligned}\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)=&\\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)\\\\\\\\&\\\\times \\\\frac{1}{\\\\bar{n}+1}\\\\sum _{n\\\\in \\\\mathbb {N}}\\\\left(\\\\frac{\\\\bar{n}}{\\\\bar{n}+1}\\\\right)^n|n\\\\rangle \\\\langle n|\\\\\\\\&\\\\times \\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)^\\\\dag ,\\\\end{aligned}$ where $\\\\bar{n}\\\\in \\\\mathbb {C}$ generalizes the notion of a mean thermal photon number and $\\\\zeta $ is the squeezing parameter.\",\n \"A full and exact analysis is provided in Appendix , and it provides the relations between the different parameter sets as $\\\\zeta =e^{i\\\\arg \\\\tau }\\\\mathrm {artanh}\\\\,|\\\\tau |\\\\quad \\\\text{and}\\\\quad \\\\bar{n}=\\\\frac{\\\\omega }{1-\\\\omega },$ with $\\\\omega =\\\\frac{r-1}{r+1},\\\\,|\\\\tau |^2=\\\\frac{s-r}{s+r},\\\\,\\\\text{and}\\\\,\\\\frac{\\\\tau }{\\\\tau ^\\\\ast }=\\\\frac{q}{p},$ where $r=\\\\sqrt{s^2-pq}$ .\",\n \"Recall that we require $|q|=|p|$ .\",\n \"Also note that the squeezing operation is unitary which implies that we have the eigenvalues $(1-\\\\omega )\\\\omega ^n$ , following a geometric distribution.\",\n \"In the following, let us study the resulting quasistates in more detail.\"\n ],\n [\n \"$s$ -Parameterized quasistates\",\n \"Arguably the most frequently used choice of parameters is obtained for $p=q=0$ , likewise $\\\\zeta =\\\\tau =0$ .\",\n \"This choice results in the $s$ -parametrized quasiprobability distributions $P_{F^{-1}\\\\Omega F}$ .\",\n \"Here, we additionally find the quasistates $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}$ .\",\n \"In particular, those quasistates have the eigenvalues $(1-\\\\omega )\\\\omega ^n=\\\\frac{\\\\bar{n}^{n}}{(\\\\bar{n}+1)^{n+1}}=\\\\frac{2}{1+s}\\\\left(-\\\\frac{1-s}{1+s}\\\\right)^n$ for $n\\\\in \\\\mathbb {N}$ ; see Eq.\",\n \"(REF ).\",\n \"For the parameters $-10$ , defining the vacuum state.\",\n \"Its $Q$ function is shown on the left in the top row of Fig.\",\n \"REF , and the corresponding quasiprobability distribution is the original Glauber-Sudarshan distribution.\",\n \"Furthermore, the case $s=0$ interestingly describes an operator with the maximally possible singularities a Glauber-Sudarshan distribution can exhibit [41]; see also the center plot in the top row of Fig.\",\n \"REF for its $Q$ function.\",\n \"There, we observe that the spread (i.e., variance) in phase space is reduced in all directions when compared to the vacuum state, further highlighting that it is a genuine quasistate that is incompatible with the uncertainty relation which holds true for any physical state.\",\n \"The underlying phase-space distribution $P_{F^{-1}\\\\Omega F}$ is the Wigner-Weyl distribution.\",\n \"Finally, the quasistates for the Husimi-Kano distribution are obtained in the limit $s\\\\rightarrow -1$ , and its corresponding $Q$ function is described by a singular Dirac distribution.\",\n \"Beyond real-valued $s$ parameters, we can additionally consider complex values as well [58].\",\n \"Then, the eigenvalues (REF ) are complex too.\",\n \"The example $s=i$ is shown on the left of the bottom row in Fig.\",\n \"REF .\",\n \"While the amplitude $|Q|$ is identical with the vacuum state, the quasistate characteristics are clearly visible through the nontrivial phases, $\\\\arg Q\\\\ne 0$ .\"\n ],\n [\n \"Non-$s$ -parametrized quasistates\",\n \"Moreover, we can also consider the quasistates for cases with $p,q\\\\ne 0$ .\",\n \"As we demand $|q|=|p|$ , we have two extremal scenarios, $q=p^\\\\ast $ and $q=-p^\\\\ast $ .\",\n \"In addition to the $s$ -parametrized quasiprobabilities in Table REF , we also listed the Agarwal-Wolf distributions.\",\n \"They correspond to the case $q=-p^\\\\ast =\\\\mp 1$ (also, $s=0$ ).\",\n \"The complex-valued $Q$ function of the resulting quasistate $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}$ is shown in the bottom-right plot of Fig.\",\n \"REF .\",\n \"The amplitude $|Q|$ is again compatible with the uncertainty principle bounded by the vacuum state, similarly to the previously discussed scenario of a complex $s$ parameter.\",\n \"Here, however, the functional behavior of the phase $\\\\arg Q$ does not possess a radial symmetry.\",\n \"Rather, the quasistates to the Agarwal-Wolf distributions exhibit hyperbolic isophase contours.\",\n \"Note that this dismisses a sometimes held believe that Agarwal-Wolf distributions are an example of a $s$ -parametrized quasiprobability.\",\n \"Finally, we can ask ourselves what happens in the case $q=p^\\\\ast $ .\",\n \"An example is shown in the top-right plot in Fig.\",\n \"REF .\",\n \"Interestingly, the resulting quasistates are squeezed.\",\n \"Specifically for a squeezing parameter $\\\\zeta $ , we can choose $s=\\\\frac{1+\\\\tanh ^2|\\\\zeta |}{1-\\\\tanh ^2|\\\\zeta |}\\\\text{ and }q=\\\\frac{-2e^{i\\\\arg \\\\zeta }\\\\tanh |\\\\zeta |}{1-\\\\tanh ^2|\\\\zeta |}=p^\\\\ast .$ This choice implies $\\\\bar{n}=0$ and means that the quasistates are pure squeezed states, cf.\",\n \"Eqs.\",\n \"(REF ) and (REF ).\",\n \"Consequently, the quasiprobability density $P_{F^{-1}\\\\Omega F}(\\\\alpha )$ now describes the decomposition of the density operator in terms of displaced squeezed states with a coherent amplitude $\\\\alpha $ and a squeezing defined by $\\\\zeta $ .\",\n \"This further implies that we have a generalized Glauber-Sudarshan quasiprobability distribution which is, however, a phase-space representation that expands the state in terms of squeezed states.\",\n \"Thus, the case $q=p^\\\\ast $ significantly extends the notion of $s$ -parametrized distributions to additionally include arbitrary squeezed states.\"\n ],\n [\n \"Quantum correlations\",\n \"In a recent work [19], we generalized the concept of quasiprobability representations to more general notions of quantum coherence [62], beyond the specific example of harmonic oscillators studied in the previous section.\",\n \"Among the various types of quantumness, quantum correlations between multiple degrees of freedom play an outstanding role for applications in quantum technologies [1], [63].\",\n \"For this reason, let us study the entanglement of quantum systems within the framework of quasistates.\",\n \"A comprehensive introduction to the theory of entanglement (likewise, inseparability) can be found in Ref.\",\n \"[63].\"\n ],\n [\n \"Entanglement quasiprobabilities\",\n \"A pure separable state in a multipartite system is described by a tensor-product vector, $|(a,b,\\\\ldots )\\\\rangle =|a\\\\rangle \\\\otimes |b\\\\rangle \\\\otimes \\\\cdots ,$ where $a\\\\in \\\\mathcal {S}_A$ , $b\\\\in \\\\mathcal {S}_B$ , etc.\",\n \"In addition, the inclusion of classical correlations in terms of statistical mixtures, given by a classical probability density $P$ , yields the definition of a mixed separable state [64], $\\\\hat{\\\\rho }{=}\\\\int _{S_A{\\\\times } S_B{\\\\times }\\\\cdots } da\\\\,db\\\\cdots P(a,b,\\\\ldots )|(a,b,\\\\ldots )\\\\rangle \\\\langle (a,b,\\\\ldots )|.$ A state is defined to be inseparable (i.e., entangled) if it cannot be written in the form of Eq.\",\n \"(REF ).\",\n \"However, when we allow for $P$ to be a quasiprobability, even entangled states can be expanded in terms of separable states using Eq.\",\n \"(REF ) [21].\",\n \"A construction approach for bipartite entanglement quasiprobabilities was introduced in Ref.\",\n \"[22].\",\n \"As this approach can be generalized from the bipartite scenario to the multipartite case [19], we restrict ourselves to bipartite systems in the following.\",\n \"The optimal decomposition of entangled states in terms of separable ones warrants that the entanglement is uniquely identified through negativities in $P(a,b)$ [22], [19].\",\n \"The separable states $|(a,b)\\\\rangle $ needed for the decomposition are obtained from the solution of the so-called separability eigenvalue equations (SEEs), $\\\\hat{\\\\rho }_a|b\\\\rangle =g|b\\\\rangle \\\\text{ and }\\\\hat{\\\\rho }_b|a\\\\rangle =g|a\\\\rangle ,$ with $\\\\hat{\\\\rho }_a=\\\\mathrm {tr}_A[\\\\hat{\\\\rho }(|a\\\\rangle \\\\langle a|\\\\otimes \\\\hat{1}_B)]$ and $\\\\hat{\\\\rho }_b=\\\\mathrm {tr}_B[\\\\hat{\\\\rho }(\\\\hat{1}_A\\\\otimes |b\\\\rangle \\\\langle b|)]$ .\",\n \"The SEEs approach generalizes the eigenvalue problem to composite systems while respecting the tensor-product structure of the corresponding eigenvectors and was initially introduced to construct entanglement witnesses [65], [66].\",\n \"The solutions $(a,b)\\\\in \\\\mathcal {S}$ allow us to construct the distribution $P$ by solving the linear problem $G\\\\vec{p}=\\\\vec{g},$ where $G=[|\\\\langle (a,b)|(a^{\\\\prime },b^{\\\\prime })\\\\rangle |^2]_{(a,b),(a^{\\\\prime },b^{\\\\prime })\\\\in \\\\mathcal {S}}$ is the Gram-Schmidt matrix of separability eigenvectors.\",\n \"Furthermore, the solution vector $\\\\vec{p}=[P(a,b)]_{(a,b)\\\\in \\\\mathcal {S}}$ defines the quasiprobabilities, and $\\\\vec{g}=[g]_{(a,b)\\\\in \\\\mathcal {S}}$ is the vector of separability eigenvalues, cf.\",\n \"Eq.\",\n \"(REF ).\",\n \"Let us point out that the here-used Gram-Schmidt matrix also relates to the reconstruction approach in Sec.\",\n \"[Eq.\",\n \"(REF )] and is a result of a underlying principle of convex decomposition, discussed in greater detail in Ref.\",\n \"[19].\",\n \"One can always find a set $\\\\mathcal {S}_0$ such that $\\\\mathcal {S}\\\\subset \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0$ ; we then set $P(a,b)=0$ for all $(a,b)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0\\\\setminus \\\\mathcal {S}$ .\",\n \"Using the construction to find optimal entanglement quasiprobabilities via the SEEs, we can decompose any state, be it entangled or separable [22], [19], as $\\\\hat{\\\\rho }=\\\\sum _{(a,b)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P(a,b)|(a,b)\\\\rangle \\\\langle (a,b)|.$ It is worth mentioning that the entanglement quasiprobability is always real valued and normalized.\"\n ],\n [\n \"Quasistate representation\",\n \"A universally applicable approach to diminish the negativities in quasiprobabilities for a single system is a convolution of the form $P_K(\\\\chi )=rP(\\\\chi )+\\\\frac{1-r}{|\\\\mathcal {S}_0|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}_0}P(\\\\psi ),$ with $0< r\\\\le 1$ and $|\\\\mathcal {S}_0|$ being the cardinality of the set; see Appendix for details.\",\n \"The kernel under study consequently leads to quasistates of the form $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\frac{1}{r}|\\\\chi \\\\rangle \\\\langle \\\\chi |-\\\\frac{1-r}{r|\\\\mathcal {S}_0|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}_0}|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ We emphasize that quasistates defined in this manner are both Hermitian, $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\hat{\\\\Delta }_K(\\\\chi )^\\\\dag $ , and normalized, $\\\\mathrm {tr}[\\\\hat{\\\\Delta }_K(\\\\chi )]=1$ .\",\n \"Thus, the only distinction to true states is the fact that $\\\\hat{\\\\Delta }_K(\\\\chi )$ is, in general, not a positive semidefinite operator.\",\n \"The convolution can be generalized to composite systems via product kernels.\",\n \"Choosing, for example, the same $r$ for both systems, we can reformulate Eq.\",\n \"(REF ) as $\\\\hat{\\\\rho }=\\\\sum _{(\\\\tilde{a},\\\\tilde{b})\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P_{K\\\\otimes K}(\\\\tilde{a},\\\\tilde{b})\\\\,\\\\hat{\\\\Delta }_K(\\\\tilde{a})\\\\otimes \\\\hat{\\\\Delta }_K(\\\\tilde{b}).$ In particular, we have $\\\\begin{aligned}P_{K\\\\otimes K}(\\\\tilde{a},\\\\tilde{b})=&r^2P(\\\\tilde{a},\\\\tilde{b})+\\\\frac{(1-r)^2}{|\\\\mathcal {S}_0|^2}\\\\\\\\&+\\\\frac{r(1-r)}{|\\\\mathcal {S}_0|}\\\\left[P(\\\\tilde{a})+P(\\\\tilde{b})\\\\right],\\\\end{aligned}$ using the marginal distributions $P(\\\\tilde{a})=\\\\sum _{b\\\\in \\\\mathcal {S}_0}P(\\\\tilde{a},b)$ and $P(\\\\tilde{b})=\\\\sum _{a\\\\in \\\\mathcal {S}_0}P(a,\\\\tilde{b})$ .\",\n \"Eventually, we get a nonnegative distribution $P_{K\\\\otimes K}$ for a sufficiently large $r$ , which is also demonstrated later.\",\n \"This means that an entangled state can be decomposed according to Eq.\",\n \"(REF ) in terms of a classical joint probability distribution $P_{K\\\\otimes K}(a,b)\\\\ge 0$ and tensor-product quasistates $\\\\hat{\\\\Delta }_K(a)\\\\otimes \\\\hat{\\\\Delta }_K(b)\\\\ngeq 0$ [cf.\",\n \"Eq.\",\n \"(REF )].\",\n \"Since we can chose $r=0$ for separable states, we find that quasistates are not required in this case.\",\n \"Therefore, we are able to conclude that entanglement is unambiguously identified in terms of classical joint probability distributions if and only if the tensor-product operators of the decomposition have to be unphysical quasistates.\"\n ],\n [\n \"Example\",\n \"As a proof of concept, let us consider an entangled two-qubit quantum state parametrized as $\\\\hat{\\\\rho }=\\\\frac{1}{4}\\\\left[\\\\hat{1}\\\\otimes \\\\hat{1}+\\\\sum _{j\\\\in \\\\lbrace x,y,z\\\\rbrace }\\\\rho (j)\\\\,\\\\hat{\\\\sigma }_j\\\\otimes \\\\hat{\\\\sigma }_j\\\\right].$ This corresponds to a physical density operator if and only if the real coefficients satisfy $[\\\\rho (x),\\\\rho (y),\\\\rho (z)]\\\\in \\\\mathrm {conv}\\\\lbrace [-1,1,1],[1,-1,1],[1,1,-1],[-1,-1,-1]\\\\rbrace $ , where the extremal points represent Bell states.\",\n \"In Ref.\",\n \"[19], we solved the SEEs [Eq.\",\n \"(REF )] for this family of states.\",\n \"In particular, the separability eigenvectors are tensor products of eigenstates of Pauli operators ($\\\\hat{\\\\sigma }_x$ , $\\\\hat{\\\\sigma }_y$ , and $\\\\hat{\\\\sigma }_z$ ).\",\n \"This implies that we have $\\\\mathcal {S}_0=\\\\lbrace x_{+},x_{-},y_{+},y_{-},z_{+},z_{-}\\\\rbrace ,$ where $\\\\hat{\\\\sigma }_w|w_s\\\\rangle =s|w_{s}\\\\rangle $ for $w_s\\\\in \\\\mathcal {S}_0$ .\",\n \"The exact entanglement quasiprobability reads [19] $P(a_s,b_t)=\\\\delta (a,b)\\\\left[\\\\frac{q}{12}+\\\\frac{|\\\\rho (a)|+st\\\\rho (a)}{4}\\\\right],$ with $a,b\\\\in \\\\lbrace x,y,z\\\\rbrace $ and $s,t\\\\in \\\\lbrace +1,-1\\\\rbrace $ , identifying the separability eigenvectors, $(a_s, b_t)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0$ .\",\n \"Of particular importance is the parameter $q=1-|\\\\rho (x)|-|\\\\rho (y)|-|\\\\rho (z)|$ as it determines if the state is separable; namely, $\\\\hat{\\\\rho }$ in Eq.\",\n \"(REF ) is separable if and only if $q\\\\ge 0$ [19].\",\n \"Conversely, the negativity of the quasiprobability in the case of entanglement is given by this quantity as well; it reads $\\\\min _{(u, v)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P(u,v)=q/12\\\\ge -1/6$ , and the lower bound is attained for Bell states.\",\n \"Figure: (Color online)The top panel depicts the entanglement quasiprobabilities [Eq.\",\n \"()] for a Bell state with ρ(x)=ρ(y)=ρ(z)=-1\\\\rho (x)=\\\\rho (y)=\\\\rho (z)=-1.Each patch in the AA-BB plane corresponds to a product state |a〉〈a|⊗|b〉〈b||a\\\\rangle \\\\langle a|\\\\otimes |b\\\\rangle \\\\langle b|, with a,b∈𝒮 0 a,b\\\\in \\\\mathcal {S}_0 according to Eq.\",\n \"().The bottom panel shows the eigenvalues of those separable-state projectors.The negativities in the quasiprobabilities PP identify the entanglement.Now we can apply the kernel in Eq.\",\n \"(REF ).\",\n \"Inserting the distribution in Eq.\",\n \"(REF ) yields $\\\\begin{aligned}&P_{K\\\\otimes K}(a_s,b_t)\\\\\\\\= & r^2\\\\delta (a,b)\\\\left[\\\\frac{q}{12}+\\\\frac{|\\\\rho (a)|+st\\\\rho (a)}{4}\\\\right]+\\\\frac{(1-r)^2}{36}\\\\\\\\&+\\\\frac{r(1-r)}{6}\\\\left[\\\\frac{q}{3}+\\\\frac{|\\\\rho (a)|}{2}+\\\\frac{|\\\\rho (b)|}{2}\\\\right].\\\\end{aligned}$ After some straightforward algebra, we find that $P_{K\\\\otimes K}$ is always nonnegative for $00$ and $\\\\mathrm {Re}[z^2-4xy]>0$ .\",\n \"We use the branch with a nonnegative real part for the square root of a complex number.\",\n \"The above relation can be extended to integrals of normally ordered expressions because under this prescription, we have an algebra of commuting operators.\",\n \"In the case that operator ordering becomes relevant, let us formulate additional relations.\",\n \"From the observations that $\\\\hat{a}^mf(\\\\hat{n})|n\\\\rangle =f(\\\\hat{n}+m)\\\\hat{a}^m|n\\\\rangle $ and $f(\\\\hat{n})\\\\hat{a}^{\\\\dag m}|n\\\\rangle =\\\\hat{a}^{\\\\dag m}f(\\\\hat{n}+m)|n\\\\rangle $ hold true, we conclude that $e^{x\\\\hat{a}^2}\\\\omega ^{\\\\hat{n}}=\\\\omega ^{\\\\hat{n}}e^{\\\\omega ^2x\\\\hat{a}^2}\\\\text{ and }\\\\omega ^{\\\\hat{n}}e^{y\\\\hat{a}^{\\\\dag 2}}=e^{\\\\omega ^2y\\\\hat{a}^{\\\\dag 2}}\\\\omega ^{\\\\hat{n}}.$ In the same manner, we get $\\\\exp (u\\\\hat{a})\\\\omega ^{\\\\hat{n}}=\\\\omega ^{\\\\hat{n}}\\\\exp (\\\\omega u\\\\hat{a})$ and $\\\\omega ^{\\\\hat{n}}\\\\exp (v\\\\hat{a}^\\\\dag )=\\\\exp (\\\\omega v\\\\hat{a}^\\\\dag )\\\\omega ^{\\\\hat{n}}$ .\",\n \"Furthermore, using the above integral, $|\\\\alpha \\\\rangle \\\\langle \\\\alpha |={:}\\\\exp (-[\\\\hat{a}-\\\\alpha ]^\\\\dag [\\\\hat{a}-\\\\alpha ]){:}$ , and $\\\\pi \\\\hat{1}=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,|\\\\alpha \\\\rangle \\\\langle \\\\alpha |$ , we obtain $\\\\begin{aligned}&e^{x\\\\hat{a}^2}e^{y\\\\hat{a}^{\\\\dag 2}}=\\\\exp [x\\\\hat{a}^2]\\\\hat{1}\\\\exp [y\\\\hat{a}^{\\\\dag 2}]\\\\\\\\=&\\\\frac{1}{\\\\sqrt{1-4xy}}{:}\\\\exp \\\\left(\\\\frac{4xy\\\\hat{a}^\\\\dag \\\\hat{a}+x\\\\hat{a}^2+y\\\\hat{a}^{\\\\dag 2}}{1-4uv}\\\\right){:}\\\\\\\\=&e^{v\\\\hat{a}^{\\\\dag 2}/(1-4uv)}\\\\left(\\\\frac{1}{1-4uv}\\\\right)^{\\\\hat{n}+1/2}e^{u\\\\hat{a}^2/(1-4uv)}.\\\\end{aligned}$ The same approach yields $\\\\exp [u\\\\hat{a}]\\\\exp [y\\\\hat{a}^{\\\\dag 2}]=\\\\exp [y(\\\\hat{a}^\\\\dag +u)^2]\\\\exp [u\\\\hat{a}]$ and $\\\\exp [x\\\\hat{a}^2]\\\\exp [v\\\\hat{a}^\\\\dag ]=\\\\exp [v\\\\hat{a}^\\\\dag ]\\\\exp [x(\\\\hat{a}+v)^2]$ .\",\n \"Finally, let us also mention the well-known formula $\\\\exp (u\\\\hat{a})\\\\exp (v\\\\hat{a}^\\\\dag )=\\\\exp (uv)\\\\exp (v\\\\hat{a}^\\\\dag )\\\\exp (u\\\\hat{a})$ .\"\n ],\n [\n \"Spectral decomposition of Gaussian quasistates\",\n \"In the main part of this contribution, we consider filter functions of a Gaussian form [cf.\",\n \"Eq.\",\n \"(REF )].\",\n \"The exact analysis for the resulting quasistates is performed here.\",\n \"For zero displacement, the integral in Eq.\",\n \"(REF ) yields $\\\\begin{aligned}\\\\hat{\\\\Delta }=&\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}e^{-\\\\beta \\\\hat{a}^\\\\dag }e^{\\\\beta ^\\\\ast \\\\hat{a}}\\\\\\\\=&\\\\frac{2\\\\,{:}\\\\exp \\\\left(\\\\frac{-2[1+s]\\\\hat{a}^\\\\dag \\\\hat{a} +q\\\\hat{a}^{\\\\dag 2}+p\\\\hat{a}^2}{(1+s)^2-pq}\\\\right){:}}{\\\\sqrt{(1+s)^2-pq}}.\\\\end{aligned}$ To characterize this quasistate, let us consider its decomposition in terms of exponential operators.\",\n \"First, the squeezing operator $\\\\hat{S}=\\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)$ can be equivalently given as [58] $\\\\hat{S}=\\\\exp \\\\left(-\\\\frac{\\\\tau }{2}\\\\hat{a}^{\\\\dag 2}\\\\right)\\\\sqrt{1-|\\\\tau |^2}^{\\\\hat{n}+1/2}\\\\exp \\\\left(\\\\frac{\\\\tau ^\\\\ast }{2}\\\\hat{a}^2\\\\right),$ where $\\\\tau =e^{i\\\\arg \\\\zeta }\\\\tanh |\\\\zeta |$ .\",\n \"Note that $|\\\\tau |<1$ , and we have infinite squeezing for $|\\\\tau |\\\\rightarrow 1$ .\",\n \"In addition, we consider thermal-state-like operator $\\\\begin{aligned}\\\\hat{W}=&(1-\\\\omega )\\\\omega ^{\\\\hat{n}}=\\\\sum _{n\\\\in \\\\mathbb {N}}(1-\\\\omega )\\\\omega ^{n}|n\\\\rangle \\\\langle n|\\\\\\\\=&(1-\\\\omega ){:}e^{-(1-\\\\omega )\\\\hat{n}}{:}\\\\\\\\=&\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,\\\\frac{1-\\\\omega }{\\\\pi \\\\omega }\\\\exp \\\\left(-\\\\frac{(1-\\\\omega )|\\\\alpha |^2}{\\\\omega }\\\\right)|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.\\\\end{aligned}$ See Ref.\",\n \"[68] for a derivation and additional considerations.\",\n \"Also note that the vacuum state $\\\\hat{W}=|0\\\\rangle \\\\langle 0|$ is obtained in the limit $\\\\omega \\\\rightarrow 0$ .\",\n \"Now, we defined an operator $\\\\hat{T}$ to be compared with the quasistate $\\\\hat{\\\\Delta }$ , $\\\\hat{T}=\\\\hat{S}\\\\hat{W}\\\\hat{S}^\\\\dag .$ Using the exchange relations in Eqs.\",\n \"(REF ) and (REF ), we find after some algebra a normally ordered expression, $\\\\begin{aligned}\\\\hat{T}=&(1-\\\\omega )\\\\sqrt{\\\\frac{1-|\\\\tau |^2}{1-|\\\\tau |^2\\\\omega ^2}}\\\\\\\\&\\\\times \\\\exp \\\\left(-\\\\frac{\\\\tau }{2}\\\\frac{1-\\\\omega ^2}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{a}^{\\\\dag 2}\\\\right)\\\\\\\\&\\\\times {:}\\\\exp \\\\left(-\\\\frac{[1-\\\\omega ][1+\\\\omega |\\\\tau |^2]}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{n}\\\\right){:}\\\\\\\\&\\\\times \\\\exp \\\\left(-\\\\frac{\\\\tau ^\\\\ast }{2}\\\\frac{1-\\\\omega ^2}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{a}^2\\\\right).\\\\end{aligned}$ Finally, we can compare this expression with Eq.\",\n \"(REF ), $\\\\hat{\\\\Delta }=\\\\hat{T}$ .\",\n \"Namely, equating coefficient yields $\\\\omega =\\\\frac{r-1}{r+1},\\\\,|\\\\tau |^2=\\\\frac{s-r}{s+r},\\\\,\\\\text{and}\\\\,\\\\frac{\\\\tau }{\\\\tau ^\\\\ast }=\\\\frac{q}{p},$ where $r=\\\\sqrt{s^2-pq}$ .\",\n \"This also implies $\\\\tau =-q/(s+r)$ and $\\\\tau ^\\\\ast =-p/(s+r)$ .\",\n \"Conversely, we can deduce parameters $s$ , $p$ , and $q$ from given values of $\\\\omega $ and $\\\\tau $ , $\\\\begin{aligned}p=\\\\frac{-2(1+\\\\omega )\\\\tau ^\\\\ast }{(1-\\\\omega )(1-|\\\\tau |^2)},\\\\quad q=\\\\frac{-2(1+\\\\omega )\\\\tau }{(1-\\\\omega )(1-|\\\\tau |^2)},\\\\\\\\\\\\text{and}\\\\quad s+1=\\\\frac{2(1+\\\\omega |\\\\tau |^2)}{(1-\\\\omega )(1-|\\\\tau |^2)}.\\\\end{aligned}$\"\n ],\n [\n \"Uniform attenuation\",\n \"In the main text, we consider a specific example of a kernel.\",\n \"For studying its properties, let us consider the corresponding convolution $P_K(\\\\chi )=\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )K(\\\\psi ,\\\\chi )=a\\\\,P(\\\\chi )+b\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )$ for a set $\\\\mathcal {S}$ of states.\",\n \"Note that $\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )=1$ is the normalization condition.\",\n \"To ensure that the result is normalized as well, $1=\\\\sum _{\\\\chi \\\\in \\\\mathcal {S}}P_K(\\\\chi )=a+b|\\\\mathcal {S}|$ , we find $b=\\\\frac{1-a}{|\\\\mathcal {S}|},$ using the cardinality $|\\\\mathcal {S}|$ of the set of states.\",\n \"Further, when $0\\\\le a\\\\le 1$ is satisfied, the non-negativity of $P_K$ is guaranteed for any nonnegative $P$ .\",\n \"The considered convolution can be formulated via the $|\\\\mathcal {S}|\\\\times |\\\\mathcal {S}|$ map $K=a\\\\,\\\\mathrm {id}+b\\\\,nn^\\\\mathrm {T},$ using the identity $\\\\mathrm {id}$ and the $|\\\\mathcal {S}|$ -dimensional vector $n=[1,\\\\ldots ,1]^\\\\mathrm {T}$ .\",\n \"Then the inverse takes a similar form, $K^{-1}=\\\\frac{1}{a}\\\\mathrm {id}-\\\\frac{b/a}{a+b|\\\\mathcal {S}|}nn^\\\\mathrm {T},$ where $a+b|\\\\mathcal {S}|=1$ for the given $b$ .\",\n \"Using the normalization, the convolution kernel $K$ corresponds to uniform addition of noise to the probabilities.\",\n \"On the level of the operators, we get $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\frac{1}{a}|\\\\chi \\\\rangle \\\\langle \\\\chi |-\\\\frac{1-a}{a|\\\\mathcal {S}|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ In the continuous case, we can generalize the above relations as $P_K(\\\\chi )=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\,P(\\\\psi )\\\\left[a\\\\delta (\\\\psi ,\\\\chi )+\\\\frac{1-a}{|\\\\mathcal {S}|}\\\\right],$ where the volume $|\\\\mathcal {S}|=\\\\int _{\\\\mathcal {S}}d\\\\psi $ is used instead.\",\n \"Furthermore, the projection operators transform as $\\\\hat{\\\\Delta }_{K}(\\\\chi )=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\left[\\\\frac{1}{a}\\\\delta (\\\\psi ,\\\\chi )-\\\\frac{1-a}{a|\\\\mathcal {S}|}\\\\right]|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ As a discrete-variable example, we study a qubit system with orthogonal basis $\\\\lbrace |0\\\\rangle ,|1\\\\rangle \\\\rbrace $ .\",\n \"We can identify the Pauli matrices as $\\\\hat{\\\\sigma }_z=|1\\\\rangle \\\\langle 1|-|0\\\\rangle \\\\langle 0|$ , $\\\\hat{\\\\sigma }_x=|0\\\\rangle \\\\langle 1|+|1\\\\rangle \\\\langle 0|$ , and $\\\\hat{\\\\sigma }_y=i|0\\\\rangle \\\\langle 1|-i|1\\\\rangle \\\\langle 0|$ .\",\n \"The Hermitian operator basis can be completed with the operator $\\\\hat{1}=|0\\\\rangle \\\\langle 0|+|1\\\\rangle \\\\langle 1|$ .\",\n \"The eigenvectors $|j_{\\\\pm 1}\\\\rangle $ of the Pauli matrices $\\\\hat{\\\\sigma }_j$ to the eigenvalues $\\\\pm 1$ for $j\\\\in \\\\lbrace z,x,y\\\\rbrace $ form the set $\\\\mathcal {S}$ .\",\n \"Then we can decompose any density operator as $\\\\hat{\\\\rho }=\\\\sum _{j\\\\in \\\\lbrace x,y,z\\\\rbrace ,s\\\\in \\\\lbrace +1,-1\\\\rbrace }P(j_s)|j_{s}\\\\rangle \\\\langle j_{s}|.$ Conversely, the attenuated distribution reads $P_{K}(j_s)=aP(j_s)+(1-a)/6$ , and the transformed operators take the form $\\\\hat{\\\\Delta }_K(j_s)=a^{-1}|j_s\\\\rangle \\\\langle j_s|-(1-a)(2a)^{-1}\\\\hat{1}$ .\"\n ]\n]"},"id":{"kind":"string","value":"1808.08471"}}},{"rowIdx":196,"cells":{"text":{"kind":"list like","value":[["Invariant Sets in Quasiperiodically Forced Dynamical Systems"],["Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.","We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps.","The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set.","We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction.","We provide a characterization of invariant sets in the quasiperiodically forced systems.","A theoretical result on uniform boundedness of the invariant sets is presented.","The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages.","Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability.","We show that our theoretical results can be used to understand stability regions in such complex systems."],["Introduction","Methods of ergodic theory [1], [2] have been applied for solving a number of problems in science and technology such as control of mixing of fluid flows [3], analysis of traffic jams [4] and of quantum many-body models [5], performance evaluation of ecosystem models [6].","A visualization method for invariant sets of discrete-time dynamical systems (or maps) based on ergodic theory was developed in [7].","This method is based on the ergodic partition or ergodic decomposition [8] of state space in a given dynamical system, which is associated with the eigenspace at eigenvalue 1 of the Koopman operator [9].","This method partitions a (compact and metric) state space $X$ using joint level sets of time-averages of a basis of functions defined on $X$ .","This leads to an approximation of invariant sets on which the dynamics of the system are ergodic.","In an ergodic invariant set, almost all points are accessible in the sense that the initial conditions in this set thoroughly sample the set.","The method enables the visualization of low-dimensional (in practice usually two-dimensional) slices through high-dimensional invariant structures and offers a computationally tractable method for analyzing global structures of state space in discrete-time nonlinear models.","The numerical methods associated with the ergodic partition theory have been developed: see [9] and references therein.","The purpose of this paper is to extend the ergodic partition theory and exploit it to provide global analysis of state space in quasiperiodically forced dynamical systems.","Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see, e.g., [10], [11].","Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [12], analysis of dynamics described by the Schrödinger equations [13] and of quantum chaos [14], and dynamical systems analysis with applications to fluid mixing [15], [16].","In the present paper, we study the global structure of state space in quasiperiodically forced systems from the viewpoint of ergodic partition.","First, we develop a theory of ergodic partition of state space in measure-preserving and dissipative, continuous-time dynamical systems or flows.","This is a natural extension of the existing theory for measure-preserving maps in [7].","The ergodic partition is again based on eigenspace at eigenvalue 0 of the associated Koopman group.","Related analyses of time-periodic flows via spectral properties of linear operators are reported in [17], [18].","Since an arbitrary quasiperiodically forced system with a smooth invariant measure is transformed into an autonomous system determining a measure-preserving flow (see Section ), the ergodic partition theory is used for visualization of invariant sets for the quasiperiodically forced system.","Next, we provide a characterization of invariant sets in quasiperiodically forced systems.","In general, understanding invariant structures in the quasiperiodically forced systems is not easy because the associated portraits of state space change with time in an aperiodic manner.","Here, we introduce a notion of uniform boundedness of invariant sets.","A theoretical result on uniform boundedness of invariant sets is presented in this paper: see Theorem REF , and Corollaries REF and REF .","This clarifies a dynamical feature of invariant sets in the quasiperiodically forced systems that is directly determined by the ergodic partition and its associated visualization technique.","Also, we apply the theory developed above to analysis of a nonlinear model of complex power grids that represents the short-term swing instability, which we name the Coherent Swing Instability (CSI) [19].","Preliminary results in this paper were published in the conference proceedings [20], [21].","This paper contains detailed discussion of the theory with a new proof and a set of new numerical simulations.","The rest of this paper is organized as follows.","In Section we introduce set-up and mathematical preliminaries from dynamical systems theory.","A theory of ergodic partition in measure-preserving flows is developed in Section .","Based on the result, the basin of attraction in the dissipative case is explored in ().","In Section we provide a new characterization of invariant sets in the quasiperiodically forced dynamical systems.","The developed theory is applied in Section to two simple examples of the quasiperiodically forced system and to a nonlinear model of the CSI phenomenon of a power grid.","Conclusions of this paper are presented in Section ."],["Quasiperiodically Forced Dynamical Systems","Throughout this paper, we address the following quasiperiodically forced dynamical system that evolves on a finite-dimensional metric space $M$ : for ${m}\\in {M}$ and $t\\in \\mathbb {R}$ , $\\frac{{d}{m}}{{d} t}={g}({m},t).$ The function $g$ is assumed to be quasiperiodic on $t$ in the sense of Moser [22], that is, ${g}({m},t)={G}({m},\\mathit {\\Omega }_1t,\\mathit {\\Omega }_2t,\\ldots ,\\mathit {\\Omega }_Nt),$ where ${G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)$ is assumed to be a smooth vector-valued function and of period $2\\pi $ in $\\theta _1,\\theta _2,\\ldots ,\\theta _N$ .","The real numbers $\\mathit {\\Omega }_1,\\mathit {\\Omega }_2,\\ldots ,\\mathit {\\Omega }_N$ are the $N$ basic angular frequencies and assumed to be rationally independent: there exists no $N$ -dimensional integer vector $(k_1,k_2,\\ldots ,k_N)^\\top $ ($\\top $ stands for the transpose operation of real-valued vectors) in which the entries do not all vanish, satisfying the resonance relation: $\\sum ^{N}_{i=1}k_i\\mathit {\\Omega }_i=0.$ The system (REF ) is non-autonomous and transformed into an autonomous system defined on the augmented state space $X:=M\\times \\mathbb {T}^N$ , in the same manner as in [16].","By introducing the $N$ variables $\\theta _i=\\mathit {\\Omega }_it\\in \\mathbb {T}$ ($i=1,2,\\ldots ,N$ ), we have $\\frac{d{m}}{dt}={G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N), \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N.$ In this way, by taking trajectories of the augmented system (REF ) on the augmented state variable ${x}:=({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)\\in X$ , a flow, that is, one-parameter group of diffeomorphisms is defined for the quasiperiodically forced system (REF )."],["Measure-Preserving Flows and Partition of State Space","In Section , as one type of flows induced by (REF ), we will investigate a measure-preserving flow defined on a class of probability spaces, where ergodic theory has been developed [8], [23].","The probability space considered here corresponds to the tuple $({X},\\mathfrak {B}_{X},\\mu )$ , where ${X}$ is a compact metric space, $\\mathfrak {B}_{X}$ is the Borel $\\sigma $ -algebra of ${X}$ , and $\\mu $ is a probability measure.","A measure-preserving flow of the probability space is a one-parameter group of measure-preserving diffeomorphisms ${S}^t: {X}\\rightarrow {X}$ , $t\\in \\mathbb {R}$ such that ${S}^t$ is measure-preserving (for all $A\\in \\mathfrak {B}_{X}$ we have $\\mu ({S}^{-t}(A))=\\mu (A)$ ), ${S}^0$ the identity, and ${S}^{t_1+t_2}={S}^{t_1}\\circ {S}^{t_2}$ for $t_1,t_2\\in \\mathbb {R}$ .","The important notion which we utilize throughout this paper is the partition of state space in dynamical systems.","One motivation behind the following definitions is that we aim to locate such a partition that plays a role in the data-driven estimation of certain statistical properties of non-ergodic measure-preserving dynamical systems; we will show this later as the ergodic partition.","The definitions of partition and measurable partition from [7] are the following.","Definition 1 A partition of ${X}$ is a family $\\zeta $ of sets satisfying $A,B\\in \\zeta \\quad \\Rightarrow \\quad \\mu (A\\cap B)=0, \\quad \\mu \\left({X}\\setminus \\bigcup _{A\\in \\zeta }A\\right)=0.$ The element of $\\zeta $ containing a point ${x}\\in {X}$ is denoted by $\\zeta ({x})$ .","A partition $\\zeta $ of ${X}$ is said to be measurable if there exists a countable family $\\mathfrak {D}$ of measurable sets $\\lbrace D_i\\rbrace $ such that every $D_i$ is a union of elements of $\\zeta $ , and for any pair $A_1,A_2$ of elements of $\\zeta $ there exists $D_j\\in \\mathfrak {D}$ such that $A_1\\subset D_j$ and $A_2\\subset D_j^{\\rm c}$ , where $D_j^{\\rm c}$ stands for the complement of $D_j$ in ${X}$ .","The family $\\mathfrak {D}$ is called a basis for $\\zeta $ .","In Section we use a product operation on the set of partitions of ${X}$ .","The product operation is defined in [23] as follows.","Definition 2 If $\\zeta _n$ , $n=1,\\ldots ,N$ are partitions, we then define their product $\\vee ^{N}_{n=1}\\zeta _n$ as the partition whose elements are the sets of the form $\\cap ^{N}_{i=1}{A_i}$ , for $A_i\\in \\zeta _i$ , $i=1,\\ldots ,N$ , satisfying $\\mu \\left(\\cap ^{N}_{i=1}A_i\\right)\\ne 0$ .","For a countable sequence of partitions, the notation $\\vee ^{\\infty }_{n=1}\\zeta _n$ will be used for the $\\sigma $ -algebra generated by $\\cup ^\\infty _{n=1}\\zeta _n$ ."],["Koopman Group","Consider a flow ${S}^t: {X}\\rightarrow {X}$ ($t\\in \\mathbb {R}$ ) on a finite-dimensional space ${X}$ and denote by $\\mathcal {F}$ a space of scalar-valued functions defined on ${X}$ : $\\mathcal {F}\\ni f: {X}\\rightarrow \\mathbb {C}$ , which we call the space of observables.","In the following, we suppose that the existence and uniqueness of solutions associated with the flow hold for all $t$ .","Then, we define the one-parameter group of linear operators $U^t$ ($t\\in \\mathbb {R}$ ) as a group of composition operators with ${S}^t$ : for $f\\in \\mathcal {F}$ , $(U^tf)({x}):=f({S}^t({x}))=(f\\circ {S}^t)({x}).$ where it is known as the Koopman group [24], [9].","For spaces as $\\mathcal {F}=\\mathcal {C}^0(X)$ and $\\mathcal {L}^p(X)$ , the Koopman group is strongly continuous.","Although the original flow ${S}^t$ is possibly described by a nonlinear differential equation and evolves on the finite-dimensional space ${X}$ , the operators $U^t$ are linear but evolve on the infinite-dimensional space $\\mathcal {F}$ .","The eigenvalue $\\lambda \\in \\mathbb {C}$ and eigenfunction $\\phi _\\lambda \\in \\mathcal {F}\\setminus \\lbrace 0\\rbrace $ of the Koopman group are defined as follows: $(U^t\\phi _\\lambda )({x})=\\exp (\\lambda t)\\phi _\\lambda ({x}).$ The notion of the Koopman group and its spectral characterization will be used in the following sections."],["Ergodic Partition in Measure-Preserving Flows","For completeness, in this section we extend the theory of ergodic partition in [7] to the measure-preserving flow ${S}^t$ on the probability space $(X,\\mathfrak {B}_{X},\\mu )$ , which is introduced in Section REF .","This section consists of a few lemmas and a theorem.","Their proofs are almost identical to the proofs for maps [7] and appear in Appendices and .","We denote by $\\mathcal {L}^1_\\mu ({X})$ the space of all $\\mu $ -integrable functions on ${X}$ and by $\\mathcal {C}({X})$ the space of all real-valued continuous functions on ${X}$ endowed with the sup norm.","Roughly speaking, an ergodic partition is a partition of the state space ${X}$ into invariant sets on which the dynamics are ergodic.","The precise definition of ergodic partition is the following.","Definition 3 A measurable partition $\\zeta $ of ${X}$ is said to be ergodic under the flow ${S}^t$ if for any element $A$ of $\\zeta $ , (i) $A$ is invariant under ${S}^t$ , and (ii) there exists an invariant probability measure $\\mu _A$ on $A$ such that the restriction of ${S}^t$ to $A$ , denoted by ${S}^t|_A$ , is an ergodic diffeomorphism on $A$ , and for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , $\\int _{X} f{d}\\mu =\\int _{X}\\left[\\int _{A=:\\zeta ({x})} f|_A{d}\\mu _A\\right]{d}\\mu ({x}),$ where $f|_A$ stands for the restriction of $f$ to the ergodic element $A$ , and $\\mu ({x})$ is again a probability measure on $X$ .","It is remarked that the ergodic partition is defined for the entire state space $X$ with $\\sigma $ -algebra, but it can be relaxed in context of the $\\sigma $ -algebra of invariant sets which can be parts of the state space.","Here, we denote by $f^\\ast $ the time-average of $f$ under the flow ${S}^t$ if the right-hand side of $f^\\ast ({x}):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t({x})){d} t,$ exists for almost every (a.e.)","point ${x}\\in {X}$ with respect to the measure $\\mu $ .","Birkhoff's Ergodic Theorem [25], [1] shows that for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , (i) $f^\\ast ({x})$ exists for a.e.","point ${x}\\in {X}$ (with respect to $\\mu $ ); (ii) $f^\\ast ({S}^t({x}))=f^\\ast ({x})$ for a.e.","point ${x}\\in {X}$ ; and (iii) $\\int _{X}f{d}\\mu =\\int _{X}f^\\ast {d}\\mu $ .","Here, we let $\\mathit {\\Sigma }$ be the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for all $f\\in \\mathcal {C}({X})$ , and $\\mathit {\\Sigma }(f)$ the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for a particular $f\\in \\mathcal {C}({X})$ .","The following lemma is standard (see page 129 of [23] for discrete-time dynamical systems (maps): it is obvious for flows).","Lemma 4 For a countable and dense set $S$ in $\\mathcal {C}({X})$ , $\\mathit {\\Sigma }=\\bigcap _{f\\in S}\\mathit {\\Sigma }(f).$ Now, we have a set $\\mathit {\\Sigma }$ such that the time-averages of all continuous functions on ${X}$ exist.","Then, the complimentary set $\\mathit {\\Sigma }^{\\rm c}$ is of measure zero.","This is because according to Birkhoff's Ergodic Theorem (i), each $\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is of measure zero, and thus $\\mathit {\\Sigma }^{\\rm c}:=\\cup _{f\\in S}\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is a countable union of sets with zero measure, which is again of measure zero.","The next lemma shows that the time-average of a continuous function induces a measurable partition on ${X}$ .","Lemma 5 Let $f$ be a continuous function on ${X}$ .","The family of level sets of $f^\\ast $ , $A_\\alpha :=\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace ,~~~\\alpha \\in \\mathbb {R},$ is a measurable partition of ${X}$ .","We denote by $\\zeta _f$ this partition and call it the partition induced by the function $f$ .","See Appedix .","Now, we can prove the first theorem stating that $\\zeta _f$ induces the ergodic partition of ${X}$ , which holds for general measure-preserving flows including the augmented system (REF ) for the quasiperiodically forced system (REF ).","Theorem 6 Consider the measure-preserving flow ${S}^t: X\\rightarrow X$ ($t\\in \\mathbb {R}$ ) associated with $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0} ={F}({x}) \\quad \\textrm {for each}~{x}\\in X$ where ${F}: X\\rightarrow \\mathrm {T}X$ (tangent bundle of $X$ ) is a nonlinear vector field.","Let $\\zeta {e}$ be the product of measurable partitions of ${X}$ induced by every $f\\in S$ : $\\zeta {e}=\\bigvee _{f\\in S}\\zeta _f.$ Then, $\\zeta {e}$ is the ergodic partition of ${X}$ .","See Appendix .","By combining the above theorem and lemma 20 in [26], we have the following corollary based on a finite number of basis functions.","Corollary 7 Assume there exists a complete system of functions $\\lbrace f_i\\rbrace $ , $f_i\\in \\mathcal {C}(X)$ , $i\\in \\mathbb {N}_{>0}$ (set of all natural numbers except for 0) i.e.","finite linear combinations of $f_i$ are dense in $\\mathcal {C}(X)$ .","The ergodic partition $\\zeta {e}$ is $\\zeta {e}=\\bigvee _{i\\in \\mathbb {N}_{>0}}\\zeta _{f_i}.$ For a given function $f$ , it is obvious that each level set $A_\\alpha =\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace $ ($\\alpha \\in \\mathbb {R}$ ) as an element of $\\zeta _f$ is invariant under ${S}^t$ .","In [7] the authors proposed to use the level sets $A_\\alpha $ , which are directly computed with the time-averaging of $f$ under ${S}^t$ , for identifying and visualizing invariant sets in discrete-time dynamical systems possessing a smooth invariant measure.","Theorem REF implies that the same computation can be used for visualization of invariant sets for the measure-preserving flow, namely, continuous-time dynamical systems with a smooth invariant measure.","Since, as shown in Section , the quasiperiodically forced system (REF ) defined on the state space $M$ is transformed to the flow defined on the augmented state space $X=M\\times \\mathbb {T}^N$ , the ergodic partition theory is applicable to visualization of invariant sets in $X$ for the original quasiperiodically forced system (REF ) possessing a smooth invariant measure.","However, in the quasiperiodically forced system (REF ) we are interested in dynamics on $M$ not on $X=M\\times \\mathbb {T}^N$ .","While for $N=1$ it is clear that every intersection of the invariant set in $M\\times \\mathbb {T}^1$ with $M$ is an invariant set of the associated Poincaré map.","The relationship in the quasiperiodically forced system (namely, $N\\ge 2$ ) is much less clear.","We make it precise in Section .","Here, let us introduce the connection of time-average $f^\\ast $ and the Koopman group.","It is obvious that for any $f\\in \\mathcal {L}^1_\\mu (X)$ its time-average $f^\\ast $ is an eigenfuction of the operators $U^t$ at eigenvalue 0: for a.e.","point ${x}\\in {X}$ with respect to $\\mu $ , $(U^tf^\\ast )({x})=f^\\ast ({S}^t({x}))=f^\\ast ({x})=\\exp (0t)f^\\ast ({x}).$ Since the eigenfunctions at 0 are invariant under the flow, we have a complete characterization of (possibly non-smooth) invariant sets."],["Basin of Attraction in Dissipative Flows","In the last of the previous section, we indicate that the time-average of an observable enables the visualization of low-dimensional slices through high-dimensional invariant sets of the quasiperiodically forced system with a smooth invariant measure.","Here, we consider a more general class of the quasiperiodically forced systems with dissipation and will point out that the use of time-average works for characterizing an important invariant set—basin of attraction.","In Section we considered the measure-preserving flow on a compact metric space.","Every continuous flow on a compact metric space preserves a measure.","This is the content of the so-called Krylov-Bogolyubov theorem [27].","However, this might be a useless statement if our intent is to study the behavior of trajectories from their time-averages, by the method used in the measure-preserving cases in Section .","For example, let us take a simple dissipative flow, $\\frac{{d} x}{{d} t}=-\\lambda x \\qquad x\\in \\mathbb {R}, \\qquad \\lambda >0.$ All of the initial conditions converge to $x=0$ as $t\\rightarrow \\infty $ along the flow $S^t(x)=\\exp (-\\lambda t)x$ .","The invariant measure clearly exists and is the Dirac measure supported at $x=0$ (to which any “initial\" measure defined on $\\mathbb {R}$ evolves).","Thus, the following statement holds: for a function $f:\\mathbb {R}\\rightarrow \\mathbb {C}$ , $f^\\ast (x)=\\int _{\\mathbb {R}}f{d}\\mu ,$ for a.e.","point $x\\in \\mathbb {R}$ with respect to the Dirac measure.","However, that excludes the whole real line except for the origin itself.","The situation feels better though.","For example, it is easy to see that the time-average of any continuous function on $\\mathbb {R}$ is just its value at 0: $f^*(x)=f(0)$ .","Note that for the measure-preserving case in Section , we basically considered integrable functions for which time-averages exist.","We can not do that in the dissipative case.","A function that is 1 everywhere on a finite interval $(\\underline{a},\\overline{a})$ containing 0, except at 0 where its value is 0, is clearly integrable (being a union of two simple functions).","Then, its time-average under the dissipative flow is identically 1, and its integral with respect to the invariant Dirac measure is 0.","It turns out that in dissipative systems, it is best for the time-average to work with continuous functions.","Adopting that idea, the whole method of ergodic partitioning can be extended to capture basins of attraction for continuous flows that are not necessarily measure-preserving.","Consider a continuous flow ${S}^t: X\\rightarrow X$ , where $X$ is not necessary compact, and label by $F$ the set on which time averages of continuous functions do not exist.","On the set $X\\setminus F$ , consider a set $C$ on which time-averages of continuous functions (or, better, a countable, separating set of continuous functions) are constant.There can be uncountably many such sets inside $X\\setminus F$ : for example, let ${S}^t$ be the identity map, mapping every point into itself.","Then, by the same construction described in Section and Appendix , there is an invariant measure $\\mu _C$ such that $f^*({x})=\\int _C f d\\mu _C$ for any continuous function $f$ , ${x}\\in C$ .","Such a measure is called a physical measure [28], provided $M$ is equipped with a measure $\\mu $ that ${S}^t$ does not preserve (but we are interested in)—say Lebesgue measure—and $C$ contains an open (and thus positive measure) set in $M$ .In light of this discussion, one might say that the notion of “ergodic measure\" should be preserved for such constructs even in dissipative systems.","Namely, many ergodic measures we obtain in measure-preserving system are certainly physically important, although their support does not contain an open set.","The tricky part is that due to Birkhoff's Ergodic Theorem, ergodic measures are associated with time averages of integrable functions.","In dissipative systems, this does not work (see the above example).","We could keep the requirement of equality of space and time averages, like in (REF ), but for continuous functions only.","Physical measures would then be ergodic measures with support that contains an open set.","Suppose that the state space of a continuous flow admits a finite number of attractors, and that the union of their basins of attraction is of full measure.","We state the result for flows (continuous-time systems) to emphasize that considerations here work for both discrete and continuous time.","Theorem 8 Let the system $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0}={F}({x}), \\quad {x}\\in X\\subset \\mathbb {R}^n$ with a flow ${S}^t$ have a finite number $N$ of attractors with basins of attraction $A_j, j=1,...,N$ , such that $\\mu (\\cup _j A_j)=\\mu (X)$ , where $\\mu $ is the Lebesgue measure.","Also, let the time averages of continuous functions exist everywhere on a set $X\\setminus F$ , where $\\mu (F)=0$ .","Then, the time-average $h^*({x})$ of a continuous function $h\\in \\mathcal {C}(X)$ is a piecewise constant eigenfunction of the Koopman operators $U^t$ at eigenvalue 0 defined a.e.","point with respect to $\\mu $ .","For any point ${x}\\in A_j,$ there exists a set $C$ such that $(U^t h^*)({x})&=U^t \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int _0^{T} h({S}^\\tau ({x}))d\\tau \\nonumber \\\\&= \\int _C (U^t h) d\\mu _C\\nonumber \\\\&=\\int _C (h\\circ {S}^t) d\\mu _C\\nonumber \\\\&=\\int _C h d\\mu _C \\nonumber \\\\&=h^*({x}), \\nonumber $ where in the second line we utilized (REF ), and transitioning from line 3 to 4 we used the fact that $\\mu _C$ is invariant under ${S}^t$ .","Since $(U^t h^*)({x})=h^*({x})$ is exactly the equation which $h^*({x})$ has to satisfy in order to be an eigenfunction of $U^t$ , and the basin of attraction $A_j$ is arbitrary, by taking into account that $\\mu (\\cup _j A_j)=\\mu (X)$ the proposition is proven.","The concept of the ergodic partition in dissipative systems is applied to a more general class of systems than those treated in Theorem REF .","It provides ergodic measures on invariant sets that are not necessarily attractors—because they do not have an open set of initial conditions converging to them.","Yet, these sets are dynamically important and can not be refined without losing the equivalence of space and time averages.","More precisely, the ergodic partition $\\zeta {e}$ of $M$ under ${S}^t$ (not necessarily measure-$\\mu $ preserving) is a partition into sets $C_{\\alpha }$ , where $\\alpha $ is a member of an indexing set, such that on each set $C_{\\alpha }$ (ergodic set) there exists an ergodic measure $\\mu _{C_\\alpha }$ such that $\\mu _{C_{\\alpha }}(C_{\\alpha })=1,$ For every $f\\in \\mathcal {C}(M),f^{\\ast }({x}\\in C_{\\alpha })=\\int _{C_{\\alpha }}fd\\mu _{C_{\\alpha }},$ for a.e ${x}$ point with respect to $\\mu $ .","The last condition emphasizes the role of the measure $\\mu $ —the time averages are equal to space averages with respect to $\\mu _{C_\\alpha }$ , but their equality is almost everywhere with respect to measure $\\mu $ that might be of our interest.","In this way, we have escaped the realm of the Krylov-Bogolyubov Theorem, which, in the case when dynamics are not measure-preserving, neglects dynamics on possibly large swaths of the state space."],["Sample-Based Characterization of Invariant Sets","In this section, we provide a characterization of invariant sets in the quasiperiodically forced system (REF ).","Generally speaking, understanding invariant structures of (REF ) is not easy because the associated portrait of $M$ changes with time in an aperiodic manner.","As shown in Section , the ergodic partition theory enables one to visualize an invariant set as its low-dimensional slice in $M$ by the time-averaging technique.","Also, in () it was shown that the time-averaging technique works for dissipative case to visualize the basin of attraction.","The obtained slice here is just a sample of the invariant set at a particular initial time (in other words, an initial condition in $\\mathbb {T}^N$ ).","Because of the aperiodic nature, such a sample does not seem to provide complete information on the entire structure of invariant set in the augmented state space $M\\times \\mathbb {T}^N$ .","However, we will prove that a boundedness property of the invariant set in $M\\times \\mathbb {T}^N$ is captured by means of one sample of it in $M$ (see Theorem REF ).","In the following, in order to encompass both the cases in Section and (), we consider the original state space $M$ that is metric and not necessarily compact.","A continuous, linear skew-product flow ${S}^t$ (diffeomorphism) on the augmented state space $X:=M\\times \\mathbb {T}^N$ derived from the quasiperiodically forced system (REF ) is described below: for ${x}=({m},{\\theta })\\in M\\times \\mathbb {T}^N$ , ${S}^t({x}):=\\left({S}^t_{M}({m},{\\theta }),{S}^t_\\mathit {\\Omega }({\\theta })\\right),$ where ${S}^t_{M}: M\\times \\mathbb {T}^N\\rightarrow M$ is the continuous map defined by trajectories of the original system (REF ), and ${S}^t_\\mathit {\\Omega }: \\mathbb {T}^N\\rightarrow \\mathbb {T}^N$ the linear (continuous) flow on the torus $\\mathbb {T}^N$ .","For a fixed ${\\theta }_0\\in \\mathbb {T}^N$ , we will write the orbit on $\\mathbb {T}^N$ through ${\\theta }_0$ as $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0):=\\lbrace {\\theta } : {\\theta }={S}^t_\\mathit {\\Omega }({\\theta }_0)\\in \\mathbb {T}^N, t\\in \\mathbb {R}\\rbrace $ .","Since the $N$ basic frequencies in the original system (REF ) are rationally independent, the following lemma is obvious.","Lemma 9 For any ${\\theta }_0\\in \\mathbb {T}^N$ , $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ is dense in $\\mathbb {T}^N$ .","Now, we investigate an invariant set in the augmented system (REF ).","Let $I\\subseteq X$ be a positively invariant set of (REF ) that is supposed to be closedIf $I$ is invariant and open, then its closure $\\mathrm {cl}(I)$ is still invariant..","It follows from Lemma REF and by closure that the following decomposition of $I$ holds: $I=\\bigcup _{\\theta \\in \\mathbb {T}^N} A_\\theta \\times \\lbrace {\\theta }\\rbrace ,$ where $A_{\\theta }$ is a closed subset of $M$ .","Let us denote by $A_{\\theta _0}$ an intersection or sample of the invariant set $I$ at ${\\theta }={\\theta }_0$ .","The next lemma provides a topological property of the intersections $A_\\theta $ .","Lemma 10 For each $\\epsilon >0$ there exists a positive constant $\\delta $ so that $|{\\theta }-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $d(A_\\theta ,A_{\\theta _0})<\\epsilon $ , where $d(A_{\\theta },A_{\\theta _0})$ is the Hausdorff distance that induces a topology on the family of all closed subsets of $M$ .","Assume not.","Then, for each $\\delta >0$ there exist a sequence of times $\\lbrace t_j\\rbrace $ , where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ , and a positive constant $\\epsilon _1$ such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $|{\\theta }_j-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ for every $j>J$ ($J$ is an integer), while $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ holds for some $k>J$ .","Here, from the continuity of ${S}^t_M$ in $t$ , for each ${m}\\in A_{\\theta _0}$ and $\\epsilon >0$ , there exists a positive constant $\\delta $ such that $|{\\theta }_k-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $|{m}_k-{m}|_{M}<\\epsilon $ , where ${m}_k:={S}^{t_k}_{M}({m},{\\theta }_0)\\in A_{\\theta _k}$ .","This gives us a contradiction of $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ by taking $\\epsilon =\\epsilon _1$ and from the definition of Hausdorff distance.","This lemma suggests that the invariant set $I$ is topologically connected with respect to ${\\theta }$ .","One trivial example is provided by taking as $I$ the closure of a single trajectory starting at a point $({m},{\\theta }_0)$ , for which $A_{\\theta _0}$ consists of a single point.","In this case, $I$ is topologically regarded as a product set of a single point in $M$ and the torus $\\mathbb {T}^N$ .","This is rigorously stated in the next theorem.","Theorem 11 Suppose that $A_{\\theta _0}$ is bounded in $M$ .","Then, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .","Now, let us construct the mapping $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ in the following manner.","For each $({m},{\\theta })\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , by choosing time $\\tau _\\theta $ that satisfies ${\\theta }={S}^{\\tau _\\theta }_{\\it \\Omega }({\\theta }_0)$ , we define $h_{\\theta _0}({m},{\\theta })$ as $({S}^{\\tau _\\theta }_M({m},{\\theta }_0),{\\theta })$ .","We here prove the continuity of $h_{\\theta _0}$ .","For each $\\epsilon >0$ and ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $\\left|{\\theta }_j-{\\theta }\\right|_{\\mathbb {T}^N}<\\epsilon /2$ for every $j>J$ ($J$ is an integer).","Furthermore, because of the continuity of ${S}^t_{M}$ in $t$ , for each ${m}_j\\in A_{\\theta _0}$ there exists a positive constant $\\delta _j$ such that $\\left|{m}-{m}_j\\right|_M<\\delta _j$ implies $\\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M<\\epsilon /2$ .","Here, because for every $j>J$ $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\le \\left|{m}-{m}_j\\right|_{M}+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}< \\delta _j+\\frac{\\epsilon }{2},$ we set $\\delta :=\\delta _j+\\epsilon /2$ .","Thus, if $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ , then $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}&\\le \\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}\\nonumber \\\\& <\\frac{\\epsilon }{2}+\\frac{\\epsilon }{2}=\\epsilon .\\nonumber $ This shows that $h_{\\theta _0}$ is continuous.","We next prove that $h_{\\theta _0}$ is uniformly continuous.","Assume not.","Then, for each $\\delta >0$ there exists a positive constant $\\epsilon _1$ such that $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ implies $\\left|h_{\\theta _0}({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ for some $({m}_j,{\\theta }_j) $ and $({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .","We here set $\\delta =1/j$ .","Since it is supposed that $A_{\\theta _0}$ is bounded and closed (and $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\subset \\mathbb {T}^N$ ), the two sequences of points $\\lbrace ({m}_j,{\\theta }_j)\\rbrace $ and $\\lbrace ({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\rbrace $ have convergent subsequences, denoted as $\\lbrace ({m}_{j_k},{\\theta }_{j_k})\\rbrace $ and $\\lbrace ({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})\\rbrace $ .","Because of $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<1/j$ , both the subsequences have the same convergent point, denoted as $({m}^\\ast ,{\\theta }^\\ast )$ .","That is, for each $\\delta _1>0$ , there exists a positive integer $J_k$ such that both the inequalities $\\left|({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ and $\\left|({m}_{j_k},{\\theta }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ hold for every $j_k\\ge J_k$ .","Here, since $h_{\\theta _0}$ is proven to be continuous at $({m}^\\ast ,{\\theta }^\\ast )$ , for each $\\epsilon >0$ , there exists a positive constant $\\delta _2$ such that $\\left|({m},{\\theta })-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _2$ implies $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon /2$ .","By taking $\\epsilon =\\epsilon _1$ and $\\delta _1=\\delta _2$ thus choosing $J_k$ appropriately, we see that both the inequalities $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^M}<\\epsilon _1/2$ and $\\left|h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon _1/2$ hold for every $j_k\\ge J_k$ , and $\\begin{aligned}\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^M}\\le &\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N} \\\\& + \\left|h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N} \\\\< & \\frac{\\epsilon _1}{2}+\\frac{\\epsilon _1}{2} = \\epsilon _1,\\end{aligned}$ implying the contradiction for $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ .","Thus, $h_{\\theta _0}$ is uniformly continuous.","By virtue of Theorem 3.45 in page 136 of [29], the uniformly-continuous $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ is extended to the mapping $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ that is also (uniformly) continuous.","We now prove that $\\hat{h}_{\\theta _0}$ is a bijection.","For each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , the statement is obvious from the construction of $h_{\\theta _0}$ (see its dependence on ${\\theta }$ ).","It is thus enough to check the case ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .","For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of times $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ converges to ${\\theta }$ .","Because $\\hat{h}_{\\theta _0}$ is continuous, the sequence $\\lbrace \\hat{h}_{\\theta _0}({m},{\\theta }_j)=h_{\\theta _0}({m},{\\theta }_j)\\rbrace $ converges to $\\hat{h}_{\\theta _0}({m},{\\theta })$ , which is represented as $\\displaystyle \\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}({m},{\\theta }_0),{\\theta }\\right)$ .","Namely, the limit exists for every ${m}\\in A_{\\theta _0}$ .","Assume that $\\hat{h}_{\\theta _0}$ is not an injection.","Then, there exist ${m},{m}^{\\prime }\\in A_{\\theta _0}$ satisfying ${m}\\ne {m}^{\\prime }$ such that for each $\\epsilon >0$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M<\\epsilon $ holds for every $j>J$ ($J$ is an integer and depends on $\\epsilon $ ).","Here, the fundamental property of uniqueness of trajectories in (REF ) (or ${S}^t$ diffeomorphsim) implies that for each $t_j$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M\\ge \\epsilon _j$ holds ($\\epsilon _j$ is a positive constant and depends on $t_j$ ), showing the contradiction by taking $\\epsilon =\\epsilon _j$ .","Hence, $\\hat{h}_{\\theta _0}$ is an injection.","Regarding $\\hat{h}_{\\theta _0}$ surjective, it is obvious that for each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , $\\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=h_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left({S}_M^{\\tau _\\theta }(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)=(A_\\theta ,{\\theta })$ .","For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ we see $\\displaystyle \\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)$ , and the limit converges to $A_\\theta $ from Lemma REF .","Hence, from (REF ), $\\hat{h}_{\\theta _0}$ is a surjection.","Finally, the inverse of $\\hat{h}_{\\theta _0}$ exists according to the above construction and is continuous.","Therefore, $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ is a homeomorphism, namely, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .","The following two corollaries provide a way of characterization of boundedness of $I$ by means of one sample of it.","Corollary 12 Suppose that $A_{\\theta _0}$ is bounded in $M$ .","Then, $I$ is bounded in $X$ .","Corollary 13 Suppose that a closed subset $B_{\\theta _0}$ of $A_{\\theta _0}$ is bounded in $M$ .","Then, $\\hat{h}_{\\theta _0}(B_{\\theta _0}\\times \\mathbb {T}^N)$ is a subset of $I$ and bounded in $X$ .","Corollaries REF and REF imply that by computing a slice of the invariant set in $M$ at one sample onset, it is possible to determine whether the invariant set or its subset is bounded in the augmented state space $X$ .","For a bounded invariant set $I$ in $X$ , any cross-section $I_{\\theta _0}=A_{\\theta _0}\\times \\lbrace {\\theta }_0\\rbrace $ (${\\theta }_0\\in \\mathbb {T}^N$ ) is bounded in terms of $M$ , in other words, the boundedness property of $A_{\\theta _0}$ does not depend on the choice of ${\\theta }_0$ .","In this way, we call the bounded invariant set $I$ in $X$ uniformly bounded invariant set."],["Example Studies","In this section, we provide three examples of the application of the above theoretical results.","The last example is related to practical problems on stability of power grids."],["Forced Linear Harmonic Oscillator","First, we will consider the following one-degree-of-freedom linear harmonic oscillator with quasi-periodic forcing of (multiple) $N$ frequencies: $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}),$ or $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $F_i$ ($>0$ ) ($i=1,2,\\ldots ,N$ ) are the amplitudes of periodic forces, $\\mathit {\\Omega }_i$ ($>0$ ) the angular frequencies of the forces, and $\\theta _{i0}\\in \\mathbb {T}$ the initial phases.","We assume there is no resonance: the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent; and $\\mathit {\\Omega }_i\\ne 1$ ($i=1,2,\\ldots ,N$ ).","The solution of (REF ) with initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{aligned}m_1(t) &=C_1\\cos t+C_2\\sin t+\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos (\\mathit {\\Omega }_it+\\theta _{i0}), \\\\m_2(t) &=\\displaystyle \\frac{{d}}{{d} t}m_1(t), \\\\{\\vspace{14.22636pt}}\\theta _i(t) &= \\mathit {\\Omega }_it+\\theta _{i0} \\qquad i=1,2,\\ldots ,N,\\end{aligned}\\right\\rbrace $ with $C_1:=m_{10}-\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0}, \\qquad C_2:=m_{20}+\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}.$ Now, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).","We define the following quadratic observable $f: \\mathbb {R}^2\\times \\mathbb {T}^N\\rightarrow \\mathbb {R}$ , related to the potential energy $m^2_1/2$ : $f(m_1,m_2,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2_1.$ Then, we calculate its time-average under the solution as $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\left\\lbrace C^2_1+C^2_2+\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}\\right\\rbrace .$ Here, as shown in (REF ), the constants $C_1$ and $C_2$ are determined by the initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","The level set, parameterized by a real-valued constant $c\\in \\mathbb {R}$ , of the time-average in the two-dimensional initial plane $(m_{10},m_{20})$ at fixed $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ becomes a circle with center of $\\left(\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0},-\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}\\right),$ if the following inequality holds: $r^2:=2c-\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}>0$ Thus, the constant $r$ corresponds to the radius of the circle of the level set for $c$ .","The center of the level set seems to stay close to the origin and is shifted with the choice of initial phases $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","Numerical results of the level sets for $N=2$ , $\\mathit {\\Omega }_1=\\pi /3$ , $\\mathit {\\Omega }_2=11/10$ , and $F_1=F_2=0.2$ are presented in Figure REF .","Figure: Level sets of the quasiperiodically-forced one-degree-of-freedom harmonic oscillator () for different (θ 10 ,θ 20 )(\\theta _{10},\\theta _{20}):(a) (0,0)(0,0), (b) (π/2,π/2)(\\pi /2,\\pi /2), (c) (π,π)(\\pi ,\\pi ), and (d) (3π/2,3π/2)(3\\pi /2,3\\pi /2).The horizontal (or vertical) axis for each figure is x 1 ∈[-10,10]x_1\\in [-10,10] (or x 2 ∈[-10,10]x_2\\in [-10,10])."],["One-Dimensional Dissipative System","Next, we consider the following linear dissipative system with quasi-periodic forcing: $\\frac{{d} m}{{d} t}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}).$ or $\\frac{dm}{dt}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $\\lambda >0$ .","In the same manner as above, we suppose the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent.","The solution of (REF ) with initial condition $(m_{0},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{array}{ccl}m(t) &=& \\displaystyle C^{-\\lambda t}+\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin (\\mathit {\\Omega }_it+\\theta _{i0})-\\mathit {\\Omega }_i\\cos (\\mathit {\\Omega }_it+\\theta _{i0})\\rbrace ,\\\\{\\vspace{11.38109pt}}\\theta _i(t) &=& \\mathit {\\Omega }_it+\\theta _{i0}~~~(i=1,2,\\ldots ,N),\\end{array}\\right\\rbrace $ with $C:=m_0-\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin \\theta _{i0}-\\mathit {\\Omega }_i\\cos \\theta _{i0}\\rbrace .$ Here, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).","We consider the quadratic observable as $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2.$ Then, its time-average under the solution is directly calculated as follows: $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\sum ^{N}_{i=1}\\frac{F^2_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}.$ Although this is trivial, since the transient term in the solution is filtered out, the value of the time-average is constant in the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ , in other words, does not depend on the initial condition $(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","This implies that for any choice of sample at $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})^\\top \\in \\mathbb {T}^N$ , the partition of $M=\\mathbb {R}$ based on the level set of the time-averages is just one and corresponds to $\\mathbb {R}$ itself.","Also, the above observation on the partition holds when we pick up a polynomial observable like $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=\\sum ^{n}_{j=0}a_jm^j,$ with coefficients $a_j\\in \\mathbb {R}$ and finite integer $n$ .","For every continuous function defined on a closed interval in $\\mathbb {R}$ , from the Weierstrass Approximation Theorem, it can be uniformly approximated on that interval by polynomials to any degree of accuracy.","This implies that the above observation of the partition holds for every continuous function, which is an example of Theorem REF .","Thus, the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ is the basin of attraction for the system (REF ), that is to say, the torus $\\mathbb {T}^N$ is the unique attractor for the system."],["Two-Dimensional Nonlinear Model of a Power Grid","To show the effectiveness of the theory beyond the basic examples, we apply the developed theory for analyzing the CSI phenomenon in a loop power grid.","CSI is a undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance [19].","In the case of small dissipation, this phenomenon generally does not happen upon an infinitesimally small perturbation around a steady operating state (stable equilibrium).If we have no dissipation, nonlinear stability properties are not understood in the high-dimensional cases.","However, it encompasses the situation when the grid's operating state escapes a predefined, positive-measure set around the equilibrium.","In this way, the notion of instability that we address here is non-local.","In [19], we derived a reduced-order dynamical system that described averaged dynamics of machines in a simple loop power grid and explained the non-local instability.","The reduced-order system has an quasiperiodic forcing (so it is non-autonomous), and its solutions define a measure-preserving flow.","The goal of this section is to characterize the non-local instability by analyzing invariant sets of the quasiperiodically forced system, which is introduced in (REF ) below."],["Mathematical Model","Consider short-term (zero to ten seconds) swing dynamics of a rudimentary power grid with the loop topology shown in Figure REF , where the small gray circles denote synchronous generators.","The loop part of the grid consists of $N{G}$ small, identical generators, encompassed by the dotted box, which operate in the grid and are connected to the infinite busAn ideal voltage source of constant voltage and constant frequency.","The loss-less transmission lines joining the infinite bus and a generator are much longer than those joining two generators in the loop part.","Thus, the magnitude of interaction between the infinite bus and a generator is much smaller than that between two neighboring generators on the loop part.","We call the model of Figure REF the loop power grid in the following.","Here, we assume that the lengths of transmission lines between two neighboring generators are identical.","In [19], we showed that the CSI phenomenon in the loop power grid was accurately captured by the following dynamical system defined on the cylindrical state space $\\mathbb {T}^1\\times \\mathbb {R}$ : $\\frac{d\\delta }{dt}=\\omega , \\qquad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right)$ where $e_{ij}=\\sqrt{\\frac{2}{N{G}}}\\cos \\left(\\frac{2\\pi ij}{N{G}}+\\frac{\\pi }{4}\\right),\\qquad \\mathit {\\Omega }_j=2\\sqrt{|b{int}|}\\left|\\sin \\frac{\\pi j}{N{G}}\\right|.$ The system (REF ) represents spatially-averaged dynamics of the $N{G}$ generators in the loop power grid.","The variable $\\delta \\in \\mathbb {T}^1$ is the average of angular positions of rotors (with respect to the infinite bus) of the $N{G}$ generators, and $\\omega \\in \\mathbb {R}$ is the average of deviations of rotor speeds in the $N{G}$ generators relative to the system angular frequency.","The parameter $p{m}$ stands for the mechanical input power to a generator, $b$ for the maximum transmission power between the infinite bus and a generator, and $b{int}$ for the maximum transmission power between two neighboring generators in the loop power grid.","The constants $e_{ij}$ are the eigenfunctions of linear modal oscillations between coupled generators in the loop part, $\\mathit {\\Omega }_j$ their eigen-(angular) frequency, and $c_j$ the strengths of modal oscillations.","The finite index set $\\mathcal {J}$ determines which modes are excited in the loop part.","The system (REF ) is derived under the observation that the linear modal oscillations in the loop part act as perturbations on the spatially-averaged dynamics of the $N{G}$ generators: see [19] and references therein.","Note that (REF ) is the Hamiltonian system: $\\frac{d\\delta }{dt}=\\frac{}{\\omega }{H}(\\delta ,\\omega ,t), \\qquad \\frac{d\\omega }{dt}=-\\frac{}{\\delta }{H}(\\delta ,\\omega ,t),$ with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ , given by ${H}(\\delta ,\\omega ,t):=\\frac{1}{2}\\omega ^2-p{m}\\delta -\\frac{b}{N{G}}\\sum ^{N{G}}_{i=1}\\cos \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right).$ Because the flow defined here is divergence-free, i.e.","$({H}/\\delta )({d}\\delta /{d} t)+({H}/\\omega )({d}\\omega /{d} t)=0$ , the system (REF ) preserves the Liouville measure ${d}\\delta {d}\\omega $ .","Note that it does not conserve the Hamiltonian function ${H}$ , because of ${d}{H}/{d} t\\ne 0$ if $c_j\\ne 0$ .","In this way, the augmented system for (REF ), namely $\\frac{d\\delta }{dt}=\\omega , \\quad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\theta _j+\\delta \\right), \\quad \\frac{d\\theta _j}{dt}=\\mathit {\\Omega }_j \\qquad j\\in \\mathcal {J},$ defines a measure-preserving flow on $M\\times \\mathbb {T}^{|{\\cal J}|}$ with $M=\\mathbb {T}^1\\times \\mathbb {R}$ , where $|{\\cal J}|$ stands for the cardinality of ${\\cal J}$ ."],["Results and Implications","It was shown in [19] that the unbounded motion in the quasiperiodically forced system (REF ) corresponds to the CSI phenomenon.","Because of $\\delta \\in \\mathbb {T}^1$ , the unbounded motion involves the unbounded trajectory in the $\\omega $ -direction, that is, the average of deviation of rotor speeds.","We use Corollaries REF and REF to analyze invariant sets of the flow (defined by the system (REF )), in which all the generators show bounded deviation of rotor speeds in time.","The analysis is crucial to understanding the so-called stability region of the loop power grid, i.e.","how the grid's behavior depends on initial conditions representing failures in the grid.","Numerical simulations are performed for analysis of invariant sets.","To do so, we need to fix (i) a function $f$ , (ii) the subset of state space on which we identify invariant sets, and (iii) the exit time $T{ex}$ to obtain an approximation of each time-average $f^\\ast $ .","We use the function $f(\\delta )=\\sin 2\\delta $ and the grid of $401\\times 401$ of initial conditions $(\\delta ,\\omega )$ on $[1,2]\\times [-0.15,0.15]$ .","The averaging operation of a single function can be used for the identification of invariant sets.","Numerical integration of the system (REF ) is performed with the 4th-order symplectic integrator [30] with time step $h$ : see Appendix for details.","The parameter settings are the following: $p{m}=0.95, \\quad b=1, \\quad N{G}=20, \\quad b{int}=100, \\quad h=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\frac{1}{N},\\quad T{ex}=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\times 2000,$ where $N=8$ or 16 depending on the setting of $\\mathcal {J}$ .","The values of $p{m}$ , $b$ , $N{G}$ , and $ b{int}$ are the same as in [19].","Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–I: Initial phase θ 0 =0{\\theta }_0={0} and multiple setting of 𝒥\\mathcal {J} i.e.","(a) 𝒥={1}\\mathcal {J}=\\lbrace 1\\rbrace , (b) 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace , (c) 𝒥={1,2,3}\\mathcal {J}=\\lbrace 1,2,3\\rbrace , (d) 𝒥={1,2,3,4}\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace , (e) 𝒥={1,2,3,4,5}\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace , and (f) 𝒥={1,2,3,4,5,6}\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace .The horizontal axis for each figure is δ∈[1,2]\\delta \\in [1,2], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–II: 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace and multiple setting of initial phase θ 0 =(θ 10 ,θ 20 ) ⊤ =(2πkΩ 1 /Ω 2 ,0) ⊤ {\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)^\\top for (a) k=0k=0 (same as Figure (b)), (b) k=1k=1, (c) k=2k=2, (d) k=3k=3, (e) k=4k=4, and (f) k=5k=5.The horizontal axis for each figure is δ∈[0,π]\\delta \\in [0,\\pi ], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure REF shows numerical results on analysis of invariant sets of the flow defined by the quasiperiodically forced system (REF ).","In this figure we change the number of excitation modes, i.e.","$\\mathcal {J}$ ; (a) $\\mathcal {J}=\\lbrace 1\\rbrace $ , (b) $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , (c) $\\mathcal {J}=\\lbrace 1,2,3\\rbrace $ , (d) $\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace $ , (e) $\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace $ , and (f) $\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace $ .","The amplitude $c_j$ in the figure is common and satisfies $\\displaystyle \\sqrt{\\sum _{j\\in \\mathcal {J}}c_j^2}=1.5$ , implying that the root-means-square of the forcing term does not change for any setting of $\\mathcal {J}$ .","All the initial phases ${\\theta }_0$ are set to zero.","For the outer dark-green region in each figure, every trajectory starting from it is unbounded in time.","Except for the dark-green on the bottom, the color bar attached to each figure denotes the value of time-average $f^\\ast (\\delta )$ .","The level sets of $f^\\ast (\\delta )$ are colored by the same color.","That is, the set of the same color belongs to one invariant set.","By Corollary REF , the fact that the level set is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space $M\\times \\mathbb {T}^{|\\mathcal {J}|}$ .","In the figures (a,b,c), we see that the color plot of the level sets forms concentric rings.","However, in the figures (d,e,f), we see that the color plot does does not necessarily exhibit concentric rings and does become scattered in the bands close to the outer dark-green regions.","This implies that the structure of invariant sets is complicated in the bands.","We anticipate this results from the so-called resonance phenomenon (see, e.g., [31], [32]) as an interaction between a family of bounded oscillations in the unforced system and quasiperiodic forcing.","Figure REF shows other numerical results on analysis of invariant sets.","In this figure, we consider the forcing term with two frequencies, $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , and we change the initial phases ${\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)$ where $k=0,1,\\ldots ,5$ .","The color plots here are conducted in the same manner as in Figure REF , and for the outer dark-green regions every trajectory starting from them is unbounded in time.","By Corollary REF , the fact that the level set with same color is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space.","Here, under the current setting of parameters, it is conjectured that outside the outer dark-green regions (i.e., outside the computational domain of the analysis), there exists no state from which trajectory is bounded in time.","This is true in the unperturbed case because there exists one homoclinic orbit separating the bounded and unbounded trajectories inside the computational domain.","Thus, it can be inferred that the level sets discussed above are bounded in $M=\\mathbb {T}^1\\times \\mathbb {R}$ , implying by Corollary REF that the whole of the corresponding invariant sets are uniformly bounded in the augmented state space.","The intersection of uniformly bounded invariant sets for all ${\\theta }$ corresponds to the stability region of the loop power grid, in which all the generators show bounded deviation of rotor speeds in time."],["Conclusions","In this paper, we studied the ergodic partition and invariant sets of the quasiperiodically forced dynamical system (REF ).","The main theoretical contributions of this paper are twofold.","One is to provide a theory of ergodic partition of state space for smooth flows.","The theory is a natural extension of that in [7] and is applicable to measure-preserving and dissipative flows arising in various physical and engineering systems.","Examples of them include dynamical systems induced by time-dependent Hamiltonians and incompressible fluid flows with time-dependent velocity profiles.","The other is to provide a new characterization of invariant sets in the the quasiperiodically forced system (REF ), in which we introduced a concept of uniformly bounded invariant sets.","The developed theory was applied to characterize the CSI phenomenon of a rudimentary power grid.","We have speculated that the phenomenon can be characterized, in particular, the stability region corresponds to the intersection of uniformly bounded sets for all initial phases; or a sufficient condition for the phenomenon is that the operating state of the grid is placed outside of the bounded sets at a particular initial phase or time like $t=0$ ."],["Acknowledgments","Y.S.","thanks Dr. Marko $\\rm Budi\\check{s}i\\acute{c}$ for his introduction to theory and computation of ergodic partition and fruitful discussions.","The authors also appreciate the reviewers for their valuable suggestion of the manuscript.","During part of the work on this paper, Y.S.","was at the Department of Mechanical Engineering, University of California, Santa Barbara, United States, and at the Department of Electrical Engineering, Kyoto University, Japan."],["Proof of Lemma ","A continuous function $f$ on $X$ is measurable and, from the assumption that $X$ is compact for the theoretical analysis in Section , $f$ is bounded on $X$ .","Here, we note that the time-average $f^\\ast $ of the measurable function $f$ is measurable as a limit of measurable functions $f_T$ on $X$ , defined as $f_T(x):=\\frac{1}{T}\\int ^T_0 f({S}^t(x)){d}t \\qquad T>0.$ Since we consider the sets $A_\\alpha $ on $\\mathit {\\Sigma }\\subset X$ , the fact that the family of $A_\\alpha $ is a partition of $\\mathit {\\Sigma }$ is obvious.","Next, the fact that the partition, denoted by $\\zeta _f$ , is measurable follows by taking $\\mathfrak {D}_f$ to be the collection of pre-images under $f^\\ast $ of open intervals with rational endpoints in $\\mathbb {R}$ .","Because $f^\\ast $ is measurable, each pre-image $(f^\\ast )^{-1}([a,b])$ , where $a$ and $b$ are rational numbers, is measurable.","Every set of this type is clearly separated into sets of the form $(f^\\ast )^{-1}(\\lbrace c\\rbrace )$ , $c\\in \\mathbb {R}$ .","This implies that every element of $\\mathfrak {D}_f$ is a union of elements of $\\zeta _f$ .","Furthermore, because the set of all rational numbers is dense in $\\mathbb {R}$ , for any pair $\\alpha ,\\beta \\in \\mathbb {R}$ satisfying $\\alpha <\\beta $ , there exist two rational numbers $\\underline{a},\\overline{a}$ such that $\\alpha <\\underline{a}<\\beta <\\overline{a}$ .","The pre-image $(f^\\ast )^{-1}([\\underline{a},\\overline{a}])$ is an element of $\\mathfrak {D}_f$ which we denote by $D$ .","Obviously, we see $A_\\alpha \\subset {D}^{\\rm c}$ and $A_\\beta \\subset {D}$ .","Thus, it follows that $\\mathfrak {D}_f$ is a basis for $\\zeta _f$ , and we conclude that $\\zeta _f$ is measurable."],["Proof of Theorem ","Let $A$ be an element of $\\zeta {e}$ .","For a.e.","point $x\\in A$ , the time-average $f^\\ast (x)$ exists for all $f\\in \\mathcal {C}(X)$ .","Thus, the following linear functional $L_{A}$ on $\\mathcal {C}(X)$ is well-defined: $L_{A}(f):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t (x)){d} t \\qquad x\\in A.$ Then, because $L_{A}$ is a positive linear functional and $L_{A}(1)=1$ , by Riesz's Representation Theorem (I.8.4 in [23]) there exists a unique probability measure $\\mu _{A}$ on $X$ such that $\\int _{X} f{d}\\mu _A=L_{A}(f),$ for all $f\\in \\mathcal {C}(X)$ .","Note that $\\mu _{A}$ is invariant for ${S}^t$ .","To prove this, for all $t\\in \\mathbb {R}$ we have $\\int _{X} f\\circ {S}^t\\,{d}\\mu _{A}=L_{A}(f\\circ {S}^t)=L_{A}(f)=\\int _{X} f{d}\\mu _{A}.$ The second equality is a consequence of (REF ).","For the above operation, the continuity of ${S}^t$ is required.","Because $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _A}(X)$ , $\\mu _{A}$ is invariant.","Now, we prove that $\\mu _{A}$ is a probability measure on $A$ .","There is a sequence of compact sets $A^{\\rm c}_n$ , subsets of $A^{\\rm c}$ , such that ${A}^{\\rm c}_1\\subset \\cdots \\subset {A}^{\\rm c}_{n}\\subset {A}^{\\rm c}_{n+1}\\subset \\cdots , \\qquad \\mu _{A}\\left(A^{\\rm c}\\setminus \\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.","\\qquad $ Here, we can show $\\mu _{A}(A^{\\rm c}_n)=0$ for every $A^{\\rm c}_n$ .","To do this, note that by Urysohn's Lemma, for every $A^{\\rm c}_n$ , there is a continuous, positive function $f_n$ on $X$ that is equal (i) to one on $A^{\\rm c}_n$ and (ii) to zero outside of $A^{\\rm c}_{n+1}$ .","Clearly, we see $f_n=0$ on $A$ .","Therefore, because of $\\int _{X}f_n{d}\\mu _{A}=\\int _{X\\setminus A^{\\rm c}_{n+1}}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_{n+1}\\setminus A^{\\rm c}_n}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_n}f_n{d}\\mu _{A}$ and the positiveness of $f_n$ , we have $0\\le \\mu _{A}(A^{\\rm c}_n)\\le \\int _{X} f_n{d}\\mu _{A}=L_{A}(f_n)=0.$ The measure of a union of the countable number of sets with measure zero is zero: $\\mu _{A}\\left(\\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.$ Therefore, by (REF ) and (REF ), we have $\\mu _{A}(A^{\\rm c})=0$ .","It follows from $\\mu _{A}(X)=1$ that $\\mu _{A}$ is a probability measure on $A$ .","Next, let us prove that $\\mu _{A}$ is an ergodic measure on $A$ .","First, observe that the set of all restrictions of functions in $\\mathcal {C}(X)$ to $A$ , denoted by $\\mathcal {C}(X)|_{A}$ , is dense in the set of all $\\mu _{A}$ -integrable functions on $A$ , denoted by $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","To show this, note that $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ .","Let $f$ be an element of $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","Consider the extension of $f$ to $X$ , $\\bar{f}$ , such that $\\bar{f}=f$ on $A$ and $\\bar{f}=0$ elsewhere.","Then, we have $\\bar{f}\\in \\mathcal {L}^1_{\\mu _{A}}(X)$ because the following integral exists: $\\int _{X}\\bar{f}{d}\\mu _{A}=\\int _{A}f{d}\\mu _{A}.$ Here, since $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ , there is a sequence of functions in $\\mathcal {C}(X)$ , $\\lbrace f_n\\rbrace $ , converging to $\\bar{f}$ .","Thus, the corresponding sequence of restrictions, $\\lbrace f_n|_{A}\\rbrace $ , converges to $f$ .","Therefore, we observe that $\\mathcal {C}(X)|_{A}$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","Now, by the same argument as (REF ), for all $f\\in \\mathcal {C}(X)|_{A}$ we have $\\int _{A}f{d}\\mu _{A}&= L_A(f) \\nonumber \\\\&= f^\\ast (x) \\qquad x\\in A.$ Since (REF ) holds for the dense set $\\mathcal {C}(X)|_A$ in $\\mathcal {L}^1_{\\mu _{A}}(A)$ , ${S}^t|_{A}$ is ergodic: see Proposition 2.2 in Chapter II of [23] for discrete-time systems.","This proposition can be naturally extended to continuous-time systems.","Hence, we complete the proof that there indeed exists an ergodic measure $\\mu _{A}$ for any element $A$ of the partition $\\zeta {e}$ .","Finally, we consider (REF ) and that $A$ is invariant.","The equality (REF ) is obtained with the proof of Theorem 6.4 in Chapter II of [23].","The proof is obtained for discrete-time systems and is extended to continuous-time systems.","By construction, the fact that $A$ is invariant is obvious.","This completes the proof of Theorem REF ."],["Symplectic Integration of Time-Dependent Hamiltonian Systems","In Section REF , it is required to numerically simulate the Hamiltonian system (REF ) with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ .","Symplectic integrator [30] is normally formulated in the case of time-independent Hamiltonian functions.","However, one can exploit the integrator in the case of time-dependent Hamiltonian functions by augmenting the original Hamiltonian system.","Consider the $N$ degree-of-freedom Hamiltonian system with the Hamiltonian function ${H}(q,p,t)$ : for $i=1,2,\\ldots ,N$ , $\\frac{dq_i}{dt}=\\frac{}{p_i}{H}(q,p,t), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}{H}(q,p,t)$ where $q=(q_1,q_2,\\ldots ,q_N)^\\top $ , $p=(p_1,p_2,\\ldots ,p_N)^\\top $ , and $t\\in \\mathbb {R}$ .","Now, by replacing the time variable $t$ with one new variable $q_0$ and defining the other new variable $dp_0/dt:=-H/t$ , we have the augmented Hamiltonian function $\\bar{H}(q_0,p_0,q,p)$ as follows: $\\bar{H}(q_0,p_0,q,p):=p_0+{H}(q,p,q_0).$ Thus, the augmented Hamiltonian system of the time-independent Hamiltonian function $\\bar{H}$ is derived as $\\frac{dq_i}{dt}=\\frac{}{p_i}\\bar{H}(q_0,p_0,q,p), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}\\bar{H}(q_0,p_0,q,p)$ where $i=0,1,\\ldots ,N$ .","The flow induced by trajectories of the augmented system (REF ) is divergence-free and conserves the value of the Hamiltonian function $\\bar{H}$ .","Thus, by using the integrator for the augmented system, numerical simulations of the original system (REF ) are indirectly performed.","Note that the accuracy of numerical integration of (REF ) is checked by estimating the value of $\\bar{H}$ .","This idea is applicable to the case of non-periodic time-dependent Hamiltonian functions."]],"string":"[\n [\n \"Invariant Sets in Quasiperiodically Forced Dynamical Systems\"\n ],\n [\n \"Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.\",\n \"We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps.\",\n \"The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set.\",\n \"We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction.\",\n \"We provide a characterization of invariant sets in the quasiperiodically forced systems.\",\n \"A theoretical result on uniform boundedness of the invariant sets is presented.\",\n \"The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages.\",\n \"Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability.\",\n \"We show that our theoretical results can be used to understand stability regions in such complex systems.\"\n ],\n [\n \"Introduction\",\n \"Methods of ergodic theory [1], [2] have been applied for solving a number of problems in science and technology such as control of mixing of fluid flows [3], analysis of traffic jams [4] and of quantum many-body models [5], performance evaluation of ecosystem models [6].\",\n \"A visualization method for invariant sets of discrete-time dynamical systems (or maps) based on ergodic theory was developed in [7].\",\n \"This method is based on the ergodic partition or ergodic decomposition [8] of state space in a given dynamical system, which is associated with the eigenspace at eigenvalue 1 of the Koopman operator [9].\",\n \"This method partitions a (compact and metric) state space $X$ using joint level sets of time-averages of a basis of functions defined on $X$ .\",\n \"This leads to an approximation of invariant sets on which the dynamics of the system are ergodic.\",\n \"In an ergodic invariant set, almost all points are accessible in the sense that the initial conditions in this set thoroughly sample the set.\",\n \"The method enables the visualization of low-dimensional (in practice usually two-dimensional) slices through high-dimensional invariant structures and offers a computationally tractable method for analyzing global structures of state space in discrete-time nonlinear models.\",\n \"The numerical methods associated with the ergodic partition theory have been developed: see [9] and references therein.\",\n \"The purpose of this paper is to extend the ergodic partition theory and exploit it to provide global analysis of state space in quasiperiodically forced dynamical systems.\",\n \"Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see, e.g., [10], [11].\",\n \"Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [12], analysis of dynamics described by the Schrödinger equations [13] and of quantum chaos [14], and dynamical systems analysis with applications to fluid mixing [15], [16].\",\n \"In the present paper, we study the global structure of state space in quasiperiodically forced systems from the viewpoint of ergodic partition.\",\n \"First, we develop a theory of ergodic partition of state space in measure-preserving and dissipative, continuous-time dynamical systems or flows.\",\n \"This is a natural extension of the existing theory for measure-preserving maps in [7].\",\n \"The ergodic partition is again based on eigenspace at eigenvalue 0 of the associated Koopman group.\",\n \"Related analyses of time-periodic flows via spectral properties of linear operators are reported in [17], [18].\",\n \"Since an arbitrary quasiperiodically forced system with a smooth invariant measure is transformed into an autonomous system determining a measure-preserving flow (see Section ), the ergodic partition theory is used for visualization of invariant sets for the quasiperiodically forced system.\",\n \"Next, we provide a characterization of invariant sets in quasiperiodically forced systems.\",\n \"In general, understanding invariant structures in the quasiperiodically forced systems is not easy because the associated portraits of state space change with time in an aperiodic manner.\",\n \"Here, we introduce a notion of uniform boundedness of invariant sets.\",\n \"A theoretical result on uniform boundedness of invariant sets is presented in this paper: see Theorem REF , and Corollaries REF and REF .\",\n \"This clarifies a dynamical feature of invariant sets in the quasiperiodically forced systems that is directly determined by the ergodic partition and its associated visualization technique.\",\n \"Also, we apply the theory developed above to analysis of a nonlinear model of complex power grids that represents the short-term swing instability, which we name the Coherent Swing Instability (CSI) [19].\",\n \"Preliminary results in this paper were published in the conference proceedings [20], [21].\",\n \"This paper contains detailed discussion of the theory with a new proof and a set of new numerical simulations.\",\n \"The rest of this paper is organized as follows.\",\n \"In Section we introduce set-up and mathematical preliminaries from dynamical systems theory.\",\n \"A theory of ergodic partition in measure-preserving flows is developed in Section .\",\n \"Based on the result, the basin of attraction in the dissipative case is explored in ().\",\n \"In Section we provide a new characterization of invariant sets in the quasiperiodically forced dynamical systems.\",\n \"The developed theory is applied in Section to two simple examples of the quasiperiodically forced system and to a nonlinear model of the CSI phenomenon of a power grid.\",\n \"Conclusions of this paper are presented in Section .\"\n ],\n [\n \"Quasiperiodically Forced Dynamical Systems\",\n \"Throughout this paper, we address the following quasiperiodically forced dynamical system that evolves on a finite-dimensional metric space $M$ : for ${m}\\\\in {M}$ and $t\\\\in \\\\mathbb {R}$ , $\\\\frac{{d}{m}}{{d} t}={g}({m},t).$ The function $g$ is assumed to be quasiperiodic on $t$ in the sense of Moser [22], that is, ${g}({m},t)={G}({m},\\\\mathit {\\\\Omega }_1t,\\\\mathit {\\\\Omega }_2t,\\\\ldots ,\\\\mathit {\\\\Omega }_Nt),$ where ${G}({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)$ is assumed to be a smooth vector-valued function and of period $2\\\\pi $ in $\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N$ .\",\n \"The real numbers $\\\\mathit {\\\\Omega }_1,\\\\mathit {\\\\Omega }_2,\\\\ldots ,\\\\mathit {\\\\Omega }_N$ are the $N$ basic angular frequencies and assumed to be rationally independent: there exists no $N$ -dimensional integer vector $(k_1,k_2,\\\\ldots ,k_N)^\\\\top $ ($\\\\top $ stands for the transpose operation of real-valued vectors) in which the entries do not all vanish, satisfying the resonance relation: $\\\\sum ^{N}_{i=1}k_i\\\\mathit {\\\\Omega }_i=0.$ The system (REF ) is non-autonomous and transformed into an autonomous system defined on the augmented state space $X:=M\\\\times \\\\mathbb {T}^N$ , in the same manner as in [16].\",\n \"By introducing the $N$ variables $\\\\theta _i=\\\\mathit {\\\\Omega }_it\\\\in \\\\mathbb {T}$ ($i=1,2,\\\\ldots ,N$ ), we have $\\\\frac{d{m}}{dt}={G}({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N), \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N.$ In this way, by taking trajectories of the augmented system (REF ) on the augmented state variable ${x}:=({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)\\\\in X$ , a flow, that is, one-parameter group of diffeomorphisms is defined for the quasiperiodically forced system (REF ).\"\n ],\n [\n \"Measure-Preserving Flows and Partition of State Space\",\n \"In Section , as one type of flows induced by (REF ), we will investigate a measure-preserving flow defined on a class of probability spaces, where ergodic theory has been developed [8], [23].\",\n \"The probability space considered here corresponds to the tuple $({X},\\\\mathfrak {B}_{X},\\\\mu )$ , where ${X}$ is a compact metric space, $\\\\mathfrak {B}_{X}$ is the Borel $\\\\sigma $ -algebra of ${X}$ , and $\\\\mu $ is a probability measure.\",\n \"A measure-preserving flow of the probability space is a one-parameter group of measure-preserving diffeomorphisms ${S}^t: {X}\\\\rightarrow {X}$ , $t\\\\in \\\\mathbb {R}$ such that ${S}^t$ is measure-preserving (for all $A\\\\in \\\\mathfrak {B}_{X}$ we have $\\\\mu ({S}^{-t}(A))=\\\\mu (A)$ ), ${S}^0$ the identity, and ${S}^{t_1+t_2}={S}^{t_1}\\\\circ {S}^{t_2}$ for $t_1,t_2\\\\in \\\\mathbb {R}$ .\",\n \"The important notion which we utilize throughout this paper is the partition of state space in dynamical systems.\",\n \"One motivation behind the following definitions is that we aim to locate such a partition that plays a role in the data-driven estimation of certain statistical properties of non-ergodic measure-preserving dynamical systems; we will show this later as the ergodic partition.\",\n \"The definitions of partition and measurable partition from [7] are the following.\",\n \"Definition 1 A partition of ${X}$ is a family $\\\\zeta $ of sets satisfying $A,B\\\\in \\\\zeta \\\\quad \\\\Rightarrow \\\\quad \\\\mu (A\\\\cap B)=0, \\\\quad \\\\mu \\\\left({X}\\\\setminus \\\\bigcup _{A\\\\in \\\\zeta }A\\\\right)=0.$ The element of $\\\\zeta $ containing a point ${x}\\\\in {X}$ is denoted by $\\\\zeta ({x})$ .\",\n \"A partition $\\\\zeta $ of ${X}$ is said to be measurable if there exists a countable family $\\\\mathfrak {D}$ of measurable sets $\\\\lbrace D_i\\\\rbrace $ such that every $D_i$ is a union of elements of $\\\\zeta $ , and for any pair $A_1,A_2$ of elements of $\\\\zeta $ there exists $D_j\\\\in \\\\mathfrak {D}$ such that $A_1\\\\subset D_j$ and $A_2\\\\subset D_j^{\\\\rm c}$ , where $D_j^{\\\\rm c}$ stands for the complement of $D_j$ in ${X}$ .\",\n \"The family $\\\\mathfrak {D}$ is called a basis for $\\\\zeta $ .\",\n \"In Section we use a product operation on the set of partitions of ${X}$ .\",\n \"The product operation is defined in [23] as follows.\",\n \"Definition 2 If $\\\\zeta _n$ , $n=1,\\\\ldots ,N$ are partitions, we then define their product $\\\\vee ^{N}_{n=1}\\\\zeta _n$ as the partition whose elements are the sets of the form $\\\\cap ^{N}_{i=1}{A_i}$ , for $A_i\\\\in \\\\zeta _i$ , $i=1,\\\\ldots ,N$ , satisfying $\\\\mu \\\\left(\\\\cap ^{N}_{i=1}A_i\\\\right)\\\\ne 0$ .\",\n \"For a countable sequence of partitions, the notation $\\\\vee ^{\\\\infty }_{n=1}\\\\zeta _n$ will be used for the $\\\\sigma $ -algebra generated by $\\\\cup ^\\\\infty _{n=1}\\\\zeta _n$ .\"\n ],\n [\n \"Koopman Group\",\n \"Consider a flow ${S}^t: {X}\\\\rightarrow {X}$ ($t\\\\in \\\\mathbb {R}$ ) on a finite-dimensional space ${X}$ and denote by $\\\\mathcal {F}$ a space of scalar-valued functions defined on ${X}$ : $\\\\mathcal {F}\\\\ni f: {X}\\\\rightarrow \\\\mathbb {C}$ , which we call the space of observables.\",\n \"In the following, we suppose that the existence and uniqueness of solutions associated with the flow hold for all $t$ .\",\n \"Then, we define the one-parameter group of linear operators $U^t$ ($t\\\\in \\\\mathbb {R}$ ) as a group of composition operators with ${S}^t$ : for $f\\\\in \\\\mathcal {F}$ , $(U^tf)({x}):=f({S}^t({x}))=(f\\\\circ {S}^t)({x}).$ where it is known as the Koopman group [24], [9].\",\n \"For spaces as $\\\\mathcal {F}=\\\\mathcal {C}^0(X)$ and $\\\\mathcal {L}^p(X)$ , the Koopman group is strongly continuous.\",\n \"Although the original flow ${S}^t$ is possibly described by a nonlinear differential equation and evolves on the finite-dimensional space ${X}$ , the operators $U^t$ are linear but evolve on the infinite-dimensional space $\\\\mathcal {F}$ .\",\n \"The eigenvalue $\\\\lambda \\\\in \\\\mathbb {C}$ and eigenfunction $\\\\phi _\\\\lambda \\\\in \\\\mathcal {F}\\\\setminus \\\\lbrace 0\\\\rbrace $ of the Koopman group are defined as follows: $(U^t\\\\phi _\\\\lambda )({x})=\\\\exp (\\\\lambda t)\\\\phi _\\\\lambda ({x}).$ The notion of the Koopman group and its spectral characterization will be used in the following sections.\"\n ],\n [\n \"Ergodic Partition in Measure-Preserving Flows\",\n \"For completeness, in this section we extend the theory of ergodic partition in [7] to the measure-preserving flow ${S}^t$ on the probability space $(X,\\\\mathfrak {B}_{X},\\\\mu )$ , which is introduced in Section REF .\",\n \"This section consists of a few lemmas and a theorem.\",\n \"Their proofs are almost identical to the proofs for maps [7] and appear in Appendices and .\",\n \"We denote by $\\\\mathcal {L}^1_\\\\mu ({X})$ the space of all $\\\\mu $ -integrable functions on ${X}$ and by $\\\\mathcal {C}({X})$ the space of all real-valued continuous functions on ${X}$ endowed with the sup norm.\",\n \"Roughly speaking, an ergodic partition is a partition of the state space ${X}$ into invariant sets on which the dynamics are ergodic.\",\n \"The precise definition of ergodic partition is the following.\",\n \"Definition 3 A measurable partition $\\\\zeta $ of ${X}$ is said to be ergodic under the flow ${S}^t$ if for any element $A$ of $\\\\zeta $ , (i) $A$ is invariant under ${S}^t$ , and (ii) there exists an invariant probability measure $\\\\mu _A$ on $A$ such that the restriction of ${S}^t$ to $A$ , denoted by ${S}^t|_A$ , is an ergodic diffeomorphism on $A$ , and for all $f\\\\in \\\\mathcal {L}^1_\\\\mu ({X})$ , $\\\\int _{X} f{d}\\\\mu =\\\\int _{X}\\\\left[\\\\int _{A=:\\\\zeta ({x})} f|_A{d}\\\\mu _A\\\\right]{d}\\\\mu ({x}),$ where $f|_A$ stands for the restriction of $f$ to the ergodic element $A$ , and $\\\\mu ({x})$ is again a probability measure on $X$ .\",\n \"It is remarked that the ergodic partition is defined for the entire state space $X$ with $\\\\sigma $ -algebra, but it can be relaxed in context of the $\\\\sigma $ -algebra of invariant sets which can be parts of the state space.\",\n \"Here, we denote by $f^\\\\ast $ the time-average of $f$ under the flow ${S}^t$ if the right-hand side of $f^\\\\ast ({x}):=\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0 f({S}^t({x})){d} t,$ exists for almost every (a.e.)\",\n \"point ${x}\\\\in {X}$ with respect to the measure $\\\\mu $ .\",\n \"Birkhoff's Ergodic Theorem [25], [1] shows that for all $f\\\\in \\\\mathcal {L}^1_\\\\mu ({X})$ , (i) $f^\\\\ast ({x})$ exists for a.e.\",\n \"point ${x}\\\\in {X}$ (with respect to $\\\\mu $ ); (ii) $f^\\\\ast ({S}^t({x}))=f^\\\\ast ({x})$ for a.e.\",\n \"point ${x}\\\\in {X}$ ; and (iii) $\\\\int _{X}f{d}\\\\mu =\\\\int _{X}f^\\\\ast {d}\\\\mu $ .\",\n \"Here, we let $\\\\mathit {\\\\Sigma }$ be the set of points ${x}\\\\in {X}$ such that $f^\\\\ast ({x})$ exists for all $f\\\\in \\\\mathcal {C}({X})$ , and $\\\\mathit {\\\\Sigma }(f)$ the set of points ${x}\\\\in {X}$ such that $f^\\\\ast ({x})$ exists for a particular $f\\\\in \\\\mathcal {C}({X})$ .\",\n \"The following lemma is standard (see page 129 of [23] for discrete-time dynamical systems (maps): it is obvious for flows).\",\n \"Lemma 4 For a countable and dense set $S$ in $\\\\mathcal {C}({X})$ , $\\\\mathit {\\\\Sigma }=\\\\bigcap _{f\\\\in S}\\\\mathit {\\\\Sigma }(f).$ Now, we have a set $\\\\mathit {\\\\Sigma }$ such that the time-averages of all continuous functions on ${X}$ exist.\",\n \"Then, the complimentary set $\\\\mathit {\\\\Sigma }^{\\\\rm c}$ is of measure zero.\",\n \"This is because according to Birkhoff's Ergodic Theorem (i), each $\\\\lbrace \\\\mathit {\\\\Sigma }(f)\\\\rbrace ^{\\\\rm c}$ is of measure zero, and thus $\\\\mathit {\\\\Sigma }^{\\\\rm c}:=\\\\cup _{f\\\\in S}\\\\lbrace \\\\mathit {\\\\Sigma }(f)\\\\rbrace ^{\\\\rm c}$ is a countable union of sets with zero measure, which is again of measure zero.\",\n \"The next lemma shows that the time-average of a continuous function induces a measurable partition on ${X}$ .\",\n \"Lemma 5 Let $f$ be a continuous function on ${X}$ .\",\n \"The family of level sets of $f^\\\\ast $ , $A_\\\\alpha :=\\\\lbrace {x} : {x}\\\\in \\\\mathit {\\\\Sigma }, f^\\\\ast ({x})=\\\\alpha \\\\rbrace ,~~~\\\\alpha \\\\in \\\\mathbb {R},$ is a measurable partition of ${X}$ .\",\n \"We denote by $\\\\zeta _f$ this partition and call it the partition induced by the function $f$ .\",\n \"See Appedix .\",\n \"Now, we can prove the first theorem stating that $\\\\zeta _f$ induces the ergodic partition of ${X}$ , which holds for general measure-preserving flows including the augmented system (REF ) for the quasiperiodically forced system (REF ).\",\n \"Theorem 6 Consider the measure-preserving flow ${S}^t: X\\\\rightarrow X$ ($t\\\\in \\\\mathbb {R}$ ) associated with $\\\\left.\\\\frac{d{S}^t({x})}{dt}\\\\right|_{t=0} ={F}({x}) \\\\quad \\\\textrm {for each}~{x}\\\\in X$ where ${F}: X\\\\rightarrow \\\\mathrm {T}X$ (tangent bundle of $X$ ) is a nonlinear vector field.\",\n \"Let $\\\\zeta {e}$ be the product of measurable partitions of ${X}$ induced by every $f\\\\in S$ : $\\\\zeta {e}=\\\\bigvee _{f\\\\in S}\\\\zeta _f.$ Then, $\\\\zeta {e}$ is the ergodic partition of ${X}$ .\",\n \"See Appendix .\",\n \"By combining the above theorem and lemma 20 in [26], we have the following corollary based on a finite number of basis functions.\",\n \"Corollary 7 Assume there exists a complete system of functions $\\\\lbrace f_i\\\\rbrace $ , $f_i\\\\in \\\\mathcal {C}(X)$ , $i\\\\in \\\\mathbb {N}_{>0}$ (set of all natural numbers except for 0) i.e.\",\n \"finite linear combinations of $f_i$ are dense in $\\\\mathcal {C}(X)$ .\",\n \"The ergodic partition $\\\\zeta {e}$ is $\\\\zeta {e}=\\\\bigvee _{i\\\\in \\\\mathbb {N}_{>0}}\\\\zeta _{f_i}.$ For a given function $f$ , it is obvious that each level set $A_\\\\alpha =\\\\lbrace {x} : {x}\\\\in \\\\mathit {\\\\Sigma }, f^\\\\ast ({x})=\\\\alpha \\\\rbrace $ ($\\\\alpha \\\\in \\\\mathbb {R}$ ) as an element of $\\\\zeta _f$ is invariant under ${S}^t$ .\",\n \"In [7] the authors proposed to use the level sets $A_\\\\alpha $ , which are directly computed with the time-averaging of $f$ under ${S}^t$ , for identifying and visualizing invariant sets in discrete-time dynamical systems possessing a smooth invariant measure.\",\n \"Theorem REF implies that the same computation can be used for visualization of invariant sets for the measure-preserving flow, namely, continuous-time dynamical systems with a smooth invariant measure.\",\n \"Since, as shown in Section , the quasiperiodically forced system (REF ) defined on the state space $M$ is transformed to the flow defined on the augmented state space $X=M\\\\times \\\\mathbb {T}^N$ , the ergodic partition theory is applicable to visualization of invariant sets in $X$ for the original quasiperiodically forced system (REF ) possessing a smooth invariant measure.\",\n \"However, in the quasiperiodically forced system (REF ) we are interested in dynamics on $M$ not on $X=M\\\\times \\\\mathbb {T}^N$ .\",\n \"While for $N=1$ it is clear that every intersection of the invariant set in $M\\\\times \\\\mathbb {T}^1$ with $M$ is an invariant set of the associated Poincaré map.\",\n \"The relationship in the quasiperiodically forced system (namely, $N\\\\ge 2$ ) is much less clear.\",\n \"We make it precise in Section .\",\n \"Here, let us introduce the connection of time-average $f^\\\\ast $ and the Koopman group.\",\n \"It is obvious that for any $f\\\\in \\\\mathcal {L}^1_\\\\mu (X)$ its time-average $f^\\\\ast $ is an eigenfuction of the operators $U^t$ at eigenvalue 0: for a.e.\",\n \"point ${x}\\\\in {X}$ with respect to $\\\\mu $ , $(U^tf^\\\\ast )({x})=f^\\\\ast ({S}^t({x}))=f^\\\\ast ({x})=\\\\exp (0t)f^\\\\ast ({x}).$ Since the eigenfunctions at 0 are invariant under the flow, we have a complete characterization of (possibly non-smooth) invariant sets.\"\n ],\n [\n \"Basin of Attraction in Dissipative Flows\",\n \"In the last of the previous section, we indicate that the time-average of an observable enables the visualization of low-dimensional slices through high-dimensional invariant sets of the quasiperiodically forced system with a smooth invariant measure.\",\n \"Here, we consider a more general class of the quasiperiodically forced systems with dissipation and will point out that the use of time-average works for characterizing an important invariant set—basin of attraction.\",\n \"In Section we considered the measure-preserving flow on a compact metric space.\",\n \"Every continuous flow on a compact metric space preserves a measure.\",\n \"This is the content of the so-called Krylov-Bogolyubov theorem [27].\",\n \"However, this might be a useless statement if our intent is to study the behavior of trajectories from their time-averages, by the method used in the measure-preserving cases in Section .\",\n \"For example, let us take a simple dissipative flow, $\\\\frac{{d} x}{{d} t}=-\\\\lambda x \\\\qquad x\\\\in \\\\mathbb {R}, \\\\qquad \\\\lambda >0.$ All of the initial conditions converge to $x=0$ as $t\\\\rightarrow \\\\infty $ along the flow $S^t(x)=\\\\exp (-\\\\lambda t)x$ .\",\n \"The invariant measure clearly exists and is the Dirac measure supported at $x=0$ (to which any “initial\\\" measure defined on $\\\\mathbb {R}$ evolves).\",\n \"Thus, the following statement holds: for a function $f:\\\\mathbb {R}\\\\rightarrow \\\\mathbb {C}$ , $f^\\\\ast (x)=\\\\int _{\\\\mathbb {R}}f{d}\\\\mu ,$ for a.e.\",\n \"point $x\\\\in \\\\mathbb {R}$ with respect to the Dirac measure.\",\n \"However, that excludes the whole real line except for the origin itself.\",\n \"The situation feels better though.\",\n \"For example, it is easy to see that the time-average of any continuous function on $\\\\mathbb {R}$ is just its value at 0: $f^*(x)=f(0)$ .\",\n \"Note that for the measure-preserving case in Section , we basically considered integrable functions for which time-averages exist.\",\n \"We can not do that in the dissipative case.\",\n \"A function that is 1 everywhere on a finite interval $(\\\\underline{a},\\\\overline{a})$ containing 0, except at 0 where its value is 0, is clearly integrable (being a union of two simple functions).\",\n \"Then, its time-average under the dissipative flow is identically 1, and its integral with respect to the invariant Dirac measure is 0.\",\n \"It turns out that in dissipative systems, it is best for the time-average to work with continuous functions.\",\n \"Adopting that idea, the whole method of ergodic partitioning can be extended to capture basins of attraction for continuous flows that are not necessarily measure-preserving.\",\n \"Consider a continuous flow ${S}^t: X\\\\rightarrow X$ , where $X$ is not necessary compact, and label by $F$ the set on which time averages of continuous functions do not exist.\",\n \"On the set $X\\\\setminus F$ , consider a set $C$ on which time-averages of continuous functions (or, better, a countable, separating set of continuous functions) are constant.There can be uncountably many such sets inside $X\\\\setminus F$ : for example, let ${S}^t$ be the identity map, mapping every point into itself.\",\n \"Then, by the same construction described in Section and Appendix , there is an invariant measure $\\\\mu _C$ such that $f^*({x})=\\\\int _C f d\\\\mu _C$ for any continuous function $f$ , ${x}\\\\in C$ .\",\n \"Such a measure is called a physical measure [28], provided $M$ is equipped with a measure $\\\\mu $ that ${S}^t$ does not preserve (but we are interested in)—say Lebesgue measure—and $C$ contains an open (and thus positive measure) set in $M$ .In light of this discussion, one might say that the notion of “ergodic measure\\\" should be preserved for such constructs even in dissipative systems.\",\n \"Namely, many ergodic measures we obtain in measure-preserving system are certainly physically important, although their support does not contain an open set.\",\n \"The tricky part is that due to Birkhoff's Ergodic Theorem, ergodic measures are associated with time averages of integrable functions.\",\n \"In dissipative systems, this does not work (see the above example).\",\n \"We could keep the requirement of equality of space and time averages, like in (REF ), but for continuous functions only.\",\n \"Physical measures would then be ergodic measures with support that contains an open set.\",\n \"Suppose that the state space of a continuous flow admits a finite number of attractors, and that the union of their basins of attraction is of full measure.\",\n \"We state the result for flows (continuous-time systems) to emphasize that considerations here work for both discrete and continuous time.\",\n \"Theorem 8 Let the system $\\\\left.\\\\frac{d{S}^t({x})}{dt}\\\\right|_{t=0}={F}({x}), \\\\quad {x}\\\\in X\\\\subset \\\\mathbb {R}^n$ with a flow ${S}^t$ have a finite number $N$ of attractors with basins of attraction $A_j, j=1,...,N$ , such that $\\\\mu (\\\\cup _j A_j)=\\\\mu (X)$ , where $\\\\mu $ is the Lebesgue measure.\",\n \"Also, let the time averages of continuous functions exist everywhere on a set $X\\\\setminus F$ , where $\\\\mu (F)=0$ .\",\n \"Then, the time-average $h^*({x})$ of a continuous function $h\\\\in \\\\mathcal {C}(X)$ is a piecewise constant eigenfunction of the Koopman operators $U^t$ at eigenvalue 0 defined a.e.\",\n \"point with respect to $\\\\mu $ .\",\n \"For any point ${x}\\\\in A_j,$ there exists a set $C$ such that $(U^t h^*)({x})&=U^t \\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int _0^{T} h({S}^\\\\tau ({x}))d\\\\tau \\\\nonumber \\\\\\\\&= \\\\int _C (U^t h) d\\\\mu _C\\\\nonumber \\\\\\\\&=\\\\int _C (h\\\\circ {S}^t) d\\\\mu _C\\\\nonumber \\\\\\\\&=\\\\int _C h d\\\\mu _C \\\\nonumber \\\\\\\\&=h^*({x}), \\\\nonumber $ where in the second line we utilized (REF ), and transitioning from line 3 to 4 we used the fact that $\\\\mu _C$ is invariant under ${S}^t$ .\",\n \"Since $(U^t h^*)({x})=h^*({x})$ is exactly the equation which $h^*({x})$ has to satisfy in order to be an eigenfunction of $U^t$ , and the basin of attraction $A_j$ is arbitrary, by taking into account that $\\\\mu (\\\\cup _j A_j)=\\\\mu (X)$ the proposition is proven.\",\n \"The concept of the ergodic partition in dissipative systems is applied to a more general class of systems than those treated in Theorem REF .\",\n \"It provides ergodic measures on invariant sets that are not necessarily attractors—because they do not have an open set of initial conditions converging to them.\",\n \"Yet, these sets are dynamically important and can not be refined without losing the equivalence of space and time averages.\",\n \"More precisely, the ergodic partition $\\\\zeta {e}$ of $M$ under ${S}^t$ (not necessarily measure-$\\\\mu $ preserving) is a partition into sets $C_{\\\\alpha }$ , where $\\\\alpha $ is a member of an indexing set, such that on each set $C_{\\\\alpha }$ (ergodic set) there exists an ergodic measure $\\\\mu _{C_\\\\alpha }$ such that $\\\\mu _{C_{\\\\alpha }}(C_{\\\\alpha })=1,$ For every $f\\\\in \\\\mathcal {C}(M),f^{\\\\ast }({x}\\\\in C_{\\\\alpha })=\\\\int _{C_{\\\\alpha }}fd\\\\mu _{C_{\\\\alpha }},$ for a.e ${x}$ point with respect to $\\\\mu $ .\",\n \"The last condition emphasizes the role of the measure $\\\\mu $ —the time averages are equal to space averages with respect to $\\\\mu _{C_\\\\alpha }$ , but their equality is almost everywhere with respect to measure $\\\\mu $ that might be of our interest.\",\n \"In this way, we have escaped the realm of the Krylov-Bogolyubov Theorem, which, in the case when dynamics are not measure-preserving, neglects dynamics on possibly large swaths of the state space.\"\n ],\n [\n \"Sample-Based Characterization of Invariant Sets\",\n \"In this section, we provide a characterization of invariant sets in the quasiperiodically forced system (REF ).\",\n \"Generally speaking, understanding invariant structures of (REF ) is not easy because the associated portrait of $M$ changes with time in an aperiodic manner.\",\n \"As shown in Section , the ergodic partition theory enables one to visualize an invariant set as its low-dimensional slice in $M$ by the time-averaging technique.\",\n \"Also, in () it was shown that the time-averaging technique works for dissipative case to visualize the basin of attraction.\",\n \"The obtained slice here is just a sample of the invariant set at a particular initial time (in other words, an initial condition in $\\\\mathbb {T}^N$ ).\",\n \"Because of the aperiodic nature, such a sample does not seem to provide complete information on the entire structure of invariant set in the augmented state space $M\\\\times \\\\mathbb {T}^N$ .\",\n \"However, we will prove that a boundedness property of the invariant set in $M\\\\times \\\\mathbb {T}^N$ is captured by means of one sample of it in $M$ (see Theorem REF ).\",\n \"In the following, in order to encompass both the cases in Section and (), we consider the original state space $M$ that is metric and not necessarily compact.\",\n \"A continuous, linear skew-product flow ${S}^t$ (diffeomorphism) on the augmented state space $X:=M\\\\times \\\\mathbb {T}^N$ derived from the quasiperiodically forced system (REF ) is described below: for ${x}=({m},{\\\\theta })\\\\in M\\\\times \\\\mathbb {T}^N$ , ${S}^t({x}):=\\\\left({S}^t_{M}({m},{\\\\theta }),{S}^t_\\\\mathit {\\\\Omega }({\\\\theta })\\\\right),$ where ${S}^t_{M}: M\\\\times \\\\mathbb {T}^N\\\\rightarrow M$ is the continuous map defined by trajectories of the original system (REF ), and ${S}^t_\\\\mathit {\\\\Omega }: \\\\mathbb {T}^N\\\\rightarrow \\\\mathbb {T}^N$ the linear (continuous) flow on the torus $\\\\mathbb {T}^N$ .\",\n \"For a fixed ${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ , we will write the orbit on $\\\\mathbb {T}^N$ through ${\\\\theta }_0$ as $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0):=\\\\lbrace {\\\\theta } : {\\\\theta }={S}^t_\\\\mathit {\\\\Omega }({\\\\theta }_0)\\\\in \\\\mathbb {T}^N, t\\\\in \\\\mathbb {R}\\\\rbrace $ .\",\n \"Since the $N$ basic frequencies in the original system (REF ) are rationally independent, the following lemma is obvious.\",\n \"Lemma 9 For any ${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ , $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ is dense in $\\\\mathbb {T}^N$ .\",\n \"Now, we investigate an invariant set in the augmented system (REF ).\",\n \"Let $I\\\\subseteq X$ be a positively invariant set of (REF ) that is supposed to be closedIf $I$ is invariant and open, then its closure $\\\\mathrm {cl}(I)$ is still invariant..\",\n \"It follows from Lemma REF and by closure that the following decomposition of $I$ holds: $I=\\\\bigcup _{\\\\theta \\\\in \\\\mathbb {T}^N} A_\\\\theta \\\\times \\\\lbrace {\\\\theta }\\\\rbrace ,$ where $A_{\\\\theta }$ is a closed subset of $M$ .\",\n \"Let us denote by $A_{\\\\theta _0}$ an intersection or sample of the invariant set $I$ at ${\\\\theta }={\\\\theta }_0$ .\",\n \"The next lemma provides a topological property of the intersections $A_\\\\theta $ .\",\n \"Lemma 10 For each $\\\\epsilon >0$ there exists a positive constant $\\\\delta $ so that $|{\\\\theta }-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ implies $d(A_\\\\theta ,A_{\\\\theta _0})<\\\\epsilon $ , where $d(A_{\\\\theta },A_{\\\\theta _0})$ is the Hausdorff distance that induces a topology on the family of all closed subsets of $M$ .\",\n \"Assume not.\",\n \"Then, for each $\\\\delta >0$ there exist a sequence of times $\\\\lbrace t_j\\\\rbrace $ , where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ , and a positive constant $\\\\epsilon _1$ such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ satisfies $|{\\\\theta }_j-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ for every $j>J$ ($J$ is an integer), while $d(A_{\\\\theta _k},A_{\\\\theta _0})>\\\\epsilon _1$ holds for some $k>J$ .\",\n \"Here, from the continuity of ${S}^t_M$ in $t$ , for each ${m}\\\\in A_{\\\\theta _0}$ and $\\\\epsilon >0$ , there exists a positive constant $\\\\delta $ such that $|{\\\\theta }_k-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ implies $|{m}_k-{m}|_{M}<\\\\epsilon $ , where ${m}_k:={S}^{t_k}_{M}({m},{\\\\theta }_0)\\\\in A_{\\\\theta _k}$ .\",\n \"This gives us a contradiction of $d(A_{\\\\theta _k},A_{\\\\theta _0})>\\\\epsilon _1$ by taking $\\\\epsilon =\\\\epsilon _1$ and from the definition of Hausdorff distance.\",\n \"This lemma suggests that the invariant set $I$ is topologically connected with respect to ${\\\\theta }$ .\",\n \"One trivial example is provided by taking as $I$ the closure of a single trajectory starting at a point $({m},{\\\\theta }_0)$ , for which $A_{\\\\theta _0}$ consists of a single point.\",\n \"In this case, $I$ is topologically regarded as a product set of a single point in $M$ and the torus $\\\\mathbb {T}^N$ .\",\n \"This is rigorously stated in the next theorem.\",\n \"Theorem 11 Suppose that $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N$ is homeomorphic to $I$ .\",\n \"Now, let us construct the mapping $h_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\rightarrow I$ in the following manner.\",\n \"For each $({m},{\\\\theta })\\\\in A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , by choosing time $\\\\tau _\\\\theta $ that satisfies ${\\\\theta }={S}^{\\\\tau _\\\\theta }_{\\\\it \\\\Omega }({\\\\theta }_0)$ , we define $h_{\\\\theta _0}({m},{\\\\theta })$ as $({S}^{\\\\tau _\\\\theta }_M({m},{\\\\theta }_0),{\\\\theta })$ .\",\n \"We here prove the continuity of $h_{\\\\theta _0}$ .\",\n \"For each $\\\\epsilon >0$ and ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , there exists a sequence of $\\\\lbrace t_j\\\\rbrace $ (where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ ) such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ satisfies $\\\\left|{\\\\theta }_j-{\\\\theta }\\\\right|_{\\\\mathbb {T}^N}<\\\\epsilon /2$ for every $j>J$ ($J$ is an integer).\",\n \"Furthermore, because of the continuity of ${S}^t_{M}$ in $t$ , for each ${m}_j\\\\in A_{\\\\theta _0}$ there exists a positive constant $\\\\delta _j$ such that $\\\\left|{m}-{m}_j\\\\right|_M<\\\\delta _j$ implies $\\\\left|{S}_M^{\\\\tau _\\\\theta }({m},{\\\\theta }_0)-{S}_M^{\\\\tau _{\\\\theta _j}}({m}_j,{\\\\theta }_0)\\\\right|_M<\\\\epsilon /2$ .\",\n \"Here, because for every $j>J$ $\\\\left|({m},{\\\\theta })-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\le \\\\left|{m}-{m}_j\\\\right|_{M}+\\\\left|{\\\\theta }-{\\\\theta }_j\\\\right|_{\\\\mathbb {T}^N}< \\\\delta _j+\\\\frac{\\\\epsilon }{2},$ we set $\\\\delta :=\\\\delta _j+\\\\epsilon /2$ .\",\n \"Thus, if $\\\\left|({m},{\\\\theta })-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta $ , then $\\\\left|h_{\\\\theta _0}({m},{\\\\theta })-h_{\\\\theta _0}({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}&\\\\le \\\\left|{S}_M^{\\\\tau _\\\\theta }({m},{\\\\theta }_0)-{S}_M^{\\\\tau _{\\\\theta _j}}({m}_j,{\\\\theta }_0)\\\\right|_M+\\\\left|{\\\\theta }-{\\\\theta }_j\\\\right|_{\\\\mathbb {T}^N}\\\\nonumber \\\\\\\\& <\\\\frac{\\\\epsilon }{2}+\\\\frac{\\\\epsilon }{2}=\\\\epsilon .\\\\nonumber $ This shows that $h_{\\\\theta _0}$ is continuous.\",\n \"We next prove that $h_{\\\\theta _0}$ is uniformly continuous.\",\n \"Assume not.\",\n \"Then, for each $\\\\delta >0$ there exists a positive constant $\\\\epsilon _1$ such that $\\\\left|({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta $ implies $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-h_{\\\\theta _0}({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\ge \\\\epsilon _1$ for some $({m}_j,{\\\\theta }_j) $ and $({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)\\\\in A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ .\",\n \"We here set $\\\\delta =1/j$ .\",\n \"Since it is supposed that $A_{\\\\theta _0}$ is bounded and closed (and $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\subset \\\\mathbb {T}^N$ ), the two sequences of points $\\\\lbrace ({m}_j,{\\\\theta }_j)\\\\rbrace $ and $\\\\lbrace ({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)\\\\rbrace $ have convergent subsequences, denoted as $\\\\lbrace ({m}_{j_k},{\\\\theta }_{j_k})\\\\rbrace $ and $\\\\lbrace ({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})\\\\rbrace $ .\",\n \"Because of $\\\\left|({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<1/j$ , both the subsequences have the same convergent point, denoted as $({m}^\\\\ast ,{\\\\theta }^\\\\ast )$ .\",\n \"That is, for each $\\\\delta _1>0$ , there exists a positive integer $J_k$ such that both the inequalities $\\\\left|({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _1$ and $\\\\left|({m}_{j_k},{\\\\theta }_{j_k})-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _1$ hold for every $j_k\\\\ge J_k$ .\",\n \"Here, since $h_{\\\\theta _0}$ is proven to be continuous at $({m}^\\\\ast ,{\\\\theta }^\\\\ast )$ , for each $\\\\epsilon >0$ , there exists a positive constant $\\\\delta _2$ such that $\\\\left|({m},{\\\\theta })-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _2$ implies $\\\\left|h_{\\\\theta _0}({m},{\\\\theta })-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\epsilon /2$ .\",\n \"By taking $\\\\epsilon =\\\\epsilon _1$ and $\\\\delta _1=\\\\delta _2$ thus choosing $J_k$ appropriately, we see that both the inequalities $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^M}<\\\\epsilon _1/2$ and $\\\\left|h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\epsilon _1/2$ hold for every $j_k\\\\ge J_k$ , and $\\\\begin{aligned}\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^M}\\\\le &\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N} \\\\\\\\& + \\\\left|h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^N} \\\\\\\\< & \\\\frac{\\\\epsilon _1}{2}+\\\\frac{\\\\epsilon _1}{2} = \\\\epsilon _1,\\\\end{aligned}$ implying the contradiction for $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\ge \\\\epsilon _1$ .\",\n \"Thus, $h_{\\\\theta _0}$ is uniformly continuous.\",\n \"By virtue of Theorem 3.45 in page 136 of [29], the uniformly-continuous $h_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\rightarrow I$ is extended to the mapping $\\\\hat{h}_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N\\\\rightarrow I$ that is also (uniformly) continuous.\",\n \"We now prove that $\\\\hat{h}_{\\\\theta _0}$ is a bijection.\",\n \"For each ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , the statement is obvious from the construction of $h_{\\\\theta _0}$ (see its dependence on ${\\\\theta }$ ).\",\n \"It is thus enough to check the case ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ .\",\n \"For each ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , there exists a sequence of times $\\\\lbrace t_j\\\\rbrace $ (where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ ) such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ converges to ${\\\\theta }$ .\",\n \"Because $\\\\hat{h}_{\\\\theta _0}$ is continuous, the sequence $\\\\lbrace \\\\hat{h}_{\\\\theta _0}({m},{\\\\theta }_j)=h_{\\\\theta _0}({m},{\\\\theta }_j)\\\\rbrace $ converges to $\\\\hat{h}_{\\\\theta _0}({m},{\\\\theta })$ , which is represented as $\\\\displaystyle \\\\left(\\\\lim _{t_j\\\\rightarrow \\\\infty }{S}_M^{t_j}({m},{\\\\theta }_0),{\\\\theta }\\\\right)$ .\",\n \"Namely, the limit exists for every ${m}\\\\in A_{\\\\theta _0}$ .\",\n \"Assume that $\\\\hat{h}_{\\\\theta _0}$ is not an injection.\",\n \"Then, there exist ${m},{m}^{\\\\prime }\\\\in A_{\\\\theta _0}$ satisfying ${m}\\\\ne {m}^{\\\\prime }$ such that for each $\\\\epsilon >0$ , $\\\\left|{S}_M^{t_j}({m},{\\\\theta }_0)-{S}_M^{t_j}({m}^{\\\\prime },{\\\\theta }_0)\\\\right|_M<\\\\epsilon $ holds for every $j>J$ ($J$ is an integer and depends on $\\\\epsilon $ ).\",\n \"Here, the fundamental property of uniqueness of trajectories in (REF ) (or ${S}^t$ diffeomorphsim) implies that for each $t_j$ , $\\\\left|{S}_M^{t_j}({m},{\\\\theta }_0)-{S}_M^{t_j}({m}^{\\\\prime },{\\\\theta }_0)\\\\right|_M\\\\ge \\\\epsilon _j$ holds ($\\\\epsilon _j$ is a positive constant and depends on $t_j$ ), showing the contradiction by taking $\\\\epsilon =\\\\epsilon _j$ .\",\n \"Hence, $\\\\hat{h}_{\\\\theta _0}$ is an injection.\",\n \"Regarding $\\\\hat{h}_{\\\\theta _0}$ surjective, it is obvious that for each ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , $\\\\hat{h}_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=h_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=\\\\left({S}_M^{\\\\tau _\\\\theta }(A_{\\\\theta _0},{\\\\theta }_0),{\\\\theta }\\\\right)=(A_\\\\theta ,{\\\\theta })$ .\",\n \"For each ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ we see $\\\\displaystyle \\\\hat{h}_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=\\\\left(\\\\lim _{t_j\\\\rightarrow \\\\infty }{S}_M^{t_j}(A_{\\\\theta _0},{\\\\theta }_0),{\\\\theta }\\\\right)$ , and the limit converges to $A_\\\\theta $ from Lemma REF .\",\n \"Hence, from (REF ), $\\\\hat{h}_{\\\\theta _0}$ is a surjection.\",\n \"Finally, the inverse of $\\\\hat{h}_{\\\\theta _0}$ exists according to the above construction and is continuous.\",\n \"Therefore, $\\\\hat{h}_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N\\\\rightarrow I$ is a homeomorphism, namely, $A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N$ is homeomorphic to $I$ .\",\n \"The following two corollaries provide a way of characterization of boundedness of $I$ by means of one sample of it.\",\n \"Corollary 12 Suppose that $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $I$ is bounded in $X$ .\",\n \"Corollary 13 Suppose that a closed subset $B_{\\\\theta _0}$ of $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $\\\\hat{h}_{\\\\theta _0}(B_{\\\\theta _0}\\\\times \\\\mathbb {T}^N)$ is a subset of $I$ and bounded in $X$ .\",\n \"Corollaries REF and REF imply that by computing a slice of the invariant set in $M$ at one sample onset, it is possible to determine whether the invariant set or its subset is bounded in the augmented state space $X$ .\",\n \"For a bounded invariant set $I$ in $X$ , any cross-section $I_{\\\\theta _0}=A_{\\\\theta _0}\\\\times \\\\lbrace {\\\\theta }_0\\\\rbrace $ (${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ ) is bounded in terms of $M$ , in other words, the boundedness property of $A_{\\\\theta _0}$ does not depend on the choice of ${\\\\theta }_0$ .\",\n \"In this way, we call the bounded invariant set $I$ in $X$ uniformly bounded invariant set.\"\n ],\n [\n \"Example Studies\",\n \"In this section, we provide three examples of the application of the above theoretical results.\",\n \"The last example is related to practical problems on stability of power grids.\"\n ],\n [\n \"Forced Linear Harmonic Oscillator\",\n \"First, we will consider the following one-degree-of-freedom linear harmonic oscillator with quasi-periodic forcing of (multiple) $N$ frequencies: $\\\\frac{dm_1}{dt}=m_2, \\\\qquad \\\\frac{dm_2}{dt}=-m_1+\\\\sum ^{N}_{i=1}F_i\\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}),$ or $\\\\frac{dm_1}{dt}=m_2, \\\\qquad \\\\frac{dm_2}{dt}=-m_1+\\\\sum ^{N}_{i=1}F_i\\\\sin \\\\theta _i, \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N$ where $F_i$ ($>0$ ) ($i=1,2,\\\\ldots ,N$ ) are the amplitudes of periodic forces, $\\\\mathit {\\\\Omega }_i$ ($>0$ ) the angular frequencies of the forces, and $\\\\theta _{i0}\\\\in \\\\mathbb {T}$ the initial phases.\",\n \"We assume there is no resonance: the $N$ angular frequencies $\\\\mathit {\\\\Omega }_i$ are rationally independent; and $\\\\mathit {\\\\Omega }_i\\\\ne 1$ ($i=1,2,\\\\ldots ,N$ ).\",\n \"The solution of (REF ) with initial condition $(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ is analytically represented as follows: $\\\\left.\\\\begin{aligned}m_1(t) &=C_1\\\\cos t+C_2\\\\sin t+\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}), \\\\\\\\m_2(t) &=\\\\displaystyle \\\\frac{{d}}{{d} t}m_1(t), \\\\\\\\{\\\\vspace{14.22636pt}}\\\\theta _i(t) &= \\\\mathit {\\\\Omega }_it+\\\\theta _{i0} \\\\qquad i=1,2,\\\\ldots ,N,\\\\end{aligned}\\\\right\\\\rbrace $ with $C_1:=m_{10}-\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos \\\\theta _{i0}, \\\\qquad C_2:=m_{20}+\\\\sum ^{N}_{i=1}\\\\frac{F_i\\\\mathit {\\\\Omega }_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\sin \\\\theta _{i0}.$ Now, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).\",\n \"We define the following quadratic observable $f: \\\\mathbb {R}^2\\\\times \\\\mathbb {T}^N\\\\rightarrow \\\\mathbb {R}$ , related to the potential energy $m^2_1/2$ : $f(m_1,m_2,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=m^2_1.$ Then, we calculate its time-average under the solution as $\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0(U^tf)(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0}){d} t=\\\\frac{1}{2}\\\\left\\\\lbrace C^2_1+C^2_2+\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{(1-\\\\mathit {\\\\Omega }^2_i)^2}\\\\right\\\\rbrace .$ Here, as shown in (REF ), the constants $C_1$ and $C_2$ are determined by the initial condition $(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"The level set, parameterized by a real-valued constant $c\\\\in \\\\mathbb {R}$ , of the time-average in the two-dimensional initial plane $(m_{10},m_{20})$ at fixed $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ becomes a circle with center of $\\\\left(\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos \\\\theta _{i0},-\\\\sum ^{N}_{i=1}\\\\frac{F_i\\\\mathit {\\\\Omega }_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\sin \\\\theta _{i0}\\\\right),$ if the following inequality holds: $r^2:=2c-\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{(1-\\\\mathit {\\\\Omega }^2_i)^2}>0$ Thus, the constant $r$ corresponds to the radius of the circle of the level set for $c$ .\",\n \"The center of the level set seems to stay close to the origin and is shifted with the choice of initial phases $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"Numerical results of the level sets for $N=2$ , $\\\\mathit {\\\\Omega }_1=\\\\pi /3$ , $\\\\mathit {\\\\Omega }_2=11/10$ , and $F_1=F_2=0.2$ are presented in Figure REF .\",\n \"Figure: Level sets of the quasiperiodically-forced one-degree-of-freedom harmonic oscillator () for different (θ 10 ,θ 20 )(\\\\theta _{10},\\\\theta _{20}):(a) (0,0)(0,0), (b) (π/2,π/2)(\\\\pi /2,\\\\pi /2), (c) (π,π)(\\\\pi ,\\\\pi ), and (d) (3π/2,3π/2)(3\\\\pi /2,3\\\\pi /2).The horizontal (or vertical) axis for each figure is x 1 ∈[-10,10]x_1\\\\in [-10,10] (or x 2 ∈[-10,10]x_2\\\\in [-10,10]).\"\n ],\n [\n \"One-Dimensional Dissipative System\",\n \"Next, we consider the following linear dissipative system with quasi-periodic forcing: $\\\\frac{{d} m}{{d} t}=-\\\\lambda m+\\\\sum ^{N}_{i=1}F_i\\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}).$ or $\\\\frac{dm}{dt}=-\\\\lambda m+\\\\sum ^{N}_{i=1}F_i\\\\sin \\\\theta _i, \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N$ where $\\\\lambda >0$ .\",\n \"In the same manner as above, we suppose the $N$ angular frequencies $\\\\mathit {\\\\Omega }_i$ are rationally independent.\",\n \"The solution of (REF ) with initial condition $(m_{0},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ is analytically represented as follows: $\\\\left.\\\\begin{array}{ccl}m(t) &=& \\\\displaystyle C^{-\\\\lambda t}+\\\\sum ^{N}_{i=1}\\\\frac{F_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}\\\\lbrace \\\\lambda \\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0})-\\\\mathit {\\\\Omega }_i\\\\cos (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0})\\\\rbrace ,\\\\\\\\{\\\\vspace{11.38109pt}}\\\\theta _i(t) &=& \\\\mathit {\\\\Omega }_it+\\\\theta _{i0}~~~(i=1,2,\\\\ldots ,N),\\\\end{array}\\\\right\\\\rbrace $ with $C:=m_0-\\\\sum ^{N}_{i=1}\\\\frac{F_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}\\\\lbrace \\\\lambda \\\\sin \\\\theta _{i0}-\\\\mathit {\\\\Omega }_i\\\\cos \\\\theta _{i0}\\\\rbrace .$ Here, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).\",\n \"We consider the quadratic observable as $f(m,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=m^2.$ Then, its time-average under the solution is directly calculated as follows: $\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0(U^tf)(m_0,\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0}){d} t=\\\\frac{1}{2}\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}.$ Although this is trivial, since the transient term in the solution is filtered out, the value of the time-average is constant in the augmented state space $\\\\mathbb {R}\\\\times \\\\mathbb {T}^N$ , in other words, does not depend on the initial condition $(m_0,\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"This implies that for any choice of sample at $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})^\\\\top \\\\in \\\\mathbb {T}^N$ , the partition of $M=\\\\mathbb {R}$ based on the level set of the time-averages is just one and corresponds to $\\\\mathbb {R}$ itself.\",\n \"Also, the above observation on the partition holds when we pick up a polynomial observable like $f(m,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=\\\\sum ^{n}_{j=0}a_jm^j,$ with coefficients $a_j\\\\in \\\\mathbb {R}$ and finite integer $n$ .\",\n \"For every continuous function defined on a closed interval in $\\\\mathbb {R}$ , from the Weierstrass Approximation Theorem, it can be uniformly approximated on that interval by polynomials to any degree of accuracy.\",\n \"This implies that the above observation of the partition holds for every continuous function, which is an example of Theorem REF .\",\n \"Thus, the augmented state space $\\\\mathbb {R}\\\\times \\\\mathbb {T}^N$ is the basin of attraction for the system (REF ), that is to say, the torus $\\\\mathbb {T}^N$ is the unique attractor for the system.\"\n ],\n [\n \"Two-Dimensional Nonlinear Model of a Power Grid\",\n \"To show the effectiveness of the theory beyond the basic examples, we apply the developed theory for analyzing the CSI phenomenon in a loop power grid.\",\n \"CSI is a undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance [19].\",\n \"In the case of small dissipation, this phenomenon generally does not happen upon an infinitesimally small perturbation around a steady operating state (stable equilibrium).If we have no dissipation, nonlinear stability properties are not understood in the high-dimensional cases.\",\n \"However, it encompasses the situation when the grid's operating state escapes a predefined, positive-measure set around the equilibrium.\",\n \"In this way, the notion of instability that we address here is non-local.\",\n \"In [19], we derived a reduced-order dynamical system that described averaged dynamics of machines in a simple loop power grid and explained the non-local instability.\",\n \"The reduced-order system has an quasiperiodic forcing (so it is non-autonomous), and its solutions define a measure-preserving flow.\",\n \"The goal of this section is to characterize the non-local instability by analyzing invariant sets of the quasiperiodically forced system, which is introduced in (REF ) below.\"\n ],\n [\n \"Mathematical Model\",\n \"Consider short-term (zero to ten seconds) swing dynamics of a rudimentary power grid with the loop topology shown in Figure REF , where the small gray circles denote synchronous generators.\",\n \"The loop part of the grid consists of $N{G}$ small, identical generators, encompassed by the dotted box, which operate in the grid and are connected to the infinite busAn ideal voltage source of constant voltage and constant frequency.\",\n \"The loss-less transmission lines joining the infinite bus and a generator are much longer than those joining two generators in the loop part.\",\n \"Thus, the magnitude of interaction between the infinite bus and a generator is much smaller than that between two neighboring generators on the loop part.\",\n \"We call the model of Figure REF the loop power grid in the following.\",\n \"Here, we assume that the lengths of transmission lines between two neighboring generators are identical.\",\n \"In [19], we showed that the CSI phenomenon in the loop power grid was accurately captured by the following dynamical system defined on the cylindrical state space $\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ : $\\\\frac{d\\\\delta }{dt}=\\\\omega , \\\\qquad \\\\frac{d\\\\omega }{dt}=p{m}-\\\\frac{b}{N{G}}\\\\sum _{i=1}^{N{G}}\\\\sin \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\mathit {\\\\Omega }_jt+\\\\delta \\\\right)$ where $e_{ij}=\\\\sqrt{\\\\frac{2}{N{G}}}\\\\cos \\\\left(\\\\frac{2\\\\pi ij}{N{G}}+\\\\frac{\\\\pi }{4}\\\\right),\\\\qquad \\\\mathit {\\\\Omega }_j=2\\\\sqrt{|b{int}|}\\\\left|\\\\sin \\\\frac{\\\\pi j}{N{G}}\\\\right|.$ The system (REF ) represents spatially-averaged dynamics of the $N{G}$ generators in the loop power grid.\",\n \"The variable $\\\\delta \\\\in \\\\mathbb {T}^1$ is the average of angular positions of rotors (with respect to the infinite bus) of the $N{G}$ generators, and $\\\\omega \\\\in \\\\mathbb {R}$ is the average of deviations of rotor speeds in the $N{G}$ generators relative to the system angular frequency.\",\n \"The parameter $p{m}$ stands for the mechanical input power to a generator, $b$ for the maximum transmission power between the infinite bus and a generator, and $b{int}$ for the maximum transmission power between two neighboring generators in the loop power grid.\",\n \"The constants $e_{ij}$ are the eigenfunctions of linear modal oscillations between coupled generators in the loop part, $\\\\mathit {\\\\Omega }_j$ their eigen-(angular) frequency, and $c_j$ the strengths of modal oscillations.\",\n \"The finite index set $\\\\mathcal {J}$ determines which modes are excited in the loop part.\",\n \"The system (REF ) is derived under the observation that the linear modal oscillations in the loop part act as perturbations on the spatially-averaged dynamics of the $N{G}$ generators: see [19] and references therein.\",\n \"Note that (REF ) is the Hamiltonian system: $\\\\frac{d\\\\delta }{dt}=\\\\frac{}{\\\\omega }{H}(\\\\delta ,\\\\omega ,t), \\\\qquad \\\\frac{d\\\\omega }{dt}=-\\\\frac{}{\\\\delta }{H}(\\\\delta ,\\\\omega ,t),$ with the time-dependent Hamiltonian function ${H}(\\\\delta ,\\\\omega ,t)$ , given by ${H}(\\\\delta ,\\\\omega ,t):=\\\\frac{1}{2}\\\\omega ^2-p{m}\\\\delta -\\\\frac{b}{N{G}}\\\\sum ^{N{G}}_{i=1}\\\\cos \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\mathit {\\\\Omega }_jt+\\\\delta \\\\right).$ Because the flow defined here is divergence-free, i.e.\",\n \"$({H}/\\\\delta )({d}\\\\delta /{d} t)+({H}/\\\\omega )({d}\\\\omega /{d} t)=0$ , the system (REF ) preserves the Liouville measure ${d}\\\\delta {d}\\\\omega $ .\",\n \"Note that it does not conserve the Hamiltonian function ${H}$ , because of ${d}{H}/{d} t\\\\ne 0$ if $c_j\\\\ne 0$ .\",\n \"In this way, the augmented system for (REF ), namely $\\\\frac{d\\\\delta }{dt}=\\\\omega , \\\\quad \\\\frac{d\\\\omega }{dt}=p{m}-\\\\frac{b}{N{G}}\\\\sum _{i=1}^{N{G}}\\\\sin \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\theta _j+\\\\delta \\\\right), \\\\quad \\\\frac{d\\\\theta _j}{dt}=\\\\mathit {\\\\Omega }_j \\\\qquad j\\\\in \\\\mathcal {J},$ defines a measure-preserving flow on $M\\\\times \\\\mathbb {T}^{|{\\\\cal J}|}$ with $M=\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ , where $|{\\\\cal J}|$ stands for the cardinality of ${\\\\cal J}$ .\"\n ],\n [\n \"Results and Implications\",\n \"It was shown in [19] that the unbounded motion in the quasiperiodically forced system (REF ) corresponds to the CSI phenomenon.\",\n \"Because of $\\\\delta \\\\in \\\\mathbb {T}^1$ , the unbounded motion involves the unbounded trajectory in the $\\\\omega $ -direction, that is, the average of deviation of rotor speeds.\",\n \"We use Corollaries REF and REF to analyze invariant sets of the flow (defined by the system (REF )), in which all the generators show bounded deviation of rotor speeds in time.\",\n \"The analysis is crucial to understanding the so-called stability region of the loop power grid, i.e.\",\n \"how the grid's behavior depends on initial conditions representing failures in the grid.\",\n \"Numerical simulations are performed for analysis of invariant sets.\",\n \"To do so, we need to fix (i) a function $f$ , (ii) the subset of state space on which we identify invariant sets, and (iii) the exit time $T{ex}$ to obtain an approximation of each time-average $f^\\\\ast $ .\",\n \"We use the function $f(\\\\delta )=\\\\sin 2\\\\delta $ and the grid of $401\\\\times 401$ of initial conditions $(\\\\delta ,\\\\omega )$ on $[1,2]\\\\times [-0.15,0.15]$ .\",\n \"The averaging operation of a single function can be used for the identification of invariant sets.\",\n \"Numerical integration of the system (REF ) is performed with the 4th-order symplectic integrator [30] with time step $h$ : see Appendix for details.\",\n \"The parameter settings are the following: $p{m}=0.95, \\\\quad b=1, \\\\quad N{G}=20, \\\\quad b{int}=100, \\\\quad h=\\\\frac{2\\\\pi }{\\\\mathit {\\\\Omega }_1}\\\\frac{1}{N},\\\\quad T{ex}=\\\\frac{2\\\\pi }{\\\\mathit {\\\\Omega }_1}\\\\times 2000,$ where $N=8$ or 16 depending on the setting of $\\\\mathcal {J}$ .\",\n \"The values of $p{m}$ , $b$ , $N{G}$ , and $ b{int}$ are the same as in [19].\",\n \"Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–I: Initial phase θ 0 =0{\\\\theta }_0={0} and multiple setting of 𝒥\\\\mathcal {J} i.e.\",\n \"(a) 𝒥={1}\\\\mathcal {J}=\\\\lbrace 1\\\\rbrace , (b) 𝒥={1,2}\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace , (c) 𝒥={1,2,3}\\\\mathcal {J}=\\\\lbrace 1,2,3\\\\rbrace , (d) 𝒥={1,2,3,4}\\\\mathcal {J}=\\\\lbrace 1,2,3,4\\\\rbrace , (e) 𝒥={1,2,3,4,5}\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5\\\\rbrace , and (f) 𝒥={1,2,3,4,5,6}\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5,6\\\\rbrace .The horizontal axis for each figure is δ∈[1,2]\\\\delta \\\\in [1,2], and the vertical axis is ω∈[-0.15,0.15]\\\\omega \\\\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–II: 𝒥={1,2}\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace and multiple setting of initial phase θ 0 =(θ 10 ,θ 20 ) ⊤ =(2πkΩ 1 /Ω 2 ,0) ⊤ {\\\\theta }_0=(\\\\theta _{10},\\\\theta _{20})^\\\\top =(2\\\\pi k\\\\mathit {\\\\Omega }_1/\\\\mathit {\\\\Omega }_2,0)^\\\\top for (a) k=0k=0 (same as Figure (b)), (b) k=1k=1, (c) k=2k=2, (d) k=3k=3, (e) k=4k=4, and (f) k=5k=5.The horizontal axis for each figure is δ∈[0,π]\\\\delta \\\\in [0,\\\\pi ], and the vertical axis is ω∈[-0.15,0.15]\\\\omega \\\\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure REF shows numerical results on analysis of invariant sets of the flow defined by the quasiperiodically forced system (REF ).\",\n \"In this figure we change the number of excitation modes, i.e.\",\n \"$\\\\mathcal {J}$ ; (a) $\\\\mathcal {J}=\\\\lbrace 1\\\\rbrace $ , (b) $\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace $ , (c) $\\\\mathcal {J}=\\\\lbrace 1,2,3\\\\rbrace $ , (d) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4\\\\rbrace $ , (e) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5\\\\rbrace $ , and (f) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5,6\\\\rbrace $ .\",\n \"The amplitude $c_j$ in the figure is common and satisfies $\\\\displaystyle \\\\sqrt{\\\\sum _{j\\\\in \\\\mathcal {J}}c_j^2}=1.5$ , implying that the root-means-square of the forcing term does not change for any setting of $\\\\mathcal {J}$ .\",\n \"All the initial phases ${\\\\theta }_0$ are set to zero.\",\n \"For the outer dark-green region in each figure, every trajectory starting from it is unbounded in time.\",\n \"Except for the dark-green on the bottom, the color bar attached to each figure denotes the value of time-average $f^\\\\ast (\\\\delta )$ .\",\n \"The level sets of $f^\\\\ast (\\\\delta )$ are colored by the same color.\",\n \"That is, the set of the same color belongs to one invariant set.\",\n \"By Corollary REF , the fact that the level set is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space $M\\\\times \\\\mathbb {T}^{|\\\\mathcal {J}|}$ .\",\n \"In the figures (a,b,c), we see that the color plot of the level sets forms concentric rings.\",\n \"However, in the figures (d,e,f), we see that the color plot does does not necessarily exhibit concentric rings and does become scattered in the bands close to the outer dark-green regions.\",\n \"This implies that the structure of invariant sets is complicated in the bands.\",\n \"We anticipate this results from the so-called resonance phenomenon (see, e.g., [31], [32]) as an interaction between a family of bounded oscillations in the unforced system and quasiperiodic forcing.\",\n \"Figure REF shows other numerical results on analysis of invariant sets.\",\n \"In this figure, we consider the forcing term with two frequencies, $\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace $ , and we change the initial phases ${\\\\theta }_0=(\\\\theta _{10},\\\\theta _{20})^\\\\top =(2\\\\pi k\\\\mathit {\\\\Omega }_1/\\\\mathit {\\\\Omega }_2,0)$ where $k=0,1,\\\\ldots ,5$ .\",\n \"The color plots here are conducted in the same manner as in Figure REF , and for the outer dark-green regions every trajectory starting from them is unbounded in time.\",\n \"By Corollary REF , the fact that the level set with same color is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space.\",\n \"Here, under the current setting of parameters, it is conjectured that outside the outer dark-green regions (i.e., outside the computational domain of the analysis), there exists no state from which trajectory is bounded in time.\",\n \"This is true in the unperturbed case because there exists one homoclinic orbit separating the bounded and unbounded trajectories inside the computational domain.\",\n \"Thus, it can be inferred that the level sets discussed above are bounded in $M=\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ , implying by Corollary REF that the whole of the corresponding invariant sets are uniformly bounded in the augmented state space.\",\n \"The intersection of uniformly bounded invariant sets for all ${\\\\theta }$ corresponds to the stability region of the loop power grid, in which all the generators show bounded deviation of rotor speeds in time.\"\n ],\n [\n \"Conclusions\",\n \"In this paper, we studied the ergodic partition and invariant sets of the quasiperiodically forced dynamical system (REF ).\",\n \"The main theoretical contributions of this paper are twofold.\",\n \"One is to provide a theory of ergodic partition of state space for smooth flows.\",\n \"The theory is a natural extension of that in [7] and is applicable to measure-preserving and dissipative flows arising in various physical and engineering systems.\",\n \"Examples of them include dynamical systems induced by time-dependent Hamiltonians and incompressible fluid flows with time-dependent velocity profiles.\",\n \"The other is to provide a new characterization of invariant sets in the the quasiperiodically forced system (REF ), in which we introduced a concept of uniformly bounded invariant sets.\",\n \"The developed theory was applied to characterize the CSI phenomenon of a rudimentary power grid.\",\n \"We have speculated that the phenomenon can be characterized, in particular, the stability region corresponds to the intersection of uniformly bounded sets for all initial phases; or a sufficient condition for the phenomenon is that the operating state of the grid is placed outside of the bounded sets at a particular initial phase or time like $t=0$ .\"\n ],\n [\n \"Acknowledgments\",\n \"Y.S.\",\n \"thanks Dr. Marko $\\\\rm Budi\\\\check{s}i\\\\acute{c}$ for his introduction to theory and computation of ergodic partition and fruitful discussions.\",\n \"The authors also appreciate the reviewers for their valuable suggestion of the manuscript.\",\n \"During part of the work on this paper, Y.S.\",\n \"was at the Department of Mechanical Engineering, University of California, Santa Barbara, United States, and at the Department of Electrical Engineering, Kyoto University, Japan.\"\n ],\n [\n \"Proof of Lemma \",\n \"A continuous function $f$ on $X$ is measurable and, from the assumption that $X$ is compact for the theoretical analysis in Section , $f$ is bounded on $X$ .\",\n \"Here, we note that the time-average $f^\\\\ast $ of the measurable function $f$ is measurable as a limit of measurable functions $f_T$ on $X$ , defined as $f_T(x):=\\\\frac{1}{T}\\\\int ^T_0 f({S}^t(x)){d}t \\\\qquad T>0.$ Since we consider the sets $A_\\\\alpha $ on $\\\\mathit {\\\\Sigma }\\\\subset X$ , the fact that the family of $A_\\\\alpha $ is a partition of $\\\\mathit {\\\\Sigma }$ is obvious.\",\n \"Next, the fact that the partition, denoted by $\\\\zeta _f$ , is measurable follows by taking $\\\\mathfrak {D}_f$ to be the collection of pre-images under $f^\\\\ast $ of open intervals with rational endpoints in $\\\\mathbb {R}$ .\",\n \"Because $f^\\\\ast $ is measurable, each pre-image $(f^\\\\ast )^{-1}([a,b])$ , where $a$ and $b$ are rational numbers, is measurable.\",\n \"Every set of this type is clearly separated into sets of the form $(f^\\\\ast )^{-1}(\\\\lbrace c\\\\rbrace )$ , $c\\\\in \\\\mathbb {R}$ .\",\n \"This implies that every element of $\\\\mathfrak {D}_f$ is a union of elements of $\\\\zeta _f$ .\",\n \"Furthermore, because the set of all rational numbers is dense in $\\\\mathbb {R}$ , for any pair $\\\\alpha ,\\\\beta \\\\in \\\\mathbb {R}$ satisfying $\\\\alpha <\\\\beta $ , there exist two rational numbers $\\\\underline{a},\\\\overline{a}$ such that $\\\\alpha <\\\\underline{a}<\\\\beta <\\\\overline{a}$ .\",\n \"The pre-image $(f^\\\\ast )^{-1}([\\\\underline{a},\\\\overline{a}])$ is an element of $\\\\mathfrak {D}_f$ which we denote by $D$ .\",\n \"Obviously, we see $A_\\\\alpha \\\\subset {D}^{\\\\rm c}$ and $A_\\\\beta \\\\subset {D}$ .\",\n \"Thus, it follows that $\\\\mathfrak {D}_f$ is a basis for $\\\\zeta _f$ , and we conclude that $\\\\zeta _f$ is measurable.\"\n ],\n [\n \"Proof of Theorem \",\n \"Let $A$ be an element of $\\\\zeta {e}$ .\",\n \"For a.e.\",\n \"point $x\\\\in A$ , the time-average $f^\\\\ast (x)$ exists for all $f\\\\in \\\\mathcal {C}(X)$ .\",\n \"Thus, the following linear functional $L_{A}$ on $\\\\mathcal {C}(X)$ is well-defined: $L_{A}(f):=\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0 f({S}^t (x)){d} t \\\\qquad x\\\\in A.$ Then, because $L_{A}$ is a positive linear functional and $L_{A}(1)=1$ , by Riesz's Representation Theorem (I.8.4 in [23]) there exists a unique probability measure $\\\\mu _{A}$ on $X$ such that $\\\\int _{X} f{d}\\\\mu _A=L_{A}(f),$ for all $f\\\\in \\\\mathcal {C}(X)$ .\",\n \"Note that $\\\\mu _{A}$ is invariant for ${S}^t$ .\",\n \"To prove this, for all $t\\\\in \\\\mathbb {R}$ we have $\\\\int _{X} f\\\\circ {S}^t\\\\,{d}\\\\mu _{A}=L_{A}(f\\\\circ {S}^t)=L_{A}(f)=\\\\int _{X} f{d}\\\\mu _{A}.$ The second equality is a consequence of (REF ).\",\n \"For the above operation, the continuity of ${S}^t$ is required.\",\n \"Because $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _A}(X)$ , $\\\\mu _{A}$ is invariant.\",\n \"Now, we prove that $\\\\mu _{A}$ is a probability measure on $A$ .\",\n \"There is a sequence of compact sets $A^{\\\\rm c}_n$ , subsets of $A^{\\\\rm c}$ , such that ${A}^{\\\\rm c}_1\\\\subset \\\\cdots \\\\subset {A}^{\\\\rm c}_{n}\\\\subset {A}^{\\\\rm c}_{n+1}\\\\subset \\\\cdots , \\\\qquad \\\\mu _{A}\\\\left(A^{\\\\rm c}\\\\setminus \\\\bigcup _{n\\\\ge 1}A^{\\\\rm c}_n\\\\right)=0.\",\n \"\\\\qquad $ Here, we can show $\\\\mu _{A}(A^{\\\\rm c}_n)=0$ for every $A^{\\\\rm c}_n$ .\",\n \"To do this, note that by Urysohn's Lemma, for every $A^{\\\\rm c}_n$ , there is a continuous, positive function $f_n$ on $X$ that is equal (i) to one on $A^{\\\\rm c}_n$ and (ii) to zero outside of $A^{\\\\rm c}_{n+1}$ .\",\n \"Clearly, we see $f_n=0$ on $A$ .\",\n \"Therefore, because of $\\\\int _{X}f_n{d}\\\\mu _{A}=\\\\int _{X\\\\setminus A^{\\\\rm c}_{n+1}}f_n{d}\\\\mu _{A}+\\\\int _{A^{\\\\rm c}_{n+1}\\\\setminus A^{\\\\rm c}_n}f_n{d}\\\\mu _{A}+\\\\int _{A^{\\\\rm c}_n}f_n{d}\\\\mu _{A}$ and the positiveness of $f_n$ , we have $0\\\\le \\\\mu _{A}(A^{\\\\rm c}_n)\\\\le \\\\int _{X} f_n{d}\\\\mu _{A}=L_{A}(f_n)=0.$ The measure of a union of the countable number of sets with measure zero is zero: $\\\\mu _{A}\\\\left(\\\\bigcup _{n\\\\ge 1}A^{\\\\rm c}_n\\\\right)=0.$ Therefore, by (REF ) and (REF ), we have $\\\\mu _{A}(A^{\\\\rm c})=0$ .\",\n \"It follows from $\\\\mu _{A}(X)=1$ that $\\\\mu _{A}$ is a probability measure on $A$ .\",\n \"Next, let us prove that $\\\\mu _{A}$ is an ergodic measure on $A$ .\",\n \"First, observe that the set of all restrictions of functions in $\\\\mathcal {C}(X)$ to $A$ , denoted by $\\\\mathcal {C}(X)|_{A}$ , is dense in the set of all $\\\\mu _{A}$ -integrable functions on $A$ , denoted by $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"To show this, note that $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ .\",\n \"Let $f$ be an element of $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"Consider the extension of $f$ to $X$ , $\\\\bar{f}$ , such that $\\\\bar{f}=f$ on $A$ and $\\\\bar{f}=0$ elsewhere.\",\n \"Then, we have $\\\\bar{f}\\\\in \\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ because the following integral exists: $\\\\int _{X}\\\\bar{f}{d}\\\\mu _{A}=\\\\int _{A}f{d}\\\\mu _{A}.$ Here, since $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ , there is a sequence of functions in $\\\\mathcal {C}(X)$ , $\\\\lbrace f_n\\\\rbrace $ , converging to $\\\\bar{f}$ .\",\n \"Thus, the corresponding sequence of restrictions, $\\\\lbrace f_n|_{A}\\\\rbrace $ , converges to $f$ .\",\n \"Therefore, we observe that $\\\\mathcal {C}(X)|_{A}$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"Now, by the same argument as (REF ), for all $f\\\\in \\\\mathcal {C}(X)|_{A}$ we have $\\\\int _{A}f{d}\\\\mu _{A}&= L_A(f) \\\\nonumber \\\\\\\\&= f^\\\\ast (x) \\\\qquad x\\\\in A.$ Since (REF ) holds for the dense set $\\\\mathcal {C}(X)|_A$ in $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ , ${S}^t|_{A}$ is ergodic: see Proposition 2.2 in Chapter II of [23] for discrete-time systems.\",\n \"This proposition can be naturally extended to continuous-time systems.\",\n \"Hence, we complete the proof that there indeed exists an ergodic measure $\\\\mu _{A}$ for any element $A$ of the partition $\\\\zeta {e}$ .\",\n \"Finally, we consider (REF ) and that $A$ is invariant.\",\n \"The equality (REF ) is obtained with the proof of Theorem 6.4 in Chapter II of [23].\",\n \"The proof is obtained for discrete-time systems and is extended to continuous-time systems.\",\n \"By construction, the fact that $A$ is invariant is obvious.\",\n \"This completes the proof of Theorem REF .\"\n ],\n [\n \"Symplectic Integration of Time-Dependent Hamiltonian Systems\",\n \"In Section REF , it is required to numerically simulate the Hamiltonian system (REF ) with the time-dependent Hamiltonian function ${H}(\\\\delta ,\\\\omega ,t)$ .\",\n \"Symplectic integrator [30] is normally formulated in the case of time-independent Hamiltonian functions.\",\n \"However, one can exploit the integrator in the case of time-dependent Hamiltonian functions by augmenting the original Hamiltonian system.\",\n \"Consider the $N$ degree-of-freedom Hamiltonian system with the Hamiltonian function ${H}(q,p,t)$ : for $i=1,2,\\\\ldots ,N$ , $\\\\frac{dq_i}{dt}=\\\\frac{}{p_i}{H}(q,p,t), \\\\qquad \\\\frac{dp_i}{dt}=-\\\\frac{}{q_i}{H}(q,p,t)$ where $q=(q_1,q_2,\\\\ldots ,q_N)^\\\\top $ , $p=(p_1,p_2,\\\\ldots ,p_N)^\\\\top $ , and $t\\\\in \\\\mathbb {R}$ .\",\n \"Now, by replacing the time variable $t$ with one new variable $q_0$ and defining the other new variable $dp_0/dt:=-H/t$ , we have the augmented Hamiltonian function $\\\\bar{H}(q_0,p_0,q,p)$ as follows: $\\\\bar{H}(q_0,p_0,q,p):=p_0+{H}(q,p,q_0).$ Thus, the augmented Hamiltonian system of the time-independent Hamiltonian function $\\\\bar{H}$ is derived as $\\\\frac{dq_i}{dt}=\\\\frac{}{p_i}\\\\bar{H}(q_0,p_0,q,p), \\\\qquad \\\\frac{dp_i}{dt}=-\\\\frac{}{q_i}\\\\bar{H}(q_0,p_0,q,p)$ where $i=0,1,\\\\ldots ,N$ .\",\n \"The flow induced by trajectories of the augmented system (REF ) is divergence-free and conserves the value of the Hamiltonian function $\\\\bar{H}$ .\",\n \"Thus, by using the integrator for the augmented system, numerical simulations of the original system (REF ) are indirectly performed.\",\n \"Note that the accuracy of numerical integration of (REF ) is checked by estimating the value of $\\\\bar{H}$ .\",\n \"This idea is applicable to the case of non-periodic time-dependent Hamiltonian functions.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08340"}}},{"rowIdx":197,"cells":{"text":{"kind":"list like","value":[["How do Convolutional Neural Networks Learn Design?"],["Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.","Book covers are designed in a unique way, specific to genres which convey important information to their readers.","By using Convolutional Neural Networks (CNN) to predict book genres from cover images, visual cues which distinguish genres can be highlighted and analyzed.","In order to understand these visual clues contributing towards the decision of a genre, we present the application of Layer-wise Relevance Propagation (LRP) on the book cover image classification results.","We use LRP to explain the pixel-wise contributions of book cover design and highlight the design elements contributing towards particular genres.","In addition, with the use of state-of-the-art object and text detection methods, insights about genre-specific book cover designs are discovered."],["Introduction","Visual design renders specific impressions to transmit information which enriches the product's value.","However, these visual designs despite of being important are not analyzed objectively or statistically.","Analyzing these visual designs enables us to understand the contained information carried by them.","An interesting target of visual design analysis is book cover image design where the design of a book cover can infer the genre.","Each book cover is carefully designed by typographers and their designs represent the book contents in an intuitive way for better sales.","This association of books to specific genres is based on the differences in their underlying book cover designs [1].","The slight change in book cover design can reflect changes in book genre which makes design learning a challenging task for book covers.","In order to understand the design elements used for machine aided book cover classification, we employ Convolutional Neural Networks (CNN) [2].","In recent years, CNNs have achieved state-of-the-art results in isolated character recognition [3], [4] and large-scale image recognition [5], [6].","Notably, Iwana et al.","[1] demonstrated that CNNs can be used for genre classification based on book cover image, although with a high level of difficulty.","However, that study was subjective and not enough explanation is given as to why the CNN performed as it did.","To interpret the reasoning behind a CNN's prediction we used a method called Layer-wise Relevance Propagation (LRP) [7].","LRP decomposes output function on its input variables and highlights input pixels contributing towards the network decision.","It produces a layer-wise relevance heatmap by recursively multiplying the relevance of higher layers by the normalized feature maps of the target layer.","The heatmaps can help us to discover the input image elements which have an effect on the classification result.","The main contributions of this paper are threefold.","Firstly, we classified the book cover images using one-vs-others classification with CNNs.","Secondly, the models built by the CNNs are analyzed using LRP.","With LRP, we demonstrate design elements specifically relevant to classification of the book cover images.","We show that certain objects have a strong relevance to particular genres.","Finally, we use state-of-the-art object detection and text detection methods, namely Single Shot Multibox Detector (SSD) [8] and Efficient and Accurate Scene Text Detector [9], to quantitatively enforce the results found by LRP.","This reveals the specific elements in which CNNs classify book cover images for genre classification.","The organization is as follows.","Section provides related works in design understanding and genre classification as well as feature visualization of CNNs.","Section reviews the data and tools used for understanding book cover design.","Section presents analysis of CNN's understanding of book cover design.","In Section , we demonstrate the use of LRP combined with SSD and EAST for quantitative analysis.","Finally, Section draws a conclusion.","Artistic style understanding and subjective genre classification is a budding field in machine learning.","For example, recent attempts have been done to identify artistic styles and quality of paintings and photographs [10], [11] with neural network models.","In addition, there have been trials to classify music by genre [12], [13], book covers by genre [1], movie posters by genre [14], paintings by genre [15], and text by genre [16], [17].","Also, in a general sense, document classification can be considered genre classification and deep CNNs are the state-of-the-art in the document classification domain [18], [19], [20]."],["Visualization inside of CNNs","There is a desire to visualize features and determine pixel-wise attention and relevance within the hidden layers of CNNs.","However, this is a not a straightforward task [21].","Erhan et al.","[21] proposed using gradient decent to maximize a node's activation to visualize the employed features.","Similar work has been done for large-scale image classification [22].","Zeiler and Fergus [23] used deconvolutional neural networks to visualize features learned by CNNs.","In addition, they created heatmaps by monitoring class changes systematic cover up of portions of the images.","Class Activation Maps (CAM) [24], GradCAM [25], and GradCAM++ [26] reveal the parts of images which are most important to a class using global average pooling (GAP).","Recently, LRP has been used in the fields of text [27] where classification scores were projected back to input features for extracting relevant words for a specific prediction.","The method has also shown successes in model understanding in fields of sentiment analysis [28], action recognition [29], and age and gender classification [30].","As far as the authors are aware, this is the first time LRP has been used for the understanding of genre or design classification."],["Amazon Book Cover Dataset","We used the Book Cover Image to Genre datasethttps://github.com/uchidalab/book-dataset Task 1.A.","The dataset consists of 57,000 book cover images divided into 30 classes of equal sizes.","In the experiments, we used the predefined training set and test set modified for one-vs-others classification.","In this way, genre-wise training sets were prepared with an equal distribution of positive and negative data samples."],["Convolution Neural Networks","CNNs are able to tackle image recognition by implementing convolutions of learned filter-like shared weights which maintain the structural qualities of images while acting as feature extractors [2].","For the experiment, we implement CNNs to tackle book genre classification.","To use the book cover images with a CNN, they were preprocessed by scaling them to 112$\\times $ 112 pixels by 3 color channel images and by normalizing the values to be between -1 and 1.","The CNN used for the experiments has six convolutional layers with Rectified Linear Units (ReLU) activations and a softmax output layer.","The convolutional layers consisted of three layers of 10 nodes with $5\\times 5$ convolutional filters, one layer of 25 nodes with a $4\\times 4$ filter, one layer of 50 nodes with a $3\\times 3$ filter, and one layer of 100 nodes with a $1\\times 1$ filter.","A $2\\times 2$ maxpooling layer with stride 2 was used between each convolution layer.","Finally, the CNNs were trained using gradient decent with a batch size of 25 at a learning rate of 0.001 for 50,000 iterations.","The accuracy results for each genre is summarized in Fig.","REF .","In particular, the CNNs had difficulties with the reference classes, such as \"Engineering & Transportation,\" \"Health, Fitness and Dieting,\" \"History,\" \"Medical Books,\" and \"Reference.\"","Conversely, \"Children Books,\" \"Romance,\" and \"Test Preparation\" had high accuracies.","However, more than just classification accuracy, the purpose of this paper is to understand why the CNN's performed as such and reveal the relevant parts of the images.","Figure: CNN accuracy by genre."],["Layer-wise Relevance Propagation","The LRP algorithm and the LRP toolbox [31] aims to explain the reasoning behind the decision made by a network model which allows its users to validate classifier results.","LRP is mainly derived from Deep Taylor Decomposition [32], a method of decomposing network's output predictions onto its input variable.","The results after such a decomposition is visualized in the form of a heatmap highlighting each pixel's importance for the prediction.","LRP explains output function, i.e.","classifier's decision, which helps us to derive all of the crucial pixels for a particular prediction.","In Fig.","REF , the technique is shown in which the output value given by the network is decomposed backwards layer by layer until it reaches the input.","This backward decomposition of network's prediction uses local redistribution rules for assigning relevance values $R_i$ to each neuron contributing towards the output, namely $\\sum _{i} R_i = \\sum _{j} R_j =\\dots = \\sum _{k}R_k = f(x),$ where $f(x)$ is the prediction function, $R_i$ is the relevance of node $i$ in the target layer, $R_j$ is the relevance of node $j$ of the previous layer, and $R_k$ is the relevance of node $k$ of the highest layer.","The total amount of relevance is conserved in this equation.","Figure: Feed forward neural network with the (left) forward pass and the (right) backward relevance calculation.","The function f(x)f(x) is the prediction outcome given input xx.","The variables a i a_i and a j a_j are the inputs for node ii and jj, respectively.","R i R_i is the relevance of node ii and R j R_j is the relevance of node jj.For the experiment, we used the $\\alpha -\\beta $ decomposition formula defined by $R_i = \\sum _{j} \\left(\\alpha \\frac{(a_i w_{ij})^+}{\\sum _{i} ( a_i w_{ij}) ^ +} + \\beta \\frac{(a_i w_{ij})^-}{\\sum _{i} ( a_i w_{ij}) ^ -} \\right) R_j,$ where $\\alpha $ and $\\beta $ are hyperparameters to weight the positive values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ and the negative values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ , respectively.","Furthermore, $w_{ij}$ is the weight between nodes $i$ and $j$ and $a_i$ is the input to node $i$ .","This decomposition allows for the separation of the positive connections and the negative connections.","Values inside positive bracket indicates propagation of activating input messages while negative weight connections indicate deactivating input values."],["Single-Shot Multibox Detector","To develop a better understanding of the objects within book cover images, we employed SSD [8], a state-of-the-art deep neural network based object detection method.","SSD is a feed forward CNN which produces a multi-scale collection of fixed size bounding boxes and scores for object detection within the boxes.","A final non-maximal suppression step determines the final detections.","The result of SSD is bounding box regions with object classification labels.","Using SSD, it is possible to accurately detect multiple objects of different classes within images."],["Efficient and Accurate Scene Text Detector","For humans, text is an important component of book covers; it is where the title, authors, and additional information is conveyed.","However, a CNN may place a different importance on text than humans.","Thus, to analyze the relevance of text in book covers, we use EAST [9] as a text detector.","EAST uses a multi-channel Fully Convolutional Network (FCN) and non-maximal suppression on predicted geometric shapes to detect multi-orient text-line and word boxes."],["How CNNs Understand Book Cover Design: Qualitative Analysis","In this section, we have presented LRP results from main genres.","The analysis helped us to deduce book cover design elements contributing towards a prediction by CNN.","We used $\\alpha -\\beta $ decomposition formula with values of $\\alpha =2$ and $\\beta =-1$ which is suggested for networks using ReLU activation functions because it emphasizes the positive elements and de-emphasizes the negative ones [7].","This is important due to the ReLU activation function setting negative values to zero.","In the heatmaps generated by LRP under this decomposition, pixels adding positive contribution are represented in red color and the ones adding negative contribution are represented by blue color."],["Sports & Outdoors","Under this genre, many book covers with pictures of players playing indoor and outdoor games were seen.","Figure REF (a) shows LRP results on these covers, which presents significance of player's picture on the cover.","The first image in Fig.","REF (a) supports this fact with LRP being centered on players who are either playing a sport or showing some player like gesture, with car in background adding no contribution.","The second image in Fig.","REF (a) emphasizes the animal's importance for this genre's prediction.","Figure: Correctly recognized book covers.","Object classes by SSD and text by EAST are highlighted."],["Engineering & Transportation","For this genre, almost all the covers with vehicle pictures on their covers were classified correctly by the network.","With LRP in Fig.","REF (b), part of image containing cars or motorbikes seem to add more relevance than others.","The last image in the Fig.","REF (b) presents the cases when contribution of person image was dominated by vehicle in the image."],["Romance","Its obvious from the genre name that pictures of couples on the cover are going to have more relevance and LRP results showed this fact to be true.","However, among pictures presented in Fig.","REF (c), LRP depicted girls to add more relevance than men or other things.","The reason could reside in their physical appearance, hairs, and choice of dresses.","The same was demonstrated in last picture of Fig.","REF (c) in which girl's hair are seen to add more relevance with zero relevance coming from animal part on book cover."],["Children's Books","Almost all the children book covers contain pictures of cartoon characters.","LRP on covers from this genre showed these cartoon characters to have higher relevance.","An interesting result is shown in first picture of Fig.","REF (d), where person is depicted as an adversarial identity and importance of cartoons in cover is highlighted.","Some covers showed more relevance for one object in the set of objects.","Like, in Fig.","REF (d) some cartoons in last picture have higher relevance.","It can be because of the object placement and their orientations.","With the help of this information, one can make smart choices for different characters, cartoons and color patterns."],["Cookbooks, Food & Wine Books","Book covers in this genre most commonly contained pictures of different kinds of food.","The results in Fig.","REF (e) showed these food pictures as containment of higher relevance for this genre.","However, carefully analyzing LRP results we discovered shapes of dishes like bowls or spoons adding significant relevance for the genre's prediction.","So, this marks significance of dish shape designs on covers from this genre."],["Test Preparation","The genre contained covers with both text and pictorial information as shown in Fig.","REF (f).","With most contribution coming from big text content on covers.","Images of Fig.","REF (f) presents big texts to add more relevance than images of people.","In first image of Fig.","REF (f), despite of big girl face, relevance is concentrated on text area of book cover.","Such analysis helped us to find design elements specific to the presented genres.","To get more familiar with design, we also presented some cases where the network was not able to correctly classify the genre.","Figure REF shows some of these misclassifications, mainly from the presented genres.","The correct genre names are written below the image.","From the analysis presented above, one can easily decode the reason behind their misclassification because the designs on these book covers are not aligned with their genres which makes it obvious for network to mis-classify.","Here, cover from \"Sports & Outdoors\" contains birds, \"Romance\" cover contains text, \"Cookbooks, Food & Wine Books\" contain no food picture and \"Test Preparation\" cover is also without any significant feature.","LRP justifies all these covers misclassification by highlighting these mentioned objects contributing towards the \"other\" class in one-vs-others."],["Experiment Setup","In order to quantitatively analyze LRP, we propose using SSD to detect objects and EAST to detect text within the book cover images.","We then use LRP to compare the relevance of objects bound by the detection methods.","The SSD was trained on the 2012 PASCAL Visual Object Classes (VOC) Challenge dataset [33].","The VOC dataset contains 20 classes, including \"person,\" six animal classes, eight vehicle classes, and seven indoor object classes.","While SSD trained with VOC is intended for natural scene images, it can be used with book cover images because book covers often contain many of the shared classes, such as \"person\" and \"car.\"","Similarly, EAST was trained on the 2015 ICDAR Robust Reading Competition dataset [34] meant for scene text detection.","Despite being trained for scene text, shown in Fig.","REF , EAST performs remarkably well on book covers for detecting text.","To extract object and text bounding box information, the book covers were prepared by scaling the images to $512\\times 512$ pixels by 3 color channels.","It is important to note that the images used for SSD and EAST were larger than the images used by the CNN used for genre classification.","This is due to the detection methods being much more effective at the higher resolution.","To accommodate this, the bounding boxes were scaled post detection and projected onto the LRP heatmaps.","The relevance of an object $R_{\\mathrm {obj}}$ is calculated using the sum of the relevance within the bounding box, or $R_{\\mathrm {obj}} = \\sum _{(n,m)\\in \\mathcal {B}}{R_{(n,m)}},$ where $R_{(n,m)}$ is the relevance at pixel coordinates $(n,m)$ within bounding box $\\mathcal {B}$ .","Figure: “Misclassified” book covers with correct genre names written below each book cover."],["LRP with Object Detection","A macro view of the genres can be seen by viewing the average relevance of object classes.","Figure REF illustrates the average object-wise relevance of each object class as detected by SSD and EAST for each book genre using the test set book cover images.","It should be noted that detected objects such as \"bottle\" and \"tvmonitor\" were overfit to certain book cover images because many books have plain covers which resemble bottle labels or televisions.","However, this does not mean that the information is useless.","For example, from Fig.","REF , \"bottle\" is more relevant for reference and nonfiction genres where plain covers are common.","Figure: Average object-wise relevance for text detected by EAST and each object class detected by SSD for each book genre.","Only object-genre combinations with five or more data points are shown.In addition, by examining the distribution of the $R_{\\mathrm {obj}}$ of specific object classes, such as \"person,\" it is possible to create associations between genres and detected objects.","For example, the relevance of \"person\" $R_{\\mathrm {person}}$ for each genre is shown in Fig.","REF .","The figure demonstrates that detected \"person\"s within certain genres are more relevant than other genres.","For instance, the genres of \"Romance\" and \"Mystery, Thriller & Suspense\" put a high average relevance in \"person.\"","This indicates that \"person\" is important for the CNNs of those categories.","In addition, mentioned in Section REF and shown in Fig.","REF (f), people are common in \"Test Preparation\" but are not necessarily relevant.","This is supported by Fig.","REF which indicates that on average, \"person\" has very little relevance.","Distributions for the other object classes are provided in the supplemental material.","Figure: Box plot of relevance of \"person\" R person R_{\\mathrm {person}} for each genre.","The boxes represent the first through third quartile and the mean is in red.","The whiskers mark the minimum and maximum datum."],["LRP with Text Detection","Figure REF also reveals that the average relevance of text is low.","The reasoning behind this phenomenon can be explained by Fig.","REF .","The figure shows that the majority of the detected text boxes have a very small relevance $R_{\\mathrm {text}}$ , but there are some text boxes have a higher relevance.","For most genres, the title text contains a significant amount of relevance determined by LRP, but the small descriptive text carries very little relevance.","Figure REF (f) in particular demonstrates this with the large title text having a high relevance and much of the smaller descriptive text having near zero relevance.","Figure: Box plot of relevance of \"text\" R text R_{\\mathrm {text}} for each genre.","The boxes represent the first through third quartile and the mean is in red.","The whiskers mark the minimum and maximum datum."],["Conclusion","In this paper, we presented importance of design in book covers belonging to a specific genre.","The application of LRP on the book cover dataset showed genre specific book cover features.","The method described most relevant parts of input book cover contributing towards a genre prediction by CNN.","We also presented quantitative analysis of LRP using an object detection method, SSD, and a text detection method, EAST.","The analysis further demonstrates that genre classification heavily relies on specific objects for each genres."],["ACKNOWLEDGEMENT","This research was partially supported by MEXT-Japan (Grant No.J17H06100)."]],"string":"[\n [\n \"How do Convolutional Neural Networks Learn Design?\"\n ],\n [\n \"Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.\",\n \"Book covers are designed in a unique way, specific to genres which convey important information to their readers.\",\n \"By using Convolutional Neural Networks (CNN) to predict book genres from cover images, visual cues which distinguish genres can be highlighted and analyzed.\",\n \"In order to understand these visual clues contributing towards the decision of a genre, we present the application of Layer-wise Relevance Propagation (LRP) on the book cover image classification results.\",\n \"We use LRP to explain the pixel-wise contributions of book cover design and highlight the design elements contributing towards particular genres.\",\n \"In addition, with the use of state-of-the-art object and text detection methods, insights about genre-specific book cover designs are discovered.\"\n ],\n [\n \"Introduction\",\n \"Visual design renders specific impressions to transmit information which enriches the product's value.\",\n \"However, these visual designs despite of being important are not analyzed objectively or statistically.\",\n \"Analyzing these visual designs enables us to understand the contained information carried by them.\",\n \"An interesting target of visual design analysis is book cover image design where the design of a book cover can infer the genre.\",\n \"Each book cover is carefully designed by typographers and their designs represent the book contents in an intuitive way for better sales.\",\n \"This association of books to specific genres is based on the differences in their underlying book cover designs [1].\",\n \"The slight change in book cover design can reflect changes in book genre which makes design learning a challenging task for book covers.\",\n \"In order to understand the design elements used for machine aided book cover classification, we employ Convolutional Neural Networks (CNN) [2].\",\n \"In recent years, CNNs have achieved state-of-the-art results in isolated character recognition [3], [4] and large-scale image recognition [5], [6].\",\n \"Notably, Iwana et al.\",\n \"[1] demonstrated that CNNs can be used for genre classification based on book cover image, although with a high level of difficulty.\",\n \"However, that study was subjective and not enough explanation is given as to why the CNN performed as it did.\",\n \"To interpret the reasoning behind a CNN's prediction we used a method called Layer-wise Relevance Propagation (LRP) [7].\",\n \"LRP decomposes output function on its input variables and highlights input pixels contributing towards the network decision.\",\n \"It produces a layer-wise relevance heatmap by recursively multiplying the relevance of higher layers by the normalized feature maps of the target layer.\",\n \"The heatmaps can help us to discover the input image elements which have an effect on the classification result.\",\n \"The main contributions of this paper are threefold.\",\n \"Firstly, we classified the book cover images using one-vs-others classification with CNNs.\",\n \"Secondly, the models built by the CNNs are analyzed using LRP.\",\n \"With LRP, we demonstrate design elements specifically relevant to classification of the book cover images.\",\n \"We show that certain objects have a strong relevance to particular genres.\",\n \"Finally, we use state-of-the-art object detection and text detection methods, namely Single Shot Multibox Detector (SSD) [8] and Efficient and Accurate Scene Text Detector [9], to quantitatively enforce the results found by LRP.\",\n \"This reveals the specific elements in which CNNs classify book cover images for genre classification.\",\n \"The organization is as follows.\",\n \"Section provides related works in design understanding and genre classification as well as feature visualization of CNNs.\",\n \"Section reviews the data and tools used for understanding book cover design.\",\n \"Section presents analysis of CNN's understanding of book cover design.\",\n \"In Section , we demonstrate the use of LRP combined with SSD and EAST for quantitative analysis.\",\n \"Finally, Section draws a conclusion.\",\n \"Artistic style understanding and subjective genre classification is a budding field in machine learning.\",\n \"For example, recent attempts have been done to identify artistic styles and quality of paintings and photographs [10], [11] with neural network models.\",\n \"In addition, there have been trials to classify music by genre [12], [13], book covers by genre [1], movie posters by genre [14], paintings by genre [15], and text by genre [16], [17].\",\n \"Also, in a general sense, document classification can be considered genre classification and deep CNNs are the state-of-the-art in the document classification domain [18], [19], [20].\"\n ],\n [\n \"Visualization inside of CNNs\",\n \"There is a desire to visualize features and determine pixel-wise attention and relevance within the hidden layers of CNNs.\",\n \"However, this is a not a straightforward task [21].\",\n \"Erhan et al.\",\n \"[21] proposed using gradient decent to maximize a node's activation to visualize the employed features.\",\n \"Similar work has been done for large-scale image classification [22].\",\n \"Zeiler and Fergus [23] used deconvolutional neural networks to visualize features learned by CNNs.\",\n \"In addition, they created heatmaps by monitoring class changes systematic cover up of portions of the images.\",\n \"Class Activation Maps (CAM) [24], GradCAM [25], and GradCAM++ [26] reveal the parts of images which are most important to a class using global average pooling (GAP).\",\n \"Recently, LRP has been used in the fields of text [27] where classification scores were projected back to input features for extracting relevant words for a specific prediction.\",\n \"The method has also shown successes in model understanding in fields of sentiment analysis [28], action recognition [29], and age and gender classification [30].\",\n \"As far as the authors are aware, this is the first time LRP has been used for the understanding of genre or design classification.\"\n ],\n [\n \"Amazon Book Cover Dataset\",\n \"We used the Book Cover Image to Genre datasethttps://github.com/uchidalab/book-dataset Task 1.A.\",\n \"The dataset consists of 57,000 book cover images divided into 30 classes of equal sizes.\",\n \"In the experiments, we used the predefined training set and test set modified for one-vs-others classification.\",\n \"In this way, genre-wise training sets were prepared with an equal distribution of positive and negative data samples.\"\n ],\n [\n \"Convolution Neural Networks\",\n \"CNNs are able to tackle image recognition by implementing convolutions of learned filter-like shared weights which maintain the structural qualities of images while acting as feature extractors [2].\",\n \"For the experiment, we implement CNNs to tackle book genre classification.\",\n \"To use the book cover images with a CNN, they were preprocessed by scaling them to 112$\\\\times $ 112 pixels by 3 color channel images and by normalizing the values to be between -1 and 1.\",\n \"The CNN used for the experiments has six convolutional layers with Rectified Linear Units (ReLU) activations and a softmax output layer.\",\n \"The convolutional layers consisted of three layers of 10 nodes with $5\\\\times 5$ convolutional filters, one layer of 25 nodes with a $4\\\\times 4$ filter, one layer of 50 nodes with a $3\\\\times 3$ filter, and one layer of 100 nodes with a $1\\\\times 1$ filter.\",\n \"A $2\\\\times 2$ maxpooling layer with stride 2 was used between each convolution layer.\",\n \"Finally, the CNNs were trained using gradient decent with a batch size of 25 at a learning rate of 0.001 for 50,000 iterations.\",\n \"The accuracy results for each genre is summarized in Fig.\",\n \"REF .\",\n \"In particular, the CNNs had difficulties with the reference classes, such as \\\"Engineering & Transportation,\\\" \\\"Health, Fitness and Dieting,\\\" \\\"History,\\\" \\\"Medical Books,\\\" and \\\"Reference.\\\"\",\n \"Conversely, \\\"Children Books,\\\" \\\"Romance,\\\" and \\\"Test Preparation\\\" had high accuracies.\",\n \"However, more than just classification accuracy, the purpose of this paper is to understand why the CNN's performed as such and reveal the relevant parts of the images.\",\n \"Figure: CNN accuracy by genre.\"\n ],\n [\n \"Layer-wise Relevance Propagation\",\n \"The LRP algorithm and the LRP toolbox [31] aims to explain the reasoning behind the decision made by a network model which allows its users to validate classifier results.\",\n \"LRP is mainly derived from Deep Taylor Decomposition [32], a method of decomposing network's output predictions onto its input variable.\",\n \"The results after such a decomposition is visualized in the form of a heatmap highlighting each pixel's importance for the prediction.\",\n \"LRP explains output function, i.e.\",\n \"classifier's decision, which helps us to derive all of the crucial pixels for a particular prediction.\",\n \"In Fig.\",\n \"REF , the technique is shown in which the output value given by the network is decomposed backwards layer by layer until it reaches the input.\",\n \"This backward decomposition of network's prediction uses local redistribution rules for assigning relevance values $R_i$ to each neuron contributing towards the output, namely $\\\\sum _{i} R_i = \\\\sum _{j} R_j =\\\\dots = \\\\sum _{k}R_k = f(x),$ where $f(x)$ is the prediction function, $R_i$ is the relevance of node $i$ in the target layer, $R_j$ is the relevance of node $j$ of the previous layer, and $R_k$ is the relevance of node $k$ of the highest layer.\",\n \"The total amount of relevance is conserved in this equation.\",\n \"Figure: Feed forward neural network with the (left) forward pass and the (right) backward relevance calculation.\",\n \"The function f(x)f(x) is the prediction outcome given input xx.\",\n \"The variables a i a_i and a j a_j are the inputs for node ii and jj, respectively.\",\n \"R i R_i is the relevance of node ii and R j R_j is the relevance of node jj.For the experiment, we used the $\\\\alpha -\\\\beta $ decomposition formula defined by $R_i = \\\\sum _{j} \\\\left(\\\\alpha \\\\frac{(a_i w_{ij})^+}{\\\\sum _{i} ( a_i w_{ij}) ^ +} + \\\\beta \\\\frac{(a_i w_{ij})^-}{\\\\sum _{i} ( a_i w_{ij}) ^ -} \\\\right) R_j,$ where $\\\\alpha $ and $\\\\beta $ are hyperparameters to weight the positive values of $\\\\frac{(a_i w_{ij})}{\\\\sum _{i} ( a_i w_{ij})}$ and the negative values of $\\\\frac{(a_i w_{ij})}{\\\\sum _{i} ( a_i w_{ij})}$ , respectively.\",\n \"Furthermore, $w_{ij}$ is the weight between nodes $i$ and $j$ and $a_i$ is the input to node $i$ .\",\n \"This decomposition allows for the separation of the positive connections and the negative connections.\",\n \"Values inside positive bracket indicates propagation of activating input messages while negative weight connections indicate deactivating input values.\"\n ],\n [\n \"Single-Shot Multibox Detector\",\n \"To develop a better understanding of the objects within book cover images, we employed SSD [8], a state-of-the-art deep neural network based object detection method.\",\n \"SSD is a feed forward CNN which produces a multi-scale collection of fixed size bounding boxes and scores for object detection within the boxes.\",\n \"A final non-maximal suppression step determines the final detections.\",\n \"The result of SSD is bounding box regions with object classification labels.\",\n \"Using SSD, it is possible to accurately detect multiple objects of different classes within images.\"\n ],\n [\n \"Efficient and Accurate Scene Text Detector\",\n \"For humans, text is an important component of book covers; it is where the title, authors, and additional information is conveyed.\",\n \"However, a CNN may place a different importance on text than humans.\",\n \"Thus, to analyze the relevance of text in book covers, we use EAST [9] as a text detector.\",\n \"EAST uses a multi-channel Fully Convolutional Network (FCN) and non-maximal suppression on predicted geometric shapes to detect multi-orient text-line and word boxes.\"\n ],\n [\n \"How CNNs Understand Book Cover Design: Qualitative Analysis\",\n \"In this section, we have presented LRP results from main genres.\",\n \"The analysis helped us to deduce book cover design elements contributing towards a prediction by CNN.\",\n \"We used $\\\\alpha -\\\\beta $ decomposition formula with values of $\\\\alpha =2$ and $\\\\beta =-1$ which is suggested for networks using ReLU activation functions because it emphasizes the positive elements and de-emphasizes the negative ones [7].\",\n \"This is important due to the ReLU activation function setting negative values to zero.\",\n \"In the heatmaps generated by LRP under this decomposition, pixels adding positive contribution are represented in red color and the ones adding negative contribution are represented by blue color.\"\n ],\n [\n \"Sports & Outdoors\",\n \"Under this genre, many book covers with pictures of players playing indoor and outdoor games were seen.\",\n \"Figure REF (a) shows LRP results on these covers, which presents significance of player's picture on the cover.\",\n \"The first image in Fig.\",\n \"REF (a) supports this fact with LRP being centered on players who are either playing a sport or showing some player like gesture, with car in background adding no contribution.\",\n \"The second image in Fig.\",\n \"REF (a) emphasizes the animal's importance for this genre's prediction.\",\n \"Figure: Correctly recognized book covers.\",\n \"Object classes by SSD and text by EAST are highlighted.\"\n ],\n [\n \"Engineering & Transportation\",\n \"For this genre, almost all the covers with vehicle pictures on their covers were classified correctly by the network.\",\n \"With LRP in Fig.\",\n \"REF (b), part of image containing cars or motorbikes seem to add more relevance than others.\",\n \"The last image in the Fig.\",\n \"REF (b) presents the cases when contribution of person image was dominated by vehicle in the image.\"\n ],\n [\n \"Romance\",\n \"Its obvious from the genre name that pictures of couples on the cover are going to have more relevance and LRP results showed this fact to be true.\",\n \"However, among pictures presented in Fig.\",\n \"REF (c), LRP depicted girls to add more relevance than men or other things.\",\n \"The reason could reside in their physical appearance, hairs, and choice of dresses.\",\n \"The same was demonstrated in last picture of Fig.\",\n \"REF (c) in which girl's hair are seen to add more relevance with zero relevance coming from animal part on book cover.\"\n ],\n [\n \"Children's Books\",\n \"Almost all the children book covers contain pictures of cartoon characters.\",\n \"LRP on covers from this genre showed these cartoon characters to have higher relevance.\",\n \"An interesting result is shown in first picture of Fig.\",\n \"REF (d), where person is depicted as an adversarial identity and importance of cartoons in cover is highlighted.\",\n \"Some covers showed more relevance for one object in the set of objects.\",\n \"Like, in Fig.\",\n \"REF (d) some cartoons in last picture have higher relevance.\",\n \"It can be because of the object placement and their orientations.\",\n \"With the help of this information, one can make smart choices for different characters, cartoons and color patterns.\"\n ],\n [\n \"Cookbooks, Food & Wine Books\",\n \"Book covers in this genre most commonly contained pictures of different kinds of food.\",\n \"The results in Fig.\",\n \"REF (e) showed these food pictures as containment of higher relevance for this genre.\",\n \"However, carefully analyzing LRP results we discovered shapes of dishes like bowls or spoons adding significant relevance for the genre's prediction.\",\n \"So, this marks significance of dish shape designs on covers from this genre.\"\n ],\n [\n \"Test Preparation\",\n \"The genre contained covers with both text and pictorial information as shown in Fig.\",\n \"REF (f).\",\n \"With most contribution coming from big text content on covers.\",\n \"Images of Fig.\",\n \"REF (f) presents big texts to add more relevance than images of people.\",\n \"In first image of Fig.\",\n \"REF (f), despite of big girl face, relevance is concentrated on text area of book cover.\",\n \"Such analysis helped us to find design elements specific to the presented genres.\",\n \"To get more familiar with design, we also presented some cases where the network was not able to correctly classify the genre.\",\n \"Figure REF shows some of these misclassifications, mainly from the presented genres.\",\n \"The correct genre names are written below the image.\",\n \"From the analysis presented above, one can easily decode the reason behind their misclassification because the designs on these book covers are not aligned with their genres which makes it obvious for network to mis-classify.\",\n \"Here, cover from \\\"Sports & Outdoors\\\" contains birds, \\\"Romance\\\" cover contains text, \\\"Cookbooks, Food & Wine Books\\\" contain no food picture and \\\"Test Preparation\\\" cover is also without any significant feature.\",\n \"LRP justifies all these covers misclassification by highlighting these mentioned objects contributing towards the \\\"other\\\" class in one-vs-others.\"\n ],\n [\n \"Experiment Setup\",\n \"In order to quantitatively analyze LRP, we propose using SSD to detect objects and EAST to detect text within the book cover images.\",\n \"We then use LRP to compare the relevance of objects bound by the detection methods.\",\n \"The SSD was trained on the 2012 PASCAL Visual Object Classes (VOC) Challenge dataset [33].\",\n \"The VOC dataset contains 20 classes, including \\\"person,\\\" six animal classes, eight vehicle classes, and seven indoor object classes.\",\n \"While SSD trained with VOC is intended for natural scene images, it can be used with book cover images because book covers often contain many of the shared classes, such as \\\"person\\\" and \\\"car.\\\"\",\n \"Similarly, EAST was trained on the 2015 ICDAR Robust Reading Competition dataset [34] meant for scene text detection.\",\n \"Despite being trained for scene text, shown in Fig.\",\n \"REF , EAST performs remarkably well on book covers for detecting text.\",\n \"To extract object and text bounding box information, the book covers were prepared by scaling the images to $512\\\\times 512$ pixels by 3 color channels.\",\n \"It is important to note that the images used for SSD and EAST were larger than the images used by the CNN used for genre classification.\",\n \"This is due to the detection methods being much more effective at the higher resolution.\",\n \"To accommodate this, the bounding boxes were scaled post detection and projected onto the LRP heatmaps.\",\n \"The relevance of an object $R_{\\\\mathrm {obj}}$ is calculated using the sum of the relevance within the bounding box, or $R_{\\\\mathrm {obj}} = \\\\sum _{(n,m)\\\\in \\\\mathcal {B}}{R_{(n,m)}},$ where $R_{(n,m)}$ is the relevance at pixel coordinates $(n,m)$ within bounding box $\\\\mathcal {B}$ .\",\n \"Figure: “Misclassified” book covers with correct genre names written below each book cover.\"\n ],\n [\n \"LRP with Object Detection\",\n \"A macro view of the genres can be seen by viewing the average relevance of object classes.\",\n \"Figure REF illustrates the average object-wise relevance of each object class as detected by SSD and EAST for each book genre using the test set book cover images.\",\n \"It should be noted that detected objects such as \\\"bottle\\\" and \\\"tvmonitor\\\" were overfit to certain book cover images because many books have plain covers which resemble bottle labels or televisions.\",\n \"However, this does not mean that the information is useless.\",\n \"For example, from Fig.\",\n \"REF , \\\"bottle\\\" is more relevant for reference and nonfiction genres where plain covers are common.\",\n \"Figure: Average object-wise relevance for text detected by EAST and each object class detected by SSD for each book genre.\",\n \"Only object-genre combinations with five or more data points are shown.In addition, by examining the distribution of the $R_{\\\\mathrm {obj}}$ of specific object classes, such as \\\"person,\\\" it is possible to create associations between genres and detected objects.\",\n \"For example, the relevance of \\\"person\\\" $R_{\\\\mathrm {person}}$ for each genre is shown in Fig.\",\n \"REF .\",\n \"The figure demonstrates that detected \\\"person\\\"s within certain genres are more relevant than other genres.\",\n \"For instance, the genres of \\\"Romance\\\" and \\\"Mystery, Thriller & Suspense\\\" put a high average relevance in \\\"person.\\\"\",\n \"This indicates that \\\"person\\\" is important for the CNNs of those categories.\",\n \"In addition, mentioned in Section REF and shown in Fig.\",\n \"REF (f), people are common in \\\"Test Preparation\\\" but are not necessarily relevant.\",\n \"This is supported by Fig.\",\n \"REF which indicates that on average, \\\"person\\\" has very little relevance.\",\n \"Distributions for the other object classes are provided in the supplemental material.\",\n \"Figure: Box plot of relevance of \\\"person\\\" R person R_{\\\\mathrm {person}} for each genre.\",\n \"The boxes represent the first through third quartile and the mean is in red.\",\n \"The whiskers mark the minimum and maximum datum.\"\n ],\n [\n \"LRP with Text Detection\",\n \"Figure REF also reveals that the average relevance of text is low.\",\n \"The reasoning behind this phenomenon can be explained by Fig.\",\n \"REF .\",\n \"The figure shows that the majority of the detected text boxes have a very small relevance $R_{\\\\mathrm {text}}$ , but there are some text boxes have a higher relevance.\",\n \"For most genres, the title text contains a significant amount of relevance determined by LRP, but the small descriptive text carries very little relevance.\",\n \"Figure REF (f) in particular demonstrates this with the large title text having a high relevance and much of the smaller descriptive text having near zero relevance.\",\n \"Figure: Box plot of relevance of \\\"text\\\" R text R_{\\\\mathrm {text}} for each genre.\",\n \"The boxes represent the first through third quartile and the mean is in red.\",\n \"The whiskers mark the minimum and maximum datum.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we presented importance of design in book covers belonging to a specific genre.\",\n \"The application of LRP on the book cover dataset showed genre specific book cover features.\",\n \"The method described most relevant parts of input book cover contributing towards a genre prediction by CNN.\",\n \"We also presented quantitative analysis of LRP using an object detection method, SSD, and a text detection method, EAST.\",\n \"The analysis further demonstrates that genre classification heavily relies on specific objects for each genres.\"\n ],\n [\n \"ACKNOWLEDGEMENT\",\n \"This research was partially supported by MEXT-Japan (Grant No.J17H06100).\"\n ]\n]"},"id":{"kind":"string","value":"1808.08402"}}},{"rowIdx":198,"cells":{"text":{"kind":"list like","value":[["Towards Tight Approximation Bounds for Graph Diameter and Eccentricities"],["Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.","There are no known very efficient algorithms for computing the Diameter exactly.","Thus, much research has been devoted to how fast this parameter can be approximated.","Chechik et al.","showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\\tilde{O}(m^{3/2})$ time.","Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs.","Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$.","It was, however, completely plausible that a $3/2$-approximation is possible in linear time.","In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$.","Another fundamental set of graph parameters are the Eccentricities.","The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$.","Chechik et al.","showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\\tilde{O}(m^{3/2})$ time and Abboud et al.","showed that no $O(n^{2-\\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs.","We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH.","We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\\delta$ factor in $\\tilde{O}(m/\\delta)$ time for any $0<\\delta<1$.","This beats all Eccentricity algorithms in Cairo et al."],["Introduction","Among the most important graph parameters are the graph's Diameter and the Eccentricities of its vertices.","The Eccentricity of a vertex $v$ is the (shortest path) distance to the furthest vertex from $v$ , and the Diameter is the largest Eccentricity over all vertices in the graph.","The Eccentricities and Diameter measure how fast information can spread in networks.","Efficient algorithms for their computation are highly desired (see e.g.","[40], [9], [33]).","Unfortunately, the fastest known algorithms for these parameters are very slow on large graphs.","For unweighted graphs on $n$ vertices and $m$ edges, the fastest Diameter algorithm runs in $\\tilde{O}(\\min \\lbrace mn,n^\\omega \\rbrace )$ $\\tilde{O}$ notation hides polylogarithmic factors time [16] where $\\omega <2.373$ is the exponent of square matrix multiplication [52], [32], [45].","For weighted graphs, the fastest Eccentricity and Diameter algorithms actually compute all distances in the graph, i.e.","they solve the All-Pairs Shortest Paths (APSP) problem.","The fastest known algorithms for APSP in weighted graphs run in $\\min \\lbrace \\tilde{O}(mn),n^3/\\exp (\\sqrt{\\log n})\\rbrace $ [53], [35], [37].","Whether one can solve Diameter faster than APSP is a well-known open problem (e.g.","see Problem 6.1 in [20] and [2], [17]).","Whether one can solve Eccentricities faster than APSP was addressed by [50] (for dense graphs) and by [34] (for sparse graphs).","Vassilevska W. and Williams [50] showed that Eccentricities and APSP are equivalent under subcubic reductions, so that either both of them admit $O(n^{3-\\varepsilon })$ time algorithms for $\\varepsilon >0$ , or neither of them do.","Lincoln et al.","[34] proved that under a popular conjecture about the complexity of weighted Clique, the $O(mn)$ runtime for Eccentricities cannot be beaten by any polynomial factor for any sparsity of the form $m=\\Theta (n^{1+1/k})$ for integer $k$ .","Due to the hardness of exact computation, efficient approximation algorithms are sought.","A folklore $\\tilde{O}(m+n)$ time algorithm achieves a 2-approximation for Diameter in directed weighted graphs and a 3-approximation for Eccentricities in undirected weighted graphs.","Aingworth et al.","[2] presented an almost-$3/2$ approximation An almost-$c$ approximation of $X$ is an estimate $X^{\\prime }$ so that $X\\le X^{\\prime }\\le cX+O(1)$ .","algorithm for Diameter running in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] improved the result of [2] with an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","Chechik et al.","[21] obtained a (genuine) $3/2$ approximation algorithm for Diameter (in directed graphs) and a (genuine) $5/3$ -approximation algorithm for Eccentricities (in undirected graphs), running in $\\tilde{O}(\\min \\lbrace m^{3/2},mn^{2/3}\\rbrace )$ time.","These are the only known non-trivial approximation algorithms for Diameter in directed graphs.","So far, there are no known faster than $mn$ algorithms for approximating Eccentrities in directed graphs within any constant factor.","Cairo et al.","[15] generalized the above results for undirected graphs and obtained a time-approximation tradeoff: for every $k\\ge 1$ they obtained an $\\tilde{O}(mn^{1/(k+1)})$ time algorithm that achieves an almost-$2-1/2^k$ approximation for Diameter and an almost $3-4/(2^k+1)$ -approximation for Eccentricities."],["Our contributions.","We address the following natural question: Main Question: Are the known approximation algorithms for Diameter and Eccentricities optimal?","A partial answer is known.","Under the Strong Exponential Time Hypothesis (SETH), every $3/2-\\varepsilon $ approximation algorithm (for $\\varepsilon >0$ ) for Diameter in undirected unweighted graphs with $O(n)$ nodes and edges must use $n^{2-o(1)}$ time [41].","Similarly, every $5/3-\\varepsilon $ approximation algorithm for the Eccentricities of undirected unweighted graphs with $O(n)$ nodes and edges must use use $n^{2-o(1)}$ time [7].","This however does not answer the question of whether the runtimes of the known $3/2$ and $5/3$ approximation algorithms can be improved.","It is completely plausible that there is a $3/2$ -approximation algorithm for Diameter or a $5/3$ -approximation for Eccentricities running in linear time.","We address our Main Question for both sparse and dense graphs.","Our results are shown in Table REF .","Table: Our results.","All of the lower bounds hold even for sparse graphs.","SS-TT Diameter is a variant of Diameter introduced later in this section."],["Sparse graphs.","Our first result (restated as Theorem REF ) regards approximating Diameter in undirected unweighted sparse graphs.","Theorem 1 ($3/2$ -Diameter Approx.","is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $8/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Diameter of an undirected unweighted sparse graph.","In particular, any $3/2$ -approximation algorithm in sparse graphs must take $n^{3/2-o(1)}$ time.","Hence the $\\tilde{O}(m^{3/2})$ time $3/2$ -approximation algorithm of [41], [21] is optimal in two ways: improving the approximation ratio to $3/2-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([41]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $8/5$ .","Our lower bound instance says that in $O(m^{3/2-\\delta })$ time one cannot return 6 when the Diameter is 8.","One may be tempted to extend the above lower bound, by showing that, say, in $O(m^{4/3-\\delta })$ time one cannot even return 5 when the Diameter is 8.","This approach, however fails: in Theorem REF we give an $O(m^2/n)$ time algorithm that does return 5 in this case, and in general when the Diameter is $2h$ , it returns at least $h+1$ .","Notice that when the Diameter is $2h$ , the folklore linear time algorithm returns an estimate of only $h$ .","Hence for sparse graphs, our algorithm runs in linear time and outperforms the folklore algorithm.","Also, for constant even Diameter, it gives a better than 2 approximation.","We obtain stronger Diameter hardness results for weighted graphs and for directed unweighted graphs.","In particular, assuming SETH: For weighted sparse graphs, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $5/3-\\varepsilon $ Diameter approximation (for $\\varepsilon >0$ ) (Theorem REF ).","For directed unweighted sparse graphs, using a general time-accuracy tradeoff lower bound (Theorem REF ), we show that no near-linear time algorithm can achieve an approximation factor better than $5/3$ .","We summarize our Diameter lower bounds and compare them to the known upper bounds in Figure REF .","Figure: Our hardness results for Diameter.","The xx-axis is the approximation factor and the yy-axis is the runtime exponent.","Black lines represent lower bounds.","Black dots represent existing algorithms.","Blue dots represent existing algorithms whose approximation is potentially off by an additive term (the algorithms of ).","Transparent dots represent algorithms that might exist and would be tight with our lower bounds.We address our Main Question for Eccentricities as well.","Our main result for Eccentricities is Theorem REF .","Its first consequence is as follows: Theorem 2 ($5/3$ -Eccentricities Alg.","is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $9/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.","In other words, the $\\tilde{O}(m^{3/2})$ time $5/3$ -approximation algorithm of [41], [21] is tight in two ways.","Improving the approximation ratio to $5/3-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([7]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $9/5$ .","More generally, we prove (in Theorem REF ): for every $k\\ge 2$ , under SETH, distinguishing between Eccentricities $2k-1$ and $4k-3$ in unweighted undirected sparse graphs requires $n^{1+1/(k-1)-o(1)}$ time.","Thus, no near-linear time algorithm can achieve a $2-\\varepsilon $ -approximation for Eccentricities for $\\varepsilon >0$ .","The best (folklore) near-linear time approximation algorithm for Eccentricities currently only achieves a 3-approximation, and only in undirected graphs.","There is no known constant factor approximation algorithm for directed graphs!","Is our limitation result for linear time Eccentricity algorithms far from the truth?","We show that our lower bound result is essentially tight, for both directed and undirected graphs by producing the first non-trivial near-linear time approximation algorithm for the Eccentricities in weighted directed graphs (Theorem REF ).","Theorem 3 (2-Approx.","for Eccentricities in near-linear time.)","Under SETH, no $n^{1+o(1)}$ time algorithm can output a $2-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.","For every $\\delta >0$ , there is an $\\tilde{O}(m/\\delta )$ time algorithm that produces a $(2+\\delta )$ -approximation for the Eccentricities of any directed weighted graph.","The approximation hardness result is the first result within fine-grained complexity that gives tight hardness for near linear time algorithms.","The $2+\\delta $ approximation ratio that our algorithm produces beats all approximation ratios for Eccentricities given by Cairo et al. [15].","It also constitutes the first known constant factor approximation algorithm for Eccentricities in directed graphs.","Our approximation algorithm also implies as a corollary an approximation algorithm for the Source Radius problemThe Source Radius problem is a natural extension of the undirected Radius definition.","The goal is to return $\\min _x\\max _vd(x,v)$ .","studied in [7] with the same runtime and approximation factor ($2+\\delta $ ).","Abboud et al.","[7] showed that, under the Hitting Set Conjecture, any $(2-\\varepsilon )$ -approximation algorithm (for $\\varepsilon >0$ ) for Source Radius requires $n^{2-o(1)}$ time, and hence our Source Radius algorithm is also essentially tight.","Our lower bound in Theorem REF holds already for undirected unweighted graphs, and the upper bound works even for directed weighted graphs.","The algorithm produces a $(2+\\delta )$ -approximation, which while close, is not quite a 2-approximation.","We design (in Theorem REF ) a genuine 2-approximation algorithm running in $\\tilde{O}(m\\sqrt{n})$ time that also works for directed weighted graphs.","We then complement it (in Theorem REF ) with a tight lower bound under SETH: in sparse directed graphs, if you go below factor 2 in the accuracy, the runtime blows up to quadratic.","Theorem 4 (Tight 2-Approx.","for Eccentricities) Under SETH, no $n^{2-\\delta }$ time algorithm for $\\delta >0$ can output a $2-\\varepsilon $ approximation for the Eccentricities of a directed unweighted sparse graph.","There is an $\\tilde{O}(m\\sqrt{n})$ time algorithm that produces a 2-approximation for the Eccentricities of any directed weighted graph.","We thus give an essentially complete answer to our Main Question for Eccentricities.","Our results are summarized in Figures REF and REF .","Figure: Our algorithms and hardness results for Eccentricities.","The lower bounds are for unweighted graphs and the upper bounds are for weighted graphs.","The xx-axis is the approximation factor and the yy-axis is the runtime exponent.","Black lines represent lower bounds.","Black dots represent existing algorithms (including our algorithm at (2,3/2)(2,3/2) in figure b).","Blue dots represent existing algorithms whose position may not be exactly as it appears in the figure.","Here, the blue dots represent our (2+δ)(2+\\delta )-approximation algorithm running in O ˜(m/δ)\\tilde{O}(m/\\delta ) time.","Transparent dots represent algorithms that might exist and would be tight with our lower bounds.Our conditional lower bounds for both Diameter and Eccentricities are all based on a common construction: a conditional lower bound for a problem called $S$ -$T$ Diameter.","In $S$ -$T$ Diameter, the input is a graph $G=(V,E)$ and two subsets $S,T\\subseteq V$ , not necessarily disjoint, and the output is $D_{S,T}:=\\max _{s\\in S,t\\in T} d(s,t)$ .","$S$ -$T$ Diameter is a problem of independent interest.","It is related to the bichromatic furthest pair problem studied in geometry (e.g.","as in [29]), but for graphs (if we set $T=V\\setminus S$ ).","It is easy to see that if one can compute the $S$ -$T$ Diameter, then one can also compute the Diameter in the same time: just set $S=T=V$ .","We show that actually, when it comes to exact computation, the $S$ -$T$ Diameter and Diameter in weighted graphs are computationally equivalent (Theorem REF ).","We show that $S$ -$T$ Diameter also has similar approximation algorithms to Diameter.","We give a 3-approximation running in linear time (Claim REF based on the folklore Diameter 2-approximation algorithm), and a 2-approximation running in $\\tilde{O}(m^{3/2})$ time (Theorem REF based on the $3/2$ -approximation algorithm of [41], [21]).","We prove the following lower bound for $S$ -$T$ Diameter (restated as Theorem REF ), the proof of which is the starting point for all of our conditional lower bounds.","Theorem 5 Under SETH, for every $k\\ge 2$ , every algorithm that can distinguish between $S$ -$T$ Diameter $k$ and $3k-2$ in undirected unweighted graphs requires $n^{1+1/(k-1)-o(1)}$ time.","Theorem REF implies that under SETH, our aforementioned 2 and 3-approximation algorithms are optimal.","For all of our lower bounds, we also address the question of whether they can be extended to higher values of Diameter and Eccentricities.","All of our lower bounds, with the exception of directed Eccentricities, are of the form “any algorithm that can distinguish between Diameter (or Eccentricity) $a$ and $b$ requires a certain amount of time\" for small values of $a$ and $b$ .","This doesn't exclude the possibility of an algorithm that distinguishes between higher Diameters (or Eccentricities) of the same ratio i.e.","between $a\\ell $ and $b\\ell $ for some $\\ell $ .","For weighted Diameter, our lower bound easily extends to higher values of Diameter by simply scaling up the edge weights.","For $S-T$ Diameter and undirected Eccentricities, our lower bounds easily extend to higher values of Diameter and Eccentricities by simply subdividing the edges.","For unweighted directed Diameter, our lower bound extends to higher values of Diameter with a slight loss in approximation factor by subdividing some of the edges.","For unweighted undirected Diameter, our lower bound does not seem to easily extend to higher values of Diameter."],["Dense graphs.","Can we address our Main Question for dense graphs as well?","In particular, can we extend our runtime lower bounds of the form $n^{1+1/\\ell -o(1)}$ to $mn^{1/\\ell -o(1)}$ , thus matching the known algorithms for larger values of $m$ ?","We show that the answer is “no”.","For undirected unweighted graphs, we obtain $\\tilde{O}(n^2)$ time algorithms for Diameter achieving an almost $3/2$ -approximation (Theorem REF ), and for all Eccentricities achieving an almost $5/3$ -approximation algorithm (Theorem REF ).","These algorithms run in near-linear time in dense graphs, improving the previous best runtime of $\\tilde{O}(m\\sqrt{n})$ by Roditty and Vassilevska W. [41], and subsuming (for dense unweighted graphs) the results of Cairo et al. [15].","Theorem 6 There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph with Diameter $D=3h+z$ for $h\\ge 0,z\\in \\lbrace 0,1,2\\rbrace $ , returns an extimate $D^{\\prime }$ such that $2h-1\\le D^{\\prime }\\le D$ if $z=0,1$ and $2h\\le D^{\\prime }\\le D$ if $z=2$ .","There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph returns estimates $\\varepsilon ^{\\prime }(v)$ of the Eccentricities $\\varepsilon (v)$ of all vertices such that $3\\varepsilon (v)/5-1\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v$ .","We also show (in Theorem REF ) that one can improve the estimates slightly with an $O(n^{2.05})$ time algorithm."],["Related work","The fastest known algorithm for APSP in dense weighted graphs is by R. Williams [53] and runs in $O(n^3/2^{\\Theta (\\sqrt{\\log n})})$ time.","For sparse undirected graphs, the fastest known APSP algorithm is by Pettie [35] running in $O(mn+n^2\\log \\log n)$ time.","The fastest APSP algorithm for sparse undirected weighted graphs is by Pettie and Ramachandran [37] and runs in $O(mn\\log \\alpha (m,n))$ time.","For APSP on undirected unweighted graphs with $m>n\\log \\log n$ , Chan [17] presented an $O(mn \\log \\log n/\\log n)$ time algorithm.","In graphs with small integer edge weights bounded in absolute value by $M$ , APSP can be computed in $\\tilde{O}(Mn^\\omega )$ time (by Shoshan and Zwick [46] building upon Seidel [43] and Alon, Galil and Margalit [6]) in undirected graphs and in $\\tilde{O}(M^{0.681}n^{2.5302})$ time (by Zwick [55]) in directed graphs.","Zwick [55] also showed that APSP in directed weighted graphs admits an $(1+\\varepsilon )$ -approximation algorithm for any $\\varepsilon >0$ , running in time $\\tilde{O}(n^\\omega /\\varepsilon \\log (M/\\varepsilon ))$ .","For Diameter in graphs with integer edge weights bounded by $M$ , Cygan et al.","[16] obtained an algorithm running in time $\\tilde{O}(Mn^\\omega )$ .","The pioneering work of Aingworth et al.","[2] on Diameter and shortest paths approximation was the root to many subsequent works.","Building upon Aingworth et al.","[2], Dor, Halperin and Zwick [24] presented additive approximation algorithms for APSP in undirected unweighted graphs, achieving among other things, an additive 2-approximation in $\\tilde{O}(n^{7/3})$ time (notably, the best known bound on $\\omega $ is $>7/3$ ).","They also presented an $\\tilde{O}(n^2)$ time additive $O(\\log n)$ -approximation algorithm.","These algorithms were generalized by Cohen and Zwick [23] who showed that in undirected weighted graphs APSP has a (multiplicative) 3-approximation in $\\tilde{O}(n^2)$ time, a $7/3$ -approximation in $\\tilde{O}(n^{7/3})$ time, and a 2-approximation in $\\tilde{O}(n \\sqrt{mn})$ time.","Baswana and Kavitha [11] presented an $\\tilde{O}(m\\sqrt{n}+n^2)$ time multiplicative 2-approximation algorithm and an $\\tilde{O}(m^{2/3} n+n^2)$ time $7/3$ -approximation algorithm for APSP in weighted undirected graphs.","Spanners are closely related to shortest paths approximation.","A subgraph $H$ is an $(\\alpha , \\beta )$ -spanner of $G=(V,E)$ if for every $u,v\\in V$ , $d_H(u,v) \\le \\alpha \\cdot d_G(u,v) + \\beta $ , where $d_{G^{\\prime }}(u,v)$ is the distance between $u$ and $v$ in $G^{\\prime }$ .","Any weighted undirected graph has a $(2k-1,0)$ -spanner with $O(n^{1+1/k})$ edges [3].","Baswana and Sen [14] presented a randomized linear time algorithm for constructing a $(2k-1,0)$ -spanner with $O(kn^{1+1/k})$ edges.","Dor, Halperin and Zwick [24] showed that a $(1,2)$ -spanner with $O(n^{1.5})$ edges can be constructed in $\\tilde{O}(n^2)$ time.","Elkin and Peleg [25] showed that for every integer $k\\ge 1$ and $\\varepsilon >0$ there is a $(1+\\varepsilon , \\beta )$ -spanner with $O(\\beta n^{1+1/k})$ edges, where $\\beta $ depends on $k$ and $\\varepsilon $ but is independent of $n$ .","Baswana et.","al [12] presented a $(1,6)$ -spanner with $O(n^{4/3})$ edges.","Woodruff [54] presented an $\\tilde{O}(n^2)$ time algorithm that computes a $(1,6)$ -spanner with $O(n^{4/3})$ edges.","Chechik [18] presented a $(1,4)$ -spanner with $O(n^{7/5})$ edges.","Recently, Abboud and Bodwin[1] showed that there is no additive spanner with constant error and $O(n^{4/3-\\varepsilon })$ edges.","Thorup and Zwick [48] introduced the notion of distance oracles, a data structure that stores approximate distances for a weighted undirected graph.","Thorup and Zwick designed a distance oracle that for any $k$ takes $O(mn^{1/k})$ time to construct and, is of size is $O(kn^{1+1/k})$ , and given a pair of vertices $u,v \\in V$ it returns in $O(k)$ time a $(2k-1)$ -approximation for $d(u,v)$ .","Baswana and Sen [13] improved the construction time to $O(n^2)$ for unweighted graphs.","Baswana and Kavitha [11] extended the $O(n^2)$ construction time to weighted graphs.","Subsequently, Baswana, Gaur, Sen, and Upadhyay [10] obtained subquadratic construction time in unweighted graphs, at the price of having additive constant error in addition to the $2k-1$ multiplicative error.","Chechik [19] gave an oracle with space $O(n^{1+1/k})$ and $O(1)$ query time, which like previous work, returns a $(2k-1)$ -approximation.","Pǎtraşcu and Roditty [38] obtained a distance oracle that uses $\\tilde{O}(n^{5/3})$ space, has $O(1)$ query time, and returns a $(2k+1)$ -approximation.","Sommer [44] presented an $\\tilde{O}(n^2)$ time algorithm that constructs such a distance oracle.","The construction time was recently improved to $O(n^2)$ by Knudsen [30].","Pǎtraşcu et.","al [39] presented infinity many distance oracles with fractional approximation factors that for graphs with $m=\\tilde{O}(n)$ converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.","Thorup and Zwick [47] also extended their techniques from [48] to compact routing schemes.","The lower bounds presented in this paper were inspired by a lower bound by Pǎtraşcu and Roditty [38] who showed conditional hardness based on a conjecture on the hardness of a set intersection problem for the space usage of any distance oracle that can distinguish between distances 3 and 7."],["Organization","In Section we prove our lower bounds for $S$ -$T$ Diameter which serve as a basis for the rest of our lower bounds.","We also show equivalence between Diameter and $S$ -$T$ Diameter.","In Section we prove our lower bounds for Eccentricities: one for directed graphs and one for undirected graphs.","In Section we prove our lower bounds for Diameter.","This section is divided into four subsections, one for each of the results in Table REF .","In Section we describe our algorithms for sparse graphs: our 2-approximation and $(2+\\delta )$ -approximation for Eccentricities, our 2-approximation and 3-approximation for $S$ -$T$ Diameter, and our less than 2-approximation for Diameter.","In Section we describe our algorithms for dense graphs: our nearly $3/2$ -approximations for Diameter and our nearly $5/3$ -approximations for Eccentricities."],["Preliminaries","Let $G=(V,E)$ be a graph, where $|V|=n$ and $|E|=m$ .","For every $u,v\\in V$ let $d_G(u,v)$ be the length of the shortest path from $u$ to $v$ .","When the graph $G$ is clear from the context we omit the subscript $G$ .","The eccentricity $\\varepsilon (v)$ of a vertex $v$ is defined as $\\max _{u\\in V} d(v,u)$ .","The diameter $D$ of a graph is $\\max _{v\\in V}\\varepsilon (v)$ .","In a directed graph we have $\\varepsilon ^{out}(v)=\\max _{u\\in V} d(v,u)$ (resp., $\\varepsilon ^{in}(v)=\\max _{u\\in V} d(u,v)$ ).","Let $deg(v)$ be the degree of $v$ and let $N_s(u)$ be the set of the $s$ closest vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.","In a directed graph let $deg^{out}(v)$ (resp., $deg^{in}(v)$ ) be the outgoing (incoming) degree of $v$ .","Let $N^{\\text{out}}_s(v)$ (resp., $N^{\\text{in}}_s(v)$ ) be the set of the $s$ closest outgoing (incoming) vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.","For a subset $S\\subseteq V$ of vertices and a vertex $v \\in V$ we write $d(S,v):=\\min _{s \\in S}d(s,v)$ to denote the distance from the set $S$ to the vertex $v$ .","Let $k\\ge 2$ .","The $k$ -Orthogonal Vectors Problem ($k$ -OV) is as follows: given $k$ sets $S_1,\\ldots ,S_k$ , where each $S_i$ contains $n$ vectors in $\\lbrace 0,1\\rbrace ^d$ , determine whether there exist $v_1\\in S_1,\\ldots ,v_k\\in S_k$ so that their generalized inner product is 0, i.e.","$\\sum _{i=1}^d \\prod _{j=1}^k v_j[i]=0$ .","R. Williams [51] (see also [49]) showed that if for some $\\varepsilon >0$ there is an $n^{k-\\varepsilon } \\text{\\rm poly~} (d)$ time algorithm for $k$ -OV, then CNF-SAT on formulas with $N$ variables and $m$ clauses can be solved in $2^{N(1-\\varepsilon /k)} \\text{\\rm poly~} (m)$ time.","In particular, such an algorithm would contradict the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [28] which states that for every $\\varepsilon >0$ there is a $K$ such that $K$ -SAT on $N$ variables cannot be solved in $2^{(1-\\varepsilon )N} \\text{\\rm poly~} N$ time (say, on a word-RAM with $O(\\log N)$ bit words).","This also motivates the following $k$ -OV Conjectures (implied by SETH) for all constants $k\\ge 2$ : $k$ -OV requires $n^{k-o(1)}$ time on a word-RAM with $O(\\log n)$ bit words.","Most of our conditional lower bounds are based on the $k$ -OV Conjecture for a particular constant $k$ , and thus they also hold under SETH.","A main motivation behind SETH is that despite decades of research, the best upper bounds for $K$ -SAT on $N$ variables and $M$ clauses remain of the form $2^{N(1-c/K)} \\text{\\rm poly~} {(M)}$ for constant $c$ (see e.g.","[27], [36], [42]).","The best algorithms for the $k$ -OV problem for any constant $k\\ge 2$ on $n$ vectors and dimension $c\\log n$ run in time $n^{2-1/O(\\log c)}$ (Abboud, Williams and Yu [8] and Chan and Williams [22])."],["$S$ -{{formula:678000a5-c252-477a-976e-b4f5fbe676e0}} Diameter hardness","In the following we will prove that under SETH, our $S$ -$T$ Diameter algorithms are essentially optimal.","We prove the following theorem: Theorem 7 Let $k\\ge 2$ be an integer.","There is an $O(k n^{k-1} d^{k-1})$ time reduction that transforms any instance of $k$ -OV on sets of $n$ $d$ -dimensional vectors into a graph on $O(n^{k-1}+k n^{k-2} d^{k-1})$ nodes and $O(k n^{k-1} d^{k-1})$ edges and two disjoint sets $S$ and $T$ on $n^{k-1}$ nodes each, so that if the $k$ -OV instance has a solution, then $D_{S,T}\\ge 3k-2$ , and if it does not, $D_{S,T}\\le k$ .","From Theorem REF we get that if there is some $k\\ge 2$ , $\\varepsilon >0$ and $\\delta >0$ so that there is an $O(M^{1+1/(k-1)-\\varepsilon })$ time $(3-2/k-\\delta )$ -approximation algorithm for $S$ -$T$ Diameter in $M$ -edge graphs, then $k$ -OV has an $n^{k-\\gamma } \\text{\\rm poly~} (d)$ -time algorithm for some $\\gamma >0$ and SETH is false.","We obtain an immediate corollary.","Corollary 8 For $S$ -$T$ Diameter, under SETH, there is no $(2-\\varepsilon )$ -approximation algorithm running in $O(m^{2-\\delta })$ time for any $\\varepsilon >0,\\delta >0$ , no $O(m^{3/2-\\delta })$ time $7/3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0,\\delta >0$ , no $(m+n)^{1+o(1)}$ time, $3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0$ .","We will prove the following more detailed theorem, which will be useful for our Diameter lower bounds.","Theorem 9 Let $k\\ge 2$ .","Given a $k$ -OV instance consisting of sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.","The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .","The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .","$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .","If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .","If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .","Suppose the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ .","Let $t=k-2$ .","Let $s$ be such that $0\\le s\\le t$ .","Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ ."],["Proof of Theorem ","We will prove the theorem for $k=t+2$ for any $t\\ge 0$ .","We will create a layered graph $G$ on $t+3$ layers, $L_0,\\ldots ,L_{t+2}$ , where the edges go only between adjacent layers $L_i,L_{i+1}$ .","We will set $S=L_0$ and $T=L_{t+2}$ for the $S$ -$T$ Diameter instance.","In particular, $D_{S,T}\\ge t+2$ because of the layering.","Let us describe the vertices of $G$ .","$L_0$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(a_0,a_1,\\ldots , a_t)$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $L_{t+2}$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(b_1,b_2,\\ldots , b_{t+1})$ where for each $i$ , $b_i\\in W_i$ .","Layer $L_1$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(a_0,\\ldots ,a_{t-1}, \\bar{x})$ where for each $i$ , $a_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .","Similarly, $L_{t+1}$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(b_2,\\ldots ,b_{t+1}, \\bar{x})$ where for each $i$ , $b_i\\in W_i$ and $\\bar{x}$ is a $(t+1)$ -tuple of coordinates.","For every $j\\in \\lbrace 2,\\ldots ,t\\rbrace $ , $L_j$ consists of $n^t d^{t+1}$ vertices $(a_0, \\ldots , a_{t-j},b_{t+3-j},\\ldots , b_{t+1},\\bar{x})$ , where for each $i$ , $a_i\\in W_i$ , $b_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .","In other words, there is a vector from $W_i$ for every $i\\notin \\lbrace t-j+1,t-j+2\\rbrace $ .","Now let us define the edges.","Consider a node $(a_0,\\ldots ,a_t)\\in L_0$ .","For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(a_0,\\ldots ,a_t)$ to $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ if and only if for every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , $a_j$ is 1 in coordinates $x_0,\\ldots ,x_{t-j}$ .","For any $i\\in \\lbrace 1,\\ldots ,t\\rbrace $ let's define the edges between $L_i$ and $L_{i+1}$ .","For $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_i$ Here if $i=1$ , there are no $b$ 's in the tuple.","and for any $c_{t+2-i}\\in W_{t+2-i}$ , add an edge to $(a_0,\\ldots ,a_{t-i-1},c_{t+2-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_{i+1}$ .","Here we “forget” vector $a_{t-i}$ and replace it with $c_{t+2-i}$ , leaving everything else the same.","Finally, the edges between $L_{t+1}$ and $L_{t+2}$ are as follows.","Consider some $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .","For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(b_1,\\ldots ,b_{t+1})$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ if and only if for every $j\\in \\lbrace 1,\\ldots ,{t+1}\\rbrace $ , $b_j$ is 1 in coordinates $x_{t+1-j},\\ldots ,x_t$ .","Figure REF shows the construction of the graph for $t=2$ .","Figure: The reduction graph from (t+2)(t+2)-OV for t=2t=2.","The figure depicts when a path of length t+2t+2 exists between arbitrary a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and b 1 b 2 b 3 ∈L t+2 b_1b_2b_3\\in L_{t+2}.","It also shows that when there is a path of length t+2t+2 between a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and a 1 a 2 a 3 ∈L t+2 a_1a_2a_3\\in L_{t+2}, a 0 ,a 1 ,a 2 ,a 3 a_0,a_1,a_2,a_3 cannot be an orthogonal 4-tuple.An important claim is as follows: Claim 10 For every $\\bar{x}$ , each $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ is at distance $t$ to every $(b_2,\\ldots ,b_t,\\bar{x})\\in L_{t+1}$ .","Consider the path starting from $(a_0,\\ldots ,a_{t-1},\\bar{x})$ , and then for each $i\\ge 1$ following the edges $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1},\\bar{x})\\in L_i$ to $(a_0,\\ldots ,a_{t-1-i},b_{t+2-i},\\ldots ,b_{t+1},\\bar{x})\\in L_{i+1}$ , until we reach $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ .","This path exists by construction and has length $t$ .","Now we proceed to prove the bounds on the $S$ -$T$ Diameter.","Lemma 11 (Property 3 of Theorem REF ) If the $(t+2)$ -OV instance has no solution, then $D_{S,T}=t+2$ .","If the $(t+2)$ -OV instance has no solution, then for every $c_0\\in W_0,c_1\\in W_1,\\ldots ,c_{t+1}\\in W_{t+1}$ , there is some coordinate $x$ such that $c_0[x]=c_1[x]=\\ldots =c_{t+1}[x]=1$ .","Now consider the graph and any $(a_0,\\ldots ,a_t)\\in L_0$ , $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .","For every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , let $x_j$ be a coordinate so that $a_0,\\ldots ,a_{t-j},b_{t-j+1},\\ldots ,b_{t+1}$ are all 1 in $x_j$ .","Let $\\bar{x}=(x_0,\\ldots ,x_{t})$ .","By construction, $(a_0,\\ldots ,a_t)$ has an edge to $(a_0,\\ldots ,a_{t-1},\\bar{x})$ and $(b_2,\\ldots ,b_{t+1},\\bar{x})$ has an edge to $(b_1,\\ldots ,b_{t+1})$ .","Also, by Claim REF , $(a_0,\\ldots ,a_{t-1},\\bar{x})$ has a path of length $t$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})$ .","This shows that $D_{S,T}\\le t+2$ ; equality follows because the graph is layered.","Now we prove the guarantee for the case when an orthogonal tuple exists.","Lemma 12 (Property 4 of Theorem REF ) If there exist $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ that are orthogonal, then $D_{S,T}\\ge 3t+4$ .","To prove the lemma, we will actually prove a more general claim: Property 5 of Theorem REF .","Claim 13 (Property 5 of Theorem REF ) Suppose that $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ are orthogonal.","Let $s$ be such that $0\\le s\\le t$ .","Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","If the claim is true, then using $s=0$ we get that the Diameter is at least $3t+4$ so Lemma REF is true.","The claim for $s>0$ is useful for the rest of our constructions.","We will show that the distance between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ is strictly more than $3t+2-2s$ .","Because the graph is layered and hence bipartite and $t+2\\equiv 3t+2\\mod {2}$ , the distance must be at least $3t-2s+4$ .","Let's assume for contradiction that the shortest path $P$ between $\\alpha $ and $\\beta $ is of length $\\le 3t+2-2s$ .","First let's look at any subpath $P^{\\prime }$ of $P$ strictly within $M=L_1\\cup \\ldots \\cup L_{t+1}$ .","All nodes on $P^{\\prime }$ must share the same $\\bar{x}$ .","Furthermore, if $P^{\\prime }$ starts with a node of $L_1$ and ends with a node of $L_{t+1}$ , as $P^{\\prime }$ needs to be a shortest path and by Claim REF , $P^{\\prime }$ must be of length exactly $t$ .","Next, notice that $P$ cannot go from $L_0$ to $L_{t+2}$ and then back to $L_0$ .","This is because it needs to end up in $L_{t+2}$ and any time it crosses over $M$ , it would need to pay a distance of $t+2$ , so $P$ would have to have length at least $3t+6>3t+2-2s$ .","Hence, $P$ must be of the following form: a path from $\\alpha $ through $L_0\\cup M$ back to $L_0$ (possibly containing only $\\alpha $ ), followed by a path crossing $M$ to reach $L_{t+2}$ , followed by a path through $L_{t+2}\\cup M$ to $L_{t+2}$ (possibly empty).","We will show that if $P$ has length $\\le 3t+2-2s$ then $P$ must contain a length $t+2$ subpath $Q$ between a node $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)\\in L_0$ , for some choices of the $w$ 's and some $q\\le t-s$ , and a node $(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})\\in L_{t+2}$ , for some choices of $v$ 's.","That is, this path traverses $M$ without weaving, by following $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_{t-1},\\bar{x})\\in L_1$ , $(a_0, \\ldots , a_{q},w_{q+1},\\ldots , w_{t-2},a_{t+1},\\bar{x})\\in L_2$ , $\\ldots $ , $(v_2, \\ldots , v_q, a_{q+1}, \\ldots , a_{t-s}, \\ldots , a_{t+1}, \\bar{x})\\in L_{t+1}$ .","Suppose we show that such a subpath exists.","Then by the construction of our graph we have that for every $i\\in \\lbrace 0,\\ldots ,q\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace 0,\\ldots ,t-i\\rbrace $ , and that for all $i\\in \\lbrace q+1,\\ldots ,t+1\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace t+1-i,\\ldots ,t\\rbrace $ .","That is, for all $i$ , $a_i[x_{t-q}]=1$ , and we get a contradiction since the $a_i$ were supposed to be orthogonal.","Now let $\\alpha ^*$ be the last node from $L_0$ on $P$ and let $\\beta ^*$ be the first node of $L_{t+1}$ of $P$ .","Let $a^*\\in L_1$ be the node right after $\\alpha ^*$ and let $b^*\\in L_{t+1}$ be the node right before $\\beta ^*$ .","Since the subpath of $P$ between $a^*$ and $b^*$ is within $M$ , it must share the same $\\bar{x}$ , and it must have length exactly $t$ by Claim REF .","We will show that the subpath $Q$ that we are looking for is the subpath of $P$ between $\\alpha ^*$ and $\\beta ^*$ .","Its length is exactly what we want: $t+2$ .","It remains to show that for some $q\\le t-s$ and some choices of $w$ 's and $v$ 's, $\\alpha ^*=(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)$ and $\\beta ^*=(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})$ .","Consider the path $P_1$ between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ and $\\alpha ^*$ and the path $P_2$ between $\\beta =(a_1,\\ldots ,a_{t+1})$ and $\\beta ^*$ .","Let $L_i$ be the layer in $M$ with largest $i$ that $P_1$ touches and let $L_j$ be the layer in $M$ with smallest $j$ that $P_2$ touches.","For convenience, let us define $j^{\\prime }=t+2-j$ .","The length of $P_1$ is then at least $2i$ and the length of $P_2$ is at least $2j^{\\prime }$ .","The length $|P|$ of $P$ equals $t+2+|P_1|+|P_2|\\ge t+2+2i+2j^{\\prime }=t+2+2(i+j^{\\prime })$ .","Since we have assumed that $|P|\\le 3t+2-2s$ , we must have that $t+2+2(i+j^{\\prime })\\le 3t+2-2s$ and hence $i+j^{\\prime }\\le t-s$ .","Now, since $P_1$ goes at most to $L_i$ , then from getting from $\\alpha $ to $\\alpha ^*$ , at most the last $i$ elements of $(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ can have been “forgotten”.","Hence, $\\alpha ^*=(a_0,\\ldots ,a_{t-\\max \\lbrace s,i\\rbrace },b_{t-\\max \\lbrace s,i\\rbrace +1},\\ldots ,b_{t-i},w_{t-i+1},\\ldots ,w_{t})$ for some $w$ 's.","(If $i\\ge s$ , the $b$ 's do not appear.)","Similarly, between $\\beta $ and $\\beta ^*$ , at most the first $j^{\\prime }$ elements of $\\beta $ can have been forgotten.","Thus, we have that $\\beta ^*=(v_1,\\ldots ,v_{j^{\\prime }},a_{j^{\\prime }+1},\\ldots ,a_{t+1})$ for some $v$ 's.","Now, since $i+j^{\\prime }\\le t-s$ , we must have that $j^{\\prime }\\le t-s-i\\le t-\\max \\lbrace s,i\\rbrace $ , and hence the path between $\\alpha ^*$ and $\\beta ^*$ is the path $Q$ we are searching for.","See Figure REF for an illustration of $L_i$ , $L_j$ etc.","in the case when $s=0$ .","Figure: Here PP contains at least 2 nodes in L 0 L_0 and at least 2 in L t+2 L_{t+2}, and s=0s=0."],["Equivalence between Diameter and $S$ -{{formula:bdb52e68-e562-4944-b84c-7c1359b488ce}} Diameter","Here we will prove that when it comes to exact computation, $S$ -$T$ -Diameter and Diameter in weighted graphs are equivalent.","The proof for directed graphs is much simpler, so we focus on the equivalence for undirected graphs.","Also, it is clear that if one can solve $S$ -$T$ Diameter, one can also solve Diameter in the same running time since one can simply set $S=T=V$ .","We prove: Theorem 14 Suppose that there is a $T(n,m)$ time algorithm that can compute the Diameter of an $n$ node, $m$ edge graph with nonnegative integer edge weights.","Then, the $S$ -$T$ Diameter of any $n$ node $m$ edge graph with nonnegative integer edge weights can be computed in $T(O(n),O(m))$ time.","Let $G=(V,E),S,T$ be the $S$ -$T$ Diameter instance; let $w:E\\rightarrow \\lbrace 0,\\ldots ,M\\rbrace $ be the edge weights.","First, we can always assume that $M$ is even: if it is not, multiply all edge weights by 2; all distances (and hence also the $S$ -$T$ Diameter) double.","Let $S=\\lbrace s_1,\\ldots ,s_k\\rbrace $ and $T=\\lbrace t_1,\\ldots ,t_{\\ell }\\rbrace $ .","Now, let $W=Mn$ .","First add $|S|=k$ new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ .","For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .","Let $G_S$ be this new graph.","Let's consider the Diameter of $G_S$ .","For every pair of nodes $u,v\\notin S^{\\prime }$ , the distance is the same as in $G$ .","For $v_i\\in S^{\\prime }$ and $x\\notin S^{\\prime }$ , the distance is $W+d_G(s_i,x)\\le W+M(n-1)<2W$ .","For $v_i,v_j\\in S^{\\prime }$ , the distance is $2W+d(s_i,s_j)$ .","Hence the Diameter of $G_S$ is $2W+\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .","Hence by computing the Diameter of $G_S$ , we can compute $D_S=\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .","We can create a similar graph $G_T$ whose Diameter will allow us to compute $D_T=\\max _{t_i,t_j\\in S} d(t_i,t_j)$ .","After this, let's create a graph $G_{S,T}$ as follows.","Add new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ and $T^{\\prime }=\\lbrace u_1,\\ldots ,u_\\ell \\rbrace $ .","For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .","For each $j\\in \\lbrace 1,\\ldots ,\\ell \\rbrace $ add a new edge $(u_j,t_j)$ of weight $W$ .","With a similar argument as above, the Diameter $D^{\\prime }$ of $G_{S,T}$ is $D^{\\prime }=2W+\\max _{u,v\\in S\\cup T} d_G(u,v)$ .","Let's assume without loss of generality that $D_S\\ge D_T$ .","If $D^{\\prime }> D_S$ , then $D^{\\prime }=2W+\\max _{u\\in S,v\\in T} d_G(u,v)$ , and we can compute the $S$ -$T$ Diameter of $G$ by subtracting $2W$ .","Now suppose that we get $D^{\\prime }\\le D_S$ ; we must have then actually gotten $D^{\\prime }=D_S$ .","The $S$ -$T$ Diameter of $G$ might be strictly smaller than $D_S$ .","We add two new nodes $x$ and $y$ to $G_{S,T}$ .","We add an edge $(x,y)$ of weight $2W$ , edges $(x,v_i)$ for every $v_i\\in S^{\\prime }$ of weight $D_S/2$ and (symmetrically) edges $(y,u_j)$ for every $u_j\\in T^{\\prime }$ of weight $D_S/2$ .","The construction of $G^{\\prime }$ is depicted in Figure REF .","Figure: A depiction of the construction of G ' G^{\\prime }.Let us consider the distances in this new graph $G^{\\prime }$ .","For every $a,b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(a,b)=d_G(a,b)d(a,b)$ .","For every $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(x,b)=W+D_S/2 +\\min _{a\\in S} d_G(a,b)\\le 2W+D_S/2$ .","Similarly, $d(y,b)=W+D_S/2 +\\min _{a\\in T} d_G(a,b)\\le 2W+D_S/2$ .","For every $v_i\\in S^{\\prime }$ , $d(x,v_i)=D_S/2$ , and $d(y,v_i)=2W+D_S/2$ .","For every $u_i\\in T^{\\prime }$ , $d(y,u_i)=D_S/2$ , and $d(x,u_i)=2W+D_S/2$ .","For every $v_i,v_j\\in S^{\\prime }$ , $d(v_i,v_j)=D_S$ .","For every $u_i,u_j\\in T^{\\prime }$ , $d(u_i,u_j)=D_S$ .","For every $v_i\\in S^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(v_i,b)\\le W+d(s_i,b)\\le 2W.$ For every $u_i\\in T^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(u_i,b)\\le W+d(t_i,b)\\le 2W.$ For every $v_i\\in S^{\\prime }$ and $u_j\\in T^{\\prime }$ , $d(v_i,u_j)$ is the minimum of $D_S+2W, D_S+2W+\\min _{s\\in S} d_G(s,t_j), D_S+2W+\\min _{t\\in t} d_G(t,s_i)$ and $2W+d_G(s_i,t_j)$ .","The middle two terms are $\\ge D_S+2W$ , and hence $d(v_i,u_j)=2W+\\min \\lbrace D_S,d_G(s_i,t_j)\\rbrace $ .","Consider $s_i\\in S, t_j\\in T$ that are the end points of the $S$ -$T$ Diameter $D$ in $G$ .","Then $D=d_G(s_i,t_j)$ .","Now, we have from before that $D\\le D_S$ , as otherwise we have computed $D$ already.","Hence in $G^{\\prime }$ , the distance $d(u_i,v_j)$ equals $2W+\\min \\lbrace D_S,D\\rbrace =2W+D$ We note that for any $s_i,s_j\\in S$ , and any $t\\in T$ , $d_G(s_i,s_j)\\le d_G(s_i,t)+d_G(t,s_j)\\le 2\\max _{s\\in S,t\\in T} d_G(s,t) = 2D$ .","Thus, $D\\ge D_S/2$ .","The distances in cases (1) to (5) are all $\\le 2W+D_S/2\\le 2W+D$ .","Hence the Diameter of $G^{\\prime }$ is actually exactly $2W+D$ ."],["Undirected graphs","Theorem 15 Let $k\\ge 2$ .","Under the $k$ -OV conjecture, every algorithm that can distinguish between eccentricity at most $2k-1$ and eccentricity at least $4k-3$ for every vertex in an $O(n)$ edge and node undirected graph, requires at least $n^{1+1/(k-1)-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.","Figure: The undirected Eccentricities lower bound for k=5k=5.Let's start with the $S$ -$T$ -diameter construction for $k$ obtained from a given $k$ -OV instance.","We remove any internal nodes if they don't have edges to one of their adjacent layers - they don't hurt the instance.","We have a graph on $O(n^{k-1}d^{k-2})$ vertices and edges with the following properties: (1) Suppose that the $k$ -OV instance has no $k$ -OV solution.","Then for every $s\\in S,t\\in T$ , $d(s,t)=k$ .","Also, for every $s\\in S$ and $u\\notin S\\cup T$ , $d(s,u)\\le (k-1)+k=2k-1$ since we can take a $\\le (k-1)$ length path from $u$ to some node $t\\in T$ and since $d(s,t)=k$ .","(2) If there is a $k$ -OV solution, there are two nodes $s\\in S,t\\in T$ with $d(s,t)\\ge 3k-2$ .","We modify the construction as follows.","For every $s\\in S$ , we create an undirected path on $k-2$ new vertices $s_1\\rightarrow s_2\\rightarrow \\ldots \\rightarrow s_{k-2}$ and add an edge $(s,s_1)$ ; let's call $s$ by $s_0$ .","Now, the distance between $s_0$ and $s_i$ is $i$ .","Add a new node $y$ and create edges $(s_{k-2},y)$ for every $s\\in S$ .","Now, $d(y,s_0)=k-1$ for every $s\\in S$ , and also for every $s,s^{\\prime }\\in S$ and all $i,j\\in \\lbrace 0,\\ldots ,k-2\\rbrace $ , we have that $d(s_i,s^{\\prime }_j)\\le 2k-2$ .","For every $s\\in S$ , $t\\in T$ , there is now potentially a new path between them, from $s$ to $y$ in $k-1$ steps, then to some other $s^{\\prime }$ in $k-1$ steps and then to $t$ using $\\ge k$ steps.","The length is $\\ge 2(k-1)+k=3k-2$ , so when there is a $k$ -OV solution, there is still a pair $s,t$ at distance at least $3k-2$ .","Now, we also attach paths to the nodes in $T$ .","In particular, for each $t$ , add an undirected path $t\\rightarrow t_1\\rightarrow \\ldots \\rightarrow t_{k-1}$ .","The distance between any $s\\in S$ and any $t_i$ is $i+d(s,t)$ .","Hence when there is no $k$ -OV solution, the Eccentricities of all $s_0$ for $s\\in S$ are $\\le k+(k-1)=2k-1$ , and when there is a $k$ -OV solution, there is $s\\in S, t\\in T$ so that $d(s_0,t_{k-1})\\ge (3k-2)+(k-1)=4k-3$ ."],["Directed graphs","Theorem 16 Under the 2-OV conjecture, for any $\\delta >0$ , any $(2-\\delta )$ -approximation algorithm for all Eccentricities in an $n$ node, $O(n)$ -edge directed unweighted graph, requires $n^{2-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.","Figure: The directed Eccentricities lower bound.Suppose we are given an instance of 2-OV: two sets of vectors $U,V$ over $\\lbrace 0,1\\rbrace ^d$ and we want to know whether there are $u\\in U,v\\in V$ with $u\\cdot v=0$ .","Let $L\\ge 1$ be any integer.","Let us create a directed unweighted graph $G$ ; an illustration can be found in Figure REF .","$G$ will have a vertex $u$ for every $u\\in U$ and a vertex $c$ for every $c\\in [d]$ .","Every $v\\in V$ will be represented by a directed path $v_0\\rightarrow v_1\\rightarrow \\ldots \\rightarrow v_L$ .","In addition, there is a directed path $P_x$ on $L$ extra nodes, $x_1\\rightarrow \\ldots \\rightarrow x_L$ so that every $u\\in U$ has directed edges $(u,x_1)$ and $(x_L,u)$ .","For every $u\\in U$ and every $c$ for which $u[c]=1$ , we add a directed edge $(u,c)$ .","For every $v$ and every $c$ for which $v[c]=1$ , we add a directed edge $(c,v_0)$ .","Remove any $c$ that does not have at least one edge coming from $U$ .","Let us consider the eccentricity of any node $u\\in U$ .","First, for all $u^{\\prime }\\in U$ , $d(u,u^{\\prime })\\le L+1$ since one can go through the path $P_x$ .","For every $c\\in [d]$ , there is at least one edge coming from some $u^{\\prime }\\in U$ , and so one can reach $c$ from $u$ by first taking $P_x$ to $u^{\\prime }$ and then using the edge $(u^{\\prime },c)$ .","Hence, $d(u,c)\\le L+2$ for all $c\\in [d]$ .","For any $v\\in V$ and $i\\in \\lbrace 1,\\ldots , L\\rbrace $ , the distance $d(u,v_i)=i+d(u,v_0)$ , and so we consider $d(u,v_0)$ .","If there is a $c$ for which $u[c]=v[c]=1$ , then $d(u,v_0)=2$ , and hence for all $i$ , $d(u,v_i)\\le L+2$ .","If no such $c$ exists and so if $u$ and $v$ are orthogonal, the only way to reach $v_0$ is potentially via $P_x$ to some other $u^{\\prime }\\in U$ which is at distance 2 to $v_0$ .","Hence if $u$ and $v$ are orthogonal, $d(u,v_0)=L+3$ , and hence $d(u,v_L)=2L+3$ .","In other words, the eccentricity of $u$ is $L+2$ if it is not orthogonal to any vectors in $V$ and it is $\\ge 2L+3$ if there is some $v$ that is orthogonal to $u$ .","The number of vertices in the graph is $O(nL+d)$ and the number of edges is $O(nL+nd)$ .","Suppose that there is a $(2-\\varepsilon )$ -approximation algorithm for all Eccentricities in graphs with $O(m)$ nodes and edges running in $O(m^{2-\\delta })$ time for some $\\varepsilon ,\\delta >0$ .","Then, we construct the above instance for $L=\\lceil 1/\\varepsilon \\rceil $ and run the algorithm on it.","The approximation returned is at least as good as a $(2-1/L)$ -approximation.","Hence if the diameter is at least $2L+3$ , the algorithm will return an estimate that is at least $L(2L+3)/(2L-1) > L+2$ .","Thus the algorithm can solve 2-OV in time $O((nL+nd)^{2-\\delta }) = O(n^{2-\\delta }d^{2-\\delta })$ , contradicting the 2-OV conjecture."],["Diameter lower bounds","For all of our constructions we begin with the $S$ -$T$ diameter lower bound construction from Theorem REF .","Here, if the $k$ -OV instance has no solution, $D_{S,T}\\le k$ and if the instance has a solution $D_{S,T}\\ge 3k-2$ .","To adapt this construction to Diameter, we need to ensure that if the OV instance has no solution then all pairs of vertices have small enough distance.","We begin by augmenting the $S$ -$T$ Diameter construction by adding a matching between $S$ and a new set $S^{\\prime }$ as well as a matching between $T$ and a new set $T^{\\prime }$ .","Without any further modifications, pairs of vertices $u,v\\in S\\cup S^{\\prime }$ (or $u,v\\in T\\cup T^{\\prime }$ ) could be far from one another.","The challenge is to add extra gadgetry to make these pairs close for “no\" instances while maintaining that in “yes\" instances the distance between the diameter endpoints $s^{\\prime }\\in S^{\\prime },t^{\\prime }\\in T^{\\prime }$ is large.","That is, for “yes\" instances, we want a shortest path between the diameter endpoints $s^{\\prime }$ and $t^{\\prime }$ to contain the vertex $s\\in S$ matched to $s^{\\prime }$ and the vertex $t\\in T$ matched to $t^{\\prime }$ so that we can use use the fact that $d(s,t)\\ge 3k-2$ .","In other words, we do not want there to be a shortcut from $s^{\\prime }$ to some vertex in $S$ that allows us to use a path of length $k$ from $S$ to $T$ .","For example, we cannot simply create a vertex $x$ and connect it to all vertices in $S\\cup S^{\\prime }$ because this would introduce shortcuts from $S^{\\prime }$ to $S$ .","We will describe some intuition for the augmentations to the graph regarding 3-OV for simplicity.","Recall that $s^{\\prime }\\in S^{\\prime }, t^{\\prime }\\in T^{\\prime }$ are the endpoints of the diameter and let $t$ be the vertex matched to $t^{\\prime }$ .","To solve the problem outlined in the above paragraph, we observe that in the “yes\" case there are three types of vertices $s\\in S$ .","(1) close: $d(s,t)=3$ , (2) far: $d(s,t)\\ge 7$ (property 4 of Theorem REF ), and (3) intermediate: $d(s,t)\\ge 5$ (property 5 of Theorem REF ).","For close $s$ , we need $d(s^{\\prime },s)$ to be large so that there is no shortcut from $s^{\\prime }$ to $t^{\\prime }$ through $s$ .","For far $s$ , it is ok if $d(s^{\\prime },s)$ is small because $d(s,t)$ is large enough to ensure that paths from $s^{\\prime }$ to $t^{\\prime }$ through $s$ are still long enough.","For intermediate $s$ , $d(s^{\\prime },s)$ cannot be small, but it also need not be large.","To fulfill these specifications, we add a small clique (the graph is still sparse) and connect each of its vertices to only some of the vertices in $S$ and/or $S^{\\prime }$ according to the implications of property 5 of Theorem REF .","When $s$ is close, we ensure that $d(s^{\\prime },s)$ is large by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique, uses an edge inside of the clique, and then goes from the clique to $s$ .","When $s$ is intermediate, we ensure that $d(s^{\\prime },s)$ is not too small by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique and then from the clique to $s$ (without using an edge inside of the clique).","These intermediate $s$ are important as they allow every vertex in the clique to have an edge to some vertex in $S$ and thus be close enough to the $T$ side of the graph in the “no\" case."],["5 vs 8 unweighted undirected construction","In this section we show that under the 3-OV Hypothesis, any algorithm that can distinguish between Diameter 5 and 8 in sparse undirected unweighted graphs, requires $\\Omega (n^{3/2-o(1)})$ time.","Theorem REF gives us the following theorem.","Theorem 17 Given a 3-OV instance consisting of three sets $A,B,C \\subseteq \\lbrace 0,1\\rbrace ^d$ , $|A|=|B|=|C|=n$ , we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following properties.","The graph consists of 4 layers of vertices $S,L_1,L_2,T$ .","The number of nodes in the sets is $|S|=|T|=n^2$ and $|L_1|,|L_2|\\le nd^2$ .","$S$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .","Similarly, $T$ consists of all tuples $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .","If the 3-OV instance has no solution, then $d(u,v)=3$ for all $u \\in S$ and $v \\in T$ .","If the 3-OV instance has a solution $a \\in A, b \\in B, c \\in C$ with $a,b,c$ orthogonal, then $d((a,b)\\in S,(b,c) \\in T)\\ge 7$ .","Setting $k=3,s=1$ in Property 5 of Theorem REF : for any $b^{\\prime } \\in B$ we have $d((a,b) \\in S,(b^{\\prime },c) \\in T)\\ge 5$ and $d((a,b^{\\prime }) \\in S,(b,c) \\in T)\\ge 5$ .","For any vertex $u \\in L_1$ there exists a vertex $s \\in S$ that is adjacent to $u$ .","Similarly, for any vertex $v \\in L_2$ there exists a vertex $t \\in T$ that is adjacent to $v$ .","We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the other properties.","In the rest of the section we use Theorem REF to prove the following result.","Theorem 18 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.","If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .","If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 8$ ."],["Construction of the graph","We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.","Figure REF illustrates the construction of the graph.","Figure: The illustration for the 5 vs 8 construction.","The edges between sets S,L 1 ,L 2 S,L_1,L_2 and TT are not depicted.","The edges between vertices in S ' S^{\\prime } and SS (TT and T ' T^{\\prime }) form a matching.","Vertices in S '' S^{\\prime \\prime } (T '' T^{\\prime \\prime }) form a clique.We start by adding a set $S^{\\prime }$ of $n^2$ vertices.","$S^{\\prime }$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .","We connect every $(a,b) \\in S^{\\prime }$ to its counterpart $(a,b) \\in S$ .","Thus, there is a matching between the sets of vertices $S$ and $S^{\\prime }$ .","We also add another set $S^{\\prime \\prime }$ of $n$ vertices.","$S^{\\prime \\prime }$ contains one vertex $a$ for every $a \\in A$ .","For every pair of vertices from $S^{\\prime \\prime }$ we add an edge between the vertices.","Thus, the $n$ vertices form a clique.","Furthermore, for every vertex $a \\in S^{\\prime \\prime }$ we add an edge to $(a,b) \\in S$ for all $b \\in B$ .","In total we added $n^2+n=O(n^2)$ vertices and $\\binom{n}{2}+2n^2=O(n^2)$ edges.","We do a similar construction for the set $T$ of vertices.","We add a set $T^{\\prime }$ of $n^2$ vertices - one vertex for every tuple $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .","We connect every $(b,c)\\in T^{\\prime }$ to $(b,c) \\in T$ .","Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.","$T^{\\prime \\prime }$ contains one vertex for every vector $c \\in C$ .","For every pair of vertices from $T^{\\prime \\prime }$ we add an edge between the vertices.","We connect every $c \\in T^{\\prime \\prime }$ to $(b,c) \\in T$ for all $b \\in B$ .","This finishes the construction of the graph.","In the rest of the section we show that the construction satisfies the promised two properties."],["Correctness of the construction","We need to consider two cases."],["Case 1: the 3-OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .","We consider three subcases."],["Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$ and {{formula:e49ba0d1-2dce-4e1e-bc34-0f08bcbafce8}}","We observe that there exists $s \\in S$ that has $d(u,s)\\le 1$ .","Indeed, if $u \\in S$ , then $s=u$ works.","If $u \\in S^{\\prime } \\cup S^{\\prime \\prime }$ , then we are done by the construction.","On the other hand, if $u \\in L_1$ , then there exists such an $s \\in S$ by property 6 from Theorem REF .","Similarly we can show that there exists $t \\in T$ such that $d(v,t)\\le 1$ .","Finally, by property 3 we have that $d(s,t)=3$ .","Thus, we can upper bound the distance between $u$ and $v$ by $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le 1+3+1=5$ as required."],["Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$","From the previous case we know that there are two vertices $s_1,s_2 \\in S$ such that $d(u,s_1)\\le 1$ and $d(s_2,v)\\le 1$ .","To show that $d(u,v)\\le 5$ it is sufficient to show that $d(s_1,s_2)\\le 3$ .","This is indeed true since both vertices $s_1$ and $s_2$ are connected to some two vertices in $S^{\\prime \\prime }$ and every two vertices in $S^{\\prime \\prime }$ are at distance at most 1 from each other."],["Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup L_2$","The case is analogous to the previous case."],["Case 2: the 3-OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 8$ .","Let $a \\in A, b \\in B, c \\in C$ be a solution to the 3-OV instance.","We claim that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 8$ .","Let $P$ be an optimal path between $u=((a,b) \\in S^{\\prime })$ and $v=((b,c) \\in T^{\\prime })$ that achieves the smallest distance.","We want to show that $P$ uses at least 8 edges.","Let $t \\in T$ be the first vertex from the set $T$ that is on path $P$ .","Let $s \\in S$ be the last vertex on path $P$ that belongs to $S$ and precedes $t$ in $P$ .","We can easily check that, if $s \\ne ((a,b) \\in S)$ , then $d(u,s)\\ge 3$ and, similarly, if $t \\ne ((b,c) \\in T)$ , then $d(t,v)\\ge 3$ .","We consider three subcases."],["Case 2.1: $s \\ne ((a,b) \\in S)$ and {{formula:bf5aaac7-5b1b-4304-83ad-0a1fd19bda3a}}","Since $s$ and $t$ are separated by two layers of vertices, we must have $d(s,t)\\ge 3$ .","Thus we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+3+3=9>8$ as required."],["Case 2.2: $s = ((a,b) \\in S)$ and {{formula:a28d868f-f38c-4534-b9c1-df1ca7ddd683}}","In this case we use property 4 and conclude $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)=1+d((a,b) \\in S,(b,c) \\in T)+1\\ge 1+7+1=9>8$ as required."],["Case 2.3: either $s = ((a,b) \\in S)$ or {{formula:1d8b5c54-e4d4-4cf8-8777-ab4e928d855f}} holds but not both","W.l.o.g.","$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .","If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 4$ and we get that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 4+3+1=8$ .","On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .","By property 5 we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 5$ .","We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+5+1=9>8$ as required."],["6 vs 10 weighted undirected construction","In this section we change the construction from Theorem REF to show that under the 3-OV Hypothesis, any algorithm that can distinguish between diameter 6 and 10 in sparse undirected weighted graphs requires $\\Omega (n^{3/2-o(1)})$ time.","We get the following theorem.","Theorem 19 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an weighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.","If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .","If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 10$ .","Each edge of the graph has weight either 1 or 2."],["Construction of the graph","The construction of the graph is the same as in Theorem REF except all edges connecting vertices between sets $L_1$ and $L_2$ have weight 2 and all edges inside the cliques on nodes $S^{\\prime \\prime }$ and $T^{\\prime \\prime }$ have weight 2.","All the remaining edges have weight 1."],["Correctness of the construction","The correctness proof is essentially the same as for Theorem REF .","As before we consider two cases."],["Case 1: the 3-OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .","In the analysis of Case 1 in Theorem REF we show a path between $u$ and $v$ such that the path involves at most one edge from the cliques or between sets $L_1$ and $L_2$ .","Since we added weight 2 to the latter edges, the length of the path increased by at most 1 as a result.","So we have upper bound $d(u,v)\\le 6$ for all pairs $u$ and $v$ of vertices."],["Case 2: the 3-OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 10$ .","Similarly to Theorem REF we will show that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 10$ , where $a \\in A, b \\in B, c \\in C$ is a solution to the 3-OV instance.","The analysis of the subcases is essentially the same as in Theorem REF .","For cases 2.1 and 2.2 in the proof of Theorem REF we had $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 9$ .","Since we increased edge weights between $L_1$ and $L_2$ to 2 and every path from $(a,b) \\in S^{\\prime }$ to $(b,c) \\in T^{\\prime }$ must cross the layer between $L_1$ and $L_2$ , we also increased the lower bound of the length of the path from 9 to 10 for cases 2.1 and 2.2.","It remains to consider Case 2.3.","As in the proof of Theorem REF , w.l.o.g.","$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .","If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 5$ and we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 5+4+1=10$ .","On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .","By property 5 and because we increased edge weights between $L_1$ and $L_2$ to 2, we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 6$ .","We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+6+1=10$ as required."],["$3k-4$ vs {{formula:cef9204d-0a6e-4579-9476-41520f256b04}} unweighted directed construction","In this section, we show that under SETH, for every $k\\ge 3$ , every algorithm that can distinguish between Diameter $3k-4$ and $5k-7$ in directed unweighted graphs requires $\\Omega (n^{1+1/(k-1)-o(1)})$ time.","Theorem REF gives us the following theorem.","Theorem 20 Given a $k$ -OV instance consisting of $k\\ge 2$ sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.","The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .","The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .","$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .","If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .","If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .","Setting $s=k-2$ in Property 5 of Theorem REF : If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then for any tuple $(b_1,\\dots ,b_{k-2})$ , if $\\alpha =(a_0,b_1,\\dots ,b_{k-2})\\in S$ and $\\beta =(a_1,\\dots ,a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .","Symmetrically, if $\\alpha =(a_0,a_1,\\dots ,a_{k-2})\\in S$ and $\\beta =(b_1,\\dots ,b_{k-2},a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .","For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .","We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the previous three properties.","In the rest of the section we use Theorem REF to prove the following result.","Theorem 21 Given a $k$ -OV instance, we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, directed graph with $O(k n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(kn^{k-1}d^{k-1})$ edges that satisfies the following two properties.","If the $k$ -OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .","If the $k$ -OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 5k-7$ ."],["Construction of the graph","We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.","First we will construct a weighted graph and then we will make it unweighted.","Figure REF illustrates the construction of the graph for the special case $k=4$ .","Figure: The 3k-43k-4 vs 5k-75k-7 construction for the special case k=4k=4.","The edges between sets S,L 1 ,L 2 ,L 3 S,L_1,L_2,L_3 and TT are not depicted.","The matching between sets SS and S ' S^{\\prime } consists of unweighted paths of length k-2=2k-2=2.","The edges between sets SS and S '' S^{\\prime \\prime } consists of unweighted paths of length k-2=2k-2=2.","Similarly for the right side.We start by adding a set $S^{\\prime }$ of $n^{k-1}$ vertices.","$S^{\\prime }$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","We connect every $(a_0,a_1,\\ldots , a_{k-2}) \\in S^{\\prime }$ to its counterpart $(a_0,a_1,\\ldots , a_{k-2}) \\in S$ with an undirected edge of weight $k-2$ to form a matching.","We also add another set $S^{\\prime \\prime }$ of $n$ vertices.","$S^{\\prime \\prime }$ contains one vertex $a_0$ for every $a_0 \\in W_0$ .","For every pair of vertices in $S^{\\prime \\prime }$ we add an undirected edge of weight 1 between the vertices.","Thus, the $n$ vertices form a clique.","Furthermore, for every vertex $a_0 \\in S^{\\prime \\prime }$ we add an undirected edge of weight $k-2$ to $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .","Finally for every vertex $a_0 \\in S^{\\prime \\prime }$ we add a directed edge of weight 1 towards $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .","Some of the edges that we added have weight $k-2$ .","We make those unweighted by subdividing them into edges of weight 1.","Let $S^{\\prime \\prime \\prime }$ be the set of newly added vertices.","In total we added $O(kn^{k-1})$ vertices and $O(kn^{k-1})$ edges.","We do a similar construction for the set $T$ of vertices.","We add a set $T^{\\prime }$ of $n^{k-1}$ vertices — one vertex for every tuple $(a_1,\\dots ,a_{k-1})$ where for each $i$ , $a_i\\in W_i$ .","We connect every $(a_1,\\dots ,a_{k-1})\\in T^{\\prime }$ to $(a_1,\\dots ,a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ .","Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.","$T^{\\prime \\prime }$ contains one vertex for every vector $a_{k-1} \\in W_{k-1}$ .","We connect every pair of vertices in $T^{\\prime \\prime }$ by an undirected edge of weight 1.","We connect every vertex $a_{k-1} \\in T^{\\prime \\prime }$ to $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ for all $b_1,\\dots ,b_{k-2}$ .","Also, for every vertex $a_{k-1} \\in T^{\\prime \\prime }$ we add a directed edge of weight 1 from $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T^{\\prime }$ to $a_{k-1}$ for all $b_1,\\dots ,b_{k-2}$ .","Some of the edges that we just added have weight $k-2$ .","We make those unweighted by subdividing them into edges of weight 1.","Let $T^{\\prime \\prime \\prime }$ be the set of newly added vertices.","This finishes the construction of the graph.","In the rest of the section we show that the construction satisfies the promised two properties stated in Theorem REF ."],["Correctness of the construction","We need to consider two cases."],["Case 1: the $k$ -OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .","We consider subcases."],["Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_i$ for {{formula:341c2d42-7167-4db5-bd6a-4b4996ef5fb1}} and {{formula:b26c11ee-44d0-4a3f-8d06-2943a8aaf115}} for {{formula:b2ebe27d-5f57-42a2-b20b-50932506480e}}","We observe that there exists $s \\in S$ that has $d(u,s)\\le k-2$ .","Similarly, there exists $t \\in T$ with $d(t,v)\\le k-2$ .","By property 3 from Theorem REF we have that $d(s,t)\\le k$ .","This gives us upper bound $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le (k-2)+k+(k-2)=3k-4$ as required.","The proof when the sets for $u$ and $v$ are swapped is identical since we only use paths on unweighted edges."],["Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_1$","We note that there is some vertex $s\\in S^{\\prime \\prime }$ with $d(u,s)\\le 2(k-2)$ (via undirected edges).","Also, there is some vertex $s^{\\prime }\\in S^{\\prime \\prime }$ with $d(s^{\\prime },v)\\le k-1$ (possibly using directed edges).","$S^{\\prime \\prime }$ is a clique so $d(s,s^{\\prime })\\le 1$ .","Thus, $d(u,v)\\le d(u,s)+d(s,s^{\\prime })+d(s^{\\prime },v)\\le 2(k-2)+1+(k-1)=3k-4$ ."],["Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup T^{\\prime \\prime \\prime } \\cup L_{k-1}$","This case is similar to the previous case.","We note that there is some vertex $t\\in T^{\\prime \\prime }$ with $d(t,v)\\le 2(k-2)$ (via undirected edges).","Also, there is some vertex $t^{\\prime }\\in T^{\\prime \\prime }$ with $d(u,t^{\\prime })\\le k-1$ (possibly using directed edges).","$S^{\\prime \\prime }$ is a clique so $d(t^{\\prime },t)\\le 1$ .","Thus, $d(u,v)\\le d(u,t^{\\prime })+d(t^{\\prime },t)+d(t,v)\\le (k-1)+1+2(k-2)=3k-4$ ."],["Case 2: the $k$ -OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 5k-7$ .","Let $(a_0,a_1,\\ldots , a_{k-1})$ be a solution to the $k$ -OV instance where for each $i$ , $a_i\\in W_i$ .","We claim that $d((a_0,\\dots ,a_{k-2}) \\in S^{\\prime },(a_1,\\dots ,a_{k-1}) \\in T^{\\prime })\\ge 5k-7$ .","Let $P$ be an shortest path between $u=((a_0,\\dots ,a_{k-2}) \\in S^{\\prime })$ and $v=((a_1,\\dots ,a_{k-1}) \\in T^{\\prime })$ .","We want to show that $P$ uses at least $5k-7$ edges.","Let $s \\in S$ be the first vertex on path $P$ that belongs to $S$ and let $t \\in T$ be the last vertex from the set $T$ that is on path $P$ .","We observe that due to the directionality of the edges, $s$ and $t$ must be the counterparts of $u$ and $v$ respectively; that is, $s=((a_0,\\dots ,a_{k-2}) \\in S)$ and $t=((a_1,\\dots ,a_{k-1}) \\in T)$ .","Note that these definitions of $s$ and $t$ differ from the definitions of $s$ and $t$ in previous proofs.","We consider three subcases."],["Case 2.1: A vertex in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ appears after {{formula:6406b760-7ede-4dc4-b0f6-a59f46a09387}} on the path {{formula:a6c998cf-9396-4ea3-832a-1c6dca88cdb2}}","We observe that if $s_1,s_2\\in S$ is a pair of vertices on the path $P$ such that no vertex in $S$ appears between them on $P$ , then the portion of $P$ between $s_1$ and $s_2$ either contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ or contains no vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .","Let $s_1,s_2\\in S$ be such that the portion of $P$ between them contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .","Such $s_1,s_2$ exist by the specification of this case.","If $s_1=s_2$ then $P$ is not a shortest path.","Otherwise, the portion of $P$ between $s_1$ and $s_2$ must include a vertex in $S^{\\prime \\prime }$ .","Thus, $d(s_1,s_2)\\ge 2(k-2)$ .","We consider three subcases.","$s_1\\ne s$ .","The distance between any pair of vertices in $S$ is at least 2 so $d(s,s_1)\\ge 2$ .","Then, $d(u,v)\\ge d(u,s)+d(s,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2+2(k-2)+k+(k-2)=5k-6$ .","$s_1=s$ and $s_2=((a_0,b_1,\\dots ,b_{k-2}) \\in S)$ for some $b_1,\\dots ,b_{k-2}$ .","In this case, by property 5 we have $d(s_2,t)\\ge k+2$ .","Thus, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2(k-2)+(k+2)+(k-2)=5k-6$ .","$s_1=s$ and $s_2=((b_0,\\dots ,b_{k-2} \\in S)$ for some with $b_0\\ne a_0$ .","In this case, the path from $s_1$ to $s_2$ must include an edge in the clique $S^{\\prime \\prime }$ since these are the only edges among vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ for which adjacent tuples can differ with respect to their first element.","Thus, $d(s_1,s_2)\\ge 2(k-2)+1\\ge 2k-3$ .","Therefore, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+(2k-3)+k+(k-2)=5k-7$ ."],["Case 2.2: A vertex in $T^{\\prime }\\cup T^{\\prime \\prime }\\cup T^{\\prime \\prime \\prime }$ appears before {{formula:2e52092d-2023-4c53-a764-a41805329e43}} on the path {{formula:d86d9c20-8168-4be0-8d10-99eb7834ec04}}","This case is analogous to the previous case."],["Case 2.3: The portion of the path $P$ between {{formula:558fd34a-9236-4d24-8a2c-f5022f0a499a}} and {{formula:c0c71b02-54cb-42b0-8842-f210ddc962a7}} contains no vertices in {{formula:7c23b5b2-7499-4d20-8946-fe3d3dacb80d}}","By property 4, $d(s,t)\\ge 3k-2$ .","Thus, $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge (k-2)+(3k-2)+(k-2)=5k-6$ .","We note that a slight modification of this construction gives a lower bound for higher values of Diameter.","For any L, we can get an $L(3k-4)$ vs $L(5k-8)+1$ construction by subdividing all of the edges in the construction (into paths of length $L$ ) except for the directed edges and the edges within the cliques."],["2-Approximation for Eccentricities in $\\tilde{O}(m\\sqrt{n})$ time","Theorem 22 Given a weighted, directed $m$ edge $n$ node graph, in $\\tilde{O}(m\\sqrt{n})$ time we can output quantities $\\epsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have $\\epsilon (v)/2\\le \\epsilon ^{\\prime }(v)\\le \\epsilon (v)$ .","The algorithm is inspired by the 2-approximation algorithm for directed radius of Abboud, Vassilevska W., and Wang [7].","We claim that the following algorithm achieves the above guarantees.","Sample a random subset $S \\subset V$ of size $|S|=\\Theta (\\sqrt{n} \\log n)$ .","With high probability for every $u \\in V$ we have $N^{\\text{in}}_{\\sqrt{n}}(u)\\cap S \\ne \\emptyset $ .","Let $w$ be a vertex that maximizes $d(S,w)$ , which we find using Dijkstra's algorithm.","Let $S^{\\prime }:=N^{\\text{in}}_{\\sqrt{n}}(w)$ .","For every vertex $v \\in S^{\\prime }$ we output $\\epsilon ^{\\prime }(v)=\\epsilon (v)$ by running Dijkstra's algorithm and following the outgoing edges.","For every vertex $v \\notin S^{\\prime }$ we output the estimate $\\epsilon ^{\\prime }(v)=\\max _{s \\in S\\cup \\lbrace w\\rbrace }d(v,s)$ .","We can determine all these quantities by running Dijkstra's algorithm out of all vertices in $S \\cup \\lbrace w\\rbrace $ and following the incoming edges."],["Correctness","Consider an arbitrary vertex $v\\notin S^{\\prime }$ (if $v \\in S^{\\prime }$ , then we are done by the third step).","If there exists $s \\in S$ such that $d(v,s)\\ge \\epsilon (v)/2$ , then we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,s)\\ge \\epsilon (v)/2$ .","Otherwise, we have $d(v,s)<\\epsilon (v)/2$ for all $s \\in S$ .","Let $v^{\\prime }$ be a vertex that achieves $d(v,v^{\\prime })=\\epsilon (v)$ .","By the triangle inequality we have $d(s,v^{\\prime })>\\epsilon (v)/2$ for all $s \\in S$ .","Equivalently, $d(S,v^{\\prime })>\\epsilon (v)/2$ .","This implies that $d(S,w)>\\epsilon (v)/2$ by our choice of $w$ .","Since $d(S,w)>\\epsilon (v)/2$ and $S^{\\prime }=N^{\\text{in}}_{\\sqrt{n}}(w)$ intersects $S$ , we must have that $S^{\\prime }$ contains all vertices $u$ with $d(u,w)\\le \\epsilon (v)/2$ .","Since $v \\notin S^{\\prime }$ , we must have $d(v,w)>\\epsilon (v)/2$ and we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,w)>\\epsilon (v)/2$ ."],["Almost 2-Approximation for Eccentricities in almost linear time","In contrast to our $\\tilde{O}(m\\sqrt{n})$ time algorithm from the previous section, our near-linear time $(2+\\delta )$ -approximation algorithm is very different from all previously known algorithms.","Our algorithm proceeds in iterations and maintains a set $S$ of nodes for which we still do not have a good eccentricity estimate.","In each iteration either we get a good estimate for many new vertices and hence remove them from $S$ , or we remove all vertices from $S$ that have large eccentricities, and for the remaining nodes in $S$ we have a better upper bound on their eccentricities.","After a small number of iterations we have a good estimate for all vertices of the graph.","Theorem 23 Suppose that we are given a weighted, directed $m$ edge $n$ node graph.","The weights of all edge are non-negative integers bounded by $n^{O(1)}$ .","For any $1>\\tau >0$ we can in $\\tilde{O}(m/\\tau )$ time output quantities $\\varepsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have ${1-\\tau \\over 2}\\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ .","We maintain a subset $S\\subseteq V$ of vertices $v$ for which we still do not have an estimate $\\varepsilon ^{\\prime }(v)$ .","Initially $S=V$ and we will end with $|S|\\le O(1)$ .","When $|S|\\le O(1)$ we can evaluate $\\varepsilon (v)$ for all $v \\in S$ in the total time of $O(m)$ .","Also we maintain a value $D$ that upper bounds the largest eccentricity of a vertex in $S$ .","That is, $\\varepsilon (v)\\le D$ for all $v \\in S$ .","Initially we set $D=n^C$ for some large enough constant $C>0$ (we assume that the input graph is strongly connected).","The algorithm proceeds in phases.","Each phase takes $\\tilde{O}(m)$ time and either $|S|$ decreases by a factor of at least 2 or $D$ decreases by a factor of at least $1/(1-\\tau )$ .","After $O(\\log (n)/\\tau )$ phases either $|S|\\le O(1)$ or $D<1$ .","For a subset $S \\subseteq V$ of vertices and a vertex $x \\in V$ we define a set $S_x\\subseteq S$ to contain those $|S_x|=|S|/2$ vertices from $S$ that are closest to $x$ (according to distance $d(\\cdot ,x)$ ).","The ties are broken by taking the vertex with the smaller id.","Given a subset $S \\subseteq V$ of vertices and a threshold $D$ , a phase proceeds as follows.","[itemsep=0mm] We sample a set $A \\subseteq S$ of $O(\\log n)$ random vertices from the set $S$ .","With high probability for all $x \\in V$ we have $A \\cap S_x \\ne \\emptyset $ .","Let $w \\in V$ be a vertex that maximizes $d(A,w)$ .","We can find it using Dijkstra's algorithm.","We consider two cases.","Case $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ .","For all $x \\in S\\setminus S_w$ we have ${1-\\tau \\over 2} D \\le \\varepsilon (x)\\le D$ and we assign the estimate $\\varepsilon ^{\\prime }(x)={1-\\tau \\over 2}D$ .","This gives us that ${1-\\tau \\over 2} \\varepsilon (x)\\le {1-\\tau \\over 2}D=\\varepsilon ^{\\prime }(x)\\le \\varepsilon (x)$ for all $x\\in S\\setminus S_w$ .","We update $S$ to be $S_w$ .","This decreases the size of $S$ by a factor of 2 as required.","Case $d(S\\setminus S_w,w)<{1-\\tau \\over 2}D$ .","Set $S^{\\prime }=S$ .","For every vertex $v \\in S$ evaluate $r_v:=\\max _{x \\in A}d(v,x)$ .","We can evaluate these quantities by running Dijkstra's algorithm from every vertex in $A$ and following the incoming edges.","If $r_v\\ge {1-\\tau \\over 2}D$ , then assign the estimate $\\varepsilon ^{\\prime }(v)={1-\\tau \\over 2}D$ and remove $v$ from $S^{\\prime }$ .","Similarly as in the previous case we have ${1-\\tau \\over 2} \\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v \\in S\\setminus S^{\\prime }$ .","Below we will show that for every $v \\in S^{\\prime }$ we have $\\varepsilon (v)\\le (1-\\tau )D$ .","Thus we can update $S=S^{\\prime }$ and decrease the threshold $D$ to $(1-\\tau )D$ as required.","Correctness We have to show that, if there exists $v \\in S^{\\prime }$ such that $\\varepsilon (v)>(1-\\tau )D$ , then we will end up in the first case (this is the contrapositive of the claim in the second case).","Since $v \\in S^{\\prime }$ we must have that $d(v,x)\\le {1-\\tau \\over 2}D$ for all $x \\in A$ .","Since $\\varepsilon (v)>(1-\\tau )D$ , we must have that there exists $v^{\\prime }$ such that $d(v,v^{\\prime })>(1-\\tau )D$ .","By the triangle inequality we get that $d(x,v^{\\prime })>{1 - \\tau \\over 2}D$ for every $x \\in A$ .","By choice of $w$ we have $d(A,w)>{1-\\tau \\over 2}D$ .","Since $A \\cap S_w \\ne \\emptyset $ , we have $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ and we will end up in the first case.","The guarantee on the approximation factor follows from the description.","As a corollary, we get an algorithm for Source Radius with the same runtime and approximation ratio as Theorem REF .","First, run the Eccentricities algorithm and let $v$ be the vertex with the smallest estimated eccentricity $\\epsilon ^{\\prime }(v)$ .","Then run Dijkstra's algorithm from $v$ and report $\\epsilon (v)$ as the Radius estimate $R^{\\prime }$ .","Let $R$ be the true radius of the graph and let $x$ be the vertex with minimum Eccentricity i.e.","$\\epsilon (x)=R$ .","If $\\alpha $ is the approximation ratio for the Eccentricities algorithm then $\\epsilon (v)\\le \\alpha \\epsilon ^{\\prime }(v)\\le \\alpha \\epsilon (v)$ and $\\epsilon (x)\\le \\alpha \\epsilon ^{\\prime }(x)\\le \\alpha \\epsilon (x)$ .","By choice of $v$ , $\\epsilon ^{\\prime }(v)\\le \\epsilon ^{\\prime }(x)$ .","Thus, $\\alpha R=\\alpha \\epsilon (x)\\ge \\alpha \\epsilon ^{\\prime }(x)\\ge \\alpha \\epsilon ^{\\prime }(v)\\ge \\epsilon (v)=R^{\\prime }$ .","Clearly $R^{\\prime }\\ge R$ , so $R\\le R^{\\prime }\\le \\alpha R$ .","$S$ -$T$ Diameter algorithms Recall that the $S$ -$T$ diameter problem is as follows: Given an undirected graph $G=(V,E)$ and two sets $S\\subseteq V, T\\subseteq V$ , determine $D_{S,T}=\\max _{s\\in S, t\\in T} d(s,t)$ .","Here we will outline two algorithms for the problem.","Let us first consider a fast 3-approximation algorithm.","Claim 24 There is an $O(m+n)$ time algorithm that for any $n$ node $m$ edge graph $G=(V,E)$ and $S\\subseteq V, T\\subseteq V$ , computes an estimate $D^{\\prime }$ such that $D_{S,T}/3\\le D^{\\prime }\\le D_{S,T}$ and two nodes $s\\in S$ , $t\\in T$ such that $d(s,t)=D^{\\prime }$ .","In graphs with nonnegative weights, the same estimate can be achieved in $O(m+n\\log n)$ time.","The algorithm is extremely simple: pick arbitrary nodes $s\\in S$ and $t\\in T$ , compute BFS($s$ ) and BFS($t$ ) and return $\\max \\lbrace \\max _{t^{\\prime }\\in T} d(s,t^{\\prime }),\\max _{s^{\\prime }\\in S} d(s^{\\prime },t)\\rbrace $ (also returning the two nodes achieving the maximum).","For weighted graphs, run Dijkstra's algorithm instead of BFS.","Let's see why this algorithm provides the promised guarantee.","Suppose that for every $t^{\\prime }\\in T$ , $d(s,t^{\\prime })d$ .","Let $a$ be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d$ .","We thus have that $a\\in Y$ .","Also, since $d(\\bar{t},s^*)\\le D/2$ , $d(a,s^*)\\le D/2-d$ and hence $d(a,s_a)\\le D/2-d$ , so that $d(s_a,t^*)\\ge D-2(D/2-d)\\ge 2d$ .","This finishes the argument that 2-Approx returns an estimate $D^{\\prime }$ with $2\\lfloor D/4\\rfloor \\le D^{\\prime }\\le D$ .","It is not too hard to see that the only time that we might get an estimate that is less than $D/2$ is in the last part of the argument and only if the diameter is of the form $4d+3$ .","(We will prove the algorithm guarantees formally soon.)","The analysis fails to work in that case because $Y$ is guaranteed to contain only the nodes at distance $d$ from $\\bar{t}$ .","In particular, if $Y$ contains all nodes at distance $d+1$ from $\\bar{t}$ instead of just those at distance at most $d$ , we could consider $a$ to be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d+1$ , and $a\\in Y$ .","Now since $d(\\bar{t},s^*)\\le 2d+1$ (as otherwise we'd be done), $d(a,s_a)\\le d(a,s^*)\\le 2d+1-d-1 =d$ , so that $d(s_a,t^*)\\ge 2d+3$ .","Hence everything would work out.","We handle this issue with a trick from Chechik et al. [21].","First, we make graph have constant degree by blowing up the number of nodes and adding 0 weight edges as follows.","Let $v$ be an original node and suppose it has degree $d(v)$ .","Replace $v$ with a $d(v)$ -cycle of 0 weight edges so that each of the cycle nodes is connected to a one of the neighbors of $v$ , where each neighbor has a cycle node corresponding to it.","This makes every node have degree 3 and increases the number of vertices to $O(m)$ .","Now, we run algorithm 2-Approx with two changes.","The first is that instead of BFS we use Dijkstra's algorithm because the edges now have weights.","The second change is that we redefine $Y$ as follows.","Let $Z$ be the closest $\\sqrt{m}$ nodes $Z$ to $\\bar{t}$ .","Define $Y$ to be $Z$ , together with all nodes that have an edge of weight 1 to some node of $Z$ .","Now, consider the shortest path $P$ between $\\bar{t}$ and $s^*$ .","Consider the last node $a$ of $P$ (in the direction from $\\bar{t}$ to $s^*$ ) that has distance $\\le d$ from $\\bar{t}$ .","The node $a^{\\prime }$ after $a$ on $P$ must be connected to $a$ by an edge $(a,a^{\\prime })$ of weight 1, as otherwise $a^{\\prime }$ would be at distance at most $d$ from $\\bar{t}$ .","Since $a\\in Z$ , we must have $a^{\\prime }\\in Y$ and we know also that $d(\\bar{t},a^{\\prime })=d+1$ .","In the last case when $d(\\bar{t},s^*)\\le 2d+1$ , we get that $Y$ contains a node $a^{\\prime }$ of distance $\\le (2d+1)-(d+1) = d$ from $s^*$ and hence $d(a^{\\prime },s_{a^{\\prime }})\\le d$ and hence $d(s_{a^{\\prime }},t^*)\\ge 2d+3$ .","Since every node has degree 3, the number of nodes in $Y$ is $\\le 4|Z|\\le O(\\sqrt{m})$ and hence we can afford to run Dijkstra from each of them and complete the algorithm in $\\tilde{O}(m^{3/2})$ time.","Let us now formally analyze the guarantees of the algorithm.","Suppose that $D=4d+z$ where $z\\in \\lbrace 0,1,2,3\\rbrace $ ; we will show that the estimate that the algorithm gives is always at least $\\lceil D/2\\rceil =2d+\\lceil z/2\\rceil $ .","If some node $x\\in X$ has $d(x,t^*)\\le d$ , we get that $d(t_x,s^*)\\ge D-2d=2d+z\\ge \\lceil D/2\\rceil $ .","If we are not done, all nodes of $X$ have $d(x,\\bar{t})\\ge d(x,t^*)\\ge d+1$ and $Z$ contains all nodes at distance $\\le d$ from $\\bar{t}$ .","If $s^*\\in Z$ , we are done so we must have $d(s^*,\\bar{t})\\ge d+1$ .","Consider the last node $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path (in the direction towards $s^*$ ) for which $d(\\bar{t},a^{\\prime })\\le d$ .","We have that $a^{\\prime }\\in Z$ .","Also, the node $a$ after $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path must be in $Y$ since by the choice of $a^{\\prime }$ , $d(\\bar{t},a)=t+1$ and $(a,a^{\\prime })$ is an edge of weight 1.","If $d(\\bar{t},s^*)\\ge 2d+\\lceil z/2\\rceil $ , we are done, so we get that $d(a,s^*)\\le 2d+\\lceil z/2\\rceil -1 - (d+1) =d+\\lceil z/2\\rceil -2$ .","Hence, $d(s_a,t^*)\\ge D-2(d+\\lceil z/2\\rceil -2)=2d + (z+4-2\\lceil z/2\\rceil )\\ge 2d+z$ .","It is not hard to extend the $S$ -$T$ Diameter algorithms to work for weighted undirected graphs as well.","The basic idea is to use Dijkstra's algorithm instead of BFS in Algorithm 1.","This gives an almost 2-approximation.","In particular, let $a^{\\prime }$ be the last node on the $\\bar{t}$ to $s^*$ shortest path that is at distance $\\le d$ from $\\bar{t}$ and let $a$ be the node after $a^{\\prime }$ .","The weighted version of Algorithm 1 achieves a guarantee $D^{\\prime }$ of the $S$ -$T$ Diameter, such that $D/2-2w(a,a^{\\prime })\\le D^{\\prime }\\le D$ .","We can obtain an $\\tilde{O}(m^{3/2})$ true 2-approximation algorithm similar to the proof above.","Let $Z$ be the closest $\\sqrt{m}$ nodes to $\\bar{t}$ and extend $Z$ to $Y$ by adding all nodes that have a non-zero edge to a node of $Z$ .","With this modification, the node $a$ is guaranteed to be in $Y$ and hence the estimate for the diameter is at least $D/2$ .","Corollary 26 In $\\tilde{O}(m\\sqrt{n})$ time one can obtain an estimate $D^{\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node undirected graph with nonnegative edge weights such that $D/2 - 2w(a,a^{\\prime }) \\le D^{\\prime }\\le D$ for some edge $(a,a^{\\prime })$ .","In $\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\prime \\prime }$ such that $D/2\\le D^{\\prime \\prime }\\le D$ .","It is an easy exercise to see that when $D=2h+1$ then the value $\\max \\lbrace \\epsilon ^{in}(v), \\epsilon ^{out}(v) \\rbrace $ of an arbitrary vertex $v \\in V$ is an estimation to the diameter which is at least $h+1$ and at most $D$ .","In this section we present a deterministic algorithm that gets a directed unweighted graph $G$ with $D=2h$ and computes in $O(m^2/n)$ time an estimation $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .","The algorithm works as follows.","A variable $\\hat{D}$ is set to zero.","The algorithm searches for a vertex $v$ that its sum of in and out degree is minimal.","Then the algorithm computes the in and out eccentricity of $v$ and every vertex that has an edge with $v$ (incoming or outgoing).","The algorithm outputs the maximum of all the in and out eccentricities that were computed.","Theorem 27 Let $G=(V,E)$ an unweighted directed graph and let $D=2h$ .","Algorithm REF returns in $O(m^2/n)$ time an estimate $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .","We start with the running time analysis.","Consider the graph $G$ and ignore the edge directions.","For every $v\\in V$ let $deg(v) = deg^{in}(v)+deg^{out}(v)$ .","Since $m = \\frac{1}{2}\\sum _{v\\in V} deg(v)$ a vertex $v$ of minimal degree satisfies $deg(v)\\le 2m/n$ .","Therefore, the cost of computing in and out eccentricities for all vertices in the set $N(v) \\cup \\lbrace v \\rbrace $ is $O(\\frac{m}{n} \\times m)$ .","We now turn to bound $\\hat{D}$ .","Let $a, b\\in V$ and let $d(a,b)=2h$ .","Let $P(a,b)$ be a shortest path from $a$ to $b$ and let $v\\in P(a,b)$ .","If $d(a,v)\\le h-1$ then $\\epsilon ^{out}(v)\\ge h+1$ .","Similarly, if $d(v,b)\\le h-1$ then $\\epsilon ^{in}(v)\\ge h+1$ .","Consider now the case that $d(a,v)=h$ and $d(v,b)=h$ .","Let $u\\in P(a,b)$ be the vertex that precedes $v$ on the shortest path from $a$ to $b$ .","Since $u$ has an incoming edge to $v$ it follows that $u\\in N(v)$ and $\\epsilon ^{out}(u)$ and $\\epsilon ^{in}(u)$ are computed.","Since $d(a,v)=h$ it follows that $d(a,u)=h-1$ , $\\epsilon ^{out}(u)\\ge h+1$ and $\\hat{D}$ is at least $h+1$ .","Fast approximation of the diameter [1] Diam-Approx$G$ $\\hat{D} = 0$ $v = \\arg \\min _{x\\in V} deg^{in}(x)+deg^{out}(x)$ every $w \\in N^{\\text{in}}(v) \\cup N^{\\text{out}}(v) \\cup \\lbrace v \\rbrace $ compute $\\epsilon ^{in}(w)$ and $\\epsilon ^{out}(w)$ $\\hat{D}= \\max \\lbrace \\hat{D},\\epsilon ^{in}(w),\\epsilon ^{out}(w)\\rbrace $ $\\hat{D}$ Algorithms for dense graphs Algorithm overview The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.","[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.","Therefore, it is interesting to consider the opposite question to the one considered by [41].","Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?","We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.","For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .","Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.","As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.","To enable this approach we can no longer sample $A$ naively.","Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.","The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .","We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.","The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w) 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .","For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .","Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .","They did not provide the details and refer the reader to [48].","However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.","The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .","The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .","This can be done as follows.","Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .","The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).","However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.","To this end, more ideas are needed.","The following simple observation helps us to achieve our goal.","Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .","Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .","For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .","It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .","It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].","This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.","It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.","see [48]).","The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.","Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .","In more detail, our algorithm works as follows.","For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.","Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .","Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .","Finally, it computes the clusters and $W$ using the bunches as we described above.","The rest of the algorithm is almost identical to Algorithm REF .","The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .","The pseudo-code is given in Algorithm REF .","New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .","$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .","compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .","every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .","The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].","The cost of computing $A$ is $O(n/p)$ time [48].","Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .","To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .","As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .","Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .","This completes the analysis of the part that precedes the while loop.","Next, we analyze the cost of the while loop.","Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.","From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .","One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1}) h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .","If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).","Assume, wlog, that $d(a,A)\\le h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .","We now turn to we analyze the running time of Algorithm REF .","Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .","The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .","From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.","For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .","Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .","Step 2 takes $O(n^2)$ time as well.","In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.","Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.","We also compute $|A|$ shortest paths trees in $H$ .","As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time.","Near linear almost 5/3-approximation for Eccentricities almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.","We run lines 2-11 of Algorithm REF .","The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .","The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .","The pseudo-code is given in Algorithm REF .","We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.","Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.","The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.","Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .","We now turn to bound $\\epsilon ^{\\prime }(u)$ .","In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .","It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .","Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .","We distinguish between two cases.","Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .","It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .","From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.","Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .","Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .","Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .","Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .","From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .","As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .","Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .","As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .","Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .","This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .","Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .","Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .","If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .","Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .","As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .","Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .","From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows.","Algorithms for dense graphs using matrix multiplication Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.","The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.","In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].","To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].","The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .","After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.","The main idea of our algorithms is as follows.","If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .","Small distances are easy to compute with matrix multiplication.","Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .","Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].","If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.","[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .","The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.","A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.","Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].","Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .","Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .","We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .","Let us compare to our $O(n^2\\log n)$ time algorithms.","For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .","The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .","For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .","We will rely on two known algorithms.","The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).","Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .","The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .","The algorithm uses fast matrix multiplication and is quite straightforward.","Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.","$A$ is the adjacency matrix added to the identity matrix.","Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .","For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.","Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .","Define $A^0$ as the identity matrix.","Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).","Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .","Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .","Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .","Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.","Armed with these two algorithms, let us recap Roditty et al.","'s (and Cairo et al.","'s) approximation algorithm and see how to modify it.","The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.","RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .","Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .","Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .","The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .","Let us describe how to modify the algorithm.","We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.","Our Modified Approximation.","[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .","Let $X=3a\\log n$ .","Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .","For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .","Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .","For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .","Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.","Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .","Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .","[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .","Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .","[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .","Consider our modified algorithm, FasterApproximation.","Now we will prove several claims.","Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .","The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.","Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.","Recall that $X+a\\log n = O(\\log n)$ .","Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.","Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .","Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.","The rest of the steps run in linear time.","Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .","Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","Suppose that $\\tilde{D}_1\\ge X$ .","The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .","By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .","Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .","Suppose now that $\\tilde{D}_1< X$ .","This means that $D100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .","On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."],["Algorithm overview","The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.","[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.","Therefore, it is interesting to consider the opposite question to the one considered by [41].","Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?","We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.","For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .","Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.","As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.","To enable this approach we can no longer sample $A$ naively.","Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.","The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .","We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.","The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w) 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .","For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .","Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .","They did not provide the details and refer the reader to [48].","However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.","The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .","The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .","This can be done as follows.","Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .","The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).","However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.","To this end, more ideas are needed.","The following simple observation helps us to achieve our goal.","Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .","Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .","For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .","It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .","It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].","This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.","It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.","see [48]).","The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.","Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .","In more detail, our algorithm works as follows.","For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.","Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .","Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .","Finally, it computes the clusters and $W$ using the bunches as we described above.","The rest of the algorithm is almost identical to Algorithm REF .","The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .","The pseudo-code is given in Algorithm REF .","New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .","$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .","compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .","every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .","The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].","The cost of computing $A$ is $O(n/p)$ time [48].","Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .","To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .","As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .","Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .","This completes the analysis of the part that precedes the while loop.","Next, we analyze the cost of the while loop.","Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.","From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .","One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1}) h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .","If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).","Assume, wlog, that $d(a,A)\\le h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .","We now turn to we analyze the running time of Algorithm REF .","Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .","The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .","From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.","For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .","Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .","Step 2 takes $O(n^2)$ time as well.","In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.","Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.","We also compute $|A|$ shortest paths trees in $H$ .","As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time."],["Near linear almost 5/3-approximation for Eccentricities","almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.","We run lines 2-11 of Algorithm REF .","The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .","The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .","The pseudo-code is given in Algorithm REF .","We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.","Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.","The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.","Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .","We now turn to bound $\\epsilon ^{\\prime }(u)$ .","In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .","It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .","Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .","We distinguish between two cases.","Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .","It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .","From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.","Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .","Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .","Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .","Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .","From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .","As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .","Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .","As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .","Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .","This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .","Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .","Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .","If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .","Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .","As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .","Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .","From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows."],["Algorithms for dense graphs using matrix multiplication","Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.","The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.","In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].","To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].","The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .","After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.","The main idea of our algorithms is as follows.","If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .","Small distances are easy to compute with matrix multiplication.","Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .","Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].","If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.","[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .","The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.","A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.","Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].","Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .","Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .","We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .","Let us compare to our $O(n^2\\log n)$ time algorithms.","For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .","The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .","For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .","We will rely on two known algorithms.","The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).","Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .","The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .","The algorithm uses fast matrix multiplication and is quite straightforward.","Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.","$A$ is the adjacency matrix added to the identity matrix.","Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .","For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.","Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .","Define $A^0$ as the identity matrix.","Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).","Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .","Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .","Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .","Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.","Armed with these two algorithms, let us recap Roditty et al.","'s (and Cairo et al.","'s) approximation algorithm and see how to modify it.","The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.","RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .","Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .","Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .","The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .","Let us describe how to modify the algorithm.","We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.","Our Modified Approximation.","[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .","Let $X=3a\\log n$ .","Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .","For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .","Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .","For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .","Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.","Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .","Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .","[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .","Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .","[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .","Consider our modified algorithm, FasterApproximation.","Now we will prove several claims.","Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .","The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.","Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.","Recall that $X+a\\log n = O(\\log n)$ .","Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.","Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .","Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.","The rest of the steps run in linear time.","Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .","Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","Suppose that $\\tilde{D}_1\\ge X$ .","The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .","By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .","Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .","Suppose now that $\\tilde{D}_1< X$ .","This means that $D100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .","On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."]],"string":"[\n [\n \"Towards Tight Approximation Bounds for Graph Diameter and Eccentricities\"\n ],\n [\n \"Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.\",\n \"There are no known very efficient algorithms for computing the Diameter exactly.\",\n \"Thus, much research has been devoted to how fast this parameter can be approximated.\",\n \"Chechik et al.\",\n \"showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\\\\tilde{O}(m^{3/2})$ time.\",\n \"Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\\\\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs.\",\n \"Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$.\",\n \"It was, however, completely plausible that a $3/2$-approximation is possible in linear time.\",\n \"In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\\\\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$.\",\n \"Another fundamental set of graph parameters are the Eccentricities.\",\n \"The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$.\",\n \"Chechik et al.\",\n \"showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\\\\tilde{O}(m^{3/2})$ time and Abboud et al.\",\n \"showed that no $O(n^{2-\\\\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs.\",\n \"We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH.\",\n \"We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\\\\delta$ factor in $\\\\tilde{O}(m/\\\\delta)$ time for any $0<\\\\delta<1$.\",\n \"This beats all Eccentricity algorithms in Cairo et al.\"\n ],\n [\n \"Introduction\",\n \"Among the most important graph parameters are the graph's Diameter and the Eccentricities of its vertices.\",\n \"The Eccentricity of a vertex $v$ is the (shortest path) distance to the furthest vertex from $v$ , and the Diameter is the largest Eccentricity over all vertices in the graph.\",\n \"The Eccentricities and Diameter measure how fast information can spread in networks.\",\n \"Efficient algorithms for their computation are highly desired (see e.g.\",\n \"[40], [9], [33]).\",\n \"Unfortunately, the fastest known algorithms for these parameters are very slow on large graphs.\",\n \"For unweighted graphs on $n$ vertices and $m$ edges, the fastest Diameter algorithm runs in $\\\\tilde{O}(\\\\min \\\\lbrace mn,n^\\\\omega \\\\rbrace )$ $\\\\tilde{O}$ notation hides polylogarithmic factors time [16] where $\\\\omega <2.373$ is the exponent of square matrix multiplication [52], [32], [45].\",\n \"For weighted graphs, the fastest Eccentricity and Diameter algorithms actually compute all distances in the graph, i.e.\",\n \"they solve the All-Pairs Shortest Paths (APSP) problem.\",\n \"The fastest known algorithms for APSP in weighted graphs run in $\\\\min \\\\lbrace \\\\tilde{O}(mn),n^3/\\\\exp (\\\\sqrt{\\\\log n})\\\\rbrace $ [53], [35], [37].\",\n \"Whether one can solve Diameter faster than APSP is a well-known open problem (e.g.\",\n \"see Problem 6.1 in [20] and [2], [17]).\",\n \"Whether one can solve Eccentricities faster than APSP was addressed by [50] (for dense graphs) and by [34] (for sparse graphs).\",\n \"Vassilevska W. and Williams [50] showed that Eccentricities and APSP are equivalent under subcubic reductions, so that either both of them admit $O(n^{3-\\\\varepsilon })$ time algorithms for $\\\\varepsilon >0$ , or neither of them do.\",\n \"Lincoln et al.\",\n \"[34] proved that under a popular conjecture about the complexity of weighted Clique, the $O(mn)$ runtime for Eccentricities cannot be beaten by any polynomial factor for any sparsity of the form $m=\\\\Theta (n^{1+1/k})$ for integer $k$ .\",\n \"Due to the hardness of exact computation, efficient approximation algorithms are sought.\",\n \"A folklore $\\\\tilde{O}(m+n)$ time algorithm achieves a 2-approximation for Diameter in directed weighted graphs and a 3-approximation for Eccentricities in undirected weighted graphs.\",\n \"Aingworth et al.\",\n \"[2] presented an almost-$3/2$ approximation An almost-$c$ approximation of $X$ is an estimate $X^{\\\\prime }$ so that $X\\\\le X^{\\\\prime }\\\\le cX+O(1)$ .\",\n \"algorithm for Diameter running in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] improved the result of [2] with an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"Chechik et al.\",\n \"[21] obtained a (genuine) $3/2$ approximation algorithm for Diameter (in directed graphs) and a (genuine) $5/3$ -approximation algorithm for Eccentricities (in undirected graphs), running in $\\\\tilde{O}(\\\\min \\\\lbrace m^{3/2},mn^{2/3}\\\\rbrace )$ time.\",\n \"These are the only known non-trivial approximation algorithms for Diameter in directed graphs.\",\n \"So far, there are no known faster than $mn$ algorithms for approximating Eccentrities in directed graphs within any constant factor.\",\n \"Cairo et al.\",\n \"[15] generalized the above results for undirected graphs and obtained a time-approximation tradeoff: for every $k\\\\ge 1$ they obtained an $\\\\tilde{O}(mn^{1/(k+1)})$ time algorithm that achieves an almost-$2-1/2^k$ approximation for Diameter and an almost $3-4/(2^k+1)$ -approximation for Eccentricities.\"\n ],\n [\n \"Our contributions.\",\n \"We address the following natural question: Main Question: Are the known approximation algorithms for Diameter and Eccentricities optimal?\",\n \"A partial answer is known.\",\n \"Under the Strong Exponential Time Hypothesis (SETH), every $3/2-\\\\varepsilon $ approximation algorithm (for $\\\\varepsilon >0$ ) for Diameter in undirected unweighted graphs with $O(n)$ nodes and edges must use $n^{2-o(1)}$ time [41].\",\n \"Similarly, every $5/3-\\\\varepsilon $ approximation algorithm for the Eccentricities of undirected unweighted graphs with $O(n)$ nodes and edges must use use $n^{2-o(1)}$ time [7].\",\n \"This however does not answer the question of whether the runtimes of the known $3/2$ and $5/3$ approximation algorithms can be improved.\",\n \"It is completely plausible that there is a $3/2$ -approximation algorithm for Diameter or a $5/3$ -approximation for Eccentricities running in linear time.\",\n \"We address our Main Question for both sparse and dense graphs.\",\n \"Our results are shown in Table REF .\",\n \"Table: Our results.\",\n \"All of the lower bounds hold even for sparse graphs.\",\n \"SS-TT Diameter is a variant of Diameter introduced later in this section.\"\n ],\n [\n \"Sparse graphs.\",\n \"Our first result (restated as Theorem REF ) regards approximating Diameter in undirected unweighted sparse graphs.\",\n \"Theorem 1 ($3/2$ -Diameter Approx.\",\n \"is Tight) Under SETH, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $8/5-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Diameter of an undirected unweighted sparse graph.\",\n \"In particular, any $3/2$ -approximation algorithm in sparse graphs must take $n^{3/2-o(1)}$ time.\",\n \"Hence the $\\\\tilde{O}(m^{3/2})$ time $3/2$ -approximation algorithm of [41], [21] is optimal in two ways: improving the approximation ratio to $3/2-\\\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([41]) and improving the runtime to $O(m^{3/2-\\\\delta })$ causes an approximation ratio blow-up to $8/5$ .\",\n \"Our lower bound instance says that in $O(m^{3/2-\\\\delta })$ time one cannot return 6 when the Diameter is 8.\",\n \"One may be tempted to extend the above lower bound, by showing that, say, in $O(m^{4/3-\\\\delta })$ time one cannot even return 5 when the Diameter is 8.\",\n \"This approach, however fails: in Theorem REF we give an $O(m^2/n)$ time algorithm that does return 5 in this case, and in general when the Diameter is $2h$ , it returns at least $h+1$ .\",\n \"Notice that when the Diameter is $2h$ , the folklore linear time algorithm returns an estimate of only $h$ .\",\n \"Hence for sparse graphs, our algorithm runs in linear time and outperforms the folklore algorithm.\",\n \"Also, for constant even Diameter, it gives a better than 2 approximation.\",\n \"We obtain stronger Diameter hardness results for weighted graphs and for directed unweighted graphs.\",\n \"In particular, assuming SETH: For weighted sparse graphs, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $5/3-\\\\varepsilon $ Diameter approximation (for $\\\\varepsilon >0$ ) (Theorem REF ).\",\n \"For directed unweighted sparse graphs, using a general time-accuracy tradeoff lower bound (Theorem REF ), we show that no near-linear time algorithm can achieve an approximation factor better than $5/3$ .\",\n \"We summarize our Diameter lower bounds and compare them to the known upper bounds in Figure REF .\",\n \"Figure: Our hardness results for Diameter.\",\n \"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.\",\n \"Black lines represent lower bounds.\",\n \"Black dots represent existing algorithms.\",\n \"Blue dots represent existing algorithms whose approximation is potentially off by an additive term (the algorithms of ).\",\n \"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.We address our Main Question for Eccentricities as well.\",\n \"Our main result for Eccentricities is Theorem REF .\",\n \"Its first consequence is as follows: Theorem 2 ($5/3$ -Eccentricities Alg.\",\n \"is Tight) Under SETH, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $9/5-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.\",\n \"In other words, the $\\\\tilde{O}(m^{3/2})$ time $5/3$ -approximation algorithm of [41], [21] is tight in two ways.\",\n \"Improving the approximation ratio to $5/3-\\\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([7]) and improving the runtime to $O(m^{3/2-\\\\delta })$ causes an approximation ratio blow-up to $9/5$ .\",\n \"More generally, we prove (in Theorem REF ): for every $k\\\\ge 2$ , under SETH, distinguishing between Eccentricities $2k-1$ and $4k-3$ in unweighted undirected sparse graphs requires $n^{1+1/(k-1)-o(1)}$ time.\",\n \"Thus, no near-linear time algorithm can achieve a $2-\\\\varepsilon $ -approximation for Eccentricities for $\\\\varepsilon >0$ .\",\n \"The best (folklore) near-linear time approximation algorithm for Eccentricities currently only achieves a 3-approximation, and only in undirected graphs.\",\n \"There is no known constant factor approximation algorithm for directed graphs!\",\n \"Is our limitation result for linear time Eccentricity algorithms far from the truth?\",\n \"We show that our lower bound result is essentially tight, for both directed and undirected graphs by producing the first non-trivial near-linear time approximation algorithm for the Eccentricities in weighted directed graphs (Theorem REF ).\",\n \"Theorem 3 (2-Approx.\",\n \"for Eccentricities in near-linear time.)\",\n \"Under SETH, no $n^{1+o(1)}$ time algorithm can output a $2-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.\",\n \"For every $\\\\delta >0$ , there is an $\\\\tilde{O}(m/\\\\delta )$ time algorithm that produces a $(2+\\\\delta )$ -approximation for the Eccentricities of any directed weighted graph.\",\n \"The approximation hardness result is the first result within fine-grained complexity that gives tight hardness for near linear time algorithms.\",\n \"The $2+\\\\delta $ approximation ratio that our algorithm produces beats all approximation ratios for Eccentricities given by Cairo et al. [15].\",\n \"It also constitutes the first known constant factor approximation algorithm for Eccentricities in directed graphs.\",\n \"Our approximation algorithm also implies as a corollary an approximation algorithm for the Source Radius problemThe Source Radius problem is a natural extension of the undirected Radius definition.\",\n \"The goal is to return $\\\\min _x\\\\max _vd(x,v)$ .\",\n \"studied in [7] with the same runtime and approximation factor ($2+\\\\delta $ ).\",\n \"Abboud et al.\",\n \"[7] showed that, under the Hitting Set Conjecture, any $(2-\\\\varepsilon )$ -approximation algorithm (for $\\\\varepsilon >0$ ) for Source Radius requires $n^{2-o(1)}$ time, and hence our Source Radius algorithm is also essentially tight.\",\n \"Our lower bound in Theorem REF holds already for undirected unweighted graphs, and the upper bound works even for directed weighted graphs.\",\n \"The algorithm produces a $(2+\\\\delta )$ -approximation, which while close, is not quite a 2-approximation.\",\n \"We design (in Theorem REF ) a genuine 2-approximation algorithm running in $\\\\tilde{O}(m\\\\sqrt{n})$ time that also works for directed weighted graphs.\",\n \"We then complement it (in Theorem REF ) with a tight lower bound under SETH: in sparse directed graphs, if you go below factor 2 in the accuracy, the runtime blows up to quadratic.\",\n \"Theorem 4 (Tight 2-Approx.\",\n \"for Eccentricities) Under SETH, no $n^{2-\\\\delta }$ time algorithm for $\\\\delta >0$ can output a $2-\\\\varepsilon $ approximation for the Eccentricities of a directed unweighted sparse graph.\",\n \"There is an $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithm that produces a 2-approximation for the Eccentricities of any directed weighted graph.\",\n \"We thus give an essentially complete answer to our Main Question for Eccentricities.\",\n \"Our results are summarized in Figures REF and REF .\",\n \"Figure: Our algorithms and hardness results for Eccentricities.\",\n \"The lower bounds are for unweighted graphs and the upper bounds are for weighted graphs.\",\n \"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.\",\n \"Black lines represent lower bounds.\",\n \"Black dots represent existing algorithms (including our algorithm at (2,3/2)(2,3/2) in figure b).\",\n \"Blue dots represent existing algorithms whose position may not be exactly as it appears in the figure.\",\n \"Here, the blue dots represent our (2+δ)(2+\\\\delta )-approximation algorithm running in O ˜(m/δ)\\\\tilde{O}(m/\\\\delta ) time.\",\n \"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.Our conditional lower bounds for both Diameter and Eccentricities are all based on a common construction: a conditional lower bound for a problem called $S$ -$T$ Diameter.\",\n \"In $S$ -$T$ Diameter, the input is a graph $G=(V,E)$ and two subsets $S,T\\\\subseteq V$ , not necessarily disjoint, and the output is $D_{S,T}:=\\\\max _{s\\\\in S,t\\\\in T} d(s,t)$ .\",\n \"$S$ -$T$ Diameter is a problem of independent interest.\",\n \"It is related to the bichromatic furthest pair problem studied in geometry (e.g.\",\n \"as in [29]), but for graphs (if we set $T=V\\\\setminus S$ ).\",\n \"It is easy to see that if one can compute the $S$ -$T$ Diameter, then one can also compute the Diameter in the same time: just set $S=T=V$ .\",\n \"We show that actually, when it comes to exact computation, the $S$ -$T$ Diameter and Diameter in weighted graphs are computationally equivalent (Theorem REF ).\",\n \"We show that $S$ -$T$ Diameter also has similar approximation algorithms to Diameter.\",\n \"We give a 3-approximation running in linear time (Claim REF based on the folklore Diameter 2-approximation algorithm), and a 2-approximation running in $\\\\tilde{O}(m^{3/2})$ time (Theorem REF based on the $3/2$ -approximation algorithm of [41], [21]).\",\n \"We prove the following lower bound for $S$ -$T$ Diameter (restated as Theorem REF ), the proof of which is the starting point for all of our conditional lower bounds.\",\n \"Theorem 5 Under SETH, for every $k\\\\ge 2$ , every algorithm that can distinguish between $S$ -$T$ Diameter $k$ and $3k-2$ in undirected unweighted graphs requires $n^{1+1/(k-1)-o(1)}$ time.\",\n \"Theorem REF implies that under SETH, our aforementioned 2 and 3-approximation algorithms are optimal.\",\n \"For all of our lower bounds, we also address the question of whether they can be extended to higher values of Diameter and Eccentricities.\",\n \"All of our lower bounds, with the exception of directed Eccentricities, are of the form “any algorithm that can distinguish between Diameter (or Eccentricity) $a$ and $b$ requires a certain amount of time\\\" for small values of $a$ and $b$ .\",\n \"This doesn't exclude the possibility of an algorithm that distinguishes between higher Diameters (or Eccentricities) of the same ratio i.e.\",\n \"between $a\\\\ell $ and $b\\\\ell $ for some $\\\\ell $ .\",\n \"For weighted Diameter, our lower bound easily extends to higher values of Diameter by simply scaling up the edge weights.\",\n \"For $S-T$ Diameter and undirected Eccentricities, our lower bounds easily extend to higher values of Diameter and Eccentricities by simply subdividing the edges.\",\n \"For unweighted directed Diameter, our lower bound extends to higher values of Diameter with a slight loss in approximation factor by subdividing some of the edges.\",\n \"For unweighted undirected Diameter, our lower bound does not seem to easily extend to higher values of Diameter.\"\n ],\n [\n \"Dense graphs.\",\n \"Can we address our Main Question for dense graphs as well?\",\n \"In particular, can we extend our runtime lower bounds of the form $n^{1+1/\\\\ell -o(1)}$ to $mn^{1/\\\\ell -o(1)}$ , thus matching the known algorithms for larger values of $m$ ?\",\n \"We show that the answer is “no”.\",\n \"For undirected unweighted graphs, we obtain $\\\\tilde{O}(n^2)$ time algorithms for Diameter achieving an almost $3/2$ -approximation (Theorem REF ), and for all Eccentricities achieving an almost $5/3$ -approximation algorithm (Theorem REF ).\",\n \"These algorithms run in near-linear time in dense graphs, improving the previous best runtime of $\\\\tilde{O}(m\\\\sqrt{n})$ by Roditty and Vassilevska W. [41], and subsuming (for dense unweighted graphs) the results of Cairo et al. [15].\",\n \"Theorem 6 There is an expected $O(n^2\\\\log n)$ time algorithm that for any undirected unweighted graph with Diameter $D=3h+z$ for $h\\\\ge 0,z\\\\in \\\\lbrace 0,1,2\\\\rbrace $ , returns an extimate $D^{\\\\prime }$ such that $2h-1\\\\le D^{\\\\prime }\\\\le D$ if $z=0,1$ and $2h\\\\le D^{\\\\prime }\\\\le D$ if $z=2$ .\",\n \"There is an expected $O(n^2\\\\log n)$ time algorithm that for any undirected unweighted graph returns estimates $\\\\varepsilon ^{\\\\prime }(v)$ of the Eccentricities $\\\\varepsilon (v)$ of all vertices such that $3\\\\varepsilon (v)/5-1\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ for all $v$ .\",\n \"We also show (in Theorem REF ) that one can improve the estimates slightly with an $O(n^{2.05})$ time algorithm.\"\n ],\n [\n \"Related work\",\n \"The fastest known algorithm for APSP in dense weighted graphs is by R. Williams [53] and runs in $O(n^3/2^{\\\\Theta (\\\\sqrt{\\\\log n})})$ time.\",\n \"For sparse undirected graphs, the fastest known APSP algorithm is by Pettie [35] running in $O(mn+n^2\\\\log \\\\log n)$ time.\",\n \"The fastest APSP algorithm for sparse undirected weighted graphs is by Pettie and Ramachandran [37] and runs in $O(mn\\\\log \\\\alpha (m,n))$ time.\",\n \"For APSP on undirected unweighted graphs with $m>n\\\\log \\\\log n$ , Chan [17] presented an $O(mn \\\\log \\\\log n/\\\\log n)$ time algorithm.\",\n \"In graphs with small integer edge weights bounded in absolute value by $M$ , APSP can be computed in $\\\\tilde{O}(Mn^\\\\omega )$ time (by Shoshan and Zwick [46] building upon Seidel [43] and Alon, Galil and Margalit [6]) in undirected graphs and in $\\\\tilde{O}(M^{0.681}n^{2.5302})$ time (by Zwick [55]) in directed graphs.\",\n \"Zwick [55] also showed that APSP in directed weighted graphs admits an $(1+\\\\varepsilon )$ -approximation algorithm for any $\\\\varepsilon >0$ , running in time $\\\\tilde{O}(n^\\\\omega /\\\\varepsilon \\\\log (M/\\\\varepsilon ))$ .\",\n \"For Diameter in graphs with integer edge weights bounded by $M$ , Cygan et al.\",\n \"[16] obtained an algorithm running in time $\\\\tilde{O}(Mn^\\\\omega )$ .\",\n \"The pioneering work of Aingworth et al.\",\n \"[2] on Diameter and shortest paths approximation was the root to many subsequent works.\",\n \"Building upon Aingworth et al.\",\n \"[2], Dor, Halperin and Zwick [24] presented additive approximation algorithms for APSP in undirected unweighted graphs, achieving among other things, an additive 2-approximation in $\\\\tilde{O}(n^{7/3})$ time (notably, the best known bound on $\\\\omega $ is $>7/3$ ).\",\n \"They also presented an $\\\\tilde{O}(n^2)$ time additive $O(\\\\log n)$ -approximation algorithm.\",\n \"These algorithms were generalized by Cohen and Zwick [23] who showed that in undirected weighted graphs APSP has a (multiplicative) 3-approximation in $\\\\tilde{O}(n^2)$ time, a $7/3$ -approximation in $\\\\tilde{O}(n^{7/3})$ time, and a 2-approximation in $\\\\tilde{O}(n \\\\sqrt{mn})$ time.\",\n \"Baswana and Kavitha [11] presented an $\\\\tilde{O}(m\\\\sqrt{n}+n^2)$ time multiplicative 2-approximation algorithm and an $\\\\tilde{O}(m^{2/3} n+n^2)$ time $7/3$ -approximation algorithm for APSP in weighted undirected graphs.\",\n \"Spanners are closely related to shortest paths approximation.\",\n \"A subgraph $H$ is an $(\\\\alpha , \\\\beta )$ -spanner of $G=(V,E)$ if for every $u,v\\\\in V$ , $d_H(u,v) \\\\le \\\\alpha \\\\cdot d_G(u,v) + \\\\beta $ , where $d_{G^{\\\\prime }}(u,v)$ is the distance between $u$ and $v$ in $G^{\\\\prime }$ .\",\n \"Any weighted undirected graph has a $(2k-1,0)$ -spanner with $O(n^{1+1/k})$ edges [3].\",\n \"Baswana and Sen [14] presented a randomized linear time algorithm for constructing a $(2k-1,0)$ -spanner with $O(kn^{1+1/k})$ edges.\",\n \"Dor, Halperin and Zwick [24] showed that a $(1,2)$ -spanner with $O(n^{1.5})$ edges can be constructed in $\\\\tilde{O}(n^2)$ time.\",\n \"Elkin and Peleg [25] showed that for every integer $k\\\\ge 1$ and $\\\\varepsilon >0$ there is a $(1+\\\\varepsilon , \\\\beta )$ -spanner with $O(\\\\beta n^{1+1/k})$ edges, where $\\\\beta $ depends on $k$ and $\\\\varepsilon $ but is independent of $n$ .\",\n \"Baswana et.\",\n \"al [12] presented a $(1,6)$ -spanner with $O(n^{4/3})$ edges.\",\n \"Woodruff [54] presented an $\\\\tilde{O}(n^2)$ time algorithm that computes a $(1,6)$ -spanner with $O(n^{4/3})$ edges.\",\n \"Chechik [18] presented a $(1,4)$ -spanner with $O(n^{7/5})$ edges.\",\n \"Recently, Abboud and Bodwin[1] showed that there is no additive spanner with constant error and $O(n^{4/3-\\\\varepsilon })$ edges.\",\n \"Thorup and Zwick [48] introduced the notion of distance oracles, a data structure that stores approximate distances for a weighted undirected graph.\",\n \"Thorup and Zwick designed a distance oracle that for any $k$ takes $O(mn^{1/k})$ time to construct and, is of size is $O(kn^{1+1/k})$ , and given a pair of vertices $u,v \\\\in V$ it returns in $O(k)$ time a $(2k-1)$ -approximation for $d(u,v)$ .\",\n \"Baswana and Sen [13] improved the construction time to $O(n^2)$ for unweighted graphs.\",\n \"Baswana and Kavitha [11] extended the $O(n^2)$ construction time to weighted graphs.\",\n \"Subsequently, Baswana, Gaur, Sen, and Upadhyay [10] obtained subquadratic construction time in unweighted graphs, at the price of having additive constant error in addition to the $2k-1$ multiplicative error.\",\n \"Chechik [19] gave an oracle with space $O(n^{1+1/k})$ and $O(1)$ query time, which like previous work, returns a $(2k-1)$ -approximation.\",\n \"Pǎtraşcu and Roditty [38] obtained a distance oracle that uses $\\\\tilde{O}(n^{5/3})$ space, has $O(1)$ query time, and returns a $(2k+1)$ -approximation.\",\n \"Sommer [44] presented an $\\\\tilde{O}(n^2)$ time algorithm that constructs such a distance oracle.\",\n \"The construction time was recently improved to $O(n^2)$ by Knudsen [30].\",\n \"Pǎtraşcu et.\",\n \"al [39] presented infinity many distance oracles with fractional approximation factors that for graphs with $m=\\\\tilde{O}(n)$ converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.\",\n \"Thorup and Zwick [47] also extended their techniques from [48] to compact routing schemes.\",\n \"The lower bounds presented in this paper were inspired by a lower bound by Pǎtraşcu and Roditty [38] who showed conditional hardness based on a conjecture on the hardness of a set intersection problem for the space usage of any distance oracle that can distinguish between distances 3 and 7.\"\n ],\n [\n \"Organization\",\n \"In Section we prove our lower bounds for $S$ -$T$ Diameter which serve as a basis for the rest of our lower bounds.\",\n \"We also show equivalence between Diameter and $S$ -$T$ Diameter.\",\n \"In Section we prove our lower bounds for Eccentricities: one for directed graphs and one for undirected graphs.\",\n \"In Section we prove our lower bounds for Diameter.\",\n \"This section is divided into four subsections, one for each of the results in Table REF .\",\n \"In Section we describe our algorithms for sparse graphs: our 2-approximation and $(2+\\\\delta )$ -approximation for Eccentricities, our 2-approximation and 3-approximation for $S$ -$T$ Diameter, and our less than 2-approximation for Diameter.\",\n \"In Section we describe our algorithms for dense graphs: our nearly $3/2$ -approximations for Diameter and our nearly $5/3$ -approximations for Eccentricities.\"\n ],\n [\n \"Preliminaries\",\n \"Let $G=(V,E)$ be a graph, where $|V|=n$ and $|E|=m$ .\",\n \"For every $u,v\\\\in V$ let $d_G(u,v)$ be the length of the shortest path from $u$ to $v$ .\",\n \"When the graph $G$ is clear from the context we omit the subscript $G$ .\",\n \"The eccentricity $\\\\varepsilon (v)$ of a vertex $v$ is defined as $\\\\max _{u\\\\in V} d(v,u)$ .\",\n \"The diameter $D$ of a graph is $\\\\max _{v\\\\in V}\\\\varepsilon (v)$ .\",\n \"In a directed graph we have $\\\\varepsilon ^{out}(v)=\\\\max _{u\\\\in V} d(v,u)$ (resp., $\\\\varepsilon ^{in}(v)=\\\\max _{u\\\\in V} d(u,v)$ ).\",\n \"Let $deg(v)$ be the degree of $v$ and let $N_s(u)$ be the set of the $s$ closest vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.\",\n \"In a directed graph let $deg^{out}(v)$ (resp., $deg^{in}(v)$ ) be the outgoing (incoming) degree of $v$ .\",\n \"Let $N^{\\\\text{out}}_s(v)$ (resp., $N^{\\\\text{in}}_s(v)$ ) be the set of the $s$ closest outgoing (incoming) vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.\",\n \"For a subset $S\\\\subseteq V$ of vertices and a vertex $v \\\\in V$ we write $d(S,v):=\\\\min _{s \\\\in S}d(s,v)$ to denote the distance from the set $S$ to the vertex $v$ .\",\n \"Let $k\\\\ge 2$ .\",\n \"The $k$ -Orthogonal Vectors Problem ($k$ -OV) is as follows: given $k$ sets $S_1,\\\\ldots ,S_k$ , where each $S_i$ contains $n$ vectors in $\\\\lbrace 0,1\\\\rbrace ^d$ , determine whether there exist $v_1\\\\in S_1,\\\\ldots ,v_k\\\\in S_k$ so that their generalized inner product is 0, i.e.\",\n \"$\\\\sum _{i=1}^d \\\\prod _{j=1}^k v_j[i]=0$ .\",\n \"R. Williams [51] (see also [49]) showed that if for some $\\\\varepsilon >0$ there is an $n^{k-\\\\varepsilon } \\\\text{\\\\rm poly~} (d)$ time algorithm for $k$ -OV, then CNF-SAT on formulas with $N$ variables and $m$ clauses can be solved in $2^{N(1-\\\\varepsilon /k)} \\\\text{\\\\rm poly~} (m)$ time.\",\n \"In particular, such an algorithm would contradict the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [28] which states that for every $\\\\varepsilon >0$ there is a $K$ such that $K$ -SAT on $N$ variables cannot be solved in $2^{(1-\\\\varepsilon )N} \\\\text{\\\\rm poly~} N$ time (say, on a word-RAM with $O(\\\\log N)$ bit words).\",\n \"This also motivates the following $k$ -OV Conjectures (implied by SETH) for all constants $k\\\\ge 2$ : $k$ -OV requires $n^{k-o(1)}$ time on a word-RAM with $O(\\\\log n)$ bit words.\",\n \"Most of our conditional lower bounds are based on the $k$ -OV Conjecture for a particular constant $k$ , and thus they also hold under SETH.\",\n \"A main motivation behind SETH is that despite decades of research, the best upper bounds for $K$ -SAT on $N$ variables and $M$ clauses remain of the form $2^{N(1-c/K)} \\\\text{\\\\rm poly~} {(M)}$ for constant $c$ (see e.g.\",\n \"[27], [36], [42]).\",\n \"The best algorithms for the $k$ -OV problem for any constant $k\\\\ge 2$ on $n$ vectors and dimension $c\\\\log n$ run in time $n^{2-1/O(\\\\log c)}$ (Abboud, Williams and Yu [8] and Chan and Williams [22]).\"\n ],\n [\n \"$S$ -{{formula:678000a5-c252-477a-976e-b4f5fbe676e0}} Diameter hardness\",\n \"In the following we will prove that under SETH, our $S$ -$T$ Diameter algorithms are essentially optimal.\",\n \"We prove the following theorem: Theorem 7 Let $k\\\\ge 2$ be an integer.\",\n \"There is an $O(k n^{k-1} d^{k-1})$ time reduction that transforms any instance of $k$ -OV on sets of $n$ $d$ -dimensional vectors into a graph on $O(n^{k-1}+k n^{k-2} d^{k-1})$ nodes and $O(k n^{k-1} d^{k-1})$ edges and two disjoint sets $S$ and $T$ on $n^{k-1}$ nodes each, so that if the $k$ -OV instance has a solution, then $D_{S,T}\\\\ge 3k-2$ , and if it does not, $D_{S,T}\\\\le k$ .\",\n \"From Theorem REF we get that if there is some $k\\\\ge 2$ , $\\\\varepsilon >0$ and $\\\\delta >0$ so that there is an $O(M^{1+1/(k-1)-\\\\varepsilon })$ time $(3-2/k-\\\\delta )$ -approximation algorithm for $S$ -$T$ Diameter in $M$ -edge graphs, then $k$ -OV has an $n^{k-\\\\gamma } \\\\text{\\\\rm poly~} (d)$ -time algorithm for some $\\\\gamma >0$ and SETH is false.\",\n \"We obtain an immediate corollary.\",\n \"Corollary 8 For $S$ -$T$ Diameter, under SETH, there is no $(2-\\\\varepsilon )$ -approximation algorithm running in $O(m^{2-\\\\delta })$ time for any $\\\\varepsilon >0,\\\\delta >0$ , no $O(m^{3/2-\\\\delta })$ time $7/3-\\\\varepsilon $ -approximation algorithm for any $\\\\varepsilon >0,\\\\delta >0$ , no $(m+n)^{1+o(1)}$ time, $3-\\\\varepsilon $ -approximation algorithm for any $\\\\varepsilon >0$ .\",\n \"We will prove the following more detailed theorem, which will be useful for our Diameter lower bounds.\",\n \"Theorem 9 Let $k\\\\ge 2$ .\",\n \"Given a $k$ -OV instance consisting of sets $W_0,W_1,\\\\dots ,W_{k-1} \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.\",\n \"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\\\dots ,L_k=T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\\\dots ,|L_{k-1}|\\\\le n^{k-2}d^{k-1}$ .\",\n \"$S$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $T$ consists of all tuples $(b_1,b_2,\\\\ldots , b_{k-1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then if $\\\\alpha =(a_0,\\\\dots a_{k-2})\\\\in S$ and $\\\\beta = (a_1,\\\\dots ,a_{k-1}) \\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge 3k-2$ .\",\n \"Suppose the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Let $t=k-2$ .\",\n \"Let $s$ be such that $0\\\\le s\\\\le t$ .\",\n \"Let $b_{t-s+j}\\\\in W_{t-s+j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"Symmetrically, let $c_{j}\\\\in W_{j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t})\\\\in L_0$ and $\\\\beta =(c_{1},\\\\ldots ,c_s,a_{s+1},\\\\dots , a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"For all $i$ from 1 to $k-1$ , for all $v \\\\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .\"\n ],\n [\n \"Proof of Theorem \",\n \"We will prove the theorem for $k=t+2$ for any $t\\\\ge 0$ .\",\n \"We will create a layered graph $G$ on $t+3$ layers, $L_0,\\\\ldots ,L_{t+2}$ , where the edges go only between adjacent layers $L_i,L_{i+1}$ .\",\n \"We will set $S=L_0$ and $T=L_{t+2}$ for the $S$ -$T$ Diameter instance.\",\n \"In particular, $D_{S,T}\\\\ge t+2$ because of the layering.\",\n \"Let us describe the vertices of $G$ .\",\n \"$L_0$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(a_0,a_1,\\\\ldots , a_t)$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $L_{t+2}$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(b_1,b_2,\\\\ldots , b_{t+1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"Layer $L_1$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(a_0,\\\\ldots ,a_{t-1}, \\\\bar{x})$ where for each $i$ , $a_i\\\\in W_i$ and $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .\",\n \"Similarly, $L_{t+1}$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(b_2,\\\\ldots ,b_{t+1}, \\\\bar{x})$ where for each $i$ , $b_i\\\\in W_i$ and $\\\\bar{x}$ is a $(t+1)$ -tuple of coordinates.\",\n \"For every $j\\\\in \\\\lbrace 2,\\\\ldots ,t\\\\rbrace $ , $L_j$ consists of $n^t d^{t+1}$ vertices $(a_0, \\\\ldots , a_{t-j},b_{t+3-j},\\\\ldots , b_{t+1},\\\\bar{x})$ , where for each $i$ , $a_i\\\\in W_i$ , $b_i\\\\in W_i$ and $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .\",\n \"In other words, there is a vector from $W_i$ for every $i\\\\notin \\\\lbrace t-j+1,t-j+2\\\\rbrace $ .\",\n \"Now let us define the edges.\",\n \"Consider a node $(a_0,\\\\ldots ,a_t)\\\\in L_0$ .\",\n \"For every $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ , connect $(a_0,\\\\ldots ,a_t)$ to $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})\\\\in L_1$ if and only if for every $j\\\\in \\\\lbrace 0,\\\\ldots ,t\\\\rbrace $ , $a_j$ is 1 in coordinates $x_0,\\\\ldots ,x_{t-j}$ .\",\n \"For any $i\\\\in \\\\lbrace 1,\\\\ldots ,t\\\\rbrace $ let's define the edges between $L_i$ and $L_{i+1}$ .\",\n \"For $(a_0,\\\\ldots ,a_{t-i},b_{t+3-i},\\\\ldots ,b_{t+1}, \\\\bar{x})\\\\in L_i$ Here if $i=1$ , there are no $b$ 's in the tuple.\",\n \"and for any $c_{t+2-i}\\\\in W_{t+2-i}$ , add an edge to $(a_0,\\\\ldots ,a_{t-i-1},c_{t+2-i},b_{t+3-i},\\\\ldots ,b_{t+1}, \\\\bar{x})\\\\in L_{i+1}$ .\",\n \"Here we “forget” vector $a_{t-i}$ and replace it with $c_{t+2-i}$ , leaving everything else the same.\",\n \"Finally, the edges between $L_{t+1}$ and $L_{t+2}$ are as follows.\",\n \"Consider some $(b_1,\\\\ldots ,b_{t+1})\\\\in L_{t+2}$ .\",\n \"For every $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ , connect $(b_1,\\\\ldots ,b_{t+1})$ to $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{t+1}$ if and only if for every $j\\\\in \\\\lbrace 1,\\\\ldots ,{t+1}\\\\rbrace $ , $b_j$ is 1 in coordinates $x_{t+1-j},\\\\ldots ,x_t$ .\",\n \"Figure REF shows the construction of the graph for $t=2$ .\",\n \"Figure: The reduction graph from (t+2)(t+2)-OV for t=2t=2.\",\n \"The figure depicts when a path of length t+2t+2 exists between arbitrary a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\\\in L_0 and b 1 b 2 b 3 ∈L t+2 b_1b_2b_3\\\\in L_{t+2}.\",\n \"It also shows that when there is a path of length t+2t+2 between a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\\\in L_0 and a 1 a 2 a 3 ∈L t+2 a_1a_2a_3\\\\in L_{t+2}, a 0 ,a 1 ,a 2 ,a 3 a_0,a_1,a_2,a_3 cannot be an orthogonal 4-tuple.An important claim is as follows: Claim 10 For every $\\\\bar{x}$ , each $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})\\\\in L_1$ is at distance $t$ to every $(b_2,\\\\ldots ,b_t,\\\\bar{x})\\\\in L_{t+1}$ .\",\n \"Consider the path starting from $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ , and then for each $i\\\\ge 1$ following the edges $(a_0,\\\\ldots ,a_{t-i},b_{t+3-i},\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_i$ to $(a_0,\\\\ldots ,a_{t-1-i},b_{t+2-i},\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{i+1}$ , until we reach $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{t+1}$ .\",\n \"This path exists by construction and has length $t$ .\",\n \"Now we proceed to prove the bounds on the $S$ -$T$ Diameter.\",\n \"Lemma 11 (Property 3 of Theorem REF ) If the $(t+2)$ -OV instance has no solution, then $D_{S,T}=t+2$ .\",\n \"If the $(t+2)$ -OV instance has no solution, then for every $c_0\\\\in W_0,c_1\\\\in W_1,\\\\ldots ,c_{t+1}\\\\in W_{t+1}$ , there is some coordinate $x$ such that $c_0[x]=c_1[x]=\\\\ldots =c_{t+1}[x]=1$ .\",\n \"Now consider the graph and any $(a_0,\\\\ldots ,a_t)\\\\in L_0$ , $(b_1,\\\\ldots ,b_{t+1})\\\\in L_{t+2}$ .\",\n \"For every $j\\\\in \\\\lbrace 0,\\\\ldots ,t\\\\rbrace $ , let $x_j$ be a coordinate so that $a_0,\\\\ldots ,a_{t-j},b_{t-j+1},\\\\ldots ,b_{t+1}$ are all 1 in $x_j$ .\",\n \"Let $\\\\bar{x}=(x_0,\\\\ldots ,x_{t})$ .\",\n \"By construction, $(a_0,\\\\ldots ,a_t)$ has an edge to $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ and $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})$ has an edge to $(b_1,\\\\ldots ,b_{t+1})$ .\",\n \"Also, by Claim REF , $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ has a path of length $t$ to $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})$ .\",\n \"This shows that $D_{S,T}\\\\le t+2$ ; equality follows because the graph is layered.\",\n \"Now we prove the guarantee for the case when an orthogonal tuple exists.\",\n \"Lemma 12 (Property 4 of Theorem REF ) If there exist $a_0\\\\in W_0,\\\\ldots ,a_{t+1}\\\\in W_{t+1}$ that are orthogonal, then $D_{S,T}\\\\ge 3t+4$ .\",\n \"To prove the lemma, we will actually prove a more general claim: Property 5 of Theorem REF .\",\n \"Claim 13 (Property 5 of Theorem REF ) Suppose that $a_0\\\\in W_0,\\\\ldots ,a_{t+1}\\\\in W_{t+1}$ are orthogonal.\",\n \"Let $s$ be such that $0\\\\le s\\\\le t$ .\",\n \"Let $b_{t-s+j}\\\\in W_{t-s+j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"Symmetrically, let $c_{j}\\\\in W_{j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t})\\\\in L_0$ and $\\\\beta =(c_{1},\\\\ldots ,c_s,a_{s+1},\\\\dots , a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"If the claim is true, then using $s=0$ we get that the Diameter is at least $3t+4$ so Lemma REF is true.\",\n \"The claim for $s>0$ is useful for the rest of our constructions.\",\n \"We will show that the distance between $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ is strictly more than $3t+2-2s$ .\",\n \"Because the graph is layered and hence bipartite and $t+2\\\\equiv 3t+2\\\\mod {2}$ , the distance must be at least $3t-2s+4$ .\",\n \"Let's assume for contradiction that the shortest path $P$ between $\\\\alpha $ and $\\\\beta $ is of length $\\\\le 3t+2-2s$ .\",\n \"First let's look at any subpath $P^{\\\\prime }$ of $P$ strictly within $M=L_1\\\\cup \\\\ldots \\\\cup L_{t+1}$ .\",\n \"All nodes on $P^{\\\\prime }$ must share the same $\\\\bar{x}$ .\",\n \"Furthermore, if $P^{\\\\prime }$ starts with a node of $L_1$ and ends with a node of $L_{t+1}$ , as $P^{\\\\prime }$ needs to be a shortest path and by Claim REF , $P^{\\\\prime }$ must be of length exactly $t$ .\",\n \"Next, notice that $P$ cannot go from $L_0$ to $L_{t+2}$ and then back to $L_0$ .\",\n \"This is because it needs to end up in $L_{t+2}$ and any time it crosses over $M$ , it would need to pay a distance of $t+2$ , so $P$ would have to have length at least $3t+6>3t+2-2s$ .\",\n \"Hence, $P$ must be of the following form: a path from $\\\\alpha $ through $L_0\\\\cup M$ back to $L_0$ (possibly containing only $\\\\alpha $ ), followed by a path crossing $M$ to reach $L_{t+2}$ , followed by a path through $L_{t+2}\\\\cup M$ to $L_{t+2}$ (possibly empty).\",\n \"We will show that if $P$ has length $\\\\le 3t+2-2s$ then $P$ must contain a length $t+2$ subpath $Q$ between a node $(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_t)\\\\in L_0$ , for some choices of the $w$ 's and some $q\\\\le t-s$ , and a node $(v_1,\\\\ldots ,v_q,a_{q+1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ , for some choices of $v$ 's.\",\n \"That is, this path traverses $M$ without weaving, by following $(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_{t-1},\\\\bar{x})\\\\in L_1$ , $(a_0, \\\\ldots , a_{q},w_{q+1},\\\\ldots , w_{t-2},a_{t+1},\\\\bar{x})\\\\in L_2$ , $\\\\ldots $ , $(v_2, \\\\ldots , v_q, a_{q+1}, \\\\ldots , a_{t-s}, \\\\ldots , a_{t+1}, \\\\bar{x})\\\\in L_{t+1}$ .\",\n \"Suppose we show that such a subpath exists.\",\n \"Then by the construction of our graph we have that for every $i\\\\in \\\\lbrace 0,\\\\ldots ,q\\\\rbrace $ , $a_i[x_j]=1$ for all $j\\\\in \\\\lbrace 0,\\\\ldots ,t-i\\\\rbrace $ , and that for all $i\\\\in \\\\lbrace q+1,\\\\ldots ,t+1\\\\rbrace $ , $a_i[x_j]=1$ for all $j\\\\in \\\\lbrace t+1-i,\\\\ldots ,t\\\\rbrace $ .\",\n \"That is, for all $i$ , $a_i[x_{t-q}]=1$ , and we get a contradiction since the $a_i$ were supposed to be orthogonal.\",\n \"Now let $\\\\alpha ^*$ be the last node from $L_0$ on $P$ and let $\\\\beta ^*$ be the first node of $L_{t+1}$ of $P$ .\",\n \"Let $a^*\\\\in L_1$ be the node right after $\\\\alpha ^*$ and let $b^*\\\\in L_{t+1}$ be the node right before $\\\\beta ^*$ .\",\n \"Since the subpath of $P$ between $a^*$ and $b^*$ is within $M$ , it must share the same $\\\\bar{x}$ , and it must have length exactly $t$ by Claim REF .\",\n \"We will show that the subpath $Q$ that we are looking for is the subpath of $P$ between $\\\\alpha ^*$ and $\\\\beta ^*$ .\",\n \"Its length is exactly what we want: $t+2$ .\",\n \"It remains to show that for some $q\\\\le t-s$ and some choices of $w$ 's and $v$ 's, $\\\\alpha ^*=(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_t)$ and $\\\\beta ^*=(v_1,\\\\ldots ,v_q,a_{q+1},\\\\ldots ,a_{t+1})$ .\",\n \"Consider the path $P_1$ between $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})$ and $\\\\alpha ^*$ and the path $P_2$ between $\\\\beta =(a_1,\\\\ldots ,a_{t+1})$ and $\\\\beta ^*$ .\",\n \"Let $L_i$ be the layer in $M$ with largest $i$ that $P_1$ touches and let $L_j$ be the layer in $M$ with smallest $j$ that $P_2$ touches.\",\n \"For convenience, let us define $j^{\\\\prime }=t+2-j$ .\",\n \"The length of $P_1$ is then at least $2i$ and the length of $P_2$ is at least $2j^{\\\\prime }$ .\",\n \"The length $|P|$ of $P$ equals $t+2+|P_1|+|P_2|\\\\ge t+2+2i+2j^{\\\\prime }=t+2+2(i+j^{\\\\prime })$ .\",\n \"Since we have assumed that $|P|\\\\le 3t+2-2s$ , we must have that $t+2+2(i+j^{\\\\prime })\\\\le 3t+2-2s$ and hence $i+j^{\\\\prime }\\\\le t-s$ .\",\n \"Now, since $P_1$ goes at most to $L_i$ , then from getting from $\\\\alpha $ to $\\\\alpha ^*$ , at most the last $i$ elements of $(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})$ can have been “forgotten”.\",\n \"Hence, $\\\\alpha ^*=(a_0,\\\\ldots ,a_{t-\\\\max \\\\lbrace s,i\\\\rbrace },b_{t-\\\\max \\\\lbrace s,i\\\\rbrace +1},\\\\ldots ,b_{t-i},w_{t-i+1},\\\\ldots ,w_{t})$ for some $w$ 's.\",\n \"(If $i\\\\ge s$ , the $b$ 's do not appear.)\",\n \"Similarly, between $\\\\beta $ and $\\\\beta ^*$ , at most the first $j^{\\\\prime }$ elements of $\\\\beta $ can have been forgotten.\",\n \"Thus, we have that $\\\\beta ^*=(v_1,\\\\ldots ,v_{j^{\\\\prime }},a_{j^{\\\\prime }+1},\\\\ldots ,a_{t+1})$ for some $v$ 's.\",\n \"Now, since $i+j^{\\\\prime }\\\\le t-s$ , we must have that $j^{\\\\prime }\\\\le t-s-i\\\\le t-\\\\max \\\\lbrace s,i\\\\rbrace $ , and hence the path between $\\\\alpha ^*$ and $\\\\beta ^*$ is the path $Q$ we are searching for.\",\n \"See Figure REF for an illustration of $L_i$ , $L_j$ etc.\",\n \"in the case when $s=0$ .\",\n \"Figure: Here PP contains at least 2 nodes in L 0 L_0 and at least 2 in L t+2 L_{t+2}, and s=0s=0.\"\n ],\n [\n \"Equivalence between Diameter and $S$ -{{formula:bdb52e68-e562-4944-b84c-7c1359b488ce}} Diameter\",\n \"Here we will prove that when it comes to exact computation, $S$ -$T$ -Diameter and Diameter in weighted graphs are equivalent.\",\n \"The proof for directed graphs is much simpler, so we focus on the equivalence for undirected graphs.\",\n \"Also, it is clear that if one can solve $S$ -$T$ Diameter, one can also solve Diameter in the same running time since one can simply set $S=T=V$ .\",\n \"We prove: Theorem 14 Suppose that there is a $T(n,m)$ time algorithm that can compute the Diameter of an $n$ node, $m$ edge graph with nonnegative integer edge weights.\",\n \"Then, the $S$ -$T$ Diameter of any $n$ node $m$ edge graph with nonnegative integer edge weights can be computed in $T(O(n),O(m))$ time.\",\n \"Let $G=(V,E),S,T$ be the $S$ -$T$ Diameter instance; let $w:E\\\\rightarrow \\\\lbrace 0,\\\\ldots ,M\\\\rbrace $ be the edge weights.\",\n \"First, we can always assume that $M$ is even: if it is not, multiply all edge weights by 2; all distances (and hence also the $S$ -$T$ Diameter) double.\",\n \"Let $S=\\\\lbrace s_1,\\\\ldots ,s_k\\\\rbrace $ and $T=\\\\lbrace t_1,\\\\ldots ,t_{\\\\ell }\\\\rbrace $ .\",\n \"Now, let $W=Mn$ .\",\n \"First add $|S|=k$ new nodes $S^{\\\\prime }=\\\\lbrace v_1,\\\\ldots ,v_k\\\\rbrace $ .\",\n \"For each $i\\\\in \\\\lbrace 1,\\\\ldots ,k\\\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .\",\n \"Let $G_S$ be this new graph.\",\n \"Let's consider the Diameter of $G_S$ .\",\n \"For every pair of nodes $u,v\\\\notin S^{\\\\prime }$ , the distance is the same as in $G$ .\",\n \"For $v_i\\\\in S^{\\\\prime }$ and $x\\\\notin S^{\\\\prime }$ , the distance is $W+d_G(s_i,x)\\\\le W+M(n-1)<2W$ .\",\n \"For $v_i,v_j\\\\in S^{\\\\prime }$ , the distance is $2W+d(s_i,s_j)$ .\",\n \"Hence the Diameter of $G_S$ is $2W+\\\\max _{s_i,s_j\\\\in S} d(s_i,s_j)$ .\",\n \"Hence by computing the Diameter of $G_S$ , we can compute $D_S=\\\\max _{s_i,s_j\\\\in S} d(s_i,s_j)$ .\",\n \"We can create a similar graph $G_T$ whose Diameter will allow us to compute $D_T=\\\\max _{t_i,t_j\\\\in S} d(t_i,t_j)$ .\",\n \"After this, let's create a graph $G_{S,T}$ as follows.\",\n \"Add new nodes $S^{\\\\prime }=\\\\lbrace v_1,\\\\ldots ,v_k\\\\rbrace $ and $T^{\\\\prime }=\\\\lbrace u_1,\\\\ldots ,u_\\\\ell \\\\rbrace $ .\",\n \"For each $i\\\\in \\\\lbrace 1,\\\\ldots ,k\\\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .\",\n \"For each $j\\\\in \\\\lbrace 1,\\\\ldots ,\\\\ell \\\\rbrace $ add a new edge $(u_j,t_j)$ of weight $W$ .\",\n \"With a similar argument as above, the Diameter $D^{\\\\prime }$ of $G_{S,T}$ is $D^{\\\\prime }=2W+\\\\max _{u,v\\\\in S\\\\cup T} d_G(u,v)$ .\",\n \"Let's assume without loss of generality that $D_S\\\\ge D_T$ .\",\n \"If $D^{\\\\prime }> D_S$ , then $D^{\\\\prime }=2W+\\\\max _{u\\\\in S,v\\\\in T} d_G(u,v)$ , and we can compute the $S$ -$T$ Diameter of $G$ by subtracting $2W$ .\",\n \"Now suppose that we get $D^{\\\\prime }\\\\le D_S$ ; we must have then actually gotten $D^{\\\\prime }=D_S$ .\",\n \"The $S$ -$T$ Diameter of $G$ might be strictly smaller than $D_S$ .\",\n \"We add two new nodes $x$ and $y$ to $G_{S,T}$ .\",\n \"We add an edge $(x,y)$ of weight $2W$ , edges $(x,v_i)$ for every $v_i\\\\in S^{\\\\prime }$ of weight $D_S/2$ and (symmetrically) edges $(y,u_j)$ for every $u_j\\\\in T^{\\\\prime }$ of weight $D_S/2$ .\",\n \"The construction of $G^{\\\\prime }$ is depicted in Figure REF .\",\n \"Figure: A depiction of the construction of G ' G^{\\\\prime }.Let us consider the distances in this new graph $G^{\\\\prime }$ .\",\n \"For every $a,b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(a,b)=d_G(a,b)d(a,b)$ .\",\n \"For every $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(x,b)=W+D_S/2 +\\\\min _{a\\\\in S} d_G(a,b)\\\\le 2W+D_S/2$ .\",\n \"Similarly, $d(y,b)=W+D_S/2 +\\\\min _{a\\\\in T} d_G(a,b)\\\\le 2W+D_S/2$ .\",\n \"For every $v_i\\\\in S^{\\\\prime }$ , $d(x,v_i)=D_S/2$ , and $d(y,v_i)=2W+D_S/2$ .\",\n \"For every $u_i\\\\in T^{\\\\prime }$ , $d(y,u_i)=D_S/2$ , and $d(x,u_i)=2W+D_S/2$ .\",\n \"For every $v_i,v_j\\\\in S^{\\\\prime }$ , $d(v_i,v_j)=D_S$ .\",\n \"For every $u_i,u_j\\\\in T^{\\\\prime }$ , $d(u_i,u_j)=D_S$ .\",\n \"For every $v_i\\\\in S^{\\\\prime }$ and $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(v_i,b)\\\\le W+d(s_i,b)\\\\le 2W.$ For every $u_i\\\\in T^{\\\\prime }$ and $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(u_i,b)\\\\le W+d(t_i,b)\\\\le 2W.$ For every $v_i\\\\in S^{\\\\prime }$ and $u_j\\\\in T^{\\\\prime }$ , $d(v_i,u_j)$ is the minimum of $D_S+2W, D_S+2W+\\\\min _{s\\\\in S} d_G(s,t_j), D_S+2W+\\\\min _{t\\\\in t} d_G(t,s_i)$ and $2W+d_G(s_i,t_j)$ .\",\n \"The middle two terms are $\\\\ge D_S+2W$ , and hence $d(v_i,u_j)=2W+\\\\min \\\\lbrace D_S,d_G(s_i,t_j)\\\\rbrace $ .\",\n \"Consider $s_i\\\\in S, t_j\\\\in T$ that are the end points of the $S$ -$T$ Diameter $D$ in $G$ .\",\n \"Then $D=d_G(s_i,t_j)$ .\",\n \"Now, we have from before that $D\\\\le D_S$ , as otherwise we have computed $D$ already.\",\n \"Hence in $G^{\\\\prime }$ , the distance $d(u_i,v_j)$ equals $2W+\\\\min \\\\lbrace D_S,D\\\\rbrace =2W+D$ We note that for any $s_i,s_j\\\\in S$ , and any $t\\\\in T$ , $d_G(s_i,s_j)\\\\le d_G(s_i,t)+d_G(t,s_j)\\\\le 2\\\\max _{s\\\\in S,t\\\\in T} d_G(s,t) = 2D$ .\",\n \"Thus, $D\\\\ge D_S/2$ .\",\n \"The distances in cases (1) to (5) are all $\\\\le 2W+D_S/2\\\\le 2W+D$ .\",\n \"Hence the Diameter of $G^{\\\\prime }$ is actually exactly $2W+D$ .\"\n ],\n [\n \"Undirected graphs\",\n \"Theorem 15 Let $k\\\\ge 2$ .\",\n \"Under the $k$ -OV conjecture, every algorithm that can distinguish between eccentricity at most $2k-1$ and eccentricity at least $4k-3$ for every vertex in an $O(n)$ edge and node undirected graph, requires at least $n^{1+1/(k-1)-o(1)}$ time on a $O(\\\\log n)$ -bit word-RAM.\",\n \"Figure: The undirected Eccentricities lower bound for k=5k=5.Let's start with the $S$ -$T$ -diameter construction for $k$ obtained from a given $k$ -OV instance.\",\n \"We remove any internal nodes if they don't have edges to one of their adjacent layers - they don't hurt the instance.\",\n \"We have a graph on $O(n^{k-1}d^{k-2})$ vertices and edges with the following properties: (1) Suppose that the $k$ -OV instance has no $k$ -OV solution.\",\n \"Then for every $s\\\\in S,t\\\\in T$ , $d(s,t)=k$ .\",\n \"Also, for every $s\\\\in S$ and $u\\\\notin S\\\\cup T$ , $d(s,u)\\\\le (k-1)+k=2k-1$ since we can take a $\\\\le (k-1)$ length path from $u$ to some node $t\\\\in T$ and since $d(s,t)=k$ .\",\n \"(2) If there is a $k$ -OV solution, there are two nodes $s\\\\in S,t\\\\in T$ with $d(s,t)\\\\ge 3k-2$ .\",\n \"We modify the construction as follows.\",\n \"For every $s\\\\in S$ , we create an undirected path on $k-2$ new vertices $s_1\\\\rightarrow s_2\\\\rightarrow \\\\ldots \\\\rightarrow s_{k-2}$ and add an edge $(s,s_1)$ ; let's call $s$ by $s_0$ .\",\n \"Now, the distance between $s_0$ and $s_i$ is $i$ .\",\n \"Add a new node $y$ and create edges $(s_{k-2},y)$ for every $s\\\\in S$ .\",\n \"Now, $d(y,s_0)=k-1$ for every $s\\\\in S$ , and also for every $s,s^{\\\\prime }\\\\in S$ and all $i,j\\\\in \\\\lbrace 0,\\\\ldots ,k-2\\\\rbrace $ , we have that $d(s_i,s^{\\\\prime }_j)\\\\le 2k-2$ .\",\n \"For every $s\\\\in S$ , $t\\\\in T$ , there is now potentially a new path between them, from $s$ to $y$ in $k-1$ steps, then to some other $s^{\\\\prime }$ in $k-1$ steps and then to $t$ using $\\\\ge k$ steps.\",\n \"The length is $\\\\ge 2(k-1)+k=3k-2$ , so when there is a $k$ -OV solution, there is still a pair $s,t$ at distance at least $3k-2$ .\",\n \"Now, we also attach paths to the nodes in $T$ .\",\n \"In particular, for each $t$ , add an undirected path $t\\\\rightarrow t_1\\\\rightarrow \\\\ldots \\\\rightarrow t_{k-1}$ .\",\n \"The distance between any $s\\\\in S$ and any $t_i$ is $i+d(s,t)$ .\",\n \"Hence when there is no $k$ -OV solution, the Eccentricities of all $s_0$ for $s\\\\in S$ are $\\\\le k+(k-1)=2k-1$ , and when there is a $k$ -OV solution, there is $s\\\\in S, t\\\\in T$ so that $d(s_0,t_{k-1})\\\\ge (3k-2)+(k-1)=4k-3$ .\"\n ],\n [\n \"Directed graphs\",\n \"Theorem 16 Under the 2-OV conjecture, for any $\\\\delta >0$ , any $(2-\\\\delta )$ -approximation algorithm for all Eccentricities in an $n$ node, $O(n)$ -edge directed unweighted graph, requires $n^{2-o(1)}$ time on a $O(\\\\log n)$ -bit word-RAM.\",\n \"Figure: The directed Eccentricities lower bound.Suppose we are given an instance of 2-OV: two sets of vectors $U,V$ over $\\\\lbrace 0,1\\\\rbrace ^d$ and we want to know whether there are $u\\\\in U,v\\\\in V$ with $u\\\\cdot v=0$ .\",\n \"Let $L\\\\ge 1$ be any integer.\",\n \"Let us create a directed unweighted graph $G$ ; an illustration can be found in Figure REF .\",\n \"$G$ will have a vertex $u$ for every $u\\\\in U$ and a vertex $c$ for every $c\\\\in [d]$ .\",\n \"Every $v\\\\in V$ will be represented by a directed path $v_0\\\\rightarrow v_1\\\\rightarrow \\\\ldots \\\\rightarrow v_L$ .\",\n \"In addition, there is a directed path $P_x$ on $L$ extra nodes, $x_1\\\\rightarrow \\\\ldots \\\\rightarrow x_L$ so that every $u\\\\in U$ has directed edges $(u,x_1)$ and $(x_L,u)$ .\",\n \"For every $u\\\\in U$ and every $c$ for which $u[c]=1$ , we add a directed edge $(u,c)$ .\",\n \"For every $v$ and every $c$ for which $v[c]=1$ , we add a directed edge $(c,v_0)$ .\",\n \"Remove any $c$ that does not have at least one edge coming from $U$ .\",\n \"Let us consider the eccentricity of any node $u\\\\in U$ .\",\n \"First, for all $u^{\\\\prime }\\\\in U$ , $d(u,u^{\\\\prime })\\\\le L+1$ since one can go through the path $P_x$ .\",\n \"For every $c\\\\in [d]$ , there is at least one edge coming from some $u^{\\\\prime }\\\\in U$ , and so one can reach $c$ from $u$ by first taking $P_x$ to $u^{\\\\prime }$ and then using the edge $(u^{\\\\prime },c)$ .\",\n \"Hence, $d(u,c)\\\\le L+2$ for all $c\\\\in [d]$ .\",\n \"For any $v\\\\in V$ and $i\\\\in \\\\lbrace 1,\\\\ldots , L\\\\rbrace $ , the distance $d(u,v_i)=i+d(u,v_0)$ , and so we consider $d(u,v_0)$ .\",\n \"If there is a $c$ for which $u[c]=v[c]=1$ , then $d(u,v_0)=2$ , and hence for all $i$ , $d(u,v_i)\\\\le L+2$ .\",\n \"If no such $c$ exists and so if $u$ and $v$ are orthogonal, the only way to reach $v_0$ is potentially via $P_x$ to some other $u^{\\\\prime }\\\\in U$ which is at distance 2 to $v_0$ .\",\n \"Hence if $u$ and $v$ are orthogonal, $d(u,v_0)=L+3$ , and hence $d(u,v_L)=2L+3$ .\",\n \"In other words, the eccentricity of $u$ is $L+2$ if it is not orthogonal to any vectors in $V$ and it is $\\\\ge 2L+3$ if there is some $v$ that is orthogonal to $u$ .\",\n \"The number of vertices in the graph is $O(nL+d)$ and the number of edges is $O(nL+nd)$ .\",\n \"Suppose that there is a $(2-\\\\varepsilon )$ -approximation algorithm for all Eccentricities in graphs with $O(m)$ nodes and edges running in $O(m^{2-\\\\delta })$ time for some $\\\\varepsilon ,\\\\delta >0$ .\",\n \"Then, we construct the above instance for $L=\\\\lceil 1/\\\\varepsilon \\\\rceil $ and run the algorithm on it.\",\n \"The approximation returned is at least as good as a $(2-1/L)$ -approximation.\",\n \"Hence if the diameter is at least $2L+3$ , the algorithm will return an estimate that is at least $L(2L+3)/(2L-1) > L+2$ .\",\n \"Thus the algorithm can solve 2-OV in time $O((nL+nd)^{2-\\\\delta }) = O(n^{2-\\\\delta }d^{2-\\\\delta })$ , contradicting the 2-OV conjecture.\"\n ],\n [\n \"Diameter lower bounds\",\n \"For all of our constructions we begin with the $S$ -$T$ diameter lower bound construction from Theorem REF .\",\n \"Here, if the $k$ -OV instance has no solution, $D_{S,T}\\\\le k$ and if the instance has a solution $D_{S,T}\\\\ge 3k-2$ .\",\n \"To adapt this construction to Diameter, we need to ensure that if the OV instance has no solution then all pairs of vertices have small enough distance.\",\n \"We begin by augmenting the $S$ -$T$ Diameter construction by adding a matching between $S$ and a new set $S^{\\\\prime }$ as well as a matching between $T$ and a new set $T^{\\\\prime }$ .\",\n \"Without any further modifications, pairs of vertices $u,v\\\\in S\\\\cup S^{\\\\prime }$ (or $u,v\\\\in T\\\\cup T^{\\\\prime }$ ) could be far from one another.\",\n \"The challenge is to add extra gadgetry to make these pairs close for “no\\\" instances while maintaining that in “yes\\\" instances the distance between the diameter endpoints $s^{\\\\prime }\\\\in S^{\\\\prime },t^{\\\\prime }\\\\in T^{\\\\prime }$ is large.\",\n \"That is, for “yes\\\" instances, we want a shortest path between the diameter endpoints $s^{\\\\prime }$ and $t^{\\\\prime }$ to contain the vertex $s\\\\in S$ matched to $s^{\\\\prime }$ and the vertex $t\\\\in T$ matched to $t^{\\\\prime }$ so that we can use use the fact that $d(s,t)\\\\ge 3k-2$ .\",\n \"In other words, we do not want there to be a shortcut from $s^{\\\\prime }$ to some vertex in $S$ that allows us to use a path of length $k$ from $S$ to $T$ .\",\n \"For example, we cannot simply create a vertex $x$ and connect it to all vertices in $S\\\\cup S^{\\\\prime }$ because this would introduce shortcuts from $S^{\\\\prime }$ to $S$ .\",\n \"We will describe some intuition for the augmentations to the graph regarding 3-OV for simplicity.\",\n \"Recall that $s^{\\\\prime }\\\\in S^{\\\\prime }, t^{\\\\prime }\\\\in T^{\\\\prime }$ are the endpoints of the diameter and let $t$ be the vertex matched to $t^{\\\\prime }$ .\",\n \"To solve the problem outlined in the above paragraph, we observe that in the “yes\\\" case there are three types of vertices $s\\\\in S$ .\",\n \"(1) close: $d(s,t)=3$ , (2) far: $d(s,t)\\\\ge 7$ (property 4 of Theorem REF ), and (3) intermediate: $d(s,t)\\\\ge 5$ (property 5 of Theorem REF ).\",\n \"For close $s$ , we need $d(s^{\\\\prime },s)$ to be large so that there is no shortcut from $s^{\\\\prime }$ to $t^{\\\\prime }$ through $s$ .\",\n \"For far $s$ , it is ok if $d(s^{\\\\prime },s)$ is small because $d(s,t)$ is large enough to ensure that paths from $s^{\\\\prime }$ to $t^{\\\\prime }$ through $s$ are still long enough.\",\n \"For intermediate $s$ , $d(s^{\\\\prime },s)$ cannot be small, but it also need not be large.\",\n \"To fulfill these specifications, we add a small clique (the graph is still sparse) and connect each of its vertices to only some of the vertices in $S$ and/or $S^{\\\\prime }$ according to the implications of property 5 of Theorem REF .\",\n \"When $s$ is close, we ensure that $d(s^{\\\\prime },s)$ is large by requiring that a shortest path from $s^{\\\\prime }$ to $s$ goes from $s^{\\\\prime }$ to the clique, uses an edge inside of the clique, and then goes from the clique to $s$ .\",\n \"When $s$ is intermediate, we ensure that $d(s^{\\\\prime },s)$ is not too small by requiring that a shortest path from $s^{\\\\prime }$ to $s$ goes from $s^{\\\\prime }$ to the clique and then from the clique to $s$ (without using an edge inside of the clique).\",\n \"These intermediate $s$ are important as they allow every vertex in the clique to have an edge to some vertex in $S$ and thus be close enough to the $T$ side of the graph in the “no\\\" case.\"\n ],\n [\n \"5 vs 8 unweighted undirected construction\",\n \"In this section we show that under the 3-OV Hypothesis, any algorithm that can distinguish between Diameter 5 and 8 in sparse undirected unweighted graphs, requires $\\\\Omega (n^{3/2-o(1)})$ time.\",\n \"Theorem REF gives us the following theorem.\",\n \"Theorem 17 Given a 3-OV instance consisting of three sets $A,B,C \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , $|A|=|B|=|C|=n$ , we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following properties.\",\n \"The graph consists of 4 layers of vertices $S,L_1,L_2,T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^2$ and $|L_1|,|L_2|\\\\le nd^2$ .\",\n \"$S$ consists of all tuples $(a,b)$ of vertices $a \\\\in A$ and $b \\\\in B$ .\",\n \"Similarly, $T$ consists of all tuples $(b,c)$ of vertices $b \\\\in B$ and $c \\\\in C$ .\",\n \"If the 3-OV instance has no solution, then $d(u,v)=3$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the 3-OV instance has a solution $a \\\\in A, b \\\\in B, c \\\\in C$ with $a,b,c$ orthogonal, then $d((a,b)\\\\in S,(b,c) \\\\in T)\\\\ge 7$ .\",\n \"Setting $k=3,s=1$ in Property 5 of Theorem REF : for any $b^{\\\\prime } \\\\in B$ we have $d((a,b) \\\\in S,(b^{\\\\prime },c) \\\\in T)\\\\ge 5$ and $d((a,b^{\\\\prime }) \\\\in S,(b,c) \\\\in T)\\\\ge 5$ .\",\n \"For any vertex $u \\\\in L_1$ there exists a vertex $s \\\\in S$ that is adjacent to $u$ .\",\n \"Similarly, for any vertex $v \\\\in L_2$ there exists a vertex $t \\\\in T$ that is adjacent to $v$ .\",\n \"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the other properties.\",\n \"In the rest of the section we use Theorem REF to prove the following result.\",\n \"Theorem 18 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.\",\n \"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 5$ .\",\n \"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 8$ .\"\n ],\n [\n \"Construction of the graph\",\n \"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.\",\n \"Figure REF illustrates the construction of the graph.\",\n \"Figure: The illustration for the 5 vs 8 construction.\",\n \"The edges between sets S,L 1 ,L 2 S,L_1,L_2 and TT are not depicted.\",\n \"The edges between vertices in S ' S^{\\\\prime } and SS (TT and T ' T^{\\\\prime }) form a matching.\",\n \"Vertices in S '' S^{\\\\prime \\\\prime } (T '' T^{\\\\prime \\\\prime }) form a clique.We start by adding a set $S^{\\\\prime }$ of $n^2$ vertices.\",\n \"$S^{\\\\prime }$ consists of all tuples $(a,b)$ of vertices $a \\\\in A$ and $b \\\\in B$ .\",\n \"We connect every $(a,b) \\\\in S^{\\\\prime }$ to its counterpart $(a,b) \\\\in S$ .\",\n \"Thus, there is a matching between the sets of vertices $S$ and $S^{\\\\prime }$ .\",\n \"We also add another set $S^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$S^{\\\\prime \\\\prime }$ contains one vertex $a$ for every $a \\\\in A$ .\",\n \"For every pair of vertices from $S^{\\\\prime \\\\prime }$ we add an edge between the vertices.\",\n \"Thus, the $n$ vertices form a clique.\",\n \"Furthermore, for every vertex $a \\\\in S^{\\\\prime \\\\prime }$ we add an edge to $(a,b) \\\\in S$ for all $b \\\\in B$ .\",\n \"In total we added $n^2+n=O(n^2)$ vertices and $\\\\binom{n}{2}+2n^2=O(n^2)$ edges.\",\n \"We do a similar construction for the set $T$ of vertices.\",\n \"We add a set $T^{\\\\prime }$ of $n^2$ vertices - one vertex for every tuple $(b,c)$ of vertices $b \\\\in B$ and $c \\\\in C$ .\",\n \"We connect every $(b,c)\\\\in T^{\\\\prime }$ to $(b,c) \\\\in T$ .\",\n \"Finally, we add a set $T^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$T^{\\\\prime \\\\prime }$ contains one vertex for every vector $c \\\\in C$ .\",\n \"For every pair of vertices from $T^{\\\\prime \\\\prime }$ we add an edge between the vertices.\",\n \"We connect every $c \\\\in T^{\\\\prime \\\\prime }$ to $(b,c) \\\\in T$ for all $b \\\\in B$ .\",\n \"This finishes the construction of the graph.\",\n \"In the rest of the section we show that the construction satisfies the promised two properties.\"\n ],\n [\n \"Correctness of the construction\",\n \"We need to consider two cases.\"\n ],\n [\n \"Case 1: the 3-OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 5$ .\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 1.1: $u \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup L_1$ and {{formula:e49ba0d1-2dce-4e1e-bc34-0f08bcbafce8}}\",\n \"We observe that there exists $s \\\\in S$ that has $d(u,s)\\\\le 1$ .\",\n \"Indeed, if $u \\\\in S$ , then $s=u$ works.\",\n \"If $u \\\\in S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime }$ , then we are done by the construction.\",\n \"On the other hand, if $u \\\\in L_1$ , then there exists such an $s \\\\in S$ by property 6 from Theorem REF .\",\n \"Similarly we can show that there exists $t \\\\in T$ such that $d(v,t)\\\\le 1$ .\",\n \"Finally, by property 3 we have that $d(s,t)=3$ .\",\n \"Thus, we can upper bound the distance between $u$ and $v$ by $d(u,v)\\\\le d(u,s)+d(s,t)+d(t,v)\\\\le 1+3+1=5$ as required.\"\n ],\n [\n \"Case 1.2: $u,v \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup L_1$\",\n \"From the previous case we know that there are two vertices $s_1,s_2 \\\\in S$ such that $d(u,s_1)\\\\le 1$ and $d(s_2,v)\\\\le 1$ .\",\n \"To show that $d(u,v)\\\\le 5$ it is sufficient to show that $d(s_1,s_2)\\\\le 3$ .\",\n \"This is indeed true since both vertices $s_1$ and $s_2$ are connected to some two vertices in $S^{\\\\prime \\\\prime }$ and every two vertices in $S^{\\\\prime \\\\prime }$ are at distance at most 1 from each other.\"\n ],\n [\n \"Case 1.3: $u,v \\\\in T \\\\cup T^{\\\\prime } \\\\cup T^{\\\\prime \\\\prime } \\\\cup L_2$\",\n \"The case is analogous to the previous case.\"\n ],\n [\n \"Case 2: the 3-OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 8$ .\",\n \"Let $a \\\\in A, b \\\\in B, c \\\\in C$ be a solution to the 3-OV instance.\",\n \"We claim that $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 8$ .\",\n \"Let $P$ be an optimal path between $u=((a,b) \\\\in S^{\\\\prime })$ and $v=((b,c) \\\\in T^{\\\\prime })$ that achieves the smallest distance.\",\n \"We want to show that $P$ uses at least 8 edges.\",\n \"Let $t \\\\in T$ be the first vertex from the set $T$ that is on path $P$ .\",\n \"Let $s \\\\in S$ be the last vertex on path $P$ that belongs to $S$ and precedes $t$ in $P$ .\",\n \"We can easily check that, if $s \\\\ne ((a,b) \\\\in S)$ , then $d(u,s)\\\\ge 3$ and, similarly, if $t \\\\ne ((b,c) \\\\in T)$ , then $d(t,v)\\\\ge 3$ .\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 2.1: $s \\\\ne ((a,b) \\\\in S)$ and {{formula:bf5aaac7-5b1b-4304-83ad-0a1fd19bda3a}}\",\n \"Since $s$ and $t$ are separated by two layers of vertices, we must have $d(s,t)\\\\ge 3$ .\",\n \"Thus we get lower bound $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+3+3=9>8$ as required.\"\n ],\n [\n \"Case 2.2: $s = ((a,b) \\\\in S)$ and {{formula:a28d868f-f38c-4534-b9c1-df1ca7ddd683}}\",\n \"In this case we use property 4 and conclude $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)=1+d((a,b) \\\\in S,(b,c) \\\\in T)+1\\\\ge 1+7+1=9>8$ as required.\"\n ],\n [\n \"Case 2.3: either $s = ((a,b) \\\\in S)$ or {{formula:1d8b5c54-e4d4-4cf8-8777-ab4e928d855f}} holds but not both\",\n \"W.l.o.g.\",\n \"$s \\\\ne ((a,b) \\\\in S)$ and $t = ((b,c) \\\\in T)$ .\",\n \"If the path uses an edge in the clique on $S^{\\\\prime \\\\prime }$ before arriving at $s$ , then $d(u,s)\\\\ge 4$ and we get that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 4+3+1=8$ .\",\n \"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\\\prime }) \\\\in S)$ for some $b^{\\\\prime } \\\\in B$ .\",\n \"By property 5 we have $d(s,t)=d((a,b^{\\\\prime })\\\\in S,(b,c)\\\\in T)\\\\ge 5$ .\",\n \"We conclude that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+5+1=9>8$ as required.\"\n ],\n [\n \"6 vs 10 weighted undirected construction\",\n \"In this section we change the construction from Theorem REF to show that under the 3-OV Hypothesis, any algorithm that can distinguish between diameter 6 and 10 in sparse undirected weighted graphs requires $\\\\Omega (n^{3/2-o(1)})$ time.\",\n \"We get the following theorem.\",\n \"Theorem 19 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an weighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.\",\n \"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 6$ .\",\n \"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 10$ .\",\n \"Each edge of the graph has weight either 1 or 2.\"\n ],\n [\n \"Construction of the graph\",\n \"The construction of the graph is the same as in Theorem REF except all edges connecting vertices between sets $L_1$ and $L_2$ have weight 2 and all edges inside the cliques on nodes $S^{\\\\prime \\\\prime }$ and $T^{\\\\prime \\\\prime }$ have weight 2.\",\n \"All the remaining edges have weight 1.\"\n ],\n [\n \"Correctness of the construction\",\n \"The correctness proof is essentially the same as for Theorem REF .\",\n \"As before we consider two cases.\"\n ],\n [\n \"Case 1: the 3-OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 6$ .\",\n \"In the analysis of Case 1 in Theorem REF we show a path between $u$ and $v$ such that the path involves at most one edge from the cliques or between sets $L_1$ and $L_2$ .\",\n \"Since we added weight 2 to the latter edges, the length of the path increased by at most 1 as a result.\",\n \"So we have upper bound $d(u,v)\\\\le 6$ for all pairs $u$ and $v$ of vertices.\"\n ],\n [\n \"Case 2: the 3-OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 10$ .\",\n \"Similarly to Theorem REF we will show that $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 10$ , where $a \\\\in A, b \\\\in B, c \\\\in C$ is a solution to the 3-OV instance.\",\n \"The analysis of the subcases is essentially the same as in Theorem REF .\",\n \"For cases 2.1 and 2.2 in the proof of Theorem REF we had $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 9$ .\",\n \"Since we increased edge weights between $L_1$ and $L_2$ to 2 and every path from $(a,b) \\\\in S^{\\\\prime }$ to $(b,c) \\\\in T^{\\\\prime }$ must cross the layer between $L_1$ and $L_2$ , we also increased the lower bound of the length of the path from 9 to 10 for cases 2.1 and 2.2.\",\n \"It remains to consider Case 2.3.\",\n \"As in the proof of Theorem REF , w.l.o.g.\",\n \"$s \\\\ne ((a,b) \\\\in S)$ and $t = ((b,c) \\\\in T)$ .\",\n \"If the path uses an edge in the clique on $S^{\\\\prime \\\\prime }$ before arriving at $s$ , then $d(u,s)\\\\ge 5$ and we get lower bound $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 5+4+1=10$ .\",\n \"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\\\prime }) \\\\in S)$ for some $b^{\\\\prime } \\\\in B$ .\",\n \"By property 5 and because we increased edge weights between $L_1$ and $L_2$ to 2, we have $d(s,t)=d((a,b^{\\\\prime })\\\\in S,(b,c)\\\\in T)\\\\ge 6$ .\",\n \"We conclude that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+6+1=10$ as required.\"\n ],\n [\n \"$3k-4$ vs {{formula:cef9204d-0a6e-4579-9476-41520f256b04}} unweighted directed construction\",\n \"In this section, we show that under SETH, for every $k\\\\ge 3$ , every algorithm that can distinguish between Diameter $3k-4$ and $5k-7$ in directed unweighted graphs requires $\\\\Omega (n^{1+1/(k-1)-o(1)})$ time.\",\n \"Theorem REF gives us the following theorem.\",\n \"Theorem 20 Given a $k$ -OV instance consisting of $k\\\\ge 2$ sets $W_0,W_1,\\\\dots ,W_{k-1} \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.\",\n \"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\\\dots ,L_k=T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\\\dots ,|L_{k-1}|\\\\le n^{k-2}d^{k-1}$ .\",\n \"$S$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $T$ consists of all tuples $(b_1,b_2,\\\\ldots , b_{k-1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then if $\\\\alpha =(a_0,\\\\dots a_{k-2})\\\\in S$ and $\\\\beta = (a_1,\\\\dots ,a_{k-1}) \\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge 3k-2$ .\",\n \"Setting $s=k-2$ in Property 5 of Theorem REF : If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then for any tuple $(b_1,\\\\dots ,b_{k-2})$ , if $\\\\alpha =(a_0,b_1,\\\\dots ,b_{k-2})\\\\in S$ and $\\\\beta =(a_1,\\\\dots ,a_{k-1})\\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge k+2$ .\",\n \"Symmetrically, if $\\\\alpha =(a_0,a_1,\\\\dots ,a_{k-2})\\\\in S$ and $\\\\beta =(b_1,\\\\dots ,b_{k-2},a_{k-1})\\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge k+2$ .\",\n \"For all $i$ from 1 to $k-1$ , for all $v \\\\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .\",\n \"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the previous three properties.\",\n \"In the rest of the section we use Theorem REF to prove the following result.\",\n \"Theorem 21 Given a $k$ -OV instance, we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, directed graph with $O(k n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(kn^{k-1}d^{k-1})$ edges that satisfies the following two properties.\",\n \"If the $k$ -OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 3k-4$ .\",\n \"If the $k$ -OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 5k-7$ .\"\n ],\n [\n \"Construction of the graph\",\n \"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.\",\n \"First we will construct a weighted graph and then we will make it unweighted.\",\n \"Figure REF illustrates the construction of the graph for the special case $k=4$ .\",\n \"Figure: The 3k-43k-4 vs 5k-75k-7 construction for the special case k=4k=4.\",\n \"The edges between sets S,L 1 ,L 2 ,L 3 S,L_1,L_2,L_3 and TT are not depicted.\",\n \"The matching between sets SS and S ' S^{\\\\prime } consists of unweighted paths of length k-2=2k-2=2.\",\n \"The edges between sets SS and S '' S^{\\\\prime \\\\prime } consists of unweighted paths of length k-2=2k-2=2.\",\n \"Similarly for the right side.We start by adding a set $S^{\\\\prime }$ of $n^{k-1}$ vertices.\",\n \"$S^{\\\\prime }$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We connect every $(a_0,a_1,\\\\ldots , a_{k-2}) \\\\in S^{\\\\prime }$ to its counterpart $(a_0,a_1,\\\\ldots , a_{k-2}) \\\\in S$ with an undirected edge of weight $k-2$ to form a matching.\",\n \"We also add another set $S^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$S^{\\\\prime \\\\prime }$ contains one vertex $a_0$ for every $a_0 \\\\in W_0$ .\",\n \"For every pair of vertices in $S^{\\\\prime \\\\prime }$ we add an undirected edge of weight 1 between the vertices.\",\n \"Thus, the $n$ vertices form a clique.\",\n \"Furthermore, for every vertex $a_0 \\\\in S^{\\\\prime \\\\prime }$ we add an undirected edge of weight $k-2$ to $(a_0,b_1,\\\\ldots , b_{k-2}) \\\\in S$ for all $b_1,\\\\dots , b_{k-2}$ .\",\n \"Finally for every vertex $a_0 \\\\in S^{\\\\prime \\\\prime }$ we add a directed edge of weight 1 towards $(a_0,b_1,\\\\ldots , b_{k-2}) \\\\in S$ for all $b_1,\\\\dots , b_{k-2}$ .\",\n \"Some of the edges that we added have weight $k-2$ .\",\n \"We make those unweighted by subdividing them into edges of weight 1.\",\n \"Let $S^{\\\\prime \\\\prime \\\\prime }$ be the set of newly added vertices.\",\n \"In total we added $O(kn^{k-1})$ vertices and $O(kn^{k-1})$ edges.\",\n \"We do a similar construction for the set $T$ of vertices.\",\n \"We add a set $T^{\\\\prime }$ of $n^{k-1}$ vertices — one vertex for every tuple $(a_1,\\\\dots ,a_{k-1})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We connect every $(a_1,\\\\dots ,a_{k-1})\\\\in T^{\\\\prime }$ to $(a_1,\\\\dots ,a_{k-1}) \\\\in T$ by an undirected edge of weight $k-2$ .\",\n \"Finally, we add a set $T^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$T^{\\\\prime \\\\prime }$ contains one vertex for every vector $a_{k-1} \\\\in W_{k-1}$ .\",\n \"We connect every pair of vertices in $T^{\\\\prime \\\\prime }$ by an undirected edge of weight 1.\",\n \"We connect every vertex $a_{k-1} \\\\in T^{\\\\prime \\\\prime }$ to $(b_1,\\\\dots ,b_{k-2},a_{k-1}) \\\\in T$ by an undirected edge of weight $k-2$ for all $b_1,\\\\dots ,b_{k-2}$ .\",\n \"Also, for every vertex $a_{k-1} \\\\in T^{\\\\prime \\\\prime }$ we add a directed edge of weight 1 from $(b_1,\\\\dots ,b_{k-2},a_{k-1}) \\\\in T^{\\\\prime }$ to $a_{k-1}$ for all $b_1,\\\\dots ,b_{k-2}$ .\",\n \"Some of the edges that we just added have weight $k-2$ .\",\n \"We make those unweighted by subdividing them into edges of weight 1.\",\n \"Let $T^{\\\\prime \\\\prime \\\\prime }$ be the set of newly added vertices.\",\n \"This finishes the construction of the graph.\",\n \"In the rest of the section we show that the construction satisfies the promised two properties stated in Theorem REF .\"\n ],\n [\n \"Correctness of the construction\",\n \"We need to consider two cases.\"\n ],\n [\n \"Case 1: the $k$ -OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 3k-4$ .\",\n \"We consider subcases.\"\n ],\n [\n \"Case 1.1: $u \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup S^{\\\\prime \\\\prime \\\\prime } \\\\cup L_i$ for {{formula:341c2d42-7167-4db5-bd6a-4b4996ef5fb1}} and {{formula:b26c11ee-44d0-4a3f-8d06-2943a8aaf115}} for {{formula:b2ebe27d-5f57-42a2-b20b-50932506480e}}\",\n \"We observe that there exists $s \\\\in S$ that has $d(u,s)\\\\le k-2$ .\",\n \"Similarly, there exists $t \\\\in T$ with $d(t,v)\\\\le k-2$ .\",\n \"By property 3 from Theorem REF we have that $d(s,t)\\\\le k$ .\",\n \"This gives us upper bound $d(u,v)\\\\le d(u,s)+d(s,t)+d(t,v)\\\\le (k-2)+k+(k-2)=3k-4$ as required.\",\n \"The proof when the sets for $u$ and $v$ are swapped is identical since we only use paths on unweighted edges.\"\n ],\n [\n \"Case 1.2: $u,v \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup S^{\\\\prime \\\\prime \\\\prime } \\\\cup L_1$\",\n \"We note that there is some vertex $s\\\\in S^{\\\\prime \\\\prime }$ with $d(u,s)\\\\le 2(k-2)$ (via undirected edges).\",\n \"Also, there is some vertex $s^{\\\\prime }\\\\in S^{\\\\prime \\\\prime }$ with $d(s^{\\\\prime },v)\\\\le k-1$ (possibly using directed edges).\",\n \"$S^{\\\\prime \\\\prime }$ is a clique so $d(s,s^{\\\\prime })\\\\le 1$ .\",\n \"Thus, $d(u,v)\\\\le d(u,s)+d(s,s^{\\\\prime })+d(s^{\\\\prime },v)\\\\le 2(k-2)+1+(k-1)=3k-4$ .\"\n ],\n [\n \"Case 1.3: $u,v \\\\in T \\\\cup T^{\\\\prime } \\\\cup T^{\\\\prime \\\\prime } \\\\cup T^{\\\\prime \\\\prime \\\\prime } \\\\cup L_{k-1}$\",\n \"This case is similar to the previous case.\",\n \"We note that there is some vertex $t\\\\in T^{\\\\prime \\\\prime }$ with $d(t,v)\\\\le 2(k-2)$ (via undirected edges).\",\n \"Also, there is some vertex $t^{\\\\prime }\\\\in T^{\\\\prime \\\\prime }$ with $d(u,t^{\\\\prime })\\\\le k-1$ (possibly using directed edges).\",\n \"$S^{\\\\prime \\\\prime }$ is a clique so $d(t^{\\\\prime },t)\\\\le 1$ .\",\n \"Thus, $d(u,v)\\\\le d(u,t^{\\\\prime })+d(t^{\\\\prime },t)+d(t,v)\\\\le (k-1)+1+2(k-2)=3k-4$ .\"\n ],\n [\n \"Case 2: the $k$ -OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 5k-7$ .\",\n \"Let $(a_0,a_1,\\\\ldots , a_{k-1})$ be a solution to the $k$ -OV instance where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We claim that $d((a_0,\\\\dots ,a_{k-2}) \\\\in S^{\\\\prime },(a_1,\\\\dots ,a_{k-1}) \\\\in T^{\\\\prime })\\\\ge 5k-7$ .\",\n \"Let $P$ be an shortest path between $u=((a_0,\\\\dots ,a_{k-2}) \\\\in S^{\\\\prime })$ and $v=((a_1,\\\\dots ,a_{k-1}) \\\\in T^{\\\\prime })$ .\",\n \"We want to show that $P$ uses at least $5k-7$ edges.\",\n \"Let $s \\\\in S$ be the first vertex on path $P$ that belongs to $S$ and let $t \\\\in T$ be the last vertex from the set $T$ that is on path $P$ .\",\n \"We observe that due to the directionality of the edges, $s$ and $t$ must be the counterparts of $u$ and $v$ respectively; that is, $s=((a_0,\\\\dots ,a_{k-2}) \\\\in S)$ and $t=((a_1,\\\\dots ,a_{k-1}) \\\\in T)$ .\",\n \"Note that these definitions of $s$ and $t$ differ from the definitions of $s$ and $t$ in previous proofs.\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 2.1: A vertex in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ appears after {{formula:6406b760-7ede-4dc4-b0f6-a59f46a09387}} on the path {{formula:a6c998cf-9396-4ea3-832a-1c6dca88cdb2}}\",\n \"We observe that if $s_1,s_2\\\\in S$ is a pair of vertices on the path $P$ such that no vertex in $S$ appears between them on $P$ , then the portion of $P$ between $s_1$ and $s_2$ either contains only vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ or contains no vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ .\",\n \"Let $s_1,s_2\\\\in S$ be such that the portion of $P$ between them contains only vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ .\",\n \"Such $s_1,s_2$ exist by the specification of this case.\",\n \"If $s_1=s_2$ then $P$ is not a shortest path.\",\n \"Otherwise, the portion of $P$ between $s_1$ and $s_2$ must include a vertex in $S^{\\\\prime \\\\prime }$ .\",\n \"Thus, $d(s_1,s_2)\\\\ge 2(k-2)$ .\",\n \"We consider three subcases.\",\n \"$s_1\\\\ne s$ .\",\n \"The distance between any pair of vertices in $S$ is at least 2 so $d(s,s_1)\\\\ge 2$ .\",\n \"Then, $d(u,v)\\\\ge d(u,s)+d(s,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+2+2(k-2)+k+(k-2)=5k-6$ .\",\n \"$s_1=s$ and $s_2=((a_0,b_1,\\\\dots ,b_{k-2}) \\\\in S)$ for some $b_1,\\\\dots ,b_{k-2}$ .\",\n \"In this case, by property 5 we have $d(s_2,t)\\\\ge k+2$ .\",\n \"Thus, $d(u,v)\\\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+2(k-2)+(k+2)+(k-2)=5k-6$ .\",\n \"$s_1=s$ and $s_2=((b_0,\\\\dots ,b_{k-2} \\\\in S)$ for some with $b_0\\\\ne a_0$ .\",\n \"In this case, the path from $s_1$ to $s_2$ must include an edge in the clique $S^{\\\\prime \\\\prime }$ since these are the only edges among vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ for which adjacent tuples can differ with respect to their first element.\",\n \"Thus, $d(s_1,s_2)\\\\ge 2(k-2)+1\\\\ge 2k-3$ .\",\n \"Therefore, $d(u,v)\\\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+(2k-3)+k+(k-2)=5k-7$ .\"\n ],\n [\n \"Case 2.2: A vertex in $T^{\\\\prime }\\\\cup T^{\\\\prime \\\\prime }\\\\cup T^{\\\\prime \\\\prime \\\\prime }$ appears before {{formula:2e52092d-2023-4c53-a764-a41805329e43}} on the path {{formula:d86d9c20-8168-4be0-8d10-99eb7834ec04}}\",\n \"This case is analogous to the previous case.\"\n ],\n [\n \"Case 2.3: The portion of the path $P$ between {{formula:558fd34a-9236-4d24-8a2c-f5022f0a499a}} and {{formula:c0c71b02-54cb-42b0-8842-f210ddc962a7}} contains no vertices in {{formula:7c23b5b2-7499-4d20-8946-fe3d3dacb80d}}\",\n \"By property 4, $d(s,t)\\\\ge 3k-2$ .\",\n \"Thus, $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge (k-2)+(3k-2)+(k-2)=5k-6$ .\",\n \"We note that a slight modification of this construction gives a lower bound for higher values of Diameter.\",\n \"For any L, we can get an $L(3k-4)$ vs $L(5k-8)+1$ construction by subdividing all of the edges in the construction (into paths of length $L$ ) except for the directed edges and the edges within the cliques.\"\n ],\n [\n \"2-Approximation for Eccentricities in $\\\\tilde{O}(m\\\\sqrt{n})$ time\",\n \"Theorem 22 Given a weighted, directed $m$ edge $n$ node graph, in $\\\\tilde{O}(m\\\\sqrt{n})$ time we can output quantities $\\\\epsilon ^{\\\\prime }(v)$ such that for all $v \\\\in V$ we have $\\\\epsilon (v)/2\\\\le \\\\epsilon ^{\\\\prime }(v)\\\\le \\\\epsilon (v)$ .\",\n \"The algorithm is inspired by the 2-approximation algorithm for directed radius of Abboud, Vassilevska W., and Wang [7].\",\n \"We claim that the following algorithm achieves the above guarantees.\",\n \"Sample a random subset $S \\\\subset V$ of size $|S|=\\\\Theta (\\\\sqrt{n} \\\\log n)$ .\",\n \"With high probability for every $u \\\\in V$ we have $N^{\\\\text{in}}_{\\\\sqrt{n}}(u)\\\\cap S \\\\ne \\\\emptyset $ .\",\n \"Let $w$ be a vertex that maximizes $d(S,w)$ , which we find using Dijkstra's algorithm.\",\n \"Let $S^{\\\\prime }:=N^{\\\\text{in}}_{\\\\sqrt{n}}(w)$ .\",\n \"For every vertex $v \\\\in S^{\\\\prime }$ we output $\\\\epsilon ^{\\\\prime }(v)=\\\\epsilon (v)$ by running Dijkstra's algorithm and following the outgoing edges.\",\n \"For every vertex $v \\\\notin S^{\\\\prime }$ we output the estimate $\\\\epsilon ^{\\\\prime }(v)=\\\\max _{s \\\\in S\\\\cup \\\\lbrace w\\\\rbrace }d(v,s)$ .\",\n \"We can determine all these quantities by running Dijkstra's algorithm out of all vertices in $S \\\\cup \\\\lbrace w\\\\rbrace $ and following the incoming edges.\"\n ],\n [\n \"Correctness\",\n \"Consider an arbitrary vertex $v\\\\notin S^{\\\\prime }$ (if $v \\\\in S^{\\\\prime }$ , then we are done by the third step).\",\n \"If there exists $s \\\\in S$ such that $d(v,s)\\\\ge \\\\epsilon (v)/2$ , then we are done since $\\\\epsilon ^{\\\\prime }(v)\\\\ge d(v,s)\\\\ge \\\\epsilon (v)/2$ .\",\n \"Otherwise, we have $d(v,s)<\\\\epsilon (v)/2$ for all $s \\\\in S$ .\",\n \"Let $v^{\\\\prime }$ be a vertex that achieves $d(v,v^{\\\\prime })=\\\\epsilon (v)$ .\",\n \"By the triangle inequality we have $d(s,v^{\\\\prime })>\\\\epsilon (v)/2$ for all $s \\\\in S$ .\",\n \"Equivalently, $d(S,v^{\\\\prime })>\\\\epsilon (v)/2$ .\",\n \"This implies that $d(S,w)>\\\\epsilon (v)/2$ by our choice of $w$ .\",\n \"Since $d(S,w)>\\\\epsilon (v)/2$ and $S^{\\\\prime }=N^{\\\\text{in}}_{\\\\sqrt{n}}(w)$ intersects $S$ , we must have that $S^{\\\\prime }$ contains all vertices $u$ with $d(u,w)\\\\le \\\\epsilon (v)/2$ .\",\n \"Since $v \\\\notin S^{\\\\prime }$ , we must have $d(v,w)>\\\\epsilon (v)/2$ and we are done since $\\\\epsilon ^{\\\\prime }(v)\\\\ge d(v,w)>\\\\epsilon (v)/2$ .\"\n ],\n [\n \"Almost 2-Approximation for Eccentricities in almost linear time\",\n \"In contrast to our $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithm from the previous section, our near-linear time $(2+\\\\delta )$ -approximation algorithm is very different from all previously known algorithms.\",\n \"Our algorithm proceeds in iterations and maintains a set $S$ of nodes for which we still do not have a good eccentricity estimate.\",\n \"In each iteration either we get a good estimate for many new vertices and hence remove them from $S$ , or we remove all vertices from $S$ that have large eccentricities, and for the remaining nodes in $S$ we have a better upper bound on their eccentricities.\",\n \"After a small number of iterations we have a good estimate for all vertices of the graph.\",\n \"Theorem 23 Suppose that we are given a weighted, directed $m$ edge $n$ node graph.\",\n \"The weights of all edge are non-negative integers bounded by $n^{O(1)}$ .\",\n \"For any $1>\\\\tau >0$ we can in $\\\\tilde{O}(m/\\\\tau )$ time output quantities $\\\\varepsilon ^{\\\\prime }(v)$ such that for all $v \\\\in V$ we have ${1-\\\\tau \\\\over 2}\\\\varepsilon (v)\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ .\",\n \"We maintain a subset $S\\\\subseteq V$ of vertices $v$ for which we still do not have an estimate $\\\\varepsilon ^{\\\\prime }(v)$ .\",\n \"Initially $S=V$ and we will end with $|S|\\\\le O(1)$ .\",\n \"When $|S|\\\\le O(1)$ we can evaluate $\\\\varepsilon (v)$ for all $v \\\\in S$ in the total time of $O(m)$ .\",\n \"Also we maintain a value $D$ that upper bounds the largest eccentricity of a vertex in $S$ .\",\n \"That is, $\\\\varepsilon (v)\\\\le D$ for all $v \\\\in S$ .\",\n \"Initially we set $D=n^C$ for some large enough constant $C>0$ (we assume that the input graph is strongly connected).\",\n \"The algorithm proceeds in phases.\",\n \"Each phase takes $\\\\tilde{O}(m)$ time and either $|S|$ decreases by a factor of at least 2 or $D$ decreases by a factor of at least $1/(1-\\\\tau )$ .\",\n \"After $O(\\\\log (n)/\\\\tau )$ phases either $|S|\\\\le O(1)$ or $D<1$ .\",\n \"For a subset $S \\\\subseteq V$ of vertices and a vertex $x \\\\in V$ we define a set $S_x\\\\subseteq S$ to contain those $|S_x|=|S|/2$ vertices from $S$ that are closest to $x$ (according to distance $d(\\\\cdot ,x)$ ).\",\n \"The ties are broken by taking the vertex with the smaller id.\",\n \"Given a subset $S \\\\subseteq V$ of vertices and a threshold $D$ , a phase proceeds as follows.\",\n \"[itemsep=0mm] We sample a set $A \\\\subseteq S$ of $O(\\\\log n)$ random vertices from the set $S$ .\",\n \"With high probability for all $x \\\\in V$ we have $A \\\\cap S_x \\\\ne \\\\emptyset $ .\",\n \"Let $w \\\\in V$ be a vertex that maximizes $d(A,w)$ .\",\n \"We can find it using Dijkstra's algorithm.\",\n \"We consider two cases.\",\n \"Case $d(S\\\\setminus S_w,w)\\\\ge {1-\\\\tau \\\\over 2}D$ .\",\n \"For all $x \\\\in S\\\\setminus S_w$ we have ${1-\\\\tau \\\\over 2} D \\\\le \\\\varepsilon (x)\\\\le D$ and we assign the estimate $\\\\varepsilon ^{\\\\prime }(x)={1-\\\\tau \\\\over 2}D$ .\",\n \"This gives us that ${1-\\\\tau \\\\over 2} \\\\varepsilon (x)\\\\le {1-\\\\tau \\\\over 2}D=\\\\varepsilon ^{\\\\prime }(x)\\\\le \\\\varepsilon (x)$ for all $x\\\\in S\\\\setminus S_w$ .\",\n \"We update $S$ to be $S_w$ .\",\n \"This decreases the size of $S$ by a factor of 2 as required.\",\n \"Case $d(S\\\\setminus S_w,w)<{1-\\\\tau \\\\over 2}D$ .\",\n \"Set $S^{\\\\prime }=S$ .\",\n \"For every vertex $v \\\\in S$ evaluate $r_v:=\\\\max _{x \\\\in A}d(v,x)$ .\",\n \"We can evaluate these quantities by running Dijkstra's algorithm from every vertex in $A$ and following the incoming edges.\",\n \"If $r_v\\\\ge {1-\\\\tau \\\\over 2}D$ , then assign the estimate $\\\\varepsilon ^{\\\\prime }(v)={1-\\\\tau \\\\over 2}D$ and remove $v$ from $S^{\\\\prime }$ .\",\n \"Similarly as in the previous case we have ${1-\\\\tau \\\\over 2} \\\\varepsilon (v)\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ for all $v \\\\in S\\\\setminus S^{\\\\prime }$ .\",\n \"Below we will show that for every $v \\\\in S^{\\\\prime }$ we have $\\\\varepsilon (v)\\\\le (1-\\\\tau )D$ .\",\n \"Thus we can update $S=S^{\\\\prime }$ and decrease the threshold $D$ to $(1-\\\\tau )D$ as required.\",\n \"Correctness We have to show that, if there exists $v \\\\in S^{\\\\prime }$ such that $\\\\varepsilon (v)>(1-\\\\tau )D$ , then we will end up in the first case (this is the contrapositive of the claim in the second case).\",\n \"Since $v \\\\in S^{\\\\prime }$ we must have that $d(v,x)\\\\le {1-\\\\tau \\\\over 2}D$ for all $x \\\\in A$ .\",\n \"Since $\\\\varepsilon (v)>(1-\\\\tau )D$ , we must have that there exists $v^{\\\\prime }$ such that $d(v,v^{\\\\prime })>(1-\\\\tau )D$ .\",\n \"By the triangle inequality we get that $d(x,v^{\\\\prime })>{1 - \\\\tau \\\\over 2}D$ for every $x \\\\in A$ .\",\n \"By choice of $w$ we have $d(A,w)>{1-\\\\tau \\\\over 2}D$ .\",\n \"Since $A \\\\cap S_w \\\\ne \\\\emptyset $ , we have $d(S\\\\setminus S_w,w)\\\\ge {1-\\\\tau \\\\over 2}D$ and we will end up in the first case.\",\n \"The guarantee on the approximation factor follows from the description.\",\n \"As a corollary, we get an algorithm for Source Radius with the same runtime and approximation ratio as Theorem REF .\",\n \"First, run the Eccentricities algorithm and let $v$ be the vertex with the smallest estimated eccentricity $\\\\epsilon ^{\\\\prime }(v)$ .\",\n \"Then run Dijkstra's algorithm from $v$ and report $\\\\epsilon (v)$ as the Radius estimate $R^{\\\\prime }$ .\",\n \"Let $R$ be the true radius of the graph and let $x$ be the vertex with minimum Eccentricity i.e.\",\n \"$\\\\epsilon (x)=R$ .\",\n \"If $\\\\alpha $ is the approximation ratio for the Eccentricities algorithm then $\\\\epsilon (v)\\\\le \\\\alpha \\\\epsilon ^{\\\\prime }(v)\\\\le \\\\alpha \\\\epsilon (v)$ and $\\\\epsilon (x)\\\\le \\\\alpha \\\\epsilon ^{\\\\prime }(x)\\\\le \\\\alpha \\\\epsilon (x)$ .\",\n \"By choice of $v$ , $\\\\epsilon ^{\\\\prime }(v)\\\\le \\\\epsilon ^{\\\\prime }(x)$ .\",\n \"Thus, $\\\\alpha R=\\\\alpha \\\\epsilon (x)\\\\ge \\\\alpha \\\\epsilon ^{\\\\prime }(x)\\\\ge \\\\alpha \\\\epsilon ^{\\\\prime }(v)\\\\ge \\\\epsilon (v)=R^{\\\\prime }$ .\",\n \"Clearly $R^{\\\\prime }\\\\ge R$ , so $R\\\\le R^{\\\\prime }\\\\le \\\\alpha R$ .\",\n \"$S$ -$T$ Diameter algorithms Recall that the $S$ -$T$ diameter problem is as follows: Given an undirected graph $G=(V,E)$ and two sets $S\\\\subseteq V, T\\\\subseteq V$ , determine $D_{S,T}=\\\\max _{s\\\\in S, t\\\\in T} d(s,t)$ .\",\n \"Here we will outline two algorithms for the problem.\",\n \"Let us first consider a fast 3-approximation algorithm.\",\n \"Claim 24 There is an $O(m+n)$ time algorithm that for any $n$ node $m$ edge graph $G=(V,E)$ and $S\\\\subseteq V, T\\\\subseteq V$ , computes an estimate $D^{\\\\prime }$ such that $D_{S,T}/3\\\\le D^{\\\\prime }\\\\le D_{S,T}$ and two nodes $s\\\\in S$ , $t\\\\in T$ such that $d(s,t)=D^{\\\\prime }$ .\",\n \"In graphs with nonnegative weights, the same estimate can be achieved in $O(m+n\\\\log n)$ time.\",\n \"The algorithm is extremely simple: pick arbitrary nodes $s\\\\in S$ and $t\\\\in T$ , compute BFS($s$ ) and BFS($t$ ) and return $\\\\max \\\\lbrace \\\\max _{t^{\\\\prime }\\\\in T} d(s,t^{\\\\prime }),\\\\max _{s^{\\\\prime }\\\\in S} d(s^{\\\\prime },t)\\\\rbrace $ (also returning the two nodes achieving the maximum).\",\n \"For weighted graphs, run Dijkstra's algorithm instead of BFS.\",\n \"Let's see why this algorithm provides the promised guarantee.\",\n \"Suppose that for every $t^{\\\\prime }\\\\in T$ , $d(s,t^{\\\\prime })d$ .\",\n \"Let $a$ be the node on the shortest path between $\\\\bar{t}$ and $s^*$ with $d(\\\\bar{t},a)=d$ .\",\n \"We thus have that $a\\\\in Y$ .\",\n \"Also, since $d(\\\\bar{t},s^*)\\\\le D/2$ , $d(a,s^*)\\\\le D/2-d$ and hence $d(a,s_a)\\\\le D/2-d$ , so that $d(s_a,t^*)\\\\ge D-2(D/2-d)\\\\ge 2d$ .\",\n \"This finishes the argument that 2-Approx returns an estimate $D^{\\\\prime }$ with $2\\\\lfloor D/4\\\\rfloor \\\\le D^{\\\\prime }\\\\le D$ .\",\n \"It is not too hard to see that the only time that we might get an estimate that is less than $D/2$ is in the last part of the argument and only if the diameter is of the form $4d+3$ .\",\n \"(We will prove the algorithm guarantees formally soon.)\",\n \"The analysis fails to work in that case because $Y$ is guaranteed to contain only the nodes at distance $d$ from $\\\\bar{t}$ .\",\n \"In particular, if $Y$ contains all nodes at distance $d+1$ from $\\\\bar{t}$ instead of just those at distance at most $d$ , we could consider $a$ to be the node on the shortest path between $\\\\bar{t}$ and $s^*$ with $d(\\\\bar{t},a)=d+1$ , and $a\\\\in Y$ .\",\n \"Now since $d(\\\\bar{t},s^*)\\\\le 2d+1$ (as otherwise we'd be done), $d(a,s_a)\\\\le d(a,s^*)\\\\le 2d+1-d-1 =d$ , so that $d(s_a,t^*)\\\\ge 2d+3$ .\",\n \"Hence everything would work out.\",\n \"We handle this issue with a trick from Chechik et al. [21].\",\n \"First, we make graph have constant degree by blowing up the number of nodes and adding 0 weight edges as follows.\",\n \"Let $v$ be an original node and suppose it has degree $d(v)$ .\",\n \"Replace $v$ with a $d(v)$ -cycle of 0 weight edges so that each of the cycle nodes is connected to a one of the neighbors of $v$ , where each neighbor has a cycle node corresponding to it.\",\n \"This makes every node have degree 3 and increases the number of vertices to $O(m)$ .\",\n \"Now, we run algorithm 2-Approx with two changes.\",\n \"The first is that instead of BFS we use Dijkstra's algorithm because the edges now have weights.\",\n \"The second change is that we redefine $Y$ as follows.\",\n \"Let $Z$ be the closest $\\\\sqrt{m}$ nodes $Z$ to $\\\\bar{t}$ .\",\n \"Define $Y$ to be $Z$ , together with all nodes that have an edge of weight 1 to some node of $Z$ .\",\n \"Now, consider the shortest path $P$ between $\\\\bar{t}$ and $s^*$ .\",\n \"Consider the last node $a$ of $P$ (in the direction from $\\\\bar{t}$ to $s^*$ ) that has distance $\\\\le d$ from $\\\\bar{t}$ .\",\n \"The node $a^{\\\\prime }$ after $a$ on $P$ must be connected to $a$ by an edge $(a,a^{\\\\prime })$ of weight 1, as otherwise $a^{\\\\prime }$ would be at distance at most $d$ from $\\\\bar{t}$ .\",\n \"Since $a\\\\in Z$ , we must have $a^{\\\\prime }\\\\in Y$ and we know also that $d(\\\\bar{t},a^{\\\\prime })=d+1$ .\",\n \"In the last case when $d(\\\\bar{t},s^*)\\\\le 2d+1$ , we get that $Y$ contains a node $a^{\\\\prime }$ of distance $\\\\le (2d+1)-(d+1) = d$ from $s^*$ and hence $d(a^{\\\\prime },s_{a^{\\\\prime }})\\\\le d$ and hence $d(s_{a^{\\\\prime }},t^*)\\\\ge 2d+3$ .\",\n \"Since every node has degree 3, the number of nodes in $Y$ is $\\\\le 4|Z|\\\\le O(\\\\sqrt{m})$ and hence we can afford to run Dijkstra from each of them and complete the algorithm in $\\\\tilde{O}(m^{3/2})$ time.\",\n \"Let us now formally analyze the guarantees of the algorithm.\",\n \"Suppose that $D=4d+z$ where $z\\\\in \\\\lbrace 0,1,2,3\\\\rbrace $ ; we will show that the estimate that the algorithm gives is always at least $\\\\lceil D/2\\\\rceil =2d+\\\\lceil z/2\\\\rceil $ .\",\n \"If some node $x\\\\in X$ has $d(x,t^*)\\\\le d$ , we get that $d(t_x,s^*)\\\\ge D-2d=2d+z\\\\ge \\\\lceil D/2\\\\rceil $ .\",\n \"If we are not done, all nodes of $X$ have $d(x,\\\\bar{t})\\\\ge d(x,t^*)\\\\ge d+1$ and $Z$ contains all nodes at distance $\\\\le d$ from $\\\\bar{t}$ .\",\n \"If $s^*\\\\in Z$ , we are done so we must have $d(s^*,\\\\bar{t})\\\\ge d+1$ .\",\n \"Consider the last node $a^{\\\\prime }$ on the $\\\\bar{t}$ to $s^*$ shortest path (in the direction towards $s^*$ ) for which $d(\\\\bar{t},a^{\\\\prime })\\\\le d$ .\",\n \"We have that $a^{\\\\prime }\\\\in Z$ .\",\n \"Also, the node $a$ after $a^{\\\\prime }$ on the $\\\\bar{t}$ to $s^*$ shortest path must be in $Y$ since by the choice of $a^{\\\\prime }$ , $d(\\\\bar{t},a)=t+1$ and $(a,a^{\\\\prime })$ is an edge of weight 1.\",\n \"If $d(\\\\bar{t},s^*)\\\\ge 2d+\\\\lceil z/2\\\\rceil $ , we are done, so we get that $d(a,s^*)\\\\le 2d+\\\\lceil z/2\\\\rceil -1 - (d+1) =d+\\\\lceil z/2\\\\rceil -2$ .\",\n \"Hence, $d(s_a,t^*)\\\\ge D-2(d+\\\\lceil z/2\\\\rceil -2)=2d + (z+4-2\\\\lceil z/2\\\\rceil )\\\\ge 2d+z$ .\",\n \"It is not hard to extend the $S$ -$T$ Diameter algorithms to work for weighted undirected graphs as well.\",\n \"The basic idea is to use Dijkstra's algorithm instead of BFS in Algorithm 1.\",\n \"This gives an almost 2-approximation.\",\n \"In particular, let $a^{\\\\prime }$ be the last node on the $\\\\bar{t}$ to $s^*$ shortest path that is at distance $\\\\le d$ from $\\\\bar{t}$ and let $a$ be the node after $a^{\\\\prime }$ .\",\n \"The weighted version of Algorithm 1 achieves a guarantee $D^{\\\\prime }$ of the $S$ -$T$ Diameter, such that $D/2-2w(a,a^{\\\\prime })\\\\le D^{\\\\prime }\\\\le D$ .\",\n \"We can obtain an $\\\\tilde{O}(m^{3/2})$ true 2-approximation algorithm similar to the proof above.\",\n \"Let $Z$ be the closest $\\\\sqrt{m}$ nodes to $\\\\bar{t}$ and extend $Z$ to $Y$ by adding all nodes that have a non-zero edge to a node of $Z$ .\",\n \"With this modification, the node $a$ is guaranteed to be in $Y$ and hence the estimate for the diameter is at least $D/2$ .\",\n \"Corollary 26 In $\\\\tilde{O}(m\\\\sqrt{n})$ time one can obtain an estimate $D^{\\\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node undirected graph with nonnegative edge weights such that $D/2 - 2w(a,a^{\\\\prime }) \\\\le D^{\\\\prime }\\\\le D$ for some edge $(a,a^{\\\\prime })$ .\",\n \"In $\\\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\\\prime \\\\prime }$ such that $D/2\\\\le D^{\\\\prime \\\\prime }\\\\le D$ .\",\n \"It is an easy exercise to see that when $D=2h+1$ then the value $\\\\max \\\\lbrace \\\\epsilon ^{in}(v), \\\\epsilon ^{out}(v) \\\\rbrace $ of an arbitrary vertex $v \\\\in V$ is an estimation to the diameter which is at least $h+1$ and at most $D$ .\",\n \"In this section we present a deterministic algorithm that gets a directed unweighted graph $G$ with $D=2h$ and computes in $O(m^2/n)$ time an estimation $\\\\hat{D}$ such that $h+1\\\\le \\\\hat{D} \\\\le D$ .\",\n \"The algorithm works as follows.\",\n \"A variable $\\\\hat{D}$ is set to zero.\",\n \"The algorithm searches for a vertex $v$ that its sum of in and out degree is minimal.\",\n \"Then the algorithm computes the in and out eccentricity of $v$ and every vertex that has an edge with $v$ (incoming or outgoing).\",\n \"The algorithm outputs the maximum of all the in and out eccentricities that were computed.\",\n \"Theorem 27 Let $G=(V,E)$ an unweighted directed graph and let $D=2h$ .\",\n \"Algorithm REF returns in $O(m^2/n)$ time an estimate $\\\\hat{D}$ such that $h+1\\\\le \\\\hat{D} \\\\le D$ .\",\n \"We start with the running time analysis.\",\n \"Consider the graph $G$ and ignore the edge directions.\",\n \"For every $v\\\\in V$ let $deg(v) = deg^{in}(v)+deg^{out}(v)$ .\",\n \"Since $m = \\\\frac{1}{2}\\\\sum _{v\\\\in V} deg(v)$ a vertex $v$ of minimal degree satisfies $deg(v)\\\\le 2m/n$ .\",\n \"Therefore, the cost of computing in and out eccentricities for all vertices in the set $N(v) \\\\cup \\\\lbrace v \\\\rbrace $ is $O(\\\\frac{m}{n} \\\\times m)$ .\",\n \"We now turn to bound $\\\\hat{D}$ .\",\n \"Let $a, b\\\\in V$ and let $d(a,b)=2h$ .\",\n \"Let $P(a,b)$ be a shortest path from $a$ to $b$ and let $v\\\\in P(a,b)$ .\",\n \"If $d(a,v)\\\\le h-1$ then $\\\\epsilon ^{out}(v)\\\\ge h+1$ .\",\n \"Similarly, if $d(v,b)\\\\le h-1$ then $\\\\epsilon ^{in}(v)\\\\ge h+1$ .\",\n \"Consider now the case that $d(a,v)=h$ and $d(v,b)=h$ .\",\n \"Let $u\\\\in P(a,b)$ be the vertex that precedes $v$ on the shortest path from $a$ to $b$ .\",\n \"Since $u$ has an incoming edge to $v$ it follows that $u\\\\in N(v)$ and $\\\\epsilon ^{out}(u)$ and $\\\\epsilon ^{in}(u)$ are computed.\",\n \"Since $d(a,v)=h$ it follows that $d(a,u)=h-1$ , $\\\\epsilon ^{out}(u)\\\\ge h+1$ and $\\\\hat{D}$ is at least $h+1$ .\",\n \"Fast approximation of the diameter [1] Diam-Approx$G$ $\\\\hat{D} = 0$ $v = \\\\arg \\\\min _{x\\\\in V} deg^{in}(x)+deg^{out}(x)$ every $w \\\\in N^{\\\\text{in}}(v) \\\\cup N^{\\\\text{out}}(v) \\\\cup \\\\lbrace v \\\\rbrace $ compute $\\\\epsilon ^{in}(w)$ and $\\\\epsilon ^{out}(w)$ $\\\\hat{D}= \\\\max \\\\lbrace \\\\hat{D},\\\\epsilon ^{in}(w),\\\\epsilon ^{out}(w)\\\\rbrace $ $\\\\hat{D}$ Algorithms for dense graphs Algorithm overview The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.\",\n \"[2] runs in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] removed the $\\\\tilde{O}(n^2)$ term to obtain an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"For every graph with $\\\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.\",\n \"Therefore, it is interesting to consider the opposite question to the one considered by [41].\",\n \"Can the $\\\\tilde{O}(m\\\\sqrt{n})$ term be removed?\",\n \"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\\\log n)$ expected time algorithm.\",\n \"For a graph of Diameter $D=3h+z$ , where $z\\\\in [0,1,2]$ our algorithm returns an estimation $\\\\hat{D}$ such that $2h-1\\\\le \\\\hat{D} \\\\le D$ , when $z \\\\in [0,1]$ and $2h\\\\le \\\\hat{D} \\\\le D$ , when $z=2$ .\",\n \"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.\",\n \"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.\",\n \"To enable this approach we can no longer sample $A$ naively.\",\n \"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.\",\n \"The expected running time of their algorithm is $\\\\tilde{O}(mn/|A|)$ .\",\n \"We provide a new implementation of their algorithm that runs in expected $\\\\tilde{O}(n(n/|A|)^2)$ time.\",\n \"The set $A$ has the following important property, for every vertex $w\\\\in V$ , its cluster (see [48]) $\\\\lbrace u \\\\mid d(u,w) 4/p \\\\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\\\log n$ .\",\n \"For every $w \\\\in V$ we then have $|C_A(w)|\\\\le 4/p$ .\",\n \"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\\\log n)$ .\",\n \"They did not provide the details and refer the reader to [48].\",\n \"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.\",\n \"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\\\log n)$ expected time implementation of Algorithm REF .\",\n \"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"This can be done as follows.\",\n \"Once we have computed $B_A(v)$ for every $v\\\\in V$ , we can scan $B_A(v)$ , and for every $w\\\\in B_A(v)$ we can add $v$ to $C_A(w)$ .\",\n \"The cost of this process is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\\\in V$ and we can compute $W$ (as needed in Algorithm REF ).\",\n \"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.\",\n \"To this end, more ideas are needed.\",\n \"The following simple observation helps us to achieve our goal.\",\n \"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .\",\n \"Let $A^*$ be a set such that $A^*\\\\subseteq A_i$ , for every $i\\\\ge 1$ .\",\n \"For every $v\\\\in V$ it holds that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ .\",\n \"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\\\le O(1/p)$ for every $v\\\\in V$ .\",\n \"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].\",\n \"This implies that we can compute $N_{1/p}(v)$ for every $v\\\\in V$ in $O(n/p^2)$ time.\",\n \"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\\\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ (a greedy algorithm works; e.g.\",\n \"see [48]).\",\n \"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\\\in V$ and the hitting set $A$ , as described above.\",\n \"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"In more detail, our algorithm works as follows.\",\n \"For every $v\\\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.\",\n \"Then it finds a set $A$ such that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ .\",\n \"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\\\in V$ the bunch $B_A(v)$ .\",\n \"Finally, it computes the clusters and $W$ using the bunches as we described above.\",\n \"The rest of the algorithm is almost identical to Algorithm REF .\",\n \"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\\\in V$ .\",\n \"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\\\in V$ .\",\n \"compute $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .\",\n \"every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\\\cup X$ every $v\\\\in V$ compute $B_A(v)$ every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\\\log n)$ expected time a set $A$ of expected size $O((n/p)\\\\cdot \\\\log n)$ that guarantees for every vertex $w\\\\in V \\\\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\\\in V$ that $|B_A(v)|=O(1/p)$ .\",\n \"The cost of computing $N_{1/p}(v)$ for every $v\\\\in V$ is $O(n (1/p)^2)$ [24].\",\n \"The cost of computing $A$ is $O(n/p)$ time [48].\",\n \"Computing $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .\",\n \"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .\",\n \"As we explained earlier the cost of computing clusters using bunches is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ .\",\n \"Since for every $v\\\\in V$ we have $B_A(v)\\\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .\",\n \"This completes the analysis of the part that precedes the while loop.\",\n \"Next, we analyze the cost of the while loop.\",\n \"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.\",\n \"From Observation REF it follows that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1}) h$ or $d(a,A)>h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .\",\n \"If this is not the case then either $d(a,A)\\\\le h$ or $d(b,A)\\\\le h$ (or both).\",\n \"Assume, wlog, that $d(a,A)\\\\le h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .\",\n \"We now turn to we analyze the running time of Algorithm REF .\",\n \"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\\\log n)$ .\",\n \"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\\\sqrt{n}$ .\",\n \"From Lemma REF it follows that the size of the set $A$ is $O(\\\\sqrt{n} \\\\log n)$ and its construction time is $O(n^2\\\\log n)$ in expectation.\",\n \"For every $w\\\\in V$ the size of $C_A(w)$ is $O(\\\\sqrt{n})$ .\",\n \"Therefore, Step 1 takes $O(n \\\\times |C_A(w)|^2)=O(n^2)$ .\",\n \"Step 2 takes $O(n^2)$ time as well.\",\n \"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.\",\n \"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.\",\n \"We also compute $|A|$ shortest paths trees in $H$ .\",\n \"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\\\log n)$ time.\",\n \"Near linear almost 5/3-approximation for Eccentricities almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ every $u \\\\in V$ $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ $\\\\epsilon ^{\\\\prime }(u) = \\\\max (\\\\epsilon _1(u),\\\\epsilon _2(u),\\\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.\",\n \"We run lines 2-11 of Algorithm REF .\",\n \"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"Then, for every $u\\\\in V$ we compute $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ , which are defined as follows: $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ , $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ and $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ .\",\n \"The algorithm sets $\\\\epsilon ^{\\\\prime }(u)$ to $\\\\max \\\\lbrace \\\\epsilon _1(u), \\\\epsilon _2(u), \\\\epsilon _3(u) \\\\rbrace $ for every $u\\\\in V$ as an estimation to $\\\\epsilon (u)$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"We now prove: Theorem 35 For every $u\\\\in V$ , Algorithm REF computes in $O(n^2 \\\\log n)$ expected time a value $\\\\epsilon ^{\\\\prime }(u)$ that satisfies: $\\\\frac{3\\\\epsilon (u)}{5}-1 \\\\le \\\\epsilon ^{\\\\prime }(u)\\\\le \\\\epsilon (u).$ We start by analyzing the running time.\",\n \"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\\\log n)$ time in expectation.\",\n \"The computation of $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ for every $u\\\\in V$ costs $O(n^2)$ time in total.\",\n \"Let $u\\\\in V$ be an arbitrary vertex and let $\\\\epsilon (u)=d(u,t)$ .\",\n \"We now turn to bound $\\\\epsilon ^{\\\\prime }(u)$ .\",\n \"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\\\in V$ that $\\\\epsilon (u)\\\\ge \\\\epsilon (v)-d(u,v)$ .\",\n \"It is straightforward to see that both $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ are at most $\\\\epsilon (u)$ .\",\n \"Recall that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2\\\\le \\\\epsilon (p(u))-d(u,p(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u) = d_H(u,y)-2\\\\le d(u,y)\\\\le \\\\epsilon (u)$ .\",\n \"We distinguish between two cases.\",\n \"Case 1: $B_A(u) \\\\cap B_A(t)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that $P(u,t)\\\\cap (B_A(u) \\\\cap B_A(t)) \\\\ne \\\\emptyset $ and $M(u,t)= \\\\epsilon (u)$ .\",\n \"From Lemma REF it follows that $M(u,w)\\\\le d(u,w)$ for every $w\\\\in V$ after Step 2.\",\n \"Therefore, $\\\\epsilon _1(u)=\\\\epsilon (u)$ .\",\n \"Since $\\\\epsilon _2(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u)\\\\le \\\\epsilon (u)$ we get that $\\\\epsilon ^{\\\\prime }(u)=\\\\epsilon (u)$ .\",\n \"Case 2: $B_A(u) \\\\cap B_A(t) = \\\\emptyset $ .\",\n \"Consider first the case that $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"From Observation REF we get that $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d_H(u,p(u))$ .\",\n \"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d(u,p(u))$ .\",\n \"Hence, we get that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2 \\\\ge \\\\epsilon _H(u)-2d(u,p(u))-2\\\\ge \\\\epsilon (u)-2d(u,p(u))-2$ .\",\n \"As before we have $\\\\epsilon _2(u) \\\\le \\\\epsilon (u)$ .\",\n \"Using $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ we get that: $\\\\epsilon _2(u) \\\\ge \\\\epsilon (u)-\\\\frac{2\\\\epsilon (u)}{5}\\\\ge \\\\frac{3\\\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\\\ge \\\\frac{\\\\epsilon (u)}{5}$ .\",\n \"This means that $d(u,A)-1\\\\ge \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"Let $S$ be the set of all vertices $v\\\\in V$ such that $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ , that is, $S=V \\\\setminus \\\\cup _{w\\\\in B_A(u)} C_A(w)$ .\",\n \"Let $t^{\\\\prime } = \\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in S} d(x,A)-1$ .\",\n \"If $d(t^{\\\\prime },A)-1 \\\\ge \\\\frac{2\\\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\\\prime })\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"Assume now that $d(t^{\\\\prime },A) < \\\\frac{2\\\\epsilon (u)}{5}$ .\",\n \"As $t^{\\\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\\\frac{2\\\\epsilon (u)}{5}$ and $d(u,p(t))>\\\\frac{3\\\\epsilon (u)}{5}$ .\",\n \"Therefore, $\\\\epsilon _3(u) = d_H(u,y)-2\\\\ge d(u,p(t))-2\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"From Lemma REF and Lemma REF it follows that $\\\\epsilon _1(u)\\\\le \\\\epsilon (u)$ and the bound follows.\",\n \"Algorithms for dense graphs using matrix multiplication Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.\",\n \"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\\\log n)$ time algorithm.\",\n \"In fact, the guarantees are exactly the same as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].\",\n \"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms of [15] and [41].\",\n \"The main overhead of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\\\cup \\\\lbrace w\\\\rbrace \\\\cup T$ of $O(\\\\sqrt{n}\\\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"After one knows all distances from every $s\\\\in S$ to every $v\\\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.\",\n \"The main idea of our algorithms is as follows.\",\n \"If the Diameter is of size $\\\\le O(\\\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\\\log n)$ .\",\n \"Small distances are easy to compute with matrix multiplication.\",\n \"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .\",\n \"Then we can find the distances for all pairs in $S\\\\times V$ at distance $\\\\le t$ by computing $A_S\\\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\\\times n$ by $n\\\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].\",\n \"If on the other hand the Diameter is $D\\\\ge 100\\\\log n$ , then one can use an $\\\\tilde{O}(n^2)$ time algorithm by Dor et al.\",\n \"[24] to compute estimates of all pairwise distances with an additive error at most $4\\\\log n$ .\",\n \"The maximum distance estimate computed, minus $4\\\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.\",\n \"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.\",\n \"Below we recap the guarentees of the $\\\\tilde{O}(m\\\\sqrt{n})$ time approximation algorithms of [15], [41].\",\n \"Theorem 37 ([15], [41]) The following can be computed in $\\\\tilde{O}(m\\\\sqrt{n})$ time: an estimate $\\\\hat{D}$ of the graph Diameter $D$ , such that $\\\\frac{2}{3} D-\\\\frac{1}{3}\\\\le \\\\hat{D}\\\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\\\epsilon (v)$ , such that $\\\\frac{3}{5} \\\\epsilon (v)-\\\\frac{1}{5}\\\\le e(v)\\\\le \\\\epsilon (v)$ .\",\n \"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\\\omega )$ time for $\\\\omega < 2.373$ .\",\n \"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .\",\n \"Let us compare to our $O(n^2\\\\log n)$ time algorithms.\",\n \"For Diameter $D=3h+z$ , the $O(n^2\\\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .\",\n \"The estimate $\\\\hat{D}$ here is $\\\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\\\ge 2h$ when $z=0$ and $\\\\ge 2h+1$ when $z=1,2$ .\",\n \"For Eccentricities, the $O(n^2\\\\log n)$ time algorithm returns estimates $e(v)\\\\ge 3\\\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\\\ge (3\\\\epsilon -1)/5$ .\",\n \"We will rely on two known algorithms.\",\n \"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).\",\n \"Among many other results, [24] show that in $\\\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ for an explicit constant $a\\\\le 4$ .\",\n \"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\\\in S$ and $v\\\\in V$ for which $d(s,v)\\\\le Q$ .\",\n \"The algorithm uses fast matrix multiplication and is quite straightforward.\",\n \"Let $A$ be the $n\\\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.\",\n \"$A$ is the adjacency matrix added to the identity matrix.\",\n \"Let $A_S$ be the $|S|\\\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .\",\n \"For an integer $i\\\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.\",\n \"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .\",\n \"Define $A^0$ as the identity matrix.\",\n \"Consider $A_S \\\\cdot A^i$ for any choice of $i\\\\ge 0$ (under the Boolean matrix product).\",\n \"Here, $(A_S \\\\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .\",\n \"Thus, if we compute $D_i:=A_S\\\\cdot A^i$ for every $1\\\\le i< Q$ , we would know the distance from every $s\\\\in S$ to every $v\\\\in V$ , whenever this distance is at most $Q$ .\",\n \"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\\\cdot A$ .\",\n \"Thus, the running time is $O(Q\\\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix.\",\n \"Armed with these two algorithms, let us recap Roditty et al.\",\n \"'s (and Cairo et al.\",\n \"'s) approximation algorithm and see how to modify it.\",\n \"The algorithm proceeds as follows: Let $D$ , $R$ and $\\\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.\",\n \"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\\\epsilon (s)=\\\\max _v d(s,v)$ .\",\n \"Set $\\\\tilde{D}=\\\\max _{x\\\\in S} \\\\epsilon (x)$ .\",\n \"Set for every $v\\\\in V$ , $\\\\tilde{\\\\epsilon }(v)=\\\\max \\\\lbrace d(w,v),\\\\max _{x\\\\in W} d(x,v), \\\\max _{x\\\\in T} (\\\\epsilon (x)-d(x,v))\\\\rbrace $ .\",\n \"The runtime bottleneck in the above algorithm is step (2) which runs in $\\\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .\",\n \"Let us describe how to modify the algorithm.\",\n \"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.\",\n \"Our Modified Approximation.\",\n \"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\\\prime }(\\\\cdot ,\\\\cdot )$ so that for every $u,v\\\\in V$ , $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ .\",\n \"Let $X=3a\\\\log n$ .\",\n \"Set $\\\\tilde{D}_1 = \\\\max _{u,v\\\\in V} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"For every $v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_1(v) = \\\\max _{u} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"[t]@X@ Let $Q=2(X+a\\\\log n)=8a\\\\log n$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .\",\n \"Let $d_{\\\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\\\infty $ otherwise.\",\n \"Set $\\\\epsilon _{\\\\le }(s)=\\\\max _v d_{\\\\le }(s,v)$ .\",\n \"Set $\\\\tilde{D}_2=\\\\max _{x\\\\in S} \\\\epsilon _{\\\\le }(x)$ .\",\n \"[t]@X@ $\\\\forall v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_2(v)=\\\\max \\\\lbrace d_\\\\le (w,v), \\\\max _{x\\\\in W} d_\\\\le (x,v), \\\\max _{y\\\\in T} (\\\\epsilon _\\\\le (y)-d_\\\\le (y,v))\\\\rbrace $ .\",\n \"Third Part: Set $\\\\tilde{D},\\\\tilde{\\\\epsilon }(\\\\cdot )$: If $\\\\tilde{D}_1\\\\ge X$ , set $\\\\tilde{D}=\\\\tilde{D}_1$ , and otherwise set $\\\\tilde{D}=\\\\tilde{D}_2$ .\",\n \"[t]@X@ For every $v\\\\in V$ , if there exists some $x\\\\in S$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ , then set $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_1(v)$ , and otherwise $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ .\",\n \"Consider our modified algorithm, FasterApproximation.\",\n \"Now we will prove several claims.\",\n \"Claim 38 The running time of algorithm FasterApproximation is $\\\\tilde{O}(M(\\\\sqrt{n},n,n))$ .\",\n \"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\\\tilde{O}(n^2)$ time.\",\n \"Step 8 runs in $O((X+a\\\\log n)\\\\cdot M(|S|,n,n))$ time where $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ , using the iterated rectangular matrix product algorithm.\",\n \"Recall that $X+a\\\\log n = O(\\\\log n)$ .\",\n \"Thus Step 8 runs in $\\\\tilde{O}(M(|S|,n,n))$ time.\",\n \"Since $|S|=\\\\tilde{O}(\\\\sqrt{n})$ and we can partition an $|S|\\\\times n\\\\times n$ matrix product into $ \\\\text{\\\\rm polylog~} n$ , $n^{1/2}\\\\times n\\\\times n$ matrix products, the runtime of the step is $\\\\tilde{O}(M(n^{1/(2)},n,n))$ .\",\n \"Steps 10 and 13 run in $O(n|S|)<\\\\tilde{O}(n^2)$ time.\",\n \"The rest of the steps run in linear time.\",\n \"Since $M(n^{1/2},n,n)\\\\ge n^2$ (one must at least read the input), the total running time is $\\\\tilde{O}(M(n^{1/2},n,n))$ .\",\n \"Claim 39 $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Suppose that $\\\\tilde{D}_1\\\\ge X$ .\",\n \"The algorithm returns $\\\\tilde{D}=\\\\tilde{D}_1=\\\\max _{u,v} d^{\\\\prime }(u,v)-a\\\\log n$ .\",\n \"By the guarantee on $d^{\\\\prime }$ , we have $D-a\\\\log n \\\\le \\\\tilde{D}_1\\\\le D$ .\",\n \"Hence $\\\\tilde{D}\\\\ge D(1-(a\\\\log n)/D)\\\\ge D(1-(a\\\\log n)/X) =2D/3 \\\\ge (2D-1)/3$ .\",\n \"Suppose now that $\\\\tilde{D}_1< X$ .\",\n \"This means that $D100a\\\\log n$ ), then the $+a\\\\log n$ APSP algorithm with $a\\\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\\\log n>0.99D_{S,T}$ .\",\n \"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\\\le 100a\\\\log n$ , then we only need to compute $O(\\\\log n)$ matrix products of dimension $O(\\\\sqrt{n} \\\\log n)\\\\times n\\\\times n$ again.\"\n ],\n [\n \"Algorithm overview\",\n \"The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.\",\n \"[2] runs in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] removed the $\\\\tilde{O}(n^2)$ term to obtain an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"For every graph with $\\\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.\",\n \"Therefore, it is interesting to consider the opposite question to the one considered by [41].\",\n \"Can the $\\\\tilde{O}(m\\\\sqrt{n})$ term be removed?\",\n \"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\\\log n)$ expected time algorithm.\",\n \"For a graph of Diameter $D=3h+z$ , where $z\\\\in [0,1,2]$ our algorithm returns an estimation $\\\\hat{D}$ such that $2h-1\\\\le \\\\hat{D} \\\\le D$ , when $z \\\\in [0,1]$ and $2h\\\\le \\\\hat{D} \\\\le D$ , when $z=2$ .\",\n \"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.\",\n \"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.\",\n \"To enable this approach we can no longer sample $A$ naively.\",\n \"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.\",\n \"The expected running time of their algorithm is $\\\\tilde{O}(mn/|A|)$ .\",\n \"We provide a new implementation of their algorithm that runs in expected $\\\\tilde{O}(n(n/|A|)^2)$ time.\",\n \"The set $A$ has the following important property, for every vertex $w\\\\in V$ , its cluster (see [48]) $\\\\lbrace u \\\\mid d(u,w) 4/p \\\\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\\\log n$ .\",\n \"For every $w \\\\in V$ we then have $|C_A(w)|\\\\le 4/p$ .\",\n \"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\\\log n)$ .\",\n \"They did not provide the details and refer the reader to [48].\",\n \"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.\",\n \"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\\\log n)$ expected time implementation of Algorithm REF .\",\n \"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"This can be done as follows.\",\n \"Once we have computed $B_A(v)$ for every $v\\\\in V$ , we can scan $B_A(v)$ , and for every $w\\\\in B_A(v)$ we can add $v$ to $C_A(w)$ .\",\n \"The cost of this process is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\\\in V$ and we can compute $W$ (as needed in Algorithm REF ).\",\n \"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.\",\n \"To this end, more ideas are needed.\",\n \"The following simple observation helps us to achieve our goal.\",\n \"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .\",\n \"Let $A^*$ be a set such that $A^*\\\\subseteq A_i$ , for every $i\\\\ge 1$ .\",\n \"For every $v\\\\in V$ it holds that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ .\",\n \"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\\\le O(1/p)$ for every $v\\\\in V$ .\",\n \"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].\",\n \"This implies that we can compute $N_{1/p}(v)$ for every $v\\\\in V$ in $O(n/p^2)$ time.\",\n \"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\\\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ (a greedy algorithm works; e.g.\",\n \"see [48]).\",\n \"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\\\in V$ and the hitting set $A$ , as described above.\",\n \"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"In more detail, our algorithm works as follows.\",\n \"For every $v\\\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.\",\n \"Then it finds a set $A$ such that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ .\",\n \"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\\\in V$ the bunch $B_A(v)$ .\",\n \"Finally, it computes the clusters and $W$ using the bunches as we described above.\",\n \"The rest of the algorithm is almost identical to Algorithm REF .\",\n \"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\\\in V$ .\",\n \"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\\\in V$ .\",\n \"compute $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .\",\n \"every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\\\cup X$ every $v\\\\in V$ compute $B_A(v)$ every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\\\log n)$ expected time a set $A$ of expected size $O((n/p)\\\\cdot \\\\log n)$ that guarantees for every vertex $w\\\\in V \\\\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\\\in V$ that $|B_A(v)|=O(1/p)$ .\",\n \"The cost of computing $N_{1/p}(v)$ for every $v\\\\in V$ is $O(n (1/p)^2)$ [24].\",\n \"The cost of computing $A$ is $O(n/p)$ time [48].\",\n \"Computing $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .\",\n \"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .\",\n \"As we explained earlier the cost of computing clusters using bunches is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ .\",\n \"Since for every $v\\\\in V$ we have $B_A(v)\\\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .\",\n \"This completes the analysis of the part that precedes the while loop.\",\n \"Next, we analyze the cost of the while loop.\",\n \"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.\",\n \"From Observation REF it follows that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1}) h$ or $d(a,A)>h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .\",\n \"If this is not the case then either $d(a,A)\\\\le h$ or $d(b,A)\\\\le h$ (or both).\",\n \"Assume, wlog, that $d(a,A)\\\\le h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .\",\n \"We now turn to we analyze the running time of Algorithm REF .\",\n \"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\\\log n)$ .\",\n \"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\\\sqrt{n}$ .\",\n \"From Lemma REF it follows that the size of the set $A$ is $O(\\\\sqrt{n} \\\\log n)$ and its construction time is $O(n^2\\\\log n)$ in expectation.\",\n \"For every $w\\\\in V$ the size of $C_A(w)$ is $O(\\\\sqrt{n})$ .\",\n \"Therefore, Step 1 takes $O(n \\\\times |C_A(w)|^2)=O(n^2)$ .\",\n \"Step 2 takes $O(n^2)$ time as well.\",\n \"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.\",\n \"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.\",\n \"We also compute $|A|$ shortest paths trees in $H$ .\",\n \"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\\\log n)$ time.\"\n ],\n [\n \"Near linear almost 5/3-approximation for Eccentricities\",\n \"almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ every $u \\\\in V$ $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ $\\\\epsilon ^{\\\\prime }(u) = \\\\max (\\\\epsilon _1(u),\\\\epsilon _2(u),\\\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.\",\n \"We run lines 2-11 of Algorithm REF .\",\n \"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"Then, for every $u\\\\in V$ we compute $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ , which are defined as follows: $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ , $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ and $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ .\",\n \"The algorithm sets $\\\\epsilon ^{\\\\prime }(u)$ to $\\\\max \\\\lbrace \\\\epsilon _1(u), \\\\epsilon _2(u), \\\\epsilon _3(u) \\\\rbrace $ for every $u\\\\in V$ as an estimation to $\\\\epsilon (u)$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"We now prove: Theorem 35 For every $u\\\\in V$ , Algorithm REF computes in $O(n^2 \\\\log n)$ expected time a value $\\\\epsilon ^{\\\\prime }(u)$ that satisfies: $\\\\frac{3\\\\epsilon (u)}{5}-1 \\\\le \\\\epsilon ^{\\\\prime }(u)\\\\le \\\\epsilon (u).$ We start by analyzing the running time.\",\n \"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\\\log n)$ time in expectation.\",\n \"The computation of $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ for every $u\\\\in V$ costs $O(n^2)$ time in total.\",\n \"Let $u\\\\in V$ be an arbitrary vertex and let $\\\\epsilon (u)=d(u,t)$ .\",\n \"We now turn to bound $\\\\epsilon ^{\\\\prime }(u)$ .\",\n \"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\\\in V$ that $\\\\epsilon (u)\\\\ge \\\\epsilon (v)-d(u,v)$ .\",\n \"It is straightforward to see that both $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ are at most $\\\\epsilon (u)$ .\",\n \"Recall that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2\\\\le \\\\epsilon (p(u))-d(u,p(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u) = d_H(u,y)-2\\\\le d(u,y)\\\\le \\\\epsilon (u)$ .\",\n \"We distinguish between two cases.\",\n \"Case 1: $B_A(u) \\\\cap B_A(t)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that $P(u,t)\\\\cap (B_A(u) \\\\cap B_A(t)) \\\\ne \\\\emptyset $ and $M(u,t)= \\\\epsilon (u)$ .\",\n \"From Lemma REF it follows that $M(u,w)\\\\le d(u,w)$ for every $w\\\\in V$ after Step 2.\",\n \"Therefore, $\\\\epsilon _1(u)=\\\\epsilon (u)$ .\",\n \"Since $\\\\epsilon _2(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u)\\\\le \\\\epsilon (u)$ we get that $\\\\epsilon ^{\\\\prime }(u)=\\\\epsilon (u)$ .\",\n \"Case 2: $B_A(u) \\\\cap B_A(t) = \\\\emptyset $ .\",\n \"Consider first the case that $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"From Observation REF we get that $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d_H(u,p(u))$ .\",\n \"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d(u,p(u))$ .\",\n \"Hence, we get that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2 \\\\ge \\\\epsilon _H(u)-2d(u,p(u))-2\\\\ge \\\\epsilon (u)-2d(u,p(u))-2$ .\",\n \"As before we have $\\\\epsilon _2(u) \\\\le \\\\epsilon (u)$ .\",\n \"Using $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ we get that: $\\\\epsilon _2(u) \\\\ge \\\\epsilon (u)-\\\\frac{2\\\\epsilon (u)}{5}\\\\ge \\\\frac{3\\\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\\\ge \\\\frac{\\\\epsilon (u)}{5}$ .\",\n \"This means that $d(u,A)-1\\\\ge \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"Let $S$ be the set of all vertices $v\\\\in V$ such that $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ , that is, $S=V \\\\setminus \\\\cup _{w\\\\in B_A(u)} C_A(w)$ .\",\n \"Let $t^{\\\\prime } = \\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in S} d(x,A)-1$ .\",\n \"If $d(t^{\\\\prime },A)-1 \\\\ge \\\\frac{2\\\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\\\prime })\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"Assume now that $d(t^{\\\\prime },A) < \\\\frac{2\\\\epsilon (u)}{5}$ .\",\n \"As $t^{\\\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\\\frac{2\\\\epsilon (u)}{5}$ and $d(u,p(t))>\\\\frac{3\\\\epsilon (u)}{5}$ .\",\n \"Therefore, $\\\\epsilon _3(u) = d_H(u,y)-2\\\\ge d(u,p(t))-2\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"From Lemma REF and Lemma REF it follows that $\\\\epsilon _1(u)\\\\le \\\\epsilon (u)$ and the bound follows.\"\n ],\n [\n \"Algorithms for dense graphs using matrix multiplication\",\n \"Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.\",\n \"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\\\log n)$ time algorithm.\",\n \"In fact, the guarantees are exactly the same as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].\",\n \"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms of [15] and [41].\",\n \"The main overhead of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\\\cup \\\\lbrace w\\\\rbrace \\\\cup T$ of $O(\\\\sqrt{n}\\\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"After one knows all distances from every $s\\\\in S$ to every $v\\\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.\",\n \"The main idea of our algorithms is as follows.\",\n \"If the Diameter is of size $\\\\le O(\\\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\\\log n)$ .\",\n \"Small distances are easy to compute with matrix multiplication.\",\n \"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .\",\n \"Then we can find the distances for all pairs in $S\\\\times V$ at distance $\\\\le t$ by computing $A_S\\\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\\\times n$ by $n\\\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].\",\n \"If on the other hand the Diameter is $D\\\\ge 100\\\\log n$ , then one can use an $\\\\tilde{O}(n^2)$ time algorithm by Dor et al.\",\n \"[24] to compute estimates of all pairwise distances with an additive error at most $4\\\\log n$ .\",\n \"The maximum distance estimate computed, minus $4\\\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.\",\n \"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.\",\n \"Below we recap the guarentees of the $\\\\tilde{O}(m\\\\sqrt{n})$ time approximation algorithms of [15], [41].\",\n \"Theorem 37 ([15], [41]) The following can be computed in $\\\\tilde{O}(m\\\\sqrt{n})$ time: an estimate $\\\\hat{D}$ of the graph Diameter $D$ , such that $\\\\frac{2}{3} D-\\\\frac{1}{3}\\\\le \\\\hat{D}\\\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\\\epsilon (v)$ , such that $\\\\frac{3}{5} \\\\epsilon (v)-\\\\frac{1}{5}\\\\le e(v)\\\\le \\\\epsilon (v)$ .\",\n \"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\\\omega )$ time for $\\\\omega < 2.373$ .\",\n \"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .\",\n \"Let us compare to our $O(n^2\\\\log n)$ time algorithms.\",\n \"For Diameter $D=3h+z$ , the $O(n^2\\\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .\",\n \"The estimate $\\\\hat{D}$ here is $\\\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\\\ge 2h$ when $z=0$ and $\\\\ge 2h+1$ when $z=1,2$ .\",\n \"For Eccentricities, the $O(n^2\\\\log n)$ time algorithm returns estimates $e(v)\\\\ge 3\\\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\\\ge (3\\\\epsilon -1)/5$ .\",\n \"We will rely on two known algorithms.\",\n \"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).\",\n \"Among many other results, [24] show that in $\\\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ for an explicit constant $a\\\\le 4$ .\",\n \"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\\\in S$ and $v\\\\in V$ for which $d(s,v)\\\\le Q$ .\",\n \"The algorithm uses fast matrix multiplication and is quite straightforward.\",\n \"Let $A$ be the $n\\\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.\",\n \"$A$ is the adjacency matrix added to the identity matrix.\",\n \"Let $A_S$ be the $|S|\\\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .\",\n \"For an integer $i\\\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.\",\n \"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .\",\n \"Define $A^0$ as the identity matrix.\",\n \"Consider $A_S \\\\cdot A^i$ for any choice of $i\\\\ge 0$ (under the Boolean matrix product).\",\n \"Here, $(A_S \\\\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .\",\n \"Thus, if we compute $D_i:=A_S\\\\cdot A^i$ for every $1\\\\le i< Q$ , we would know the distance from every $s\\\\in S$ to every $v\\\\in V$ , whenever this distance is at most $Q$ .\",\n \"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\\\cdot A$ .\",\n \"Thus, the running time is $O(Q\\\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix.\",\n \"Armed with these two algorithms, let us recap Roditty et al.\",\n \"'s (and Cairo et al.\",\n \"'s) approximation algorithm and see how to modify it.\",\n \"The algorithm proceeds as follows: Let $D$ , $R$ and $\\\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.\",\n \"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\\\epsilon (s)=\\\\max _v d(s,v)$ .\",\n \"Set $\\\\tilde{D}=\\\\max _{x\\\\in S} \\\\epsilon (x)$ .\",\n \"Set for every $v\\\\in V$ , $\\\\tilde{\\\\epsilon }(v)=\\\\max \\\\lbrace d(w,v),\\\\max _{x\\\\in W} d(x,v), \\\\max _{x\\\\in T} (\\\\epsilon (x)-d(x,v))\\\\rbrace $ .\",\n \"The runtime bottleneck in the above algorithm is step (2) which runs in $\\\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .\",\n \"Let us describe how to modify the algorithm.\",\n \"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.\",\n \"Our Modified Approximation.\",\n \"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\\\prime }(\\\\cdot ,\\\\cdot )$ so that for every $u,v\\\\in V$ , $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ .\",\n \"Let $X=3a\\\\log n$ .\",\n \"Set $\\\\tilde{D}_1 = \\\\max _{u,v\\\\in V} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"For every $v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_1(v) = \\\\max _{u} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"[t]@X@ Let $Q=2(X+a\\\\log n)=8a\\\\log n$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .\",\n \"Let $d_{\\\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\\\infty $ otherwise.\",\n \"Set $\\\\epsilon _{\\\\le }(s)=\\\\max _v d_{\\\\le }(s,v)$ .\",\n \"Set $\\\\tilde{D}_2=\\\\max _{x\\\\in S} \\\\epsilon _{\\\\le }(x)$ .\",\n \"[t]@X@ $\\\\forall v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_2(v)=\\\\max \\\\lbrace d_\\\\le (w,v), \\\\max _{x\\\\in W} d_\\\\le (x,v), \\\\max _{y\\\\in T} (\\\\epsilon _\\\\le (y)-d_\\\\le (y,v))\\\\rbrace $ .\",\n \"Third Part: Set $\\\\tilde{D},\\\\tilde{\\\\epsilon }(\\\\cdot )$: If $\\\\tilde{D}_1\\\\ge X$ , set $\\\\tilde{D}=\\\\tilde{D}_1$ , and otherwise set $\\\\tilde{D}=\\\\tilde{D}_2$ .\",\n \"[t]@X@ For every $v\\\\in V$ , if there exists some $x\\\\in S$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ , then set $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_1(v)$ , and otherwise $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ .\",\n \"Consider our modified algorithm, FasterApproximation.\",\n \"Now we will prove several claims.\",\n \"Claim 38 The running time of algorithm FasterApproximation is $\\\\tilde{O}(M(\\\\sqrt{n},n,n))$ .\",\n \"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\\\tilde{O}(n^2)$ time.\",\n \"Step 8 runs in $O((X+a\\\\log n)\\\\cdot M(|S|,n,n))$ time where $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ , using the iterated rectangular matrix product algorithm.\",\n \"Recall that $X+a\\\\log n = O(\\\\log n)$ .\",\n \"Thus Step 8 runs in $\\\\tilde{O}(M(|S|,n,n))$ time.\",\n \"Since $|S|=\\\\tilde{O}(\\\\sqrt{n})$ and we can partition an $|S|\\\\times n\\\\times n$ matrix product into $ \\\\text{\\\\rm polylog~} n$ , $n^{1/2}\\\\times n\\\\times n$ matrix products, the runtime of the step is $\\\\tilde{O}(M(n^{1/(2)},n,n))$ .\",\n \"Steps 10 and 13 run in $O(n|S|)<\\\\tilde{O}(n^2)$ time.\",\n \"The rest of the steps run in linear time.\",\n \"Since $M(n^{1/2},n,n)\\\\ge n^2$ (one must at least read the input), the total running time is $\\\\tilde{O}(M(n^{1/2},n,n))$ .\",\n \"Claim 39 $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Suppose that $\\\\tilde{D}_1\\\\ge X$ .\",\n \"The algorithm returns $\\\\tilde{D}=\\\\tilde{D}_1=\\\\max _{u,v} d^{\\\\prime }(u,v)-a\\\\log n$ .\",\n \"By the guarantee on $d^{\\\\prime }$ , we have $D-a\\\\log n \\\\le \\\\tilde{D}_1\\\\le D$ .\",\n \"Hence $\\\\tilde{D}\\\\ge D(1-(a\\\\log n)/D)\\\\ge D(1-(a\\\\log n)/X) =2D/3 \\\\ge (2D-1)/3$ .\",\n \"Suppose now that $\\\\tilde{D}_1< X$ .\",\n \"This means that $D100a\\\\log n$ ), then the $+a\\\\log n$ APSP algorithm with $a\\\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\\\log n>0.99D_{S,T}$ .\",\n \"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\\\le 100a\\\\log n$ , then we only need to compute $O(\\\\log n)$ matrix products of dimension $O(\\\\sqrt{n} \\\\log n)\\\\times n\\\\times n$ again.\"\n ]\n]"},"id":{"kind":"string","value":"1808.08494"}}},{"rowIdx":199,"cells":{"text":{"kind":"list like","value":[["NavigationNet: A Large-scale Interactive Indoor Navigation Dataset"],["Abstract Indoor navigation aims at performing navigation within buildings.","In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor scenes to complete various tasks in order to serve human.","In most scenarios, we expected an intelligent device capable of navigating itself in unseen environment.","Although several solutions have been proposed to deal with this issue, they usually require pre-installed beacons or a map pre-built with SLAM, which means that they are not capable of working in novel environments.","To address this, we proposed NavigationNet, a computer vision dataset and benchmark to allow the utilization of deep reinforcement learning on scene-understanding-based indoor navigation.","We also proposed and formalized several typical indoor routing problems that are suitable for deep reinforcement learning."],["Introduction","Indoor navigation aims at preforming navigation within buildings, such as train stations, airports, shopping centers, offices and museums.","Most intelligent devices require an indoor routing functionality to guide itself precisely through rooms to complete various tasks.","In most of the scenarios, we expect an intelligent device capable of navigating itself in new environments to serve human.","In the past years, several solutions have been proposed to deal with this problem [1].","Although they can be applied in some special cases, none of them can work smartly in novel environments or are based on scene understanding.","Those solutions require either dense pre-installed beacons (e.g.","WiFi/Bluetooth chips) over the scene or a map pre-built with SLAM in which human workers are required to carry cameras to scan the scene in advance.","Such requirements significantly limit the applications of these solutions.","New scenes have to be well prepared before the robots can work properly in it, which makes it impossible to be put into large-scale application in various indoor scenarios.","Additionally, neither beacons nor maps are helpful in semantically understanding indoor scenes and thus will lead to the failure of the robot to perform many practical tasks.","For example, to complete a regular command like 'to take a cup', a housekeeping android should understand what the scene is like, what a cup is and where a cup usually be, to navigate properly.","We expected a more general solution of indoor navigation problems that requires no preparation of the scene in advance.","Such a high demand requires a more human-like robot that is much more intelligent to implement self-exploration in novel environments.","Recent years, we have witnessed the rapid development of deep reinforcement learning (DRL) [2], [3] and computer vision [4], [5].","Therefore, smart indoor navigation enabled by DRL with visual inputs has been introduced [6].","In this setting, pre-installed beacons or pre-built map are no longer required.","A smart robot with vision will be enough to do the navigation job in different scenes.","To apply reinforcement learning methods on navigation problems, massive trial-and-error is necessary in training the model.","A straight-forward way is to construct a real robot and train Reinforcement Learning (RL) models on real-world scenes.","However, such process is extremely slow and costly.","Additionally, it is often accompanied by robot collision and damage, which means that more time and resources would be wasted.","Even worse, due to the bottleneck of physical movements, to execute any action in the real world requires at least several seconds, which is unacceptable for any machine learning methods since that they usually require millions of times of trial-and-error.","Aside from those drawbacks, such a process raises the problem of reproducibility.","Even if all the background settings of a research are published in detail, it is still very hard for third parties to reproduce the experiment result since it is costly and time-consuming to find or construct a new environment that is identical to the one the original author used.","That makes it impossible for third parties to examine and evaluate a new research result.","We believe, an open-source, low-cost, large-scale dataset and corresponding benchmarks would be a key to largely advance this research field.","Some previous work has tried to train the robot in computer graphics (CG) scenes like AI2-THOR[6], SUNCG[7].","However, we do have noticed that training models on CG environments raised the problem of model transferring as the gap between hand-crafted scenes and the real world is significant.","The visual appearance of images rendered by CG system always look non-realistic.","Existing 3D models are limited in representing real-world scenes and objects.","To deal with this problem, we proposed NavigationNet, a large-scale, real-world, interactive dataset.","In each scene, we use human-size robot with 8 cameras to capture photos towards 8 different directions at all walk-able positions.","Given all the collected images, we can easily tour the room like a CG system by moving among those walk-able positions.","However, different from CG systems, what we observed from this dataset are all real images captured from the real world.","Therefore, we can virtually command a robot to walk in the scene and get the first-perspective view of the robot.","Our dataset is interactive.","That is to say, as we send a command of movement (e.g.","moving forward, turning right), the robot will perform as required and return a new view it sees from the new position.","To our best knowledge, our NavigationNet is the first real-world indoor scene dataset that is interactive in computer vision field.","Our dataset covers about 1500 $m^2$ indoor area with various bedrooms, studies, meeting room and etc.","This makes the bias less in experiments on our dataset.","Based on NavigationNet, we proposed four applications that require indoor navigation.","We will explained them in detail in chapter .","To summarize, we presented a large-scale dataset consisting of visual data collected from real scenarios to allow a low-cost, reproducible training of indoor navigation robots based on reinforcement learning methods.","We also proposed and formalized several practical indoor navigation tasks in the framework of reinforcement learning."],["Related Work","In this chapter, we will discuss some of the previous works that are related to our work or act as a tool in this project.","First we will talk about the technology of Visual Semantic Planning.","A brief introduction to the popular datasets in the field of computer vision, after which we will explain three techniques used in our work in detail: Reinforcement Learning, Convolutional Neural Network(CNN) and Simultaneous Localization and Mapping."],["Visual Semantic Planning","Fundamental tasks in mobile robot controlling include navigation, mapping, grasping, path planning and so on.","Such task of interacting with a visual world and planning a sequence of actions to achieve a certain goal is addressed as Visual Semantic Planning [8].","A series of work [9], [10], [11], [12] has addressed the task-level planning and learning from dynamics.","Classical approaches [13], [14] of motion planning require building map purely geometrically using Light Detection and Ranging (LiDAR) or Structure from Motion (SfM) [15], [16], [17].","Srivastava et al.","[18] provides an interface between task and motion planning, such that the task planner can effectively operate in an abstracted state space that ignores geometry.","Recently proposed end-to-end learning-based approaches [19], [10], [20] can go directly from pixels to actions.","For instance Zhu et al.","[19] adopt feed-forward neural network to target-driven visual navigation in indoor scenes.","Gupta et al.","[10] use online Cognitive Mapping and Planning (CMP) approach for goal direction visual navigation without requiring a pre-constructed map.","Jang et al.","[9] train a deep neural network to perform the semantic grasping task inspired by the \"two-stream hypothesis\" of human vision.","In terms of Complete Path Planning, Kollar et al.","[21] use reinforcement learning to find trajectories and the learnt policy transfers successfully to a real environment."],["Datasets for Computer Vision","Datasets and corresponding benchmarks have played a significant role in many areas such as computer vision and speech recognition.","The evolution from WordNet[22] to ImageNet[23] was one of the many successful examples that proved the power of an effective dataset to accelerate the development of one area.","On the contrary, the lack of appropriate data has become one of the most crucial challenges for deep reinforcement learning, especially when it is dealing with real world problems like visual semantic learning.","Since training and quantitatively evaluating DRL algorithms in real environments is either impractical or costly, this problem has become more urgent.","For RL algorithms dealing with virtual-world problems, the Arcade Learning Environment (ALE) [24] exposed Atari 2600 games as reinforcement learning problems.","Works such as Duan et al.","[25] and Brockman et al.","[26] propose toolkits to qualify progress in reinforcement learning.","Other benchmarks [27], [28], [29], [30], [31], simulators [32] or physics engines [33] are also designed for the development of deep reinforcement learning.","Also, scientists are trying to model the real world for DRL or other visual tasks.","Chang et al.","[34] provides 90 building-scale reconstructed scenes for supervised and self-supervised computer vision tasks such as keypoint matching, view overlap prediction semantic segmentation, scene classification and so on.","AI2-THOR[19], SUNCG[7] and House3D are all CG scenes designed especially for deep reinforcement learning.","The advanced version of AI2-THOR[35] even provides the functionality to let the robot directly interact with the objects in the scenarios.","Different from our NavigationNet, these datasets are either synthetic or reconstructed using computer graphics techniques, which introduce inconsistency distribution between the real-world scenarios and the produced ones."],["Reinforcement Learning","Reinforcement learning (RL) [36] method was proposed in the late 90s.","It provides a technique to allow the robot or any other intelligent systems to build a value function to evaluate policies in given situations from an interactive way of trail-and-error.","Pioneering works from Mnih et al.","[37], Lillicrap et al.","[38], Schulman et al.","[39] and Silver et al [40] introduced and ignited the idea of join Reinforcement Learning and Deep Learning into Deep Reinforcement Learning.","In learning complex behavior skills and solving challenging control tasks in high-dimensional raw sensory state-space, DRL methods have shown tremendous success [40], [41], [3], [42].","Trust Region Policy Optimization (TRPO) [41] makes a series of approximations to the theoretically-justified procedure and has robust performance on a wide range of tasks.","Asynchronous Advantage Actor-Critic (A3C) Algorithm [3], which uses asynchronous gradient descent for optimization, succeeds on task of navigating random 3D mazes as well as a wide variety of continuous motor control problems.","UNsupervised REinforcement and Auxiliary Learning (UNREAL) [43] brings together the A3C framework with auxiliary control tasks and auxiliary reward tasks.","Actor Critic using Kronecker-factored Trust Region (ACKTR) [42] Algorithm uses a Kronecker-factored approximation to natural policy gradient that allows the covariance of the gradient to be inverted efficiently.","Proximal Policy Optimization (PPO) [44] Algorithms use multiple epochs of stochastic gradient ascent to perform each policy update.","Reinforcement Learning (RL) provides a powerful and flexible framework to several applications.","Andrew Y. Ng et al.","[45] describe a successful application of autonomous helicopter.","Finn et al.","[12] present an approach that automates state-space construction by learning a state representation directly from camera images with deep spatial autoencoder.","Heess et al.","[46] explore a rich environment can help to promote the learning of complex behavior.","They normalize observations, scale the reward and use per-batch normalization of the advantages.","Gu et al.","[47] demonstrate a DRL algorithm based on off-policy training of deep Q-function can scale to complex 3D manipulation tasks and can learn deep neural network policies efficiently enough to train on real physical robots.","Denil et al.","[48] find DRL methods can learn to perform the experiments necessary to discover properties such as mass and cohesion of objects."],["Convolutional Neural Network","Deep convolutional neural networks [49], [50] have led to a series of breakthrough for visual tasks.","He et al.","[5] present a Residual Network (ResNet) to ease the training of deep neural network.","We extract feature map with ResNet 101.","Huang et al.","[51] presents Dense Convolutional Network (DenseNet) which connects each layer to every other layer in a feed-forward fashion.","Chen et al.","[52] propose Dual Path Net (DPN) by revealing ResNet and DenseNet within the higher order recurrent neural network framework.","Deep Learning models have been successful in learning powerful representations and understanding stereo context [53], [54].","Shaked et al.","[55] present a three-step pipeline for the stereo matching problem and a network architecture (ResMatch) for computing the matching cost at each possible disparity.","Kendall et al.","[56] propose an end-to-end model (GC-Net) for regressing disparity from a rectified pair of stereo images."],["Simultaneous Localization and Mapping","Visual SLAM plays an important role in autonomous navigation of mobile robots [57].","The emergence of MonoSLAM [58] makes the idea of utilizing one camera become popular.","Parallel Tracking and Mapping (PTaM) [59] uses an approach based on keyframes with two parallel processing threads.","ORB [60] becomes a computationally-efficient replacement to SIFT keypoint detector and descriptor that has similar matching performance.","Keyframe bundle adjustment outperforms filtering, since it gives the most accuracy per unit of computing time [61].","S-PTAM [62] allows accuracy improvement of the mapping process with respect to monocular SLAM and avoiding the well-known bootstrapping problem.","ORB-SLAM2 [63] is a complete SLAM system for monocular, stereo and RGB-D cameras.","RGB-D SLAM [64] can producing high quality globally consistent surface reconstruction with only a low-cost commodity RGB-D sensor.","Mur-Artal et al.","[65] propose a visual-inertial monocular SLAM which is able to close loops and reuse its map to achieve zero-drift localization in already mapped areas."],["NavigationNet","NavigationNet is specifically designed for applying reinforcement learning methods to indoor navigation tasks.","We collected data from the real world and organized it properly to allow the robot to roam in the dataset as if they are in the corresponding real-world scenario.","In this chapter, we will explain its organization and applications in detail to show you the world of NavigationNet."],["Data Organization","Images in NavigationNet are organized hierarchically.","The hightest node is the root of NavigationNet.","The second level nodes, also the primary elements, are scenes.","A scene is a collection of images collected from the same indoor space.","In the next level, a scene is divided into rooms, space connected with doors.","In the current version of NavigationNet, we have 15 scenes, each with 1-3 rooms.","The origin room for each scene is at least 50 $m^2$ in area.","The fourth level is called postion.","A position is from where images are collected.","A room contains hundreds of postions.","When being trained, the robot could move among the adjacent positions to 'see' from that perspective.","When constructing the dataset, our mobile robot iterated over all the walkable points in the room with the granularity of 20cm.","Hence, in the dataset, two adjacent positions are 20cm away from each other.","With such a granularity, one room usually contains thounsands of positions.","From the perspective of information, a position can be seen as a bundle of information that the robot will receive when it is placed at that position.","Usually, we offer 8 images in this bundle.","These images are divided into four directions: front, back, left and right.","Towards each direction, two parallel images are taken to complete a binocular vision system.","Such a setting enables the possibilities like stereo matching and panorama reconstruction.","In addition to the images, we also provide a ground-truth map attached to the scene.","The map can be considered as a binary map indicating where is walkable or non-walkable.","This is essential especially to the Auto-SLAM problem."],["Robot Moving Control","To satisfy the different requirements of potential tasks, we tried our best to provide a more generalizable moving control SDK for NavigationNet.","As is stated in section REF , the fundamental element in NavigationNet is scene, virtual robots are allowed to roam in scenes.","To serve well as an action in a reinforcement learning task, movements should be standard, discrete, limited and complete.","An action should be standard, so that the robot should move the same distance towards the same direction or turning the same angle when a specific instruction of a movement is given.","An action should be discrete, so that the action space can be finite and discrete.","Actions should be limited, so that the algorithm do not need to choose actions from an infinite action space.","Actions should be complete, so that all the actions joined together could describe the whole real-world action space.","In practice, it is impossible to use a finite action space to cover an infinite one.","We need some sort of compromise.","The one we take in this case is to discrete the movement length and the turning angle.","So that we defined six types of movements.","MOVE FORWARD Ask the robot to move forward one position.","For example, go from (9, 12) to (9, 13) when facing north but go from (11, 8) to (10, 8) when facing west.","Since two adjacent nodes are 20cm away, this is to move forward 20cm in real-world scenes.","MOVE BACKWARD Ask the robot to move backward one position.","This is the opposite movement of MOVE FORWARD and can revert the effect of MOVE FORWARD.","MOVE LEFT Ask the robot to move left one position.","For example, go from (5, 4) to (4, 4) when facing north but go from (7, 4) to (7, 3) when facing west.","The real world distance would be the same as MOVE FORWARD and MOVE BACKWARD.","It should be paid attention to that MOVE LEFT is very different from TURNING LEFT as it will move the position of the robot but will not change the facing direction.","MOVE RIGHT Ask the robot to move right one position.","This is the opposite movement of MOVE LEFT and can revert the effect of MOVE LEFT.","TURN LEFT Ask the robot to turn left at 90-degree.","For example, when the robot receives this request when at (5, 7) facing north, it should then facing west still at (5, 7).","It should be paid extra attention that this movement is different from MOVE LEFT that the former changes the facing while the latter change the position.","TURN RIGHT Ask the robot to turn right at 90-degree.","This is the opposite movement of TURN LEFT.","Figure: MovementsAt the same time, we should make it clear that, in any given tasks, according to the specific setting, not all the actions (movement) need to be taken into consideration.","For example, when we would like to simulate a two-wheel one-eye robot, TURN LEFT and TURN RIGHT are a must, otherwise seventy-five percent of the images can never be perceived while MOVE LEFT and MOVE RIGHT should be eliminated since a two-wheel robot can not make the exact movement of moving left 20cm.","Meanwhile, when simulating a robot with panorama vision, we could only take MOVE LEFT and MOVE RIGHT but not TURN LEFT and TURN RIGHT as they are not necessary but can only enlarge the action space unnecessarily and make the task more difficult."],["NavigationNet Construction","NavigationNet is a large-scale dataset with hundreds of thousands of images.","The large amount makes the data collection a task of impossible.","What is worst is that, as to each image, it is required that the position, angle and height where it is taken should be exactly where it is expected so long that the simulation of reality will not offset.","These two factors together make the task of collection much more difficult than expected.","In this chapter, we will talk about how we constructed this dataset efficiently and accurately."],["Collector Mobile Robot","Robots are designed to liberate human beings from the repetitive boring work.","NavigationNet is built for intelligent robots, also built with the help of intelligent robots.","Collecting data for NavigationNet requires heavy labor and high accuracy but little flexibility (most of the possible conditions are under control), which is surprisingly suitable for robots.","In order to reduce the labor requirement, we exploited the possibilities of smart hardware.","Our team developed a dedicated data-collecting mobile robot with Arduino Mega2560 and Raspberry Pi 3 Model B codenamed GoodCar.","Arduino[66] is an open-source physical computing platform based on a simple I/O board and a development environment that implements the Processing/Wiring language.","Arduino boards can be programmed in the same name program language Arduino, which is a C-like area-specific language to receive and send signals from tens of low-voltage I/O ports.","It is specifically suitable for robot controlling.","In this project, we use a Arduino Mega2560 to control the movement of the robot.","Raspberry Pi[67] is a cheap single-board computer system running a specificialy made Linux distribution.","It is originally built for computer programming education but we find it suitable for robot controlling.","It consumes little electricity which allows us to strip the 220V power wires.","It contains all kinds of I/O ports such that we could plug it with Arduinos, cameras, other full-sized computers and many other modules to make it the center of the system.","It runs standard linux distribution such that the possibility of software extension is beyond imagine.","Last but not least, it is cheap so that we could us as many as we want on the robot.","In practice, we use two Raspberry Pi 3B for controlling all other modules and communicating with the upper computer.","To build this robot, we used a two-level structure.","The upper-level is as high as 1.4m to simulate the the height of the eyes of an average adult.","We plugged-in eight cameras on this level, two for each direction to make a stereo vision system and to avoid unnecessary turning-around.","The lower-level is for the motion systems.","The Raspberry Pi and Arduino mentioned above are all on this level.","The Arduino is to control the mobile devices and sensors.","It is linked with four motors to control the movement.","In the meantime, it is linked with a sonar and code plate counters to avoid collision and measure the distance of movement.","Also it is linked with a Raspberry Pi and under its control.","We used two Raspberry Pi 3B in each robot, one is master and the other is slave.","The master computer controls many things.","First it communicates with the Arduino to control the movement indirectly.","Second it is responsible for evaluating the signal Arduino sent back to avoid collision.","Third it is master computer's duty to control four of the cameras to take photo at given position and store the images on the SD card.","Fourth the slave computer is also under its control to take photos when required.","Lastly it communicates with the upper computer, which is controlled by human, via Wi-Fi connection.","That is the control core of our robot.","Under the lower-level is the mobile devices.","We have tried on many types of mobile devices and at last we used four track structures to reduce error.","We had constructed six mobile robots before moving on to collect data.","Figure: Robot"],["Data Collection","To make the data collecting process efficient and accurate, also to make the whole process manageable and under control, we designed and followed the following instructions: Step 1 Measure the size of the target room(s).","Find the most southwestern corner as the starting point.","Step 2 Move the robot to the starting point.","Evaluate and record its distance towards the borders.","Step 3 Let the robot face north and start to collect required photos at this point.","Step 4 After photos are taken, let the robot run 20cm towards north and collect data.","Step 5 Keep iterating over Step 3-4 until the robot reaches the end of the line.","Step 6 Move the robot to the starting point of the line and then move 20cm towards east.","Step 7 Iterating over Step3-6 until the robot reaches the east corner.","Step 8 Complete this scenario by taking photos from those points that the robot did not reach.","Data Processing After the collection, the original data should be pre-processed before taken into production.","The data processing steps we taken includes: Data Cleaning When collecting data with robot, there are always inevitable errors occurring especially when dealing with room corners.","For each room we collect, we will scan over all the collected photos and find errors.","If necessary, we would go and retry collecting data to complete the set.","Map Building When collecting data, we also recorded whether a point is accessible.","With this information, we are able to rebuild a walkable-or-not map for reference.","Stereo Vision For some of the task, to simulate a two-eye system or a RGB-D system, it would be of much help if we could produce depth channel in the very beginning.","To serve this purpose, we use MC-CNN[54] to produce some pre-processed depth channel data.","Applications NavigationNet is designed for indoor navigation problems.","In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.","Apart from these four applications, we believe our dataset will spawn more applications.","Target-driven Navigation Motivation In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.","For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.","Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.","To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.","This usually falls in the field of visual semantic learning.","That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.","Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based) Formalization The formalization of this issue is inspired by Yuke Zhu et al.[6].","To complete the task of target-driven navigation, a robot control system is given two images.","One is the target image, the other is the current first-person perspective image perceived from the environment.","The robot can choose to walk forward, backward, turn left or right according to the input images.","The environment will update the states accordingly and give out a reward.","This process loops until the target image is the same with the perceived one.","To formalize that: Goal The goal is straight-forward, to find the given object.","In practice, it can be converted into make the perceived image and the given image identical.","Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.","One thing that is sure is that a large positive reward will be given when the goal is achieved.","To improve time efficiency, each step it taken should be given a small negative reward.","In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.","Action Space To find an object, the understanding of the scene is the most important factor of the task.","Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.","Hence the State space would be the set of all possible image pairs.","Evaluation To evaluate such a system, many aspects should be taken into consideration.","The most important two are, trajectory should be short and the robot should not run into the obstacles.","Collisions should be limited strictly as only one collision could greatly damage the robot.","To make a trajectory finite, we consider a trajectory longer than 10000 a loop.","To calculate a overall score on all the test results, we utilize the idea of histogram.","A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .","A overall score is calculated by finding the average over this histogram.","Multi-target Navigation Motivation The target-driven navigation task is focused on finding one object efficiently and safely.","However, in practical scenarios, this is often not enough.","In many cases, we need to find more than one items and we do not care about the order.","For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.","This has become a rather different problem since it usually requires the knowledge of adjacency.","For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.","Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based) Formalization We also formalized this problem into a reinforcement learning problem.","To find all the given items, the robot is given a list of items to collect, each one is represented by an image.","The rest will be similar with the target-driven task.","A first-person perspective image is given, an action is executed and then state is updated, reward is given.","To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.","Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.","We believe that such a design would be better for deep reinforcement training[43].","Also a minor negative time time punishment is necessary.","Action Space Since the task type is similar with the target-driven problem.","The same action space remains.","{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.","However, a set images with unknown-length is hard to be processed.","Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Evaluation Although the task is different from the previous task, the two major meters are the same.","Hence, we use the same metric system in this task as is in the target-driven navigation task.","Sweeper Route Planning Motivation Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.","One widely-discussed but never well-addressed problem is the sweeper route planning problem.","Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.","The current solutions often lead to stuck at some small space.","Besides sweeper robot, it is general problem we will meet in many scenarios.","This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.","However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.","The most important factor is that the robot will not know that some movement is blocked until the collision really happens.","It must see and understand to avoid collisions.","Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based) Formalization Again, we formalized this problem into a reinforcement learning task.","At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.","The environment should give out the reward then according to how much area has been covered by the sweeper.","Typically, the robot should maintain a map of the room inside its system.","To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.","However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.","Reward Reward design is simple in this problem.","Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.","That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.","Action Space In this task, a sweeper does not really care about which direction it is facing.","The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).","So we will not give the robot the ability to turn left/right.","As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.","State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.","Evaluation The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.","To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.","For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.","If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.","An average over scores of all steps in a trajectory is the final score of the trajectory.","Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Auto-SLAM with Deep Reinforcement Learning Motivation Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.","Many solutions have been proposed to increase the mapping accuracy.","However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.","Additionally, applying SLAM for some large-scale scenes (e.g.","Museum) is effort consuming, which requires us to carry camera walked about all corners.","So, we hope that robots can preform SLAM autonomously.","With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.","Formalization This task is also formalized as a reinforcement learning problem.","On each step, the robot is fed with images perceived at its current location.","The system could use the visual inputs and history information to improve the map.","Reward should be calculated according to the difference between the constructed map and the ground truth.","An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.","To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.","Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.","Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.","By talking about progress, we are assuming total trust in legacy SLAM algorithms.","We would like to give a positive reward for any new reconstructed area after each steps.","In the meantime, a fruitless step is also punished with a minor negative reward.","Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.","In the meantime, we need to add the function of moving left/right to the robot.","Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.","Most of it is about 3D reconstruction.","Thus only a first-person single-eye perception is far from enough.","In this setting, the environment should provide stereo vision from the four direction with depth information.","Evaluation SLAM tasks are divided into two part, mapping and positioning.","Building maps is a way for positioning.","Hence, we use the accuracy of positioning but not the of mapping as the benchmark.","To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.","After that, 10 images are given as query to the positioning system.","The score is calculated by average over the positioning success rate.","Conclusion We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.","In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.","By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.","We extract robot controlling system out from robot itself.","We hope to eliminate the long and high-cost preparation process before really training the system.","Also, by constructing this dataset, we hope to make deep learning methods work on real robots.","On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.","We also proposed four possible tasks that can be conducted on our dataset.","We are hoping that more tasks can be proposed to exploit its maximum potential.","Our future work includes increasing the number of scenes in the dataset.","Also we will construct more attributes in the dataset (object segmentation for example)."],["Applications","NavigationNet is designed for indoor navigation problems.","In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.","Apart from these four applications, we believe our dataset will spawn more applications."],["Motivation","In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.","For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.","Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.","To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.","This usually falls in the field of visual semantic learning.","That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.","Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based)The formalization of this issue is inspired by Yuke Zhu et al.[6].","To complete the task of target-driven navigation, a robot control system is given two images.","One is the target image, the other is the current first-person perspective image perceived from the environment.","The robot can choose to walk forward, backward, turn left or right according to the input images.","The environment will update the states accordingly and give out a reward.","This process loops until the target image is the same with the perceived one.","To formalize that: Goal The goal is straight-forward, to find the given object.","In practice, it can be converted into make the perceived image and the given image identical.","Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.","One thing that is sure is that a large positive reward will be given when the goal is achieved.","To improve time efficiency, each step it taken should be given a small negative reward.","In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.","Action Space To find an object, the understanding of the scene is the most important factor of the task.","Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.","Hence the State space would be the set of all possible image pairs.","To evaluate such a system, many aspects should be taken into consideration.","The most important two are, trajectory should be short and the robot should not run into the obstacles.","Collisions should be limited strictly as only one collision could greatly damage the robot.","To make a trajectory finite, we consider a trajectory longer than 10000 a loop.","To calculate a overall score on all the test results, we utilize the idea of histogram.","A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .","A overall score is calculated by finding the average over this histogram.","The target-driven navigation task is focused on finding one object efficiently and safely.","However, in practical scenarios, this is often not enough.","In many cases, we need to find more than one items and we do not care about the order.","For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.","This has become a rather different problem since it usually requires the knowledge of adjacency.","For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.","Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based)We also formalized this problem into a reinforcement learning problem.","To find all the given items, the robot is given a list of items to collect, each one is represented by an image.","The rest will be similar with the target-driven task.","A first-person perspective image is given, an action is executed and then state is updated, reward is given.","To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.","Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.","We believe that such a design would be better for deep reinforcement training[43].","Also a minor negative time time punishment is necessary.","Action Space Since the task type is similar with the target-driven problem.","The same action space remains.","{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.","However, a set images with unknown-length is hard to be processed.","Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Although the task is different from the previous task, the two major meters are the same.","Hence, we use the same metric system in this task as is in the target-driven navigation task.","Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.","One widely-discussed but never well-addressed problem is the sweeper route planning problem.","Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.","The current solutions often lead to stuck at some small space.","Besides sweeper robot, it is general problem we will meet in many scenarios.","This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.","However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.","The most important factor is that the robot will not know that some movement is blocked until the collision really happens.","It must see and understand to avoid collisions.","Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based)Again, we formalized this problem into a reinforcement learning task.","At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.","The environment should give out the reward then according to how much area has been covered by the sweeper.","Typically, the robot should maintain a map of the room inside its system.","To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.","However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.","Reward Reward design is simple in this problem.","Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.","That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.","Action Space In this task, a sweeper does not really care about which direction it is facing.","The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).","So we will not give the robot the ability to turn left/right.","As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.","State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.","The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.","To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.","For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.","If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.","An average over scores of all steps in a trajectory is the final score of the trajectory.","Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.","Many solutions have been proposed to increase the mapping accuracy.","However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.","Additionally, applying SLAM for some large-scale scenes (e.g.","Museum) is effort consuming, which requires us to carry camera walked about all corners.","So, we hope that robots can preform SLAM autonomously.","With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.","This task is also formalized as a reinforcement learning problem.","On each step, the robot is fed with images perceived at its current location.","The system could use the visual inputs and history information to improve the map.","Reward should be calculated according to the difference between the constructed map and the ground truth.","An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.","To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.","Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.","Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.","By talking about progress, we are assuming total trust in legacy SLAM algorithms.","We would like to give a positive reward for any new reconstructed area after each steps.","In the meantime, a fruitless step is also punished with a minor negative reward.","Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.","In the meantime, we need to add the function of moving left/right to the robot.","Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.","Most of it is about 3D reconstruction.","Thus only a first-person single-eye perception is far from enough.","In this setting, the environment should provide stereo vision from the four direction with depth information.","SLAM tasks are divided into two part, mapping and positioning.","Building maps is a way for positioning.","Hence, we use the accuracy of positioning but not the of mapping as the benchmark.","To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.","After that, 10 images are given as query to the positioning system.","The score is calculated by average over the positioning success rate."],["Conclusion","We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.","In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.","By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.","We extract robot controlling system out from robot itself.","We hope to eliminate the long and high-cost preparation process before really training the system.","Also, by constructing this dataset, we hope to make deep learning methods work on real robots.","On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.","We also proposed four possible tasks that can be conducted on our dataset.","We are hoping that more tasks can be proposed to exploit its maximum potential.","Our future work includes increasing the number of scenes in the dataset.","Also we will construct more attributes in the dataset (object segmentation for example)."]],"string":"[\n [\n \"NavigationNet: A Large-scale Interactive Indoor Navigation Dataset\"\n ],\n [\n \"Abstract Indoor navigation aims at performing navigation within buildings.\",\n \"In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor scenes to complete various tasks in order to serve human.\",\n \"In most scenarios, we expected an intelligent device capable of navigating itself in unseen environment.\",\n \"Although several solutions have been proposed to deal with this issue, they usually require pre-installed beacons or a map pre-built with SLAM, which means that they are not capable of working in novel environments.\",\n \"To address this, we proposed NavigationNet, a computer vision dataset and benchmark to allow the utilization of deep reinforcement learning on scene-understanding-based indoor navigation.\",\n \"We also proposed and formalized several typical indoor routing problems that are suitable for deep reinforcement learning.\"\n ],\n [\n \"Introduction\",\n \"Indoor navigation aims at preforming navigation within buildings, such as train stations, airports, shopping centers, offices and museums.\",\n \"Most intelligent devices require an indoor routing functionality to guide itself precisely through rooms to complete various tasks.\",\n \"In most of the scenarios, we expect an intelligent device capable of navigating itself in new environments to serve human.\",\n \"In the past years, several solutions have been proposed to deal with this problem [1].\",\n \"Although they can be applied in some special cases, none of them can work smartly in novel environments or are based on scene understanding.\",\n \"Those solutions require either dense pre-installed beacons (e.g.\",\n \"WiFi/Bluetooth chips) over the scene or a map pre-built with SLAM in which human workers are required to carry cameras to scan the scene in advance.\",\n \"Such requirements significantly limit the applications of these solutions.\",\n \"New scenes have to be well prepared before the robots can work properly in it, which makes it impossible to be put into large-scale application in various indoor scenarios.\",\n \"Additionally, neither beacons nor maps are helpful in semantically understanding indoor scenes and thus will lead to the failure of the robot to perform many practical tasks.\",\n \"For example, to complete a regular command like 'to take a cup', a housekeeping android should understand what the scene is like, what a cup is and where a cup usually be, to navigate properly.\",\n \"We expected a more general solution of indoor navigation problems that requires no preparation of the scene in advance.\",\n \"Such a high demand requires a more human-like robot that is much more intelligent to implement self-exploration in novel environments.\",\n \"Recent years, we have witnessed the rapid development of deep reinforcement learning (DRL) [2], [3] and computer vision [4], [5].\",\n \"Therefore, smart indoor navigation enabled by DRL with visual inputs has been introduced [6].\",\n \"In this setting, pre-installed beacons or pre-built map are no longer required.\",\n \"A smart robot with vision will be enough to do the navigation job in different scenes.\",\n \"To apply reinforcement learning methods on navigation problems, massive trial-and-error is necessary in training the model.\",\n \"A straight-forward way is to construct a real robot and train Reinforcement Learning (RL) models on real-world scenes.\",\n \"However, such process is extremely slow and costly.\",\n \"Additionally, it is often accompanied by robot collision and damage, which means that more time and resources would be wasted.\",\n \"Even worse, due to the bottleneck of physical movements, to execute any action in the real world requires at least several seconds, which is unacceptable for any machine learning methods since that they usually require millions of times of trial-and-error.\",\n \"Aside from those drawbacks, such a process raises the problem of reproducibility.\",\n \"Even if all the background settings of a research are published in detail, it is still very hard for third parties to reproduce the experiment result since it is costly and time-consuming to find or construct a new environment that is identical to the one the original author used.\",\n \"That makes it impossible for third parties to examine and evaluate a new research result.\",\n \"We believe, an open-source, low-cost, large-scale dataset and corresponding benchmarks would be a key to largely advance this research field.\",\n \"Some previous work has tried to train the robot in computer graphics (CG) scenes like AI2-THOR[6], SUNCG[7].\",\n \"However, we do have noticed that training models on CG environments raised the problem of model transferring as the gap between hand-crafted scenes and the real world is significant.\",\n \"The visual appearance of images rendered by CG system always look non-realistic.\",\n \"Existing 3D models are limited in representing real-world scenes and objects.\",\n \"To deal with this problem, we proposed NavigationNet, a large-scale, real-world, interactive dataset.\",\n \"In each scene, we use human-size robot with 8 cameras to capture photos towards 8 different directions at all walk-able positions.\",\n \"Given all the collected images, we can easily tour the room like a CG system by moving among those walk-able positions.\",\n \"However, different from CG systems, what we observed from this dataset are all real images captured from the real world.\",\n \"Therefore, we can virtually command a robot to walk in the scene and get the first-perspective view of the robot.\",\n \"Our dataset is interactive.\",\n \"That is to say, as we send a command of movement (e.g.\",\n \"moving forward, turning right), the robot will perform as required and return a new view it sees from the new position.\",\n \"To our best knowledge, our NavigationNet is the first real-world indoor scene dataset that is interactive in computer vision field.\",\n \"Our dataset covers about 1500 $m^2$ indoor area with various bedrooms, studies, meeting room and etc.\",\n \"This makes the bias less in experiments on our dataset.\",\n \"Based on NavigationNet, we proposed four applications that require indoor navigation.\",\n \"We will explained them in detail in chapter .\",\n \"To summarize, we presented a large-scale dataset consisting of visual data collected from real scenarios to allow a low-cost, reproducible training of indoor navigation robots based on reinforcement learning methods.\",\n \"We also proposed and formalized several practical indoor navigation tasks in the framework of reinforcement learning.\"\n ],\n [\n \"Related Work\",\n \"In this chapter, we will discuss some of the previous works that are related to our work or act as a tool in this project.\",\n \"First we will talk about the technology of Visual Semantic Planning.\",\n \"A brief introduction to the popular datasets in the field of computer vision, after which we will explain three techniques used in our work in detail: Reinforcement Learning, Convolutional Neural Network(CNN) and Simultaneous Localization and Mapping.\"\n ],\n [\n \"Visual Semantic Planning\",\n \"Fundamental tasks in mobile robot controlling include navigation, mapping, grasping, path planning and so on.\",\n \"Such task of interacting with a visual world and planning a sequence of actions to achieve a certain goal is addressed as Visual Semantic Planning [8].\",\n \"A series of work [9], [10], [11], [12] has addressed the task-level planning and learning from dynamics.\",\n \"Classical approaches [13], [14] of motion planning require building map purely geometrically using Light Detection and Ranging (LiDAR) or Structure from Motion (SfM) [15], [16], [17].\",\n \"Srivastava et al.\",\n \"[18] provides an interface between task and motion planning, such that the task planner can effectively operate in an abstracted state space that ignores geometry.\",\n \"Recently proposed end-to-end learning-based approaches [19], [10], [20] can go directly from pixels to actions.\",\n \"For instance Zhu et al.\",\n \"[19] adopt feed-forward neural network to target-driven visual navigation in indoor scenes.\",\n \"Gupta et al.\",\n \"[10] use online Cognitive Mapping and Planning (CMP) approach for goal direction visual navigation without requiring a pre-constructed map.\",\n \"Jang et al.\",\n \"[9] train a deep neural network to perform the semantic grasping task inspired by the \\\"two-stream hypothesis\\\" of human vision.\",\n \"In terms of Complete Path Planning, Kollar et al.\",\n \"[21] use reinforcement learning to find trajectories and the learnt policy transfers successfully to a real environment.\"\n ],\n [\n \"Datasets for Computer Vision\",\n \"Datasets and corresponding benchmarks have played a significant role in many areas such as computer vision and speech recognition.\",\n \"The evolution from WordNet[22] to ImageNet[23] was one of the many successful examples that proved the power of an effective dataset to accelerate the development of one area.\",\n \"On the contrary, the lack of appropriate data has become one of the most crucial challenges for deep reinforcement learning, especially when it is dealing with real world problems like visual semantic learning.\",\n \"Since training and quantitatively evaluating DRL algorithms in real environments is either impractical or costly, this problem has become more urgent.\",\n \"For RL algorithms dealing with virtual-world problems, the Arcade Learning Environment (ALE) [24] exposed Atari 2600 games as reinforcement learning problems.\",\n \"Works such as Duan et al.\",\n \"[25] and Brockman et al.\",\n \"[26] propose toolkits to qualify progress in reinforcement learning.\",\n \"Other benchmarks [27], [28], [29], [30], [31], simulators [32] or physics engines [33] are also designed for the development of deep reinforcement learning.\",\n \"Also, scientists are trying to model the real world for DRL or other visual tasks.\",\n \"Chang et al.\",\n \"[34] provides 90 building-scale reconstructed scenes for supervised and self-supervised computer vision tasks such as keypoint matching, view overlap prediction semantic segmentation, scene classification and so on.\",\n \"AI2-THOR[19], SUNCG[7] and House3D are all CG scenes designed especially for deep reinforcement learning.\",\n \"The advanced version of AI2-THOR[35] even provides the functionality to let the robot directly interact with the objects in the scenarios.\",\n \"Different from our NavigationNet, these datasets are either synthetic or reconstructed using computer graphics techniques, which introduce inconsistency distribution between the real-world scenarios and the produced ones.\"\n ],\n [\n \"Reinforcement Learning\",\n \"Reinforcement learning (RL) [36] method was proposed in the late 90s.\",\n \"It provides a technique to allow the robot or any other intelligent systems to build a value function to evaluate policies in given situations from an interactive way of trail-and-error.\",\n \"Pioneering works from Mnih et al.\",\n \"[37], Lillicrap et al.\",\n \"[38], Schulman et al.\",\n \"[39] and Silver et al [40] introduced and ignited the idea of join Reinforcement Learning and Deep Learning into Deep Reinforcement Learning.\",\n \"In learning complex behavior skills and solving challenging control tasks in high-dimensional raw sensory state-space, DRL methods have shown tremendous success [40], [41], [3], [42].\",\n \"Trust Region Policy Optimization (TRPO) [41] makes a series of approximations to the theoretically-justified procedure and has robust performance on a wide range of tasks.\",\n \"Asynchronous Advantage Actor-Critic (A3C) Algorithm [3], which uses asynchronous gradient descent for optimization, succeeds on task of navigating random 3D mazes as well as a wide variety of continuous motor control problems.\",\n \"UNsupervised REinforcement and Auxiliary Learning (UNREAL) [43] brings together the A3C framework with auxiliary control tasks and auxiliary reward tasks.\",\n \"Actor Critic using Kronecker-factored Trust Region (ACKTR) [42] Algorithm uses a Kronecker-factored approximation to natural policy gradient that allows the covariance of the gradient to be inverted efficiently.\",\n \"Proximal Policy Optimization (PPO) [44] Algorithms use multiple epochs of stochastic gradient ascent to perform each policy update.\",\n \"Reinforcement Learning (RL) provides a powerful and flexible framework to several applications.\",\n \"Andrew Y. Ng et al.\",\n \"[45] describe a successful application of autonomous helicopter.\",\n \"Finn et al.\",\n \"[12] present an approach that automates state-space construction by learning a state representation directly from camera images with deep spatial autoencoder.\",\n \"Heess et al.\",\n \"[46] explore a rich environment can help to promote the learning of complex behavior.\",\n \"They normalize observations, scale the reward and use per-batch normalization of the advantages.\",\n \"Gu et al.\",\n \"[47] demonstrate a DRL algorithm based on off-policy training of deep Q-function can scale to complex 3D manipulation tasks and can learn deep neural network policies efficiently enough to train on real physical robots.\",\n \"Denil et al.\",\n \"[48] find DRL methods can learn to perform the experiments necessary to discover properties such as mass and cohesion of objects.\"\n ],\n [\n \"Convolutional Neural Network\",\n \"Deep convolutional neural networks [49], [50] have led to a series of breakthrough for visual tasks.\",\n \"He et al.\",\n \"[5] present a Residual Network (ResNet) to ease the training of deep neural network.\",\n \"We extract feature map with ResNet 101.\",\n \"Huang et al.\",\n \"[51] presents Dense Convolutional Network (DenseNet) which connects each layer to every other layer in a feed-forward fashion.\",\n \"Chen et al.\",\n \"[52] propose Dual Path Net (DPN) by revealing ResNet and DenseNet within the higher order recurrent neural network framework.\",\n \"Deep Learning models have been successful in learning powerful representations and understanding stereo context [53], [54].\",\n \"Shaked et al.\",\n \"[55] present a three-step pipeline for the stereo matching problem and a network architecture (ResMatch) for computing the matching cost at each possible disparity.\",\n \"Kendall et al.\",\n \"[56] propose an end-to-end model (GC-Net) for regressing disparity from a rectified pair of stereo images.\"\n ],\n [\n \"Simultaneous Localization and Mapping\",\n \"Visual SLAM plays an important role in autonomous navigation of mobile robots [57].\",\n \"The emergence of MonoSLAM [58] makes the idea of utilizing one camera become popular.\",\n \"Parallel Tracking and Mapping (PTaM) [59] uses an approach based on keyframes with two parallel processing threads.\",\n \"ORB [60] becomes a computationally-efficient replacement to SIFT keypoint detector and descriptor that has similar matching performance.\",\n \"Keyframe bundle adjustment outperforms filtering, since it gives the most accuracy per unit of computing time [61].\",\n \"S-PTAM [62] allows accuracy improvement of the mapping process with respect to monocular SLAM and avoiding the well-known bootstrapping problem.\",\n \"ORB-SLAM2 [63] is a complete SLAM system for monocular, stereo and RGB-D cameras.\",\n \"RGB-D SLAM [64] can producing high quality globally consistent surface reconstruction with only a low-cost commodity RGB-D sensor.\",\n \"Mur-Artal et al.\",\n \"[65] propose a visual-inertial monocular SLAM which is able to close loops and reuse its map to achieve zero-drift localization in already mapped areas.\"\n ],\n [\n \"NavigationNet\",\n \"NavigationNet is specifically designed for applying reinforcement learning methods to indoor navigation tasks.\",\n \"We collected data from the real world and organized it properly to allow the robot to roam in the dataset as if they are in the corresponding real-world scenario.\",\n \"In this chapter, we will explain its organization and applications in detail to show you the world of NavigationNet.\"\n ],\n [\n \"Data Organization\",\n \"Images in NavigationNet are organized hierarchically.\",\n \"The hightest node is the root of NavigationNet.\",\n \"The second level nodes, also the primary elements, are scenes.\",\n \"A scene is a collection of images collected from the same indoor space.\",\n \"In the next level, a scene is divided into rooms, space connected with doors.\",\n \"In the current version of NavigationNet, we have 15 scenes, each with 1-3 rooms.\",\n \"The origin room for each scene is at least 50 $m^2$ in area.\",\n \"The fourth level is called postion.\",\n \"A position is from where images are collected.\",\n \"A room contains hundreds of postions.\",\n \"When being trained, the robot could move among the adjacent positions to 'see' from that perspective.\",\n \"When constructing the dataset, our mobile robot iterated over all the walkable points in the room with the granularity of 20cm.\",\n \"Hence, in the dataset, two adjacent positions are 20cm away from each other.\",\n \"With such a granularity, one room usually contains thounsands of positions.\",\n \"From the perspective of information, a position can be seen as a bundle of information that the robot will receive when it is placed at that position.\",\n \"Usually, we offer 8 images in this bundle.\",\n \"These images are divided into four directions: front, back, left and right.\",\n \"Towards each direction, two parallel images are taken to complete a binocular vision system.\",\n \"Such a setting enables the possibilities like stereo matching and panorama reconstruction.\",\n \"In addition to the images, we also provide a ground-truth map attached to the scene.\",\n \"The map can be considered as a binary map indicating where is walkable or non-walkable.\",\n \"This is essential especially to the Auto-SLAM problem.\"\n ],\n [\n \"Robot Moving Control\",\n \"To satisfy the different requirements of potential tasks, we tried our best to provide a more generalizable moving control SDK for NavigationNet.\",\n \"As is stated in section REF , the fundamental element in NavigationNet is scene, virtual robots are allowed to roam in scenes.\",\n \"To serve well as an action in a reinforcement learning task, movements should be standard, discrete, limited and complete.\",\n \"An action should be standard, so that the robot should move the same distance towards the same direction or turning the same angle when a specific instruction of a movement is given.\",\n \"An action should be discrete, so that the action space can be finite and discrete.\",\n \"Actions should be limited, so that the algorithm do not need to choose actions from an infinite action space.\",\n \"Actions should be complete, so that all the actions joined together could describe the whole real-world action space.\",\n \"In practice, it is impossible to use a finite action space to cover an infinite one.\",\n \"We need some sort of compromise.\",\n \"The one we take in this case is to discrete the movement length and the turning angle.\",\n \"So that we defined six types of movements.\",\n \"MOVE FORWARD Ask the robot to move forward one position.\",\n \"For example, go from (9, 12) to (9, 13) when facing north but go from (11, 8) to (10, 8) when facing west.\",\n \"Since two adjacent nodes are 20cm away, this is to move forward 20cm in real-world scenes.\",\n \"MOVE BACKWARD Ask the robot to move backward one position.\",\n \"This is the opposite movement of MOVE FORWARD and can revert the effect of MOVE FORWARD.\",\n \"MOVE LEFT Ask the robot to move left one position.\",\n \"For example, go from (5, 4) to (4, 4) when facing north but go from (7, 4) to (7, 3) when facing west.\",\n \"The real world distance would be the same as MOVE FORWARD and MOVE BACKWARD.\",\n \"It should be paid attention to that MOVE LEFT is very different from TURNING LEFT as it will move the position of the robot but will not change the facing direction.\",\n \"MOVE RIGHT Ask the robot to move right one position.\",\n \"This is the opposite movement of MOVE LEFT and can revert the effect of MOVE LEFT.\",\n \"TURN LEFT Ask the robot to turn left at 90-degree.\",\n \"For example, when the robot receives this request when at (5, 7) facing north, it should then facing west still at (5, 7).\",\n \"It should be paid extra attention that this movement is different from MOVE LEFT that the former changes the facing while the latter change the position.\",\n \"TURN RIGHT Ask the robot to turn right at 90-degree.\",\n \"This is the opposite movement of TURN LEFT.\",\n \"Figure: MovementsAt the same time, we should make it clear that, in any given tasks, according to the specific setting, not all the actions (movement) need to be taken into consideration.\",\n \"For example, when we would like to simulate a two-wheel one-eye robot, TURN LEFT and TURN RIGHT are a must, otherwise seventy-five percent of the images can never be perceived while MOVE LEFT and MOVE RIGHT should be eliminated since a two-wheel robot can not make the exact movement of moving left 20cm.\",\n \"Meanwhile, when simulating a robot with panorama vision, we could only take MOVE LEFT and MOVE RIGHT but not TURN LEFT and TURN RIGHT as they are not necessary but can only enlarge the action space unnecessarily and make the task more difficult.\"\n ],\n [\n \"NavigationNet Construction\",\n \"NavigationNet is a large-scale dataset with hundreds of thousands of images.\",\n \"The large amount makes the data collection a task of impossible.\",\n \"What is worst is that, as to each image, it is required that the position, angle and height where it is taken should be exactly where it is expected so long that the simulation of reality will not offset.\",\n \"These two factors together make the task of collection much more difficult than expected.\",\n \"In this chapter, we will talk about how we constructed this dataset efficiently and accurately.\"\n ],\n [\n \"Collector Mobile Robot\",\n \"Robots are designed to liberate human beings from the repetitive boring work.\",\n \"NavigationNet is built for intelligent robots, also built with the help of intelligent robots.\",\n \"Collecting data for NavigationNet requires heavy labor and high accuracy but little flexibility (most of the possible conditions are under control), which is surprisingly suitable for robots.\",\n \"In order to reduce the labor requirement, we exploited the possibilities of smart hardware.\",\n \"Our team developed a dedicated data-collecting mobile robot with Arduino Mega2560 and Raspberry Pi 3 Model B codenamed GoodCar.\",\n \"Arduino[66] is an open-source physical computing platform based on a simple I/O board and a development environment that implements the Processing/Wiring language.\",\n \"Arduino boards can be programmed in the same name program language Arduino, which is a C-like area-specific language to receive and send signals from tens of low-voltage I/O ports.\",\n \"It is specifically suitable for robot controlling.\",\n \"In this project, we use a Arduino Mega2560 to control the movement of the robot.\",\n \"Raspberry Pi[67] is a cheap single-board computer system running a specificialy made Linux distribution.\",\n \"It is originally built for computer programming education but we find it suitable for robot controlling.\",\n \"It consumes little electricity which allows us to strip the 220V power wires.\",\n \"It contains all kinds of I/O ports such that we could plug it with Arduinos, cameras, other full-sized computers and many other modules to make it the center of the system.\",\n \"It runs standard linux distribution such that the possibility of software extension is beyond imagine.\",\n \"Last but not least, it is cheap so that we could us as many as we want on the robot.\",\n \"In practice, we use two Raspberry Pi 3B for controlling all other modules and communicating with the upper computer.\",\n \"To build this robot, we used a two-level structure.\",\n \"The upper-level is as high as 1.4m to simulate the the height of the eyes of an average adult.\",\n \"We plugged-in eight cameras on this level, two for each direction to make a stereo vision system and to avoid unnecessary turning-around.\",\n \"The lower-level is for the motion systems.\",\n \"The Raspberry Pi and Arduino mentioned above are all on this level.\",\n \"The Arduino is to control the mobile devices and sensors.\",\n \"It is linked with four motors to control the movement.\",\n \"In the meantime, it is linked with a sonar and code plate counters to avoid collision and measure the distance of movement.\",\n \"Also it is linked with a Raspberry Pi and under its control.\",\n \"We used two Raspberry Pi 3B in each robot, one is master and the other is slave.\",\n \"The master computer controls many things.\",\n \"First it communicates with the Arduino to control the movement indirectly.\",\n \"Second it is responsible for evaluating the signal Arduino sent back to avoid collision.\",\n \"Third it is master computer's duty to control four of the cameras to take photo at given position and store the images on the SD card.\",\n \"Fourth the slave computer is also under its control to take photos when required.\",\n \"Lastly it communicates with the upper computer, which is controlled by human, via Wi-Fi connection.\",\n \"That is the control core of our robot.\",\n \"Under the lower-level is the mobile devices.\",\n \"We have tried on many types of mobile devices and at last we used four track structures to reduce error.\",\n \"We had constructed six mobile robots before moving on to collect data.\",\n \"Figure: Robot\"\n ],\n [\n \"Data Collection\",\n \"To make the data collecting process efficient and accurate, also to make the whole process manageable and under control, we designed and followed the following instructions: Step 1 Measure the size of the target room(s).\",\n \"Find the most southwestern corner as the starting point.\",\n \"Step 2 Move the robot to the starting point.\",\n \"Evaluate and record its distance towards the borders.\",\n \"Step 3 Let the robot face north and start to collect required photos at this point.\",\n \"Step 4 After photos are taken, let the robot run 20cm towards north and collect data.\",\n \"Step 5 Keep iterating over Step 3-4 until the robot reaches the end of the line.\",\n \"Step 6 Move the robot to the starting point of the line and then move 20cm towards east.\",\n \"Step 7 Iterating over Step3-6 until the robot reaches the east corner.\",\n \"Step 8 Complete this scenario by taking photos from those points that the robot did not reach.\",\n \"Data Processing After the collection, the original data should be pre-processed before taken into production.\",\n \"The data processing steps we taken includes: Data Cleaning When collecting data with robot, there are always inevitable errors occurring especially when dealing with room corners.\",\n \"For each room we collect, we will scan over all the collected photos and find errors.\",\n \"If necessary, we would go and retry collecting data to complete the set.\",\n \"Map Building When collecting data, we also recorded whether a point is accessible.\",\n \"With this information, we are able to rebuild a walkable-or-not map for reference.\",\n \"Stereo Vision For some of the task, to simulate a two-eye system or a RGB-D system, it would be of much help if we could produce depth channel in the very beginning.\",\n \"To serve this purpose, we use MC-CNN[54] to produce some pre-processed depth channel data.\",\n \"Applications NavigationNet is designed for indoor navigation problems.\",\n \"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.\",\n \"Apart from these four applications, we believe our dataset will spawn more applications.\",\n \"Target-driven Navigation Motivation In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.\",\n \"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.\",\n \"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.\",\n \"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.\",\n \"This usually falls in the field of visual semantic learning.\",\n \"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.\",\n \"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based) Formalization The formalization of this issue is inspired by Yuke Zhu et al.[6].\",\n \"To complete the task of target-driven navigation, a robot control system is given two images.\",\n \"One is the target image, the other is the current first-person perspective image perceived from the environment.\",\n \"The robot can choose to walk forward, backward, turn left or right according to the input images.\",\n \"The environment will update the states accordingly and give out a reward.\",\n \"This process loops until the target image is the same with the perceived one.\",\n \"To formalize that: Goal The goal is straight-forward, to find the given object.\",\n \"In practice, it can be converted into make the perceived image and the given image identical.\",\n \"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.\",\n \"One thing that is sure is that a large positive reward will be given when the goal is achieved.\",\n \"To improve time efficiency, each step it taken should be given a small negative reward.\",\n \"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.\",\n \"Action Space To find an object, the understanding of the scene is the most important factor of the task.\",\n \"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.\",\n \"Hence the State space would be the set of all possible image pairs.\",\n \"Evaluation To evaluate such a system, many aspects should be taken into consideration.\",\n \"The most important two are, trajectory should be short and the robot should not run into the obstacles.\",\n \"Collisions should be limited strictly as only one collision could greatly damage the robot.\",\n \"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.\",\n \"To calculate a overall score on all the test results, we utilize the idea of histogram.\",\n \"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .\",\n \"A overall score is calculated by finding the average over this histogram.\",\n \"Multi-target Navigation Motivation The target-driven navigation task is focused on finding one object efficiently and safely.\",\n \"However, in practical scenarios, this is often not enough.\",\n \"In many cases, we need to find more than one items and we do not care about the order.\",\n \"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.\",\n \"This has become a rather different problem since it usually requires the knowledge of adjacency.\",\n \"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.\",\n \"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based) Formalization We also formalized this problem into a reinforcement learning problem.\",\n \"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.\",\n \"The rest will be similar with the target-driven task.\",\n \"A first-person perspective image is given, an action is executed and then state is updated, reward is given.\",\n \"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.\",\n \"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.\",\n \"We believe that such a design would be better for deep reinforcement training[43].\",\n \"Also a minor negative time time punishment is necessary.\",\n \"Action Space Since the task type is similar with the target-driven problem.\",\n \"The same action space remains.\",\n \"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.\",\n \"However, a set images with unknown-length is hard to be processed.\",\n \"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Evaluation Although the task is different from the previous task, the two major meters are the same.\",\n \"Hence, we use the same metric system in this task as is in the target-driven navigation task.\",\n \"Sweeper Route Planning Motivation Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.\",\n \"One widely-discussed but never well-addressed problem is the sweeper route planning problem.\",\n \"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.\",\n \"The current solutions often lead to stuck at some small space.\",\n \"Besides sweeper robot, it is general problem we will meet in many scenarios.\",\n \"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.\",\n \"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.\",\n \"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.\",\n \"It must see and understand to avoid collisions.\",\n \"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based) Formalization Again, we formalized this problem into a reinforcement learning task.\",\n \"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.\",\n \"The environment should give out the reward then according to how much area has been covered by the sweeper.\",\n \"Typically, the robot should maintain a map of the room inside its system.\",\n \"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.\",\n \"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.\",\n \"Reward Reward design is simple in this problem.\",\n \"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.\",\n \"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.\",\n \"Action Space In this task, a sweeper does not really care about which direction it is facing.\",\n \"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).\",\n \"So we will not give the robot the ability to turn left/right.\",\n \"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.\",\n \"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.\",\n \"Evaluation The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.\",\n \"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.\",\n \"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.\",\n \"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.\",\n \"An average over scores of all steps in a trajectory is the final score of the trajectory.\",\n \"Mean Average Precision (mAP) [H] mAPsteps $score \\\\leftarrow 0$ $n \\\\leftarrow 0$ $m \\\\leftarrow 0$ $step$ in $steps$ $n \\\\leftarrow n + 1$ $step$ is $NEW\\\\_POSITION$ $m \\\\leftarrow m + 1$ $score \\\\leftarrow score + m/n$ $score \\\\leftarrow score + 0$ $score \\\\leftarrow score / n$ $score$ Auto-SLAM with Deep Reinforcement Learning Motivation Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.\",\n \"Many solutions have been proposed to increase the mapping accuracy.\",\n \"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.\",\n \"Additionally, applying SLAM for some large-scale scenes (e.g.\",\n \"Museum) is effort consuming, which requires us to carry camera walked about all corners.\",\n \"So, we hope that robots can preform SLAM autonomously.\",\n \"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.\",\n \"Formalization This task is also formalized as a reinforcement learning problem.\",\n \"On each step, the robot is fed with images perceived at its current location.\",\n \"The system could use the visual inputs and history information to improve the map.\",\n \"Reward should be calculated according to the difference between the constructed map and the ground truth.\",\n \"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.\",\n \"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.\",\n \"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.\",\n \"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.\",\n \"By talking about progress, we are assuming total trust in legacy SLAM algorithms.\",\n \"We would like to give a positive reward for any new reconstructed area after each steps.\",\n \"In the meantime, a fruitless step is also punished with a minor negative reward.\",\n \"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.\",\n \"In the meantime, we need to add the function of moving left/right to the robot.\",\n \"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.\",\n \"Most of it is about 3D reconstruction.\",\n \"Thus only a first-person single-eye perception is far from enough.\",\n \"In this setting, the environment should provide stereo vision from the four direction with depth information.\",\n \"Evaluation SLAM tasks are divided into two part, mapping and positioning.\",\n \"Building maps is a way for positioning.\",\n \"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.\",\n \"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.\",\n \"After that, 10 images are given as query to the positioning system.\",\n \"The score is calculated by average over the positioning success rate.\",\n \"Conclusion We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.\",\n \"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.\",\n \"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.\",\n \"We extract robot controlling system out from robot itself.\",\n \"We hope to eliminate the long and high-cost preparation process before really training the system.\",\n \"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.\",\n \"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.\",\n \"We also proposed four possible tasks that can be conducted on our dataset.\",\n \"We are hoping that more tasks can be proposed to exploit its maximum potential.\",\n \"Our future work includes increasing the number of scenes in the dataset.\",\n \"Also we will construct more attributes in the dataset (object segmentation for example).\"\n ],\n [\n \"Applications\",\n \"NavigationNet is designed for indoor navigation problems.\",\n \"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.\",\n \"Apart from these four applications, we believe our dataset will spawn more applications.\"\n ],\n [\n \"Motivation\",\n \"In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.\",\n \"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.\",\n \"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.\",\n \"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.\",\n \"This usually falls in the field of visual semantic learning.\",\n \"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.\",\n \"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based)The formalization of this issue is inspired by Yuke Zhu et al.[6].\",\n \"To complete the task of target-driven navigation, a robot control system is given two images.\",\n \"One is the target image, the other is the current first-person perspective image perceived from the environment.\",\n \"The robot can choose to walk forward, backward, turn left or right according to the input images.\",\n \"The environment will update the states accordingly and give out a reward.\",\n \"This process loops until the target image is the same with the perceived one.\",\n \"To formalize that: Goal The goal is straight-forward, to find the given object.\",\n \"In practice, it can be converted into make the perceived image and the given image identical.\",\n \"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.\",\n \"One thing that is sure is that a large positive reward will be given when the goal is achieved.\",\n \"To improve time efficiency, each step it taken should be given a small negative reward.\",\n \"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.\",\n \"Action Space To find an object, the understanding of the scene is the most important factor of the task.\",\n \"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.\",\n \"Hence the State space would be the set of all possible image pairs.\",\n \"To evaluate such a system, many aspects should be taken into consideration.\",\n \"The most important two are, trajectory should be short and the robot should not run into the obstacles.\",\n \"Collisions should be limited strictly as only one collision could greatly damage the robot.\",\n \"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.\",\n \"To calculate a overall score on all the test results, we utilize the idea of histogram.\",\n \"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .\",\n \"A overall score is calculated by finding the average over this histogram.\",\n \"The target-driven navigation task is focused on finding one object efficiently and safely.\",\n \"However, in practical scenarios, this is often not enough.\",\n \"In many cases, we need to find more than one items and we do not care about the order.\",\n \"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.\",\n \"This has become a rather different problem since it usually requires the knowledge of adjacency.\",\n \"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.\",\n \"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based)We also formalized this problem into a reinforcement learning problem.\",\n \"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.\",\n \"The rest will be similar with the target-driven task.\",\n \"A first-person perspective image is given, an action is executed and then state is updated, reward is given.\",\n \"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.\",\n \"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.\",\n \"We believe that such a design would be better for deep reinforcement training[43].\",\n \"Also a minor negative time time punishment is necessary.\",\n \"Action Space Since the task type is similar with the target-driven problem.\",\n \"The same action space remains.\",\n \"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.\",\n \"However, a set images with unknown-length is hard to be processed.\",\n \"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Although the task is different from the previous task, the two major meters are the same.\",\n \"Hence, we use the same metric system in this task as is in the target-driven navigation task.\",\n \"Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.\",\n \"One widely-discussed but never well-addressed problem is the sweeper route planning problem.\",\n \"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.\",\n \"The current solutions often lead to stuck at some small space.\",\n \"Besides sweeper robot, it is general problem we will meet in many scenarios.\",\n \"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.\",\n \"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.\",\n \"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.\",\n \"It must see and understand to avoid collisions.\",\n \"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based)Again, we formalized this problem into a reinforcement learning task.\",\n \"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.\",\n \"The environment should give out the reward then according to how much area has been covered by the sweeper.\",\n \"Typically, the robot should maintain a map of the room inside its system.\",\n \"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.\",\n \"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.\",\n \"Reward Reward design is simple in this problem.\",\n \"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.\",\n \"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.\",\n \"Action Space In this task, a sweeper does not really care about which direction it is facing.\",\n \"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).\",\n \"So we will not give the robot the ability to turn left/right.\",\n \"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.\",\n \"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.\",\n \"The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.\",\n \"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.\",\n \"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.\",\n \"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.\",\n \"An average over scores of all steps in a trajectory is the final score of the trajectory.\",\n \"Mean Average Precision (mAP) [H] mAPsteps $score \\\\leftarrow 0$ $n \\\\leftarrow 0$ $m \\\\leftarrow 0$ $step$ in $steps$ $n \\\\leftarrow n + 1$ $step$ is $NEW\\\\_POSITION$ $m \\\\leftarrow m + 1$ $score \\\\leftarrow score + m/n$ $score \\\\leftarrow score + 0$ $score \\\\leftarrow score / n$ $score$ Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.\",\n \"Many solutions have been proposed to increase the mapping accuracy.\",\n \"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.\",\n \"Additionally, applying SLAM for some large-scale scenes (e.g.\",\n \"Museum) is effort consuming, which requires us to carry camera walked about all corners.\",\n \"So, we hope that robots can preform SLAM autonomously.\",\n \"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.\",\n \"This task is also formalized as a reinforcement learning problem.\",\n \"On each step, the robot is fed with images perceived at its current location.\",\n \"The system could use the visual inputs and history information to improve the map.\",\n \"Reward should be calculated according to the difference between the constructed map and the ground truth.\",\n \"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.\",\n \"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.\",\n \"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.\",\n \"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.\",\n \"By talking about progress, we are assuming total trust in legacy SLAM algorithms.\",\n \"We would like to give a positive reward for any new reconstructed area after each steps.\",\n \"In the meantime, a fruitless step is also punished with a minor negative reward.\",\n \"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.\",\n \"In the meantime, we need to add the function of moving left/right to the robot.\",\n \"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.\",\n \"Most of it is about 3D reconstruction.\",\n \"Thus only a first-person single-eye perception is far from enough.\",\n \"In this setting, the environment should provide stereo vision from the four direction with depth information.\",\n \"SLAM tasks are divided into two part, mapping and positioning.\",\n \"Building maps is a way for positioning.\",\n \"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.\",\n \"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.\",\n \"After that, 10 images are given as query to the positioning system.\",\n \"The score is calculated by average over the positioning success rate.\"\n ],\n [\n \"Conclusion\",\n \"We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.\",\n \"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.\",\n \"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.\",\n \"We extract robot controlling system out from robot itself.\",\n \"We hope to eliminate the long and high-cost preparation process before really training the system.\",\n \"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.\",\n \"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.\",\n \"We also proposed four possible tasks that can be conducted on our dataset.\",\n \"We are hoping that more tasks can be proposed to exploit its maximum potential.\",\n \"Our future work includes increasing the number of scenes in the dataset.\",\n \"Also we will construct more attributes in the dataset (object segmentation for example).\"\n 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[
"Graph Learning for Anomaly Analytics: Algorithms, Applications, and\n Challenges"
],
[
"Abstract Anomaly analytics is a popular and vital task in various research contexts, which has been studied for several decades.",
"At the same time, deep learning has shown its capacity in solving many graph-based tasks like, node classification, link prediction, and graph classification.",
"Recently, many studies are extending graph learning models for solving anomaly analytics problems, resulting in beneficial advances in graph-based anomaly analytics techniques.",
"In this survey, we provide a comprehensive overview of graph learning methods for anomaly analytics tasks.",
"We classify them into four categories based on their model architectures, namely graph convolutional network (GCN), graph attention network (GAT), graph autoencoder (GAE), and other graph learning models.",
"The differences between these methods are also compared in a systematic manner.",
"Furthermore, we outline several graph-based anomaly analytics applications across various domains in the real world.",
"Finally, we discuss five potential future research directions in this rapidly growing field."
],
[
"Introduction",
"Anomalies, which are also known as outliers, commonly exist in various real-world networks [12], such as fake reviews in opinion networks [121], fake news in social networks [119], outlier members in collaboration networks [118], [94], flash crowds in traffic networks [50], socially selfish nodes in mobile networks [110], and network intrusions in computer networks [24].",
"The exploration of anomaly detection research is dating back to 1960s and it has been a popular research field for several decades [35].",
"With the increasing demand and broad applications in different domains, anomaly analytic plays an increasingly important role in various communities such as data mining and machine learning.",
"With the advancement of deep learning, graph learning is proposed subsequently, which is coined for deep learning-based models that are applied into graph-structured data [128], [112].",
"Due to its convincing performance and explainability, recent years have witnessed, in varied disciplines, an increasing number of studies focusing on anomaly detection and prediction tasks by utilizing deep graph models [99], [137], which is not limited to shallow network embedding such as random walks [40], [111].",
"As a unique non-Euclidean data structure, graphs are able to represent entities and their relationships in different kinds of scenarios.",
"However, this research direction faces several inevitable problem complexities to all detection methods when applying deep learning and artificial intelligence in real-world networks [57], [105].",
"Irregularity of graph structure.",
"Unlike other regular structured data, such as text, sequences, and images, nodes in a graph are unordered and can have distinct neighborhoods, which makes the structure of graphs irregular.",
"Therefore, some traditional deep learning architectures cannot be directly applied, such as convolution and pooling operation in convolutional neural networks (CNNs) [72].",
"Heterogeneous anomaly classes.",
"The types of nodes and links are generally not unitary in a graph, which leads to the emergence of heterogeneous information networks (HINs).",
"HINs usually incorporate more complex information among entities and relationships, especially those containing different modalities [85], which are very important in identifying different types of anomalies in a specific graph.",
"Scalability to real-world networks.",
"Nowadays, real-world networks such as social networks are composed of millions or even billions of nodes, edges, and attribute information [113].",
"This kind of large-scale network definitely increases computational complexity.",
"Therefore, it is imperative to devise scalable models having a linear time complexity with respect to the graph size.",
"Label scarcity.",
"Compared with manually generated graph data, there are mainly two reasons for the sparsity of real-world networks.",
"The first one is the scale-free network structure nature that the degree of nodes in most real-world networks follows long-tailed distribution [123].",
"The other one is limited by the collection technology and privacy protection in the process of crawling data.",
"Moreover, due to the lack of labeled datasets, devising unsupervised anomaly detection models is becoming important.",
"Diverse types of anomalies.",
"Several types of anomalies have been explored such as node, edge, subgraph, and path (shown in Fig REF ).",
"Node anomalies are entities that show anomalous behaviours in the whole graph compared with other nodes, e.g., users who spread fake news in social networks.",
"Other types of anomalies have similar concepts and their own real-world applications.",
"Here, subgraph anomaly is difficult to detect because the individual nodes could show normal behaviours when extracted from an anomalous subgraph.",
"There have been a line of deep anomaly detection research demonstrating significantly better performance than conventional models on solving the above-mentioned challenges.",
"Despite the fact that the adopted technologies vary from Graph Convolution Networks (GCNs) to Graph AutoEncoder (GAEs), most methods focus on detecting or predicting an anomaly in a specific situation due to the complexity of existing anomalies.",
"To the best of our knowledge, little attention has been devoted to summarizing these methods in a comprehensive way and clearly analyzing how they are applied to solve real-world application scenarios.",
"Figure: Classification of deep graph methods in solving anomaly analytics tasks and real-world applications.There are several surveys related to our work.",
"Zamini et al.",
"[122] summarized the anomaly detection techniques in four real-world application scenarios, namely banking, wireless sensor networks, social networks, and healthcare.",
"Akoglu et al.",
"[3] reviewed the anomaly detection methods using graph metric-based techniques, Ranshous et al.",
"[78] only focused on anomaly detection methods in dynamic networks, while Bilgin et al.",
"[8] briefly reviewed some non-deep learning methods of detecting anomalies in dynamic networks.",
"Both Chalapathy et al.",
"[11] and Pang et al.",
"[75] concentrated on deep learning enabled anomaly detection in different kinds of data, which is not limited to graph data.",
"[67] reviewed the contemporary deep learning methods for graph anomaly detection and categorized existing work according to the anomalous graph objects.",
"Actually, there are also some surveys focusing on introducing the main concepts and frameworks of Graph Neural Networks (GNNs) [137], [109], and divided the corresponding methods according to the type of GNN models.",
"Inspired by this classification strategy, we also divided the graph learning models in terms of the model type when introducing the specific anomaly detection tasks.",
"However, the main focus of our survey and GNN survey are totally different in spite of the similar classification strategy.",
"This work is different from previous studies in that we aim to summarize the graph learning methods systematically and comprehensively for detecting anomalies in various graphs, ranging from homogeneous to heterogeneous, non-attributed to attributed, undirected to directed, rather than focusing on only one specific kind of graph.",
"To fill this gap, we divide the existing methods into four categories based on their model architectures and training strategies: Graph Convolutional Networks (GCNs), Graph Attention Networks (GATs), Graph AutoEncoders (GAEs), and other GNN-based methods (shown in Figure REF ).",
"The main characteristics of these methods are compared and summarized in Table REF .",
"The characteristics of these basic models are briefly introduced in Section .",
"In summary, the contributions of this work are outlined as follows: A systematic summarization and comparison of graph learning methods for anomaly analytics is presented.",
"Specifically, we delineate their capabilities in addressing the existing problem complexities among all categories of the methods.",
"An overview of major anomaly analysis tasks in various application domains is given.",
"Insights into future research directions in this field are provided."
],
[
"Organization",
"The rest of this survey is structured as follows.",
"Section presents the notations and preliminaries of graph learning models, which will be used in the subsequent sections.",
"The anomaly analytics methods are reviewed in Sections to .",
"In Section , we outline several real-world applications of anomaly analytics that can be solved with deep graph models, and discuss some future research directions and challenges in Section .",
"Finally, we briefly conclude this survey in Section .",
"Figure: Illustration of the whole process of detecting anomalies in graph data with deep graph models.",
"The models are mainly divided into two parts according to whether anomaly score is calculated by latent representation or directly generated by end-to-end models.",
"There are mainly four types of graph anomalies, namely node, edge, (sub)graph, and path anomaly.Table: Summary of graph learning models in detecting and predicting anomalies[1]http://www-personal.umich.edu/~mejn/netdata/ & http://snap.stanford.edu/ [2]https://www.ipd.kit.edu/˜muellere/consub/ [3]https://www.cs.cmu.edu/˜./enron/ [4]https://github.com/KaiDMML/FakeNewsNet/tree/old-version [5]http://networkrepository.com/email-dnc [6]http://networkrepository.com/tech-as-topology [7]https://linqs.org/datasets/ [8]https://portal.311.nyc.gov/"
],
[
"Notations",
"A graphGraph and network are used interchangeably in this paper.",
"is represented as $G = (V,E)$ , where $V = \\lbrace v_1, ..., v_n\\rbrace $ is a set of $n$ nodes and $E \\subseteq V \\times V$ is a set of $m$ edges between nodes.",
"A graph may have different types, such as weighted or unweighted, directed or undirected.",
"Here, if a graph is directed, then $e_{ij} = (v_i,v_j) \\in E$ denotes an edge pointing from $v_i$ to $v_j$ .",
"The neighborhood set of a node $v_i$ is defined as $N(v_i) = \\lbrace v_j \\in V|(v_i,v_j) \\in E\\rbrace $ .",
"The adjacency matrix of a graph is a $n \\times n$ matrix, which is denoted as $\\mathbf {A}$ .",
"We use $\\mathbf {A}(i,:), \\mathbf {A}(:,j), \\mathbf {A}(i,j)$ to denote the $i^{th}$ row, $j^{th}$ column, and an element of $\\mathbf {A}$ , respectively.",
"For an unweighted graph, the element of its adjacency matrix is defined as: $\\textbf {A}(i,j)=\\left\\lbrace \\begin{array}{rl}1 & \\mbox{if $~e_{ij} \\in E$} \\\\0 & \\mbox{otherwise.",
"}\\end{array} \\right.$ For a weighted graph, $\\mathbf {A}(i,j)$ is defined as the weight of edge $e_{ij}$ .",
"A graph may have node attributes $\\mathbf {X^V}$ and edge attributes $\\mathbf {X^E}$ , where $\\mathbf {X^V}$ is the node attributes matrix and $\\mathbf {X^E}$ is the edge attributes matrix, respectively.",
"If the feature matrix is used as $\\mathbf {X}$ for convenience, the default setting $\\mathbf {X}$ refers to node attributes matrix.",
"Functions are marked with curlicues, e.g., $\\mathcal {F}(\\cdot )$ .",
"Throughout this paper, we use bold uppercase characters denoting matrices and bold lowercase characters representing vectors, like a matrix $\\mathbf {A}$ and a vector $\\mathbf {a}$ .",
"Unless particularly specified, the notations used in this paper are illustrated in Table REF .",
"Then, we provide a formal definition and brief introduction of some predefined matrices to better understand the concepts described in this paper."
],
[
"Preliminaries",
"Given an undirected graph, the Laplacian matrix is defined as $\\mathbf {L} = \\mathbf {D} - \\mathbf {A}$ , where $\\mathbf {D} \\in \\mathbb {R}^{n \\times n} $ is a diagonal degree matrix with $\\mathbf {D}_{ii} = \\sum _{j=1}^{n}\\mathbf {A}_{ij}$ .",
"$\\mathbf {L} = \\mathbf {Q\\Lambda Q^T}$ denotes eigendecomposition, where $\\mathbf {\\Lambda } \\in \\mathbb {R}^{n \\times n}$ is a diagonal matrix of eigenvalues in ascending order and $\\mathbf {Q} \\in \\mathbb {R}^{n \\times n}$ is composed of corresponding eigenvectors.",
"The element $\\mathbf {P}(i,j)$ of transition matrix $\\mathbf {P} = \\mathbf {D}^{-1}\\mathbf {A}$ represents the probability of a random walk from node $v_i$ to node $v_j$ .",
"As mentioned previously, this survey aims to introduce existing research on graph anomaly detection and prediction.",
"A graph is an abstract data type consisting of a set of nodes (a.k.a.",
"vertices) representing entities, with edges between nodes representing relations or connections.",
"Anomalies in graph data fall into four main categories: node anomaly, edge anomaly, path anomaly, and (sub)graph anomaly.",
"The whole process of detecting anomalies with deep graph models is briefly illustrated in Figure REF .",
"When learning a deep model on graphs for anomaly analytics tasks, we divid the models into four categories based on their model architectures and training strategies.",
"Here, we briefly introduce the process and potential mechanism of these graph neural network models.",
"Graph Convolutional Networks (GCNs) Considering that graphs lack a grid structure like image and text, it is impractical to directly apply standard convolution operation on graphs.",
"Graph convolution is generally divided into two categories, spectral convolution, which performs Fourier transform on graph signals, and spatial convolution, which learns structural information by aggregating node neighbors [107].",
"The graph signal $\\mathbf {X}$ in spectral methods is filtered by: $\\mathbf {Z}=f(\\mathbf {X},\\mathbf {A})=\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}\\widetilde{\\mathbf {A}}\\widetilde{\\mathbf {D}}^{-\\frac{1}{2}}\\mathbf {X}\\mathbf {\\Theta }$ where $\\mathbf {\\Theta }$ is a matrix of learnable parameters and $\\mathbf {Z}$ is the convolved signal matrix.",
"In addition, the equation of learning node representation of node $i$ in a Graph Convolution Network (GCN) can be written as: $\\mathbf {h_i}=\\sigma \\left( \\sum _{j\\in N(v_i)}\\alpha _{ij}\\mathbf {W} \\mathbf {h_j} \\right).$ where $\\mathbf {W}$ is weight matrix to be learned, and $\\alpha _{ij}$ is set as 1 in GCN.",
"The calculation formula of $\\alpha _{ij}$ will be introduced in GAT.",
"Graph Attention Networks (GATs) It is acknowledged that spatial convolution is to aggregate features from node neighbors to update the hidden state of this node in the next layer.",
"The aggregation operation could be (weighted) summarization, averaging, and maximization.",
"Graph Attention Network (GAT) is a special type of spatial convolution methods.",
"Although some spatial methods also consider the node importance and allocate a predefined weight for every neighbor, a Graph Attention Network is proposed so that the weight of nodes can be learned automatically by applying the attention mechanism to neighbors in the model [92].",
"The influence $\\alpha _{ij}$ of node $v_j$ on node $v_i$ in GAT is calculated as: $\\alpha _{ij}=\\frac{exp(\\sigma (\\mathbf {a}^T[\\mathbf {W}\\mathbf {h}_i\\parallel \\mathbf {W}\\mathbf {h}_j]))}{\\sum _{k\\in {N(i)}} exp(\\sigma (\\mathbf {a}^T[\\mathbf {W}\\mathbf {h}_i\\parallel \\mathbf {W}\\mathbf {h}_k]))}.$ Here, $\\mathbf {a}$ denotes a weight vector and the symbol $\\parallel $ is the concatenation operation on two vectors.",
"Graph AutoEncoder (GAE) Graph autoencoder is a popular model used in unsupervised learning tasks [4], [30].",
"Similar to the general autoencoder model, GAE is composed of an encoder compressing the sparse node vector (input) into a low-dimensional representation through learning node structural features, and a decoder reconstructing the dense vector into a high-dimensional vector similar to the input as much as possible.",
"Based on this mechanism, an essential part of loss function in GAE models is to minimize the difference between the input and output vectors: $\\mathop {min}\\limits _{\\Theta } \\mathcal {L}_2= \\parallel \\mathbf {A}-\\mathbf {\\hat{A}} \\parallel _2 + \\parallel \\mathbf {X}-\\mathbf {\\hat{X}} \\parallel _2,$ where $\\mathbf {A}$ and $\\mathbf {X}$ are the input node adjacency and attribute matrix, and $\\mathbf {\\hat{A}}$ and $\\mathbf {\\hat{X}}$ are the reconstructed node structure and attribute matrix.",
"It should be noted that the encoder could be any kind of neural network like MLP, Recurent Neural Network (RNN), and GCN.",
"Therefore, there are a line of anomaly detection methods combining GAE and GCN.",
"We refer readers to [128] for more details about deep graph models and their applications.",
"Table: Commonly used notations"
],
[
"GCN-based Methods",
"As the most popular structure among the deep graph models, Graph Convolutional Networks (GCNs) can learn and generate node embeddings through the operation of convolution, which refers to the process of aggregating information from the nodes' local neighborhoods.",
"In this section, we introduce the GCN-based anomaly detection and prediction methods, which is also the most popular model type among all anomaly analytics models.",
"The methods are divided into two classifications according to whether the methods are devised for specific anomaly detection tasks or not, namely, general models and task-driven models.",
"A toy model of how anomalous users are detected in social networks with spatial convolution operation is shown in Figure REF .",
"The main characteristics of these methods are summarized in Table REF .",
"Figure: An illustration of applying spatial convolution operation in anomalous user detection in social networks, where nodes are affected only by their immediate neighbors.",
"Both attribute feature and structure feature could be learned with a GCN model to get the final anomaly ranking list."
],
[
"General models",
"In [56], the authors defined two types of node anomaly according to its global location and topological network structure, named global anomaly and community anomaly, respectively.",
"When learning global anomaly node embeddings, an autoencoder is applied to extract node attributes $X$ .",
"As for community anomaly representation, the authors designed a convolutional encoder and deconvolutional decoder networks based on their neighborhoods.",
"Then, the anomalous node could be detected by measuring the embedding's energy in the Gaussian Mixture Model.",
"Similarly, Ding et al.",
"[22] and Zhu et al.",
"[140] proposed to learn node embeddings by combining AutoEncoder with Graph Convolutional Networks in attributed networks.",
"Specifically, the encoder module extends the operation of convolution in the spectral domain and learns a layer-wise new latent representation.",
"Then, the structure reconstruction decoder $\\mathbf {A}-\\hat{\\mathbf {A}}$ and attribute reconstruction decoder $\\mathbf {X}-\\hat{\\mathbf {X}} $ are jointly learned to compute the anomaly score, which can be formulated as: $score(v_i)=(1-\\alpha )\\Vert \\mathbf {a_i}-\\hat{\\mathbf {a_i}} \\Vert _2 + \\alpha \\Vert \\mathbf {x_i}-\\hat{\\mathbf {x_i}} \\Vert _2,$ where $\\alpha $ is a hyper-parameter balancing the importance of reconstructed structure and attribute information.",
"Instead of detecting anomalous nodes, Duan et al.",
"[29] generated node embeddings by combining GAE and GCN to detect anomalous links.",
"They assumed that a link with a lower value of predicted presence probability is regarded as anomalous, which is calculated as: $\\mathbf {P}_{u,v} < \\underset{v^{\\prime }\\in N_u}{\\rm MEAN}\\mathbf {P}_{u,v^{\\prime }}-\\mu \\cdot \\underset{v^{\\prime }\\in N_u}{\\rm STD}\\mathbf {P}_{u,v^{\\prime }},$ where MEAN and STD represent the mean and standard operations respectively, and $\\mu $ is a parameter.",
"$\\mathbf {P}$ is the predicted presence probability matrix.",
"In [131], the authors aimed to incorporate all possible features in the proposed framework AddGraph, including structural, content, and temporal features.",
"In AddGraph, they used a GCN incorporating content and structural features, with an attention-based GRU (Gated Recurrent Unit), which can combine short-term and long-term states.",
"After obtaining the hidden states of nodes at timestamp $t$ , the anomalous score of an edge is computed as: $\\mathcal {F}(i,j,w)=w \\cdot \\sigma (\\beta \\cdot (\\parallel \\mathbf {a}\\odot \\mathbf {h}_i + \\mathbf {b}\\odot \\mathbf {h}_j\\parallel _2^2-\\mu )),$ where $\\mathbf {h}_i$ and $\\mathbf {h}_j$ are the hidden states of the $i$ -th node and $j$ -th node, respectively.",
"Other characteristics are parameters to be adjusted.",
"$\\mathbf {a}$ and $\\mathbf {b}$ are parameters to optimize the output layer, and $\\beta $ and $\\mu $ are hyper-parameters.",
"Jin et al.",
"[45] leveraged a multi-scale contrastive learning technique to capture node anomalies in multiple scales.",
"ANEMONE simultaneously performs patch- and context-level contrastive learning via two GCN models.",
"Anomaly is identified by leveraging the statistics of multi-round contrastive scores.",
"Similarly, Liu et al.",
"[60] also proposed to detect node anomalies in attributed network in a contrastive learning way.",
"The objective of their model CoLA is to discriminate the agreement between the elements within the selected instance pairs, which is finally used to calculate the anomalous scores of nodes.",
"The difference between CoLA and ANEMONE is the process of sampling.",
"CoLA selected the local subgraph including the target node as positive sample, while local graph without target node is negative sample.",
"Differing from existing graph contrastive learning frameworks for GNN pre-training, Chen et al.",
"[13] performed contrastive learning in a supervised learning way.",
"In other words, the negative samples are constrained to anomalous nodes instead of being constructed according to some rules.",
"Considering that the bottleneck of anomaly detection tasks is the lack of sufficient anomaly labels, they proposed to construct pseudo anomalies via corrupting the original graph."
],
[
"Task-driven models",
"To detect malicious account at a mobile cashless payment platform, Liu et al.",
"[62] jointly learned the topology of a heterogeneous graph and the features of local structures of the nodes.",
"Specifically, they constructed \"homogeneous connected subgraph\" based on an assumption that an edge $(i, i^{\\prime })$ is added if both account $i$ and $i^{\\prime }$ login to the same device in the original heterogeneous graph.",
"This subgraph is composed of only accounts as nodes.",
"The function to learn the embeddings of nodes is defined as: $\\mathbf {H}^{(l+1)}\\leftarrow \\sigma (\\mathbf {X} \\cdot \\mathbf {W} + \\sum _{d=1}^{|D|} softmax(\\alpha _{d})\\cdot \\mathbf {A}^{(d)} \\cdot \\mathbf {H}^{(l)} \\cdot \\mathbf {V}_d),$ where $|D|$ is the number of subgraphs extracted from the original graph, and $\\mathbf {V}_{d}$ is a parameter controlling the shape of the function.",
"Moreover, an attention mechanism is also utilized in the learning process of different types of subgraphs, i.e., $softmax(\\alpha _d)=\\frac{exp~\\alpha _{d}}{\\sum _{i}exp~\\alpha _{i}}$ , and $\\alpha =[\\alpha _1, ...,\\alpha _{|D|}]^T$ is a free parameter to be learned.",
"Based on an assumption that representing nodes with the information of its neighbors will effectively improve the performance of the source node detection task, Dong et al.",
"[26] designed a model GCNSI to locate multiple rumor sources without prior knowledge of the underlying propagation model.",
"This model learns node embeddings by adopting convolution in the spectral domain, which considers its multi-hop neighbors’ information.",
"The propagation strategy of GCNSI is modified based on LPSI [106].",
"Concerning the task of spam review detection, Li et al.",
"[54] aimed to capture the local context and global context of a comment.",
"The proposed model GAS simultaneously integrates a heterogeneous bipartite graph and a homogeneous comment graph.",
"The comment edge embedding in bipartite graph is to aggregate the hidden states of three variables in the previous layer, i.e., the edge itself and its linked two nodes: $\\mathbf {H}_e^{l}=\\sigma (\\mathbf {W}_E^l \\cdot AGG_E^l(\\mathbf {H}_e^{l-1}, \\mathbf {H}_{U(e)}^{l-1}, \\mathbf {H}_{I(e)}^{l-1})),$ where $AGG_E^l(\\mathbf {H}_e^{l-1}, \\mathbf {H}_{U(e)}^{l-1}, \\mathbf {H}_{I(e)}^{l-1})=\\mathbf {H}_e^{l-1} \\parallel \\mathbf {H}_{U(e)}^{l-1} \\parallel \\mathbf {H}_{I(e)}^{l-1}.$ Here, $U(e)$ and $I(e)$ are user node set and item node set linked by edge $e$ , respectively.",
"Similarly, the user and item embedding are calculated in the same way and the comment embedding in the comment graph can be obtained from a general GCN model.",
"Finally, the classification result $y$ can be calculated according to: $y=classifier(z_i\\parallel z_u\\parallel z_c \\parallel p_c),$ where $z_i$ , $z_u$ , and $z_c$ are item, user, and comment embeddings obtained from bipartite graph, and $p_c$ is the comment embedding from comment graph, respectively.",
"With the aim of detecting a rumor on social media, Bian et al.",
"[7] proposed a top-down GCN (TD-GCN) to model the rumor propagation features, and a bottom-up GCN (BU-GCN) to model the rumor dispersion features, respectively.",
"The node representations are learned over a two-layer GCN: $\\mathbf {H}_1^{TD} \\leftarrow \\sigma (\\hat{\\mathbf {A}}^{TD} \\mathbf {X}\\mathbf {W}_0^{TD}),$ $\\mathbf {H}_2^{TD} \\leftarrow \\sigma (\\hat{\\mathbf {A}}^{TD} \\mathbf {H}_1^{TD} \\mathbf {W}_1^{TD}),$ where $\\mathbf {H}_1^{TD}$ and $\\mathbf {H}_2^{TD}$ refer to the two-layer hidden features of the TD-GCN.",
"The bottom-up features of BU-GCN are calculated in the same manner as Eq.",
"REF and REF , while the adjacency matrix should be transposed.",
"Another similar topic in social media is fake news detection.",
"Lu et al.",
"[63] aimed to model the interactions among users by creating a propagation graph as a part of the proposed model.",
"The propagation graph $G_i=(U_i,E_i)$ is constructed by the set of users $U_i$ who share or retweet the topic $s_i$ , and the edge is weighted by the cosine similarity between the feature vectors of users.",
"Then, the user embeddings will be learned by a GCN model based on this weighted propagation graph.",
"Similarly, Zhong et al.",
"[134] created a Topic-Post-Comment graph for target posts in the task of controversy detection, where the nodes represent topic, post, or comment, and the edges refer to the corresponding interactions between two nodes.",
"The node representations are obtained through a two-layer GCN, the same as Eq.",
"REF and Eq.",
"REF .",
"As the first application of deep graph model in the task of fraud invitation detection, Zhu et al.",
"[142] proposed HMGNN model by dividing the whole network into $|D|$ mini-graphs, which were represented by hypernodes.",
"The hypergraph is generated by adding edges between mini-graphs.",
"Based on this graph, the convolution for hypernodes is defined as: $\\mathbf {H}^{l+1} \\leftarrow \\sigma (\\mathbf {X}_h \\mathbf {U}^l + \\sum _{d\\in D}ATT_d(\\widetilde{\\mathbf {A}}_h^d\\mathbf {H}^l\\mathbf {W}_d^l+\\mathbf {b}_d^l)),$ where $ATT_d$ is the attention mechanism, and $\\mathbf {U}^l$ are free parameters to be trained.",
"Here, $\\mathbf {H}^0=[\\mathbf {X};\\mathbf {X}_h^1; ...; \\mathbf {X}_h^{|D|}]$ is the initial representation of the whole graph which concatenates the feature matrix of normal- and hyper- nodes.",
"To detect social spammers in a semi-supervised way, Wu et al.",
"[108] combined Graph Convolutional Networks (GCNs) and Markov Random Field (MRF) on directed social networks.",
"The layer-wise propagation rule of GCN is defined as: $\\mathbf {H}^{(l+1)}= \\sigma (\\mathbf {D}_i^{-1}\\mathbf {A}_i\\mathbf {H}^{(l)}\\mathbf {W}_i^{(l)}+\\mathbf {D}_o^{-1}\\mathbf {A}_o\\mathbf {H}^{(l)}\\mathbf {W}_o^{(l)}) +\\mathbf {\\tilde{D}}_b^{-\\frac{1}{2}}\\mathbf {\\tilde{A}}_b\\mathbf {\\tilde{D}}_b^{-\\frac{1}{2}}\\mathbf {H}^{(l)}\\mathbf {W}_b^{(l)},$ where $A_i$ , $A_o$ , and $A_b$ are three types of adjacency matrices constructed according to three different definitions of neighbors.",
"Considering that different characteristics of pairwise nodes can have different influences on social networks, the authors proposed to use MRF modeling for the joint probability distribution of users' identities.",
"The MRF is formulated as a RNN in this paper to perform multi-step inference when computing the posterior distribution.",
"Since deliberately inserting fake feedback will cause the recommender system bias to the malicious users' favor, Zhang et al.",
"[127] presented a GCN-based user representation learning framework to perform robust recommendation and fraudster detection in a unified way.",
"Given a weighted bipartite rating graph $G=(U\\cup V,E)$ , GCN is adopted to capture topological neighborhood information and side information of nodes.",
"The user and item embedding are calculated as: $\\mathbf {z}_u=\\sigma (\\mathbf {W} \\cdot AGG(\\mathbf {h}_k,\\forall k \\in N(u)) + \\mathbf {b}),$ $\\mathbf {z}_v=\\sigma (\\mathbf {W} \\cdot AGG(\\mathbf {h}_q,\\forall q \\in N(v)) + \\mathbf {b}),$ where $\\mathbf {h}_k$ and $\\mathbf {h}_q$ are the neighbor information for each node.",
"Here, the attention mechanism is incorporated into the aggregation function.",
"It is acknowledged that noisy labels will influence the results of anomaly detection algorithms in some degree.",
"Then, instead of directly generating latent representations, Zhong et al.",
"[133] designed a GCN-based model to correct noisy labels before detecting anomalous actions in videos.",
"Here, the feature similarity graph is constructed with nodes denoting snippets and edges referring to the similarity between two snippets.",
"Another temporal consistency graph module is directly built upon the temporal structure of a video.",
"Pose estimation is the first step of detecting anomalous actions in videos, and the extracted poses can be embedded by deep graph models.",
"In [68], the authors proposed spatio-temporal graph convolution block, which is composed of a spatial-attention graph convolution, a temporal convolution, and a batch normalization.",
"The generated latent vector is fed into a cluster layer to obtain a normality score.",
"Here, the normality score is calculated by a Dirichlet Process Mixture Model (DPMM) for evaluating the distribution of proportional data.",
"Table: A comparison of the GCN-based models"
],
[
"Discussion",
"As it can be found from the GCN-based anomaly detection models we have discussed above, the modern GCN model could learn both local and global structure features of a graph with convolution and pooling operations.",
"To improve the training efficiency when imposing GCN on large-scale graphs, neighborhood samplings and layer-wise samplings are two common strategies to deal with the phenomenon that some nodes have high degrees (too many neighbors).",
"In addition to node and edge anomaly, a characteristic of GCN is that it is more suitable to detect (sub)graph or group anomaly when compared with other GNN models.",
"The aforementioned models mostly focus on learning node features and graph structures, ignoring another important element consisting of a graph, i.e., edge.",
"In some real-world networks, edges generally contain abundant information, such as edge types and corresponding attributes, which could play a key role in anomaly detection tasks.",
"Therefore, incorporating edge features into graph anomaly detection models could be considered as a future work [14].",
"Besides, although applying GCN to an inductive setting is verified [36], conducting inductive GCN for graphs without explicit features remains an open problem."
],
[
"GAT-based Methods",
"In deep graph models, the weights of node neighbors are defined as an equal or default setting.",
"However, the importance of neighbors is mostly different in terms of their attribute and structural features.",
"Motivated by the attention mechanism, Velivckovic et al.",
"[92] proposed a graph attention network (GAT) by applying the attention mechanism to the spatial convolution operation of GCN.",
"A toy example of how attention mechanism is applied into cyberbullying detection is shown in Figure REF .",
"In this section, we summarize and introduce the anomaly analytics algorithms using graph attention networks.",
"The methods are divided into 2 subsections in terms of the anomaly type, i.e., node anomaly detection and (sub)graph anomaly detection.",
"The main characteristics of these methods are summarized in Table REF .",
"Figure: An illustration of how attention mechanism is applied into cyberbullying detection.",
"Each comment is first encoded by a RNN framework as the initial vector, and the comments are constructed as a temporal graph where nodes represent user comments and edges represent time intervals between two comments.",
"Then, the attention mechanism is applied to learn the temporal information among these comments for final anomaly detection."
],
[
"Node anomaly detection",
"To detect anomalous nodes in an Attributed Heterogeneous Information Network (AHIN), Hu et al.",
"[41] applied feature and path attention mechanism to differentiate the importance of meta-paths as well as attribute information.",
"As a basic analysis tool for heterogeneous graph, a meta-path captures the proximity among multiple nodes from a specific semantic perspective, which could be seen as a high-order structure.",
"For example, the meta path “Author-Paper-Author” (APA) describes that two authors collaborated with each other in a particular paper.",
"The feature attention of neighbor node $i$ on node $u$ in path $\\rho $ is calculated as: $\\hat{\\alpha }_{u,i}^{\\rho }=\\frac{exp(\\alpha _{u,i}^{\\rho })}{\\sum _{j=1}^{K}exp(\\alpha _{u,j}^{\\rho })}.$ The attention weight of path $\\rho $ for node $u$ is defined as: $\\beta _{u,\\rho }=\\frac{exp(z^{\\rho ^{T}}\\cdot \\tilde{f}_u^C)}{\\sum _{\\rho ^{\\prime } \\in \\mathbb {P}}exp(z^{\\rho ^{\\prime T}}\\cdot \\tilde{f}_u^C)}.$ Here, $z^{\\rho }$ is the attention vector of meta-path $\\rho $ , and $\\tilde{f}_u^C$ is the collection of user representations w.r.t.",
"all meta-paths.",
"The cash-out probability (i.e., anomalous score) is calculated via a regression layer with a sigmoid unit.",
"Table: A comparison among different GAT-based modelsTo detect user fraud in financial networks, Wang et al.",
"[95] designed a hierarchical attention structure in graph neural network from node-level attention to view-level attention when generating graph embeddings.",
"View-attention mechanism is applied to fuse multiple views of data information into user embeddings.",
"Finally, a softmax function is used on the representations of the embedding layer to get the classification result.",
"In [81], the authors constructed a news-oriented heterogeneous information network with nodes of creators, subjects, and articles, and two links of write and belong-to.",
"Based on this network, they proposed AA-HGNN to solve the problem of fake news detection.",
"From the perspective of node-level attention, the model first aggregates the importance of the same-type neighbors for each news node and generates an integrated embedding of schema node.",
"By using a transformation matrix, the embeddings of the nodes can be mapped into the same dimension.",
"The logistic regression layer works as the classification layer to generate the detection results.",
"To give direct explanations like how anomalies deviate from normal behaviors, [19] proposed to use graph attention mechanism to predict the future behavior of a node.",
"The anomaly score of node $i$ is defined as the difference between the expected behavior and observed behavior at time $t$ : $\\mathbf {Err}_i(t)=|\\mathbf {s}_i^{(t)}-\\hat{\\mathbf {s}}_i^{(t)}|.$ For session-level cyberbullying detection, the final embedding is fed into a single-layer dense network and predict its label.",
"In financial default user detection over online credit payment service, Zhong et al.",
"[135] devised a meta-path-based encoder to capture local structural feature of nodes and links.",
"The path representation is defined as the concatenation of node and link embeddings.",
"Moreover, attention mechanism is applied to capture different importance of nodes/links of a path.",
"After modeling the node and link interactions above, the learned representation is fed into several fully connected neural networks and a regression layer with a sigmoid unit for anomaly classification."
],
[
"(Sub)graph anomaly detection",
"Wang et al.",
"[100] proposed HAGNE to detect unknown malicious programs in computer systems.",
"Instead of setting one-hop nodes as neighbors, the authors construct a contextual neighborhood set by searching for meta-paths.",
"Then, three kinds of aggregators are applied to generate graph embeddings based on the generated meta-path set $\\mathbb {M}=\\lbrace M_1,M_2,...,M_{|\\mathbb {M}|}\\rbrace $ , namely, node-wise attentional neural aggregator, which is defined as: $\\mathbf {h}_v^{(i)(k)}= AGG_{node}(\\mathbf {h}_v^{(i)(k-1)},\\lbrace \\mathbf {h}_u^{(i)(k-1)}\\rbrace _{u\\in \\mathcal {N}_v^{i}}),$ where $i\\in \\lbrace 1,2,...,|\\mathbb {M}|\\rbrace $ , $k\\in \\lbrace 1,2,...,K\\rbrace $ denotes the layer index, and $\\mathbf {h}_v^{(i)(k)}$ is the feature vector of node $v$ at the $k$ -th layer in meta-path $M_i$ ; layer-wise dense-connected neural aggregator, which is inspired by DENSENET [42]: $\\mathbf {h}_v^{(i)(K+1)}=AGG_{layer}(\\mathbf {h}_v^{(0)},\\mathbf {h}_v^{(1)},...,\\mathbf {h}_v^{(K)});$ and path-wise attentional neural aggregator, whose attentional weight is defined as: $\\alpha _{ij}=\\frac{exp(\\sigma (\\mathbf {b}[\\mathbf {W}_b \\mathbf {h}_v^{(i)(K+1)} \\parallel \\mathbf {W}_b \\mathbf {h}_v^{(j)(K+1)}]))}{\\sum _{j^{\\prime }\\in {|\\mathbb {M}|}} exp(\\sigma (\\mathbf {b}[\\mathbf {W}_b \\mathbf {h}_v^{(i)(K+1)} \\parallel \\mathbf {W}_b \\mathbf {h}_v^{(j^{\\prime })(K+1)}]))},$ Then, the graph embedding can be calculated from the joint representation of all meta-paths: $\\mathbf {h}_G=AGG_{path}=\\sum _{i=1}^{|M|}ATT(\\mathbf {h}_v^{(i)(K+1)})\\mathbf {h}_v^{(i)(K+1)}.$ Graph matching is used to measure the anomalous level of a program [80].",
"An alert will be triggered if the highest similarity score among all the existing programs is below the threshold.",
"The similarity score is calculated as: $Sim(G_{i(1)},G_{i(2)})=\\frac{\\mathbf {h}_{G_{i(1)}} \\cdot \\mathbf {h}_{G_{i(2)}}}{\\parallel \\mathbf {h}_{G_{i(1)}} \\parallel \\cdot \\parallel \\mathbf {h}_{G_{i(2)}} \\parallel }.$ Subsequently, Fan et al.",
"[32] identified the illicit traded product in underground market with a similar process.",
"After constructing the neighbors set by the meta path-based method, the authors applied an attention mechanism to learn product and buyer embeddings, respectively.",
"Finally, their embeddings are generated by concatenating each embedding based on a specific metagraph.",
"Social media contains multi-modal information such as comment, user, time, and history.",
"Ge et al.",
"[34] proposed to use temporal graph interaction learning module as a building block to detect cyberbullying in social networks.",
"In this work, the authors incorporated GATs to automatically aggregate information from neighbor nodes to the central node in a temporal graph.",
"Edge in the temporal graph denotes time dynamics, and the weight of the node pair $(i,j)$ is defined as: $\\alpha (\\mathbf {z}_i,\\mathbf {z}_j,t_{i},t_{j})=tanh((\\mathbf {W}_o\\mathbf {z}_i)^{T}\\mathbf {z}_j+W_t(t_j-t_i)).$"
],
[
"Discussion",
"As we have explained, GAT is a branch of GCN.",
"To improve GCN, GAT-based methods are separated as a unique section in which the importance of different neighbors on the central node is considered.",
"The difference between these two sections is that the utilized traditional attention mechanism of Section is only applied on nodes, while models in Section are either using attention mechanism on other parts of the framework or using simple GCN function without any attention mechanism."
],
[
"GAE-based Methods",
"Graph AutoEncoder (GAE) is an unsupervised structure to generate low-dimensional representations, with the aim of minimizing the loss between the input of encoder and the output of decoder [91].",
"In this section, we present the GAE-based algorithms that are applied to anomaly analytics.",
"The methods are classified into three types according to the training and learning schema, namely, General AutoEncoder, Adversarial Training, and Hypersphere Learning.",
"The main characteristics of these methods are summarized in Table REF .",
"In figure REF , we present a GAE-based model for detecting anomalous citation behaviors in a heterogeneous network.",
"Figure: An illustration of combining graph autoencoder (GAE) with contrastive learning for anomalous academic’s detection in heterogeneous networks.",
"After selecting a target node, the second step is to sample positive and negative instances from the network for contrastive learning.",
"Then, different kinds of nodes are encoded with different encoders and a common decoder.",
"The encoder aims to learn structure and attribute feature of nodes and generate low-dimensional vectors, and the decoder aims to reconstruct the input vector as similar as possible.",
"The anomaly score is calculated by combining the discrimination score generated by the discriminator and the reconstruction loss generated by the AutoEncoder."
],
[
"General AutoEncoder",
"Only considering the structure of a heterogeneous network is not sufficient for abnormal event detection due to the sparsity of a network.",
"Fan et al.",
"[31] proposed AEHE to learn both attribute embedding and the second-order structure-preserving node embedding.",
"The heterogeneous attribute embedding of a node is generated by a Multilayer Perceptron (MLP) component, which consists of two hidden layers with ReLU as the activation function.",
"As for the second-order structure embedding, the authors constructed a homogeneous graph by extracting symmetry meta-paths.",
"Autoencoder is used to model the neighborhood structures, which is composed by an encoder: $\\mathbf {r}_i^t=\\sigma (\\mathbf {W}_1^t \\cdot \\mathbf {s}_i^t+\\mathbf {b}_1^t),$ and a decoder: $\\mathbf {\\hat{s}}_i^t= \\sigma (\\mathbf {\\hat{W}}_1^t \\cdot \\mathbf {r}_i^t+\\mathbf {\\hat{b}}_1^t),$ where $\\mathbf {r}_i^t$ is the latent representation of entity $a_i^t$ , and $\\mathbf {\\hat{s}}_i^t$ is the reconstructed representation of $\\mathbf {s}_i^t$ .",
"It should be noted that $\\mathbf {s}_i^t$ is the $i$ th row of the adjacency matrix, not just a node feature vector.",
"Table: A comparison among different GAE-based modelsResearch have shown that human behaviors reflect self-selection bias and peer influence in online social network, which is closely associated with cyberbullying behaviors [33].",
"In this regard, Cheng et al.",
"[17] used a GCN encoder and an inner product decoder to learn a latent matrix $\\mathbf {Z}$ by minimizing the following reconstruction error: $\\mathcal {F}(v_i)=\\parallel \\mathbf {A}-\\mathbf {\\hat{A}}\\parallel _2^2,$ where $\\mathbf {\\hat{A}}=\\sigma (\\mathbf {Z}\\mathbf {Z}^T)$ , and $\\mathbf {Z}=GCN(\\mathbf {X},\\mathbf {A})$ .",
"Then, the anomalous session could be detected by measuring the embedding's energy in the Gaussian Mixture Model, which follows [56]."
],
[
"Adversarial training",
"Adversarial methods such as GAN and adversarial attacks are popular in the machine learning community in recent years.",
"In [74], the authors incorporated an adversarial training scheme into GAEs as an additional regularization term.",
"Motivated by this work, Ding et al.",
"[21] proposed AEGIS to learn anomaly-aware node representations through graph differentiation networks (GDNs) for inductive anomaly detection.",
"AEGIS is composed of a GAE to learn node embeddings for training new networks, and a GAN to calculate the anomaly scores of nodes.",
"The autoencoder network is built with the graph differentiative layers.",
"Specifically, a GDN layer has a hierarchical attention structure from node level: $\\mathbf {h}_i^{(l)}=\\sigma (\\mathbf {W}_1\\mathbf {h}_i^{(l-1)}+\\sum _{j\\in N_i}\\alpha _{ij}\\mathbf {W}_2\\Delta _{i,j}^{(l-1)}),$ where $\\Delta _{i,j}$ denotes the embedding difference between nodes $i$ and $j$ ; and neighbor level: $\\mathbf {h}_i^l=\\sum _{k=1}^{K}\\beta _{i}^k\\mathbf {h}_i^{(l,k)}.$ Finally, the anomaly score of node $i$ is computed according to the output of a discriminator: $score(\\mathbf {z}_i)=1-D(\\mathbf {z}_i^{^{\\prime }}).$ In real-world networks, community outliers deviate significantly from other nodes in the same community in terms of link structures and attributes.",
"To alleviate the influence of these outliers and generate robust node embeddings, Bandyopadhyay et al.",
"[6] mapped every vertex to a low-dimensional vector and detected outliers via a deep autoencoder-based architecture.",
"Moreover, the authors introduced adversarial learning for outlier resistant network embedding.",
"Here, a discriminator is combined with two parallel autoencoders to align the embeddings in terms of link structure and node attributes."
],
[
"Hypersphere learning",
"Inspired by hypersphere learning, Wang et al.",
"[102] proposed One-Class Graph Neural Network (OCGNN) with the aim of minimizing the volume of a hypersphere that encloses normal nodes as much as possible.",
"Then, the nodes out of the hypersphere are regarded as abnormal.",
"With the aim of identifying anomalous sample cases and nested anomalies within the anomalous tensors, Teng et al.",
"[90] proposed DeepSphere by incorporating hypersphere learning into a LSTM Autoencoder model in a mutual supportive manner.",
"Here, attention mechanism is also applied to differentiate and aggregate different neighbors.",
"The motivation of DeepSphere is that the learned representations at large distance from the outside of hypersphere are regarded as anomalous, while the ones with small distances from the inside of the hypersphere tend to be normal."
],
[
"Discussion",
"GAE is the most popular model in tackling unsupervised graph learning tasks, which can only consider the structural patterns by using graph adjacency matrix.",
"However, GCN-based models are semi-supervised and could capture both node attributes and graph structures.",
"Despite the different architectures between GAE and GCN, existing research have shown that it is possible to combine them together in a unified framework [9].",
"When applying GCN as the encoder, GAE could be applied to the inductive learning settings where node attributes are incorporated.",
"Considering that the aim of GAE is to reconstruct the input embedding as similar as possible, it should be cautious when selecting the appropriate similarity metrics which have significant influence on subsequent anomaly detection results.",
"Figure: The Generative Adversarial Network (GAN)-based anomaly detection model is composed of three main parts: a Generator sampling similar node attributes, an Encoder generating low-dimensional node representations, and a Discriminator differentiating real nodes embeddings from generated nodes embeddings."
],
[
"Other Methods",
"Apart from the deep graph models mentioned above, there are many other popular deep learning models that can be used for anomaly analytics tasks, such as Generative Adversarial Methods [96], Meta-learning [136], and Graph Reinforcement Learning [25].",
"In this section, we summarize these different deep graph models that are utilized to solve anomaly analytics tasks.",
"The process of detecting anomalies with a Generative Adversarial Network (GAN) is shown in Figure REF .",
"The main characteristics of these methods are summarized in Table REF ."
],
[
"GAN-based methods",
"With the rapid growth of research in Generative Adversarial Network (GAN) for high-dimensional data distribution approximation, Chen et al.",
"[15] proposed to detect anomalies with a Generative Adversarial Attributed Network (GAAN), which is composed of three parts: a Generator sampling similar node attributes, an Encoder generating low-dimensional node representations, and a Discriminator differentiating real nodes embeddings from generated nodes embeddings.",
"The anomalous score is defined based on a context reconstruction loss $L_G$ and a structure discriminator loss $L_D$ : $\\mathcal {F}(v_i)=\\alpha \\mathcal {L}_G(v_i) + (1-\\alpha )\\mathcal {L}_D(v_i),$ where $\\mathcal {L}_G(v_i)=\\parallel \\mathbf {x}_i-\\mathbf {x}_i^{\\prime } \\parallel _2$ , and $\\mathcal {L}_D(v_i)$ is defined as: $\\mathcal {L}_D(v_i)=\\sum _{j=1}^{n}\\mathbf {A}_{ij}\\cdot \\sigma (\\mathbf {\\hat{A}}_{ij},1) / \\sum _{j=1}^{n}\\mathbf {A}_{ij}.$ Larger value of $\\mathcal {F}(v_i)$ indicates the node $v_i$ is more likely to be anomalous."
],
[
"Reinforcement learning-based method",
"In [130], the authors divided the graph anomaly detection tasks into two classifications, i.e., outlier detection and unexpected dense block detection.",
"When applying graph learning models to generate embeddings of nodes, a new loss function was designed as: $\\mathcal {L}(\\mathbf {u})=\\mathbb {E}_{u_+\\sim U_{u_+},u_-\\sim U_{u_-}}max\\lbrace 0,g(u, u_-)-g(u,u_+)+\\bigtriangleup _{y_u}\\rbrace ,$ where $\\bigtriangleup _{y_u}=\\frac{C}{n_{y_u}^{1/4}}$ .",
"$g()$ is a function to denote the similarity of the representations between any two user nodes.",
"Here, $U_{u_+}$ denotes the set of user nodes that has the same label as $u$ , $U_{u_-}$ refers to $ U\\backslash U_{u_+}$ , and $n_{y_u}=|U_{u_+}|$ .",
"The construction of sets $U_{u_+}$ and $U_{u_-}$ have different strategies for corresponding tasks.",
"Table: A comparison among other deep graph modelsIt is impractical to exactly detect camouflaged fraudsters with graph learning detectors.",
"Dou et al.",
"[27] proposed three neural models to enhance the deep graph models against two kinds of camouflages, i.e.",
"feature camouflage and relationship camouflage.",
"Because camouflaged nodes should be filtered when selecting similarity-aware neighbors, a reinforcement learning process is formulated as a Bernoulli Multiarmed Bandit (BMAB) to find the optimal thresholds.",
"The reward mechanism of epoch $e$ is defined as: $\\mathcal {R}(p_r^{(l)},a_r^{(l)})^{(e)}=\\left\\lbrace \\begin{aligned}+1,& G(\\mathcal {D}_r^{(l)})^{(e-1)}-G(\\mathcal {D}_r^{(l)})^{(e)} \\ge 0, \\\\-1,& G(\\mathcal {D}_r^{(l)})^{(e-1)}-G(\\mathcal {D}_r^{(l)})^{(e)} < 0.\\end{aligned}\\right.$ Here, $G(\\mathcal {D}_r^{(l)})^{(e)}$ refers to the average neighbor distances for relationship $r$ at the $l$ -th layer for epoch $e$ .",
"If the average distance of newly selected neighbors at epoch $e$ is less than that of epoch $e-1$ , then the reward is positive."
],
[
"Few-shot learning-based method",
"To investigate the novel problem of few-shot network anomaly detection under the cross-network setting, Ding et al.",
"[23] designed a new graph learning architecture, namely Graph Deviation Networks (GDNs).",
"GDN in this paper is composed of three key building blocks: an encoder to generate node embeddings, an abnormality valuator to compute the anomaly score of nodes, and a deviation loss for optimization.",
"Concretely, the GDN model can be formally represented as: $\\mathcal {F}_{\\theta }(\\mathbf {A},\\mathbf {X})=\\mathcal {F}_{\\theta _s}(\\mathcal {F}_{\\theta _e}(\\mathbf {A},\\mathbf {X})),$ which directly maps the input network to anomaly scores (scalar).",
"After detecting anomalies in arbitrary networks, a meta-learner is learned to initialize GDN from multiple auxiliary networks, which possesses the ability to distill comprehensive knowledge of anomalies."
],
[
"Discussion",
"There are also many other neural network structures and learning strategies being applied to detect graph anomalies, such as adversarial learning and reinforcement learning.",
"Considering that the number of these methods is relatively less, we summarize all these methods into one section instead of different sections.",
"Other aspects of common deep graph learning models include but not limited to graph reinforcement learning and graph adversarial learning.",
"It is well-known that the advantage of reinforcement learning is to actively learn from the feedback.",
"In graph anomaly detection tasks, reinforcement learning could help in optimal selection of neighbors and aggregating them together for more informative node embeddings.",
"Adversarial methods have shown its capacity in generating realistic entities, which improve the detection performance of anomalies that are hardly reconstructed from the latent space.",
"However, this kind of anomaly detection methods faces multiple problems during the training process, such as failure to converge and mode collapse."
],
[
"Applications",
" Thus far, we have reviewed different graph learning methods in anomaly analytics tasks.",
"In this section, we briefly introduce their applications in different kinds of networks."
],
[
"Fake news",
"With the rapid growth of the Internet, social media provides a platform for people to participate and discuss online news more conveniently, like communicating news without the physical distance barrier among individuals and acquiring news at an unprecedented rate.",
"In general, fake news in social media is defined as the verifiably false information that is generated by malicious users or social bots intentionally, with the aim of misleading the public.",
"There have been research showing that fake news spread more quickly and broadly than true news [93].",
"Detecting fake news, especially at an early stage, is complex and challenging due to the characteristics of fake news.",
"Various types of information are integrated when designing detection strategies, including news-related and social-related features [139].",
"Among the whole process of detection algorithms, extracting information from network-based features is a procedure to improve the performance of detection results.",
"In social media, users form different kinds of networks in terms of interests, topics, and relations.",
"For example, [71] proposed a heterogeneous graph to incorporate all major social actors and their interactions into node representations, which is constructed by user, news, and sources.",
"Other types of networks also exist, for instance, co-occurrence network indicating user engagements, friendship network showing the following relationships, and diffusion network tracking the source of fake news.",
"We refer readers to [138], [86] for more information about the research on fake news."
],
[
"Cyberbullying",
"Based on the definition of bullying, cyberbullying is, by extension, defined as an aggressive act intentionally carried out by a group or individual using an electronic device, against people who cannot easily defend themselves.",
"Research have shown that cyberbullying is quite prevalent on social media with 54% of young people reportedly cyberbullied on Facebook [82].",
"Apart from the traditional research using merely content-based features, recent years have witnessed a proliferation of research focusing on incorporating network-based features (e.g., number of friends, uploads, likes and so on) in detection systems [87].",
"For example, Cheng et al.",
"[16] refined the cyberbullying detection problem within a multi-modal context.",
"Then, the problem is a process of multiple modalities exploited in a collaborative fashion."
],
[
"Fake reviews",
"Rating platforms require aggregating a large-scale collection of user reviews and ratings about items (e.g., products, movies, or other users), which play a central role in deciding what service to purchase, restaurant to patronize, and movie to watch, to name but a few.",
"However, fraudulent users give fake ratings and even malicious reviews out of personal interest or prejudice.",
"Therefore, it is necessary to detect such users and eliminate the influence of malicious competition among peers on rating systems.",
"The algorithm of detecting fraudulent users is formulized as a process to calculate the trustworthiness of a user and its ratings.",
"Networks used in this line of research are diverse, ranging from homogeneous to heterogeneous, such as user-product bipartite network with signed edges [2], homogeneous co-review graph with weighted or unweighted edges [47], and a bipartite rating graph with directed and weighted edges [51]."
],
[
"Electrical grid",
"A power grid, also known as an electrical grid, can be constructed as a graph, with nodes denoting generators and edges indicating power lines.",
"Several research questions about anomaly detection or prediction need to be solved in an electrical system.",
"For example, when an electrical component has failed or is going to fail, how could we detect or predict it accurately?",
"Another more challenging problem is to determine the locations of a limited budget of sensors, then it is easier to detect and predict grid anomalies in advance.",
"Existing anomaly detection algorithms mainly focus on graph theory-based measures instead of graph learning methods.",
"For example, Hooi et al.",
"[38] detected sensor-level anomalies by designing detectors for three common types of anomalies, and constructed an optimization strategy for sensor placement, with the aim of maximizing the probability of detecting an anomaly.",
"Li et al.",
"[55] proposed an index to measure the distance between each past graph and the current graph, thereby generating anomaly scores of a graph in a specific timestamp."
],
[
"Financial defaulter",
"Despite the huge benefits created by online financial services to the society, we have been witnessing a huge growth in financial frauds.",
"The types of frauds in financial scenarios include cash-out behaviors [41], insurance fraud [95], and default users [135].",
"These frauds severely damage the security of users and service providers, which is a serious problem that needs to be solved.",
"In financial systems, users engage in interactions and have multiple sources of information.",
"These data form a large multi-view network that conventional methods cannot fully exploit.",
"By integrating the features of various kinds of objects and their interactions, [41] aims to identify whether a user is a cash-out user or not.",
"Default user is defined as a user who is likely to fail to make required payments on time in the future [135].",
"Hence, these kinds of research questions are generally formulated as binary classification problems."
],
[
"Anomalous citations and co-authors",
"In the context of big scholarly data, the concept of Academic Social Networks (ASNs) is created.",
"ASNs are complex heterogeneous networks formed by academic entities and their relationships [49], such as co-authorship network, co-citation network, and co-word network.",
"Among these complex relationships and interactions, abnormal academic behaviors (e.g., citations and collaborations) commonly and implicitly exist in different kinds of ASNs [5].",
"In [46], the authors proposed five heuristic rules to define five types of anomalous citation from the perspective of journal-level citation count.",
"Another kind of anomalous citation is defined in terms of citation context, which is identified by analyzing the context similarity between two publications.",
"As for academic collaboration, Fan et al.",
"[31] analyzed the author-paper-author meta-path (co-authorship) to discover rare pattern events in a heterogeneous information network, where each event is denoted by a specific meta-path.",
"To detect anomalous citations, Liu et al.",
"[58] first applied transfer learning to automatically identify unmarked citation purposes and then, applied a deep graph learning framework for anomalous citation detection."
],
[
"Urban computing and mobile sensing",
"In the process of constructing a smart city, urban anomalies like traffic congestion widely occur and sometimes it may bring serious environmental, economic, and social threats to the public [126].",
"To prevent tragedies, the use of smart devices and sensors to detect urban anomalies is of great value.",
"Since the urban data are collected in real time through mobile devices or distributed sensors, they are generally modeled as spatial-temporal graphs that have timesteps and location tags.",
"The emergence of urban big data inspired many novel research on anomaly detection and prediction, such as air quality prediction [132], traffic speed prediction [115], and crime detection [104].",
"By modeling the urban data as a global cross-region hypergraph, [114] proposed to encode crime dependent representations and spatial temporal dynamics for crime prediction.",
"As for intelligent transportation system, [98] proposed a model based on integration of a modified GCN and LSTM to predict anomaly distribution and duration."
],
[
"Future directions",
" There are several ongoing or future research directions that are worthy of discussion.",
"In this section, we summarize five potential research directions of anomaly analytics on graph data."
],
[
"Anomaly detection on graphs with complex types",
"Most of the existing research focus on detecting anomalies on simple graphs, while real-world networks are more complicated and have different types, such as heterogeneous graph with multiple node types [124], [129], spatio-temporal graph evolving with time [43], and hypergraph with relations not limited to pairwise relations [97].",
"Detecting and predicting anomalies on these complex graphs involve technical challenges.",
"For example, as nodes and links which are representing entities and relationships in real-world networks are constantly evolving over time, anomalous entities/relationships might sometimes present normal behaviors as other entities in static networks.",
"This will decrease the accuracy of anomaly detection methods [8].",
"So, how to model the temporal characteristic of dynamic networks and update real-time graph embeddings remain as important challenges.",
"As for heterogeneous graph anomaly detection, how to incorporate both attribute and structure information of various types of nodes and edges into the graph learning model is a key problem [101].",
"Therefore, anomaly detection and prediction on complex graphs is still a potential research direction need to be further explored."
],
[
"Interpretable and robust anomaly detection algorithms",
"Despite the fact that representation learning methods relieve much of the cost of handling features manually, a major limitation of current graph embedding approaches is the lack of interpretability.",
"Unlike the general tasks, an interpretable model for anomaly analytics can help people to understand the results, thereby avoiding the potential model risks and human bias.",
"Apart from result visualization and benchmark evaluation, efforts must be devoted to improving the interpretability of graph learning methods from the perspective of neural network structures.",
"Interpretable models for anomaly analytics can be presented clearly and are likely to be accepted by the public.",
"For example, [71] could identify which neighbor of an anomalous node influenced most by differentiating the edge weights generated by the attention mechanism.",
"Moreover, it is acknowledged that adversarial attacks will influence the model's accuracy and performance.",
"Therefore, how to enhance the robustness of a model is another challenge.",
"Several studies regarding interpretability and robustness can be found in [117], [141], [28]."
],
[
"Anomalous subgraph detection",
"Recent years have substantially witnessed superior performance on detecting point anomalies, while users in real-world tend to carry out abnormal behaviors in groups, such as spreading rumors and telecommunications fraud.",
"Apart from this, graph data have diverse structures and forms, while existing methods are not available for all situations.",
"Methods regarding group or subgraph anomaly detection have been less explored [120], especially for complex network structures like hypergraph and multi-modal graph."
],
[
"Novel applications of anomaly prediction",
"While most of the works we reviewed aim to detect existing anomalies, there are still significant works to be done in predicting anomalies in advance.",
"For example, predicting traffic jams ahead of time in transportation networks can help people map out another travel route and avoid congestion situations [48].",
"Therefore, developing representation learning frameworks that are truly appropriate to anomaly prediction settings in a timely manner is necessary to prevent accidents, huge financial loss, or even deaths.",
"As a special data structure, graphs are often employed as an auxiliary tool to combine with many research fields, such as biology, chemistry, and social science.",
"Considering that anomalies are defined quite different in various scenarios, domain knowledge is thereby necessary when applying anomaly prediction models into novel applications."
],
[
"Fairness in anomaly analysis",
"Recent years have witnessed a surge of attention in fair machine learning models [59].",
"Consequently, several fairness metrics have been proposed as the constraints of objective function in various machine learning models to guarantee the equality of the prediction results.",
"As for anomaly detection tasks, whether users can trust the detection results of the models is still a significant problem [125].",
"It is due to the fact that incorrect anomaly detection results may sometimes lead to serious consequences, such as wrong object detection when dealing with criminals and fraudsters.",
"In [84], the authors formally defined fairness-aware outlier detection problem and proposed a model to satisfy the fairness criteria.",
"However, fairness on graph anomaly detection is still of concern and deserve further attention."
],
[
"Conclusion",
" In this survey, we comprehensively reviewed anomaly analytics methods using graph learning models.",
"The algorithms are divided into four classifications: graph convolutional network-based methods, graph attention network-based methods, graph autoencoder-based methods, and other graph learning models.",
"A thorough comparison and summarization of these methods are provided in this paper.",
"Then, we enumerated and briefly introduced several real-world applications of graph anomaly analytics.",
"Finally, we discussed five future research directions when applying deep learning methods into graph anomaly analytics."
]
] | 2212.05532 |
[
[
"Near-Field Enhancement of Optical Second Harmonic Generation in Hybrid\n Gold-Lithium Niobate Nanostructures"
],
[
"Abstract Nanophotonics research has focused recently on the ability of non-linear optical processes to mediate and transform optical signals in a myriad of novel devices, including optical modulators, transducers, color filters, photodetectors, photon sources, and ultrafast optical switches.",
"The inherent weakness of optical nonlinearities at smaller scales has, however, hindered the realization of efficient miniaturized devices, and strategies for enhancing both device efficiencies and synthesis throughput via nanoengineering remain limited.",
"Here, we demonstrate a novel mechanism by which second harmonic generation, a prototypical non-linear optical phenomenon, from individual lithium niobate particles can be significantly enhanced through nonradiative coupling to the localized surface plasmon resonances of embedded gold nanoparticles.",
"A joint experimental and theoretical investigation of single mesoporous lithium niobate particles coated with aispersed layer of $\\sim$10-nm diameter gold nanoparticles shows that a $\\sim$32-fold enhancement of second harmonic generation can be achieved without introducing finely tailored radiative nanoantennas to mediate photon transfer to or from the non-linear material.",
"This work highlights the limitations of current strategies for enhancing non-linear optical phenomena and proposes a route through which a new class of subwavelength nonlinear optical platforms can be designed to maximize non-linear efficiencies through near-field energy exchange."
],
[
"Synthesis of Mesoporous Monodisperse Lithium Niobtate Particles",
"All chemicals were used without further purification.",
"Monodisperse lithium niobate particles were synthesized using our previously reported method.",
"In brief, 0.8 mM of niobium n-butoxide [Nb(OBu)$_5$ , 99%, Alfa Aesar] was dissolved in 4 mL of ethanol.",
"This solution was aged for 36 h in a desiccator in the presence of an open glass vial containing 25 mL of water to generate a humid atmosphere.",
"The resulting gel-like precursor was mixed with 20 mL of an aqueous solution of 0.1 M lithium hydroxide monohydrate (LiOH$\\cdot $ H$_2$ O, 99%, Alfa Aesar) and sonicated for 20 min.",
"A 10 mL aliquot of the resulting suspension was transferred to a 23 mL Teflon-lined autoclave (Model No.",
"4749, Parr Instruments Co., Moline, IL USA) and heated at 200 $^\\circ $ C for 48 h. After cooling to room temperature, white precipitates were isolated from the solution via a process of centrifugation (Model No.",
"AccuSpin 400, Fisher Scientific) at 8,000 rpm for 15 min and decanting of the solution.",
"These solids were washed by re-suspending them in 10 mL deionized water (18 M$\\Omega \\cdot $ cm, produced using a Barnstead NANOpure DIamond water filtration system).",
"This purification process was repeated for a total of three times.",
"The purified product was dried at 70 $^\\circ $ C for 10 h to remove residual water prior to further analyses.",
"The dried precipitates were calcined by heating from room temperature to 600 $^\\circ $ C at a rate of 5 $^\\circ $ C/min and held at 600 $^\\circ $ C for 45 min to induce complete crystallization."
],
[
"Synthesis of Gold-Lithium Niobate Hybrid Nanostructures",
"An aqueous suspension of porous, monodisperse LiNbO$_3$ particles (0.5 mg/mL) was added and dispersed in 5 mL of water via sonication.",
"Into this suspension, either 1 mL (for a higher loading of gold nanoparticles) or 0.1 mL (for a lower laoding of gold nanoparticles) of a 5 mM aqueous solution of gold(III) chloride trihydrate (HAuCl$_4\\cdot 3$ H$_2$ O, 99.9%, Sigma Aldrich) was added, and the mixture stirred at 70 $^\\circ $ C for 3 h. This step was followed by the addition of 1 mL or 0.01 mL of 0.5% (w/v) trisodium citrate dihydrate (C$_6$ H$_5$ Na$_3$ O$_7\\cdot 2$ H$_2$ O, Sigma Aldrich, $\\ge $ 99%) aqueous solution into the reaction mixture.",
"The reaction mixture was stirred further at 70 $^\\circ $ C for 1 h. The precipitates were washed two times with water.",
"The as-synthesized hybrid nanostructures of LiNbO$_3$ with gold (Au) nanoparticles (NPs) were isolated from the unreacted excess gold salts and unbound Au NPs through centrifugation (Thermo Electron Corporation, IEC microlite microcentrifuge) at 2,000 rpm for 3 min, decanting of the supernatants, and re-dispersion of the isolated solids in DI water with the assistance of a vortexer for 3 min.",
"The morphology and dimensions of the LiNbO$_3$ particles were characterized using an FEI Osiris X-FEG 8 scanning/transmission electron microscope (TEM/STEM) operated at an accelerating voltage of 200 kV.",
"Samples for TEM/STEM analyses were prepared by dispersing the purified products in ethanol followed by drop-casting 5 $\\mu $ L of each suspension onto separate TEM grids (300 mesh copper grids coated with Formvar/carbon) purchased from Cedarlane Labs.",
"Each TEM grid was dried at $\\sim $ 230 Torr for at least 20 min prior to analysis.",
"Energy dispersive X-ray spectroscopy (EDS) analyses were performed using the FEI Osiris scanning/TEM, which was equipped with a Super-X EDS system with ChemiSTEM Technology integrating the signal from four spectrometers.",
"Purity, crystallinity, and phase of the LiNbO$_3$ particles were characterized using Raman spectroscopy and powder X-ray diffraction (XRD) techniques.",
"Raman spectra were collected using a Renishaw inVia Raman microscope with a 50$\\times $ SWD objective lens (Leica, 0.5 NA), and a 514 nm laser (argon-ion laser, Model No.",
"Stellar-Pro 514/50) set to 100% laser power with an exposure time of 30 s. The Raman spectrometer was calibrated by collecting the Raman spectrum of a polished silicon (Si) standard with a distinct peak centred at 520 cm$^{-1}$ .",
"The Raman spectra for the samples were acquired from 100 to 1,000 cm$^{-1}$ using a grating with 1,800 lines/mm.",
"The XRD patterns of the samples were acquired with a Rigaku R-Axis Rapid diffractometer equipped with a 3 kW sealed tube copper source (K$\\alpha $ radiation, $\\lambda $ = 0.15418 nm) collimated to 0.5 mm.",
"The samples were packed into a cylindrical recess drilled into glass microscope slides (Leica 1 mm Surgipath Snowcoat X-tra Micro Slides) for acquiring XRD patterns for the products.",
"The optical absorption spectra of the products were measured using an Agilent Technologies ultraviolet-visible spectrophotometer (Agilent 8453, Model No.",
"G1103).",
"For these measurements, the samples were suspended in water and held in 1 cm path length poly(methyl methacrylate) cuvettes (VWR$\\mathrm {TM}$ , Catalog No.",
"634-8537).",
"The linear spectra of individual pristine LiNbO$_3$ and hybrid Au-LiNbO$_3$ particles were measured using a Zeiss M1m optical microscope operating in a dark-field spectroscopy setup.",
"The light from a halogen lamp was focused onto the sample through a dark field objective (50$\\times $ , Zeiss Epiplan Neofluar).",
"The scattered light was collected by the same objective and detected by an imaging spectrometer (Princeton Instrument Acton spectrometer equipped with a PIXIS 400 CCD detector cooled to ‒72°C).",
"The linear scattering cross section spectra were obtained by normalizing the signal from a single particle to the signal coming directly from the lamp, after subtraction of the background in proximity to the particle.",
"The second harmonic generation (SHG) activity of the individual pristine LiNbO$_3$ and hybrid Au-LiNbO$_3$ particles was assessed using a Leica SP5 laser scanning confocal two photon microscope equipped with a Coherent Chameleon Vision II laser and a Zeiss LSM 510 MP confocal microscope.",
"A dilute dispersion of nanostructures was drop-cast onto glass coverslips and brought into the focal point of the microscope.",
"The SHG response was characterized by locating individual particles using a 63$\\times $ oil immersion objective aperture and scanning the laser excitation from 850 nm to 1,070 nm as the fundamental wavelength."
],
[
"Bare Lithium Niobate Sphere Characterization",
"Modeling the microsphere as a smooth, isotropic spherical particle, the nearly constant dielectric function of LiNbO$_3$ in the optical region of interest [1] allows the microsphere's linear response near the SH to be described by the excitation of a set of 21 Mie resonances (SI, Section REF ).",
"We have established $\\mathbf {\\beta } = \\lbrace T^{\\prime },p^{\\prime },\\ell ^{\\prime },m^{\\prime }\\rbrace $ as a collective index to describe these modes such that the scattered electric field of each mode can be expanded as $\\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r},\\omega ) = \\sum _{\\mathbf {\\beta }}a_2^{-\\ell - 2} \\left[ \\rho _{\\mathbf {\\beta }}(\\omega ) \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }}) + \\rho _{\\mathbf {\\beta }}^*(-\\omega ) \\mathbf {X}^*_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }}) \\right]$ [2].",
"Using $\\mathbf {\\mathcal {E}}_\\mathrm {sca}$ along with the magnetic fields of each mode, $\\mathbf {\\mathcal {B}}_\\mathrm {sca}(\\mathbf {r},\\omega ) = (c/i\\omega )\\nabla \\times \\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r},\\omega )$ , the scattered power averaged over a period of the SH frequency can be calculated from Poynting's theorem as (Section REF ): $P^{(1)}(2\\omega _0) = \\frac{c^3E_0^2}{4\\pi \\omega _0}\\sum _{\\mathbf {\\beta }}\\frac{1 - \\delta _{p1}\\delta _{m0}}{\\omega _{\\mathbf {\\beta }}a_2^{2\\ell + 4}}|C_{\\mathbf {\\beta }}(\\omega _0)|^2|\\alpha _{\\mathbf {\\beta }}(2\\omega _0)|^2.$ Here, $\\alpha _{\\mathbf {\\beta }}(\\omega ) = (a_2^{\\ell + 2} f_{\\mathbf {\\beta }}/ 2\\omega _{\\mathbf {\\beta }})\\exp (i\\psi _{\\mathbf {\\beta }})/(\\omega _{\\mathbf {\\beta }} - i\\gamma _{\\mathbf {\\beta }}/2 - \\omega )$ is the linear polarizability of the mode $\\mathbf {\\beta }$ , with $C_{\\mathbf {\\beta }}(\\omega _0)\\sim E_0\\chi _2^{(2)}(\\omega _0,\\omega _0)$ a unitless overlap coefficient.",
"In general, $C_{\\mathbf {\\beta }}(\\omega _0)$ quantifies the efficiency of photon upconversion between the fundamental resonances of the system at $\\omega _0$ and a resonance $\\mathbf {\\beta }$ at the SH, and thus is large only if the fundamental resonances are driven strongly by the incident light.",
"Also, $C_{\\mathbf {\\beta }}(\\omega _0)$ is zero unless the fundamental and SH resonances satisfy specific symmetry requirements, which are detailed below.",
"Finally, the explicit form of this overlap coefficient is complicated, but described in full in Section REF , Eq.",
"(REF ).",
"In general, the resonance frequencies of the Mie modes are independent of $p$ and $m$ , although these indices do determine the strength of the response of a mode to a driving source of a given spatial symmetry.",
"In detail, if we allow $\\mathbf {\\alpha } = \\lbrace T,p,\\ell ,m\\rbrace $ to be the collective index of one fundamental resonance of the LiNbO$_3$ particle and $\\mathbf {\\alpha }^{\\prime } = \\lbrace T^{\\prime \\prime },p^{\\prime \\prime },\\ell ^{\\prime \\prime },m^{\\prime \\prime }\\rbrace $ to be the index of another, in the case where the sphere has a weak, isotropic, and frequency-dependent nonlinear susceptibility $\\mathbf {\\chi }_2^{(2)}(\\omega ^{\\prime },\\omega - \\omega ^{\\prime }) = \\mathbf {1}_3\\chi _2^{(2)}(\\omega ^{\\prime },\\omega - \\omega ^{\\prime })$ only modes $\\mathbf {\\beta }$ for which $p + p^{\\prime } + p^{\\prime \\prime }$ is even and $m^{\\prime } \\pm m \\pm m^{\\prime \\prime } = 0$ are driven by SHG.",
"For example, with the microsphere driven by an $x$ -polarized plane wave at the fundamental frequency such that $\\mathbf {\\alpha } = \\lbrace E,0,\\ell ,1\\rbrace $ or $\\lbrace M,1,\\ell ,1\\rbrace $ , each of the 21 SH modes in the model has an index pair $(p^{\\prime },m^{\\prime }) = (0,0)$ , $(0,2)$ , or $(1,2)$ .",
"In contrast, the dependence of a mode's response on $T^{\\prime }$ and $\\ell ^{\\prime }$ is determined by its spectral overlap with the source.",
"In the energy window of the observed SHG enhancement, seven sets of modes can be significantly driven by the upconversion process: four sets of electric modes with $\\ell ^{\\prime } = 6$ , 7, 9, and 10, and three sets of magnetic modes with $\\ell ^{\\prime } = 7$ , 10, and 11.",
"See Figure REF for details."
],
[
"Characterization of the Lithium Niobate Radius and Dielectric Function",
"Section REF details the characterization of each of the LiNbO$_3$ Mie modes, each of which is assigned a resonance frequency $\\omega _{\\mathbf {\\beta }}$ , a damping rate $\\gamma _{\\mathbf {\\beta }}$ , and an oscillator strength $f_{\\mathbf {\\beta }}$ (equivalently, an effective mass $\\mu _{\\mathbf {\\beta }} = e^2/a_2^3f_{\\mathbf {\\beta }}$ ) that determine its spectral position and response magnitude to external stimuli.",
"This characterization is performed using a dielectric function $\\epsilon _2 = 5.5 + 0.035i$ and a radius $a_2 = 500$ nm estimated from the experiment.",
"The real part of $\\epsilon _2$ is taken from estimates extracted from a set of nine single-particle scattering experiments conducted on bare LiNbO$_3$ spheres of radii 350 to 1000 nm.",
"Figure c shows results from one representative scattering experiment, with the remainder of the results given Figure REF and in Table REF .",
"Further, in conjunction with the choice of radius, the choice $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace = 5.5$ provides the best overlap between the spectral locations of the Mie resonances and the SHG enhancement peaks seen in Figure c, as well as the best agreement between the relative peak heights.",
"We note that small ($\\sim 5\\%$ ) increases (decreases) to $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace $ and $a_2$ lead to red (blue) shifts in the Mie resonance positions that can nullify each other, such appropriate choices with the ranges $a_2\\pm 20$ nm and $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace \\pm 0.1$ are likely to lead to similar results.",
"The inset of Figure c shows the range of Mie resonance energies available with $\\pm 4\\%$ changes to $a_2$ , and analogous results when varying $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace $ are shown in Figure REF c. The imaginary part of $\\epsilon _2$ , which is expected to be small [1], is chosen to agree well with the linewidths of the modes observed in the second harmonic data of Figure c. The exact value has yet to be characterized via e.g.",
"ellipsometry [3] and is difficult to estimate from the scattering data due to substrate effects (see Figure REF a).",
"The linewidths of the Mie resonances depend sensitively on the rate of internal material losses (see Figure REF d) and the heights of the narrow Mie modes in both scattering and SHG enhancement spectra can vary noticeably with $\\sim 5\\%$ changes to $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ .",
"However, small changes to $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ do not modify the peak positions within the SH enhancement spectrum such that we take $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ to be a fitting parameter that affects the overall magnitude of the theoretical SHG enhancements, but does not qualitatively affect the results.",
"Further details are given in Section REF ."
],
[
"Superradiant Nanoparticle Scattering",
"Each NP is assumed to be spherical and isotropic with a Drude-Lorentz dielectric function $\\epsilon _1(\\omega )$ fit to the Au dielectric data of Ref.",
"olmon2012optical (Section REF , Figure REF , and Table REF ) and an associated set of surface plasmon resonances within the spectral window of the observed SH emission.",
"Figure a demonstrates the excellent agreement between the resulting theoretical absorption cross section of the NP dipole plasmons using this model and extinction measurements performed on single hybrid nanostructures, highlighting the validity of the model and the uniformity of the NP responses in the experiment.",
"We thus restrict our analysis to include only the dipole resonances of the NPs, which we label $\\mathbf {d}_i(\\omega )$ , and allow each particle to lie at a position $\\mathbf {r}_i$ on the surface of the LiNbO$_3$ microsphere.",
"The dynamics of the plasmon dipoles are inferred from numerical scattering calculations using both multiple-Mie scattering and boundary element method Maxwell solvers [5], [6].",
"These simulations were performed on arrays of 10-nm diameter Au NPs in square grids with a 20 nm center-to-center spacing, as complete layers on spherical templates containing the $N\\approx 1000$ Au NPs present in the experiment are too large to compute directly.",
"Moreover, grids reliably approximate the effects of interparticle coupling in the limit that the radii of the Au NPs and their separations are much smaller than the radius of the LiNbO$_3$ microsphere.",
"Using these scaled down simulations, we extrapolated from smaller ensembles ($N\\le 225$ ) the behaviors of larger collections of Au NPs.",
"For comparative reasons, we used both regular grids and randomized grids of Au NPs to evaluate the effects of disorder on the amount of light scattered by each particle.",
"Our results (Figure REF a) show that the maximum scattering cross section of each dipole grows roughly two orders of magnitude from $7.75\\times 10^{-3}$ nm$^2$ as $N$ varies from 1 to 1000, and that the exact magnitude of the growth depends on whether the dipole is oriented perpendicular (maximum of 0.355 nm$^2$ ) or parallel (1.18 nm$^2$ ) to the plane of the ensemble.",
"Conversely, the absorption cross section maxima of each dipole converge quickly with ensemble size, as shown in Figure REF a, moving from a single particle value of 13.5 nm$^2$ to perpendicular and the parallel values of 10.3 nm$^2$ and 15.7 nm$^2$ , respectively, for ensembles of $N>50$ NPs.",
"In addition, while randomness in the positions of the particles can modulate the per-particle scattering cross section maxima by $\\pm 20\\%$ in comparison to a regular grid (Figure REF b), an extreme value distribution fit to the recorded scattering cross section maxima of 100 simulations with randomized grids of $N = 16$ NPs shows that modifications beyond this range occur in $\\sim $ 0.4% of NPs.",
"The regular-grid simulations are, thus, taken to be representative of the ensemble as a whole.",
"Further, Figure REF b shows that while the peak maxima of the scattering observables depend strongly on $N$ , their lineshapes do not.",
"Therefore, with the NPs' scattering strongly modified and their absorption changed relatively little with increasing $N$ , we conclude that the dipole oscillator strengths are best estimated from the absorption data.",
"Explicitly, each dipole is then treated quasistatically with a position- and orientation-dependent polarizability (see Section REF ), while the ensemble-induced scattering enhancements are included phenomenologically through modification of the Rayleigh scattering rate of each dipole by a positive factor $A(N)$ .",
"When driven by the scattered fields of a bare LiNbO$_3$ sphere, the ensemble then radiates a time-averaged scattered power $P_\\mathrm {pl}(2\\omega _0) \\approx A(N)\\frac{\\gamma _1^\\mathrm {rad}f_1a_1^3E_0^2}{2\\gamma _1^2}\\sum _{\\mathbf {\\beta }}\\frac{K_{\\mathbf {\\beta }}}{a_2^{2\\ell + 4}}\\left|C_{\\mathbf {\\beta }}(\\omega _0)\\right|^2\\left|\\alpha _{\\mathbf {\\beta }}(2\\omega _0)\\right|^2.$ Here, $f_1$ is the orientation-averaged oscillator strength of the NP dipoles (Eq.",
"[REF ]), and $K_{\\mathbf {\\beta }}$ (Eq.",
"[REF ]) is an overlap coefficient that determines the strength of the ensemble's response to the LiNbO$_3$ fields.",
"See Section REF for further details."
],
[
"SHG Enhancement via Near-Field Coupling",
"The first and second-order terms in the solution of the Mie resonance equations of motion in Eq.",
"() are $\\begin{split}\\rho _{\\mathbf {\\beta }}^{(1)}(\\omega ) &= \\frac{f_{\\mathbf {\\beta }}a_2^{\\ell + 2}(\\Omega _{\\mathbf {\\beta }}^* + \\omega )\\mathrm {e}^{-i\\psi _{\\mathbf {\\beta }}}}{2\\omega _{\\mathbf {\\beta }}e\\left(|\\Omega _{\\mathbf {\\beta }}|^2 - \\omega ^2 - i\\omega \\gamma _{\\mathbf {\\beta }}\\right)}F_{2\\mathbf {\\beta }}(\\omega ),\\\\\\rho _{\\mathbf {\\beta }}^{(2)}(\\omega ) &= \\frac{f_{\\mathbf {\\beta }}a_2^{\\ell + 2}(\\Omega _{\\mathbf {\\beta }}^* + \\omega )\\mathrm {e}^{-i\\psi _{\\mathbf {\\beta }}}}{2\\omega _{\\mathbf {\\beta }}e^2\\left(|\\Omega _{\\mathbf {\\beta }}|^2 - \\omega ^2 - i\\omega \\gamma _{\\mathbf {\\beta }}\\right)}\\\\&\\times \\frac{ef_\\nu a_1^3\\eta _1(\\omega )}{2\\omega _1e^2\\left(|\\Omega _1|^2 - \\omega ^2 - i\\omega \\gamma _1\\right)}\\sum _\\nu \\sigma _{\\mathbf {\\beta }\\nu }(\\mathbf {r}_0)F_{1\\nu }(\\mathbf {r}_0,\\omega ),\\end{split}$ which detail the driving of each mode $\\mathbf {\\beta }$ through direct upconversion and through excitation of the NP by the upconversion process, respectively.",
"Higher-order terms build in the consequences of multiple transfers of energy between the microsphere and the NP and become increasingly tedious to write explicitly, but can be analyzed compactly with $\\begin{split}\\rho _{\\mathbf {\\beta }}^{(n > 2)}(\\omega ) &= \\sum _{\\nu }\\sum _{\\mathbf {\\beta }^{\\prime }\\ne \\mathbf {\\beta }}\\frac{f_{\\mathbf {\\beta }}f_\\nu a_2^{\\ell + 2} a_1^3}{4e^4\\omega _{\\mathbf {\\beta }}\\omega _1}\\frac{\\eta _1(\\omega )\\eta _{\\mathbf {\\beta }}(\\omega )}{\\left(|\\Omega _{\\mathbf {\\beta }}|^2 - \\omega ^2 - i\\omega \\gamma _{\\mathbf {\\beta }}\\right)\\left(|\\Omega _1|^2 - \\omega ^2 - i\\omega \\gamma _1\\right)}\\\\&\\times \\frac{\\sigma _{\\mathbf {\\beta }\\nu }(\\mathbf {r}_0)}{a_2^{\\ell ^{\\prime } - 1}}\\left[\\sigma _{\\mathbf {\\beta }^{\\prime }\\nu }(\\mathbf {r}_0)\\rho _{\\mathbf {\\beta }^{\\prime }}^{(n - 2)}(\\omega ) +\\sigma _{\\mathbf {\\beta }^{\\prime }\\nu }^*(\\mathbf {r}_0)\\rho _{\\mathbf {\\beta }^{\\prime }}^{(n - 2)*}(-\\omega )\\right],\\end{split}$ from which it is clear that, with $|\\rho _{\\mathbf {\\beta }}^{(2)}(\\omega )| \\ll |\\rho _{\\mathbf {\\beta }}^{(1)}(\\omega )|$ , terms beyond the third order provide only small corrections.",
"To account for all $N$ NPs, we label each NP with the index $i \\in [1,N]$ such that $\\mathbf {r}_0\\rightarrow \\mathbf {r}_i$ and $d_\\nu (\\omega ) \\rightarrow d_{i\\nu }(\\omega )$ , but allow the NPs to have identical radii and their plasmons to have identical resonance frequencies, linewidths, phase offsets, and oscillator strengths.",
"Therefore, the coupling term in Eq.",
"() is replaced by the sum $\\sum _{i\\nu }\\sigma _{\\mathbf {\\beta }\\nu }(\\mathbf {r}_i)d_{i\\nu }(\\omega )$ and similar sums in each successive term of the perturbation expansion must be adjusted accordingly.",
"In this work, these sums are carried out by placing three dipoles $d_{i\\nu }(\\omega )$ at $N = \\text{1,000}$ points arranged on a sphere of radius $a_1 + a_2$ in a Fibonacci lattice [7].",
"Calculations with randomized points display only small variations from the results of the regular Fibonacci array in good agreement with the random-ensemble scattering results of Figure b and are thus not shown."
],
[
"Construction of the Material Models from Data",
"In order to maximize the agreement between our theoretical models and the SH enhancement data, we construct a model dielectric function for Au and a dielectric function and second-order susceptibility and LiNbO$_3$ using our own single particle scattering data, as well as from ellipsometry data and atomistic numerical simulations from Refs.",
"olmon2012optical and riefer2012linear, respectively.",
"The Au details are given in Section REF and the LiNbO$_3$ details are given in Section REF ."
],
[
"Fitting of the Au Lorentz-Drude Dielectric Model",
"The oscillator parameters of the dipole plasmons within the Au NPs modeled in this investigation are calculated using a simple Drude-Lorentz model of the dielectric function of gold.",
"This dielectric function is given by $\\epsilon _1(\\omega ) = 1 - \\frac{\\omega _{p1}^2}{\\omega ^2 + \\mathrm {i}\\omega \\Gamma _1} + \\sum _{i=2}^3\\frac{\\omega _{pi}^2}{\\Lambda _i^2 - \\omega ^2 - \\mathrm {i}\\omega \\Gamma _i},$ and its form and parameters are inferred from ellipsometry data published in Ref.",
"olmon2012optical.",
"In particular, as shown in Figure REF a, a Drude-model ($\\omega _{p2}$ , $\\omega _{p3}\\rightarrow 0$ ) fit via nonlinear least squares methods to the gold dielectric function provides a good approximation to the dielectric data for photon energies $\\hbar \\omega < 2.0$ eV, but this model cannot capture the effects of interband transitions that become increasingly important at higher energies.",
"Indeed, a Drude model is provided by Ref.",
"olmon2012optical, but it is not used in this analysis as it produces inaccurate predictions of the dipole plasmon energies of Au NPs, which exist well above the 2.0 eV threshold.",
"To produce a more accurate Au NP model in the region between 2.0 and 2.8 eV, the addition of two Lorentz oscillators to the Drude model dielectric function was found to be sufficient.",
"The NP dipole plasmons have resonance energies near 2.5 eV, well below the upper bound of the single-oscillator Drude-Lorentz model's accuracy near 2.9 eV (see Figure REF a).",
"The parameters for the Drude-Lorentz dielectric model were inferred by first fitting a simplified Drude model $\\epsilon _1(\\omega )\\approx - \\frac{\\omega _{p1}^2}{\\omega ^2 + \\Gamma _1^2} + \\mathrm {i}\\frac{\\omega _{p1}^2\\Gamma _1}{\\omega ^3 + \\omega \\Gamma _1}$ to the real and imaginary parts of single-crystal dielectric data of Ref.",
"olmon2012optical at low energies $\\omega \\ll \\Lambda _2$ .",
"These fits produced estimates $\\hbar \\omega _{p1} = 7.20\\pm 0.9$ eV and $\\hbar \\Gamma _1 = 71.1\\pm 13$ meV, respectively, which were then used as initial guesses for fits using the more robust Drude-Lorentz model.",
"Focusing on the energy window between 0.8 and 2.8 eV, the Drude-Lorentz fits were made using the regularized function $\\mathrm {Re}\\lbrace \\epsilon _1(\\omega )\\rbrace + \\eta (\\omega )\\mathrm {Im}\\lbrace \\epsilon _1(\\omega )\\rbrace $ to simultaneously fit the model to both components of the complex dielectric data while avoiding bias toward the data's much larger real part in the energy window between 0.8 and 2.4 eV.",
"Fits using simplex, differential evolution, simulated annealing, and random search fitting methods [9] produced stable results using weighting functions $\\eta (\\omega ) = \\lambda _1\\exp (-\\lambda _2\\omega )$ with $\\lambda _1 = 40\\pm 5$ and $\\lambda _2 = (6.5\\pm 1)\\times 10^{-16}$ $\\mathrm {s}/\\mathrm {rad}$ .",
"The central values for either factor are used in this work.",
"Taking the average of the results of the four fitting routines, the model parameters used in the following discussion are given in Table REF ."
],
[
"Fitting of the Lithium Niobate Dielectric Model and Nonlinear Susceptibility",
"The radii of the samples examined in Figure c have radii of $a_2 \\approx 500$ nm, and we use this value in the following analysis.",
"To estimate $\\epsilon _2$ , nine scattering spectra were collected using bare LiNbO$_3$ particles with radii between 350 nm and 1000 nm, with the radii estimated from microscopy analyses with a precision of $\\pm 50$ nm.",
"Mie theory scattering spectra were compared to the experimental data, but peak splitting and broadening generated by the substrate in the experimental data restricts the utility of the ideal substrate-free model.",
"More precisely, in agreement with Ref.",
"savo2020broadband, the presence of a substrate in the scattering spectra generates peak broadening that can in most cases be captured to acceptable approximation by a Mie model with phenomenologically added internal damping.",
"However, as is also demonstrated in Ref.",
"savo2020broadband, the substrate can also induce splitting, shifting, and even peak amplification that are not uniform and are not reproducible by simple Mie theory.",
"These effects can be seen in Figure REF , where single peaks in a Mie theory model are spectrally aligned with multiply-peaked features in the data, and more obviously in Figure REF .",
"The latter compares Mie theory to simulations that include a substrate, showing that a phenomenologically damped dielectric model reproduces some substrate effects (peak suppression, broadening) but not others (enhancements, shifts of narrow peaks).",
"Thus, estimates of the real part of $\\epsilon _2$ were performed by aligning the visible broad peaks of the theory and experiment in the spectral window between 720 nm and 380 nm, and the absolute peak widths and relative heights were ignored.",
"Narrow features in the data are also ignored as they are most likely to be shifted by the substrate in a manner not captured by Mie theory, as evidenced by the narrow feature near 2.5 eV in Figure REF a.",
"Results using these comparison guidelines are shown in Table REF .",
"Within the range of possible values of $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace $ shown therein, the value 5.5 provides the best agreement between the spectral positions and relative magnitudes of the SH peaks in the theory and experiment, assuming $a_2 = 500$ nm.",
"As is mentioned in the main text and demonstrated in Figures c (inset) and REF c, small tweaks to $\\epsilon _2$ and $a_2$ will provide similar Mie spectra such that the particular values chosen within a narrow range ($\\pm \\sim \\!5\\%$ ) are irrelevant to the conclusions of this investigation.",
"The imaginary part of $\\epsilon _2$ is taken to be $0.035$ .",
"This value is likely larger than the true value extracted from earlier ellipsometry experiments [1], as the linewidths of the Mie resonances of the LiNbO$_3$ sphere are increased by the presence of a substrate.",
"Because the magnitude of this increase is difficult to infer from first-principles models, we instead choose $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ to agree well with the linewidths of the modes observed in the second harmonic data of Figure c,b.",
"Within a range of $\\pm \\sim \\!10\\%$ of the chosen value, $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ simply scales the magnitude of the SH enhancements, such that, like the choices of $\\mathrm {Re}\\lbrace \\epsilon _2\\rbrace $ and $a_2$ , the precise value of $\\mathrm {Im}\\lbrace \\epsilon _2\\rbrace $ does not affect our conclusions.",
"The second-order susceptibility of the LiNbO$_3$ , $\\mathbf {\\chi }_2^{(2)}(\\omega ^{\\prime },\\omega - \\omega ^{\\prime })$ was fit to the simulated data from Ref.",
"riefer2012linear using an anharmonic oscillator model [11] in the approximation that the LiNbO$_3$ has an isotropic, homogeneous response within the microsphere boundaries.",
"Explicitly, the susceptibility is given by: $\\begin{split}\\mathbf {\\chi }_2^{(2)}&(\\omega ^{\\prime },\\omega - \\omega ^{\\prime }) = \\\\&\\mathbf {1}\\chi ^{(2)}_\\infty - \\mathbf {1}\\frac{s}{(\\Lambda ^2 - \\omega ^2 - \\mathrm {i}\\omega \\Gamma )(\\Lambda ^2 - \\omega ^{\\prime 2} - \\mathrm {i}\\omega ^{\\prime }\\Gamma )(\\Lambda ^2 - [\\omega - \\omega ^{\\prime }]^2 - \\mathrm {i}[\\omega - \\omega ^{\\prime }]\\Gamma )},\\end{split}$ wherein $\\Lambda $ and $\\Gamma $ are the natural frequency and linewidth of the Lorentz oscillator, respectively.",
"These features are used to characterize the response of the LiNbO$_3$ carriers.",
"$\\chi ^{(2)}_\\infty $ is a constant offset that approximates the shift of the real part of $\\mathbf {\\chi }_2^{(2)}$ at low energies by higher-energy transitions.",
"Further, $s$ is the characteristic anharmonic oscillator strength with units $\\mathrm {cm}\\cdot \\mathrm {s}^6/\\mathrm {statV}$ and is taken to be very small such that $sE_0\\ll \\Lambda ^6$ with $E_0$ the characteristic strength of the laser field.",
"Finally, $\\mathbf {1}_3$ is the $3\\times 3$ identity matrix.",
"To fit the data, we evaluated the second-order susceptibility in the limit that $\\omega = 2\\omega ^{\\prime }$ , i.e.",
"with the assumption that the driving laser at $\\omega ^{\\prime } = \\omega _0$ is very narrow and the observation frequency is always at $2\\omega _0$ .",
"With this restriction, the expression for the susceptibility simplifies to $\\mathbf {\\chi }_2^{(2)}(\\omega _0,\\omega _0) = \\mathbf {1}\\chi _\\infty - \\mathbf {1}s/(\\Lambda ^2 - 4\\omega _0^2 - 2\\mathrm {i}\\omega _0\\Gamma )(\\Lambda ^2 - \\omega _0^2 - \\mathrm {i}\\omega _0\\Gamma )^2$ and can be easily visualized as is shown in Figure REF .",
"As was done with the dielectric function of Au in Section REF , the model was fit to the real part, imaginary part, and absolute value of the data using simplex, differential evolution, simulated annealing, and random search nonlinear least squares methods.",
"An average of the results of each of the four methods was collected for the fit to each function of the data, and the parameter average that produced the best fit to the data in the experimentally relevant energy range was selected.",
"In this case, the fits to the absolute value of the data were superior in the region between $\\sim $ 1.0 eV – 1.5 eV analyzed in Figure , returning parameter values of $\\hbar \\Lambda = 4.00$ eV, $\\hbar \\Gamma = 748$ meV, $s = 4.08\\times 10^{-3}$ $\\mathrm {cm}\\cdot \\mathrm {s^6}$ /statV, and $\\chi _\\infty ^{(2)} = -9.97\\times 10^{-7}$ cm/statV.",
"The estimation of the dielectric function of the LiNbO$_3$ is described in the main text.",
"We note here that the corresponding linear susceptibility model of LiNbO$_3$ to the second-order function described above is a Lorentz-model dielectric $\\chi ^{(1)}(\\omega ) = \\chi _\\infty ^{(1)} + f/(\\Lambda ^2 - \\omega ^2 - \\mathrm {i}\\omega \\Gamma )$ .",
"However, in the case where $\\Lambda $ is sufficiently detuned from the fundamental and second harmonic frequencies, $\\chi ^{(1)}(\\omega )$ is well-approximated as a dielectric.",
"In our case, $\\hbar \\Lambda - \\hbar \\Gamma > 2\\omega _0$ such that the constant-dielectric approximation is appropriate and in agreement with Ref.",
"palik1998handbook."
],
[
"Estimation of the Mode Oscillator Parameters",
"In this section, we use the dielectric models we have constructed to infer the parameters for oscillators models of the plasmon and Mie resonances of the coupled Au-LiNbO$_3$ nanostructures.",
"Bare Au and LiNbO$_3$ particles are discussed in Sections REF and REF , while the effects of weak coupling between members of the NP ensemble are discussed in Section REF ."
],
[
"Bare Au NP Dipoles",
"The relation of the oscillator parameters of the Mie and plasmon resonances to the dielectric parameters of their respective particles is done via the particles' response functions.",
"These are shown explicitly in Eqs.",
"(REF )–(REF ) (see Section REF ), and provide a direct link between the Au and LiNbO$_3$ dielectric functions $\\epsilon _1(\\omega )$ and $\\epsilon _2$ , respectively, the resonance freqeuncies $\\omega $ , damping rates $\\gamma $ , masses $\\mu $ , and phase offsets $\\psi $ of each mode, and the physical observables of the particles.",
"For example, with $\\epsilon _1(\\omega )$ characterized in Section REF , the response function of Eq.",
"(REF ) can be used used to construct the absorption cross section, $\\begin{split}\\sigma _1^\\mathrm {abs}(\\omega ) &= \\frac{4\\pi \\omega }{c}\\mathrm {Im}\\left\\lbrace \\frac{e^2}{2\\omega _1\\mu _1}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _1}}{\\Omega _1 - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _1}}{\\Omega _1^* + \\omega }\\right) + \\sum _{i = 1}^2\\frac{e^2}{2\\omega _{L_i}\\mu _{L_i}}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{L_i}}}{\\Omega _{L_i} - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{L_i}}}{\\Omega _{L_i}^* + \\omega }\\right)\\right\\rbrace ,\\end{split}$ of each of the Au NP dipole modes, in which the plasmon oscillator parameters (subscript 1) and interband resonance parameters (subscript $L_i$ ) can be exactly expressed as functions of the dielectric parameters of $\\epsilon _1(\\omega )$ .",
"These functions are impossible to write explicitly, as the eigenfrequencies $\\Omega _{1,L_i} = \\omega _{1,L_i} - \\mathrm {i}\\gamma _{1,L_i}/2$ are solutions to a sextic polynomial.",
"Nevertheless, the oscillator parameters are simple to infer numerically by fitting Eq.",
"(REF ) to the standard form $\\sigma _1^\\mathrm {abs}(\\omega ) = (4\\pi \\omega /c)a_1^3\\mathrm {Im}\\lbrace [\\epsilon _1(\\omega ) - 1]/[\\epsilon _1(\\omega ) + 2]\\rbrace $ as shown in Figure REF b with the NP radius $a_1$ set to 5 nm.",
"It is well-known that plasmon resonance energies are redshifted when the metal nanoparticle supporting them comes in close contact with a dielectric substrate with a large refractive index [12].",
"As the LiNbO$_3$ spheres in this investigation are much larger than the Au NPs and have a constant dielectric function larger than 5 (see Section REF ), some plasmon redshifting is expected.",
"We incorporate the redshifts by lowering the Au NP dipole by 0.23 eV to align the NP absorption maximum with the observed maximum of hybrid-particle extinction measurements in Figure a.",
"The oscillator parameters used in this work are given in Table REF .",
"However, to model the effects of radiation on the plasmon motion, one must modify the results extracted from Eq.",
"(REF ).",
"More concretely, in the formal quasistatic approximation under which the absorption cross section is derived, only the rate of Au carrier losses to heat will appear in the cross section expression and any back-action of radiation on the dipole plasmon will be ignored.",
"This back-action generally results in broadening of the plasmon lineshape, such that one can phenomenologically expand $\\gamma _1 \\rightarrow \\gamma _1^\\mathrm {rad} + \\gamma _1^\\mathrm {NR}$ to provide the plasmon damping rate with both a radiative (rad) and nonradiative (NR) component.",
"While approximate, this simple formulation of scattering loss works well in the limit $\\gamma _1^\\mathrm {rad}\\ll \\gamma _1^\\mathrm {NR}$ .",
"The validity of the approximation can be demonstrated by comparing the Rayleigh scattering cross section $\\begin{split}\\sigma _1^\\mathrm {sca}(\\omega ) &= \\frac{8\\pi a_1^6 \\omega ^4}{3c^4}\\left|\\frac{e^2}{2\\omega _1\\mu _1}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _1}}{\\omega _1 - \\mathrm {i}(\\gamma _1^\\mathrm {NR} + \\gamma _1^\\mathrm {rad})/2 - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _1}}{\\omega _1 + \\mathrm {i}(\\gamma _1^\\mathrm {NR} + \\gamma _1^\\mathrm {rad})/2 + \\omega }\\right)\\right.\\\\[0.5em]&\\left.+ \\sum _{i = 1}^2\\frac{e^2}{2\\omega _{L_i}\\mu _{L_i}}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{L_i}}}{\\omega _{L_i} - \\mathrm {i}(\\gamma _{L_i}^\\mathrm {NR} + \\gamma _{L_i}^\\mathrm {rad})/2 - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{L_i}}}{\\omega _{L_i} + \\mathrm {i}(\\gamma _{L_i}^\\mathrm {NR} + \\gamma _{L_i}^\\mathrm {rad})/2 + \\omega }\\right) \\right|^2,\\end{split}$ in which the interband damping rates have been similarly transformed as $\\gamma _{L_i}\\rightarrow \\gamma _{L_i}^\\mathrm {NR} + \\gamma _{L_i}^\\mathrm {rad}$ to account for radiation broadening and red shifting, to the exact Mie theory scattering cross section of a spherical Au NP with a 5-nm radius.",
"Numerical fits of the modified oscillator model to the Mie scattering lineshape are generally unstable and suggest radiation-induced changes to both $\\gamma _1$ and $\\gamma _{L_i}$ are roughly $1\\%$ or smaller, so we instead approximate the radiative damping rates of the dipole plasmons and interband resonances using their Larmor formulae.",
"Assuming $\\gamma _1^\\mathrm {NR}\\approx \\gamma _1$ , we have $\\gamma _1^\\mathrm {rad} = 2e^2\\omega _1^2/3c^3\\mu _1 = 1.01\\times 10^{-4}\\gamma _1^\\mathrm {NR}$ .",
"The radiation from the Lorentz oscillator resonances is similarly minimal, with $\\gamma _{L_1}^\\mathrm {rad} = 7.15\\times 10^{-5}\\gamma _{L_1}^\\mathrm {NR}$ , $\\gamma _{L_1}^\\mathrm {rad} = 3.67\\times 10^{-2}\\gamma _{L_2}^\\mathrm {NR}$ , and $\\gamma _{L_i}^\\mathrm {NR}\\approx \\gamma _{L_i}$ .",
"Figure REF a shows the excellent agreement between the Larmor-modified oscillator model and Mie scattering calculations."
],
[
"Ensemble-Modified Au NP Dipoles",
"To describe in a concise way the scattering observables of the dipole plasmons in the NPs surrounding the LiNbO$_3$ sphere, it is first necessary to quantify the alterations to the scattering behaviors of each NP caused by its neighbors.",
"In other words, due to inter-NP interactions, we cannot take the masses $\\mu _i$ (or, equivalently, the oscillator strengths $f_i = e^2/a_1^3\\mu _i$ ) of the $N$ NPs of the ensemble to be their bare values $\\mu _1$ (or $e^2/a_1^3\\mu _1$ ).",
"We can begin this process by letting the polarizability of each NP be a tensor dependent on its position, such that differences between the ensemble perpendicular and parallel dipole oscillations of each NP can be captured.",
"Explicitly, we let $\\alpha _1(\\omega )\\rightarrow \\mathbf {\\alpha }_i(\\omega ) = \\alpha _\\perp (\\omega )\\hat{\\mathbf {r}}_i\\hat{\\mathbf {r}}_i + \\alpha _\\parallel (\\omega )(\\hat{\\mathbf {\\theta }}_i\\hat{\\mathbf {\\theta }}_i + \\hat{\\mathbf {\\phi }}_i\\hat{\\mathbf {\\phi }}_i)$ , wherein the unit vectors $\\hat{\\mathbf {r}}_i = \\hat{\\mathbf {r}}(\\theta _i,\\phi _i)$ , $\\hat{\\mathbf {\\theta }}_i = \\hat{\\mathbf {\\theta }}(\\theta _i,\\phi _i)$ , and $\\hat{\\mathbf {\\phi }}_i = \\hat{\\mathbf {\\phi }}(\\theta _i,\\phi _i)$ are simply the spherical unit vectors evaluated at the angular position of the $i^\\mathrm {th}$ dipole.",
"The tensor elements $\\alpha _{\\perp ,\\parallel }(\\omega ) = f_{\\perp ,\\parallel }(a_1^3/2\\omega _1)(\\exp {\\mathrm {i}\\psi _{\\perp ,\\parallel }}/[\\Omega _1 - \\omega ] + \\exp {-\\mathrm {i}\\psi _{\\perp ,\\parallel }}/[\\Omega _1^* + \\omega ])$ are the polarizabilities of the dipole components at $\\mathbf {r}_i$ that are oriented perpendicular and parallel to the shell of Au NPs, respectively, and have magnitudes characterized by the oscillator strengths $f_{\\perp ,\\parallel }$ and phases determined by the angles $\\psi _{\\perp ,\\parallel }$ .",
"We can define the terms of $\\mathbf {\\alpha }_i(\\omega )$ with poles at $\\pm \\Omega _1$ as $\\mathbf {\\alpha }_i^{(+)}(\\omega ) = \\left(\\frac{f_\\perp a_1^3}{2\\omega _1}\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _\\perp }}{\\Omega _1 - \\omega } \\right)\\hat{\\mathbf {r}}_i\\hat{\\mathbf {r}}_i+ \\left(\\frac{f_\\parallel a_1^3}{2\\omega _1}\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _\\parallel }}{\\Omega _1 - \\omega }\\right)(\\hat{\\mathbf {\\theta }}_i\\hat{\\mathbf {\\theta }}_i + \\hat{\\mathbf {\\phi }}_i\\hat{\\mathbf {\\phi }_i})$ and $\\mathbf {\\alpha }_i^{(-)}(\\omega ) = \\mathbf {\\alpha }_i^{(+)*}(-\\omega )$ and further define the components of either as $\\alpha _{\\perp ,\\parallel }^{(\\pm )}(\\omega ) = a_1^3f_{\\perp ,\\parallel }\\mathrm {e}^{\\mathrm {i}\\psi _{\\perp ,\\parallel }}/2\\omega _1(\\Omega _1 \\mp \\omega )$ , respectively.",
"Therefore, upon excitation by an impinging field $\\mathbf {E}(\\mathbf {r}_i,\\omega )$ , the dipole set up in the $i^\\mathrm {th}$ NP is $\\mathbf {d}_i(\\omega ,\\hat{\\mathbf {\\epsilon }}_i) = \\mathbf {\\alpha }_i(\\omega )\\cdot \\mathbf {E}(\\mathbf {r}_i,\\omega )\\approx E(\\mathbf {r}_i,\\omega )\\Theta (\\omega )\\mathbf {\\alpha }_i^{(+)}(\\omega )\\cdot \\hat{\\mathbf {\\epsilon }}_i + E(\\mathbf {r}_i,\\omega )\\Theta (-\\omega )\\mathbf {\\alpha }_i^{(-)}(\\omega )\\cdot \\hat{\\mathbf {\\epsilon }}_i^*$ .",
"Here, $E(\\mathbf {r}_i,\\omega )$ is the (real) magnitude of the electric field at $\\mathbf {r}_i$ , $\\hat{\\mathbf {\\epsilon }}_i$ is the complex polarization unit vector that describes the phases and orientations of the field components, and $\\Theta (\\omega )$ is the Heaviside function.",
"Letting $\\mathbf {d}^{(\\pm )}(\\omega ,\\hat{\\mathbf {\\epsilon }}_i)$ be the dipole terms valid at positive and negative frequencies, one finds the magnitude of $\\mathbf {d}^{(+)}$ is ${\\mathbf {d}_i^{(+)}(\\omega ,\\hat{\\mathbf {\\epsilon }}_i)} \\approx \\frac{f_i(\\hat{\\mathbf {\\epsilon }}_i)a_1^3}{2\\omega _1}\\frac{1}{|\\Omega _1 - \\omega |}\\Theta (\\omega )E(\\mathbf {r}_i,\\omega ).$ We can, therefore, define $f_i(\\hat{\\mathbf {\\epsilon }}_i) = \\sqrt{f_\\perp ^2|\\hat{\\mathbf {r}}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2 + f_\\parallel ^2(|\\hat{\\mathbf {\\theta }}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2 + |\\hat{\\mathbf {\\phi }}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2)}$ as the orientation-dependent dipole plasmon oscillator strength.",
"With $\\hat{\\mathbf {\\epsilon }}_i$ oriented strictly perpendicular to the ensemble, one can see that $f_i(\\hat{\\mathbf {r}}_i) = f_\\perp $ .",
"Similarly, $f_i(\\hat{\\mathbf {\\theta }}_i) = f_\\parallel $ .",
"Figure REF c shows the simulated absorption cross section maxima of NPs in ensembles excited with these two choices of $\\hat{\\mathbf {\\epsilon }}_i$ , where it can be seen that for $N \\gtrsim 50$ the absorption cross section of each particle is independent of $N$ .",
"Analytically, the absorption cross sections of ensemble perpendicular and parallel dipoles are simply $\\sigma ^\\mathrm {abs}_{\\perp ,\\parallel }(\\omega ) = (4\\pi \\omega /c)R_\\mathrm {abs}\\mathrm {Im}\\lbrace \\alpha _{\\perp ,\\parallel }(\\omega )\\rbrace $ , where $R_\\mathrm {abs} = 1.16$ phenomenologically accounts for the absorption contribution of the Lorentz oscillators in the region $\\omega < 2.8$ eV.",
"With $\\cos \\psi _{\\perp ,\\parallel }\\approx \\cos \\psi _1\\approx 1$ , the cross section maxima are then $\\sigma ^\\mathrm {abs}_{\\perp ,\\parallel }(\\omega _1)\\approx 4\\pi R_\\mathrm {abs} f_{\\perp ,\\parallel }a_1^3/c\\gamma _1$ such that the oscillator strengths implied by Figure REF c are $f_\\parallel = 2.37\\times 10^{30}$ s$^{-2}$ and $f_\\perp /f_\\parallel = 0.668$ .",
"Finally, to define the orientation-averaged plasmon oscillator strength $f_1$ used in the main text, we let an electric field $\\mathbf {E}_0(\\mathbf {r},\\omega ) = E_0\\pi [\\exp (2\\mathrm {i}k_0x)\\delta (\\omega - 2\\omega _0) + \\exp (-2\\mathrm {i}k_0x)\\delta (\\omega + 2\\omega _0)]\\hat{\\mathbf {z}}$ excite the NP ensemble, producing $f_i(\\hat{\\mathbf {z}}) = \\sqrt{f_\\perp ^2\\cos ^2\\theta _i + f_\\parallel ^2\\sin ^2\\theta _i}$ .",
"We then take a formal average $\\begin{split}\\langle f_i(\\hat{\\mathbf {z}})\\rangle _i = \\frac{\\int _0^{2\\pi }\\int _0^\\pi f_i(\\hat{\\mathbf {z}})(t)\\sin \\theta _i\\;\\mathrm {d}\\theta _i\\mathrm {d}\\phi _i}{\\int _0^{2\\pi }\\int _0^\\pi \\sin \\theta _i\\;\\mathrm {d}\\theta _i\\mathrm {d}\\phi _i}\\end{split}$ over all of the possible $\\hat{\\mathbf {r}}_i$ and define $f_1 = \\langle f_i(\\hat{\\mathbf {z}}) \\rangle _i$ , giving $f_1 = \\frac{1}{2}\\left(f_\\perp + \\frac{f_\\parallel ^2\\cos ^{-1}\\left\\lbrace \\frac{f_\\perp }{f_\\parallel }\\right\\rbrace }{\\sqrt{f_\\parallel ^2 - f_\\perp ^2}}\\right) = 2.13\\times 10^{30} \\;\\,\\mathrm {s}^{-2}.$"
],
[
"Bare Lithium Niobate Mie Resonances",
"Inference of the oscillator parameters of the Mie resonances $\\mathbf {\\beta }$ of the LiNbO$_3$ microsphere is more straightforward, as their damping rates $\\gamma _{\\mathbf {\\beta }}$ contain only losses to radiation.",
"Thus, only a single observable is needed to extract the full set of parameters from each mode.",
"We use the response functions shown in Eq.",
"(REF ) to fit $\\gamma _{\\mathbf {\\beta }}$ as well as the resonance frequencies $\\omega _{\\mathbf {\\beta }}$ , effective masses $\\mu _{\\mathbf {\\beta }}$ (equivalently, the oscillator strenghts $f_{\\mathbf {\\beta }} = e^2/a_2^3\\mu _{\\mathbf {\\beta }}$ ), and phase offsets $\\psi _{\\mathbf {\\beta }}$ .",
"Explicitly, we let $\\begin{split}(\\sqrt{\\epsilon _2})^\\ell A_{p\\ell m}^<(\\omega ) - 1 &\\approx -\\frac{e^2}{a_2^3}\\frac{1}{2\\omega _{M\\ell }\\mu _{M\\ell }}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{M\\ell }}}{\\Omega _{M\\ell } - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{M\\ell }}}{\\Omega _{M\\ell }^* + \\omega }\\right), \\\\(\\sqrt{\\epsilon _2})^\\ell B_{p\\ell m}^<(\\omega ) - 1 &\\approx -\\frac{e^2}{a_2^3}\\frac{1}{2\\omega _{E\\ell }\\mu _{E\\ell }}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{E\\ell }}}{\\Omega _{E\\ell } - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{E\\ell }}}{\\Omega _{E\\ell }^* + \\omega }\\right).\\end{split}$ Note that, in contrast with Eq.",
"(REF ), the above expressions only consider a single resonance for a given $\\mathbf {\\beta }$ .",
"See Section REF for details.",
"The results of the fits are shown graphically in Figure REF b,c and are tabulated in Table REF ."
],
[
"Derivation of the Scattering Observables",
"We provide here a more thorough derivation of the superradiant scattering enhancement ratio.",
"First, in Section REF , we describe the scattered fields and power of the modes of a dielectric sphere with a diameter on the order of a wavelength of the scattered light.",
"Second, in sections Section REF we develop the scattering observables of the ensemble of NPs surrounding a dielectric sphere.",
"In Section REF , we complete the calculation of the ratio of scattered powers of a NP-dressed and bare LiNbO$_3$ dielectric sphere."
],
[
"Mie Modes of the Lithium Niobate Sphere Driven by a Plane Wave",
"The time averaged radiated power from each of the LiNbO$_3$ SH modes can be calculated from Poynting's theorem.",
"Beginning with $\\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r},\\omega ) = \\sum _{\\mathbf {\\beta }}\\frac{1}{a_2^{\\ell + 2}}\\left(\\rho _{\\mathbf {\\beta }}(\\omega )\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }}) + \\rho _{\\mathbf {\\beta }}^*(-\\omega )\\mathbf {X}^*_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }})\\right),$ wherein $(2)$ signifies that the fields arise from a second-order scattering process, $\\rho _{\\mathbf {\\beta }}(\\omega )$ are the magnitudes of the moments of the modes $\\mathbf {\\beta }$ , and $\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},\\omega )$ are the regularized vector spherical harmonics $\\begin{split}\\mathbf {M}_{p\\ell m}(\\mathbf {r},k) &= \\sqrt{(2 - \\delta _{m0})\\frac{2\\ell + 1}{\\ell (\\ell + 1)}\\frac{(\\ell - m)!",
"}{(\\ell + m)!",
"}}\\left[\\frac{(-1)^{p + 1}m}{\\sin \\theta }h_\\ell ^{(1)}(kr)P_{\\ell m}(\\cos \\theta )S_{p + 1}(m\\phi )\\hat{\\mathbf {\\theta }}\\right.\\\\&\\left.- h_\\ell ^{(1)}(kr)\\frac{\\partial P_{\\ell m}(\\cos \\theta )}{\\partial \\theta }S_p(m\\phi )\\hat{\\mathbf {\\phi }}\\right],\\\\[0.5em]\\mathbf {N}_{p\\ell m}(\\mathbf {r},k) &= \\sqrt{(2 - \\delta _{m0})\\frac{2\\ell + 1}{\\ell (\\ell + 1)}\\frac{(\\ell - m)!",
"}{(\\ell + m)!",
"}}\\left[\\frac{\\ell (\\ell + 1)}{kr}h_\\ell ^{(1)}(kr)P_{\\ell m}(\\cos \\theta )S_p(m\\phi )\\hat{\\mathbf {r}}\\right.\\\\&\\hspace{-42.7026pt}\\left.+ \\frac{1}{kr}\\frac{\\partial \\lbrace rh_\\ell ^{(1)}(kr)\\rbrace }{\\partial r}\\left(\\frac{\\partial P_{\\ell m}(\\cos \\theta )}{\\partial \\theta }S_p(m\\phi )\\hat{\\mathbf {\\theta }} + \\frac{(-1)^{p+1}m}{\\sin \\theta }P_{\\ell m}(\\cos \\theta )S_{p+1}(m\\phi )\\hat{\\mathbf {\\phi }}\\right)\\right]\\end{split}$ for $T = M$ and $E$ , respectively, the calculation of the Poynting vector is straightforward.",
"Here, $h_\\ell ^{(1)}(x)$ are the spherical Hankel functions of the first kind, $P_{\\ell m}(x)$ are the associated Legendre polynomials, $S_p(x) = \\cos (x)\\delta _{p\\;\\mathrm {even}} + \\sin (x)\\delta _{p\\;\\mathrm {odd}}$ , and $k = \\omega /c$ .",
"Further, with $\\mathbf {\\mathcal {B}}_\\mathrm {sca}(\\mathbf {r},\\omega ) = \\sum _{\\mathbf {\\beta }}\\frac{c}{\\mathrm {i}\\omega a_2^{\\ell + 2}}\\left[\\rho _{\\mathbf {\\beta }}(\\omega )\\nabla \\times \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }}) + \\rho _{\\mathbf {\\beta }}^*(-\\omega )\\nabla \\times \\mathbf {X}^*_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }})\\right]$ from Faraday's law, $\\begin{split}\\lim _{r\\rightarrow \\infty }\\int _0^{2\\pi }\\int _0^\\pi &\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }})\\times \\nabla \\times \\mathbf {X}_{\\mathbf {\\beta }^{\\prime }}^*(\\mathbf {r},k_{\\mathbf {\\beta }^{\\prime }})\\cdot \\hat{\\mathbf {r}}r^2\\sin \\theta \\;\\mathrm {d}\\theta \\,\\mathrm {d}\\phi \\\\&= 4\\pi (1 - \\delta _{p1}\\delta _{m0})\\frac{(-1)^{\\ell + 1}\\mathrm {i}^{2\\ell + 1}}{k_{\\mathbf {\\beta }}}\\delta _{\\mathbf {\\beta }\\mathbf {\\beta }^{\\prime }},\\end{split}$ and $\\frac{\\omega _0}{\\pi }\\int _{-\\pi /2\\omega _0}^{\\pi /2\\omega _0}\\mathrm {e}^{\\mathrm {i}(\\omega - \\omega ^{\\prime })t}\\;\\mathrm {d}t = \\frac{2\\omega _0}{\\pi }\\frac{\\sin \\!\\left(\\frac{\\pi }{2\\omega _0}[\\omega - \\omega ^{\\prime }]\\right)}{\\omega - \\omega ^{\\prime }},$ the time-averaged scattered power from the dielectric sphere can be rapidly simplified from $\\begin{split}&\\bar{P}_2(2\\omega _0) \\equiv \\langle P_{\\mathbf {\\beta }}(t) \\rangle _{2\\pi /2\\omega _0}\\\\&= \\frac{2\\omega _0}{2\\pi }\\frac{c}{4\\pi }\\int _{-\\pi /2\\omega _0}^{\\pi /2\\omega _0}\\oint \\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r},t)\\times \\mathbf {\\mathcal {B}}_\\mathrm {sca}(\\mathbf {r},t)\\cdot \\mathrm {d}\\mathbf {a}\\,\\mathrm {d}t\\\\&= \\lim _{r\\rightarrow \\infty }\\frac{\\omega _0c}{2\\pi ^2}\\int _0^{2\\pi }\\int _0^\\pi \\iint \\frac{\\sin \\left(\\frac{\\pi }{2\\omega _0}[\\omega - \\omega ^{\\prime }]\\right)}{\\omega - \\omega ^{\\prime }}\\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r},\\omega )\\times \\mathbf {\\mathcal {B}}_\\mathrm {sca}^*(\\mathbf {r},\\omega ^{\\prime })r^2\\sin \\theta \\cdot \\hat{\\mathbf {r}}\\;\\frac{\\mathrm {d}\\omega \\,\\mathrm {d}\\omega ^{\\prime }}{4\\pi ^2}\\,\\mathrm {d}\\theta \\,\\mathrm {d}\\phi ,\\end{split}$ wherein the surface integral is taken to be across a sphere of radius $r\\rightarrow \\infty $ , $\\langle \\cdot \\rangle _\\tau $ is the time-average operator over a period $\\tau $ , and the Fourier transforms of the fields $\\mathbf {\\mathcal {F}}_\\mathrm {sca}(\\mathbf {r},t) = \\int \\mathbf {\\mathcal {F}}_\\mathrm {sca}(\\mathbf {r},\\omega )\\exp (-\\mathrm {i}\\omega t)\\;\\mathrm {d}\\omega /2\\pi $ have been used.",
"Letting the moments of the sphere be driven by the second-harmonic upconversion of an incoming plane wave of frequency $\\omega $ and electric field strength $E_0$ , we can say $\\rho _{\\mathbf {\\beta }}(\\omega ) = C_{\\mathbf {\\beta }}(\\omega )\\alpha _{\\mathbf {\\beta }}(\\omega )E_0\\pi [\\delta (\\omega - 2\\omega _0) + \\delta (\\omega + 2\\omega _0)]$ with $\\alpha _{\\mathbf {\\beta }}(\\omega ) = (a_2^{\\ell + 2} f_{\\mathbf {\\beta }}/ 2\\omega _{\\mathbf {\\beta }})\\exp (i\\psi _{\\mathbf {\\beta }})/(\\omega _{\\mathbf {\\beta }} - i\\gamma _{\\mathbf {\\beta }}/2 - \\omega )$ the linear polarizability of the $\\mathbf {\\beta }^\\mathrm {th}$ mode and $C_{\\mathbf {\\beta }}(\\omega )$ an overlap coefficient defined in Eq.",
"(REF ).",
"The final result is $\\bar{P}_2(2\\omega _0) = \\sum _{\\mathbf {\\beta }}(1 - \\delta _{p1}\\delta _{m0})\\frac{c^2E_0^2}{4\\pi \\omega _0k_{\\mathbf {\\beta }}a_2^{2\\ell + 4}}|C_{\\mathbf {\\beta }}(2\\omega _0)|^2|\\alpha _{\\mathbf {\\beta }}(2\\omega _0)|^2.$ Importantly, the radiated power from the Mie modes contains no cross-terms due to the orthogonality condition imposed by Eq.",
"(REF ).",
"Further, due to the good agreement between an oscillator model of each mode and its electromagnetic response (see Section REF ), the scattered power can also be modeled mechanically.",
"To do so, it is important to first define a generalized coordinate $q_{\\mathbf {\\beta }}(\\omega )$ to represent the displacement magnitude of the moments $\\rho _{\\mathbf {\\beta }}(\\omega )$ that obeys the reality condition $q_{\\mathbf {\\beta }}(-\\omega ) = q_{\\mathbf {\\beta }}^*(\\omega )$ .",
"We will choose the definition $q_{\\mathbf {\\beta }}(\\omega ) = [\\rho _{\\mathbf {\\beta }}(\\omega ) + \\rho _{\\mathbf {\\beta }}^*(-\\omega )]/2ea_2^{\\ell - 1}$ such that $q_{\\mathbf {\\beta }}(t) = \\mathrm {Re}\\lbrace \\rho _{\\mathbf {\\beta }}(t)\\rbrace /ea_2^{\\ell - 1}$ is a real quantity.",
"With $q_{\\mathbf {\\beta }}(t)$ defined, we model losses to radiation as a weak, frequency-independent linear damping process.",
"In this case, $\\begin{split}P_2(t) &= -\\sum _{\\mathbf {\\beta }}\\dot{q}_{\\mathbf {\\beta }}(t)F_{\\mathbf {\\beta }}^\\mathrm {rad}(t),\\\\&= \\sum _{\\mathbf {\\beta }}A_{\\mathbf {\\beta }}\\mu _{\\mathbf {\\beta }}\\gamma _{\\mathbf {\\beta }}\\dot{q}_{\\mathbf {\\beta }}^2(t),\\end{split}$ where $A_{\\mathbf {\\beta }}$ is a unitless proportionality constant that relates the mechanical quantities to the electromagnetics.",
"This leads directly to $\\bar{P}_2(2\\omega _0) = \\sum _{\\mathbf {\\beta }}A_{\\mathbf {\\beta }}\\gamma _{\\mathbf {\\beta }}\\mu _{\\mathbf {\\beta }}\\frac{\\omega _0^2 E_0^2}{2e^2a_2^{2\\ell - 2}}|C_{\\mathbf {\\beta }}(2\\omega _0)|^2|\\alpha _{\\mathbf {\\beta }}(2\\omega _0)|^2$ via Eq.",
"(REF ) and the identity $q_{\\mathbf {\\beta }}(t) = \\int (-\\mathrm {i}\\omega )\\exp (-\\mathrm {i}\\omega t)q_{\\mathbf {\\beta }}(\\omega )\\;\\mathrm {d}\\omega /2\\pi $ such that $A_{\\mathbf {\\beta }} = (1 - \\delta _{p1}\\delta _{m0})(e^2c^3/2\\pi \\omega _0^3a_2^6\\mu _{\\mathbf {\\beta }}\\omega _{\\mathbf {\\beta }}\\gamma _{\\mathbf {\\beta }}) = (1 - \\delta _{p1}\\delta _{m0})(f_{\\mathbf {\\beta }}c^3/2\\pi \\omega _0^3a_2^3\\omega _{\\mathbf {\\beta }}\\gamma _{\\mathbf {\\beta }})$ .",
"Finally, to arrive at Eq.",
"(REF ) and the notation of the main text we substitute the explicit form of $A_{\\mathbf {\\beta }}$ and make the simplification $\\bar{P}_2(2\\omega _0)\\rightarrow P^{(1)}(2\\omega _0)$ to the notation."
],
[
"Plasmon Dipoles Driven by the Scattered Lithium Niobate Electric Field",
"The radiated power by the ensemble of Au NPs can be straightforwardly derived using a mechanical model parameterized using the values outlined in Sections REF and REF .",
"More explicitly, the time-averaged scattered power by the $i^\\mathrm {th}$ dipole plasmon of the Au NP ensemble when the collection is driven by an external field can be calculated in a straightforward manner using a well-known mechanical model of dipole radiation $P_i(t) = -\\dot{\\mathbf {x}}_i(t)\\cdot A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\mathbf {F}_i^\\mathrm {rad}(t),$ wherein $\\mathbf {x}_i(t) = \\mathbf {d}_i(t)/e$ is the coordinate characterizing the magnitude of the dipole plasmon located at $\\mathbf {r}_i$ and oriented along $\\hat{\\mathbf {x}}_i$ and $A_i(N,\\hat{\\mathbf {\\epsilon }}_i)$ is a phenomenological enhancement factor that builds in the ensemble-induced scattering enhancements seen in Figure c. Similarly to the orientation-dependent oscillator strengths, we take $A_i(N,\\hat{\\mathbf {\\epsilon }}_i) = \\sqrt{A_\\perp ^2(N)|\\hat{\\mathbf {r}}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2 + A_\\parallel ^2(N)(|\\hat{\\mathbf {\\theta }}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2 + |\\hat{\\mathbf {\\phi }}_i\\cdot \\hat{\\mathbf {\\epsilon }}_i|^2)}$ where $A_{\\perp ,\\parallel }(N)$ are the enhancements experience by ensemble-perpendicular and parallel dipoles, respectively, in an ensemble of $N$ NPs.",
"With the radiation back-force $\\mathbf {F}_i^\\mathrm {rad}(t)$ treated as a damping force, one can let $\\mathbf {F}_i^\\mathrm {rad}(t) = -\\mu _i(\\hat{\\mathbf {\\epsilon }}_i)\\gamma _1^\\mathrm {rad}\\dot{\\mathbf {x}}_i(t)$ , with $\\mu _i(\\hat{\\mathbf {\\epsilon }}_i) = e^2/a_1^3f_i(\\hat{\\mathbf {\\epsilon }}_i)$ the mass of the $i^\\mathrm {th}$ plasmon, such that an average over a period $\\tau = 2\\pi /2\\omega _0$ of the second harmonic frequency gives: $\\begin{split}\\bar{P}_i(2\\omega _0) &\\equiv \\langle P_i(t)\\rangle _{2\\pi /2\\omega _0}\\\\&= A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\frac{\\omega _0}{\\pi }\\int _{-\\pi /2\\omega _0}^{\\pi /2\\omega _0}\\mu _i(\\hat{\\mathbf {\\epsilon }}_i)\\gamma _1^\\mathrm {rad}\\dot{x}_i^2(t)\\;\\mathrm {d}t.\\end{split}$ Inserting the identity $\\dot{x}_i(t) = \\int (-\\mathrm {i}\\omega ) x_i(\\omega )\\exp (\\mathrm {i}\\omega t)\\;\\mathrm {d}\\omega /2\\pi $ twice and letting $\\mathbf {x}_i(\\omega ) = \\mathbf {\\alpha }_i(\\omega )\\cdot \\mathbf {E}_\\mathrm {sca}^{(2)}(\\mathbf {r},\\omega )/e$ , one finds $\\begin{split}\\bar{P}_i(2\\omega _0) &= A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\frac{\\omega _0\\gamma _1^\\mathrm {rad}}{4\\pi ^3 a_1^3f_i(\\hat{\\mathbf {\\epsilon }}_i)}\\iint \\int _{-\\pi /2\\omega _0}^{\\pi /2\\omega _0}\\omega \\omega ^{\\prime }\\left[\\mathbf {\\alpha _i}(\\omega )\\cdot \\mathbf {E}_\\mathrm {sca}^{(2)}(\\mathbf {r}_i,\\omega )\\right]\\cdot \\left[\\mathbf {\\alpha _i}^*(\\omega ^{\\prime })\\cdot \\mathbf {E}_\\mathrm {sca}^{(2)*}(\\mathbf {r}_i,\\omega ^{\\prime })\\right]\\\\&\\times \\mathrm {e}^{-\\mathrm {i}(\\omega - \\omega ^{\\prime })t}\\;\\mathrm {d}t\\,\\mathrm {d}\\omega \\,\\mathrm {d}\\omega ^{\\prime }.\\end{split}$ Application of the identity of Eq.",
"(REF ) and neglect of terms proportional to $\\rho _{\\mathbf {\\beta }}(-2\\omega _0)\\ll \\rho _{\\mathbf {\\beta }}(2\\omega _0)$ provides $\\begin{split}\\bar{P}_i(2\\omega _0) &\\approx A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\frac{2\\gamma _1^\\mathrm {rad}}{ a_1^3f_i(\\hat{\\mathbf {\\epsilon }}_i)}\\sum _{\\mathbf {\\beta }\\mathbf {\\beta }^{\\prime }}E_0^2\\omega _0^2a_2^{-\\ell -\\ell ^{\\prime }-4}C_{\\mathbf {\\beta }}(2\\omega _0)C_{\\mathbf {\\beta }^{\\prime }}^*(2\\omega _0)\\alpha _{\\mathbf {\\beta }}(2\\omega _0)\\alpha _{\\mathbf {\\beta }^{\\prime }}^*(2\\omega _0)\\\\&\\times \\left[\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})\\right]\\cdot \\left[\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }^{\\prime }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }^{\\prime }})\\right]^*.\\end{split}$ Finally, we average over the power scattered from each dipole $i$ to define the power scattered from the typical dipole.",
"Numerically, one can show that terms proportional to $\\langle A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\left[\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})\\right]\\cdot \\left[\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }^{\\prime }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }^{\\prime }})\\right]^*/f_i(\\hat{\\mathbf {\\epsilon }}_i)\\rangle _i$ are at least two orders of magnitude smaller when $\\mathbf {\\beta }\\ne \\mathbf {\\beta }^{\\prime }$ than when $\\mathbf {\\beta } = \\mathbf {\\beta }^{\\prime }$ .",
"Thus, we can safely neglect the cross terms of the sum and define $\\bar{P}_1(\\omega ) = \\langle \\bar{P}_i(\\omega )\\rangle _i$ such that $\\bar{P}_1(2\\omega _0) \\approx \\frac{2\\gamma _1^\\mathrm {rad}}{a_1^3}E_0^2\\omega _0^2\\sum _{\\mathbf {\\beta }}\\frac{|C_{\\mathbf {\\beta }}(2\\omega _0)|^2}{a_2^{2\\ell + 4}}|\\alpha _{\\mathbf {\\beta }}(2\\omega _0)|^2\\left\\langle A_i(N,\\hat{\\mathbf {\\epsilon }}_i)\\frac{{\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})}^2}{f_i(\\hat{\\mathbf {\\epsilon }}_i)}\\right\\rangle _i.$ Rectification of Eq.",
"(REF ) with Eq.",
"(REF ) and the notation of the main text can be achieved by substituting appropriately the dimensionless constant $\\begin{split}&K_{\\mathbf {\\beta }} = \\frac{\\omega _1^2\\gamma _1^2}{A(N)f_1a_1^6}\\left\\langle \\frac{A_i(N,\\hat{\\mathbf {\\epsilon }}_i){\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})}^2}{f_i(\\hat{\\mathbf {\\epsilon }}_i)}\\right\\rangle _i\\\\&\\approx \\left\\langle \\frac{A_i(N,\\hat{\\mathbf {\\epsilon }}_i)}{A(N)}\\frac{f_\\perp ^2|\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})\\cdot \\hat{\\mathbf {r}}_i|^2 + f_\\parallel ^2\\left(|\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})\\cdot \\hat{\\mathbf {\\theta }}_i|^2 + |\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})\\cdot \\hat{\\mathbf {\\phi }}_i|^2\\right)}{f_1f_i(\\hat{\\mathbf {\\epsilon }}_i)}\\right\\rangle _i\\end{split}$ and simplifying the notation such that $\\bar{P}_1(2\\omega _0)\\rightarrow P_\\mathrm {pl}(2\\omega _0)$ .",
"Here, we have taken $A(N) = \\frac{1}{2}\\left(A_\\perp (N) + \\frac{A_\\parallel ^2(N)\\cos ^{-1}\\left\\lbrace \\frac{A_\\perp (N)}{A_\\parallel (N)}\\right\\rbrace }{\\sqrt{A_\\parallel ^2(N) - A_\\perp ^2(N)}}\\right)$ similar to the definition of the angle-averaged oscillator strength $f_1$ .",
"Thus, all that is left to characterize in $\\bar{P}_1(2\\omega _0)$ are the phenomenological constants $A_{\\perp ,\\parallel }(N)$ .",
"This can be achieved by replacing $\\mathbf {\\mathcal {E}}_\\mathrm {sca}(\\mathbf {r}_i,\\omega )$ with a field $\\mathbf {E}_0(\\mathbf {r}_i,\\omega ) = E_0\\pi [\\delta (\\omega - 2\\omega _0) + \\delta (\\omega + 2\\omega _0)]\\hat{\\mathbf {e}}_{\\perp ,\\parallel }$ with $\\hat{\\mathbf {e}}_{\\perp ,\\parallel } = \\hat{\\mathbf {r}}_i$ and $\\hat{\\mathbf {\\theta }}_i$ , respectively, in Eq.",
"(REF ).",
"The results can then be compared to the simulations detailed in Figure c, which provide the simulated scattering cross sections of a grid of spheres upon which a plane wave polarized parallel or perpendicular to the ensemble plane is impinged, such that only ensemble-parallel or perpendicular dipoles are excited.",
"The use of $\\mathbf {E}_0(\\mathbf {r}_i,\\omega )$ in the theory faithfully approximates the simulated arrangement in the limit where the spacing between NPs is much smaller than the impinging light wavelength and the LiNbO$_3$ sphere radius $a_2$ .",
"Explicitly, the power scattered by the $i^\\mathrm {th}$ dipole is $\\bar{P}_i^\\mathrm {PW}(2\\omega _0,\\hat{\\mathbf {e}}_{\\perp ,\\parallel }) = A_{\\perp ,\\parallel }(N)\\frac{2\\gamma _1^\\mathrm {rad}}{a_1^3f_{\\perp ,\\parallel }}E_0^2\\omega _0^2{\\mathbf {\\alpha }_i(2\\omega _0)\\cdot \\hat{\\mathbf {e}}_{\\perp ,\\parallel }}^2$ and, after an angular average, $\\bar{P}_1^\\mathrm {PW}(2\\omega _0,\\hat{\\mathbf {e}}_{\\perp ,\\parallel }) \\approx A(N)f_{\\perp ,\\parallel } \\frac{a_1^3\\gamma _1^\\mathrm {rad}}{2\\omega _1^2}\\frac{E_0^2\\omega _0^2}{|\\Omega _1 - 2\\omega _0|^2}$ where the superscript PW signifies that the driving source in a plane wave and $\\bar{P}_1^\\mathrm {PW}(2\\omega _0) = \\langle \\bar{P}_i^\\mathrm {PW}(2\\omega _0)\\rangle _i$ .",
"The associated scattering cross section to $\\bar{P}_1^\\mathrm {PW}(2\\omega _0)$ that can be directly compared to the simulation is given by $\\sigma ^\\mathrm {sca}_{\\perp ,\\parallel }(N,2\\omega _0) = R_\\mathrm {sca}A_{\\perp ,\\parallel }(N)f_{\\perp ,\\parallel } \\frac{4\\pi a_1^3\\gamma _1^\\mathrm {rad}}{c\\omega _1^2}\\frac{\\omega _0^2}{|\\Omega _1 - 2\\omega _0|^2}$ wherein $R_\\mathrm {sca} = 6.00$ is a scaling constant that builds in the contribution of the Lorentz resonances of Au at low energies.",
"With all other constants already well-characterized, we find the best fits to the simulated scattering data are $A_\\perp (N) = 2.71N^{0.489}$ and $A_\\parallel (N) = 2.43N^{0.730}$ such that $A(1000) = 301$ .",
"Finally, we find that due to the similar average magnitudes over $\\mathbf {r}_i$ of the harmonics $\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_i,k_{\\mathbf {\\beta }})$ , $K_{\\mathbf {\\beta }}$ varies little between modes, with $\\mathrm {max}_{\\mathbf {\\beta }}\\lbrace K_{\\mathbf {\\beta }}\\rbrace = K_{M,0,10,2} = 10.2\\times 10^{-3}$ and $\\mathrm {min}_{\\mathbf {\\beta }}\\lbrace K_{\\mathbf {\\beta }}\\rbrace = K_{E,0,10,0} = 5.04\\times 10^{-3}$ ."
],
[
"Ratio of Time-Averaged Scattered Powers",
"One can see from Eqs.",
"(REF ) and (REF ) that the superradiant scattering enhancement ratio $N\\bar{P}_1(2\\omega _0)/\\bar{P}_2(2\\omega _0)$ is greatly simplified by defining the spectral profile the associated spectral profile $\\kappa (\\omega ) = \\frac{\\sum _{\\mathbf {\\beta }}K_{\\mathbf {\\beta }}|C_{\\mathbf {\\beta }}(\\omega )|^2|\\alpha _{\\mathbf {\\beta }}(\\omega )|^2/a_2^{2\\ell }}{\\sum _{\\mathbf {\\beta }} f_2|C_{\\mathbf {\\beta }}(\\omega )|^2|\\alpha _{\\mathbf {\\beta }}(\\omega )|^2/\\gamma _2\\omega _{\\mathbf {\\beta }}a_2^{2\\ell }}$ wherein $f_2 = \\langle f_{\\mathbf {\\beta }}\\rangle _{\\mathbf {\\beta }}$ and $\\gamma _2 = \\langle \\gamma _{\\mathbf {\\beta }}\\rangle _{\\mathbf {\\beta }}$ are the average Mie oscillator strength and damping rate, respectively.",
"As $K_{\\mathbf {\\beta }}$ is small for all of the relevant modes of the system, so too is $\\kappa (\\omega )$ for $\\omega $ in the optical region, with a value between $1.67\\times 10^{-3}$ and $1.78\\times 10^{-3}$ ."
],
[
"Derivation of the Near-Field Enhanced Scattering",
"In this section, the derivation of the coupled-oscillator model of the hybrid Au-LiNbO$_3$ nanostructures is developed from first principles.",
"Section REF details the calculations of the relevant field quantities and Section REF translates these field quantities into an osicllator model."
],
[
"Nonlinear Optics First Principles",
"The oscillator model of the modes of the LiNbO$_3$ microsphere is generated from the solutions to the coupled nonlinear wave equation for the total electric field $\\nabla \\times \\nabla \\times \\mathbf {E}(\\mathbf {r},\\omega ) - \\frac{\\omega ^2}{c^2}\\mathbf {E}(\\mathbf {r},\\omega ) - \\frac{\\omega ^2}{c^2}\\left[\\mathbf {P}^{(1)}(\\mathbf {r},\\omega ) + \\mathbf {P}^{(2)}(\\mathbf {r},\\omega )\\right] = \\frac{4\\pi \\mathrm {i}\\omega }{c^2}\\mathbf {J}_0(\\mathbf {r},\\omega ),$ wherein $\\mathbf {J}_0(\\mathbf {r},\\omega )$ is the current density of the laser.",
"The polarization fields $\\mathbf {P}^{(1)}(\\mathbf {r},\\omega ) = \\chi _2^{(1)}(\\mathbf {r})\\mathbf {E}(\\mathbf {r},\\omega )$ and $\\mathbf {P}^{(2)}(\\mathbf {r},\\omega ) = \\int _{-\\infty }^\\infty \\mathbf {E}(\\mathbf {r},\\omega - \\omega ^{\\prime })\\cdot \\mathbf {\\chi }_2^{(2)}(\\mathbf {r};\\omega ^{\\prime },\\omega - \\omega ^{\\prime })\\cdot \\mathbf {E}(\\mathbf {r},\\omega ^{\\prime })\\;\\frac{\\mathrm {d}\\omega ^{\\prime }}{2\\pi }$ build in the first- and second-order material response of the LiNbO$_3$ sphere, respectively.",
"The sphere's electric susceptibilities are allowed to vary in space such that $\\chi _2^{(1)}(\\mathbf {r}) = \\chi _2^{(1)}\\Theta (r \\le a_2)$ and $\\mathbf {\\chi }_2^{(2)}(\\mathbf {r};\\omega ^{\\prime },\\omega - \\omega ^{\\prime }) = \\mathbf {1}_3\\chi _2^{(2)}(\\omega ^{\\prime },\\omega - \\omega ^{\\prime })\\Theta (r \\le a_2)$ with $\\mathbf {1}_3$ the $3\\times 3$ identity matrix.",
"The sphere's dielectric function is then given by $\\epsilon _2(\\mathbf {r}) = 1 + 4\\pi \\chi _2^{(1)}(\\mathbf {r})$ .",
"From here, the electric field can be perturbatively expanded with the assumption that $|\\mathbf {\\chi }_2^{(2)}(\\mathbf {r};\\omega ^{\\prime },\\omega - \\omega ^{\\prime })|$ is a small quantity.",
"Explicitly, we let $\\mathbf {E}(\\mathbf {r},\\omega ) = \\mathbf {E}_0(\\mathbf {r},\\omega ) + \\sum _{n = 2}^\\infty \\mathbf {E}_0^{(n)}(\\mathbf {r},\\omega ) + \\sum _{n = 1}^\\infty \\mathbf {E}_\\mathrm {sca}^{(n)}(\\mathbf {r},\\omega )$ , wherein the first term is the laser's electric field and the terms in the sums contribute $n^\\mathrm {th}$ -order corrections to the vacuum-like and scattered fields $\\mathbf {E}_\\mathrm {vac}(\\mathbf {r},\\omega ) = \\sum _{n = 2}^\\infty \\mathbf {E}_0^{(n)}(\\mathbf {r},\\omega )$ and $\\mathbf {E}_\\mathrm {sca}(\\mathbf {r},\\omega ) = \\sum _{n=1}^\\infty \\mathbf {E}_\\mathrm {sca}^{(n)}(\\mathbf {r},\\omega )$ , respectively, that are set up by the polarized sphere.",
"Subtracting the laser field equation $\\nabla \\times \\nabla \\times \\mathbf {E}_0(\\mathbf {r},\\omega ) - (\\omega ^2/c^2)\\mathbf {E}_0(\\mathbf {r},\\omega ) = (4\\pi \\mathrm {i}\\omega /c^2)\\mathbf {J}_0(\\mathbf {r},\\omega )$ from Eq.",
"(REF ), one finds $\\begin{split}\\left(\\lbrace \\nabla \\times \\nabla \\times \\rbrace - \\epsilon _2(\\mathbf {r})\\frac{\\omega ^2}{c^2}\\right)\\mathbf {E}_\\mathrm {sca}^{(1)}(\\mathbf {r},\\omega ) &= \\frac{4\\pi \\mathrm {i}\\omega }{c^2}\\mathbf {J}^{(1)}(\\mathbf {r},\\omega ),\\\\\\left(\\lbrace \\nabla \\times \\nabla \\times \\rbrace - \\epsilon _2(\\mathbf {r})\\frac{\\omega ^2}{c^2}\\right)\\left\\lbrace \\mathbf {E}_\\mathrm {sca}^{(2)}(\\mathbf {r},\\omega ) + \\mathbf {E}_0^{(2)}(\\mathbf {r},\\omega )\\right\\rbrace &= \\frac{4\\pi \\mathrm {i}\\omega }{c^2}\\mathbf {J}^{(2)}(\\mathbf {r},\\omega ).\\end{split}$ The currents in right-hand-sides of Eq.",
"(REF ) are bound currents that are only nonzero where the first- and second-order susceptibilities of the LiNbO$_3$ sphere are nonzero, respectively, such that $\\begin{split}\\mathbf {J}^{(1)}(\\mathbf {r},\\omega ) &= -\\mathrm {i}\\omega \\chi _2^{(1)}(\\mathbf {r})\\mathbf {E}_0(\\mathbf {r},\\omega ),\\\\\\mathbf {J}^{(2)}(\\mathbf {r},\\omega ) &= -\\mathrm {i}\\omega \\int _{-\\infty }^\\infty \\left(\\mathbf {E}_0(\\mathbf {r},\\omega ^{\\prime }) + \\mathbf {E}_\\mathrm {sca}^{(1)}(\\mathbf {r},\\omega ^{\\prime })\\right)\\\\&\\cdot \\mathbf {\\chi }_2^{(2)}(\\mathbf {r};\\omega ^{\\prime },\\omega - \\omega ^{\\prime })\\cdot \\left(\\mathbf {E}_0(\\mathbf {r},\\omega - \\omega ^{\\prime }) + \\mathbf {E}_\\mathrm {sca}^{(1)}(\\mathbf {r},\\omega - \\omega ^{\\prime })\\right)\\;\\frac{\\mathrm {d}\\omega ^{\\prime }}{2\\pi }.\\end{split}$ The former current can straightforwardly be seen to be large where $\\omega = \\pm \\omega _0$ if the laser light is a monochromatic plane wave with an electric field $\\begin{split}\\mathbf {E}_0(\\mathbf {r},\\omega ) &= E_0\\left[\\pi \\delta (\\omega - \\omega _0)\\mathrm {e}^{\\mathrm {i}\\omega _0 z/c} + \\pi \\delta (\\omega + \\omega _0)\\mathrm {e}^{-\\mathrm {i}\\omega _0 z/c}\\right]\\hat{\\mathbf {x}},\\end{split}$ while for the same incoming field the leading factor of $\\omega $ in the latter restricts the integral to be zero unless $\\omega ^{\\prime } = \\pm \\omega _0$ and $\\omega = 2\\omega ^{\\prime }$ .",
"The currents drive the total fields through the Green's function solution to either wave equation, $\\mathbf {E}_\\mathrm {sca}^{(n)}(\\mathbf {r},\\omega ) + \\mathbf {E}_0^{(n)}(\\mathbf {r},\\omega ) = \\frac{4\\pi \\mathrm {i}\\omega }{c}\\int \\mathbf {G}_\\mathrm {LNO}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\cdot \\frac{\\mathbf {J}^{(n)}(\\mathbf {r}^{\\prime },\\omega )}{c}\\;\\mathrm {d}^3\\mathbf {r}^{\\prime },$ where the spherical dyadic Green's function $\\mathbf {G}_\\mathrm {LNO}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )$ is given in the literature [13].",
"The Green's function, as we show below, is separable as $\\mathbf {G}_\\mathrm {LNO}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega ) = \\mathbf {G}_\\mathrm {sca}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega ) + \\mathbf {G}_0(\\mathbf {r},\\mathbf {r};\\omega )$ into a “scattering” part $\\mathbf {G}_\\mathrm {sca}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )$ and a “vacuum-like” part $\\mathbf {G}_0(\\mathbf {r},\\mathbf {r};\\omega )$ such that $\\begin{split}\\mathbf {E}_\\mathrm {sca}^{(n)}(\\mathbf {r},\\omega )= \\frac{4\\pi \\mathrm {i}\\omega }{c}\\int \\mathbf {G}_\\mathrm {sca}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\cdot \\frac{\\mathbf {J}^{(n)}(\\mathbf {r}^{\\prime },\\omega )}{c}\\;\\mathrm {d}^3\\mathbf {r}^{\\prime },\\\\\\mathbf {E}_0^{(n)}(\\mathbf {r},\\omega ) = \\frac{4\\pi \\mathrm {i}\\omega }{c}\\int \\mathbf {G}_0(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\cdot \\frac{\\mathbf {J}^{(n)}(\\mathbf {r}^{\\prime },\\omega )}{c}\\;\\mathrm {d}^3\\mathbf {r}^{\\prime }.\\end{split}$ A similar solution can be found for the scattered electric field of the Au NPs.",
"The incident field and second-order current $\\mathbf {J}^{(2)}(\\mathbf {r},\\omega )$ can drive the response of each NP in the same manner that they drive the scattered fields of the LiNbO$_3$ sphere, such that we can build a model that includes only these two sources and temporarily neglects interactions between the NPs and microsphere.",
"Ignoring the effects of radiation, the scattered potential of the NP is $-\\nabla \\cdot \\epsilon _1(\\mathbf {r},\\omega )\\nabla \\left\\lbrace \\Phi _\\mathrm {NP}(\\mathbf {r},\\omega ) + \\Phi _0^{(2)}(\\mathbf {r},\\omega )\\right\\rbrace = 4\\pi \\chi _1(\\mathbf {r},\\omega )\\left[\\nabla \\cdot \\nabla \\Phi _0(\\mathbf {r},\\omega ) + \\frac{4\\pi \\mathrm {i}}{\\omega }\\nabla \\cdot \\mathbf {J}^{(2)}(\\mathbf {r},\\omega )\\right]$ for $\\chi _1(\\mathbf {r},\\omega ) = \\chi _1(\\omega )\\Theta (\\mathbf {r}\\in V_1)$ and $\\mathbf {r}$ not on the boundary of the NP's volume $V_1$ .",
"Assuming the laser wavelength $\\lambda _0 = 2\\pi c/\\omega _0$ is much longer than the extent of the NP, the incident potential can be written as $\\Phi _0(\\mathbf {r},\\omega )\\approx \\mathbf {r}\\cdot \\lim _{k\\rightarrow 0}\\mathbf {E}_0(\\mathbf {r},\\omega ) = xE_0\\pi [\\delta (\\omega - \\omega _0) + \\delta (\\omega + \\omega _0)]$ , where $k = \\omega /c$ .",
"Similarly, $-\\nabla \\Phi _0^{(2)}(\\mathbf {r},\\omega ) = \\lim _{k\\rightarrow 0}\\mathbf {E}_0^{(2)}(\\mathbf {r},\\omega )$ .",
"Assuming that the solutions to Eq.",
"(REF ) are near zero at $\\pm \\omega _0$ , i.e.",
"the NP does not respond strongly near the fundamental frequency of the laser, the term proportional to $\\Phi _0(\\mathbf {r},\\omega )$ on the RHS can be dropped.",
"Thus, with the dielectric function $\\epsilon _1(\\mathbf {r},\\omega ) = \\epsilon _1(\\omega )\\Theta (r\\le a_1) + 1\\Theta (r\\ge a_1)$ defined such that the NP lies at the origin, one finds: $\\Phi _\\mathrm {NP}(\\mathrm {r},\\omega ) + \\Phi _0^{(2)}(\\mathbf {r},\\omega ) \\approx \\int G_\\mathrm {NP}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\chi _1(\\mathbf {r},\\omega )\\frac{4\\pi \\mathrm {i}}{\\omega }\\nabla \\cdot \\mathbf {J}^{(2)}(\\mathbf {r},\\omega )\\;\\mathrm {d}^3\\mathbf {r}^{\\prime },$ wherein the (scalar) Green's function $G_\\mathrm {NP}(\\mathbf {r},\\mathbf {r}^{\\prime },\\omega )$ is the standard solution to the Poisson equation in spherical coordinates [3].",
"Much like the dyadic Green's function for the LiNbO$_3$ fields, $G_\\mathrm {NP}(\\mathbf {r},\\mathbf {r}^{\\prime },\\omega ) = G_\\mathrm {sca}(\\mathbf {r},\\mathbf {r}^{\\prime },\\omega ) + G_\\mathrm {0}(\\mathbf {r},\\mathbf {r}^{\\prime },\\omega )$ is separable into a scattering part and a vacuum-like or free-space part.",
"The former encodes the resonances of the NP, while the latter serves simply to satisfy the principle of superposition and is otherwise unimportant."
],
[
"Construction of the Oscillator Model",
"The above definitions are clarified by the forms of both $\\mathbf {G}_\\mathrm {LNO}$ and $G_\\mathrm {NP}$ , which are cumbersome but manageable.",
"The former, in the case where both the source charges and the observer are outside the sphere's surface, is given by $\\begin{split}G_\\mathrm {NP}(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )&\\Theta (r > a_1)\\Theta (r^{\\prime } > a_1) = \\sum _{p = 0}^1\\sum _{\\ell = 1}^\\infty \\sum _{m = 0}^\\ell (2 - \\delta _{m0})\\frac{(\\ell - m)!",
"}{(\\ell + m)!",
"}\\frac{\\ell [1 - \\epsilon _1(\\omega )]}{\\ell \\epsilon _1(\\omega ) + (\\ell + 1)}\\\\&\\times \\frac{a_1^{2\\ell + 1}}{r^{\\ell + 1}r^{\\prime \\ell + 1}}P_{\\ell m}(\\cos \\theta )P_{\\ell m}(\\cos \\theta ^{\\prime })S_p(m\\phi )S_p(m\\phi ^{\\prime }) + \\frac{1}{|\\mathbf {r} - \\mathbf {r}^{\\prime }|}\\end{split}$ for a sphere centered at the origin.",
"Each of the mode functions $(a_1^{\\ell + 1}/r^{\\ell + 1})P_{\\ell m}(\\cos \\theta )S_p(m\\phi )$ describes the spatial variation of the observables of an electric multipole mode with a characteristic response function [14] $g_{p\\ell m}(\\omega ) = \\frac{\\ell [1 - \\epsilon _1(\\omega )]}{\\ell \\epsilon _1(\\omega ) + (\\ell + 1)}$ that describes its oscillations in time.",
"Here, $\\ell $ and $m$ give the order and degree of each mode's corresponding spherical harmonic and $p$ the reflection symmetry of each mode across the $x$ -axis.",
"The Green's function of the LiNbO$_3$ sphere can be similarly expanded using a set of so-called quasinormal geometric resonances of both magnetic and electric multipole symmetry [2] with response functions that depend on the same spherical harmonic symmetry parameters $p,\\ell ,m$ .",
"For an observer inside the sphere, this Green's function is: $\\begin{split}\\mathbf {G}_\\mathrm {LNO}&(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\Theta (r < a_2) = \\frac{\\mathrm {i}\\omega }{4\\pi c}\\sum _{p\\ell m}\\left[C_{p\\ell m}^>(\\omega )\\mathbf {\\mathcal {M}}_{p\\ell m}(\\mathbf {r},\\sqrt{\\epsilon _2}k)\\mathbf {M}_{p\\ell m}(\\mathbf {r}^{\\prime },k)\\right.\\\\&\\left.+D_{p\\ell m}^>(\\omega )\\mathbf {\\mathcal {N}}_{p\\ell m}(\\mathbf {r},\\sqrt{\\epsilon _2}k)\\mathbf {N}_{p\\ell m}(\\mathbf {r}^{\\prime },k)\\right]\\Theta (r < a_2)\\Theta (r^{\\prime } > a_2)\\\\&+\\sqrt{\\epsilon _2}\\frac{\\mathrm {i}\\omega }{4\\pi c}\\sum _{p\\ell m}\\left[C_{p\\ell m}^<(\\omega )\\mathbf {\\mathcal {M}}_{p\\ell m}(\\mathbf {r},\\sqrt{\\epsilon _2}k)\\mathbf {\\mathcal {M}}_{p\\ell m}(\\mathbf {r}^{\\prime },\\sqrt{\\epsilon _2}k) \\right.\\\\&+\\left.D_{p\\ell m}^<(\\omega )\\mathbf {\\mathcal {N}}_{p\\ell m}(\\mathbf {r},\\sqrt{\\epsilon _2}k)\\mathbf {\\mathcal {N}}_{p\\ell m}(\\mathbf {r}^{\\prime },\\sqrt{\\epsilon _2}k)\\right]\\Theta (r < a_2)\\Theta (r^{\\prime } < a_2)\\\\&+ \\mathbf {G}_0(\\mathbf {r},\\mathbf {r}^{\\prime };\\sqrt{\\epsilon _2}\\omega )\\Theta (r < a_2)\\Theta (r^{\\prime } < a_2),\\end{split}$ where again the sphere is centered at the origin.",
"The mode functions $\\mathbf {\\mathcal {M}}_{p\\ell m}$ and $\\mathbf {\\mathcal {N}}_{p\\ell m}$ are identical to the regularized vector spherical harmonics given in (REF ) but with the spherical Hankel functions replaced with spherical Bessel functions, $h_{\\ell }^{(1)}(x)\\rightarrow j_{\\ell }(x)$ .",
"The Green's function for an observer outside the sphere is, similarly: $\\begin{split}\\mathbf {G}_\\mathrm {LNO}&(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\Theta (r > a_2) = \\frac{\\mathrm {i}\\omega }{4\\pi c}\\sum _{p\\ell m}\\left[A_{p\\ell m}^>(\\omega )\\mathbf {M}_{p\\ell m}(\\mathbf {r},k)\\mathbf {M}_{p\\ell m}(\\mathbf {r}^{\\prime },k)\\right.\\\\&\\left.+B_{p\\ell m}^>(\\omega )\\mathbf {N}_{p\\ell m}(\\mathbf {r},k)\\mathbf {N}_{p\\ell m}(\\mathbf {r}^{\\prime },k)\\right]\\Theta (r > a_2)\\Theta (r^{\\prime } > a_2)\\\\&+\\sqrt{\\epsilon _2}\\frac{\\mathrm {i}\\omega }{4\\pi c}\\sum _{p\\ell m}\\left[A_{p\\ell m}^<(\\omega )\\mathbf {M}_{p\\ell m}(\\mathbf {r},k)\\mathbf {\\mathcal {M}}_{p\\ell m}(\\mathbf {r}^{\\prime },\\sqrt{\\epsilon _2}k) \\right.\\\\&+\\left.B_{p\\ell m}^<(\\omega )\\mathbf {N}_{p\\ell m}(\\mathbf {r},k)\\mathbf {\\mathcal {N}}_{p\\ell m}(\\mathbf {r}^{\\prime },\\sqrt{\\epsilon _2}k)\\right]\\Theta (r > a_2)\\Theta (r^{\\prime } < a_2)\\\\&+ \\mathbf {G}_0(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )\\Theta (r > a_2)\\Theta (r^{\\prime } > a_2),\\end{split}$ with the forms of the unwieldy response functions $A_{p\\ell m}(\\omega )$ , $B_{p\\ell m}(\\omega )$ , $C_{p\\ell m}(\\omega )$ , and $D_{p\\ell m}(\\omega )$ described in Ref.",
"tai1994dyadic.",
"Further, $\\mathbf {G}_0(\\mathbf {r},\\mathbf {r}^{\\prime };\\omega )$ is the Green's function of free space.",
"To compare the responses of the NP and microsphere, we can see that $B_{p\\ell m}^<(\\omega )$ obeys the simple relation $\\lim _{k\\rightarrow 0}(\\sqrt{\\epsilon _2})^\\ell B_{p\\ell m}(\\omega ) - 1 = \\ell (1 - \\epsilon _2)/(\\ell \\epsilon _2 + \\ell + 1)$ .",
"This gives us a foundation from which to make quantitative comparisons between the oscillator parameters assigned to each response in either particle.",
"In detail, using the Drude-Lorentz dielectric model of Au from Section REF , we can see that the dipole response function $g_1(\\omega )$ describes the oscillations of three coupled modes, one which has primarily free-electron character (the dipole plasmon) two others which are mostly comprised of the material's interband resonances.",
"Explicitly, $g_{p1m}(\\omega ) = -\\frac{e^2}{a_1^3}\\left[\\frac{1}{2\\omega _1 \\mu _1}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _1}}{\\Omega _1 - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _1}}{\\Omega _1^* + \\omega }\\right) + \\sum _{i=1}^2\\frac{1}{2\\omega _{L_i} \\mu _{L_i}}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{L_i}}}{\\Omega _{L_i} - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{L_i}}}{\\Omega _{L_i}^* + \\omega }\\right)\\right],$ wherein the mass $\\mu $ , resonance frequency $\\omega $ , complex eigenvalue $\\Omega $ , and phase offset $\\psi $ of the plasmon are labeled with subscript 1, and the oscillator parameters of the fictitious oscillator are labeled with subscripts $L_i$ .",
"Explicit values of these parameters are given in Section REF .",
"As is also shown in Section REF , there are many modes of the LiNbO$_3$ sphere that have significant response magnitudes in the range of laser energies in which SHG is observed ($\\sim $ 2.3–2.4 eV, see Figure c).",
"To simplify their description, we use an expansion of the LiNbO$_3$ mode response functions: $\\begin{split}(\\sqrt{\\epsilon _2})^\\ell A_{p\\ell m}^<(\\omega ) - 1 &\\approx -\\frac{e^2}{a_2^3}\\sum _j\\frac{1}{2\\omega _{M\\ell j}\\mu _{M\\ell j}}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{M\\ell j}}}{\\Omega _{M\\ell j} - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{M\\ell j}}}{\\Omega _{M\\ell j}^* + \\omega }\\right), \\\\(\\sqrt{\\epsilon _2})^\\ell B_{p\\ell m}^<(\\omega ) - 1 &\\approx -\\frac{e^2}{a_2^3}\\sum _j\\frac{1}{2\\omega _{E\\ell j}\\mu _{E\\ell j}}\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{E\\ell j}}}{\\Omega _{E\\ell j} - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _{E\\ell j}}}{\\Omega _{E\\ell j}^* + \\omega }\\right),\\end{split}$ where the labels $M,E$ denote magnetic- and electric-type oscillator parameters and the indices $j$ label the different resonances of common angular symmetry $p,\\ell ,m$ but different node structure along the radial coordinate.",
"Because only a single mode of the set of modes $j$ lies in the energetic range of interest, we will drop the sum over $j$ and the corresponding labels in the following discussion.",
"With explicit value of the relevant LiNbO$_3$ oscillator parameters given in Section REF , the formal definitions of the oscillator moments $d_\\nu (\\omega )$ of the dipole plasmons oriented along the three cardinal axes $\\nu = r,\\theta ,\\phi $ as well the LiNbO$_3$ Mie multipoles $\\mathbf {\\beta }$ can be defined analytically.",
"In particular, with a vector spherical harmonic expansion of the laser field[15] $\\mathbf {E}_0(\\mathbf {r},\\omega ) = E_0\\left[\\pi \\delta (\\omega - \\omega _0) + \\pi \\delta (\\omega + \\omega _0)\\right]\\sum _\\ell \\mathrm {i}^\\ell \\frac{2\\ell + 1}{\\ell (\\ell + 1)}\\left[\\mathbf {\\mathcal {M}}_{1\\ell 1}(\\mathbf {r},k) - \\mathrm {i}\\mathbf {\\mathcal {N}}_{0\\ell 1}(\\mathbf {r},k)\\right],$ the second-order fields of a bare LiNbO$_3$ microsphere centered at the origin can be written as $\\begin{split}\\mathbf {E}_\\mathrm {sca}^{(2)}(\\mathbf {r},\\omega ) &= \\sum _{\\mathbf {\\beta }}\\frac{1}{a_2^{\\ell + 2}}\\left[\\rho _{\\mathbf {\\beta }}(\\omega )\\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r},k_{\\mathbf {\\beta }}) + \\rho _{\\mathbf {\\beta }}^*(-\\omega )\\mathbf {X}_{\\mathbf {\\beta }}^*(\\mathbf {r},k_{\\mathbf {\\beta }})\\right],\\\\\\mathbf {E}_0^{(2)}(\\mathbf {r},\\omega ) &= -\\mathrm {i}\\frac{\\omega _0^9}{4c^9}\\pi \\delta (\\omega - 2\\omega _0)\\epsilon _2(\\epsilon _2 - 1)^2E_0^2\\chi _2^{(2)}(\\omega _0,\\omega _0)\\\\&\\times \\sum _{\\mathbf {\\gamma }\\mathbf {\\alpha }\\mathbf {\\alpha }^{\\prime }}w_{\\mathbf {\\alpha }}(\\omega _0)w_{\\mathbf {\\alpha }^{\\prime }}(\\omega _0)I_{\\mathbf {\\gamma }\\mathbf {\\alpha }\\mathbf {\\alpha }^{\\prime }}(k_{\\mathbf {\\gamma }},\\sqrt{\\epsilon _2}k_0,\\sqrt{\\epsilon _2}k_0;0,a_2)\\mathbf {\\mathcal {X}}_{\\mathbf {\\gamma }}(\\mathbf {r},\\omega ) + \\mathrm {c.c.r.",
"}\\end{split}$ Here, as defined in the main text, $\\rho _{\\mathbf {\\beta }}(\\omega )$ are the multipole moment magnitudes of the Mie resonances of the sphere.",
"Their explicit form is complicated and is detailed below.",
"The index $\\mathbf {\\gamma } = \\lbrace T^{\\prime \\prime \\prime },p^{\\prime \\prime \\prime },\\ell ^{\\prime \\prime \\prime },m^{\\prime \\prime \\prime }\\rbrace $ is another collective Mie index like $\\mathbf {\\alpha }$ and $\\mathbf {\\beta }$ .",
"It labels modes that contribute to the field near $2\\omega _0$ but, unlike $\\mathbf {\\beta }$ , is not restricted to counting only modes with strongly resonant behavior.",
"In the calculations of this work, it is taken to label all Mie modes with $\\ell > 1$ up to a cutoff $\\ell = 12$ after which the contribution of successive terms in the sum is negligible.",
"Further, the term $\\mathrm {c.c.r.",
"}$ is simply the frequency reversed ($\\omega \\rightarrow -\\omega $ ) complex conjugate of the first term of the second equality above such that $\\mathbf {E}_0^{(2)}(\\mathbf {r},\\omega )$ satisfies the Fourier reality condition $\\mathbf {E}_0^{(2)*}(\\mathbf {r},\\omega ) = \\mathbf {E}_0^{(2)}(\\mathbf {r},-\\omega )$ .",
"The functions $w_{\\mathbf {\\alpha }}(\\omega )$ inside the definition of the second-order vacuum-like field are the weights $w_{\\mathbf {\\alpha }}(\\omega _0) ={\\left\\lbrace \\begin{array}{ll}\\mathrm {i}^\\ell \\sqrt{\\dfrac{2\\ell + 1}{2\\ell (\\ell + 1)}\\dfrac{(\\ell + 1)!",
"}{(\\ell - 1)!",
"}}C_{p\\ell m}^<(\\omega )R_\\ell (k_0,\\sqrt{\\epsilon _2}k_0;0,a_2), & T = M;\\\\[1.0em]\\!\\begin{aligned}-\\mathrm {i}^{\\ell + 1}\\sqrt{\\frac{2\\ell + 1}{2\\ell (\\ell + 1)}\\frac{(\\ell + 1)!",
"}{(\\ell - 1)!",
"}}&D_{p\\ell m}^<(\\omega )\\left[\\frac{\\ell + 1}{2\\ell + 1}R_{\\ell - 1}(k_0,\\sqrt{\\epsilon _2}k_0;0,a_2)\\right.\\\\&+\\left.",
"\\frac{\\ell }{2\\ell + 1}R_{\\ell + 1}(k_0,\\sqrt{\\epsilon _2}k_0;0,a_2) \\right]\\end{aligned}, & T = E;\\end{array}\\right.",
"}$ wherein $\\begin{split}R_\\ell (k,k^{\\prime };a,b) &= \\int _a^b r^2j_\\ell (kr)j_\\ell (k^{\\prime }r)\\;\\mathrm {d}r\\\\&= \\frac{r^2}{k^2 - k^{\\prime 2}}\\left[k^{\\prime 2}j_\\ell (kr)j_{\\ell - 1}(k^{\\prime }r) - k^2j_{\\ell - 1}(kr)j_\\ell (k^{\\prime }r)\\right]_a^b\\end{split}$ is an overlap integral over the radial components $j_\\ell (kr)$ of the fundamental mode functions.",
"Further, the functions $I_{\\mathbf {\\alpha }_1\\mathbf {\\alpha }_2\\mathbf {\\alpha }_3}$ are overlap triple integrals $\\begin{split}I_{\\mathbf {\\alpha }_1\\mathbf {\\alpha }_2\\mathbf {\\alpha }_3}(k_1,k_2,k_3;a,b) &= \\int _0^{2\\pi }\\int _0^\\pi \\int _a^b \\mathbf {\\mathcal {X}}_{\\mathbf {\\alpha }_1}(\\mathbf {r}^{\\prime },k_1)\\cdot \\left[\\mathbf {\\mathcal {X}}_{\\mathbf {\\alpha }_2}(\\mathbf {r}^{\\prime },k_2)\\cdot \\mathbf {1}_3\\cdot \\mathbf {\\mathcal {X}}_{\\mathbf {\\alpha }_3}(\\mathbf {r}^{\\prime },k_3)\\right]\\\\&\\times r^{\\prime 2}\\sin \\theta ^{\\prime }\\;\\mathrm {d}r^{\\prime }\\,\\mathrm {d}\\theta ^{\\prime }\\,\\mathrm {d}\\phi ^{\\prime },\\end{split}$ wherein $\\mathbf {\\mathcal {X}}_{Tp\\ell m}(\\mathbf {r},k) = \\mathbf {\\mathcal {M}}_{p\\ell m}(\\mathbf {r},k)\\delta _{T,M} + \\mathbf {\\mathcal {N}}_{p\\ell m}(\\mathbf {r},k)\\delta _{T,E}$ .",
"Using the definition of the dipole polarizabilities $\\alpha _\\nu (\\omega )$ given in Section REF , we can define the motion of each component of a single Au dipole and each Mie resonance in the absence of their mutual coupling as $\\begin{split}d_\\nu (\\omega ) &= \\frac{e^2}{2\\omega _1\\mu _\\nu }\\left(\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _1}}{\\Omega _1 - \\omega } + \\frac{\\mathrm {e}^{-\\mathrm {i}\\psi _1}}{\\Omega _1^* + \\omega }\\right)\\hat{\\mathbf {e}}_\\nu \\cdot \\mathbf {E}_0^{(2)}(\\mathbf {r}_0,\\omega ),\\\\\\rho _{\\mathbf {\\beta }}(\\omega ) &= \\frac{e^2a_2^{\\ell -2}}{2\\omega _{\\mathbf {\\beta }}\\mu _{\\mathbf {\\beta }}}\\frac{\\mathrm {e}^{\\mathrm {i}\\psi _{\\mathbf {\\beta }}}}{\\Omega _{\\mathbf {\\beta }} - \\omega }E_0\\pi \\delta (\\omega - 2\\omega _0)C_{\\mathbf {\\beta }}(\\omega _0).\\end{split}$ In accordance with the main text, the dipole is assumed to exist at $\\mathbf {r}_0 = (a_1 + a_2)\\hat{\\mathbf {r}}(\\theta _0,\\phi _0)$ and to be driven by the second-order vacuum-like field.",
"We ignore any modifications to $\\mathbf {E}_0^{(2)}(\\mathbf {r},\\omega )$ that are produced by $d_\\nu (\\omega )$ .",
"The Mie resonances of the LiNbO$_3$ are driven through the upconversion process, as detailed by the constants $\\begin{split}C_{\\mathbf {\\beta }}(\\omega _0) &= \\mathrm {i}\\frac{\\omega _0^9}{4c^9}\\frac{\\epsilon _2^\\frac{3}{2}(\\epsilon _2 - 1)^2}{(\\sqrt{\\epsilon _2})^\\ell }E_0\\chi _2^{(2)}(\\omega _0,\\omega _0)\\\\&\\times \\sum _{\\mathbf {\\alpha }\\mathbf {\\alpha }^{\\prime }}w_{\\mathbf {\\alpha }}(\\omega _0)w_{\\mathbf {\\alpha ^{\\prime }}}(\\omega _0)I_{\\mathbf {\\beta }\\mathbf {\\alpha }\\mathbf {\\alpha ^{\\prime }}}(\\sqrt{\\epsilon _2}k_{\\mathbf {\\beta }},\\sqrt{\\epsilon _2}k_0,\\sqrt{\\epsilon _2}k_0;0,a_2).\\end{split}$ Finally, to reproduce the equations of motion of Eq.",
"(), we must define the external forces acting on the particle moments and introduce the coupling forces to Eq.",
"(REF ).",
"The former can be quickly written as $F_{1\\nu }(\\omega ) = e\\hat{\\mathbf {e}}_\\nu \\cdot \\mathbf {E}_0^{(2)}(\\mathbf {r}_0,\\omega )$ and $F_{2\\mathbf {\\beta }}(\\omega ) = eE_0\\pi \\delta (\\omega - 2\\omega _0)C_{\\mathbf {\\beta }}(\\omega )$ .",
"The latter arise from the interaction energy $U(t) = -\\mathbf {d}(t)\\cdot \\mathbf {E}_\\mathrm {sca}^{(2)}(\\mathbf {r}_0,t)$ , where $\\mathbf {d}(t) = \\sum _\\nu d_\\nu (t)\\hat{\\mathbf {e}}_\\nu $ .",
"With the identities $\\begin{split}-a_2^{\\ell - 1}\\frac{\\partial U(t)}{\\partial \\rho _{\\mathbf {\\beta }}(t)} & = \\sigma _{\\mathbf {\\beta }\\nu }d_\\nu (t),\\\\-a_2^{\\ell - 1}\\frac{\\partial U(t)}{\\partial d_{\\nu }(t)} &= \\sigma _{\\mathbf {\\beta }\\nu }\\rho _{\\mathbf {\\beta }}(t),\\end{split}$ one is immediately delivered the equations of motion in the main text with $\\sigma _{\\mathbf {\\beta }\\nu } = (e^2/a_2^3)\\hat{\\mathbf {e}}_\\nu \\cdot \\mathbf {X}_{\\mathbf {\\beta }}(\\mathbf {r}_0,k_{\\mathbf {\\beta }})$ .",
"A detailed description of the solutions to the equations of motion are given in Section , and an analysis of the contribution of the various modes of the LiNbO$_3$ sphere to enhancement signal is shown in Figure REF .",
"In closing, we note that, as the solutions to the equations of motion discussed in the main text and highlighted in Section involve a second perturbation expansion of the equations of motion of each Mie resonance, two sets of superscripts arise that correspond to two separate expansions.",
"In order to clarify the notation, we can see that no terms in the solution to the wave equation are kept beyond second order, such that we can replace superscripted variable names with script names.",
"Further, we can drop the superscripts of the first-order terms altogether, such that, elsewhere in the SI and main text, we let $\\mathbf {E}_\\mathrm {sca}^{(2)}\\rightarrow \\mathbf {\\mathcal {E}}_\\mathrm {sca}$ , $\\mathbf {E}_0^{(2)}\\rightarrow \\mathbf {\\mathcal {E}}_0$ , and $\\mathbf {E}_\\mathrm {sca}^{(1)}\\rightarrow \\mathbf {E}_\\mathrm {sca}$ ."
]
] | 2212.05569 |
[
[
"Designing Human-Centered Algorithms for the Public Sector: A Case Study\n of the U.S. Child-Welfare System"
],
[
"Abstract The U.S. Child Welfare System (CWS) is increasingly seeking to emulate business models of the private sector centered in efficiency, cost reduction, and innovation through the adoption of algorithms.",
"These data-driven systems purportedly improve decision-making, however, the public sector poses its own set of challenges with respect to the technical, theoretical, cultural, and societal implications of algorithmic decision-making.",
"To fill these gaps, my dissertation comprises four studies that examine: 1) how caseworkers interact with algorithms in their day-to-day discretionary work, 2) the impact of algorithmic decision-making on the nature of practice, organization, and street-level decision-making, 3) how casenotes can help unpack patterns of invisible labor and contextualize decision-making processes, and 4) how casenotes can help uncover deeper systemic constraints and risk factors that are hard to quantify but directly impact families and street-level decision-making.",
"My goal for this research is to investigate systemic disparities and design and develop algorithmic systems that are centered in the theory of practice and improve the quality of human discretionary work.",
"These studies have provided actionable steps for human-centered algorithm design in the public sector."
],
[
"Introduction",
"Over the past two decades, several high-stakes decision-making domains such as the child-welfare system (CWS), criminal justice system, education, and medical services have increasingly turned towards risk assessment algorithms as a means to standardize and improve decision-making.",
"Facing severely limited resources and new dilemmas in the form of burdensome workloads and high staff turnover, most human services agencies have also turned towards algorithms as they purportedly promise to reduce costs and provide greater efficiencies in public policy and social services delivery.",
"CWS has also been the center of public and media scrutiny because of the harm caused to children who are removed from the care of their parents [5].",
"On the other hand, CWS also receives severe criticism and media attention for child abuse tragedies where the system failed to remove and protect a child [6].",
"This has further mounted the pressure on CWS in several states in the United States (U.S.) to employ structured decision-making tools (and more recently, algorithmic decision-making) to prove that they are employing evidence-based, consistent, and objective decision-making processes [12], [9].",
"Decades of research in clinical psychology and medicine exhibit that statistical decision-making outperforms human experts in prediction tasks [8], [2] and is often cited as a justification for introducing algorithms in the public sector.",
"However, as illustrated by my CHI 2020 literature review [10], CWS poses its own challenges with respect to the technical (i.e., quality of data, reliability/validity of constructs), social and cultural (i.e., workers' interactions with algorithms, impact of systemic constraints), theoretical (i.e., what is empirical risk vs. theoretical risk?",
"), and societal (i.e., impact of algorithms on communities and decision-making ecosystem) implications of algorithmic decision-making.",
"Abebe et al.",
"[1] highlight that much of the computational research that focuses on fairness, bias, and accountability on algorithmic systems continues to formulate “fair” technical solutions while failing to address deeper systemic and structural injustices.",
"Through my dissertation work, I bring attention back to the sociotechnical and highlight social problems in child-welfare and how these problems become embedded in algorithmic systems.",
"Through the studies discussed below, my dissertation assumes the dual roles of computing as rebuttal where I highlight the technical limitations and feasibility of risk assessment algorithms, and of computing as synecdoche by uncovering systemic complexities and social problems that directly impact families.",
"This dissertation will also seek to make contributions at the intersection of gaps highlighted by the literature review and recommend solutions centered in strength- and asset-based approaches [4], [15], [3] that will improve the state of current algorithmic interventions, enhance child-welfare practice, and improve street-level decisions mediated through algorithms.",
"Therefore, my dissertation answers the following overarching research questions: [leftmargin=*] RQ1: (a) How do caseworkers interact with algorithms in their daily lives, and (b) How does the implementation of a given algorithm impact algorithmic decision-making, human discretion, and bureaucratic processes?",
"RQ2: (a) Can computational text analysis help uncover invisible patterns of human discretionary work conducted within the constraints of bureaucracy, and (b) can these theoretical signals derived from casenotes help contextualize algorithmic decisions?",
"RQ3: (a) How is \"risk\" quantified empirically within algorithmic systems as compared to how it is understood theoretically within the domain?, and (b) how do risk factors fluctuate and mediate each other throughout the child-welfare process and its implications for algorithmic decision-making?",
"To answer these questions, I will conduct the four studies described below.",
"Examining the nature of practice and street-level discretionary work as well as the impact of systemic and policy-related barriers on decision-making (human or algorithmic) will allow us to develop technical solutions that operate within these constraints and augment the quality of human discretionary work."
],
[
"Research Overview",
"In the following sections, I provide a short overview of my four dissertation studies."
],
[
"This study constitutes an in-depth ethnographic case study that I conducted at a child-welfare agency in Milwaukee, Wisconsin [11].",
"It was published at CSCW '2021 and was presented at the conference.",
"It contributes to the theoretical and social and cultural gaps highlighted by the literature review.",
"Algorithms in the public sector is a domain in its own right and requires a cohesive framework that explains how algorithms interact with bureaucracy and human discretion.",
"First, drawing upon theories from Human-Computer Interaction (HCI), Science and Technology Studies (STS), and Public Administration (PA), we propose a theoretical framework for algorithmic decision-making for the public sector (ADMAPS) which accounts for the interdependencies between human discretion, bureaucratic processes, and algorithmic decision-making.",
"The framework is then validated through a case study of algorithms in use at the agency.",
"Second, the ethnography uncovers the daily algorithmic practices of caseworkers, what causes them to (dis)trust an algorithm, and how they navigate through different algorithmic systems especially when they do not account for policy and systemic barriers or resource constraints at the agency."
],
[
"This study seeks to utilize sources of information that have been hard to quantify so far, namely, caseworker narratives.",
"Child-welfare caseworkers are trained in writing detailed casenotes about their interactions with families and case progress through the life of the case.",
"This study contributes to the technical and theoretical gaps illustrated by the literature review by deriving rich qualitative signals from casenotes using natural language processing techniques such as topic modeling.",
"We are specifically analyzing casenotes written by the Family Preservation Services (FPS) team that works closely with birth parents in their efforts to achieve reunification.",
"Casenotes offer a rich description of decisions, relationships, conflicts, personas, as well as policy-related and systemic barriers.",
"Analyzing these casenotes offers a unique lens towards understanding the workings of a child-welfare team trying to achieve reunification; one of the primary policy-mandated goals of CWS.",
"Theoretical signals derived from casenotes will also help contextualize the quantitative structured assessments [14] and highlight patterns of invisible labor conducted by caseworkers and systemic constraints and power asymmetries that impact all decisions.",
"This study was published at CHI'2022 [13]."
],
[
"Drawing upon findings from a two-year ethnography conducted at a child-welfare agency, we highlight how algorithmic systems are embedded within a complex decision-making ecosystem at critical points of the child-welfare process.",
"In our prior study [11], we focused on the micro-interactions between the dimensions of human discretion, algorithmic decision-making, and bureaucratic processes to understand why algorithms failed (or succeeded) to offer utility to child-welfare staff and their impact on the quality of human discretionary work.",
"In this study, we critically investigate the macro-interactions between these three elements to assess the impact of algorithmic decision-making on the nature of practice, the organization, as well as the interactions between human discretion and bureaucratic processes to understand how the nature of street-level decision-making is changing and whether algorithms are living up to the promises of cost-effective, consistent, and fair decision-making.",
"This study contributes to the social and cultural and societal gaps highlighted by the literature review by unpacking how the decision-making ecosystem within the public sector is changing.",
"It also depicts the case study of an algorithm that offers higher utility to caseworkers, however, required significant investments from the agency leadership to bring about that ecological change in decision-making where the algorithmic system plays an essential role.",
"This manuscript is currently under review for the ACM Journal on Responsible Computing in October 2022."
],
[
"Risk assessment algorithms have been adopted by several public sector agencies to make high-stakes decisions about human lives.",
"However, there is a mismatch between how risk is quantified empirically based on administrative data versus how it is understood theoretically within the domain.",
"Public servants such as caseworkers are essentially risk workers who are tasked with assessing and managing risks, translating risk in different contexts, and conducting care work in the context of risk [7].",
"However, this risk work is increasingly mediated through algorithmic systems with a mismatch between empirical risk and theoretical risk that leads to unreliable decision-making and conflicts in practice.",
"This study contributes to the theoretical and societal gaps highlighted by the literature review.",
"Algorithms model “risk” based on individual client characteristics to identify clients most in need.",
"However, this understanding of risk is primarily based on easily quantifiable risk factors that present an incomplete and biased perspective of clients.",
"In this study, I conducted computational narrative analysis of child-welfare casenotes and draw attention toward deeper systemic risk factors that are hard to quantify but directly impact families and street-level decision-making.",
"Beyond individual risk factors, the system itself poses a significant amount of risk to families where parents are over-surveilled by caseworkers and experience a lack of agency in decision-making.",
"I also problematize the notion of risk as a static construct by highlighting temporality and mediating effects of different risk and protective factors and show that any temporal point estimate of risk will produce biased predictions.",
"I also draw caution against using casenotes in NLP-based algorithms by unpacking their limitations and biased embedded within them.",
"This study is currently under submission at CHI'2023."
],
[
"Research Progress & GROUP 2022 DC Participation",
"All four studies have been completed with Study 1 and Study 2 published and presented at their respective conferences.",
"The manuscript for Study 3 is currently under submission and review for the ACM Journal on Responsible Computing.",
"Study 4 is currently under submission and review at CHI'2023."
],
[
"EXPECTED OUTCOMES",
"My dissertation assumes the dual roles of computing as rebuttal and computing as synecdoche and will make three contributions.",
"First, I highlight the technical limitations and feasibility of risk assessment algorithms and draw attention to the systemic complexities and structural issues that directly impact families.",
"Second, I developed a theoretical framework for algorithmic decision-making in the public sector that accounts for the complex interdependencies between human discretion, bureaucratic processes, and algorithmic decision-making.",
"Third, I show how computational narrative analysis can help uncover patterns of invisible labor, systemic constraints, and power asymmetries and problematize the empirical notion of risk by highlighting the temporality of risk as well as systemic risk factors that are hard to quantify but directly impact street-level decision-making."
]
] | 2212.05556 |
[
[
"A global adaptive velocity space for general discrete velocity framework\n in predictions of rarefied and multi-scale flows"
],
[
"Abstract The rarefied flow and multi-scale flow are crucial for the aerodynamic design of spacecraft, ultra-low orbital vehicles and plumes.",
"By introducing a discrete velocity space, the Boltzmann method, such as the discrete velocity method and unified methods, can capture complex and non-equilibrium velocity distribution functions (VDFs) and describe flow behaviors exactly.",
"However, the extremely steep slope and high concentration of the gas VDFs in a local particle velocity space make it very difficult for the Boltzmann method with structured velocity space to describe high speed flow.",
"Therefore, the adaptive velocity space (AVS) is required for the Boltzmann solvers to be practical in complex rarefied flow and multi-scale flow.",
"This paper makes two improvements to the AVS approach, which is then incorporated into a general discrete velocity framework, such as the unified gas-kinetic scheme.",
"Firstly, a global velocity mesh is used to prevent the interpolation of the VDFs at the physical interface during the calculation of the microscopic fluxes, maintaining the program's high level of parallelism.",
"Secondly, rather than utilizing costly interpolation, the VDFs on a new velocity space were reconstruction using the ``consanguinity\" relationship.",
"In other words, a split child node's VDF is the same as its parent's VDF, and a merged parent's VDF is the average of its children's VDFs.",
"Additionally, the discrete deviation of the equilibrium distribution functions is employed to maintain the proposed method's conservation.",
"Moreover, an appropriate set of adaptive parameters is established to enhance the automation of the proposed method.",
"Finally, a number of numerical tests are carried out to validate the proposed method."
],
[
"Introduction",
"For multi-scale flows, the physical scale varies over time and place.",
"The Knudsen (Kn) number, which is defined as Kn = $\\lambda /L$ , is used to determine the degree of gas rarefaction, where $\\lambda $ is the mean free path of the gas, and $L$ is the characteristic length of the object in the flow.",
"According to the Kn number, the flow can be qualitatively categorized into continuum flow regime (Kn $<$ 0.001), slip flow regime (0.001 $<$ Kn $<$ 0.1), transitional flow regime (0.1 $<$ Kn $<$ 10) and free molecular flow regime (Kn $>$ 10).",
"Multi-scale flow is a term with a broad engineering background that refers to the coexistence of different flow regimes.",
"For instance, considering the flow of near-space vehicles and micro-/nano-electro-mechanical systems (MEMS/NEMS), several flow regimes including continuum flow, slip flow, and even free molecule flow will exist in the same computational domain, and the local Kn number may fluctuate by many orders of magnitude.",
"This multi-scale nature in both time and space makes the flows very difficult to be modeled and predicted.",
"In terms of flow numerical prediction, the traditional method of computational fluid dynamics based on Navier-Stokes (N-S) equations is suitable for continuum flows in the airspace and macro scale, whereas the model molecular method based on rarefied gas dynamics, represented by the direct simulation Monte Carlo (DSMC) method, performs well for rarefied flows and micro-scale flows.",
"However, each of the two methods has its difficulties in simulating multi-scale flow numerically , , .",
"Motivated by the N-S method's and DSMC method's successes in continuum flow and rarefied flow, researchers have combined the two methods through flow-field zoning to address the multi-scale flow problem, and the overlapping hybrid particle-continuum method is proposed , , .",
"The rarefied computing domain and the continuum computing domain must overlap to improve information transmission.",
"To categorize the flow domain, empirical or semi-empirical criteria are typically used , .",
"The multi-scale problem can be partially resolved by the coupling approach, but it is still difficult to split the computational domain precisely and integrate several flow regimes reasonably .",
"In recent years, a class of multi-scale unified method, such as unified gas-kinetic scheme (UGKS) , , discrete unified gas-kinetic scheme (DUGKS) , , gas-kinetic unified algorithm (GKUA) , and the improved discrete velocity method (IDVM) , , have been proposed, making it possible to solve complex multi-scale flows by using the same numerical method.",
"With a local analytical integral solution of the kinetic model equation, the particle transport is coupled with particle collision processes in the UGKS, so that the time step and mesh size are independent of the collision time and the mean free path, respectively.",
"The DUGKS is another multi-scale method based on the same physical process as the UGKS.",
"Instead of the analytical integral solution, the characteristic difference solution of the kinetic model equation is adopted to reconstruct the multi-scale numerical flux at a cell interface.",
"Therefore, the DUGKS can be regarded as a special version of UGKS.",
"After a decade of development, many numerical techniques have been developed and implemented in the UGKS and DUGKS to increase the computational efficiency and reduce memory cost, such as unstructured mesh computation , , moving grids , , velocity space adaptation , memory reduction , wave–particle adaptation , implicit algorithms , , , parallelization algorithm , and further simplification and modification , , , .",
"With these improvement, the UGKS and DUGKS have been successfully applied to a variety of flow problems in different flow regimes, such as micro flows , compressible flows , , jet flow , , multi-phase flows , gas-solid flows , and gas mixture systems .",
"Besides flow problems, the UGKS and DUGKS were also extended to multi-scale transport problems such as radiative transfer , phonon heat transfer , plasma physics , neutron transport , granular flow , and turbulent flow , .",
"In the discrete velocity method (DVM) framework, a discrete velocity space is adopted to resolve the velocity distribution function (VDF), which contains detailed information about particle motion in a 6-dimensional phase space, for all points of physical space.",
"Because of the wide spreading of the particle distribution in high-speed flows and the narrow-kernel particle distribution in cases with a large Kn number, the velocity space must have a local high resolution and cover a huge domain.",
"The memory cost will be unbearable if a conventional structured velocity space is used .",
"The authors of , , suggested to use a unstructured velocity space that is manually refined as necessary before numerical simulation.",
"However, it could be challenging to generate an unstructured velocity space using as few elements as possible while still effectively capturing the flow behavior, especially for newcomers.",
"In recent years, the adaptive velocity space (AVS), which can automatically generate the corresponding velocity space according to the flow variables, has been proposed.",
"Depending on whether the velocity space varies over time, the AVS technique can be divided into two groups, i.e.",
"“fixed\" and “unfixed\".",
"An AVS was proposed by Aristov for the simulation of the 1D shock structure.",
"However, the technique has never been extended and is quite specific to this test case.",
"Baranger et al.",
", employ a compressible N-S pre-simulation result to locally refine the velocity mesh wherever it is necessary and to coarsen it elsewhere at the start of the numerical experiment.",
"Then the velocity mesh is fixed until the end of the simulation.",
"With the help of this technique, the amount of discrete velocity points can be significantly decreased.",
"However, the accuracy of this “fixed\" velocity mesh depends greatly on the N-S results, which are unreliable for rarefied and multi-scale flows.",
"Chen et al.",
"and Arslanbekov et al.",
"proposed a “unfixed\" local velocity mesh with a different velocity space for different time and space positions.",
"An adaptive physical mesh is also used in Arslanbekov's approach.",
"Kolobov et al.",
"simultaneously developed a similar “unfixed\" local AVS for plasma physics and rarefied gas dynamics.",
"Bernard et al.",
"proposed a simple adaptation strategy that the local velocity mesh is only a sub-set of a global velocity mesh.",
"And the domain of the velocity mesh is determined by the local temperature.",
"However, these adaptive meshes are based on the same background velocity mesh, whose bound are fixed at the beginning of the computation and are constant in time and space.",
"It is therefore attractive to allow for a dynamic adaptation of these bounds in time and space.",
"Correctly estimating the borders can be difficult, but Brull and Mieussens showed that it is achievable by using the conservation laws.",
"This method was only proposed for one dimensional velocity variable so far.",
"The exchange of information between the velocity meshes of two adjacent physical cells is one of the challenges for local velocity mesh, and interpolation is necessary in the transport step to match the velocity points in the physical interface.",
"This interpolation can lead to approximation errors and possible increasing computational time .",
"On the other hand, the interpolation of the VDFs between two adaptive velocity meshes during the particle-redistribution over different velocity level can also lead to approximation errors and possible increasing computational time, and the conservation of the solution must be carefully maintained.",
"Additionally, the parallel computing of the local velocity mesh is also more challenging.",
"In this paper, an “unfixed\" global AVS technique for the general DVM framework is proposed, which is interpolation-free and simple to implement.",
"The global velocity mesh ensures that there is no interpolation required for the calculation of microscopic flux at the physical interface in the transport step.",
"And excellent parallel performance of algorithm is maintained.",
"Furthermore, a novel “consanguinity\" relationship is employed to avoid the interpolation in the reconstruction of VDFs.",
"Thanks to this “consanguinity\" relationship, it is not necessary to know the geometric relationship between the velocity elements, which makes the data structure quite simple.",
"Finally, the discrete deviation of the equilibrium distribution functions are employed to maintain the proposed method's conservation.",
"The rest of the paper is organized as follows: The conservative implicit unified gas-kinetic scheme is introduced in section 2 along with a predicted equilibrium state and simplified multi-scale flux.",
"The details of the global AVS technique are presented in section 3.",
"In section 4, a series of numerical experiments are conducted to verify the proposed methods.",
"The conclusion is the final section."
],
[
"Conservative implicit unified gas-kinetic scheme",
"This section will provide a quick overview of the conservative implicit UGKS (IUGKS) ."
],
[
"2.1 Gas-kinetic models",
"The Boltzmann equation serves as the basis for the UGKS: $\\frac{\\partial f}{\\partial t}+\\mathbf {\\xi }\\cdot \\nabla f=\\Omega $ where $f=f\\left( \\mathbf {x},\\mathbf {\\xi },\\mathbf {\\eta },e,t \\right)$ is the VDFs for particles moving in D-dimensional physical space with a velocity of $\\mathbf {\\xi }=\\left( \\xi _{1}^{{}},...,\\xi _{D}^{{}} \\right)$ at position $\\mathbf {x}=\\left( x_{1}^{{}},...,x_{D}^{{}} \\right)$ and time t. Here, $\\mathbf {\\eta }=\\left( \\xi _{D+1}^{{}},...,\\xi _{3}^{{}} \\right)$ is the dummy velocity (with the degree of freedom $L=3-D$ ) consisting of the remaining components of the translational velocity of particles in three-dimensional space; $e$ is a vector of $K$ elements representing the internal degree of freedom of molecules; $\\Omega $ is the collision operator.",
"Many kinetic models, including the Bhatnagar-Gross-Krook (BGK) collision model , the Shakhov model , the ellipsoidal statistical model (ES model) , and the Rykov model , have been proposed and used in the research of rarefied flows to simplify the collisional model of the full Boltzmann equation.",
"To build kinetic models and develop the corresponding gas-kinetic schemes, numerous attempts have been made.",
"The modeling of monatomic gases only considers the non-equilibrium transitional energy.",
"In addition to the three translational degrees of freedom for the diatomic molecule, there are internal degrees of freedom, i.e., two degrees of freedom for rotation at room temperature.",
"The degrees of freedom associated with vibrations begin to arise at a temperature higher than 1000 K. Thus, only the degrees of freedom for translation and rotation are considered in this study, and the UGKS is established by using the Shakhov model for monatomic gas and the Rykov model for diatomic gas.",
"The following presents the expression for the control equation of the Boltzmann-BGK type: $\\frac{\\partial f}{\\partial t}+\\mathbf {\\xi }\\cdot \\nabla f\\equiv \\Omega =\\frac{{{g}^{*}}-f}{\\tau }$ where $\\tau $ is the relaxation time for the translational degree of freedom and can be calculated as $\\tau =\\mu /p_t$ , where $\\mu $ and $p_t$ are the viscosity and pressure determined by the translational temperature $T_t$ instead of the equilibrium temperature $T$ .",
"The energy equalization theorem is satisfied by the equilibrium temperature $T$ , the translational temperature $T_t$ , and the rotation temperature $T_r$ .",
"${g}^{*}$ represents the Maxwell equilibrium VDFs, Shakhov equilibrium VDFs, or Rykov equilibrium VDFs.",
"The variable hard sphere (VHS) model is adopted in this study to determine viscosity $\\tau =\\frac{\\mu }{p_t}=\\frac{{{\\mu }}}{\\rho R{{T}_{t}}}=\\frac{1}{\\rho R{{T}_{t}}}{{\\mu }_{0}}{{\\left( \\frac{{{T}_{t}}}{{{T}_{0}}} \\right)}^{\\omega }}$ where $\\rho $ and $R$ are the density and the specific gas constant, respectively.",
"The viscosity of the freestream flow $\\mu _0$ is correlated with the gas mean free path $\\lambda $ in the following way: $\\lambda =\\frac{2\\mu \\left( 5-2\\omega \\right)\\left( 7-2\\omega \\right)}{15\\rho {{\\left( 2\\pi RT \\right)}^{1/2}}}$ where T is temperature.",
"In the non-dimensional system, the Kn number is defined as: $Kn=\\frac{\\lambda }{{{L}_{ref}}}\\text{=}\\frac{2\\mu \\left( 5-2\\omega \\right)\\left( 7-2\\omega \\right)}{15\\rho {{L}_{ref}}{{\\left( 2\\pi RT \\right)}^{1/2}}}=\\sqrt{\\frac{\\gamma }{\\pi }}\\frac{\\sqrt{2}\\left( 5-2\\omega \\right)\\left( 7-2\\omega \\right)}{15}\\frac{Ma}{\\operatorname{Re}}$ where Ma = $U/\\sqrt{\\gamma RT}$ and Re = $\\rho UL_{ref}/\\mu $ are the Mach (Ma) number and Reynolds (Re) number, respectively, and $\\gamma $ is the specific heat ratio.",
"${L}_{ref}$ is the characteristic length.",
"According to the inter-molecular interaction model, $\\omega $ is set to 0.5, 0.81 and 0.74 for the hard sphere model, ideal argon and nitrogen, respectively."
],
[
"2.2 Reduced gas-kinetic models",
"The discrete velocity space should be used to record the free transit of molecules, which is dependent only on the D-dimensional particle velocity $\\mathbf {\\xi }$ and is not related to $\\mathbf {\\eta }$ and $e$ (for diatomic gas).",
"The following reduced VDFs are used in the current numerical scheme to prevent discretizing $\\mathbf {\\eta }$ and $e$ : $\\begin{aligned}& G\\left( t,\\mathbf {x},\\mathbf {\\xi } \\right)=m\\int {fded\\mathbf {\\eta }} \\\\& H\\left( t,\\mathbf {x},\\mathbf {\\xi } \\right)=m\\int {{{\\eta }^{2}}fded\\mathbf {\\eta }} \\\\& R\\left( t,\\mathbf {x},\\mathbf {\\xi } \\right)=\\int {efded\\mathbf {\\eta }} \\\\\\end{aligned} $ The physical meaning of $G$ , $H$ , and $R$ is the distribution of the mass, translational internal energy, and rotating energy in the dummy velocity space and rotational energy space, respectively.",
"The quasi-linear feature of the model equation is advantageous for this condensed treatment.",
"Only the simplified VDFs $G$ and $H$ are required for monatomic gases.",
"The reduced VDF $R$ for rotational energy will also be introduced for diatomic gases.",
"Note that $R$ will inevitably vanish in three dimensions.",
"Then, the macroscopic variables can be expressed as: $\\mathbf {W}=\\left( \\begin{matrix}\\rho \\\\\\rho \\mathbf {U} \\\\\\rho E \\\\\\rho {{E}_{rot}} \\\\\\end{matrix} \\right)=\\int {\\left( \\begin{matrix}G \\\\\\mathbf {\\xi }G \\\\\\frac{1}{2}\\left( {{\\xi }^{2}}G+H \\right)\\text{+}R \\\\R \\\\\\end{matrix} \\right)}d\\mathbf {\\xi } $ where, $\\rho E=\\rho u_{{}}^{2}+\\rho \\varepsilon $ is the total energy density, $\\rho \\varepsilon =\\rho c_{V}^{{}}T$ is the inertial energy density, and $\\rho {{E}_{rot}}$ is the rotational energy density.",
"The translational heat flux, rotational heat flux, and total heat flux are expressed as: $\\begin{aligned}& {{\\mathbf {q}}_{t}}=\\frac{1}{2}\\int {\\mathbf {c}\\left( {{c}^{2}}G+H \\right)}d\\mathbf {\\xi } \\\\& {{\\mathbf {q}}_{r}}=\\int {\\mathbf {c}R}d\\mathbf {\\xi } \\\\& \\mathbf {q}={{\\mathbf {q}}_{t}}+{{\\mathbf {q}}_{r}} \\\\\\end{aligned} $ where $\\mathbf {c}=\\mathbf {\\xi }-\\mathbf {U}$ is the peculiar velocity.",
"The governing equation of the reduced VDF is: $\\begin{aligned}& \\frac{\\partial G}{\\partial t}+\\mathbf {\\xi }\\cdot \\frac{\\partial G}{\\partial \\mathbf {x}}=\\frac{{{g}^{G}}-G}{\\tau } \\\\& \\frac{\\partial H}{\\partial t}+\\mathbf {\\xi }\\cdot \\frac{\\partial H}{\\partial \\mathbf {x}}=\\frac{{{g}^{H}}-H}{\\tau } \\\\& \\frac{\\partial R}{\\partial t}+\\mathbf {\\xi }\\cdot \\frac{\\partial R}{\\partial \\mathbf {x}}=\\frac{{{g}^{R}}-R}{\\tau } \\\\\\end{aligned}$ For simplicity, Eq.",
"REF can be rewritten as $\\frac{\\partial \\phi }{\\partial t}+\\mathbf {\\xi }\\cdot \\frac{\\partial \\phi }{\\partial \\mathbf {x}}=\\frac{{{g}^{\\phi }}-\\phi }{\\tau }$ where $\\phi $ stands for $G$ , $H$ and $R$ .",
"For monatomic gas flow, the Shakhov equilibrium is: $\\begin{aligned}& {{g}^{G}}=g_{{}}^{eq}+G_{\\Pr }^{{}} \\\\& {{H}^{G}}=H_{{}}^{eq}+H_{\\Pr }^{{}} \\\\\\end{aligned}$ with $G_{\\Pr }^{{}}=\\left( 1-\\Pr \\right)\\frac{\\mathbf {c}\\cdot \\mathbf {q}}{5pRT}\\left( \\frac{c_{{}}^{2}}{RT}-D-2 \\right)g_{{}}^{eq}$ $H_{{}}^{eq}=\\left( K+3-D \\right)RTg_{{}}^{eq}$ $H_{\\Pr }^{{}}=\\left( 1-\\Pr \\right)\\frac{\\mathbf {c}\\cdot \\mathbf {q}}{5pRT}\\left[ \\left( \\frac{c_{{}}^{2}}{RT}-D \\right)\\left( K+3-D \\right)-2K \\right]RTg_{{}}^{eq}$ where the Prandtl (Pr) number equals 2/3, and ${g}^{eq}$ represents the Maxwell equilibrium: ${{g}^{eq}\\left( {{T}} \\right)}=\\frac{\\rho }{{{\\left( 2\\pi RT \\right)}^{D/2}}}\\exp \\left[ -\\frac{{(\\mathbf {\\xi }-\\mathbf {U})^{2}}}{2RT} \\right]$ For diatomic gas flow, the Rykov equilibrium is: $\\begin{aligned}& {{g}^{G}}=\\left( 1-\\frac{1}{{Z}_{rot}} \\right){{G}^{t}}+\\frac{1}{{Z}_{rot}}{{G}^{r}} \\\\& {{g}^{H}}=\\left( 1-\\frac{1}{{Z}_{rot}} \\right){{H}^{t}}+\\frac{1}{{Z}_{rot}}{{H}^{r}} \\\\& {{g}^{R}}=\\left( 1-\\frac{1}{{Z}_{rot}} \\right){{R}^{t}}+\\frac{1}{{Z}_{rot}}{{R}^{r}} \\\\\\end{aligned}$ with $\\begin{aligned}& {{G}^{t}}={{g}^{eq}}\\left( {{T}_{t}} \\right)\\left[ 1+\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{t}}}{15{{p}_{t}}R{{T}_{t}}}\\left( \\frac{{{c}^{2}}}{R{{T}_{t}}}-D-2 \\right) \\right] \\\\& {{G}^{r}}={{g}^{eq}}\\left( T \\right)\\left[ 1+{{\\omega }_{0}}\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{t}}}{15pRT}\\left( \\frac{{{c}^{2}}}{RT}-D-2 \\right) \\right] \\\\\\end{aligned} $ $ \\begin{aligned}& {{H}^{t}}=R{{T}_{t}}{{g}^{eq}}\\left( {{T}_{t}} \\right) \\left( 3-D \\right)\\left[\\left( 1+\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{t}}}{15{{p}_{t}}R{{T}_{t}}}\\left( \\frac{{{c}^{2}}}{R{{T}_{t}}}-D \\right) \\right) \\right] \\\\& {{H}^{r}}=RT{{g}^{eq}}\\left( T \\right) \\left( 3-D \\right)\\left[\\left( 1+{{\\omega }_{0}}\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{t}}}{15pRT}\\left( \\frac{{{c}^{2}}}{RT}-D \\right) \\right) \\right] \\\\\\end{aligned}$ $\\begin{aligned}& {{R}^{t}}=R{{T}_{r}}\\left[ {{G}^{t}}+\\left( 1-\\delta \\right)\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{r}}}{{{p}_{t}}R{{T}_{t}}}{{g}^{eq}}\\left( {{T}_{t}} \\right) \\right] \\\\& {{R}^{r}}=RT\\left[ {{G}^{r}}+{{\\omega }_{1}}\\left( 1-\\delta \\right)\\frac{\\mathbf {c}\\cdot {{\\mathbf {q}}_{r}}}{pRT}{{g}^{eq}}\\left( T \\right) \\right] \\\\\\end{aligned} $ where the coefficients are set as: $\\delta =1/1.55$ , ${{\\omega }_{0}}=0.2354$ , and ${{\\omega }_{1}}=0.3049$ for nitrogen , in this study.",
"${Z}_{rot}$ is the rotational relaxation collision number accounting for the ratio of the slower inelastic translation–rotation energy relaxation relative to the elastic translational relaxation.",
"All the above equilibrium VDFs can be obtained by macroscopic variables."
],
[
"2.3.1 Macroscopic implicit governing equations",
"The finite volume method is used in physical space, the implicit backward Euler method is used in time, and then the implicit discrete macroscopic governing equation can be written as: $\\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}\\left( \\text{ }\\mathbf {{W}}_{i}^{n+1}-\\mathbf {W}_{i}^{n} \\right)+\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {F}_{ij}^{n+1}}=\\left| {{V}_{i}} \\right|\\mathbf {S}_{i}^{n+1}$ where $\\left|V_{i} \\right|$ and ${\\Delta t=t_{n+1}-t_{n}}$ denote the volume of $V_{i}$ and the time interval, respecitvly.",
"$j$ denotes the neighboring cell of cell $i$ and $N (i)$ is the set of all of the neighbors of $i$ .",
"$ij$ denotes the variable at the interface between cell $i$ and $j$ .",
"${A}_{ij}$ is the interface area.",
"The source term $\\mathbf {S}_{i}^{n+1}$ is expressed as $\\mathbf {S}_{i}^{n+1}={{(0,\\mathbf {0},0,\\frac{\\rho {{E}_{rot,eq}}-\\rho {{E}_{rot}}}{{{Z}_{rot}}\\tau })}^{\\operatorname{T}}}$ where $\\rho {{E}_{rot,eq}}$ is the rotational energy density at the thermal equilibrium state.",
"Replace $\\mathbf {{W}}_{i}^{n+1}$ with the predicted $\\mathbf {\\tilde{W}}_{i}^{n+1}$ , and rearrange Eq.",
"REF into the incremental form $\\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}\\Delta \\mathbf {\\tilde{W}}_{i}^{n+1}+\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\Delta \\mathbf {\\tilde{F}}_{ij}^{n+1}}=\\left| {{V}_{i}} \\right|\\tilde{\\mathbf {S}}_{i}^{n+1}-\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {F}_{ij}^{n}}$ where the symbol $\\tilde{ }$ denotes the predicted variables for the next time level.",
"The flux $\\mathbf {F}_{ij}^{n}$ is determined $\\text{ }\\mathbf {F}_{ij}^{n}=\\int {\\left( \\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}} \\right)\\left( \\begin{matrix}& G_{ij}^{n} \\\\& \\mathbf {\\xi }G_{ij}^{n} \\\\& \\frac{1}{2}\\left( {{\\left| \\mathbf {\\xi } \\right|}^{2}}G_{ij}^{n}+H_{ij}^{n} \\right)+R_{ij}^{n} \\\\& R_{ij}^{n} \\\\\\end{matrix} \\right)}d\\mathbf {\\xi }$ where the construction of $\\phi _{ij}^{n}$ (i.e.",
"$G_{ij}^{n}$ , $H_{ij}^{n}$ and $R_{ij}^{n}$ ) will be detailed in Section 2.3.3.",
"The variation of the flux $\\Delta \\mathbf {\\tilde{F}}_{ij}^{n+1}$ is approximated by $\\Delta \\mathbf {\\tilde{F}}_{ij}^{n+1}=\\mathbf {\\tilde{R}}_{ij}^{n+1}-\\mathbf {\\tilde{R}}_{ij}^{n}\\ $ where $\\mathbf {\\tilde{R}}_{ij}$ has the form of the well-known Roe’s flux function ${{\\mathbf {\\tilde{R}}}_{ij}}=\\frac{1}{2}\\left[ {{\\mathbf {G}}_{ij}}\\left( {{\\mathbf {W}}_{i}} \\right)+{{\\mathbf {G}}_{ij}}\\left( {{\\mathbf {W}}_{j}} \\right)+{{r}_{ij}}\\left( {{\\mathbf {W}}_{i}}-{{\\mathbf {W}}_{j}} \\right) \\right]$ Here ${\\mathbf {G}}_{ij}\\left( {{\\mathbf {W}}} \\right)$ is the Euler flux ${{\\mathbf {G}}_{ij}}\\left( \\mathbf {W} \\right)=\\left( \\begin{matrix}& \\rho \\mathbf {U}\\cdot {{\\mathbf {n}}_{ij}} \\\\& \\left( \\rho \\mathbf {UU}+p\\mathbf {I} \\right)\\cdot {{\\mathbf {n}}_{ij}} \\\\& \\left( E+p \\right)\\mathbf {U}\\cdot {{\\mathbf {n}}_{ij}} \\\\& {{E}_{rot}}\\mathbf {U}\\cdot {{\\mathbf {n}}_{ij}} \\\\\\end{matrix} \\right) $ and ${r}_{ij}$ is ${{r}_{ij}}=\\left| {{\\mathbf {U}}_{ij}}\\cdot {{\\mathbf {n}}_{ij}} \\right|+{{a}_{ij}}+2\\frac{{{\\mu }_{ij}}}{{{\\rho }_{ij}}\\Delta {{l}_{ij}}}$ where ${a}_{ij}$ is the acoustic speed at the interface, $\\mathbf {n}_{ij}$ is the external normal unit vector of interface, and $\\Delta {{l}_{ij}}$ is the distance between cell center $i$ and $j$ .",
"Substituting Eqs.",
"REF and REF into Eq.",
"REF , and noting that $\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {G}_{ij}\\left( {{\\mathbf {W}}_{i}} \\right)}=\\mathbf {0}$ holds, then we can get the expression $\\begin{aligned}& \\left( \\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}+\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}{{r}_{ij}}} \\right)\\Delta \\mathbf {\\tilde{W}}_{i}^{n+1}-\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}{{r}_{ij}}\\Delta \\mathbf {\\tilde{W}}_{j}^{n+1}}=\\operatorname{Res}_{i}^{n}\\left( \\mathbf {W} \\right) \\\\& \\operatorname{Res}_{i}^{n}\\left( \\mathbf {W} \\right)=\\left| {{V}_{i}} \\right|\\mathbf {S}_{i}^{n+1}-\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {F}_{ij}^{n}}-\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\left[ {{\\mathbf {G}}_{ij}}\\left( \\mathbf {\\tilde{W}}_{j}^{n+1} \\right)-{{\\mathbf {G}}_{ij}}\\left( \\mathbf {W}_{j}^{n} \\right) \\right]} \\\\\\end{aligned}$ Note that the source terms of the conserved variables $\\rho $ , $\\rho \\mathbf {U}$ and $\\rho E$ are zero, and the source term of rotational energy $\\tilde{{S}}_{rot,i}^{n+1}$ is handled as $\\tilde{{S}}_{rot,i}^{n+1}=\\frac{1}{ {Z}_{rot}{\\tau }_{i}^{n+1}}\\left[ \\left( \\rho RT \\right)_{i}^{n+1}-{\\rho E}_{rot,i}^{n+1} \\right]=\\frac{1}{ {Z}_{rot}{\\tau }_{i}^{n+1}}\\left[ \\left( \\rho RT \\right)_{i}^{n+1}-\\Delta {\\tilde{\\rho E}}_{rot,i}^{n+1}-{\\rho E}_{rot,i}^{n} \\right]$ Therefore, for $\\rho E_{rot}$ , it has $\\begin{aligned}& \\left( \\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}+\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}{{r}_{ij}}}+\\frac{\\left| {{V}_{i}} \\right|}{ {Z}_{rot}{\\tau }_{i}^{n+1}} \\right)\\Delta \\mathbf {\\tilde{W}}_{i}^{n+1}-\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}{{r}_{ij}}\\Delta \\mathbf {\\tilde{W}}_{j}^{n+1}}=\\operatorname{Res}_{i}^{n}\\left( \\mathbf {W} \\right) \\\\& \\operatorname{Res}_{i}^{n}\\left( \\mathbf {W} \\right)=\\left| {{V}_{i}} \\right|\\frac{\\left( \\rho RT \\right)_{i}^{n+1}-\\mathbf {W}_{i}^{n}}{ {Z}_{rot}{\\tau }_{i}^{n+1}}-\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {F}_{ij}^{n}}-\\frac{1}{2}\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\left[ {{\\mathbf {G}}_{ij}}\\left( \\mathbf {\\tilde{W}}_{j}^{n+1} \\right)-{{\\mathbf {G}}_{ij}}\\left( \\mathbf {W}_{j}^{n} \\right) \\right]} \\\\\\end{aligned}$ Eqs.",
"REF and REF are solved by the Symmetric Gauss–Seidel (SGS) method, or also known as the Point Relaxation Symmetric Gauss–Seidel (PRSGS) method , ."
],
[
"2.3.1 Microscopic implicit governing equations",
"Since we have obtained the predicted macroscopic variable vector $\\mathbf {\\tilde{W}}_{i}^{n+1}$ , it is the time to deal with the microscopic implicit discrete Eq.",
"REF for $\\phi _{i}^{n+1}$ .",
"Similarly, rearrange Eq.",
"REF into the incremental form $\\left( \\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}+\\frac{\\left| {{V}_{i}} \\right|}{\\tilde{\\tau }_{i}^{n+1}} \\right)\\text{ }\\Delta \\phi _{i}^{n+1}+\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}}\\Delta \\phi _{ij}^{n+1}}=\\left| {{V}_{i}} \\right|\\frac{\\tilde{g}_{i}^{\\phi ,n+1}-\\phi _{i}^{n}}{\\tilde{\\tau }_{i}^{n+1}}-\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}}\\phi _{ij}^{n}}$ where $\\tilde{g}_{i}^{\\phi ,n+1}$ and $\\tilde{\\tau }_{i}^{n+1}$ are determined by the predicted $\\mathbf {\\tilde{W}}_{i}^{n+1}$ .",
"$\\phi _{ij}^{n}$ will be detailed in Section 2.3.3.",
"$\\Delta \\phi _{ij}^{n}$ is simply handled by the upwind scheme and Eq.",
"REF is turned into $\\begin{aligned}& \\left( \\frac{\\left| {{V}_{i}} \\right|}{\\Delta t}+\\frac{\\left| {{V}_{i}} \\right|}{\\tilde{\\tau }_{i}^{n+1}}+\\sum \\limits _{j\\in {{N}^{+}}\\left( i \\right)}{{{A}_{ij}}\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}}} \\right)\\text{ }\\Delta \\phi _{i}^{n+1}+\\left( \\sum \\limits _{j\\in {{N}^{-}}\\left( i \\right)}{{{A}_{ij}}\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}}} \\right)\\Delta \\phi _{j}^{n+1}=\\operatorname{Res}_{i}^{n} \\\\& \\operatorname{Res}_{i}^{n}=\\left| {{V}_{i}} \\right|\\frac{\\tilde{g}_{i}^{\\phi ,n+1}-\\phi _{i}^{n}}{\\tilde{\\tau }_{i}^{n+1}}-\\sum \\limits _{j\\in N\\left( i \\right)}{{{A}_{ij}}\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}}\\phi _{ij}^{n}} \\\\\\end{aligned} $ where ${{N}^{+}}\\left( i \\right)$ is the set of $i$ ’s neighboring cells satisfying $\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}} \\ge 0$ while for ${{N}^{-}}\\left( i \\right)$ it satisfies $\\mathbf {\\xi }\\cdot {{\\mathbf {n}}_{ij}} < 0$ .",
"Eq.",
"REF is solved by the SGS method to obtain $\\phi _{ij}^{n+1}$ .",
"More information about the IUGKS is provided in reference , which recommends 2 times’ SGS iterations for the microscopic equation and 60 times’ SGS iterations for the macroscopic equation per time step."
],
[
"2.3.3 Simplified multi-scale numerical fluxes",
"In this study, the simplified multi-scale flux is built utilizing the numerical quadrature solution, following the lead of DUGKS.",
"As seen in Fig.",
"REF , the VDF at the interface can be obtained by integrating the kinetic Eq.",
"REF along its characteristic line from $t_{n}$ to $t_{n+1/2}$ ${{\\phi }}\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)-{{\\phi }}\\left( {{\\mathbf {x}}_{ij}}-\\mathbf {\\xi }h,\\mathbf {\\xi },{{t}_{n}} \\right)=h\\frac{{{g}^{\\phi }}\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)-{{\\phi }}\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)}{{{\\tau }^{n+1/2}}}$ where $h={\\Delta t}/{2}$ denotes a half-time step, and $\\mathbf {x}_{ij}$ denotes the midpoint of the interface.",
"Finally, the VDF $\\phi \\left( {{x}_{ij}},\\mathbf {\\xi },t_{n+1/2}^{{}} \\right)$ at the interface can be expressed as: ${{\\phi }}\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)=\\frac{{{\\tau }^{n+1/2}}}{{{\\tau }^{n+1/2}}+h}{{\\phi }}\\left( {{\\mathbf {x}}_{ij}}-\\mathbf {\\xi }h,\\mathbf {\\xi },{{t}_{n}} \\right)+\\frac{h}{{{\\tau }^{n+1/2}}+h}{{g}^{\\phi }}\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)$ where ${g}^{\\phi }\\left( {{\\mathbf {x}}_{ij}},\\mathbf {\\xi },{{t}_{n}}+h \\right)$ can be derived from the macroscopic variables ${\\mathbf {W}}^{n+1/2}_{{\\mathbf {x}}_{ij}}$ of the interface.",
"Since the collision operator conserves mass, momentum, and energy, the conserved variables ${\\mathbf {W}}^{n+1/2}_{{\\mathbf {x}}_{ij}}$ can be computed from ${\\mathbf {W}}^{n+1/2}_{{\\mathbf {x}}_{ij}}=\\int { {\\mathbf {\\varphi }}{{\\phi }}\\left( {{\\mathbf {x}}_{ij}}-\\mathbf {\\xi }h,\\mathbf {\\xi },{{t}_{n}} \\right) }d\\mathbf {\\xi } $ where ${\\mathbf {\\varphi }}=\\left(1, \\mathbf {\\xi }, \\frac{1}{2}|\\xi |^2\\right)$ is vector of the elementary collision invariants.",
"And the rotational ernergy can be computed from $\\left({\\rho R T_{rot}} \\right)^{n+1/2}_{{\\mathbf {x}}_{ij}}=\\left(1+\\frac{h}{ Z_{rot}{\\tau }^{n+1/2}_{{\\mathbf {x}}_{ij}}}\\right)\\left(\\int { {{R }}\\left( {{\\mathbf {x}}_{ij}}-\\mathbf {\\xi }h,\\mathbf {\\xi },{{t}_{n}} \\right) }d\\mathbf {\\xi } + \\frac{h}{ Z_{rot}{\\tau }^{n+1/2}_{{\\mathbf {x}}_{ij}}} \\left( {\\rho R T}\\right)^{n+1/2}_{{\\mathbf {x}}_{ij}} \\right)$ Fianlly, ${{\\phi }}\\left( {{\\mathbf {x}}_{ij}}-\\mathbf {\\xi }h,\\mathbf {\\xi },{{t}_{n}} \\right)$ can be calculated through the Taylor expansion at the control volume: $\\phi \\left( \\mathbf {x}_{ij}^{{}}-\\mathbf {\\xi }h,\\mathbf {\\xi },t_{n}^{{}} \\right)={{\\phi }_{C}}\\left( \\mathbf {x}_{C}^{{}},\\mathbf {\\xi },t_{n}^{{}} \\right)+ L \\left( \\mathbf {x}_{C}^{{}},\\mathbf {\\xi },t_{n}^{{}} \\right)\\nabla {{\\phi }_{C}}\\left( \\mathbf {x}_{C}^{{}},\\mathbf {\\xi },t_{n}^{{}} \\right)\\cdot \\left( \\mathbf {x}_{ij}^{{}}-\\mathbf {\\xi }h-\\mathbf {x}_{C}^{{}} \\right), \\mathbf {x}_{ij}^{{}}-\\mathbf {\\xi }h\\in V_{C}^{{}}$ where ${{V}_{c}}$ represents the control volume which is centered at point C (Fig.",
"REF ).",
"If $\\mathbf {\\xi }\\cdot \\mathbf {n}_{ij}^{{}}\\ge 0$ , point C is $i$ (the center of the left cell) in Fig.",
"REF ; otherwise, point C is $j$ (the center of the right cell).",
"$\\nabla {\\mathop {\\phi _{C}}}\\,\\left( \\mathbf {x}_{C}^{{}},\\mathbf {\\xi },t_{n}^{{}} \\right)$ is the gradient of the reduced VDFs at point C, which is calculated by the least-square method in this study, and $L \\left( \\mathbf {x}_{C}^{{}},\\mathbf {\\xi },t_{n}^{{}} \\right)$ is the gradient limiter used to suppress the numerical oscillations.",
"Besides, the Venkatakrishnan limiter is chosen in this paper.",
"It is necessary to employ an AVS to improve the computational efficiency of UGKS, particularly for hypersonic non-equilibrium flows.",
"Similar to the unstructured velocity space, the AVS in this study also uses the mid-point quadrature formula.",
"Therefore, the quadrature point (i.e.",
"discrete velocity point) and the quadrature weight are the center and volume (area for 2-dimensions case) of the velocity mesh, respectively.",
"Because of the usage of the global velocity mesh and the “consanguinity\" relationship during the reconstruction of VDFs in a new velocity space, the AVS technique described in this paper does not require knowledge of the geometric relationship between the elements.",
"As a result, the data structure is quite simple and the proposed method is highly practical."
],
[
"Tree data structure for adaptive velocity space",
" In this work, the quadtree and octree data structures are used to describe the two-dimensional (2D) and three-dimensional (3D) AVS, respectively.",
"And the merging and splitting steps of the adaptation are illustrated using quadtree data for simplicity of description.",
"In Fig.",
"REF , the level of root node is 0, and it can only be split, not merged.",
"The root node splits once would produce four child nodes ($k_1–k_4$ ) with level 1.",
"Once the child node $k_4$ meets the split condition, it splits again and produce four child nodes ($k_{41}–k_{44}$ ) with level 2.",
"The root node is the parent of nodes $k_1–k_4$ , while node k4 is the parent of nodes $k_{41}–k_{44}$ .",
"The four child nodes that share the same parent node are brother nodes.",
"Like a leaf on a tree, a node with no child is called leaf node.",
"The leaf nodes in Fig.",
"REF are connected by a red lines and the corresponding velocity mesh is given in Fig.",
"REF .",
"As can be observed, the velocity mesh only has the leaf nodes $k_{1}$ , $k_{2}$ , $k_{3}$ , $k_{41}$ , $k_{42}$ , $k_{43}$ and $k_{44}$ .",
"In addition, the leaf nodes are saved as double link lists in the computer code.",
"The merging process, which can be regarded of as the inverse of the splitting process, can only take place when all four of the brother nodes are leaf nodes and the merging condition are satisfied.",
"For instance, the child nodes $k_{41}–k_{44}$ can only be merged into the node $k_4$ if they all satisfy the merging condition.",
"As a result of this merge step, the parent node $k_4$ becomes a leaf node and has no child."
],
[
"Reconstruction of distribution functions",
" Following the adaptation of the velocity space, the VDF must be recreated on the new velocity space.",
"In this study, the reconstruction of VDFs is done using a special “consanguinity\" relationship rather than interpolation.",
"Without losing generality, Fig.",
"REF is also used as an example to show the reconstruction of VDFs.",
"For instance, the VDFs of nodes $k_1–k_4$ are equal to thier parent's when the root node splits into four child nodes $f^{^{\\prime }}_{k_{i}}=f_{root}$ where $f$ and $f^{^{\\prime }}$ are the VDF for the old velocity space before adaptation and the new velocity space after adaptation, respectively.",
"The subscript $i$ is $1,2,3,4$ .",
"Similarly, when node $k_4$ splits into four child nodes ($k_{41}–k_{44}$ ), the VDFs of the child nodes $k_{41}–k_{44}$ are identical to that of node $k_4$ $f^{^{\\prime }}_{k_{4i}}=f_{4}$ For the merging step, the VDF of the merge node is equal to the average of the VDFs of its child nodes.",
"For instance, the VDF of node $k_4$ can be calculated as the average of the VDFs of nodes $k_{41}–k_{44}$ when their four child nodes $k_{41}–k_{44}$ merge to thier parent node $k_4$ $f^{^{\\prime }}_{k_{4}}=\\frac{1}{4} \\sum _{i=1}^{4}f_{4i}$ Similar to this, when nodes $k_1–k_4$ merge to form root node, root node's VDF is equal to the average of those nodes' VDFs $f^{^{\\prime }}_{root}=\\frac{1}{4} \\sum _{i=1}^{4}f_{i}$ Without a doubt, the \"consanguinity\" relationship that has been proposed is quite simple and effective.",
"Furthermore, since interpolation is not used, the velocity space data structure is not required, making the double link list fairly simple."
],
[
"Conservation correction",
" The conservation of mass is met during the reconstruction of the VDFs in the preceding subsection, but it is not yet possible to guarantee that momentum and energy are also conserved.",
"It is possible that some deviation exist in the VDFs obtained from the “consanguinity\" relationship and need to be fixed to preserve conservation.",
"The deviation physical variables can be computed from $f^{^{\\prime }}$ $\\mathbf {W}^{^{\\prime }}=\\int {{\\mathbf {\\varphi }}f^{^{\\prime }}}d\\mathbf {\\xi } $ where ${\\mathbf {\\varphi }}$ is the collision invariant.",
"In line with the literature , , the discrete deviation of the equilibrium VDF is used to guarantee conservation.",
"Consequently, the adaptive VDF can be described as $f=f^{^{\\prime }} + g^{eq}(\\mathbf {W}) - g^{eq}(\\mathbf {W}^{^{\\prime }}) $ where ${\\mathbf {W}}$ is the macroscopic variables updated by Eq.",
"REF and $g^{eq}\\left( {{\\mathbf {W}}} \\right)$ is the Maxwellian VDF corresponding to ${\\mathbf {W}}$ .",
"For weakly nonequilibrium flows, the reconstruction of VDFs mentioned previously can also be bypassed, and the adaptive VDF can be represented as $f=g^{eq}(\\mathbf {W})$"
],
[
"Adaptive criterion",
" The AVS is significantly influenced by the adaptive criteria, which guide the merging and splitting process.",
"In this work, the density criterion and the internal energy criterion, also known as the $M_1$ and $M_2$ criterion, respectively, are applied in line with literature .",
"Following are the definitions for $M_1$ and $M_2$ ${{M}_{1,i,{k}}}={{A}_{k}}{{f}_{i,{k}}}/{{\\rho }_{i}}$ ${{M}_{2,i,{k}}}=\\frac{1}{2}{{A}_{{k}}}{{f}_{i,{k}}}{{\\left( {{\\mathbf {\\xi }}_{k}}-{{\\mathbf {U}}_{i}} \\right)}^{2}}/\\left[\\left( {{\\rho }}{{E}}\\right)_{i}-\\frac{1}{2}{{\\rho }_{i}}\\mathbf {U}_{i}^{2} \\right]$ where $i$ and $k$ are the element index of physical mesh and velocity mesh, respectively.",
"${A}_{{k}}$ is the area (2D) or volume (3D) of the $k$ -th velocity mesh element.",
"We can infer from the definition of $M_1$ and $M_2$ that they stand for the density and internal energy ratios to the total density and internal energy at the current velocity element, respectively.",
"The terms $M_1$ and $M_2$ refer to the density and internal energy ratios to the total density and internal energy at the present velocity element, respectively.",
"Since a global velocity mesh is adopted, a maximum value $M_k$ from $M_1$ and $M_2$ in the whole physical mesh must be selected ${{M}_{{k}}}=\\max \\left( {{M}_{1,i,{k}}},{{M}_{2,i,{k}}} \\right)$ Following is a description of the adaptive rules using $C_1$ and $C_2$ ($C_1>C_2$ ) as the splitting and merging thresholds, respectively (a) When $M_k$ is larger than $C_1$ , the velocity mesh has to be split.",
"(b) When $M_k$ is smaller than $C_2$ , the velocity mesh speed needs to be merged.",
"(c) Besides (a) and (b), the velocity element does not need to be changed.",
"It is clear that $C_1$ is in charge of splitting, whereas $C_2$ is in charge of merging.",
"When the contribution of density or interal energy of $k$ -th velocity element is large enough, i.e.",
"$M_k>C_1$ , the $k$ -th velocity element is marked and are going to split.",
"When the contribution of density or internal energy of $k$ -th velocity element is small enough, i.e.",
"$M_k<C_2$ , the $k$ -th velocity element is marked and are going to merge."
],
[
"Configuration of adaptive parameters",
" The potential to describe the behavior of flow depends on the domain and resolution of velocity space, which are controlled by adaptive parameters.",
"Consequently, it is crucial to choose these adaptive parameters.",
"A set of adaptive parameters are provided in this work to improve the automation of the AVS.",
"Since the Maxwell distribution and the normal distribution are relatively similar, the characteristics of the normal distribution can be used to predict the distribution of the VDFs.",
"The expression of the normal distribution is ${f(x)}=\\frac{1}{{\\sqrt{2\\pi }\\sigma }}\\exp \\left[ -\\frac{{(x-\\mu )^{2}}}{2{\\sigma }^2} \\right]$ where $\\mu $ is the mean or expectation of the distribution, while $\\sigma $ and ${\\sigma }^2$ are the standard deviation and variance.",
"When comparing Eq.",
"REF (Maxwell equilibrium distribution) with Eq.",
"REF (normal distribution), it is discovered that the expectation and standard deviation of Maxwell distribution are $\\mathbf {U}$ and $\\sqrt{RT}$ , respectively.",
"In numerical simulation, there are typically three standard deviation $\\begin{matrix}{\\sigma }_{\\infty } = \\sqrt{RT_{\\infty }} \\\\{\\sigma }_{0} = \\sqrt{RT_{0}} \\\\{\\sigma }_{w} = \\sqrt{RT_{w}}\\end{matrix}$ where $T_{\\infty }$ , $T_{0}$ and $T_{w}$ are static temperature (also known as the temperature of freestream), total temperature and wall temperature, respectively.",
"And the total temperature is related to static temperature via $T_{0}=T_{\\infty }(1+\\frac{\\gamma - 1}{{2}} Ma^2)$ According to the features of the normal distribution, the probability of the $3\\sigma $ and $4\\sigma $ intervals with the expectation as the center is known to be 99.73$\\%$ and 99.99$\\%$ , respectively, which can be viewed as 100$\\%$ .",
"Therefore, the $3\\sigma $ and $4\\sigma $ are chosen as critical value for this work.",
"Fig.",
"REF presents diagram of the AVS.",
"The radius of the velocity domain is $R_{dv} = 4 \\sigma _{0}$ .",
"The center of the velocity domain is $(0.4U_{\\infty }, 0.4V_{\\infty })$ rather than zero because there are three refined zones: the stationary zone (red A zone in Fig.",
"REF ), the freestream zone (blue A zone in Fig.",
"REF ) and the separated zone (orange C zone in Fig.",
"REF ).",
"The radius and center of the stationary zone are $3\\sigma _{w}$ and $(0, 0)$ , respectively.",
"The radius and center of the freestream zone are $3\\sigma _{\\infty }$ and $(U_{\\infty }, 0)$ , respectively.",
"And the radius and center of the separated zone are $5\\sigma _{w}$ and $(0, 0)$ , respectively.",
"The segregated zone is designed for the bottom of the vehicle's separate flow.",
"Most of the time, $R_{dv} = 4 \\sigma _{0}$ is enough because the wall temperature is significantly lower than the total temperature.",
"However, the radius of the velocity domain should be larger, such as $R_{dv} = 7 \\sigma _{0}$ , to cover a larger stationary zone and separated zone when the wall temperature is close to the total temperature.",
"In general, the freestream zone has the highest resolution, followed by the stationary zone.",
"Assuming that the velocity mesh's standard interval is $h_{std}$ , the corresponding level $L_{std}$ of the mesh can be determined as $2^{L_{std}} \\ge \\frac{2R_{dv}}{h_{std}}$ Consequently, the level of the freestream zone, or the maximum level of the AVS, is chosen as $L_{\\infty } = L_{max} = min(L_{std})$ And the interval of the freestream zone, or the minimum interval, is $h_{\\infty } = h_{min} = \\frac{2R_{dv}}{2^{L_{max}}}$ It can be observed that $h_{\\infty }$ is not larger than $h_{std}$ and not less than $0.5h_{std}$ .",
"In this work, $h_{std}$ is set as 0.6.",
"Once the level of the freestream zone has been determined, the level of the stationary zone can be estimated as follows $\\begin{aligned} L_{w}= \\left\\lbrace \\begin{array}{ll} L_{max}, & \\frac{T_{w}}{T_{\\infty }} < 4;\\\\L_{max} - 1, & otherwise.\\end{array}\\right.\\end{aligned}$ And the following is the minimum level of the AVS $\\begin{aligned} L_{min}= \\left\\lbrace \\begin{array}{ll} 4, & D = 2, Ma > 5;\\\\3, & otherwise.\\end{array}\\right.\\end{aligned}$ Additionally, the separated zone should be refined if there is a separated flow $L_{separated} = max\\left(L_{min}, L_{w}-1\\right), 3\\sigma _{w} < |\\xi | < 5\\sigma _{w}$ where $|\\xi |$ is the magnitude of the discrete velocity.",
"Now that the geometric shape of the velocity space and the level scope have been presented, all that is left to do is choose the appropriate adaptive thresholds $C_1$ and $C_2$ .",
"When $\\mu =0$ and $\\sigma =1$ , the normal distribution (Eq.",
"REF ) turns to the standard normal distribution ${f(x)}=\\frac{1}{{\\sqrt{2\\pi }}}\\exp \\left( -\\frac{{x^{2}}}{2} \\right)$ The maximum value of the standard normal distribution $f_{max} = 0.3989$ is found at $x = 0$ , and the critical value $f_{4\\sigma } = 1.3383\\times 10^{-4}$ is located where x equals to $-4\\sigma $ or $4\\sigma $ .",
"As was already discussed, the probability of the normal distribution falling inside the interval $[-4\\sigma ,4\\sigma ]$ is around 100$\\%$ , hence the portion outside the interval $[-4\\sigma ,4\\sigma ]$ can be neglected.",
"With the help of this feature, the merging threshold $C_2$ is related to the splitting threshold $C_1$ via $C_2=k_{\\sigma } C_1$ where $k_{\\sigma } = \\frac{ f_{max}}{f_{4\\sigma }} = 3.3546\\times 10^{-4}$ Eq.",
"REF suggests that only when $M_k$ is less than $k_{\\sigma } C_1$ will the velocity element be marked and begin to merge.",
"Therefore, only the parameter $C_1$ needs to be determined, and the law of $C_1$ will be investigated in a later simulation of flow over a cylinder.",
"Once every adaptive parameter has been determined, an AVS can be created at step 0 (1) Create an element with just a root node, whose level is 0.",
"The center and the radius of this element are $(0.4U_{\\infty }, 0.4V_{\\infty })$ and $4\\sigma _{0}$ .",
"(2) Split the single element $N$ times results in a unified mesh of $2^{N} \\times 2^{N}$ elements, where $N$ can be set to $L_{w}-1$ .",
"(3) Perform once adaption based on freestream point and static point, whose physical varibles are $\\mathbf {{W}}_{\\infty }=(\\rho _{\\infty }, U_{\\infty }, V_{\\infty }, T_{\\infty })$ and $\\mathbf {{W}}_{0}=(\\rho _{\\infty }, 0, 0, T_{\\infty })$ .",
"And the VDFs of $\\mathbf {{W}}_{\\infty }$ and $\\mathbf {{W}}_{0}$ are equal to $g^{eq}(\\mathbf {{W}}_{\\infty })$ and $g^{eq}(\\mathbf {{W}}_{0})$ .",
"Therefore, $M_k$ can be computed according to Eqs.",
"REF , REF and REF .",
"As a result, an initial AVS is obtained.",
"To efficiently match the flow field, adaptation is done in the first $n$ steps (like steps 1, 2, and 3).",
"Following that, adaptation takes place at a frequency of $\\nu _{adaption}$ .",
"In general, $\\nu _{adaption}$ can be set to 50 or 100 steps."
],
[
"Numerical experiments",
"Several test cases are performed in this section to validate the proposed method, and the results of the AVS and the Cartesian velocity space (CVS) are compared to benchmark results."
],
[
"Supersonic flow passing a circular cylinder",
" In this case, the relationship between $C_1$ and Mach number will be investigated.",
"The Kn number of freestream with the cylinder radius as characteristic length is 1.0.",
"The working gas is argon and the gas constant of argon is ${{R}_{{{Ar}}}}=208J/\\left( kg\\cdot K \\right)$ ).",
"The VHS model with $\\omega =0.81$ is employed.",
"The specific heat ratio and the Prandtl number are 5/3 and 2/3, respectively.",
"The temperature of the freestream and the surface of the cylinder both are 273 K. In this work, the characteristic length serves as the reference length, and the freestream's density and temperature serve as the reference density and temperature.",
"As a result, freestream's dimensionless density and temperature are both equal to 1.0.",
"The radius of the cylinder is 1.0 and the physical domain, which is a circular region with a circle center at (0,0) and a radius of 15, is discretized by a mesh with 64x61 cells.",
"To accurately capture the heat flux on the cylinder surface, the height of the first mesh layer at the surface is 0.01 in this instance.",
"Simulations were conducted in both CVS and AVS in order to compare the results.",
"When Ma is 2 to 6, 8 to 20, and 22 to 30, the CVS with discrete points of $89\\times 89$ , $101\\times 101$ , and $201\\times 201$ are used, respectively.",
"The details of the AVS for supersonic flow passing a circular cylinder are presented in Table REF .",
"And a curve was fitted based on the discrete data in Table REF to indicate the relationship between $C_1$ and Ma $C_1 = 13.749Ma^{-1.465}$ The discrete data of Table REF are plotted in Fig.",
"REF , together with Eq.",
"REF , making it clear that the parameter obtained from Eq.",
"REF is greater than zero.",
"Consequently, the adaptive parameter $C_1$ will then be determined by Eq.",
"REF .",
"Table: Adaptive velocity test for supersonic flow passing a circular cylinder.The results of Ma 5 supersonic flow around a cylinder in literature will be employed as a benchmark.",
"In literature , an $89\\times 89$ CVS with the range of $[-15\\sqrt{2RT_{\\infty }}, 15\\sqrt{2RT_{\\infty }}] \\times [-15\\sqrt{2RT_{\\infty }}, 15\\sqrt{2RT_{\\infty }}]$ is adopted.",
"According to Eq.",
"REF , the parameter $C_1$ is 1.30, and Fig.",
"REF shows the corresponding AVS with a maximum level of 5 and 232 elements.",
"There will be 1024 elements in the complete mesh of level 5, which is just 12.9$\\%$ of the CVS's elements and 4.4 times that of AVS.",
"Furthermore, the elements of AVS is only 2.9$\\%$ of the CVS's.",
"Fig.",
"REF presents the density, pressure, horizontal velocity (U) and temperature along the central line in front of the cylinder.",
"Fig.",
"REF presents the pressure, shear stress and heat fluxes on the surface of cylinder.",
"It is clear that the results from AVS, CVS and DSMC match one another.",
"This indicates that the AVS can accurately describe the behavior of cylinder flow."
],
[
"Lid-driven cavity flow",
" Although AVS is typically employed for high-speed non-equilibrium flows, it is interesting to find out if it is suitable for low-speed continuum flows.",
"The test case of lid-driven cavity flow is very suitable to test whether the proposed method can accurately simulate the viscosity effect of the flow or not.",
"Here, the lid-driven cavity flow with Ma = 0.16 and Re = 400 and 1000 are carried out.",
"The top wall is moving with a constant velocity $U_w$ while the other walls are static.",
"The working gas is argon and the VHS molecular model is employed with $\\omega =0.81$ .",
"The temperature of all walls are 273 K, then the sonic speed and velocity of the moving wall are $a=\\sqrt{\\gamma RT}=307.64$ m/s and $U_w$ = 49.22 m/s, respectively.",
"The physical domain is discretized by a uniform mesh with 100×100 cells, and the AVS with $C_1$ equals to 201.5 is shown in Fig.",
"REF , which contains 184 elements.",
"Fig.",
"REF presents the vertical velocity $V$ along the horizontal central line and the horizontal velocity $U$ along the vertical central line.",
"The numerical results are in good agreement with the benchmark data .",
"It demonstrates that the AVS is capable of simulating low-speed continuum flows."
],
[
"Hypersonic flow over a blunt wedge",
" It is crucial to accurately predict the hypersonic bottom flow for near-space vehicles, particularly when Reaction Control System (RCS) is installed on the bottom of some vehicles.",
"With reference to the configuration of the literature , the current method simulates the same issue to verify its performance for the hypersonic dilute expansion flow.",
"Fig.",
"REF presents the geometry of the blunt wedge, which has a length of $L$ =120 mm, a head radius of $R$ =20 mm, a bottom height of $H$ =74.72 mm, and a body slope of $\\theta $ =10°.",
"Here, the working gas is argon and the VHS model is employed.",
"The Ma number and angle of attack of freestream are 8.1 and 0°, respectively.",
"The temperature of freestream is 189 K (equivalent to an altitude of 85 km), while the temperature of the surface of wedge is fixed at 273 K. The Kn number with $R$ and $H$ as the characteristic length are 0.338 and 0.090, respectively.",
"In this simulation, the unstructured physical mesh, which has 11905 cells, is shown in Fig.",
"REF .",
"The parameter $C_1$ according to Eq.",
"REF is 0.64, and the AVS, which include 568 elements, is shown in Fig.",
"REF .",
"In addition, the CAS with 7921 ($89\\times 89$ ) elements was also adopted to simulate the flow around the blunt wedge.",
"Figs.",
"REF present the density, pressure, temperature and horizontal velocity (U) contours around the blunt wedge.",
"The background and white solid lines are the results of the CVS, whereas the black long dash line is the result of the AVS.",
"It is obvious that the results of AVS and CVS are in good agreement with one another.",
"The pressure, shear stress and heat flux along the surface of the blunt wedge are present in REF , REF and REF , respectively.",
"As can be observed, the results of AVS and CVS are consistent and in good agreement with those of DS2V."
],
[
"Supersonic flow passing a sharp flat plate",
" To further verify the proposed approach, diatomic gas (i.e.",
"nitrogen) will then be employed as working gas.",
"Additionally, the computational efficiency of AVS will be investigated in the simulation of supersonic flow across a sharp flat plate.",
"The configuration is the same with the run34 case in Ref.",
".",
"Fig.",
"REF presents the physical mesh used for calculation and the flat plate's geometric shape.",
"With a thickness of 15 mm and an upper surface length of 100 mm, the flat plate has a sharp angle of 30 degree.",
"The Rykov model is employed with nitrogen serving as the working gas (${{R}_{{{N}_{2}}}}=297J/\\left( kg\\cdot K \\right)$ ).",
"And the VHS molecular model is employed with $\\omega =0.81$ .",
"The surface temperature of the flat plate is fixed at 290 K, and the freestream's Ma number and temperature are 4.89 and 116 K, respectively.",
"The Kn number of freestream with the plate's length as characteristic length is 0.0078.",
"A uniform freestream flow is employed as the initial flow field in this simulation, and the splitting threshold $C_1$ is 1.34.",
"The AVS is shown in Fig.",
"REF .",
"The AVS in Fig.",
"REF has only 376 elements, while the CVS of Reference has 4800 elements.",
"Fig.",
"REF presents the horizontal velocity, translational, rotational, and equilibrium temperature contours around the flat plate.",
"The background and white solid lines are the results of the CVS, whereas the black long dash line is the result of the AVS.",
"It is obvious that the results of AVS and CVS are in good agreement with one another.",
"The temperature distributions along the vertical line above the flat plate at the locations of $x=5$ mm and $x=20$ mm from the leading edge are presented in Fig.",
"REF .",
"The numerical results match well with the experimental data.",
"The AVS's capability for capturing flow behavior around the flat plate is demonstrated by the strong agreement between the results from the AVS and the CVS.",
"The proposed AVS is more attractive since it has fewer elements than CVS, which lowers memory costs and speeds up computation.",
"It should be noted that the sharp plate simulation had an adaptive frequency of $\\nu _{adaption} = 100$ and just one adaptation was done at step 0, indicating that the initial AVS at step 0 is suitable for this simulation.",
"Table REF shows the comparison of details between CVS and AVS for the sharp flat plate simulation, which were performed on 2 and 22 cores, respectively.",
"Where “NC\" in Table REF means the number of elements per core.",
"Using the results of 2 cores as a reference, the parallel efficiency for CVS and AVS are 77.90$\\%$ and 45.56$\\%$ , respectively.",
"It makes sense that when “NC\" decreases, the proportion of calculation time to communication time decreases, resulting in a decrease of parallel efficiency.",
"Anyway, compared to the CVS, the AVS costs much less time.",
"The ratio of computation time for two cores is 12.06, which is nearly equal to the ratio of elements (12.77).",
"Table: Physical variables on the z = 0 surface of the sphere at Ma = 4.25 and Re = 210."
]
] | 2212.05645 |
[
[
"Reaction Nanoscopy of Ion Emission from Sub-wavelength Propanediol\n Droplets"
],
[
"Abstract Droplets provide unique opportunities for the investigation of laser-induced surface chemistry.",
"Chemical reactions on the surface of charged droplets are ubiquitous in nature and can provide critical insight into more efficient processes for industrial chemical production.",
"Here, we demonstrate the application of the reaction nanoscopy technique to strong-field ionized nanodroplets of propanediol (PDO).",
"The technique's sensitivity to the near-field around the droplet allows for the in-situ characterization of the average droplet size and charge.",
"The use of ultrashort laser pulses enables control of the amount of surface charge by the laser intensity.",
"Moreover, we demonstrate the surface chemical sensitivity of reaction nanoscopy by comparing droplets of the isomers 1,2-PDO and 1,3-PDO in their ion emission and fragmentation channels.",
"Referencing the ion yields to gas-phase data, we find an enhanced production of methyl cations from droplets of the 1,2-PDO isomer.",
"Density functional theory simulations support that this enhancement is due to the alignment of 1,2-PDO molecules on the surface.",
"The results pave the way towards spatio-temporal observations of charge dynamics and surface reactions on droplets in pump-probe studies."
],
[
"Simulation geometry",
"The interface between the nanodroplet and vacuum has been modeled using a slab of molecules in the $x$ -$z$ plane consisting of six layers as shown in fig:simgeom.",
"For an unambiguous definition of the droplet-vacuum interface, the bottom three layers are a rotated image of the top layers.",
"The formation enthalpy for the 1,2-PDO and 1,3-PDO slab from isolated gaseous molecule is -16.14 kcal/mol (0.7 eV) and -31.14 kcal/mol (1.35 eV), respectively.",
"This suggests that simulation configurations are energetically stable.",
"Figure: Side view of the simulation geometry for 1,2-PDO (left) 1,3-PDO (right).The surface normal is along the yy-axis.",
"Periodic boundary conditions were applied along xx and zz.",
"The dotted lines represent hydrogen bonds."
],
[
"Ion fragment selection",
"The measurement of the electron signal in reaction nanoscopy enables a significant reduction of the single molecule ionization signal in the droplet ion data.",
"As shown in Figure 1 of the main text, droplet events cause a much larger electron signal than gas-phase events, which allows filtering the data.",
"Despite the filtering, we always observe some contribution of gas-phase, single molecule ionization to the droplet data.",
"From a comparison of the gas-phase data and the droplet data presented in Fig:FragmentSelectionSI, it becomes clear that the peaks close to $y=0$ in the droplet data (which show up identically in the gas-phase data) are ions emitted from single gas-phase molecules which were ionized independently from the droplet but by the same laser pulse.",
"For this case of simultaneous ionization of a droplet and single-molecules, the electron signal is still large and the whole laser shot is assigned to the droplet data.",
"This effect is further confirmed by the 1D time-of-flight histograms in fig:1DtofSI, where data for $|y|<11$ mm is shown, corresponding to the white-shaded regions of Figure 2 in the main text and Fig:FragmentSelectionSI.",
"Similar but fragment-specific filters were applied for the ion data shown in Figure 5 of the main text.",
"For every ion species under investigation, we defined a polygon in the space of time-of-flight and y-position.",
"These polygons are shown in Fig:FragmentSelectionSI.",
"For the droplet data, all the polygons were additionally mirrored around $y=0$ in order to include both lobes of the dipolar ion emission pattern.",
"Figure: Filtering ion signals for gas-phase and droplet data.Two measurements are shown, one for 1,2-propanediol (1,2-PDO, panels a and b) and one for 1,3-propanediol (1,3-PDO, panels c and d).Both measurements were split into gas-phase data and droplet based on the strength of the electron signal which is recorded in coincidence with the ions.Gas-phase data (first column, panels a and c) are defined by a small or entirely absent electron signal.",
"Droplet data (second column, panels b and d) are defined as the complementary set.The ion species listed in the middle were identified after calibrating the time-of-flight.",
"The polygons indicated by the white dashed lines were used for the yield comparison in the main text (Figure 5).",
"The white shading corresponds to |y|<11|y|<11 mm.Figure: Time-of-flight histograms for |y|<11|y|<11 mm.",
"The histograms were obtained by projecting the corresponding parts of the position-resolved spectra of Fig:FragmentSelectionSI (for the droplet data, we have indicated |y|<11|y|<11 mm with the white shading)."
],
[
"Steric effects on the surface of 1,2-PDO droplets",
"To further confirm our understanding about the influence of steric effects between methyl groups on the minimum energy configuration, we performed single point energy calculations of 1,2-PDO by changing the configuration of the molecules from equilibrium.",
"We rotate the molecules located at the interface inward and outward from the minimum energy state by an angle $\\theta $ around the C2-C3 bond (Fig:FigureSteric, shown by blue arrows), resulting in the methyl groups of neighboring molecules coming close to each other and thereby raising the energy of the system.",
"This corroborates the fact that the steric effect plays an important role in the surface structure of 1,2-PDO droplets.",
"Figure: Energy of 1,2-propanediol with respect to angle of rotation θ\\theta .",
"The rotation axis is along the C2-C3 bond and θ=0 ∘ \\theta =0^\\circ corresponds to the equilibrium configuration obtained via energy minimization.",
"The rotations are depicted by the blue arrows."
]
] | 2212.05587 |
[
[
"Blockchain Network Analysis: A Comparative Study of Decentralized Banks"
],
[
"Abstract Decentralized finance (DeFi) is known for its unique mechanism design, which applies smart contracts to facilitate peer-to-peer transactions.",
"The decentralized bank is a typical DeFi application.",
"Ideally, a decentralized bank should be decentralized in the transaction.",
"However, many recent studies have found that decentralized banks have not achieved a significant degree of decentralization.",
"This research conducts a comparative study among mainstream decentralized banks.",
"We apply core-periphery network features analysis using the transaction data from four decentralized banks, Liquity, Aave, MakerDao, and Compound.",
"We extract six features and compare the banks' levels of decentralization cross-sectionally.",
"According to the analysis results, we find that: 1) MakerDao and Compound are more decentralized in the transactions than Aave and Liquity.",
"2) Although decentralized banking transactions are supposed to be decentralized, the data show that four banks have primary external transaction core addresses such as Huobi, Coinbase, Binance, etc.",
"We also discuss four design features that might affect network decentralization.",
"Our research contributes to the literature at the interface of decentralized finance, financial technology (Fintech), and social network analysis and inspires future protocol designs to live up to the promise of decentralized finance for a truly peer-to-peer transaction network."
],
[
"Introduction",
"Blockchain technology is notable for its security, transparency, and reliability worldwide [15], [45].",
"DeFi (Decentralized Finance) is one important blockchain application with over ten billion U.S. dollar market value [15].",
"According to Werner et al.",
"'s research, DeFi is a peer-to-peer financial system [40].",
"DeFi has great potential to replace traditional centralized finance with the help of blockchain technology by using tamper-proof smart contracts to verify peer-to-peer transactions [26].",
"Among the various DeFi programs, a class of lending agreements plays a role similar to that of traditional banks.",
"That is, decentralized banks provide lending and borrowing of on-chain assets, facilitated through protocols for loanable funds (PLFs) [40].",
"PLFs can create distributed ledger-based marketplaces for loanable funds of crypto assets by pooling deposited funds in smart contracts [40].",
"Many decentralized banking platforms have emerged in recent years, including Aave, Compound, MakerDao, and Liquity [15].",
"How do we measure the quality of a decentralized banking platform?",
"According to the study \"SoK: Blockchain Decentralization\" [44], decentralized transactions on decentralized banking platforms are important not only because of their financial connotation but also because blockchains with centralized transactions can be easily manipulated by a few individuals [44], which also threatens blockchain security.",
"Existing literature shows that network characteristics, which are indicators of trading network structures in decentralized markets, significantly affect market outcomes and performance, such as liquidity and volatility [13], [34], [39].",
"Furthermore, a more decentralized network can significantly predict higher returns and lower volatility for the associated DeFi tokens [3].",
"There is still a lack of sufficient research on blockchain decentralization, and decentralization measurement should include multiple dimensions.",
"The paper 'SoK: Blockchain Decentralization' designed a taxonomy to analyze blockchain decentralization in five dimensions: consensus, network, governance, wealth, and transactions, but they found a lack of studies on a transactional perspective [44].",
"This gap in decentralized banks was filled by a recent study in which Ao et al.",
"(2022) [3] applied social network analysis to Aave's user blockchain transaction data to capture the degree of decentralization, network dynamics, and economic performance of Aave.",
"They found that the AAVE token transaction network has a distinct core-periphery structure, with multiple network features in a decentralized dynamic state.",
"However, Aave does not represent all decentralized banks.",
"Moreover, the relationship between the mechanism design of DeFi protocols and their degrees of decentralization has not been well studied.",
"Table REF compares decentralized bank designs in terms of governance, Airdrop, Loan before the deposit, and Stablecoins, where 'Y' is short for 'Yes' and 'N' is short for 'No.'",
"For example, Aave, Compound, and MakerDao have a decentralized autonomous organization (DAO).",
"In contrast, Liquity has an ungoverned protocol that represents a more decentralized mechanism [9].",
"In addition, Liquity airdrops the native token LQTY to the lenders of the asset pool [9].",
"Before answering how these design mechanisms affect the degree of decentralization of decentralized banks, we need to first compare the degree of decentralization of the platforms and the patterns of their network characteristics.",
"Therefore, our study applies social network analysis [3] to blockchain transaction network data from leading decentralized banks, including Liquity, Aave, MakerDaok, and Compound and aims to answer the following research questions (RQ).",
"RQ1 on network decentralization and dynamics: How does network decentralization vary over time and across different decentralized bank protocols?",
"RQ2 on the core-periphery network: What are the core components of the transaction network in each decentralized bank?",
"We complete the core-periphery analysis and characterization of the transaction network using the LIP algorithm [31].",
"Our research successfully compares the network features and transaction decentralization of four major decentralized bank protocols.",
"Due to the introduction of multiple platforms in our comparison, the quantity of data in our study is tens of times larger than that in the previous study [3].",
"To solve the technical issue, we introduce the LIP algorithm, a faster core-peripheral analysis algorithm, in our study [31].",
"Our results (R) reveal that R1: MakerDao and Compound are more dispersed in transactions than AAVE and Liquity.",
"R2: The largest externally owned address cores for LQTY, LUSD, AAVE, Dai, and COMP are centralized exchanges such as Huobi, Coin base, and Binance.",
"The rest of the paper is organized as follows.",
"Section 1.1 addresses the related literature.",
"Section 2 introduces the data and methods.",
"Section 3 presents the results.",
"Section 4 concludes and discusses future research.",
"Figure: Literature review flowchartTable: Mechanism design comparison between four decentralized banksOur research contributes to four lines of literature including decentralized finance (DeFi), the mechanism design of decentralized bank network analysis in finance, and blockchain network analysis.",
"Our research contributes to the decentralized bank literature in the DeFi area.",
"DeFi, decentralized finance, is one of the most discussed emerging technological evolutions in global finance [43].",
"DeFi refers to an alternative financial infrastructure built on top of the Ethereum blockchain [36].",
"It uses smart contracts on blockchain to replicate the existing financial services in a more open, interoperable, and transparent way [36].",
"DeFi brings a brand new trust mechanism in this digital era, which implies a move from trust in banks or states to trust in algorithms and encryption software, according to research [4].",
"Among decentralized finance, the decentralized banks take up a large area [44].",
"According to Ao et al.",
"(2022) [3], decentralized banks differ from centralized banks in two aspects: 1) they replace centralized credit assessments with coded collateral evaluation [25], and 2) they employ smart contracts to execute asset management automatically [5].",
"We introduce the network analysis to several decentralized bank transaction networks [3] [6] [11], including Aave, Liquity, Compound, and MakerDao.",
"Our research contributes to decentralized banks by innovatively comparing the decentralized network features across several decentralized banks."
],
[
"DAO, Airdrop, Stablecoins, Loan and Deposit Mechanism Design",
"Centralized exchanges, such as Bitfines or Poloniex, are trust-based systems and have many limitations.",
"In contrast, decentralized banks facilitate a much more convenient loan experience, according to the Compound white paper, a famous decentralized bank established in 2016 [19].",
"Decentralized banks have brought customers a brand new mechanism for lending, borrowing, and depositing.",
"In terms of platform design, different decentralized banks have different mechanisms.",
"These mechanisms are divided into four aspects: Governance, Airdrop, Loan and Deposit, and Stablecoins.",
"In Table REF , we provide the different mechanisms used by the different platforms.",
"In the governance mechanism, decentralized banks such as Compound, Aave, and MakerDao all have a decentralized autonomous organization (DAO).",
"Token holders can use their tokens to participate in community governance [16].",
"However, Liquity designs a completely autonomous DeFi protocol without a DAO or centralized governance.",
"This innovative mechanism may help it to be more decentralized and transparent [9].",
"Second, in the airdrop process, the incentive mechanism, Compound, and Liquity deliver rewards tokens to the users for liquidity mining [9] [19].",
"The incentive airdrop reward may help the decentralized bank obtain better liquidity and attract more users [41].",
"In a traditional depositing scheme, people can lend an asset only if someone deposits it in a liquid pool [27].",
"However, Liquity and MakerDao devised a new mechanism that allows customers to borrow an asset before any deposits [9].",
"This loan before the deposit mechanism of MakerDao and Liquity also introduces another stablecoin design.",
"Stablecoins are one type of decentralized finance application [26] intended to remedy cryptocurrencies' excess volatility.",
"During stablecoin development, there have been 3 main periods: fiat-backed, crypto-backed, and algorithmic stablecoins [8].",
"A major role of stablecoins is to provide security and stability to investors.",
"Compared with volatile cryptocurrencies such as Bitcoin, research has found that stablecoins act as a safe haven for bitcoin [19].",
"For stablecoins, MakerDao introduced the Dai stablecoin and Liquity introduced the stablecoin LUSD stablecoin [9].",
"Liquity designed a hard and soft peg mechanism for the LUSD stablecoin.",
"Our paper contributes to the decentralized bank mechanism design, including Governance, Airdrop, Loan and Deposit, and Stablecoins, by exploring and analyzing the potential influence of these innovative mechanism designs on the transaction networks of four decentralized banks."
],
[
"Network Analysis in Finance",
"Applying social network analysis to financial markets became popular after the financial crisis from 2008 to 2009 [3].",
"Many studies have found that it is important to explore and evaluate financial network structures to identify systemic risks.",
"Banks that are too centralized, for example, may cause a chain reaction in which their failure may destroy the wider financial system [7] [42].",
"Cong et al.",
"'s (2022) [20] research introduced the network analysis method to the decentralized financial system and successfully analyzed the Ethereum financial network.",
"By introducing network analysis methods for several decentralized banks and comparing the results, our paper further demonstrates the transaction networks and possible influences of decentralized banks, contributing to the field of the decentralized finance field."
],
[
"Network Analysis on Blockchain",
"As a newly emerged and highly concerned field, blockchain is undergoing very rapid development [15].",
"A blockchain is not only a distributed ledger but a network of transactions [37].",
"Each account address in the blockchain can be thought of as a network node.",
"Somin et al.",
"(2018) [37] started their research on network analysis research of ERC20 tokens rending on the Ethereum blockchain and demonstrated the strong power-law properties, richer get richer, in the network.",
"Then Jiang and Liu (2021) [28] spread the network analysis to the NFT area, where they analyze the CryptoKitties' transaction network.",
"Cong et al (2022) [20] provide a network analysis of decentralized finance network analysis on Ethereum.",
"Ao et al.",
"(2022) analyze the Aave decentralized bank transaction network in more detail using the core-periphery method and network feature analysis [3].",
"Our paper contributes to the field of blockchain network analysis field, delving further into existing blockchain-based analysis and exploring the transaction network of decentralized banks in a more detailed manner."
],
[
"Data and Methods",
"Data and code availability We made our code and data open-source, available on GitHubhttps://github.com/SciEcon/Blockchain-Network-Analysis.",
"Figure REF depicts the data science pipeline of our study.",
"Our study has higher automation and computational efficiency than earlier studies [3], enabling cross-sectional comparisons.",
"Figure: Blockchain network analysis methodology flowchart"
],
[
"Data Source and Preprocessing",
"The data for blockchain network analysis are derived from the transaction records of 5 tokens from 4 decentralized finance protocols: LUSD and LQTY of Liquity [9], AAVE of Aave [1], Dai of MakerDao [32], and COMP of Compound [19].",
"These transaction records were obtained from BigQuery dataset of the Ethereum blockchain via the BigQuery integration with the Kaggle kernel [17].",
"The launch dates for the DeFi protocols are summarized in Table REF .",
"Specifically, the data cover the transaction records from the genesis dates of each token to July 12, 2022.",
"The transactions whose from_address or to_address is the Ethereum null address, which is often associated with token-related events such as genesis, mint, or burns [23], were filtered out.",
"We also summarize the total transaction value involved and the number of addresses in Table REF .",
"Figures REF , REF , REF , REF and REF in the Appendix visualize the change in daily transaction value and the number of addresses over time.",
"In addition, we developed undirected daily transaction networks in which the nodes represent Ethereum addresses, and the edges represent the entire daily transaction volume between two addresses, weighted by the transaction values.",
"In other words, we aggregate the transaction values between two addresses without regard to the direction so that the transaction between two addresses will not be calculated repeatedly in the core-periphery structure analysis.",
"Table: Queried data of the DeFi tokens for blockchain network analysis."
],
[
"Network Feature Extraction",
"We extracted 4 network features for all the daily transaction networks built as described in Section REF .",
"In detail, the network features include the number of components, the relative size of the largest component, the modularity score [35], and the standard deviation in the degree centrality.",
"These network features are computed using the Python NetworkX [22] algorithms.",
"According to the conceptual framework, the network features can characterize the difference between more centralized and more decentralized networks and therefore quantify the decentralization of a specific transaction network.",
"For instance, in a 'more centralized' network, in which more vertices are connected to several central vertices, the relative size of the largest component will become larger compared to the decentralized ones."
],
[
"Core-periphery Structure Detection",
"There are two more features for blockchain network analysis that require detecting the core-periphery structure of the daily transaction network.",
"The core-periphery structure refers to a fundamental network pattern that categorizes network nodes [24].",
"Specifically, the network nodes are classified into two categories: “core” nodes, which are densely connected, and “periphery” nodes, which are weakly connected.",
"Transaction networks with a more significant core-periphery structure are more decentralized than those with a less significant structure [3].",
"There are several algorithms for detecting the core-periphery structure of a given network [12] [31], [30] [14] [21].",
"By evaluating the statistical significance of the core-periphery structure in a transaction network, we can construct two additional features, the number of detected core nodes and the average degree of the core nodes, for the blockchain network analysis.",
"The core-periphery structure construction and statistical analysis are conducted using the LIP algorithm [31] in the Python cpnet library [29].",
"Together with the four extracted features, the two new network features are formally defined in Table REF .",
"Table: Definition of the extracted network.",
"↓ means the lower the less decentralized, and ↑ means the more decentralized."
],
[
"Network Feature Dynamics and Correlation",
"Given the calculated network features for each transaction network of each token, we explore how the decentralization of the transaction network for the five DeFi currencies evolves over time using the six network features introduced in Section .",
"The time series plots of the network features for each token are illustrated in Figures REF , REF , REF , REF and REF .",
"By examining the relationship between these dynamic features and the degree of decentralization, we first validated the conceptual framework introduced in Table REF .",
"Horizontally comparing the dynamic features of these four platforms reveals that COMP from Compound and Dai from MakerDao have a substantially higher degree of decentralization than AAVE from Aave and LQTY/LUSD from Liquity.",
"Figure: Time-series plots of network features of the COMP token.Figure: Time-series plots of network features of the LQTY token.In addition to calculating the network dynamics, we measured the correlations between network characteristics to better highlight their relationship with network decentralization.",
"Figures REF , REF , REF , REF and REF in the Appendix depict the feature correlation for LUSD, LQTY, AAVE, Dai, and COMP tokens respectively.",
"Through the correlation heatmap, we can determine the degree of correlation between each platform's network properties.",
"The stronger the correlation, the darker the square.",
"The greater the degree of correlation between network variables moving in the same direction, the more effective and rational the network analysis for that bank's network.",
"AAVE and Dai have a larger correlation degree in the heatmap comparison, but LQTY and LUSD of the platform have a comparatively low correlation degree.",
"This may suggest that the existing trading network is immature and that its features are obscure.",
"In the horizontal comparison of these five tokens' network features, we selected the following aspects of the significant comparison results.",
"In this comparison, we discovered that both of the earliest established platforms, MakerDao and Compound, experienced a peak in the number of components around January 2021, followed by a continuous decline.",
"Next, the graph demonstrates that there is a significant difference in the number of components between AAVE and LQTY and that the quantity of AAVE is much greater than that of LQTY.",
"This indicates that Liquity's trading network may be more straightforward than competing platforms."
],
[
"Modularity score",
"The smaller the modularity score, the more centralized the market.",
"In the horizontal comparison, both LQTY and LUSD had lower values than the two earlier platforms, Compound and MakerDao.",
"This may suggest that Liquity's trading network is less decentralized than the two previous platforms."
],
[
"Standard of degree centrality",
"We discover that LQTY and LUSD have a greater standard degree of centrality than other platforms based on these data.",
"Both Compound and MakerDao, two older decentralized banks, have rather poor scores in this category.",
"The values of LQTY and LUSD likewise exhibit an upward trend.",
"The lower the value, the less centralization there is.",
"This conclusion is identical to the modularity finding, and it matches the results of other network dynamic properties as well.",
"This circumstance suggests that the Liquity platform is currently more centralized than other platforms."
],
[
"Core-periphery Structure Comparison Between Contract Addresses and Externally Owned Addresses",
"In addition to the network feature analysis, we conducted further exploration of the detected core-periphery structure for the transaction networks of each token.",
"As introduced in Section , we conducted the core-periphery structure analysis on the daily transaction networks for the DeFi tokens, LUSD, LQTY, AAVE, Dai, and COMP.",
"Using the Python Web3 package [33], we subsequently queried the real types of addresses detected as core nodes.",
"To further investigate the decentralization of a DeFi token, we further contrasted the core-periphery structure study results in terms of the address types.",
"On the Ethereum blockchain, there are two types of addresses: contract addresses (CA) and externally owned addresses (EOA).",
"The former represents the executable smart contract on the blockchain, whereas the latter consists of user accounts.",
"We extracted the unique addresses that were detected as core nodes for at least one day within the time span of the data source.",
"Figure REF demonstrates the distribution of the days for them being core nodes in terms of the address type.",
"Moreover, we investigated the detailed address information via Etherscan.io [23] of the outlier addresses of the distribution, which are also annotated in Figure REF .",
"Figure: Distribution of the number of core days for EOAs and CAs of the five tokens with the annotated address information of the outliers.We observe that the CA outliers with the most core days are the token contracts created by the DeFi protocol developer.",
"For instance, the CA with the highest number of core days in the AAVE transaction network is the Aave: Staked Aave (642 days), which is the token contract of the AAVE token.",
"Other outliers of CA are the decentralized cryptocurrency exchanges built on Ethereum using smart contracts.",
"For instance,Uniswap [38] is one of the automated liquidity protocols powered by smart constants that exist as outliers in all five token transaction networks, which enables peer-to-peer market making.",
"Another decentralized cryptocurrency exchange that exists as the CA outlier for all five tokens is AirSwap [2], which can also archive peer-to-peer trading of Ethereum tokens.",
"The outliers among EOAs are mostly centralized cryptocurrency exchanges.",
"The most obvious examples are Coinbase [18] and Binance [10], both of which are famous exchanges where users can trade cryptocurrencies.",
"Given the vast variety of trading tools and supporting services for users to earn interest [10], centralized cryptocurrency exchanges with a high volume of transactions have gained immense appeal, where a large number of transactions occur on these EOAs.",
"However, the centralized exchanges bring high centralization” to the decentralized bank transaction networks.",
"Table REF summarizes the first list date for the four DeFi protocols.",
"Table: The first date that the DeFi protocols were first listed on the exchanges, Coinbase and Binance"
],
[
"Conclusion and Future Research",
"According to Cong et al.",
"(2022) [20], the degree of decentralization and the stability of the trading network are both important factors in building trust for decentralized banks and increasing the inclusiveness of the decentralized bank platform.",
"We, therefore, conducted a comparative study of four major decentralized banks including Liquity, Aave, MakerDao, and Compound, evaluating transaction decentralization using social network analysis.",
"We made two major findings: first, the largest externally owned address cores for LQTY, LUSD, AAVE, Dai, and COMP mainly include exchanges such as Huobi, Coin base, and Binance, second, MakerDao and Compound are more decentralized in trading than AAVE and Liquity.",
"Future research can further study the connection between protocol designs and the decentralization level.",
"For example, the higher level of centralization on LQTY and LUSD may be due to three reasons.",
"First, as the Liquity platform has not been established for a long time, there may be fewer users on the platform.",
"Second, as LQTY has not yet been listed on some exchanges, such as Binance, the token may be less well known.",
"This may lead to distrust of the Liquity platform by other decentralized bank participants, resulting in fewer addresses participating in the trading network.",
"Third, Liquity is designed as a nongovernance system, which may leave the platform without royalty users actively interacting in the network.",
"How would the design features of governance, airdrop, loan before deposits, and stablecoins affect network decentralization and other desired properties?",
"Our study provides a direction for future exploration of transaction network analysis and mechanism design for decentralized banks."
]
] | 2212.05632 |
[
[
"Harmonic maps on weighted foliations"
],
[
"Abstract On foliations, there are two kinds of harmonic maps, that is, transversally harmonic map and $(F,F')$-harmonic map which are equivalent when the foliation is minimal.",
"In this paper, we study transversally f-harmonic and $(F,F')_f$-harmonic maps on weighted foliations."
],
[
"Introduction",
"Let $(M,g,\\mathcal {F})$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be foliated Riemannian manifolds and $\\phi :M\\rightarrow M^{\\prime }$ be a smooth foliated map (i.e., $\\phi $ is a smooth leaf-preserving map).",
"Let $Q$ be the normal bundle of $\\mathcal {F}$ and $d_{T}\\phi =d\\phi |_{Q}$ , the restriction of $d\\phi $ on the normal bundle $Q$ .",
"Then $\\phi $ is said to be transversally harmonic if $\\phi $ is a solution of the Eular-Largrange equation $\\tau _{b}(\\phi )=0$ , where $\\tau _b(\\phi )={\\rm tr}_{Q}(\\nabla _{\\rm tr} d_T\\phi )$ is the transversal tension field of $\\phi $ .",
"Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [4], [10], [13], [14], [15], [19], [20], [30].",
"However, a transversally harmonic map is not a critical point of the transversal energy functional [14] $E_{B}(\\phi )=\\frac{1}{2}\\int _{M} | d_T \\phi |^2\\mu _{M}.$ In 2013, S. Dragomir and A. Tommasoli [7] defined a new harmonic map, called $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map, which is a critical point of the transversal energy functional $E_{B}(\\phi )$ .",
"Two definitions are equivalent when $\\mathcal {F}$ is minimal.",
"On the other hand, Y. Chiang and R. Wolak [5] defined the transvesally $f$ -harmonic map.",
"Let $f$ be a basic function on $M$ .",
"The map $\\phi $ is said to be transversally $f$ -harmonic if $\\phi $ is a solution of the Eular-Largrange equation $\\tau _{b,f}(\\phi )=0$ , where $\\tau _{b,f}(\\phi )$ is the transversal $f$ -tension field of $\\phi $ defined by $\\tau _{b,f}(\\phi ) ={\\rm tr}_Q(\\nabla _{\\rm tr}(e^{-f} d_T\\phi )).$ From the first variation formula (Theorem 3.4), the transvesally $f$ -harmonic map is not a critical point of the transversal $f$ -energy functional $E_{B,f}(\\phi ) = \\frac{1}{2}\\int _M |d_T\\phi |^2 e^{-f}\\mu _M.$ Sometimes, we use a function $f$ instead of $e^{-f}$ ([5], [13]).",
"Similarly, the map $\\phi $ is said to be $(\\mathcal {F},\\mathcal {F}^{\\prime })_f $ -harmonic if $\\phi $ is a critical point of the transversal $f$ -energy functional $E_{B,f}(\\phi )$ .",
"If $f$ is constant, then transversally $f$ -harmonic (resp.",
"$(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic) map is just transversally harmonic (resp.",
"$(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic) map.",
"It is well-known [13] that a transversally harmonic map is a critical point of the transversal $f$ -energy functional for special function $f$ satisfying $df=-\\kappa _B$ ($\\kappa _B$ is the basic part of the mean curvature form $\\kappa $ of $\\mathcal {F}$ ).",
"Hence if $f_\\kappa $ is a solution of $df=-\\kappa _B$ , then $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f_\\kappa }$ -harmonic map is equivalent to the transversally harmonic map.",
"Originally, the $f$ -harmonic maps on Riemannian manifolds were studied by A. Lichnerowicz in 1969 [21], later by J.Eells and L. Lemaire in 1977 [9].",
"In this article, we study $f$ -harmonic maps (transversally $f$ -harmonic map and $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map) on weighted foliations.",
"A weighted foliation $(M,\\mathcal {F}, g,e^{-f}\\nu )$ is a Riemannian foliation endowed with a weighted transversal volume form $e^{-f}\\nu $ and some basic function $f$ , where $\\nu $ is the transversal volume form of $\\mathcal {F}$ .",
"The geometry of a weighted manifold (or a smooth metric measure space) were developed by D. Bakry and M. Émery [2] and studied by many authors [24], [25], [35], [36], [39], [41].",
"Also, the geometry of weighted manifolds is closely related with that of self-shrinkers and gradient Ricci solitons.",
"An important geometric tool is the Bakry-Émery Ricci tensor, which was first introduced by A. Lichnerowicz [22].",
"For the study of weighted foliations, we define the Bakry-Émery type Ricci tensor ${\\rm Ric}_f^Q$ on $(M,\\mathcal {F}, g,e^{-f}\\nu )$ by ${\\rm Ric}_f^Q = {\\rm Ric}^Q +{\\rm Hess}_T f,$ where ${\\rm Ric}^Q$ is the transversal Ricci tensor and ${\\rm Hess}_T f$ is the transversal Hessian [13] of $\\mathcal {F}$ .",
"We call ${\\rm Ric}_f^Q$ as transversal Bakry-Émery Ricci tensor on a weighted foliation.",
"Then we have the following Theorems.",
"Theorem 1.1 (cf.",
"Theorem 3.6) Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a closed Riemannian manifold $M$ with ${\\rm Ric}_f^Q\\ge 0$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a foliated Riemannian manifold with non-positive transversal sectional curvature $K^{Q^{\\prime }}$ .",
"Then a transversally $f$ -harmonic map $\\phi :M \\rightarrow M^{\\prime }$ is transversally totally geodesic.",
"In addition, (1) if ${\\rm Ric}_f^Q>0$ at some point, then $\\phi $ is transversally constant; (2) if $K^{Q^{\\prime }}<0$ , then $\\phi $ is transverseally constant or $\\phi (M)$ is a transversally geodesic closed curve.",
"Let $\\mathcal {K}$ be the set of all basic functions $f$ satisfying $ i(\\kappa _B^\\sharp )df \\le 0$ .",
"Remark 1.2 For $f\\in \\mathcal {K}$ , Theorem 1.1 holds for $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map.",
"(cf.",
"Theorem 3.8).",
"Moreover, we study the Liouville type theorems for $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ and transversally $f$ -harmonic map, respectively.",
"The Liouville type theorem has been studied by many researchers [11], [29], [34], [42] on Riemannian manifolds and [10], [14], [15], [30] on foliations.",
"Specially, see [33], [40] for $f$ -harmonic maps on weighted Riemannian manifolds.",
"Let $\\mu _0$ be the infimum of the spectrum of the eigenvalues of the weighted basic Laplacian $\\Delta _{B,f}$ acting on $f$ -weighted $L^2$ -basic functions on $M$ .",
"Theorem 1.3 (cf.",
"Theorem 4.2) Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a complete manifold whose all leaves are compact and the mean curvature form is bounded.",
"Let $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a foliated Riemannian manifold with non-positive transversal sectional curvature.",
"Let $f\\in \\mathcal {K}$ and $\\phi :M\\rightarrow N$ be a transversally $f$ -harmonic map with $E_{B,f}(\\phi )<\\infty $ .",
"Then (1) if ${\\rm Ric}_f^Q \\ge 0$ at all points, then $\\phi $ is transversally totally geodesic; (2) if ${\\rm Ric}_f^Q\\ge 0$ at all points and $\\int _M e^{-f}\\mu _M= \\infty $ , then $\\phi $ is transvesally constant; (3) if ${\\rm Ric}_f^Q \\ge -\\mu _0$ at all points and ${\\rm Ric}^Q_f>-\\mu _0$ at some point, then $\\phi $ is transversally constant.",
"Remark 1.4 Theorem 1.3 holds for $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic maps under the same conditions in Theorem 1.3.",
"(cf.",
"Theorem 4.4)."
],
[
"Preliminaries",
"Let $(M,g,\\mathcal {F})$ be a foliated Riemannian manifold of dimension $n$ with a foliation $\\mathcal {F}$ of codimension $q (=n-p)$ and a bundle-like metric $g$ with respect to $\\mathcal {F}$ [28], [37].",
"Let $Q=TM/T\\mathcal {F}$ be the normal bundle of $\\mathcal {F}$ , where $T\\mathcal {F}$ is the tangent bundle of $\\mathcal {F}$ .",
"Let $g_Q$ be the induced metric by $g$ on $Q$ , that is, $g_Q = \\sigma ^*(g|_{T\\mathcal {F}^\\perp })$ , where $\\sigma :Q\\rightarrow T\\mathcal {F}^\\perp $ is the canonical bundle isomorphism.",
"So we consider $Q\\cong T\\mathcal {F}^\\perp $ .",
"Then $g_Q$ is the holonomy invariant metric on $Q$ , meaning that $L_Xg_Q=0$ for $X\\in T\\mathcal {F}$ , where $L_X$ is the transverse Lie derivative with respect to $X$ .",
"Let $\\nabla ^Q$ be the transverse Levi-Civita connection on the normal bundle $Q$ [37], [38] and $R^Q$ be the transversal curvature tensor of $\\nabla ^Q\\equiv \\nabla $ , which is defined by $R^Q(X,Y)=[\\nabla _X,\\nabla _Y]-\\nabla _{[X,Y]}$ for any $X,Y\\in \\Gamma TM$ .",
"Let $K^Q$ and ${\\rm Ric}^Q $ be the transversal sectional curvature and transversal Ricci operator with respect to $\\nabla $ , respectively.",
"Let $\\kappa $ be the mean curvature form of $\\mathcal {F}$ , which is defined by $\\kappa (X) = g_Q (\\sum _{i=1}^p\\pi (\\nabla ^M_{f_j}f_j), \\pi (X))$ for any tangent vector $X\\in \\Gamma TM$ , where $\\pi :TM\\rightarrow Q$ is the natural projection and $\\lbrace f_j\\rbrace $ is a local orthonormal basis of $T\\mathcal {F}$ .",
"The foliation $\\mathcal {F}$ is said to be minimal if $\\kappa =0$ [37].",
"Let $\\Omega _B^r(\\mathcal {F})$ be the space of all basic $r$ -forms, i.e., $\\omega \\in \\Omega _B^r(\\mathcal {F})$ if and only if $i(X)\\omega =0$ and $L_X\\omega =0$ for any $X\\in \\Gamma T\\mathcal {F}$ , where $i(X)$ is the interior product.",
"Then $\\Omega ^*(M)=\\Omega _B^*(\\mathcal {F})\\oplus \\Omega _B^*(\\mathcal {F})^\\perp $ [1].",
"It is well known that $\\kappa _B$ is closed, i.e., $d\\kappa _B=0$ [1], [31], where $\\kappa _B$ is the basic part of $\\kappa $ .",
"Let $\\bar{*}:\\Omega _B^r(\\mathcal {F})\\rightarrow \\Omega _B^{q-r}(\\mathcal {F})$ be the star operator given by $\\bar{*}\\omega = (-1)^{(n-q)(q-r)} *(\\omega \\wedge \\chi _{\\mathcal {F}}),\\quad \\omega \\in \\Omega _B^r(\\mathcal {F}),$ where $\\chi _{\\mathcal {F}}$ is the characteristic form of $\\mathcal {F}$ and $*$ is the Hodge star operator associated to $g$ .",
"Let $\\langle \\cdot ,\\cdot \\rangle $ be the pointwise inner product on $\\Omega _B^r(\\mathcal {F})$ , which is given by $\\langle \\omega _1,\\omega _2\\rangle \\nu = \\omega _1\\wedge \\bar{*} \\omega _2,$ where $\\nu $ is the transversal volume form such that $*\\nu =\\chi _{\\mathcal {F}}$ .",
"Let $\\delta _B :\\Omega _B^r (\\mathcal {F})\\rightarrow \\Omega _B^{r-1}(\\mathcal {F})$ be the operator defined by $\\delta _B\\omega = (-1)^{q(r+1)+1} \\bar{*} (d_B-\\kappa _B \\wedge ) \\bar{*}\\omega ,$ where $d_B = d|_{\\Omega _B^*(\\mathcal {F})}$ .",
"Locally, $\\delta _{B}$ is expressed by $\\delta _{B} = -\\sum _a i(E_a) \\nabla _{E_a} + i (\\kappa _{B}^\\sharp ),$ where $(\\cdot )^\\sharp $ is the dual vector field of $(\\cdot )$ [16] and $\\lbrace E_a\\rbrace _{a=1,\\cdots ,q}$ is a local orthonormal basic frame on $Q$ .",
"It is well known [32] that $\\delta _B$ is the formal adjoint of $d_B$ with respect to the global inner product $\\ll \\cdot ,\\cdot \\gg $ , which is defined by $\\ll \\omega _1,\\omega _2\\gg =\\int _M \\langle \\omega _1,\\omega _2\\rangle \\mu _M$ for any compactly supported basic forms $\\omega _1$ or $\\omega _2$ , where $\\mu _M =\\nu \\wedge \\chi _{\\mathcal {F}}$ is the volume form.",
"There exists a bundle-like metric such that the mean curvature form satisfies $\\delta _B\\kappa _B=0$ on compact manifolds [6], [26], [27].",
"The basic Laplacian $\\Delta _B$ acting on $\\Omega _B^*(\\mathcal {F})$ is given by $\\Delta _B=d_B\\delta _B+\\delta _B d_B.$ Let $Y$ be a foliated vector field, i.e., $[Y,Z]\\in \\Gamma T\\mathcal {F}$ for all $Z\\in \\Gamma T\\mathcal {F}$ [18] and put $\\bar{Y} = \\pi (Y)$ .",
"Now we define the bundle map $A_Y:\\Gamma Q\\rightarrow \\Gamma Q$ for any $Y\\in TM$ by $A_Y s =L_Ys-\\nabla _Ys,$ where $L_Y s = \\pi [Y,Y_s]$ for $\\pi (Y_s)=s$ .",
"It is well-known [18] that for any foliated vector field $Y$ $A_Y s = -\\nabla _{Y_s}\\bar{Y},$ where $Y_s$ is the vector field such that $\\pi (Y_s)=s$ .",
"So $A_Y$ depends only on $\\bar{Y}=\\pi (Y)$ and is a linear operator.",
"Moreover, $A_Y$ extends in an obvious way to tensors of any type on $Q$ [18].",
"Then we have the generalized Weitzenböck formula on $\\Omega _B^*(\\mathcal {F})$ [12]: for any $\\omega \\in \\Omega _B^r(\\mathcal {F}),$ $\\Delta _B \\omega = \\nabla _{\\rm tr}^*\\nabla _{\\rm tr}\\omega +F(\\omega )+A_{\\kappa _B^\\sharp }\\omega ,$ where $F(\\omega )=\\sum _{a,b}\\theta ^a \\wedge i(E_b)R^Q(E_b,E_a)\\omega $ and $\\nabla _{\\rm tr}^*\\nabla _{\\rm tr}\\omega =-\\sum _a \\nabla ^2_{E_a,E_a}\\omega +\\nabla _{\\kappa _B^\\sharp }\\omega .$ The operator $\\nabla _{\\rm tr}^*\\nabla _{\\rm tr}$ is positive definite and formally self adjoint on the space of basic forms [12].",
"If $\\omega $ is a basic 1-form, then $F(\\omega )^\\sharp ={\\rm Ric}^Q(\\omega ^\\sharp )$ .",
"Now, let $(M,g,\\mathcal {F}, e^{-f}\\nu )$ be a weighted foliation, that is, Riemannian foliation endowed with a weighted transversal volume form $e^{-f}\\nu $ , where $f$ is a basic function.",
"The formal adjoint operator $\\delta _{B,f}$ of $d$ with respect to volume form $e^{f}\\mu _M$ is given by $\\delta _{B,f}\\omega =e^f \\delta _B (e^{-f}\\omega ) =\\delta _B\\omega + i(\\nabla _{\\rm tr} f)\\omega $ for any basic form $\\omega $ , where $\\nabla _{\\rm tr} f = \\sum _a E_a(f)E_a$ .",
"That is, for any basic forms $\\omega \\in \\Omega _B^r(\\mathcal {F})$ and $\\eta \\in \\Omega _B^{r+1}(\\mathcal {F})$ , $\\int _M \\langle d_B\\omega ,\\eta \\rangle e^{-f}\\mu _M =\\int _M \\langle \\omega ,\\delta _{B,f}\\eta \\rangle e^{-f}\\mu _M.$ The weighted basic Laplacian operator $\\Delta _{B,f}$ is defined by $\\Delta _{B,f} = d_B\\delta _{B,f} + \\delta _{B,f}d_B.$ From (REF ), we have $\\Delta _{B,f} = \\Delta _B + L_{\\nabla _{\\rm tr} f}.$ Specially, $\\Delta _{B,f} =\\Delta _B + i(\\nabla _{\\rm tr} f)d_B$ on $\\Omega _B^0(\\mathcal {F})$ .",
"Then we have the following.",
"Lemma 2.1 Let $(M,g,\\mathcal {F})$ be a closed, connected Riemannian manifold with a foliation $\\mathcal {F}$ .",
"If $(\\Delta _{B,f} -\\kappa _B)h\\ge 0$ (or $\\le 0$ ) for any basic function $h$ , then $h$ is constant.",
"The proof is similar to [17].",
"That is, let $f$ be a basic function on $M$ .",
"Since $\\Delta _B-\\kappa _B =\\Delta -\\kappa $ on $\\Omega _B^0(\\mathcal {F})$ [32], we have that on $\\Omega _B^0(\\mathcal {F})$ $\\Delta _{B,f} -\\kappa _B =\\Delta _B -\\kappa _B + i(\\nabla _{\\rm tr} f) d_B = \\Delta -\\kappa + i(\\nabla f)d,$ where $\\Delta $ is the Laplace operator on $M$ .",
"The operator of right hand side in the above is a second order elliptic operator, by the maximum principle, the proof follows.",
"And the generalized Bakry-Émery-Ricci curvature ${\\rm Ric}_f^Q$ is defined by ${\\rm Ric}_f^Q = {\\rm Ric}^Q + {\\rm Hess}_T f,$ where ${\\rm Hess}_T f =\\nabla _{tr}d_B f$ is the transversal Hessian [13].",
"Note that the Bakry-Émery-Ricci curvature ${\\rm Ric}_f $ on a Riemannian manifold is related to the Ricci soliton, specially gradient Ricci solitions.",
"On foliated Riemannian manifold, the generalized Bakry-Émery-Ricci curvature ${\\rm Ric}_f^Q$ is related to the transversal Ricci solitions, which are special solutions of the transversal Ricci flow [23].",
"For later use, we recall the transversal divergence theorem on a foliated Riemannian manifold.",
"Theorem 2.2 [44] Let $(M,g,\\mathcal {F})$ be a closed, oriented Riemannian manifold with a transversally oriented foliation $\\mathcal {F}$ and a bundle-like metric $g$ with respect to $\\mathcal {F}$ .",
"Then for a vector field $X\\in \\Gamma TM$ , $\\int _M \\operatorname{div_\\nabla }(\\bar{X}) \\mu _{M}= \\int _M g_Q(\\bar{X},\\kappa ^\\sharp )\\mu _{M},$ where $\\operatorname{div_\\nabla } (\\bar{X})$ denotes the transversal divergence of $\\bar{X}$ with respect to the connection $\\nabla $ ."
],
[
"General facts",
"Let $(M, g,\\mathcal {F})$ and $(M^{\\prime }, g^{\\prime },\\mathcal {F}^{\\prime })$ be two foliated Riemannian manifolds and let $\\phi :(M,g,\\mathcal {F})\\rightarrow (M^{\\prime }, g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map, i.e., $d\\phi (T\\mathcal {F})\\subset T\\mathcal {F}^{\\prime }$ .",
"We define $d_T\\phi :Q \\rightarrow Q^{\\prime }$ by $d_T\\phi := \\pi ^{\\prime } \\circ d \\phi \\circ \\sigma .$ Then $d_T\\phi $ is a section in $ Q^*\\otimes \\phi ^{-1}Q^{\\prime }$ , where $\\phi ^{-1}Q^{\\prime }$ is the pull-back bundle on $M$ .",
"Let $\\nabla ^\\phi $ and $\\tilde{\\nabla }$ be the connections on $\\phi ^{-1}Q^{\\prime }$ and $Q^*\\otimes \\phi ^{-1}Q^{\\prime }$ , respectively.",
"The map $\\phi :(M, g,\\mathcal {F})\\rightarrow (M^{\\prime }, g^{\\prime },\\mathcal {F}^{\\prime })$ is called transversally totally geodesic if it satisfies $\\tilde{\\nabla }_{\\rm tr}d_T\\phi =0,$ where $(\\tilde{\\nabla }_{\\rm tr}d_T\\phi )(X,Y):=(\\tilde{\\nabla }_X d_T\\phi )(Y)$ for any $X,Y\\in \\Gamma Q$ .",
"Note that if $\\phi :(M,g,\\mathcal {F})\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ is transversally totally geodesic with $d\\phi (Q)\\subset Q^{\\prime }$ , then, for any transversal geodesic $\\gamma $ on $M$ , $\\phi \\circ \\gamma $ is also transversal geodesic.",
"From now on, we use $\\nabla $ instead of all induced connections if we have no confusion.",
"The transversal tension field $\\tau _{b}(\\phi )$ of $\\phi $ is defined by $\\tau _{b}(\\phi ):={\\rm tr}_{Q}(\\nabla _{\\rm tr} d_T\\phi )=\\sum _a (\\nabla _{E_a}d_T\\phi )(E_a).$ Let $\\Omega $ be a compact domain of $M$ .",
"The transversal energy of $\\phi $ on $\\Omega \\subset M$ is defined by $E_{B}(\\phi ;\\Omega )={1\\over 2}\\int _{\\Omega } | d_T \\phi |^2\\mu _{M}.$ The map $\\phi $ is said to be $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic [7] if $\\phi $ is a critical point of the transversal energy functional $E_{B}$ .",
"Let $V\\in \\phi ^{-1}Q^{\\prime }$ and let $\\phi _t$ be a foliated variation with $\\phi _0=\\phi $ and ${d\\phi _t\\over dt}|_{t=0}=V$ .",
"Then we have the first variational formula [14].",
"That is, ${d\\over dt}E_{B}(\\phi _t;\\Omega )|_{t=0}=-\\int _{\\Omega } \\langle V,\\tau _{b}(\\phi )-d_T\\phi (\\kappa _B^\\sharp )\\rangle \\mu _{M},$ where $\\langle \\cdot ,\\cdot \\rangle $ is the pull-back metric on $\\phi ^{-1}Q^{\\prime }$ .",
"Trivially, $\\phi $ is $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map if and only if ${\\tau }_{b}(\\phi )-d_T\\phi (\\kappa _B^\\sharp )=0$ .",
"But $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map is not equivalent to the transversally harmonic map.",
"Let $\\Omega _B^r(E)$ be the space of $E$ -valued basic $r$ -forms on $M$ , where $E=\\phi ^{-1}Q^{\\prime }$ .",
"We define $d_\\nabla : \\Omega _B^r(E)\\rightarrow \\Omega _B^{r+1}(E)$ by $d_\\nabla (\\omega \\otimes s)=d_B\\omega \\otimes s+(-1)^r\\omega \\wedge \\nabla s$ for any $s\\in \\Gamma E$ and $\\omega \\in \\Omega _B^r(\\mathcal {F})$ .",
"Let $\\delta _\\nabla $ be a formal adjoint of $d_\\nabla $ with respect to the inner product induced from (REF ).",
"Trivially, we have the following remark.",
"Remark 3.1 Let $\\phi :(M,\\mathcal {F})\\rightarrow (M^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Then $d_\\nabla (d_T\\phi )=0,\\quad \\delta _\\nabla d_T\\phi =-\\tau _b (\\phi ) +d_T\\phi (\\kappa _B^\\sharp ).$ Now we define the Laplacian $\\Delta $ on $\\Omega _B^*(E)$ by $\\Delta =d_\\nabla \\delta _\\nabla +\\delta _\\nabla d_\\nabla .$ Then the generalized Weitzenböck type formula (REF ) is extended to $\\Omega _B^*(E)$ as follows [14]: for any $\\Psi \\in \\Omega _B^r(E)$ , $\\Delta \\Psi = \\nabla _{\\rm tr}^*\\nabla _{\\rm tr} \\Psi + A_{\\kappa _{B}^\\sharp } \\Psi + F(\\Psi ),$ where $ \\nabla _{\\rm tr}^*\\nabla _{\\rm tr}$ , $A_X$ and $F(\\Psi )$ are naturally extended to $\\Omega _B^r(E)$ .",
"Moreover, for any $ \\Psi \\in \\Omega _B^r(E)$ , $\\frac{1}{2}\\Delta _B|\\Psi |^{2}=\\langle \\Delta \\Psi , \\Psi \\rangle -|\\nabla _{\\rm tr} \\Psi |^2-\\langle A_{\\kappa _{B}^\\sharp }\\Psi , \\Psi \\rangle -\\langle F(\\Psi ),\\Psi \\rangle .$ Then we have the generalized Weitzenböck type formula as follows.",
"Proposition 3.2 [14] Let $\\phi :(M, g,\\mathcal {F}) \\rightarrow (M^{\\prime }, g^{\\prime }, \\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Then $\\frac{1}{2}\\Delta _B| d_T \\phi |^{2}= - |\\nabla _{\\rm tr} d_T \\phi |^2 -\\langle F(d_T\\phi ),d_T\\phi \\rangle -\\langle d_\\nabla \\tau _b(\\phi ),d_T\\phi \\rangle +\\langle \\nabla _{\\kappa _B^\\sharp }d_T\\phi ,d_T\\phi \\rangle ,$ where $\\langle F(d_T\\phi ),d_T\\phi \\rangle &=\\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric^{Q}}(E_a)),d_T \\phi (E_a)) \\\\&-\\sum _{a,b}g_{Q^{\\prime }}( R^{Q^{\\prime }}(d_T \\phi (E_b), d_T \\phi (E_a))d_T \\phi (E_a), d_T \\phi (E_b)).$ By $\\Psi =d_T\\phi $ in (REF ), it is trivial from Remark 3.1.",
"Theorem 3.3 Let $\\phi :(M,g,\\mathcal {F},e^{-f}\\nu ) \\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Then $\\frac{1}{2}\\Delta _{B,f}|d_T\\phi |^2 =-|\\nabla _{\\rm tr}d_T\\phi |^2 - \\langle F_f(d_T\\phi ),d_T\\phi \\rangle -\\langle d_\\nabla (\\bar{\\tau }_{b,f}(\\phi )),d_T\\phi \\rangle + \\frac{1}{2} \\kappa _B^\\sharp (|d_T\\phi |^2),$ where $\\bar{\\tau }_{b,f}(\\phi ) =\\tau _b(\\phi ) -d_T\\phi (\\nabla _{\\rm tr}f)$ and $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle &=\\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a)) \\\\&-\\sum _{a,b}g_{Q^{\\prime }}( R^{Q^{\\prime }}(d_T \\phi (E_b), d_T \\phi (E_a))d_T \\phi (E_a), d_T \\phi (E_b)).$ From (REF ), we know $\\langle F(d_T\\phi ),d_T\\phi \\rangle = \\langle F_f (d_T\\phi ),d_T\\phi \\rangle - \\langle d_T\\phi ({\\rm Hess}_Tf),d_T \\phi \\rangle ,$ where $\\langle d_T\\phi ({\\rm Hess}_Tf),d_T \\phi \\rangle := \\sum _a g_{Q^{\\prime }} (d_T\\phi (\\nabla _{E_a} \\nabla _{\\rm tr} f),d_T\\phi (E_a))$ .",
"Hence from (REF ) and (REF ), $\\frac{1}{2}\\Delta _{B,f}|d_T\\phi |^2=&-|\\nabla _{\\rm tr}d_T\\phi |^2 - \\langle F_f(d_T\\phi ),d_T\\phi \\rangle +\\langle \\nabla _{\\kappa _B^\\sharp } d_T\\phi ,d_T\\phi \\rangle -\\langle d_\\nabla (\\tau _b(\\phi )),d_T\\phi \\rangle \\\\&+\\langle d_T\\phi ({\\rm Hess}_Tf),d_T\\phi \\rangle +\\frac{1}{2} \\langle \\nabla _{\\rm tr}|d_T\\phi |^2,\\nabla _{\\rm tr} f\\rangle .$ Note that $(\\nabla _{\\rm tr} d_T\\phi )(X,Y) =(\\nabla _{\\rm tr} d_T\\phi )(Y, X)$ for any vector fields $X,Y \\in \\Gamma Q$ .",
"Hence if we choose a local orthonormal basic frame $\\lbrace E_a\\rbrace $ such that $\\nabla E_a=0$ at $x\\in M$ , then $\\frac{1}{2} \\langle \\nabla _{\\rm tr}|d_T\\phi |^2,\\nabla _{\\rm tr} f\\rangle &=\\sum _a \\langle (\\nabla _{\\nabla _{tr} f} d_T\\phi )(E_a),d_T\\phi (E_a)\\rangle \\\\&=\\sum _a\\langle (\\nabla _{E_a} d_T\\phi )(\\nabla _{\\rm tr}f),d_T\\phi (E_a)\\rangle \\\\&=\\sum _a\\langle \\nabla _{E_a} d_T\\phi (\\nabla _{\\rm tr}f),d_T\\phi (E_a)\\rangle -\\sum _a\\langle d_T\\phi (\\nabla _{E_a}\\nabla _{\\rm tr}f),d_T\\phi (E_a)\\rangle \\\\&=\\langle d_\\nabla (d_T\\phi (\\nabla _{\\rm tr}f)),d_T\\phi \\rangle -\\langle d_T\\phi ({\\rm Hess}_Tf),d_T\\phi \\rangle .$ That is, $\\langle d_T\\phi ({\\rm Hess}_Tf),d_T\\phi \\rangle +\\frac{1}{2} \\langle \\nabla _{\\rm tr}|d_T\\phi |^2,\\nabla _{\\rm tr} f\\rangle =\\langle d_\\nabla (d_T\\phi (\\nabla _{\\rm tr}f)),d_T\\phi \\rangle .$ From (REF ) and (REF ), the proof follows."
],
[
"$f$ -harmonic maps",
"Let $f$ be a basic function on $M$ and let $\\phi :(M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Let $\\tau _{b,f}:={\\rm tr}_Q (\\nabla _{\\rm tr}( e^{-f} d_T\\phi ))$ be the transversal $f$ -tension field.",
"Then $\\tau _{b,f}(\\phi ) =( \\tau _b(\\phi ) - d_T\\phi (\\nabla _{\\rm tr}f))e^{-f}=\\bar{\\tau }_{b,f}(\\phi )e^{-f}.$ The transversally $f$ -harmonic map is a solution of the Eular-Largrange equation $\\tau _{b,f}(\\phi )=0$ (equivalently, $\\bar{\\tau }_{b,f}(\\phi )=0$ ).",
"The map $\\phi $ is said to be $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map if $\\phi $ is a critical point of the transversal $f$ -energy functional $E_{B,f}(\\phi )$ given by $E_{B,f}(\\phi ) = \\frac{1}{2}\\int _\\Omega |d_T\\phi |^2 e^{-f}\\mu _M.$ Remark that if $f$ is constant, then a transversally $f$ -harmonic and $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map are transversally harmonic and $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map, respectively.",
"Theorem 3.4 $($ The first variational formula$)$ Let $\\phi :(M, g, \\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime }, g^{\\prime }, \\mathcal {F}^{\\prime })$ be a smooth foliated map and $\\lbrace \\phi _t\\rbrace $ be a smooth foliated variation of $\\phi $ supported in a compact domain $\\Omega $ .",
"Then ${d\\over dt}E_{B,f}(\\phi _t;\\Omega )|_{t=0}=-\\int _{\\Omega } \\langle V, {\\bar{\\tau }}_{b,f}(\\phi )-d_T\\phi (\\kappa _B^\\sharp )\\rangle e^{-f} \\mu _{M},$ where $V$ is the variation vector field of $\\phi _t$ .",
"It is trivial from [13].",
"From Theorem 3.4, the map $\\phi :M\\rightarrow M^{\\prime }$ is $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map if and only if $\\tilde{\\tau }_{b,f}(\\phi ) := \\bar{\\tau }_{b,f}(\\phi ) -d_T\\phi (\\kappa _B^\\sharp ) =0.$ In general, $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map and transversally $f$ -harmonic map are not equivalent unless $\\mathcal {F}$ is minimal.",
"For more information about the transversally $f$ -harmonic map, see [5].",
"Lemma 3.5 Let $(M,\\mathcal {F},g,e^{-f}\\nu )$ be a weighted foliation and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a Riemannian foliation.",
"(1) If $\\phi :M\\rightarrow M^{\\prime }$ is a transversally $f$ -harmonic map, then $|d_T\\phi |\\Delta _{B,f}|d_T\\phi |=|d_{B}|d_T\\phi ||^{2}-|\\nabla _{\\rm tr}d_T\\phi |^{2}-\\langle F_f(d_T\\phi ),d_T\\phi \\rangle +|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |).$ (2) If $\\phi :M\\rightarrow M^{\\prime }$ be a $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map, then $|d_T\\phi |\\Delta _{B,f}|d_T\\phi |=&|d_{B}|d_T\\phi ||^{2}-|\\nabla _{\\rm tr}d_T\\phi |^{2}-\\langle F_f(d_T\\phi ),d_T\\phi \\rangle -\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle \\\\&+|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |).$ By a simple calculation, we have $\\frac{1}{2}\\Delta _{B,f}| d_T \\phi |^{2}=|d_T\\phi |\\Delta _{B,f}|d_T\\phi |-|d_{B}|d_T\\phi ||^{2}.$ Hence the proofs follow from Theorem 3.3 and (REF ).",
"Then we have the following.",
"Theorem 3.6 Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a closed manifold $M$ with ${\\rm Ric}_{f}^Q\\ge 0$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a Riemannian foliation with $K^{Q^{\\prime }}\\le 0$ .",
"Then any transversally $f$ -harmonic map $\\phi :M\\rightarrow M^{\\prime }$ is always transversally totally geodesic.",
"In addition, (1) if ${\\rm Ric}_f^Q>0$ at some point, then $\\phi $ is transversally constant.",
"(2) if $K^{Q^{\\prime }}<0$ , then $\\phi $ is transversally constant or $\\phi (M)$ is a transversally geodesic closed curve.",
"By the first Kato's inequality [3], we have $|\\nabla _{\\rm tr}d_T\\phi |\\ge |d_{B}|d_T\\phi ||.$ Since $\\phi :M\\rightarrow M^{\\prime }$ is a transversally $f$ -harmonic map, $\\tau _{b,f}(\\phi )=0$ .",
"From (REF ) and (REF ), we have $|d_T\\phi | (\\Delta _{B,f}-\\kappa _B^\\sharp )|d_T\\phi |\\le -\\langle F_f(d_T\\phi ),d_T\\phi \\rangle .$ From the assumptions ${\\rm Ric}_f^Q\\ge 0$ and $K^{Q^{\\prime }}\\le 0$ , $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ and so $(\\Delta _{B,f} -\\kappa _B^\\sharp )|d_T\\phi |\\le 0.$ By Lemma 2.1, $|d_T\\phi |$ is constant.",
"Again, from (REF ), we have $|\\nabla _{\\rm tr} d_T \\phi |^2+\\langle F_f(d_T\\phi ),d_T\\phi \\rangle =0.$ Since $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ , from (REF ), we have $|\\nabla _{\\rm tr} d_T \\phi |^2=0 \\quad \\textrm {and}\\quad \\langle F_f(d_T\\phi ),d_T\\phi \\rangle =0.$ Thus, $\\nabla _{\\rm tr}d_T\\phi =0$ , i.e., $\\phi $ is transversally totally geodesic.",
"Furthermore, from (REF ) and (REF ), we get $\\left\\lbrace \\begin{array}{ll}g_{Q^{\\prime }}(d_T\\phi ({\\rm Ric}_f^{Q}(E_a)),d_T\\phi (E_a))= 0,\\\\\\\\g_{Q^{\\prime }}(R^{Q^{\\prime }}(d_T\\phi (E_a),d_T\\phi (E_b))d_T\\phi (E_a),d_T\\phi (E_b))= 0\\end{array}\\right.$ for any indices $a$ and $b$ .",
"If ${\\rm Ric}_f^{Q}$ is positive at some point, then $d_T\\phi =0$ , i.e., $\\phi $ is transversally constant, which proves (1).",
"For the statement (2), if the rank of $d_T\\phi \\ge 2$ , then there exists a point $x\\in M$ such that at least two linearly independent vectors at $\\phi (x)$ , say, $d_T\\phi (E_1)$ and $d_T\\phi (E_2)$ .",
"Since $K^{Q^{\\prime }}<0$ , $g_{Q^{\\prime }}(R^{Q^{\\prime }}(d_T\\phi (E_1),d_T\\phi (E_2))d_T\\phi (E_2),d_T\\phi (E_1))<0,$ which contradicts (REF ).",
"Hence the rank of $d_T\\phi <2$ , that is, the rank of $d_T\\phi $ is zero or one everywhere.",
"If the rank of $d_T\\phi $ is zero, then $\\phi $ is transversally constant.",
"If the rank of $d_T\\phi $ is one, then $\\phi (M)$ is a transversally geodesic closed curve.",
"Remark 3.7 Note that if $\\phi $ is $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic, then $\\delta _\\nabla (e^{-f}d_T\\phi ) =-\\tilde{\\tau }_{b,f}(\\phi )=0.$ Now, we restrict a basic function $f$ to study $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map.",
"That is, let $\\mathcal {K}$ be the set of basic functions such that $i(\\kappa _B^\\sharp )d_B f\\le 0$ .",
"That is, $\\mathcal {K} =\\lbrace f\\in \\Omega _B^0(\\mathcal {F})| i(\\kappa _B^\\sharp )d_Bf \\le 0\\rbrace .$ Trivially, $f_\\kappa \\in \\mathcal {K}$ because of $d_Bf_\\kappa =-\\kappa _B$ .",
"And if $\\mathcal {F}$ is taut, then $\\mathcal {K} = \\Omega _B^0(\\mathcal {F})$ .",
"Then we have the following.",
"Theorem 3.8 Let $(M,\\mathcal {F},g,e^{-f}\\nu )$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be as in Theorem 3.6.",
"If $f\\in \\mathcal {K}$ , then any $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f}$ -harmonic map $\\phi :M\\rightarrow M^{\\prime }$ is transversally totally geodesic.",
"In addition, (1) if ${\\rm Ric}_f^Q>0$ at some point, then $\\phi $ is transversally constant.",
"(2) if $K^{Q^{\\prime }}<0$ , then $\\phi $ is transverseally constant or $\\phi (M)$ is a transversally geodesic closed curve.",
"From (REF ) and (REF ), we get $|d_T\\phi | \\Delta _{B,f}|d_T\\phi | \\le -\\langle F_f(d_T\\phi ),d_T\\phi \\rangle -\\langle d_\\nabla i(\\kappa _B^\\sharp )d_T\\phi ,d_T\\phi \\rangle +|d_T\\phi | \\kappa _B^\\sharp (|d_T\\phi |).$ By the curvature assumptions, $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ .",
"So we get from (REF ), $|d_T\\phi |\\Delta _{B,f}|d_T\\phi |\\le -\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle +|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |).$ Integrating (REF ) with the weighted measure, we have $\\int _{M}\\langle &|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}\\le -\\int _{M}\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle e^{-f}\\mu _{M}+\\int _{M}|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |)e^{-f}\\mu _{M}.$ From Remark 3.7, we get $\\int _{M}\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle e^{-f}\\mu _{M}=\\int _{M}\\langle i(\\kappa _{B}^\\sharp )d_T\\phi ,\\delta _\\nabla (e^{-f}d_T \\phi )\\rangle \\mu _{M}=0.$ Now, we choose a bundle-like metric $g$ such that $\\delta _{B}\\kappa _{B}=0$ .",
"Then $\\delta _{B,f} \\kappa _B =i(\\nabla _{\\rm tr} f)\\kappa _B$ from (REF ).",
"So $\\int _{M}|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |)e^{-f}\\mu _{M}&=\\frac{1}{2}\\int _M \\langle |d_T\\phi |^2, \\delta _{B,f} \\kappa _B\\rangle e^{-f}\\mu _M\\\\&=\\frac{1}{2}\\int _M \\langle d_B f,\\kappa _B\\rangle |d_T\\phi |^2 e^{-f}\\mu _M.$ Since $f\\in \\mathcal {K}$ , that is, $\\langle d_Bf,\\kappa _B\\rangle \\le 0$ , we get $\\int _{M}|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |)e^{-f}\\mu _{M} \\le 0.$ Note that $ \\int _{M}\\langle |d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}=\\int _M |d_B|d_T\\phi ||^2 e^{-f}\\mu _M\\ge 0$ .",
"So from (REF )$\\sim $ (REF ), we get $\\int _{M}\\langle |d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}=0,$ which yields $ d_{B}|d_T\\phi |=0$ .",
"That is, $|d_T\\phi |$ is constant.",
"From (REF ), we have $0=&-|\\nabla _{\\rm tr}d_T\\phi |^{2}-\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle -\\langle F_f(d_T\\phi ),d_T\\phi \\rangle .$ From (REF ) and (REF ), by integrating we get $\\int _{M}|\\nabla _{\\rm tr}d_T\\phi |^{2} e^{-f}\\mu _{M}+\\int _{M}\\langle F_f(d_T\\phi ),d_T\\phi \\rangle e^{-f}\\mu _{M}=0.$ Since $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ , from (REF ), we have $|\\nabla _{\\rm tr} d_T \\phi |^2=0 \\quad \\textrm {and}\\quad \\langle F_f(d_T\\phi ),d_T\\phi \\rangle =0.$ Since (REF ) is same to (REF ) in Theorem 3.6, the remaining part of the proof is same to that in Theorem 3.6.",
"So we omit the remaining part of the proof.",
"Corollary 3.9 Let $(M,g,\\mathcal {F},e^{-f_\\kappa }\\nu )$ be a weighted foliation on a closed manifold $M$ with $Ric_{f_\\kappa }^Q\\ge 0$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a Riemannian foliation with $K^{Q^{\\prime }}\\le 0$ .",
"If $\\mathcal {F}$ is non-minimal or ${\\rm Ric}_{f_\\kappa }^Q >0$ at some point, then $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f_\\kappa }$ -harmonic map is always transversally constant.",
"Note that $f_\\kappa $ is a solution of $d_Bf=-\\kappa _B$ .",
"So $f_\\kappa \\in \\mathcal {K}$ and $\\int _{M}|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |)e^{-f_\\kappa }\\mu _{M}&=\\frac{1}{2}\\int _M \\langle d_B f_\\kappa ,\\kappa _B\\rangle |d_T\\phi |^2 e^{-f_\\kappa }\\mu _M\\\\& =-\\frac{1}{2}\\int |\\kappa _B|^2 |d_T\\phi |^2 e^{-f_\\kappa }\\mu _M.$ Hence from (REF ) and (REF ), we get $0\\le \\int _M \\langle |d_T\\phi |,\\Delta _{B,f} |d_T\\phi |\\rangle e^{-f_\\kappa }\\mu _M \\le -\\frac{1}{2}\\int |\\kappa _B|^2 |d_T\\phi |^2 e^{-f_\\kappa }\\mu _M\\le 0.$ That is, $|\\kappa _B| |d_T\\phi | =0.$ Hence if $\\kappa _B\\ne 0$ , then $d_T\\phi =0$ , that is, $\\phi $ is transversally constant.",
"In case of ${\\rm Ric}_f^Q >0$ at some point, the proof is trivial from Theorem 3.8.",
"Remark 3.10 Since $f_\\kappa $ satisfies $d_B f_\\kappa =-\\kappa _B$ , we have from (REF ) $\\tilde{\\tau }_{b,f_\\kappa }(\\phi )= \\tau _b(\\phi ).$ Hence a transversally harmonic map is equivalent to the $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f_\\kappa }$ -harmonic map.",
"So Corollary 3.9 holds for transversally harmonic maps."
],
[
"Liouville type theorems",
"In this section, we investigate the Liouville type theorems for transversally $f$ -harmonic map and $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f}$ -harmonic map on weighted foliations.",
"Let $B_{l}=\\lbrace y\\in M|\\rho (y)\\le l\\rbrace $ , where $\\rho (y)$ is the distance between leaves through a fixed point $x_{0}$ and $y$ .",
"Let $\\omega _{l}$ be the Lipschitz continuous basic function such that $\\left\\lbrace \\begin{array}{ll}0\\le \\omega _{l}(y)\\le 1 \\quad {\\rm for \\, any} \\, y\\in M\\\\{\\rm supp}\\, \\omega _{l}\\subset B_{2l}\\\\\\omega _{l}(y)=1 \\quad {\\rm for \\, any} \\, y\\in B_{l}\\\\\\lim \\limits _{l\\rightarrow \\infty }\\omega _{l}=1\\\\|d\\omega _{l}|\\le \\frac{C}{l} \\quad \\textrm {almost everywhere on M},\\end{array}\\right.$ where $C$ is positive constant [43].",
"Therefore, $\\omega _{l}\\eta $ has compact support for any basic form $\\eta \\in \\Omega _{B}^{*}(\\mathcal {F})$ .",
"Lemma 4.1 Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a complete Riemannian manifold whose all leaves are compact and the mean curvature form is bounded.",
"Let $f\\in \\mathcal {K}$ and $\\phi : (M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map of $E_{B,f}(\\phi ) <\\infty $ .",
"Then $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^\\sharp (|d_T\\phi |)\\rangle e^{-f}\\mu _{M}\\le 0.$ Let $g$ be a bundle-like metric such that $\\delta _{B}\\kappa _{B}=0$ .",
"Then we get $\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^{\\sharp }(|d_T\\phi |)\\rangle e^{-f}\\mu _{M}=&\\frac{1}{2}\\int _M \\langle d_B |d_T\\phi |^2,\\omega _l^2\\kappa _B\\rangle e^{-f}\\mu _M\\\\=&\\frac{1}{2}\\int _{M}\\delta _{B,f}(\\omega _l^2 \\kappa _{B})|d_T\\phi |^{2}e^{-f}\\mu _{M}\\\\=&-\\int _M \\Big (\\langle d_B\\omega _l ,\\omega _l\\kappa _B\\rangle -\\frac{1}{2} i(\\nabla _{\\rm tr}f)\\omega _l^2\\kappa _B\\Big ) |d_T\\phi |^2 e^{-f}\\mu _M.$ Since $f\\in \\mathcal {K}$ , that is, $i(\\nabla _{\\rm tr}f)\\kappa _B=i(\\kappa _B^\\sharp )d_B f\\le 0$ , then from (REF ), $\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^{\\sharp }(|d_T\\phi |)\\rangle e^{-f}\\mu _{M}\\le -\\int _{M}\\langle d_B\\omega _l,\\omega _l\\kappa _B\\rangle |d_T\\phi |^2 e^{-f}\\mu _M.$ By using the Cauchy-Schwartz inequality, we get $\\Big |\\int _M \\langle d_B\\omega _l,\\omega _l\\kappa _B\\rangle |d_T\\phi |^2 e^{-f}\\mu _M\\Big | &\\le {C\\over l}{\\rm max} |\\kappa _B|\\int _M\\omega _l |d_T\\phi |^2 e^{-f}\\mu _M.$ So $E_{B,f}(\\phi )<\\infty $ implies $\\lim _{l\\rightarrow \\infty }\\int _M\\langle d_B\\omega _l,\\omega _l\\kappa _B\\rangle |d_T\\phi |^2 e^{-f}\\mu _M =0.$ From (REF ) and (REF ), by letting $l\\rightarrow \\infty $ , $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^\\sharp (|d_T\\phi |)\\rangle e^{-f}\\mu _{M}\\le 0.$ Let $\\mu _{0}$ be the infimum of the spectrum of the eigenvalues of the weighted basic Laplacian $\\Delta _{B,f}$ acting on weighted $L^{2}$ -basic functions on $M$ .",
"Then we have the following Liouville type theorem for transversally $f$ -harmonic maps.",
"Theorem 4.2 Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a complete Riemannian manifold whose all leaves are compact and the mean curvature form is bounded.",
"Let $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a foliated Riemannian manifold with $K^{Q^{\\prime }}\\le 0$ .",
"Let $f\\in \\mathcal {K}$ and $\\phi : M \\rightarrow M^{\\prime }$ be a transversally $f$ -harmonic map of $E_{B,f}(\\phi )<\\infty $ .",
"Then (1) if ${\\rm Ric}_f^Q\\ge 0$ , then $\\phi $ is transversally totally geodesic.",
"(2) if ${\\rm Ric}_f^Q\\ge 0$ and $\\int _M e^{-f}\\mu _M= \\infty $ , then $\\phi $ is transversally constant.",
"(3) if ${\\rm Ric}_f^Q \\ge -\\mu _0$ at all $x$ and ${\\rm Ric}_f^Q >-\\mu _0$ at some point, then $\\phi $ is transversally constant.",
"(1) Since ${\\rm Ric}_f^Q \\ge 0$ and $K^{Q^{\\prime }}\\le 0$ , from (REF ), $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ .",
"Hence from Lemma 3.5 (1) and the first Kato's inequality (REF ), we have $|d_T\\phi |\\Delta _{B,f} |d_T\\phi | \\le |d_B |d_T\\phi ||^2 -|\\nabla _{tr}d_T\\phi |^2 + |d_T\\phi |\\kappa _B^\\sharp (|d_T\\phi |)\\le |d_T\\phi |\\kappa _B^\\sharp (|d_T\\phi |).$ Multiplying (REF ) by $\\omega _{l}^{2}$ and integrating by parts, from Lemma 4.1, we get $\\lim _{l\\rightarrow \\infty }\\int _{M}&\\langle \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}\\le 0.$ On the other hand, by the Cauchy-Schwarz inequality, we have $\\int _{M}\\langle \\omega _{l}^{2}&|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}\\\\=&\\int _M\\omega _l^2 |d_B|d_T\\phi ||^2 e^{-f}\\mu _M+ 2\\int _M \\omega _l\\langle |d_T\\phi |d_B\\omega _l,d_B|d_T\\phi |\\rangle e^{-f}\\mu _M\\\\\\ge & \\int _{M}\\omega _{l}^{2}|d_{B}|d_T\\phi ||^{2}e^{-f}\\mu _{M}-2\\int _{M}\\omega _{l}|d_T\\phi ||d_{B}\\omega _{l}||d_{B}|d_T\\phi ||e^{-f}\\mu _{M}.$ By using the inequality $A^2 +B^2 \\ge 2AB$ , we get $2\\int _{M}\\omega _{l}|d_{B}\\omega _{l}||d_T\\phi ||d_{B}|d_T\\phi || e^{-f}\\mu _{M}\\le {C\\over l} \\int _M\\Big (\\omega _l^2 |d_T\\phi |^2 e^{-f} +|d_B|d_T\\phi ||^2e^{-f}\\Big )\\mu _M.$ From (REF ) $\\sim $ (REF ) and Fatou's inequality, it is trivial that $|d_{B}|d_T\\phi ||\\in L^{2}(e^{-f})$ , that is, $\\int _M |d_B|d_T\\phi ||^2 e^{-f}\\mu _M<\\infty $ .",
"So letting $l\\rightarrow \\infty $ , we get from (REF ) $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\omega _{l}|d_{B}\\omega _{l}||d_T\\phi ||d_{B}|d_T\\phi || e^{-f}\\mu _{M}=0.$ From (REF ) and (REF ), by letting $l\\rightarrow \\infty $ , we have $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle & \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}\\ge \\int _{M}|d_{B}|d_T\\phi ||^{2} e^{-f}\\mu _{M}.$ From (REF ) and (REF ), we have $d_B|d_T\\phi |=0$ , that is, $|d_T\\phi |$ is constant.",
"From (REF ), we have $|d_T\\phi |\\Delta _{B,f} |d_T\\phi | \\le -|\\nabla _{\\rm tr}d_T\\phi |^2 + |d_T\\phi |\\kappa _B^\\sharp (|d_T\\phi |)$ By multiplying (REF ) by $\\omega _l^2$ and integrating by parts, we have from Lemma 4.1 together with (REF ), we get $0\\le \\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle & \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M} \\le -\\int _M |\\nabla _{\\rm tr} d_T\\phi |^2 e^{-f}\\mu _M,$ which imlies that $\\nabla _{\\rm tr}d_T\\phi =0$ , that is, $\\phi $ is transversally totally geodesic.",
"(2) From (1), we know that $|d_T\\phi |$ is constant.",
"Since $E_{B,f}(\\phi )=\\frac{1}{2} |d_T\\phi |^2 \\int _M e^{-f}\\mu _M<\\infty $ , $\\int _M e^{-f}\\mu _M=\\infty $ implies $d_T\\phi =0$ , that is, $\\phi $ is transversally constant.",
"(3) Assume ${\\rm Ric}_f^{Q}\\ge -\\mu _0$ for all $x$ and ${\\rm Ric}_f^{Q}>-\\mu _0$ at some point $x_{0}$ and $K^{Q^{\\prime }}\\le 0$ .",
"Then from (REF ) $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge \\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a))\\ge -\\mu _0|d_T\\phi |^{2}.$ Since $\\phi $ is the transversally $f$ -harmonic map, from (REF ) and (REF ), we have $|d_T\\phi |(\\Delta _{B,f}-\\kappa _{B}^\\sharp )|d_T\\phi |\\le -\\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a))\\le \\mu _0|d_T\\phi |^{2}.$ Multiplying (REF ) by $\\omega _{l}^{2}$ and integrating by parts, we have $\\int _{M}&\\langle \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}-\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^\\sharp (|d_T\\phi |)\\rangle e^{-f}\\mu _{M}\\\\&\\le -\\sum _a \\int _{M}\\omega _{l}^{2} g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a)) e^{-f}\\mu _{M} \\\\&\\le \\mu _0\\int _{M}\\omega _{l}^{2}|d_T\\phi |^{2} e^{-f}\\mu _{M}.$ From Lemma 4.1 and (REF ), we get $\\lim _{l\\rightarrow \\infty } \\int _M \\langle \\omega _l^2 |d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _M\\le \\mu _0\\int _{M}|d_T\\phi |^{2} e^{-f}\\mu _{M}.$ On the other hand, by the Rayleigh quotient theorem, we have $\\int _{M}| d_{B}|d_T\\phi ||^2 e^{-f}\\mu _{M}\\ge \\mu _0 \\int _{M} |d_T\\phi |^2 e^{-f}\\mu _{M}.$ From (REF ), (REF ), (REF ) and (REF ), by $l\\rightarrow \\infty $ , we get $\\mu _{0}\\int _{M}|d_T\\phi |^{2}e^{-f}\\mu _{M}&\\le -\\sum _a \\int _{M} g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a)) e^{-f}\\mu _{M}\\\\&\\le \\mu _0\\int _{M}|d_T\\phi |^{2} e^{-f}\\mu _{M}.$ From the above inequality, we have $\\sum _a \\int _{M}g_{Q^{\\prime }}(d_T \\phi ({\\rm (Ric}_f^{Q}+\\mu _0)(E_a)),d_T \\phi (E_a)) e^{-f}\\mu _{M}=0.$ Since ${\\rm Ric}_f^{Q}>-\\mu _0$ at some point, from (REF ), $d_T \\phi =0$ , that is, $\\phi $ is transversally constant.",
"Lemma 4.3 Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ be a weighted foliation on a complete Riemannian manifold whose all leaves are compact and the mean curvature form is bounded.",
"Let $\\phi : (M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map of $E_{B,f}(\\phi )<\\infty $ .",
"Then $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,\\omega _{l}^{2}d_T \\phi \\rangle e^{-f}\\mu _{M}=0.$ Since $\\phi $ is $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic, $ \\delta _\\nabla (e^{-f}d_T\\phi )=0$ from Remark 3.7.",
"Hence $\\delta _\\nabla (\\omega _{l}^{2}e^{-f}d_T \\phi )&=\\omega _l^2 \\delta _\\nabla (e^{-f}d_T\\phi ) -i(d_{B}\\omega _{l}^{2})e^{-f}d_T \\phi \\\\&=-i(d_{B}\\omega _{l}^{2})e^{-f}d_T \\phi \\\\&=-2\\omega _{l}e^{-f}i(d_{B}\\omega _{l}) d_T \\phi .$ By using the inequality $|X^{\\flat }\\wedge d_T\\phi |^{2}+|i(X)d_T\\phi |^{2}=|X|^{2}|d_T\\phi |^{2}$ for any vector $X$ , we get $\\bigg {|}\\int _{M}\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,\\omega _{l}^{2}d_T \\phi \\rangle e^{-f}\\mu _{M}\\bigg {|}=&\\bigg {|}\\int _{M}\\langle i(\\kappa _{B}^\\sharp )d_T\\phi ,\\delta _\\nabla (\\omega _{l}^{2}e^{-f}d_T \\phi )\\rangle \\mu _{M}\\bigg {|}\\\\=&\\bigg {|}\\int _{M}\\langle i(\\kappa _{B}^\\sharp )d_T\\phi , -2\\omega _{l}i(d_{B}\\omega _{l})d_T \\phi \\rangle e^{-f}\\mu _{M}\\bigg {|}\\\\\\le &2\\int _{M}\\omega _{l}|i(\\kappa _{B}^\\sharp )d_T\\phi ||i(d_{B}\\omega _{l})d_T \\phi | e^{-f}\\mu _{M}\\\\\\le &2\\int _{M}\\omega _{l}|\\kappa _{B}||d_{B}\\omega _{l}||d_T\\phi |^{2}e^{-f}\\mu _{M}\\\\\\le &\\frac{2C}{l}\\max |\\kappa _{B}|\\int _{M}\\omega _{l}|d_T\\phi |^{2}e^{-f}\\mu _{M}.$ By letting $l\\rightarrow \\infty $ , $E_{B,f}(\\phi )<\\infty $ implies $\\lim \\limits _{l\\rightarrow \\infty }\\int _{M}\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,\\omega _{l}^{2}d_T \\phi \\rangle e^{-f}\\mu _{M}=0,$ which finishes the proof.",
"Theorem 4.4 Let $(M,g,\\mathcal {F},e^{-f}\\nu )$ and $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be as in Theorem 4.2.",
"Let $f\\in \\mathcal {K}$ and let $\\phi : M \\rightarrow M^{\\prime }$ be a $(\\mathcal {F},\\mathcal {F}^{\\prime })_{f}$ -harmonic map of $E_{B,f}(\\phi )<\\infty $ .",
"Then $(1)\\sim (3)$ in Theorem 4.2 are satisfied.",
"(1) Since ${\\rm Ric}_f^Q \\ge 0$ and $K^{Q^{\\prime }}\\le 0$ , from (REF ), $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge 0$ .",
"Hence from Lemma 3.5 (2) and the first Kato's inequality (REF ), we have $|d_T\\phi |\\Delta _{B,f} |d_T\\phi | \\le -\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle + |d_T\\phi |\\kappa _B^\\sharp (|d_T\\phi |).$ Multiplying (REF ) by $\\omega _{l}^{2}$ and integrating by parts, from Lemma 4.1 and Lemma 4.3, we get $\\lim _{l\\rightarrow \\infty }\\int _{M}&\\langle \\omega _{l}^{2}|d_T\\phi |,\\Delta _f|d_T\\phi |\\rangle e^{-f}\\mu _{M}\\le 0.$ The equation (REF ) is same to (REF ) in Theorem 4.2.",
"Hence the next process of the proof is same to that of (1) in Theorem 4.2.",
"(2) The proof is same in Theorem 4.2.",
"(3) Assume ${\\rm Ric}_f^{Q}\\ge -\\mu _0$ for all points and ${\\rm Ric}_f^{Q}>-\\mu _0$ at some point and $K^{Q^{\\prime }}\\le 0$ .",
"Then from (REF ) $\\langle F_f(d_T\\phi ),d_T\\phi \\rangle \\ge \\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a))\\ge -\\mu _0|d_T\\phi |^{2}.$ From Lemma 3.5 (2) and the Kato's inequality, we get $|d_T&\\phi |\\Delta _{B,f}|d_T\\phi |+\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,d_T \\phi \\rangle -|d_T\\phi |\\kappa _{B}^\\sharp (|d_T\\phi |)\\\\&\\le -\\sum _a g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a))\\le \\mu _0|d_T\\phi |^{2}.$ Multiplying (REF ) by $\\omega _{l}^{2}$ and integrating by parts, we get $\\int _{M}&\\langle \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}+\\int _{M}\\Big (\\langle d_\\nabla i(\\kappa _{B}^\\sharp )d_T\\phi ,\\omega _{l}^{2}d_T \\phi \\rangle -\\langle \\omega _{l}^{2}|d_T\\phi |,\\kappa _{B}^\\sharp (|d_T\\phi |)\\rangle \\Big )e^{-f}\\mu _{M}\\\\&\\le -\\sum _a \\int _{M}\\omega _{l}^{2} g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a)) e^{-f}\\mu _{M} \\\\&\\le \\mu _0\\int _{M}\\omega _{l}^{2}|d_T\\phi |^{2} e^{-f}\\mu _{M}.$ By letting $l\\rightarrow \\infty $ , form Lemma 4.1 and Lemma 4.3, we have $\\lim _{l\\rightarrow \\infty }\\int _{M}\\langle \\omega _{l}^{2}|d_T\\phi |,\\Delta _{B,f}|d_T\\phi |\\rangle e^{-f}\\mu _{M}&\\le -\\lim _{l\\rightarrow \\infty }\\sum _a \\int _{M}\\omega _{l}^{2} g_{Q^{\\prime }}(d_T \\phi ({\\rm Ric}_f^{Q}(E_a)),d_T \\phi (E_a)) e^{-f}\\mu _{M} \\\\&\\le \\mu _0\\int _{M}|d_T\\phi |^{2} e^{-f}\\mu _{M}.$ The inequality (REF ) is same to (REF ) in Theorem 4.2.",
"So the remaining part of the proof is same to that in Theorem 4.2.",
"So we omit the next process of the proof.",
"If $f$ is constant, then $\\Delta _{B,f}=\\Delta _B$ , ${\\rm Ric}_f^Q ={\\rm Ric^Q}$ , and $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic map is junt $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map.",
"Hence we have the following corollary.",
"Corollary 4.5 Let $(M,g,\\mathcal {F})$ be a complete foliated Riemannian manifold whose all leaves are compact and the mean curvature form is bounded.",
"Let $(M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a foliated Riemannian manifold with $K^{Q^{\\prime }}\\le 0$ .",
"Assume that the transversal Ricci curvature ${\\rm Ric^{Q}}$ of $M$ satisfies ${\\rm Ric^{Q}}\\ge -\\mu _{0}$ for all points and ${\\rm Ric^{Q}}>-\\mu _{0}$ at some point.",
"Then any $(\\mathcal {F},\\mathcal {F}^{\\prime })$ -harmonic map $\\phi : (M,g,\\mathcal {F}) \\rightarrow (M^{\\prime }, g^{\\prime },\\mathcal {F}^{\\prime })$ of $E_{B}(\\phi )<\\infty $ is transversally constant.",
"Remark 4.6 (1) Theorem 4.2 and Theorem REF can be found for the point foliation in [33].",
"(2) Corollary REF for the transversally harmonic map has been studied by Fu and Jung [10].",
"(3) For the study of transversally $f$ -harmonic maps (in particular, on complete manifolds) and $(\\mathcal {F},\\mathcal {F}^{\\prime })_f$ -harmonic maps, we need the condition $f$ restricted to $f\\in \\mathcal {K}$ (cf.",
"Theorem 3.8, Theorem 4.2, Theorem 4.4) except for the transversally $f$ -harmonic maps on a closed manifold $M$ (cf.",
"Theorem 3.6)."
],
[
"The stress energy tensor",
"Let $\\phi :(M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map with $M$ compact.",
"we calculate the rate of change of the transversal energy of $\\phi $ when the metric $g_Q$ is changed.",
"Let $g_Q (t)$ be the variation of $g_Q$ with $g_Q(0)=g_Q$ .",
"We put $h= {\\partial g_Q \\over \\partial t}$ , a symmetric 2-tensor on $M$ .",
"With a transversal coordinate $\\lbrace y^a\\rbrace (a=1,\\cdots ,q)$ , the metric is written by $g_Q(t)=\\sum _{a,b} g_{ab}dy^a dy^b$ .",
"Then we have the following.",
"Lemma 5.1 Let $\\mu _M (t)$ be the volume form with respect to the metric $g(t) = g_L + g_Q(t)$ .",
"Then ${d\\over dt}\\mu _M = \\frac{1}{2} ({\\rm tr}_Q h) \\mu _M=\\frac{1}{2}\\langle g_Q,h\\rangle \\mu _M.$ Note that for a nonsingular matrix $A$ , ${d\\over dt} {\\rm det}(A) = {\\rm tr}[ {\\rm det}(A) A^{-1} A^{\\prime }].$ From (REF ), we get ${d\\over dt} \\sqrt{{\\rm det}(g_{ab})} = \\frac{1}{2} ({\\rm tr}_Q h) \\sqrt{{\\rm det}(g_{ab})}.$ Since $\\chi _{\\mathcal {F}}$ is time independent, ${d\\over dt}\\mu _M = ({d\\over dt}\\nu )\\wedge \\chi _{\\mathcal {F}}$ and the transversal volume form $\\nu $ is $\\nu =\\sqrt{{\\rm det}(g_{ab})} dy^1\\wedge \\cdots \\wedge dy^q$ .",
"Hence from (REF ) ${d\\over dt}\\mu _M = ({d\\over dt}\\nu )\\wedge \\chi _{\\mathcal {F}}=\\frac{1}{2} ({\\rm tr}_Q h) \\nu \\wedge \\chi _{\\mathcal {F}} =\\frac{1}{2} ({\\rm tr}_Q h) \\mu _M.$ Then we have the following variation formula of the transversal metric.",
"Theorem 5.2 Let $\\phi :(M,g,\\mathcal {F}, e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a fixed smooth foliated map with $M$ compact.",
"Then ${d\\over dt} E_{B,f}(\\phi ,g(t))|_{t=0} =\\frac{1}{2}\\int _M \\langle S_{T}(\\phi ), h\\rangle e^{-f}\\mu _M,$ where $S_T(\\phi ) =\\frac{1}{2}|d_T\\phi |^2 g_Q -\\phi ^* g_{Q^{\\prime }}$ is the transversal stress-energy tensor [13].",
"By Lemma 5.1, we have ${d\\over dt} E_{B,f}(\\phi ,g(t)) &= \\frac{1}{2}\\int _M( {d\\over dt}|d_T\\phi |^2) e^{-f}\\mu _M + \\frac{1}{4}\\int _M \\langle |d_T\\phi |^2 g_Q,h\\rangle e^{-f}\\mu _M.$ On the other hand, since ${d\\over dt} g^{ab} =-h^{ab} (=\\sum _{c,d} g^{ac}g^{bd} h_{cd})$ , we have ${d\\over dt}|d_T\\phi |^2 = \\sum _{a,b}{d\\over dt} g^{ab} d_T\\phi ({\\partial \\over \\partial y^a}) d_T\\phi ({\\partial \\over \\partial y^b})=- h(d_T\\phi , d_T\\phi )=-\\langle h, \\phi ^* g_{Q^{\\prime }}\\rangle .$ From (REF ), we have ${d\\over dt} E_{B,f}(\\phi ,g(t)) &= \\frac{1}{2}\\int _M \\langle \\frac{1}{2}|d_T\\phi |^2 g_Q- \\phi ^* g_{Q^{\\prime }},h\\rangle e^{-f}\\mu _M,$ which impies the proof.",
"Now, we put $S_{T,f}(\\phi ) = e^{-f} S_T(\\phi ),$ called the transversal stress $f$ -energy tensor of $\\phi $ .",
"Corollary 5.3 Let $\\phi :(M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a nonconstant foliated smooth map with $M$ compact.",
"Then $\\phi :M \\rightarrow M^{\\prime }$ is an extremal of the transversal $f$ -energy functional with respect to variations of the transversal metric $g_Q$ if and only if $S_{T,f}(\\phi ) =0$ .",
"Lemma 5.4 Let $\\phi :(M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Then $({\\rm div}_\\nabla S_{T,f}(\\phi ) )(X) = - \\langle \\tau _{b,f}(\\phi ),d_T\\phi (X)\\rangle -\\frac{1}{2} e^{-f}|d_T\\phi |^2 d_B f (X)$ for any normal vector $X\\in \\Gamma Q$ .",
"Let $\\lbrace E_a\\rbrace $ be a local orthonormal basic frame on $Q$ .",
"Then at $x$ , $({\\rm div}_\\nabla S_{T,f}(\\phi ))(X) &= \\sum _a (\\nabla _{E_a} S_{T,f}(\\phi ))(E_a,X)\\\\&=e^{-f}({\\rm div}_\\nabla S_T(\\phi ))(X) - e^{-f}S_T(\\phi )(\\nabla _{\\rm tr} f,X)\\\\&= -e^{-f} \\langle \\tau _b(\\phi ), d_T\\phi (X)\\rangle -e^{-f}\\Big (\\frac{1}{2} |d_T\\phi |^2 g_Q -\\phi ^* g_{Q^{\\prime }}\\Big )(\\nabla _{\\rm tr} f,X) \\\\&=-\\langle \\tau _b(\\phi ) - d_T\\phi (\\nabla _{\\rm tr}f), d_T\\phi (X)\\rangle e^{-f} - \\frac{1}{2} e^{-f}|d_T\\phi |^2 d_B f(X)\\\\&=-\\langle \\tau _{b,f}(\\phi ),d_T\\phi (X)\\rangle -\\frac{1}{2} e^{-f}|d_T\\phi |^2 d_Bf (X),$ which implies the proof.",
"Remark 5.5 Any transversally harmonic map satisfies the transverse conservation law, i.e, ${\\rm div}_\\nabla S_T (\\phi )=0$ [13], but any transversally $f$ -harmonic map does not satify the transverse $f$ -conservation law, i.e., ${\\rm div}_\\nabla S_{T,f}(\\phi )=0$ (cf.",
"Lemma 5.4).",
"Let $F:[0,\\infty )\\rightarrow [0,\\infty )$ be a $C^2$ -function such that $F^{\\prime }>0$ on $(0,\\infty )$ .",
"The transversally $F$ -harmonic map $\\phi :M\\rightarrow M^{\\prime }$ is a solution of the Eular-Lagrange equation $\\tau _{b,F}(\\phi )=0$ [5], where $\\tau _{b,F}(\\phi )$ is the transversal $F$ -tension field given by $\\tau _{b,F}(\\phi ) = F^{\\prime } ({|d_T\\phi |^2\\over 2})\\tau _b(\\phi ) + d_T\\phi \\Big (\\nabla _{\\rm tr} F^{\\prime }({|d_T\\phi |^2\\over 2})\\Big ).$ When $F(s) =s$ , the transversal $F$ -tension field $\\tau _{b,F}(\\phi )$ is the transversal tension field $\\tau _b(\\phi )$ .",
"Proposition 5.6 Any transversally $F$ -harmonic map $\\phi : (M,g,\\mathcal {F},e^{-f}\\nu )\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ without critical points is a transversally $f$ -harmonic map with $f= -\\ln F^{\\prime }({|d_T\\phi |^2\\over 2})$ .",
"If $f= -\\ln F^{\\prime }({|d_T\\phi |^2\\over 2})$ in (REF ), then $\\tau _{b,F}(\\phi ) =\\tau _{b,f}(\\phi )$ .",
"So the proof is trivial.",
"Now, we define the transversal $F$ -stress energy tensor $S_{T,F}(\\phi )$ by $S_{T,F}(\\phi ) = F({|d_T\\phi |^2\\over 2}) g_Q - F^{\\prime }({|d_T\\phi |^2\\over 2}) \\phi ^* g_{Q^{\\prime }}.$ Then we have the following Lemma [5].",
"Lemma 5.7 Let $\\phi : (M,g,\\mathcal {F})\\rightarrow (M^{\\prime },g^{\\prime },\\mathcal {F}^{\\prime })$ be a smooth foliated map.",
"Then ${\\rm div}_\\nabla S_{T,F}(\\phi ) = -\\langle \\tau _{b,F}(\\phi ), d_T\\phi \\rangle .$ Let $\\lbrace E_a\\rbrace $ be a local orthonormal basic frame such that $\\nabla E_a=0$ at $x$ .",
"Then at $x$ , $({\\rm div}_\\nabla S_{T,F}(\\phi ))(E_a)&=\\sum _b\\nabla _{E_b}S_{T,F}(E_b,E_a)\\\\&=\\sum _b \\nabla _{E_b}\\Big ( F({|d_T\\phi |^2\\over 2})\\delta _{ab} - F^{\\prime }({|d_T\\phi |^2\\over 2})\\phi ^*g_{Q^{\\prime }}(E_a,E_b)\\Big )\\\\&=g_Q (\\nabla _{\\rm tr} F({|d_T\\phi |^2\\over 2}),E_a) -g_{Q^{\\prime }}( d_T\\phi \\Big (\\nabla _{\\rm tr} F^{\\prime }({|d_T\\phi |^2\\over 2})\\Big ),d_T\\phi (E_a))\\\\&- g_{Q^{\\prime }} (F^{\\prime }({|d_T\\phi |^2\\over 2}) \\tau _b(\\phi ),d_T\\phi (E_a)) - \\frac{1}{2} g_Q (F^{\\prime }({|d_T\\phi |^2\\over 2})\\nabla _{\\rm tr} {|d_T\\phi |^2}, E_a).$ On the other hand, by the chain rule, we have $g_Q (\\nabla _{\\rm tr} F({|d_T\\phi |^2\\over 2}),E_a)&=E_a [F({|d_T\\phi |^2\\over 2})]\\\\& = F^{\\prime }({|d_T\\phi |^2\\over 2}) E_a ({|d_T\\phi |^2\\over 2})\\\\& = \\frac{1}{2} g_Q (F^{\\prime }({|d_T\\phi |^2\\over 2})\\nabla _{\\rm tr}{|d_T\\phi |^2},E_a).$ Hence from the above equations, we have $({\\rm div}_\\nabla S_{T,F}(\\phi ))(E_a)&= -g_{Q^{\\prime }}( d_T\\phi \\Big (\\nabla _{\\rm tr} F^{\\prime }({|d_T\\phi |^2\\over 2})\\Big ) +F^{\\prime }({|d_T\\phi |^2\\over 2}) \\tau _b(\\phi ),d_T\\phi (E_a))\\\\&=- g_{Q^{\\prime }} (\\tau _{b,F}(\\phi ), d_T\\phi (E_a)),$ which finishes the proof.",
"If $\\phi :M\\rightarrow M^{\\prime }$ satisfies ${\\rm div}_\\nabla S_{T,F}(\\phi ) =0$ , then we say $\\phi $ satisfies the transverse $F$ -conservation law.",
"Generally, a transversally $f$ -harmonic map does not satisfies the transverse $f$ -conservation law.",
"But we have the following.",
"Proposition 5.8 Any transversally $F$ -harmonic map satisfies the transverse $F$ -conservation law.",
"In particular, if $f=-\\ln F^{\\prime }({|d_T\\phi |^2\\over 2})$ , then a transversally $f$ -harmonic map satisfies the transverse $F$ -conservation law, that is, ${\\rm div}_\\nabla S_{T,F}(\\phi )=0$ .",
"From Proposition 5.6 and Lemma 5.7, the proof follows."
]
] | 2212.05639 |
[
[
"Off-Policy Deep Reinforcement Learning Algorithms for Handling Various\n Robotic Manipulator Tasks"
],
[
"Abstract In order to avoid conventional controlling methods which created obstacles due to the complexity of systems and intense demand on data density, developing modern and more efficient control methods are required.",
"In this way, reinforcement learning off-policy and model-free algorithms help to avoid working with complex models.",
"In terms of speed and accuracy, they become prominent methods because the algorithms use their past experience to learn the optimal policies.",
"In this study, three reinforcement learning algorithms; DDPG, TD3 and SAC have been used to train Fetch robotic manipulator for four different tasks in MuJoCo simulation environment.",
"All of these algorithms are off-policy and able to achieve their desired target by optimizing both policy and value functions.",
"In the current study, the efficiency and the speed of these three algorithms are analyzed in a controlled environment."
],
[
"Introduction",
"More and more uses for robots are being found, including autonomous vehicles [17], industry-based robots [4], and humanoid robots [12], where the operational landscape is not static, preset and known.",
"Manufacturing robots, in particular, need to be able to perform intelligent actions in highly variable settings.",
"The primary methods of functioning for conventional automated robots are learned imitation and programming skills.",
"Automated robots in the industry have low responsiveness once they are put in a situation with unknown variables.",
"Robots typically need the capacity to physically interact with items because they are deployed in unpredictable environments.",
"Positioning, dynamic motion plannings, and grasping are typically at the forefront of such tasks [23].",
"As a result, it is no longer feasible to program every conceivable set of circumstances in advance.",
"With the need to perceive and assess the operational environment and make judgments for the proper implementation of the tasks taking into account the safety, productivity, and manufacturing challenges, intelligence must be built into the robotic system.",
"Thus, Artificial Intelligence (AI) and Machine Learning (ML) techniques find a vast amount of application in the field of robotics to realize such lofty objectives [34].",
"Reinforcement learning (RL) [3] is a sub-field of ML and has gained huge interest, especially in control and robotics areas, due to its capability of handling complex high-dimensional environments and its successful applications can be found in the works done by [24], [37], [1].",
"RL continuously interacts with its surrounding environment and gathers data trajectories to decide about the future actions of the controller and its working principle depends on a reward-penalty logic.",
"The long-term goal of RL is to find an optimal control policy that maximizes the expectation of a given objective function.",
"RL approaches have been effectively implemented in a broad range of exciting applications in the field of robotics, such as table tennis game [11], [6], acrobatic maneuvers [41], solving Rubik' cube [31], and pancake flipping [26].",
"Basically, RL is categorized in two groups: model-based where the controller (agent) needs to learn the environment's model, and then it predicts the future data [25].",
"Some researchers believe that model-based RL could benefit more from sample efficiency than model-free algorithms.",
"While the former is promising, but it has obstacles that prevent it from being extensively used in the real life situations, such as the difficulty of systematically and effectively developing a precise model from pure data acquired.",
"In order to acquire low-dimensional implicit state and action interpretations from large-dimensional data sets, several publications [14], [15] employ representational learning techniques.",
"However, there is a chance that the trained models will not agree with the underlying dynamical mechanisms, and their efficiency can also decrease dramatically if we move beyond the range of the training sets.",
"However, with the recent developments in visualization and classical mechanics based on simulation settings [26], robots are now able to efficiently manipulate based on predetermined modeling framework via graphical algorithms.",
"The model-free concept is an alternative that seeks to discover the optimum solution without first estimating the model.",
"The model-free technique has the potential to be less space-intensive than its model-based analogue because there is no longer any need to save a detailed explanation of the system [21].",
"One disadvantage for model-based RL approaches is that they sometimes suffer from learning complex models accurately as well as the increased training time.",
"In order to boost the training procedure and obtain the optimal result quickly, off-policy model-free RL algorithms are preferable [36].",
"They are advantageous since they use batch learning from their past experiences and achieve significant success during training.",
"The agent can acquire knowledge about additional policies that are distinct from the one currently being carried out by engaging in off-policy learning.",
"[16] provides a comprehensive analysis of the benefits of off-policy optimization.",
"Leveraging the function approximation potential of deep neural networks has led to the widespread use of RL.",
"Deep reinforcement learning is a set of methods that use neural networks to learn interpretations from large feature dimensions, allowing for the learning of control in a holistic fashion.",
"Among those algorithms, Deep Deterministic Policy Gradient (DDPG) [22], Twin Delayed Deep Deterministic Policy Gradient (TD3) [9] and Soft Actor-Critic (SAC) [13], are common examples of off-policy algorithms.",
"All the above algorithms optimize both the policy and the value functions and maximize the cumulative reward to converge to the optimal result.",
"DDPG is the most used off-policy algorithm for the systems with continuous state-action spaces.",
"Apart from robotic control, DDPG is also used for energy efficiency of wind turbines [45], optimization of unmanned aerial vehicles [46], and traffic management [7] due to its fast convergence.",
"[42] controlled the bicycle without human interaction, which is one of the most challenging tasks in control area using DDPG.",
"Nevertheless, DDPG, which even exhibits a high performance on humanoid robots [20] sometimes may fail to conquer during training process due to the overestimation of the Q-value function.",
"On the other side, TD3 algorithm is presented to bring a solution for this overestimation issue by using two Q-functions.",
"TD3 looks like DDPG in terms of algorithmic structure and is frequently used for Energy Systems[43] and Robotic Control [8].",
"Unlike DDPG and SAC, TD3 updates the policy network less frequently.",
"SAC is also another off-policy algorithm which maximizes the expected return and the entropy.",
"The structure of the algorithm is different from DDPG and TD3.",
"This entropy regularization method is long ago used for optimal control [33], [40], [38] and inverse reinforcement learning problems [47] and achieved high performance.",
"This algorithm is preferred due to its exploration method and the entropy regularization.",
"Similar work has recently been done for robotic manipulations by [2], in which the Advantage Actor-Critic (A2C) [27], Actor-Critic using Kronecker-Factored Trust Region (ACKTR) [44], DDPG, TD3, PPO, SAC, and TRPO algorithms are applied and compared for target reaching tasks in Pybullet simulation environment and on a physical setup.",
"In this study, the DDPG, TD3, and SAC algorithms are applied to a 7-DoF robotic manipulator to perform four different tasks in the MuJoCo environment: Reach, Pick and Place, Slide, and Push [39].",
"This study exposes the algorithms to a more broad comparison by evaluating them on tasks with varying levels of difficulty in order to produce significantly more dependable results.",
"The following contributions can be covered under the scope of this study: Model-free off-policy RL algorithms (DDPG, SAC and TD3) are analyzed and applied to a Fetch robot manipulator in the MuJoco simulation environment to learn different robot manipulation and control tasks and their learning performance is compared.",
"In all cases, the controller agents are acquired through reinforcement learning in a continuous space defined by a custom reward function.",
"The simulation of corresponding tasks can be reached via: https://www.youtube.com/watch?v=ce-PuCThzBw&list=PLLNrNJhfBBw1SgwuUE9H-mgLtmnzt4fEx To choose the most suitable model-free algorithm for each specific task from DDPG, SAC and TD3, a comparison has been made in terms of success rate, time complexity and efficiency.x This article is organized as following: After a literature background in Section , deep RL algorithms applied to the Fetch robotic system are described in Section .",
"Section explains the Mujoco simulation environment and the related robotic manipulator.",
"Experimental results are given in Section , and finally the concluding discussions regarding the experiments are provided in Section ."
],
[
"RL structure",
"Machine learning (ML) is the process by which a computer is taught to improve its overall quality on a certain function by using the information it has already amassed.",
"Supervised, unsupervised, and reinforcement learning are the three main categories into which ML algorithms fall.",
"Unlike unsupervised learning, which makes the use of methods like cluster analysis on raw data sources, supervisory techniques are founded on inference theory, with the model being trained on labeled data to carry out regression or classification .",
"However, according to the RL concept, an autonomous operator can improve its performance on a task through observation and experimentation.",
"On the other side, agent, is anything capable of gathering information about its surroundings via sensory devices and then responding to what it finds.",
"RL structures, for formalizing the sequential inference events, can be identified as Markov decision Processes (MDP) for stochastic control problems.",
"MDP is a method of decision-making that bases its considerations primarily on the latest recent policy and behavior, rather than the complete decision-making record.",
"The decision-making dilemma is classified as a partially-observable MDP because, more significantly, in many realistic areas of application, an actor cannot see all elements of the environmental state .",
"The objective of reinforcement learning is to locate a policy that, when applied to histories in the state space, would result in the maximization of the estimated summation of (discounted) rewards.",
"MDP can be thought as a set ($\\mathcal {S},\\mathcal {A},\\mathcal {P},\\mathcal {R}$ ), where $\\mathcal {S}\\subset \\mathbb {R}^{d_s}$ constitutes the continuous state, $\\mathcal {A}\\subset \\mathbb {R}^{d_a}$ continuous action space, $\\mathcal {P}$ is the state transition function, and $\\mathcal {R}\\subset \\mathbb {R}$ determines a real-valued reward function to assess the quality of the state transitions from a present state $s$ to a new state $s^{\\prime }$ .",
"A stochastic policy $\\pi _{\\theta }(a|s)$ dependent to some parameters $\\theta $ aims at mapping the states of the system to the action space by maximizing the average of the obtained rewards in the long run.",
"We focus on recurrent encounters and gather state and action combinations using the notation $x_{t} := (s_{t}, a_{t})$ in order to create governing patterns over the course of a certain timescale.",
"The series $\\lbrace x_{t}: t \\ge 1 \\rbrace $ is an example of a Markov process that operates on the space $\\mathcal {X} = \\mathcal {S} \\times \\mathcal {A}$ .",
"The transition function for this sequence may be expressed as $ f_{\\theta }(x_{t+1} | x_{t}) := g(s_{t+1} | s_{t}, a_{t}) \\, h_{\\theta }(a_{t} | s_{t}).$ The distribution for a specific route, denoted by $x_{1:n}$ , that leads from a starting state to a final time denoted by $n$ can be expressed mathematically as follows: $ p_{\\theta }(x_{1:n}) :=f_{\\theta }(x_{1}) \\prod _{t =1}^{n-1} f_{\\theta }(x_{t+1} | x_{t}),$ where $f(x_{1}) = \\eta (s_{1}) h_{\\theta }(a_{1} | s_{1}) $ is the initial distribution for $x_{1}$ .",
"A presumed return component is allocated, and its value might be the whole or discounted sum of immediate rewards.",
"This function is utilized in the long-term to assess the efficiency of a path that was obtained.",
"The discounted form, in which the reward at every time interval is calculated by a rate of discount denoted by $\\gamma \\in (0,1]$ , is the one that will be considered in this article.",
"The discount factor is what decides how an agent should prioritize the rewards that will come in the future.",
"Agents who have a small $\\gamma $ are more probable to have a short-term focus and try to maximize their profits in the short future.",
"On the other hand, agents whose values are high are more likely to have a long-term perspective and try to maximize their gains over the course of a longer time frame.",
"The following formula can be used to formulate a specific path's return: $ R(x_{1:n}) := \\sum _{t = 1}^{n-1}\\gamma ^{t-1}r(a_{t}, s_{t}, s_{t+1}).$ The primary objective of RL policy searching is to investigate the policy regimes in order to determine the value of the policy variable $\\theta $ that optimizes a cost $J(\\theta )$ that can be defined as an expected value over the total rewards.",
"$\\theta ^{*} = \\arg \\max _{\\theta \\in \\Theta } J(\\theta ).$ $ J(\\theta ) = \\mathbb {E}_{\\theta }[R(x_{1:n})] = \\int p_{\\theta }(x_{1:n}) R(x_{1:n})dx_{1:n}.$ The integral that is engaged in the formulation of this expectation necessitates it being challenging to evaluate it since it is irresolvable.",
"In most cases, this problem emerges as a result of the undetermined sampling of the histories or the construction of the rewards.",
"Several alternative approaches based on statistical approaches have been used in an effort to locate $\\theta ^{\\ast }$ .",
"Among such efforts, policy gradient is noted for being one of the most prominent options; for an early diligent work, refer to the citation[32] which is on the basis of usual gradient updates with a learning rate of $\\beta $ as: $ \\theta ^{(k+1)} = \\theta ^{(k)} + \\beta \\nabla _{\\theta } J(\\theta ^{(k)}).$ However, the gradient is computed by a Monte Carlo method.",
"$\\nabla _{\\theta } J(\\theta ) \\approx \\frac{1}{N} \\sum _{i = 1}^{N} \\left[ \\sum _{t = 1}^{n}\\nabla _{\\theta } \\log h_{\\theta }(a_{t}^{(i)} | s_{t}^{(i)}) \\right] R(x_{1:n}^{(i)}),\\quad \\text{where} \\quad x^{(i)}_{1:n} \\sim p_{\\theta }(x_{1:n}), \\quad i = 1, \\ldots , N.$ Almost all practical applications use action domains that are continuous.",
"Deterministic policy gradient (DPG) techniques permit RL to be used in contexts where actions can be performed continuously.",
"Policy gradients with a deterministic structure can be expressed for MDPs satisfying certain requirements by adopting a model-free formulation that tracks the value function's gradient [35].",
"This means that in scenarios with larger action spaces, DPG requires fewer observations because it just integrates over the state space, as opposed to the state and action spaces as in probabilistic policy gradients.",
"Compared to the policy based RL algorithms, on the other side, there exisit the value-based ones such as Q-learning.",
"In value-based methods, the agent learns estimates of the individual state-action pairs $Q(s,a)$ .",
"If enough samples are collected for each state-action pair, Q-learning will learn (near) optimal state-action values.",
"Once a Q-learning agent has converged to the optimal $Q$ values for an MDP and made greedy action choices afterwards, it will get the same expectation of the discounted rewards' summation as estimated by the value function.",
"The updates for each state-action's $Q$ values are done according to: $Q(s,a) \\leftarrow Q(s,a) + \\alpha \\left[ r + \\gamma \\max _{a^{\\prime }\\in A} Q(s^{\\prime },a^{\\prime }) - Q(s,a) \\right]$ To combine the benefits of the policy-based and value-based RL algorithms, the actor-critic methods are proposed.",
"The 'actor' refers to the policy structure that makes decision about which actions to take.",
"In this case, the critic is the estimated value function, which provides feedback on the actor's performance.",
"Every time an action is chosen, the critic assesses the new state to determine if the outcome was satisfactory or not.",
"In this case, learning procedure demands the gradient calculations for both networks [18].",
"To extend the application of $Q$ functions from discrete state-spaces to high-dimensional continuous ones, Deep Q-Networks are introduced where they include deep neural networks for approximating the $Q$ values in complex domains.",
"The technique of experience replay is utilized by DQN in order to both improve the overall efficiency of the collected samples and to cut the correlation that exists between iterative data samples [28].",
"One of the successful actor-critic methods for high-dimensional state-spaces, is the improved and extended versions of DQN and DPG, called Deep Deterministic Policy Gradient (DDPG).",
"In this study we will apply DDPG, Twin Delayed DDPG, and Soft Actor Critic (SAC) algorithms to our experiments in the MuJoCo environment and compare the success rate of these algorithms over the different robotic tasks.",
"In the following section, we will have a brief overview to these algorithms."
],
[
"Adopted RL Algorithms",
"In this part, we briefly discuss the implemented off-policy algorithms that we have selected for our robotic manipulator."
],
[
"Deep Deterministic Policy Gradient (DDPG)",
"DDPG with a continuous action domain is a model-free reinforcement learning algorithm.",
"It is an extension of the DQN (Deep Q-Network) algorithm, which is designed for discrete action spaces.",
"DDPG uses two neural networks, called the actor and the critic, to learn a policy and a value function, respectively.",
"The actor network is used to predict the best action to take in a given state, while the critic network is used to evaluate the quality of the actor's chosen action.",
"The two networks are trained using a combination of supervised learning and reinforcement learning techniques.",
"DDPG is well-suited for tasks involving continuous control, such as robotic manipulation tasks.",
"It has been used to solve a variety of challenging problems in this domain, including reaching, picking, placing, and sliding objects.",
"DDPG is an RL algorithm that concurrently learns the optimal policy and the Q-function.",
"Q-function in DDPG is being learned using Bellman equation and off-policy memory, afterwards the learned Q-function is adopted to learn the policy.",
"The method seeks to determine the optimal action value $a^{*}$ for each given state in a continuous control environment, in a manner similar to the Q-learning method.",
"The learning of Q-function, which uses the Bellman equation as provided in Eq.",
"(REF ), is the first mathematical aspect of DDPG.",
"$Q^*(s,a) = \\mathbb {E}_{s^{\\prime }\\sim P}\\left[ r(s,a) + \\gamma \\max _{a^{\\prime }} Q^* (s^{\\prime }, a^{\\prime })\\right]$ Here, $s^{\\prime }\\sim \\mathcal {P}$ means sampling the next state $s^{\\prime }$ from target transition function $\\mathcal {P}(s,a)$ .",
"The Bellman equation's role here is to approximate optimal Q-function.",
"This Q-function can be approximated by a set of artificial neural networks $Q_\\phi (s,a)$ , where $\\phi $ is the parameters of the network.",
"The working principle of DDPG is the minimization of the mean-squared Bellman error (MSBE) function defined in Eq.",
"(REF ) for a set of data trajectories $\\mathcal {D}(s,s,r,s^{\\prime })$ : $\\begin{split}\\mathcal {L}(\\phi , D) = &\\mathbb {E}_{\\mathcal {D}}\\Bigg [\\bigg (Q_\\phi (s,a)- \\\\ &\\Big (r+\\gamma (1-d)\\max _{a^{\\prime }} Q_\\phi (s^{\\prime },a^{\\prime })\\Big )\\bigg )^2 \\Bigg ]\\end{split}$ Here $(s,s^{\\prime })\\in \\mathcal {S}$ , $(a,a^{\\prime })\\in \\mathcal {A}$ , and $r\\in \\mathcal {R}$ with a learning rate $\\gamma \\in (0,1]$ and $d$ signifying that whether a terminal state is met or not.",
"Noise should be added to the policy actions during the training in order to properly explore the policy space.",
"Normally,[22] suggests utilizing Ornstein-Uhlenbeck (OU) noise for DDPG, however Gaussian noise is used in this study because recent studies have shown that OU noise has no unique efficiency over the training data [19]."
],
[
"Twin Delayed Deep Deterministic Policy Gradient (TD3)",
"TD3, or Twin Delayed DDPG, is a model-free off-policy reinforcement learning algorithm for continuous action spaces.",
"It is an extension of the DDPG algorithm, which is designed to address some of the limitations of DDPG.",
"TD3 uses two actor networks and two critic networks, which are trained to work together to maximize the agent's performance.",
"The two actor networks are used to generate two different action outputs for each state, which are then evaluated by the two critic networks.",
"This allows TD3 to better capture the uncertainty in the environment and reduce the overestimation of value estimates that can occur in DDPG.",
"Like DDPG, TD3 is well-suited for tasks involving continuous control, such as robotic manipulation tasks.",
"It has been shown to perform well on a variety of challenging problems in this domain.",
"While the DDPG algorithm benefits from a high efficiency on performance, it is fragile with respect to hyper-parameters and substantially overestimates the Q-values, causing the policy to worsen.",
"The TD3 algorithm tackles this problem by incorporating three new approaches into the DDPG algorithm.",
"To begin, unlike DDPG, TD3 uses two Q-functions $Q_{\\phi _1}(s,a), Q_{\\phi _2}(s,a)$ and selects the smallest Q-value according to the Bellman error loss function given in Eq.",
"(REF ) and then the policy is learned as $\\max \\limits _{\\theta }\\mathbb {E}\\Big [Q_{\\phi _1}\\Big (s,\\pi _\\theta (a|s)\\Big )\\Big ]$ As a result, the new algorithm is dubbed \"twin.\"",
"In the second scenario, the target network and policy are modified less frequently.",
"[10] state that one policy update for TD3 is advised per every two Q-function updates.",
"The final option is to smooth the target policy, which involves adding noise to the target action in order to make the policy more complicated when dealing with Q-function inaccuracies.",
"Compared to the usual DDPG algorithm, all of these innovative ways enhanced the efficiency of the complicated systems."
],
[
"Soft Actor Critic (SAC)",
"SAC is an RL off-policy technique that optimizes a stochastic policy to learn an action, combining stochastic policy optimization and DDPG approaches.",
"Although it has no relation to the TD3 method, it uses a double clipped Q-function and aims for policy smoothing due to the policy's intrinsic stochasticity.",
"The entropy regularization feature is the most significant aspect of the algorithm, which is employed for continuous action space.",
"During policy training, the expected return and entropy are maximized in this algorithm.",
"The use of this strategy eliminates improper convergence.",
"The term entropy which evaluates the randomness of a variable $x$ using its density function $P$ , can be defined as: $H(P) = \\mathbb {E}_{x \\sim P}[-\\log P(x)]$ Taking into account the effect of the entropy the regularized RL problem can be modified as: $\\pi ^{*}_\\theta = \\arg \\max \\limits _{\\pi } \\mathbb {E}\\Bigg [\\sum _{t=0}^{\\infty }\\gamma ^t R(s_t,a_t) + \\alpha H\\Big ( \\pi (a_t|s_t)\\Big )\\Bigg ]$ Besides policy $\\pi _\\theta $ , two Q-functions $Q_{\\phi _1}, Q_{\\phi _2}$ are being optimized by SAC algorithm.",
"For this, the Bellman equation can be reformulated as: $\\begin{split}Q^\\pi (s,a) = &\\mathbb {E}_{s^{\\prime } \\sim P, a^{\\prime } \\sim \\pi }[R(s,a,s^{\\prime })+ \\\\&\\gamma (Q^\\pi (s^{\\prime },a^{\\prime })-\\alpha \\log \\pi (a^{\\prime }|s^{\\prime }))]\\end{split}$ where, $\\alpha > 0$ can be either a fixed or varying coefficient where for this article a fixed version is used.",
"Similar to TD3, SAC algorithm computes MSBE function given in Eq.",
"(REF ) for $Q_{\\phi _1}$ and $Q_{\\phi _2}$ and then opts for the minimum Q-function."
],
[
"Environment",
"Fetch robotic manipulator is used to test the algorithms in the simulated environment whose job is to push or pull a black cube toward a red dot located on a flat surface or in midair.",
"The gripper position is controlled in three dimensions in both cases, while the gripper's opening and closing are handled in the fourth dimension.",
"The states of the system include the linear and angular velocities of the gripper and its positions in the Cartesian space.",
"However, in the Push task, the gripper is always set to be closed, requiring the agent to push the cube on the surface, requiring the agent to learn the physical properties of the cube and surface (in this case, friction).",
"For the simulations, MuJoCo simulator is chosen which is developed by [39], and is used for Multi-joint dynamics with contact.",
"Fetch robotic manipulator has 7-DoF arm and two-fingered parallel gripper.",
"The Fetch robot has already been used by different studies in the domain of deep RL for different tasks [30], [29], [5].",
"Four different tasks can be performed by using this environment and the description of these tasks are given below.",
"Slide: The manipulator should strike the rubber across the table to slide it to the desired position.",
"Pick and Place: The manipulator should pick up the box from the table and move it to the given position.",
"Push: The manipulator should move the box to the desired position by repulsing it.",
"Reach: The manipulator should move its end effector to the desired positions.",
"Figure: (a) Pick and places, (b) Reach, (c) Push and (d) Slide from MuJoCo environmentIn this simulation environment, the states of the system consist of the Cartesian positions, the angles and the velocities of all joints.",
"Additionally, the information of achieved and desired goals is obtained from the simulation.",
"The goals of the system are described as desired positions.",
"Sparse and binary rewards are used by the simulator as a default which is -1 if the desired was not achieved and 0 for if it was achieved.",
"The initial position of the gripper is always fixed for the tasks and object location is randomly placed on the table at every episode."
],
[
"Simulation Studies",
"In this section, the results of DDPG, TD3 and SAC algorithms for four different tasks; Slide, Pick $\\&$ Place, Push and Reach, are given which are procured by testing all algorithms under the same conditions by using the MuJoCo Fetch Robot Manipulator simulation.",
"In order to determine the most ideal model-free off-policy RL algorithm based on the success rate, the epoch time and the efficiency, all three algorithms are trained for 400 epochs in total and 50 episodes for each epoch.",
"In the network models of DDPG algorithm, three fully connected layers with 256 nodes is selected.",
"The structure of actor(policy) and critic(value) networks are exactly the same, except the output layer of the actor network which tanh activation function is used.",
"The learning rate, $\\alpha $ , is accepted as 0.0001 for both actor and critic networks.",
"The discount factor, $\\gamma $ , for Bellman equation is chosen as 0.98 and polyak, $\\rho $ , is 0.05 is used to update the target networks.",
"The Gaussian noise is added to the action value during the training process for exploration.",
"The network architecture of TD3 algorithm is also similar to DDPG and consists of actor and critic networks with 3 layers and 256 nodes for each layer.",
"The discount factor, $\\gamma $ , of the Bellman equation and the learning rate, $\\alpha $ of the actor and critic networks are accepted as 0.98 and 0.0001, respectively.",
"In order to update the target networks of the algorithm, the polyak value, $\\rho $ , is chosen as 0.05 and the Gaussian noise is also added to the action value during training.",
"The neural model of SAC differs from other algorithms and consists of three neural networks, which are actor, critic and q-value.",
"The neural network of SAC has two layers with 256 nodes as recommended.",
"As DDPG and TD3, the learning rate of all networks, $\\alpha $ , is 0.0001, the discount factor, $\\gamma $ , of Bellman equation is 0.98 and the polyak, $\\rho $ , is 0.05.",
"The output layer of actor networks is parametrized with tanh function.",
"Figure REF presents the general comparison of DDPG, TD3 and SAC algorithms on Slide task which is the most challenging task in Fetch simulation environment.",
"SAC algorithm fails to solve the task successfully in 400 epochs but, unlike SAC algorithm, DDPG and TD3 solve the task over 50 percent.",
"Although DDPG has high results in the first 50 epoch, TD3 shows better and higher results at the end of the training.",
"TD3 also has less frequently fluctuation on success rate over the epochs.",
"Overall TD3 algorithm has better results on solving Slide task.",
"Figure: Slide ComparisonThe comparison of next challenging task, Pick and Place, is illustrated in Figure REF .",
"In this task, DDPG shows the highest result, almost 100 percent with some fluctuation after 250 epochs.",
"The incremental slope of success rate is also better in DDPG than others.",
"TD3 finished the task with high rate of success as well, which is around 90 percent, but the learning process is a little bit unstable even after 300 epochs.",
"On the other hand, SAC algorithm shows the lowest score on this task, about 40 percent success rate but its slope exhibits a rising trend.",
"In total, DDPG has better and faster result on this task.",
"Figure: Pick and Place ComparisonIn Figure REF , the results of three off-policy algorithms on Push are given.",
"All algorithms successfully solve the task and show high results over 80 percent.",
"DDPG has evolved to 100 percent success rate after 50th epochs but TD3 shows the same characteristic after around 80th epochs.",
"Despite the fact that DDPG has reached the maximum leve of success at early stage, the total result of TD3 is more stable.",
"Unlike DDPG and TD3, SAC algorithm has highly fluctuating character.",
"Shortly, in terms of speed, DDPG has better result but in terms of stability and success, TD3 can be chosen for Push task.",
"Figure: Push ComparisonReach task is the easiest task among the others and the comparison between the algorithms is illustrated in Figure REF .",
"DDPG and TD3 achieved 100 percent from the first epoch but SAC algorithm fluctuates around 80 percent during all training.",
"TD2 shows more stable structure from the beginning compared to DDPG and SAC.",
"Thus, TD3 can be chosen for Reach task for its stability.",
"Figure: Reach ComparisonTable REF shows the total time for DDPG, TD3 and SAC algorihtms to finish 400 epochs for Slide, Pick $\\&$ Place, Push and Reach tasks.",
"Here, it can be easily seen that DDPG is the fastest algorithm while SAC showing the slowest time result.",
"The total training time of TD3 is considerably close to the result of DDPG.",
"Table: The time comparison of algorithmsConsequently, every algorithm has different efficiency for each task.",
"In order to solve the Slide task, TD3 algorithm shows better result than DDPG and SAC, however, DDPG has the best success rate for Pick $\\&$ Place task.",
"These two tasks are the most challenging tasks because control and stability should be accurate enough in order to achieve the goal.",
"While solving the relatively easier tasks Push and Reach, TD3 also shows better and stable results than other algorithms.",
"SAC algorithm fails to solve the first two challenging tasks but gives good results for the easier tasks.",
"On the other hand, DDPG finished 400 epochs faster than TD3 and SAC.",
"In short, in order to control a robotic manipulator for different tasks, TD3 shows the best and stable results, but in terms of time, DDPG can be chosen to achieve the task.",
"Figure REF illustrates the episode times of all tasks for the Fetch Robotic.",
"As an instance, considering DDPG, as it is seen from figure REF , for the sliding task, it takes 29 seconds to finish a training epoch (overall success rate is around 50 percent).",
"For Pick and Place task the duration of the training is almost around 29 seconds.",
"Push task is relatively easier than first two tasks with a duration of almost 28 seconds for each epoch.",
"The simplest task, Reach, immediately achieved 100 percent success with small amount of fluctuation and normally takes less time than other tasks with almost 26 seconds.",
"Figure: Time complexity of the applied off-policy algorithms."
],
[
"Conclusion",
"The study specifically focused on off-policy reinforcement learning algorithms, such as DDPG, TD3, and SAC, and their effectiveness in solving complex control tasks in robotic manipulators.",
"We found that these algorithms can learn to take actions in order to maximize a reward signal, which is essential for tasks such as reaching, picking, placing, and sliding objects.",
"However, the performance and reaction of different algorithms can vary depending on the specific task, which highlights the importance of selecting the most appropriate algorithm for a given task.",
"In addition to their effectiveness in solving complex control tasks, off-policy reinforcement learning algorithms have the advantage of being model-free, which means they do not require a detailed model of the environment or the system being controlled.",
"This can make them more flexible and adaptable to a wider range of tasks and environments.",
"However, the training process for these algorithms can be time-consuming and require significant computational resources, which emphasizes the need to carefully select the most appropriate algorithm for a given task.",
"In our paper, we applied DDPG, TD3 and SAC algorithms to train 7-DoF Fetch robotic manipulator, which is based on MuJoCo simulator, in order to achieve four different tasks.",
"After training 400 epochs and 50 episodes for each epoch, DDPG and TD3 algorithms successfully solve the most challenging task, Slide, but SAC algorithm fails on this.",
"All algorithms show successful results on Pick $\\&$ Place task; however, SAC algorithm still achieves lower success rate than others.",
"Push and Reach, which are the relatively easier tasks, are solved over 90 percent success rate by all algorithms.",
"While examining all the results of the algorithms, TD3 shows better and stable result for all tasks.",
"On the other hand, in terms of time, DDPG has higher successful result but has a small difference with respect to TD3.",
"Shortly, TD3 is the most successful algorithm among the off-policy model free algorithms in order to solve robotic manipulator tasks according to this study.",
"On the other hand, in order to claim an algorithm as the best option, it needs to be tested on a real-life system.",
"As a future work, DDPG, TD3 and SAC algorithms will be tested on a 7-DoF robotic manipulator, either Fetch or Kuka, in order to physically implement the given tasks."
]
] | 2212.05572 |
[
[
"Schur-Sergeev duality for Ariki-Koike algebras"
],
[
"Abstract Let $U_q(\\mathfrak{g})$ be the quantized superalgebra of $\\mathfrak{g}=\\mathfrak{gl}(k_1|\\ell_1)\\oplus\\cdots\\oplus\\mathfrak{gl}(k_m|\\ell_m)$ and $H_{m,n}(q,\\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$.",
"We define a right $H_{m,n}(q,\\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\\mathfak{g})$ and $H_{m,n}(q,\\mathbf{Q})$."
],
[
"Introduction",
"Let $k$ be a positive integer.",
"It is known that the group $\\mathrm {GL}(k,\\mathbb {C})^{\\otimes n}$ acts on the $n$ -fold tensor space of $(\\mathbb {C}^{k})^{\\otimes n}$ of $\\mathbb {C}^k$ by means of the standard action of $\\mathrm {GL}(k,\\mathbb {C})$ on each factor: $ (g_1, \\cdots , g_n)(v_1\\otimes \\cdots \\otimes v_n)=g_1(v_1)\\otimes \\cdots \\otimes g_n(v_n) $ for $g_i\\in \\mathrm {GL}(k,\\mathbb {C})$ and $v_i\\in \\mathbb {C}^{k}$ .",
"By restricting to the diagonal subgroup, i.e., taking $g_1=\\cdots =g_n$ , we obtain the standard tensor product action of $\\mathrm {GL}(k,\\mathbb {C})$ on $(\\mathbb {C}^{k})^{\\otimes n}$ .",
"There is also natural action of the symmetric group $\\mathfrak {S}_n$ on $(\\mathbb {C}^{k})^{\\otimes n}$ , given by permuting the factors: $ s_i(v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i+1}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n)\\hspace{0.0pt}=\\hspace{0.0pt}v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i+1}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n,1\\le i\\le n-1$ for the simple transpositions $s_i=(i,i+1)\\in \\mathfrak {S}_n$ and $v_j\\in \\mathbb {C}^{k}$ .",
"Schur [37], [38] showed that $\\mathrm {GL}(k,\\mathbb {C})$ and $\\mathfrak {S}_n$ are mutual centralizers of each other in $\\mathrm {End}_{\\mathbb {C}}(V^{\\otimes n})$ , which now known as the classical Schur-Weyl reciprocity, and obtained the Frobenius formula [17] by applying this reciprocity.",
"After Schur's classical work, Schur–Weyl reciprocity has been extended to various groups and algebras.",
"Here we only review briefly the ones inspiring the present work: Let $U_q(\\mathfrak {gl}(k))$ be the quantized enveloping algebra of $\\mathfrak {gl}(k)$ and $\\mathcal {H}_n(q^2)$ the Iwahori-Hecke algebra of type $A$ .",
"In [23], Jimbo defined an $\\mathcal {H}_n(q^2)$ -action on the $n$ -fold tensor space of the natural representation of $U_q(\\mathfrak {gl}(k))$ and showed the quantum Schur-Weyl reciprocity between $U_q(\\mathfrak {gl}(k))$ and $\\mathcal {H}_n(q^2)$ .",
"Let $\\mathbb {C}^{k|\\ell }$ be the superspace with dimension $k|\\ell $ and $\\mathfrak {gl}(k|\\ell )$ the general linear Lie superalgebra, that is, $\\mathfrak {gl}(k|\\ell )=\\mathrm {End}_{\\mathbb {C}}(\\mathbb {C}^{k|\\ell })$ .",
"Then $(\\mathbb {C}^{k|\\ell })^{\\otimes n}$ is a $\\mathfrak {gl}(k|\\ell )$ -module by letting $g(\\hspace{0.0pt}v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n\\hspace{0.0pt})\\hspace{0.0pt}=\\hspace{0.0pt}g(v_1)\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n\\hspace{0.0pt}+\\hspace{0.0pt}\\sum _{i=2}^{n}(-\\!1)^{\\overline{g}\\overline{v_1\\otimes \\cdots \\otimes v_{i-1}}}v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}g(v_{i})\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{n},$ where $g\\in \\mathfrak {gl}(k|\\ell )$ and $v_i\\in \\mathbb {C}^{k|\\ell }$ are homogeneous for all $i$ .",
"There is also an $\\mathfrak {S}_n$ -action on $(\\mathbb {C}^{k|\\ell })^{\\otimes n}$ given by $ s_i(v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i+1}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n)\\hspace{0.0pt}=\\hspace{0.0pt}(-\\!1)^{\\overline{v}_i\\overline{v}_{i+1}}v_1\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i+1}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_{i}\\hspace{0.0pt}\\otimes \\hspace{0.0pt}\\cdots \\hspace{0.0pt}\\otimes \\hspace{0.0pt}v_n,1\\le i<n,$ where $v_i, v_{i+1}$ are homogeneous of $\\mathbb {C}^{k|\\ell }$ .",
"Then the super Schur-Weyl duality between $\\mathfrak {gl}(k|\\ell )$ and $\\mathbb {C}\\mathfrak {S}_n$ was established first by Sergeev in [40] and then in more detail by Berele and Regev [5], which is also called the Schur–Sergeev duality in some literatures (see e.g.",
"[8]).",
"Let $\\mathcal {H}$ be the Ariki–Koike algebras, i.e., the cyclotomic Hecke algebras of type $G(m,1,n)$ and $U_q(\\overline{\\mathfrak {g}})$ the quantized enveloping algebra of a Levi subalgebra $\\overline{\\mathfrak {g}}=\\mathfrak {gl}(k_1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m)$ of $\\mathfrak {gl}(k)$ with $k=\\sum _{i=1}^mk_i$ .",
"Based on Jimbo's work [23], Ariki et al [2] gave a Schur–Weyl reciprocity between $U_q(\\overline{\\mathfrak {g}})$ and $\\mathcal {H}$ for all $k_i=1$ ; Sakamoto and Shoji [34] and Hu [20] established independently the reciprocity for the general case by applying completely different constructions of the $\\mathcal {H}$ -action on tensor space of the natural representation of $U_q(\\overline{\\mathfrak {g}})$ and by different arguments.",
"Let $U_q(\\mathfrak {gl}(k|\\ell ))$ be the quantized enveloping superalgebra of $\\mathfrak {gl}(k|\\ell )$ .",
"The super quantum Schur-Weyl duality between $U_q(\\mathfrak {gl}(k|\\ell ))$ and $\\mathcal {H}_n(q^2)$ was shown independently by Moon [30] and by Mitsuhashi [28] via different approaches, which is a quantum analogue of the super Schur-Sergeev duality.",
"Motivated by these works, the purpose of this paper is to present a super Schur-Weyl reciprocity between the quantum superalgebra $U_q(\\mathfrak {g})$ of the Lie superalgebra $\\mathfrak {g}=\\mathfrak {gl}(k_1|\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m|\\ell _m)$ and $\\mathcal {H}$ along the line of Sakamoto and Shoji's work [34], which unifies the aforementioned works.",
"Let us say more details on the super Schur-Weyl reciprocity.",
"Let $k_i,\\ell _i (i=1, \\ldots , m)$ be non-negative integers with $(\\sum _{i=1}^mk_i|\\sum _{i=1}^m\\ell _i)=(k|\\ell )$ .",
"Let $(\\Psi ^{\\otimes n}, V^{\\otimes n})$ be the vector representations of the quantized enveloping superalgebra $U_q(\\mathfrak {gl}(k|\\ell ))$ of $\\mathfrak {gl}(k|\\ell )$ over $\\mathbb {K}=\\mathbb {C}(q,\\mathbf {Q})$ (see §REF ).",
"Note that $\\mathfrak {g}$ can be viewed as a subalgebra of Lie superalgebra $\\mathfrak {gl}(k|\\ell )$ , which enable us to yield a $U_q(\\mathfrak {g})$ -action on $V^{\\otimes n}$ via the restriction of $\\Psi ^{\\otimes n}$ , which is also denoted by $(\\Psi ^{\\otimes n}, V^{\\otimes n})$ .",
"By extending Moon and Mitsuhashi's loc. cit.",
"works, we define an $\\mathcal {H}$ -action on $V^{\\otimes n}$ , which is proved to be an $\\mathcal {H}$ -representation $(\\Phi , V^{\\otimes n}$ ) (Theorem REF ).",
"It is not hard to show that $\\Phi $ actually commutes with $\\Psi ^{\\otimes n}$ , while we have to make much efforts to show that $\\Phi (\\mathcal {H})$ and $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))$ are mutually the full centralizer algebras of each other by applying the representations of cyclotomic Hecke algebras.",
"Therefore we can prove the super Schur-Weyl reciprocity for $\\mathcal {H}$ (Theorem REF ).",
"We discuss below several interesting questions motivating the present work.",
"The classical Schur algebras appeared in an implicit form in Schur's remarkable article [38].",
"Schur's ideas were represented by J.A.",
"Green in a modern way in [19], where their significance for representation theory of general linear and symmetric groups over any infinite field was shown.",
"Most of the further generalizations follow the ideas of this engrossing book.",
"Note that the classical Schur algebras may be viewed as the algebra of endomorphisms of tensor space commuting with the action of $\\mathfrak {S}_n$ and can be defined over the integers.",
"Dipper and James introduced the $q$ -Schur algebras type $A$ as the algebra of endomorphisms of tensor space commuting with the action of $H_n(q)$ in [10] (see [26] for uniform formulation of $q$ -Schur algebras of arbitrary finite type).",
"Using the cellularity of cyclotomic Hecke algebras, Dipper, James and Mathas introduced the cyclotomic $q$ -Schur algebras related to $\\mathcal {H}$ along Dipper and James's work [10].",
"In the super setting, the Schur superalgebras were introduced in Muir's PhD thesis [27], the Schur $q$ -superalgebras were introduced Du and Rui in [14] and their representations were studied extensively by Du and his coauthors (see e.g.",
"[11], [12], [13]).",
"Therefore, it is very interesting to give a super analogue of cyclotomic $q$ -Schur algebras and study their structure and representations extensively.",
"This is one of our main motivation of this paper.",
"In a subsequent paper, we will introduce the cyclotomic $q$ -Schur superalgebras and give an alternate proof of the super Schur-Weyl reciprocity by adapting Hu's argument in [20].",
"Let us remark that Deng et al.",
"[9] recently introduce the slim cyclotomic $q$ -Schur algebras, which is a new version of cyclotomic $q$ -Schur algebras.",
"It would be very interesting to formulate a super-version of the slim cyclotomic $q$ -Schur algebras.",
"Based on the quantum Schur-Weyl reciprocity, Ram [32] gave a $q$ -analogue of Frobenius formula for the characters of the Iwahori-Hecke algebras of type $A$ .",
"A super Frobenius formula for the characters of the Iwahori-Hecke algebras of type $A$ was given by Mitsuhashi in [29] by applying the super quantum Schur-Weyl reciprocity.",
"An extension of Frobenius formula for the characters of cyclotomic Hecke algebra of type $G(m,1,n)$ is found in [42] by applying the Schur-Weyl reciprocity between cyclotomic Hecke algebras and quantum algebras given in [34].",
"In 2013, Regev [33] presented a surprising beautiful formula for the characters of the symmetric group super representations by applying the Schur–Sergeev duality and the combinatorial theory of Lie superalgebras, which is developed in [5].",
"Based on Moon's work [30] and Mitsuhashi's work [28], the author gives a quantum analogue of Regev formula and derive a simple formula for the Hecke algebra super character on the exterior algebra in [45].",
"Motivated by these works, a natural problem is to provide a cyclotomic (quantum) analogue of these formulas, which is another motivation of the present paper.",
"Base on [42], [29], we will give a super Frobenius formula for the characters of the characters of cyclotomic Hecke algebras in [46].",
"Combining the Schur-Weyl duality established Sergeev and Berele–Regev and ideas of Serganova [39], Brundan and Kujiwa [7] obtained a new proof of the Mullineux conjecture, which was first conjectured by Mullineux in [31] and proved by Ford and Kleshchev in [18].",
"Very recently, based on their study on the polynomial representations of the quantum (super) hyperalgebra associated with the quantum enveloping superalgebra of $\\mathfrak {gl}(k|\\ell )$ , Du et al.",
"[13] present a new proof of the quantum version of the Mullineux conjecture for Hecke algebra of type $A$ , which was first proved by Brundan [6] along Kleshchev's classical works.",
"Thus it would be very interesting to reinterpret the Mullineux involution for cyclotomic Hecke algebra [22] via representation theory of cyclotomic $q$ -Schur superalgebras, which is our last motivation of this paper.",
"Furthermore, one might expect that this interpretation would helpful to understand Dudas and Jacon's work [15] and to enhance our understanding on wall-crossing functors for representations of rational Cherednik algebras introduced by Losev in [25].",
"We hope to deal with this issue in the future.",
"This paper is organized as follows.",
"We begin in Section with the definition of quantized enveloping superalgebra and its vector representations, and fix some combinatoric notations.",
"Section devotes to introduce the sign $q$ -permutation representation of cyclotomic Hecke algebras on tensor product of superspace.",
"Finally, we establish the super Schur-Weyl duality between the quantum superalgebra $U(\\mathfrak {g})$ and cyclotomic Hecke algebras in last section.",
"Throughout the paper, we assume that $\\mathbb {K}=\\mathbb {C}(q,\\mathbf {Q})$ the field of rational function in indeterminates $q$ and $\\mathbf {Q}=(Q_1, \\ldots , Q_m)$ .",
"For fixed non-negative $k,\\ell $ with $k+\\ell >0$ , we define the parity function $i\\mapsto \\overline{i}$ by $ \\overline{i}=\\left\\lbrace \\begin{array}{ll}\\overline{0}, &\\hbox{ if }1\\le i\\le k;\\\\\\overline{1}, & \\hbox{ if }k<i\\le k+\\ell .\\end{array}\\right.$ Assume that $k_1, \\ldots , k_m$ , $\\ell _1, \\ldots , \\ell _m$ are non-negative integers satisfying $\\sum _{i=1}^mk_i=k$ , $\\sum _{i=1}^m\\ell _i=\\ell $ and denote by $\\mathbf {k}=(k_1, \\ldots , k_m)$ , $\\ell =(\\ell _1, \\ldots , \\ell _m)$ .",
"For $i=1, \\ldots , m$ , we define $d_i=\\sum _{j\\le i}k_j+\\ell _j$ .",
"Acknowledgements.",
"Part of this work was carried out while the author was visiting Northeastern University at Qinhuangdao and the Chern Institute of Mathematics (CIM) in Nankai University and he would like to thank Professors Chengming Bai, Ming Ding and Yanbo Li for their hospitalities during his visits.",
"A first manuscript of this paper was announced at the \"Academic Seminar on Algebra and Cryptography” at Hubei University (Wuhan, May 2018), the author would like to thank Professors Xiangyong Zeng, Yunge Xu and Yuan Chen for their hospitality."
],
[
"Preliminaries",
"In this section we begin with the definitions of the quantum superalgebra $U_q(\\mathfrak {gl}(k|\\ell ))$ , i.e., the quantized universal enveloping algebra of the general linear Lie superalgebra $\\mathfrak {gl}(k|\\ell )$ , and of its vector representations.",
"Then we introduce the Lie superalgebra $\\mathfrak {g}=\\mathfrak {gl}(k_1,\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m,\\ell _m)$ and its quantized universal enveloping algebra.",
"Finally, we fix some notations of combinatorics.",
"Note that The Serre-type presentations of the quantization of $\\mathfrak {gl}({k|\\ell })$ were obtained by various authors all roughly at about the same time (see e.g.",
"[24], [16], [35], [36], [44]).",
"In this paper we adopt a definition appeared in [44] to quote results there.",
"2.1.",
"By a superspace we means a $\\mathbb {Z}_2$ -graded vector space $U$ over $\\mathbb {C}$ , namely a $\\mathbb {C}$ -vector space with a decomposition into two subspaces $U = U_{\\bar{0}}\\oplus U_{\\bar{1}}$ .",
"A nonzero element $u$ of $U_i$ will be called homogeneous and we denote its degree by $\\overline{u}={i}\\in \\mathbb {Z}_2$ .",
"We will view $\\mathbb {C}$ as a superspace concentrated in degree 0.",
"Given superspaces $U$ and $W$ , we view the direct sum $U\\oplus W$ and the tensor product $U\\otimes _{\\mathbb {C}} W$ as superspaces with $(U\\oplus W)_i = U_i\\oplus W_i$ , and $(U\\otimes _{\\mathbb {C}} W)_i= U_{\\bar{0}}\\otimes _{\\mathbb {C}} V_i\\oplus U_{\\bar{1}}\\otimes _{\\mathbb {C}} W_{\\bar{1}-i}$ for $i\\in \\mathbb {Z}_2$ .",
"With this grading, $U\\otimes _\\mathbb {C}W$ is called the tensor space of $U$ and $W$ and is denoted by $U\\otimes W$ .",
"Also, we make the vector space $\\mathrm {Hom}_\\mathbb {C}(U, W)$ of all $\\mathbb {C}$ -linear maps from $U$ to $W$ into a superspace by setting that $\\mathrm {Hom}_{\\mathbb {C}}(U, W)_i$ consists of all the $\\mathbb {C}$ -linear maps $f: U \\rightarrow W$ with $f(U_j)\\subseteq W_{i+j}$ for $i, j\\in \\mathbb {Z}_2$ .",
"Elements of $\\mathrm {Hom}_{\\mathbb {C}}(U, W)_{\\bar{0}}$ (resp.",
"$\\mathrm {Hom}_{\\mathbb {C}}(U, W)_{\\bar{1}}$ ) will be referred to as even (resp.",
"odd) linear maps.",
"Recall that a superalgebra $A$ is both a superspace and an associative algebra with identity such that $A_iA_j\\subseteq A_{i+j}$ for $i, j \\in \\mathbb {Z}_2$ .",
"Given two superalgebras $A$ and $B$ , the tensor space $A\\otimes B$ is again a superalgebra with the inducing grading and multiplication given by $(a_1 {\\otimes } b_1)(a_2 {\\otimes } b_2) =(-1)^{\\overline{b}_1\\overline{a}_2}a_1a_2{\\otimes } b_1b_2, \\text{ for } a_i\\in A \\text{ and } b_i\\in B.$ Furthermore, if $\\phi \\in \\mathrm {End}(A)$ and $\\mathrm {End}(B)$ are homogeneous endomorphisms then the tensor $\\phi \\otimes \\psi $ is defined as follows: $(\\phi \\otimes \\psi )(a\\otimes b):=(-1)^{\\bar{a}\\bar{\\psi }}\\phi (a)\\otimes \\psi (b)$ Note thess and other such expressions only make sense for homogeneous elements.",
"Observe that the $n$ -fold tensor space $A^{\\otimes n}:=A \\otimes A \\otimes \\cdots \\otimes A$ of $A$ is well-defined for all $n$ .",
"2.3.",
"Recall that the Lie superalgebra $\\mathfrak {gl}(k|\\ell )$ is the $(k+\\ell )\\times (k+\\ell )$ matrices with $\\mathbb {Z}_2$ -gradings given by $ \\mathfrak {gl}(k|\\ell )_{\\bar{0}}&=&\\left\\lbrace \\left.\\left(\\begin{array}{cc}\\mathbf {A} & \\mathbf {0} \\\\ \\mathbf {0} & \\mathbf {D}\\end{array}\\right)\\right|\\mathbf {A}=(a_{ij})_{1\\le i,j\\le k}, \\mathbf {D}=(d_{ij})_{k< i,j\\le k+\\ell }\\right\\rbrace ,\\\\ \\mathfrak {gl}(k|\\ell )_{\\bar{1}}&=&\\left\\lbrace \\left.\\left(\\begin{array}{cc}\\mathbf {0}& \\mathbf {B} \\\\ \\mathbf {C} & \\mathbf {0}\\end{array} \\right)\\right|\\mathbf {B}=(b_{ij})_{1\\le i\\le k}^{k<j\\le k+\\ell }, \\mathbf {C}=(c_{ij})_{k<i\\le k+\\ell }^{1\\le j\\le k}\\right\\rbrace $ and Lie bracket product defined by $[\\mathbf {X},\\mathbf {Y}]:=\\mathbf {XY}-(-1)^{\\overline{\\mathbf {X}}\\,\\overline{\\mathbf {Y}}}\\mathbf {YX}$ for homogeneous $\\mathbf {X},\\mathbf {Y}$ .",
"For $a,b=1,\\ldots , k+\\ell $ , denote by $\\mathbf {E}_{a,b}$ the elementary $(k+\\ell )\\times (k+\\ell )$ matrix with 1 in the $(a,b)$ -entry and zero in all other entries.",
"Let $\\epsilon _i: \\mathfrak {gl}(k|\\ell )\\rightarrow \\mathbb {C}$ be the linear function on $\\mathfrak {gl}(k|\\ell )$ defined by $\\epsilon _i(\\mathbf {E}_{a,b})=\\delta _{i,a}\\delta _{a,b} \\text{ for }i, a,b\\in [1, k+\\ell ].$ The free abelian group $P=\\bigoplus \\limits _{i=1}^{k+\\ell }\\mathbb {Z}\\epsilon _i$ (resp.",
"$P^{\\vee }=\\bigoplus \\limits _{i=1}^{k+\\ell }\\mathbb {Z}\\mathbf {E}_{b,b}$ ) is called the weight lattice (resp.",
"dual weight lattice) of $\\mathfrak {gl}(k|\\ell )$ , and there is a symmetric bilinear form $(\\,,\\,)$ on $\\mathfrak {h}^*=\\mathbb {C}\\otimes _{\\mathbb {Z}}P$ defined by $(\\epsilon _i,\\epsilon _j)=(-1)^{\\overline{i}}\\delta _{i,j} \\text{ for }i,j\\in [1, k+\\ell ].$ Then the simple roots of $\\mathfrak {gl}(k,\\ell )$ are $\\alpha _i=\\epsilon _i-\\epsilon _{i+1}$ , $i=1, \\ldots , k+\\ell -1$ .",
"We have positive root system $\\Phi ^{+}=\\lbrace \\alpha _{i,j}=\\epsilon _i-\\epsilon _j|1\\le i<j\\le k+\\ell \\rbrace $ and negative root system $\\Phi ^{-}=-\\Phi ^{+}$ .",
"Define $\\overline{\\alpha }_{i,j}=\\overline{i}+\\overline{j}$ and call $\\alpha _{i,j}$ is an even (resp.",
"odd) root if $\\overline{\\alpha }_{i,j}=\\overline{0}$ (resp.",
"$\\overline{1}$ ).",
"Note that $\\alpha _{k}$ is the only odd simple root.",
"Denote by $\\langle \\cdot ,\\cdot \\rangle $ the natural pairing between $P$ and $P^{\\vee }$ .",
"Then the simple coroot $\\alpha ^{\\vee }_i$ corresponding to $\\alpha _i$ is the unique element in $P^{\\vee }$ satisfying $\\langle \\alpha ^{\\vee }_i,\\lambda \\rangle =(-1)^{\\overline{i}}(\\alpha _i, \\lambda )\\text{ for all }\\lambda \\in P. $ Definition 2.4 The quantum superalgebra$U_q(\\mathfrak {gl}(k|\\ell )$ , that is, the quantized universal enveloping algebra of $\\mathfrak {gl}(k|\\ell )$ is the unitary superalgebra over $\\mathbb {K}$ generated by the homogeneous elements $E_1, \\ldots , E_{k+\\ell -1}, F_1, \\ldots , F_{k+\\ell -1}, K_1^{\\pm 1}, \\ldots , K_{k+\\ell }^{\\pm 1}$ with a $\\mathbb {Z}_2$ -gradation by letting $\\overline{E}_k=\\overline{F}_k=\\overline{1}$ , $\\overline{E}_a=\\overline{F}_a=\\overline{0}$ for $a\\ne k$ , and $\\overline{{K_i}^{\\pm 1}}=\\overline{0}$ .",
"These generators satisfy the following relations: $K_aK_b=K_bK_a, K_aK_a^{-1}=K_a^{-1}K_a=1$ ; $K_aE_b=q^{\\langle \\alpha ^{\\vee }_a,\\alpha _b \\rangle }E_bK_a$ ; $E_aE_b=E_bE_a, F_aF_b=F_bF_a$ if $|a-b|>1$ ; $[E_a,F_b]=\\delta _{a,b}\\frac{\\widetilde{K}_a-\\widetilde{K}_a^{-1}}{q_a-q_a^{-1}}$ , where $q_a=q^{(-1)^{\\overline{a}}}$ and $\\widetilde{K}_a=K_aK^{-1}_{a+1}$ ; For $a\\ne k$ and $|a-b|>1$ , $&& E_a^2E_b-(q_a+q_a^{-1})E_aE_bE_a+E_bE_a^2=0,\\\\&&F_a^2F_b-(q_a+q_a^{-1})F_aF_bF_a+F_bF_a^2=0;$ $E_k^2=F^2_k=0$ , $E_k\\!\\left(\\!E_{k\\!-\\!1}E_kE_{k\\!+\\!1}\\!\\!+\\!\\!E_{k\\!+\\!1}E_{k}E_{k\\!-\\!1}\\!\\right)\\!\\!-\\!\\!\\left(\\!q\\!\\!+\\!\\!q^{-\\!1}\\!\\right)\\!E_kE_{k\\!-\\!1}E_{k\\!+\\!1}E_k\\!\\!+\\!\\!\\left(\\!E_{k\\!-\\!1}E_kE_{k\\!+\\!1}\\!\\!+\\!\\!E_{k\\!+\\!1}E_kE_{k\\!-\\!1}\\!\\right)\\!E_k$ , $F_k\\!\\left(\\!F_{k\\!-\\!1}F_kF_{k\\!+\\!1}\\!\\!+\\!\\!F_{k\\!+\\!1}F_{k}F_{k\\!-\\!1}\\!\\right)\\!\\!-\\!\\!\\left(\\!q\\!\\!+\\!\\!q^{-\\!1}\\!\\right)\\!F_kF_{k\\!-\\!1}F_{k\\!+\\!1}F_k\\!\\!+\\!\\!\\left(\\!F_{k\\!-\\!1}F_kF_{k\\!+\\!1}\\!\\!+\\!\\!F_{k\\!+\\!1}F_kF_{k\\!-\\!1}\\!\\right)\\!F_k$ .",
"It is known that $U_q(\\mathfrak {gl}(k|\\ell ))$ is a Hopf superalgebra with comultiplication $\\Delta $ defined by $&&\\Delta (K_i^{\\pm 1})=K_i^{\\pm 1}\\otimes K_i^{\\pm 1},\\\\&&\\Delta (E_i)=E_i\\otimes \\widetilde{K}_i+1\\otimes E_i, \\\\&& \\Delta (F_i)=F_i\\otimes 1+ \\widetilde{K}_i^{-1}\\otimes F_i.$ 2.5.",
"Let $V$ be a superspace over $\\mathbb {K}$ with $\\dim V= k|\\ell $ , that is, $V=\\mathbb {C}^{k|\\ell }\\otimes _{\\mathbb {C}}\\mathbb {K}$ , and let $\\mathfrak {B}=\\lbrace v_1,\\ldots , v_{k+\\ell }\\rbrace $ be its homogeneous basis.",
"The vector representation $\\Psi $ of $U_q(\\mathfrak {gl}(k|\\ell ))$ on $V$ is defined by $&&\\Psi (E_i)v_{j}=\\left\\lbrace \\begin{array}{ll}(-1)^{\\overline{v}_j}v_{j-1}, &\\quad \\hbox{ if }j=i+1; \\\\0, & \\quad \\hbox{ others.}\\end{array}\\right.",
";\\\\&&\\Psi (F_i)v_{j}=\\left\\lbrace \\begin{array}{lll}(-1)^{\\overline{v}_j}v_{j+1}, &\\quad \\hbox{ if }j=i; \\\\0, & \\quad \\hbox{ others.",
"}\\end{array}\\right.\\\\&&\\Psi (K_i^{\\pm 1})(v_j)=\\left\\lbrace \\begin{array}{ll}(-1)^{\\overline{v}_j}q^{\\pm 1}v_{j}, & \\quad \\hbox{ if }j=i; \\\\0, & \\quad \\hbox{others.",
"}\\end{array}\\right.$ For a positive integer $n$ , we can define inductively a superalgebra homomorphism $\\Delta ^{(n)}: U_q(\\mathfrak {gl}(k|\\ell ))\\rightarrow U_q(\\mathfrak {gl}(k|\\ell ))^{\\otimes n},\\quad \\Delta ^{(n)}=(\\Delta ^{(n-1)}\\otimes \\mathrm {id})\\circ \\Delta $ for each $n\\ge 3$ , where $\\Delta ^{(2)}=\\Delta $ .",
"Therefore, $\\Psi $ can be extended to the representation on tensor space $V^{\\otimes n}$ via the Hopf superalgebra structure of $U_q(\\mathfrak {gl}(k|\\ell ))$ for each $n$ , we denote it by $\\Psi ^{\\otimes n}$ .",
"More precisely, the $U_q(\\mathfrak {gl}(k|\\ell ))$ -act on $V^{\\otimes n}$ is defined as follows: $&&\\Psi ^{\\otimes n}(E_a)=\\sum _{p=0}^{n-1}\\widetilde{K}_a^{\\otimes p}\\otimes \\Psi (E_a)\\otimes \\mathrm {Id}^{\\otimes ^{n-1-p}},\\\\&&\\Psi ^{\\otimes n}(F_a)=\\sum _{p=0}^{n-1}\\mathrm {Id}^{\\otimes p}\\otimes \\Psi (F_a)\\otimes (\\widetilde{K}_a^{-1})^{\\otimes ^{n-1-p}},\\\\&&\\Psi ^{\\otimes n}(K_{a})=K_a\\otimes \\cdots \\otimes K_a.$ According to [3], the vector representation is an irreducible highest weight module $V(\\epsilon _1)$ with highest weight $\\epsilon _1$ and $V^{\\otimes n}$ is complete reducible for all $n$ .",
"2.6.",
"Now assume that $V=V^{(1)}\\oplus \\cdots \\oplus V^{(m)}$ , where $V^{(i)}$ is a subsuperspace of $V$ with $\\dim V^{(i)}=k_i|\\ell _i$ and homogeneous basis $\\mathfrak {B}^{(i)}=\\left\\lbrace v^{(i)}_1, \\ldots , v_{k_i+\\ell _i}^{(i)}\\right\\rbrace , \\quad 1\\le i\\le m,$ where $v^{(i)}_1, \\ldots , v^{(i)}_{k_i}$ is even and $v^{(i)}_{k_i+1}, \\ldots , v^{(i)}_{k_i+\\ell _i}$ is odd for $i=1, \\ldots , m$ .",
"In this way ,we obtain that $\\mathfrak {B}=\\mathfrak {B}^{(1)}\\sqcup \\cdots \\sqcup \\mathfrak {B}^{(m)}$ and the vectors in $\\mathfrak {B}^{(i)}$ are said to be of color $i$ .",
"Further we linearly order the vectors $v_1^{(1)}, \\ldots , v_{m}^{k_m+\\ell _m}$ by the rule $v_{a}^{(i)}<v_{b}^{(j)}&\\text{ if and only if }& i<j\\text{ or }i=j\\text{ and }a<b.",
"$ We may identify the vectors $v_1^{(1)}$ , $\\ldots $ , $v^{(m)}_{k_m+\\ell _m}$ with the vectors $v_1$ , $\\cdots $ , $v_{k+\\ell }$ as follows: $\\begin{array}{ccccccccccccc}v_{1}^{(1)}& \\cdots & v_{k_1}^{(1)}& v_{k_1\\hspace{0.0pt}+\\hspace{0.0pt}1}^{(1)}&\\cdots & v_{k_1\\hspace{0.0pt}+\\hspace{0.0pt}\\ell _1}^{(1)}&\\cdots &v_{1}^{(m)}& \\cdots & v_{k_m}^{(m)}& v_{k_m+1}^{(m)}&\\cdots & v_{k_m+\\ell _m}^{(m)} \\\\\\updownarrow & \\vdots & \\updownarrow & \\updownarrow & \\vdots & \\updownarrow &\\vdots &\\updownarrow &\\vdots &\\updownarrow &\\updownarrow &\\vdots &\\updownarrow \\\\v_1& \\cdots & v_{k_1} & v_{k+1} & \\cdots & v_{k\\hspace{0.0pt}+\\hspace{0.0pt}\\ell _1}&\\cdots &v_{k\\hspace{0.0pt}-\\hspace{0.0pt}k_m\\hspace{0.0pt}+\\hspace{0.0pt}1}&\\cdots &v_k&v_{d_m\\hspace{0.0pt}-\\hspace{0.0pt}\\ell _m\\hspace{0.0pt}+\\hspace{0.0pt}1}&\\cdots &v_{k+\\ell },\\end{array}$ Let $\\mathcal {I}(k,\\ell ;n)=\\lbrace \\mathbf {i}=(i_1, \\ldots , i_n)|1\\le i_t\\le k+\\ell , 1\\le t\\le n\\rbrace $ .",
"For $\\mathbf {i}=(i_1, \\ldots , i_n)\\in \\mathcal {I}(k,\\ell ;n)$ , we write $v_{\\mathbf {i}}=v_{i_1}\\otimes \\cdots \\otimes v_{i_n}$ and put $c_a(v_{\\mathbf {i}})=b$ if $v_{i_a}$ is of color $b$ .",
"Then $\\mathfrak {B}^{\\otimes n}=\\lbrace v_{\\mathbf {i}}|\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)\\rbrace $ is a homogeneous basis of $V^{\\otimes n}$ .",
"We may and will identify $\\mathfrak {B}^{\\otimes n}$ with $\\mathcal {I}(k,\\ell ;n)$ , that is, we will write $v_{\\mathbf {i}}$ by $\\mathbf {i}$ , $\\overline{v}_i$ by $\\overline{i}$ , $c_a(v_{\\mathbf {i}})$ by $c_a(\\mathbf {i})$ , etc., if there are no confusions.",
"Clearly, $\\overline{\\mathbf {i}}=\\overline{i}_1+\\cdots +\\bar{i}_n$ .",
"Clearly, the Lie superalgebra $\\mathfrak {gl}(k_i|\\ell _i)$ can be viewed as a subalgebra of $\\mathfrak {gl}(k|\\ell )$ for all $i=1, \\ldots , m$ .",
"Therefore the Lie superalgebra $\\mathfrak {g}=\\mathfrak {gl}(k_1|\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m|\\ell _m)$ is a subalgebra of $\\mathfrak {gl}(k|\\ell )$ and its quantum superalgebra $U_q(\\mathfrak {g})$ can be naturally embedded in $U_q(\\mathfrak {gl}(k,\\ell ))$ as a $\\mathbb {K}$ -subalgebra generated by ${G}=\\left\\lbrace E_a, F_a, K_{b}^{\\pm 1}\\mid a\\in \\lbrace 1,2, \\ldots , d_m\\rbrace \\backslash \\lbrace d_1, d_2, \\ldots ,d_m\\rbrace , 1\\le b\\le d_m\\right\\rbrace .$ Hence the restriction of $U_q(\\mathfrak {gl}(k|\\ell ))$ -representation $(\\Psi ^{\\otimes n}, V^{\\otimes n})$ gives a $U_q(\\mathfrak {g})$ -representation, we denote it by $(\\Psi ^{\\otimes n}, V^{\\otimes n})$ .",
"2.8.",
"Recall that a composition (resp.",
"partition) $\\lambda =(\\lambda _1, \\lambda _2, \\ldots )$ of $n$ , denote $\\lambda \\models n$ (resp.",
"$\\lambda \\vdash n$ ), is a sequence (resp.",
"weakly decreasing sequence) of nonnegative integers such that $|\\lambda |=\\sum _{i\\ge 1}\\lambda _i=n$ and write $\\ell (\\lambda )$ the length of $\\lambda $ , i.e., the number of nonzero parts of $\\lambda $ .",
"A multipartition of $n$ is an ordered $m$ -tuple $\\lambda =(\\lambda ^{(1)}; \\ldots ; \\lambda ^{(m)})$ of partitions $\\lambda ^{i}$ such that $n=\\sum _{i=1}^m|\\lambda ^{i}|$ .",
"We denote by ${P}_{m,n}$ the set of all multipartitions of $n$ .",
"Then ${P}_{m,n}$ is a poset under dominance $\\unrhd $ , where $\\lambda \\unrhd \\mu \\Longleftrightarrow \\displaystyle \\sum _{k=1}^{i-1}|\\lambda ^{k}|+\\sum _{\\ell =1}^j\\lambda _l^{i}\\ge \\sum _{k=\\ell }^{i-1}|\\mu ^{k}|+\\sum _{\\ell =1}^j\\mu _\\ell ^{i}\\quad \\text{ for all }1\\le i\\le m \\text{ and }j\\ge 1.$ We write $\\lambda \\rhd \\mu $ if $\\lambda \\unrhd \\mu $ and $\\lambda \\ne \\mu $ .",
"A partition $\\lambda =(\\lambda _1, \\lambda _2, \\cdots )\\vdash n$ is said to be a $(k, \\ell )$ -hook partition of $n$ if $\\lambda _{k+1}\\le \\ell $ .",
"We let $H(k,\\ell ;n)$ denote the set of all $(k,\\ell )$ -hook partitions of $n$ , that is $H(k,\\ell ;n)=\\lbrace \\lambda =(\\lambda _1,\\lambda _2,\\cdots )\\vdash n\\mid \\lambda _{k+1}\\le \\ell \\rbrace .$ A multipartition $\\lambda =(\\lambda ^{(1)}; \\ldots ; \\lambda ^{(m)})$ of $n$ is said to be a $(\\mathbf {k},\\ell )$ -hook multipartition of $n$ if $\\lambda ^{(i)}$ is a $(k_i,\\ell _i)$ -hook partition for all $i=1, \\ldots ,m$ .",
"We denote by $H(\\mathbf {k}|\\ell ; m,n)$ the set of all $(\\mathbf {k},\\ell )$ -hook multipartitions of $n$ .",
"Thanks to [40] and [5], the irreducible representations of $U_q(\\mathfrak {gl}(k,\\ell ))$ occurring in $V^{\\otimes n}$ are parameterized by the $(k,\\ell )$ -hook partitions of $n$ .",
"Note that $U_q(\\mathfrak {g})=U_q(\\mathfrak {gl}(k_1,\\ell _1))\\otimes \\cdots \\otimes U_q(\\mathfrak {gl}(k_m,\\ell _m))$ .",
"As a consequence, the irreducible representations of $U_q(\\mathfrak {g})$ occurring in $V^{\\otimes n}$ are parameterized by the $(\\mathbf {k},\\ell )$ -hook multipartitions of $n$ .",
"2.9.",
"The diagram of an $m$ -multipartition $\\lambda $ is the set $ [\\lambda ]:=\\lbrace (i,j,c)\\in \\mathbb {Z}_{>0}\\times \\mathbb {Z}_{>0}\\times \\mathbf {m}|1\\le j\\le \\lambda ^c_i\\rbrace , \\quad \\text{ where }\\mathbf {m}=\\lbrace 1, \\dots , m\\rbrace .$ The elements of $[\\lambda ]$ are the nodes of $\\lambda $ .",
"By a $\\lambda $ -tableau, we mean a bijection $\\mathfrak {t}: [\\lambda ]\\rightarrow \\lbrace 1,2,\\dots \\rbrace $ and write $\\text{Shape}(\\mathfrak {t})=\\lambda $ if $\\mathfrak {t}$ is a $\\lambda $ -tableau.",
"If its entries are from the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ then it is called an $n$ -tableau.",
"Of course an $n$ -tableau is also an $n+1$ -tableau, etc.",
"We may and will identify a tableau $\\mathfrak {t}$ with an $m$ -tuple of tableaux $\\mathfrak {t}=(\\mathfrak {t}^1; \\dots ; \\mathfrak {t}^m)$ , where $\\mathfrak {t}^{c}$ is a $\\lambda ^{c}$ -tableau, $c=1, \\cdots , m$ , which is called the $c$ -component of $\\mathfrak {t}$ .",
"A tableau is (semi) standard if in each component the entries (weakly) increase along the rows and strictly down along the columns and denote by $\\mathrm {Std}(\\lambda )$ the set of all standard $\\lambda $ -tableaux.",
"Given a standard tableau $\\mathfrak {t}$ and an integer $i$ , we define the residue of $i$ in $\\mathfrak {t}$ to be $\\mathrm {res}_\\mathfrak {t}(i)=Q_cq^{2(b-a)}$ if $i$ appears in the node $(a, b,c)$ of $\\mathfrak {t}$ .",
"Let $\\bar{\\mathbf {0}}=\\lbrace 0_1, \\cdots , 0_k\\rbrace $ and $\\bar{\\mathbf {1}}=\\lbrace 1_1, \\cdots , 1_{\\ell }\\rbrace $ with $0_1<\\cdots <0_k<1_1<\\cdots <1_{\\ell }$ .",
"Then a tableau $\\mathfrak {t}$ of shape $\\lambda \\vdash n$ is said to be $(k,\\ell )$ -semistandard if the $\\bar{\\mathbf {0}}$ part (i.e.",
"the boxes filled with entries $0_i$ 's) of $\\mathfrak {t}$ is a tableau, the $0_i$ 's are nondecreasing in row, strictly increasing in columns, the $1_i$ 's are nondecreasing in columns, strictly increasing in rows.",
"The sign $q$ -permutation representation This section devotes to introduce an $\\mathcal {H}$ -action on $V^{\\otimes n}$ and prove that it is a (super) representation of $\\mathcal {H}$ by adapting the ideas of [34], [30], [28].",
"3.1.",
"Let $W_{m,n}$ be the complex reflection group of type $G(m,1,n)$ .",
"According to [41], $W_{m,n}$ has a presentation with generators $s_0, s_1, \\dots , s_{n-1}$ where the defining relations are $s_0^m=1, s_1^2=\\cdots =s_{n-1}^2=1$ and the homogeneous relations $&s_0 s_1s_0 s_1=s_1s_0 s_1s_0,&&\\\\& s_is_j=s_js_i,&&\\text{ if } |i-j|>1,\\\\&s_is_{i+1}s_i=s_{i+1}s_{i}s_{i+1},&& \\text{ for }1\\le i\\le n-2.$ It is well-known that $W_{m,n}\\cong (\\mathbb {Z}/m\\mathbb {Z})^{n}\\rtimes \\mathfrak {S}_{n}$ , where $s_1, \\dots , s_{n-1}$ are generators of the symmetric group $\\mathfrak {S}_{n}$ of degree $n$ corresponding to transpositions $(1\\,2)$ , $\\ldots $ , $(n\\!-\\!1\\,n)$ .",
"For $a=1, \\ldots , n-1$ and $\\mathbf {i}=(i_1, \\ldots , i_a,i_{a+1},\\ldots , i_n)$ , we define the following right action $\\mathbf {i}s_a:=(i_1, \\ldots , i_{a-1}, i_{a+1}, i_a, i_{a+2}, \\ldots , i_n).$ Following Sergeev [40] or Berele-Regev [5], there is a right action $\\phi $ of $\\mathbb {C}\\mathfrak {S}_n$ on $V^{\\otimes n}$ defined on generators by $s_a(\\mathbf {i})&:=&\\left\\lbrace \\begin{array}{ll}\\vspace{3.0pt}(-1)^{\\overline{i}_a}\\mathbf {i},& \\text{if }i_a=i_{a+1};\\\\(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,& \\text{if }i_a\\ne i_{a+1}.\\end{array}\\right.$ 3.3.",
"The Ariki-Koike algebra [1] or the cyclotomic Hecke algebra $\\mathcal {H}$ associated to $W_{m,n}$ [21], is the unital associative $\\mathbb {K}$ -algebra generated by $g_0,g_1,\\dots ,g_{n-1}$ and subject to relations $&(g_0-Q_1)\\dots (g_0-Q_m)=0,&&\\\\&g_0g_1g_0g_1=g_1g_0g_1g_0,&&\\\\&g_i^2=(q-q^{-1})g_i+1, &&\\text{ for }1\\le i<n,\\\\&g_ig_j=g_jg_i, &&\\text{ for }|i-j|>2,\\\\&g_ig_{i+1}g_i=g_{i+1}g_{i}g_{i+1}, &&\\text{ for }1\\le i<n-1.$ Let $w\\in \\mathfrak {S}_n$ and let $s_{i_1}s_{i_2}\\cdots s_{i_k}$ be a reduced expression for $w$ .",
"Then $g_{w}:=g_{i_1}g_{i_2}\\cdots g_{i_k}$ is independent of the choice of reduced expression and $\\lbrace g_{w}|w\\in \\mathfrak {S}_n\\rbrace $ is linear basis of the subalgebra $\\mathcal {H}_n(q)$ of $\\mathcal {H}$ generated by $g_1, \\ldots , g_{n-1}$ , that is, $\\mathcal {H}_n(q)$ is the Iwahori-Hecke algebra associated to $\\mathfrak {S}_n$ .",
"For $a=1, \\ldots , n-1$ , we define the endomorphisms $T_a, S_a\\in \\mathrm {End}_K(V^{\\otimes n})$ as follows: $T_a(\\mathbf {i}):=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}(q-q^{-1})\\mathbf {i}+(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}\\frac{(q-q^{-1})+(-1)^{\\overline{i}_a}(q+q^{-1})}{2} \\mathbf {i},& \\text{if }i_a=i_{a+1}; \\\\(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ $S_a(\\mathbf {i})&:=&\\left\\lbrace \\begin{array}{ll}T_a(\\mathbf {i}), & \\hbox{ if } c_a(\\mathbf {i})= c_{a+1}(\\mathbf {i});\\\\s_a(\\mathbf {i}), & \\hbox{ if } c_a(\\mathbf {i})\\ne c_{a+1}(\\mathbf {i}).\\end{array}\\right.$ The following easy verified facts will be used latter.",
"Lemma 3.6 For all $\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)$ and $1\\le a<n$ , we have $T_a(\\mathbf {i})=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}(q-q^{-1})\\mathbf {i}+s_a(\\mathbf {i}),& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}\\frac{q-q^{-1}}{2}\\mathbf {i}+\\frac{q+q^{-1}}{2}s_a(\\mathbf {i}),& \\text{if }i_a=i_{a+1}; \\\\s_a(\\mathbf {i}),&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ $T_a$ is invertible and $T_a^{-1}(\\mathbf {i}):=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}s_a(\\mathbf {i}),& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}-\\frac{q-q^{-1}}{2}\\mathbf {i}+\\frac{q+q^{-1}}{2}s_a(\\mathbf {i}),& \\text{if }i_a=i_{a+1}; \\\\(q-q^{-1})\\mathbf {i}+ s_a(\\mathbf {i}),&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ (i) follows directly by applying Eq.",
"(REF ).",
"Since $T^2_a=(q-q^{-1})T_a+1$ , $T_a^{-1}=T_a-(q-q^{-1})$ .",
"Thus (ii) follows directly by applying (i).",
"Now let $S_0(\\mathbf {i}):=Q_{c_1(\\mathbf {i})}\\mathbf {i}$ and $\\theta =S_{n-1}\\cdots S_{1}$ .",
"We define $T_0\\in \\mathrm {End}_{\\mathbb {K}}(V^{\\otimes n})$ as following $T_0(\\mathbf {i}):&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}\\theta S_0(\\mathbf {i}).$ Thanks to [30], [28], Eq.",
"(REF ) defines a (super) representation of $\\mathcal {H}_n(q)$ .",
"The remainder of this section devotes to show that Eqs.",
"(REF ) and (REF ) define a (super) representation of $\\mathcal {H}$ .",
"Lemma 3.8 For $j,p\\ge 1$ , we denote by $V_{j,p}$ the subspace of $V^{\\otimes n}$ spanned by basis elements $\\mathbf {i}$ such that $c_p(\\mathbf {i})\\ge j$ .",
"If $\\mathbf {i}\\in V_{j,p}$ then $T_{p}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p}(\\mathbf {i})\\in \\mathbf {i}+V_{j+1,p}$ .",
"We use the backward induction on $p$ to prove the claim.",
"Note that for all $p=1, \\ldots , n-1$ , we have $&&T_{p}^{-1}S_{p}(\\mathbf {i})=\\left\\lbrace \\begin{array}{ll}\\mathbf {i}+(q-q^{-1})s_{p}(\\mathbf {i}), & \\hbox{ if }c_{p}(\\mathbf {i})> c_{p+1}(\\mathbf {i});\\\\\\mathbf {i}, & \\hbox{ others}.\\end{array}\\right.$ In particular, the lemma holds for $p=n-1$ .",
"Now assume that for all $p$ and $\\mathbf {i}^{\\prime }\\in V_{j,p}$ , $T_{p}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p}(\\mathbf {i}^{\\prime })\\in \\mathbf {i}^{\\prime }+V_{j+1,p}.$ Thanks to Lemma REF (i), $T_{p-1}(V_{j,p-1})=V_{j,p}$ for all $j\\ge 1$ , which implies $S_{p-1}(V_{j,p-1})\\in V_{j,p}$ due to Eq.",
"(REF ).",
"For any $\\mathbf {i}\\in V_{j,p-1}$ , the induction argument shows $(T_{p-1}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p-1})(\\mathbf {i})&=& T_{p-1}^{-1}(T_p^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_p)(S_{p-1}(\\mathbf {i}))\\\\ &\\in &T_{p-1}^{-1}(S_{p-1}(\\mathbf {i})+V_{j+1,p})\\\\&=&T_{p-1}^{-1}S_{p-1}(\\mathbf {i})+V_{j+1,p}\\\\&\\in & \\mathbf {i}+V_{j+1,p},$ where the last inclusion follows by Eq.",
"(REF ).",
"The lemma is proved.",
"We will need the following facts.",
"Lemma 3.10 For all $j\\ge 2$ , we have the following facts: $S_jS_{j-1}T_j=T_{j-1}S_jS_{j-1},$ $S_jS_{j-1}S_j S_{j-1}^{-1}T_{j-1}=T_jS_jS_{j-1}S_jS_{j-1}^{-1}, \\\\$ $S_jS_{j-1}S_j S_{j-1}T_{j-1}=T_jS_jS_{j-1}S_jS_{j-1}.$ Let $q_+=q+q^{-1}$ and $q_{*}=q-q^{-1}$ .",
"Without loss of generality, we may assume that $j=2$ and $\\mathbf {i}=(i_1, i_2, i_3)$ .",
"Therefore we have the following five cases: If $c_1(\\mathbf {i})=c_2(\\mathbf {i})=c_{3}(\\mathbf {i})$ then $S_1=T_1$ , $S_2=T_2$ .",
"Furthermore Eq.",
"(REF ) follows owing to [30].",
"If $c_1(\\mathbf {i})$ , $c_2(\\mathbf {i})$ and $c_3(\\mathbf {i})$ are pairwise different then $S_1(\\mathbf {i})= s_1(\\mathbf {i})$ , $S_2(\\mathbf {i})=s_2(\\mathbf {i})$ .",
"Thus we only need to consider the following cases: (a) $i_1<i_2<i_3$ ; (b) $i_1<i_2>i_3$ ; (c) $i_1>i_2>i_3$ .",
"Apply Lemma REF (i) and Eq.",
"(REF ), we obtain that $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=(q-q^{-1})s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=(q-q^{-1})s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=(q-q^{-1})s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=s_2s_1s_2(\\mathbf {i})=s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=s_2s_1s_2(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=s_1s_2T_1(\\mathbf {i})=T_2s_1s_2(\\mathbf {i})=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ Therefore, in this case Eq.",
"(REF ) hold.",
"If $c_1(\\mathbf {i})=c_2(\\mathbf {i})\\ne c_3(\\mathbf {i})$ then we only need to check the following six cases: $i_1=i_2<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*S_2S_1(\\mathbf {i})+S_2S_1s_2(\\mathbf {i})\\\\&=&q_*s_2T_1(\\mathbf {i})+T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+q_*s_2s_1(\\mathbf {i})\\!+\\!\\frac{1}{2}q_+s_1s_2s_1(\\mathbf {i})\\!-\\!\\frac{1}{2}q_*^2s_2(\\mathbf {i})\\!-\\!\\frac{1}{2}q_*s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+T_1s_2s_1(\\mathbf {i})-\\frac{1}{2}q_*T_1s_2(\\mathbf {i})\\\\&=&T_1S_2T_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i}); $ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&S_2S_1S_2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+s_2s_1s_2(\\mathbf {i})-\\frac{1}{2}q_*s_1s_2(\\mathbf {i}),$ $T_2S_2S_1S_2S_1^{-\\!1}(\\mathbf {i})&=&T_2S_2S_1S_2T_1^{-1}(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+T_2^2s_1s_2s_1(\\mathbf {i})-\\frac{1}{2}q_*T_2^2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})\\!+\\!\\frac{q_*^2}{2}T_2s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+q_*}{2}T_2s_1s_2s_1(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}); $ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&S_2S_1S_2(\\mathbf {i})+q_*S_2S_1S_2T_1(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_1=i_2>i_3$ : $S_2S_1T_2(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}),$ $T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})&=&T_2^2s_1s_2T_1^{-1}(\\mathbf {i})\\\\&=&q_*T_2s_1s_2T_1^{-1}(\\mathbf {i})+s_1s_2T_1^{-1}(\\mathbf {i})\\\\&=&\\frac{q_*^2}{2}T_2s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+q_*}{2}T_2s_1s_2s_1(\\mathbf {i})\\!+\\!\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}); $ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_1<i_2<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_2T_1(\\mathbf {i})+T_2s_1s_2(\\mathbf {i})\\\\&=&q_*^2s_2(\\mathbf {i})+q_*(s_1s_2+s_2s_1)(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})\\\\&=&S_2S_1S_2(\\mathbf {i})\\\\&=& S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_3<i_1<i_2$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i}).$ $i_2<i_1<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2+s_2s_1s_2\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_3<i_2<i_1$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2+s_2s_1s_2\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i}).$ Therefore Eq.",
"(REF ) hold in this case.",
"The remainder cases $c_1(\\mathbf {i})\\ne c_2(\\mathbf {i})=c_3(\\mathbf {i})$ and $c_1(\\mathbf {i})=c_3(\\mathbf {i})\\ne c_2(\\mathbf {i})$ can be verified by a similar way.",
"As a consequence, we prove the lemma.",
"Now we can prove the main result of this section.",
"Theorem 3.12 Keeping notation as above, then the $\\mathbb {K}$ -linear map $\\Phi : \\mathcal {H}\\rightarrow \\mathrm {End}_{\\mathbb {K}}(V^{\\otimes n})$ defined by $g_a\\mapsto T_a$ ($a=0, \\ldots , n-1$ ) is a (super) representation of $\\mathcal {H}$ .",
"Thanks to [30] or [28], it suffices to show that the following three relations hold: $ &&(T_0-Q_1)\\cdots (T_0-Q_m)=0; \\\\ &&T_0T_1T_0T_1=T_1T_0T_1T_0;\\\\ &&T_0T_i=T_iT_0, \\hbox{ for }i\\ge 2.$ Applying Eq.",
"(REF ) and Lemma REF , $T_0(\\mathbf {i})=Q_{c_1(\\mathbf {i})}\\mathbf {i}+V_{j+1,1}$ for any $\\mathbf {i}\\in V_{j,1}$ .",
"It follows that $(T_0-Q_j)(\\mathbf {i})&=&(Q_{c_1(\\mathbf {i})}-Q_j)\\mathbf {i}+V_{j+1,1}.$ Since $\\mathbf {i}\\in V_{j,1}$ , we have $c_1(\\mathbf {i})\\ge j$ .",
"Therefore $(Q_{c_1(\\mathbf {i})}-Q_j)(\\mathbf {i})=0$ if $c_1(\\mathbf {i})=j$ , and $(Q_{c_1(\\mathbf {i})}-Q_j)(\\mathbf {i})\\in V_{j+1,1}$ if $c_1(\\mathbf {i})>j$ , that is, $(T_0-Q_j)(\\mathbf {i})\\in V_{j+1,1}$ .",
"Finally notice that $V_{1,1}=V^{\\otimes n}$ and $V_{n+1,1}=\\lbrace 0\\rbrace $ .",
"As a consequence, Eq.",
"(REF ) holds.",
"Now note that $S_0$ commutes with $T_2, \\cdots , T_{n-1}$ and $S_iT_j=T_jS_i$ for $|i-j|>2$ .",
"Lemma REF implies $\\theta T_j=T_{j-1}\\theta ,\\qquad j=2, \\ldots , n-1.$ Since $S_iS_j=S_jS_i$ for $|i-j|\\ge 2$ , we have $\\theta ^2T_1 &=& (S_{n-1}\\cdots S_{1})(S_{n-1}\\cdots S_{1})T_1\\\\&=&S_{n-1}(S_{n-2}S_{n-1})\\cdots (S_2S_3)(S_1S_2)S_1T_1\\\\&=&(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})(S_2S_1S_2S_1)T_1\\\\&=&(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})T_2(S_2S_1S_2S_1)\\\\&=&T_{n-1}(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})(S_2S_1S_2S_1)\\\\&=&T_{n-1}\\theta ^2.$ Now we show $S_1^{-1}S_0S_1S_0T_1=T_1S_1^{-1}S_0S_1 S_0$ .",
"To do this, we may assume $\\mathbf {i}=(i_1,i_2)$ .",
"According to Eq.",
"(REF ), $S_1^{-1}S_0S_1S_0T_1(\\mathbf {i}) &=& \\left\\lbrace \\begin{array}{ll}Q_{c_1(\\mathbf {i})}^2T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})=c_2(\\mathbf {i}); \\\\Q_{c_1(\\mathbf {i})}Q_{c_2(\\mathbf {i})}T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})\\ne c_2(\\mathbf {i}).",
"\\end{array} \\right.\\\\T_1S_1^{-1}S_0S_1S_0(\\mathbf {i}) &=& \\left\\lbrace \\begin{array}{ll}Q_{c_1(\\mathbf {i})}^2T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})=c_2(\\mathbf {i}); \\\\Q_{c_1(\\mathbf {i})}Q_{c_2(\\mathbf {i})}T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})\\ne c_2(\\mathbf {i}).\\end{array}\\right.$ Combing the above two equalities, we get $(\\theta S_0)^2T_1=T_{n-1}(\\theta S_0)^2$ .",
"As a consequence, we yield that $T_0T_1T_0T_1&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0T_1\\\\&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2T_1\\\\&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})T_{n-1}(\\theta S_0)^2;\\\\T_1T_0T_1T_0&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0\\\\&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2\\\\&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2.$ Thanks to [2]), we yield that $T_0T_1T_0T_1=T_1T_0T_1T_0$ , i.e., Eq.",
"() holds.",
"Finally, thanks to Eq.",
"(REF ), for all $j\\ge 2$ , we have $T_0T_j &=& T_{1}^{-1}\\cdots T_{n-1}^{-1}\\theta S_0T_j\\\\&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{j-2}^{-1}(T_{j-1}^{-1}T_j^{-1}T_{j-1})T_{j+1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{j-2}^{-1}(T_{j}T_{j-1}^{-1}T_{j}^{-1})T_{j+1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_jT_0.$ It completes the proof.",
"Remark 3.15 If $\\ell =0$ then the representation $(\\Phi , V^{\\otimes n})$ reduces to representation defined by Sakamoto and Shoji [34].",
"If $m=1$ then the representation $(\\Phi , V^{\\otimes n})$ reduces to sign $q$ -permutation representation of $\\mathcal {H}_n(q)$ defined by Moon [30] and Mitsuhashi [28].",
"Remark 3.16 It is known that cyclotomic Hecke algebras are cyclotomic quotients of affine Hecke algebras, it would be interesting to define an affine Hecke action on superspaces such that it is stable under the quotient.",
"Moreover, this action would give another way to construct the $\\mathcal {H}$ -action on $V^{\\otimes n}$ .",
"Remark 3.17 Let $q=1$ and $Q_i=\\xi ^i$ , where $\\xi $ is a fixed primitive $m$ -th root of unity.",
"Then $\\mathcal {H}$ reduces to the group algebra $\\mathbb {C}W_{m,n}$ of $W_{m,n}$ and the (super) representation $(\\Phi , V^{\\otimes n})$ of $\\mathcal {H}$ reduces to a (super) representation of $\\mathbb {C}W_{m,n}$ .",
"Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and $\\mathcal {H}$ In this section, we establish the Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and $\\mathcal {H}$ , which can be viewed as a super analogue of the Schur-Weyl reciprocity for cyclotomic Hecke algebras given independently by Sakamoto-Shoji [34] and Hu [20], or as a cyclotomic version of the Schur-Weyl reciprocity between the quantum superalgebra and the Iwahori-Hecke algebra obtained by Moon [30] and Mitsuhashi [28].",
"Lemma 4.1 For any $X\\in \\mathcal {H}$ and $Y\\in U_q(\\mathfrak {g})$ , $\\Phi (X)\\Psi ^{\\otimes n}(Y)=\\Psi ^{\\otimes n}(Y)\\Phi (X)$ .",
"Thanks to [30] or [28], it is enough to show that the $T_0$ -action defined by Eq.",
"(REF ) commutes with the generators of $U_q(\\mathfrak {g})$ , i.e., commutes with those elements listed in the set ${G}$ defined in Eq.",
"(REF ).",
"Clearly $S_0$ commutes with $U_q(\\mathfrak {g})$ .",
"It reduces to show $S_a$ commutes with elements of ${G}$ for all $a\\ge 1$ .",
"It is easy to check that $S_a$ commutes with $K_j^{\\pm 1}$ for all $1\\le j\\le d_m$ .",
"Given $\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)$ , we show that $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)S_a(\\mathbf {i})$ for all $E_b\\in {G}$ .",
"Note that if $c_{a}(\\mathbf {i})=c_{a+1}(\\mathbf {i})$ then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))=c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ .",
"Thus $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=T_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)T_a(\\mathbf {i})$ .",
"If $c_{a}(\\mathbf {i})\\ne c_{a+1}(\\mathbf {i})$ ) then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))\\ne c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ ).",
"In this case, we need to show $s_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)s_a(\\mathbf {i})$ .",
"For $p=0,1,\\ldots , n-1$ , we let $&X_p=\\widetilde{K}_b^{\\otimes p}\\otimes \\Psi (E_b)\\otimes \\mathrm {Id}^{n-1-p}.$ Then $s_a$ commutes with $X_p$ unless $p=a-1,a$ , which implies we only need to show $&&s_a(X_{a-1}+X_{a})(\\mathbf {i})=(X_{a-1}+X_{a})s_a(\\mathbf {i}).$ Since $s_a$ affects only the $a$ and $a\\!+\\!1$ -th factors of $\\mathbf {i}$ and the remaining parts are the same for $X_{a}(\\mathbf {i})$ and $X_{a+1}(\\mathbf {i})$ .",
"We may only consider the $a$ and $a\\!+\\!1$ -th factors, that is, it is enough to verify that $s_a(\\Psi (E_b)\\!\\otimes \\!\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!\\Psi (E_b))(i_{a}\\otimes i_{a\\!+\\!1})\\!=\\!",
"(\\Psi (E_b)\\!\\otimes \\!",
"\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!",
"\\Psi (E_b))s_a(i_{a}\\!\\otimes \\!",
"i_{a\\!+\\!1}).$ Assume that $d_{r-1}<b<d_r$ for some $1\\le r\\le m$ with $d_0=0$ .",
"Thus if $c_{a}(\\mathbf {i})$ , $r$ , $c_{a+1}(\\mathbf {i})$ are pairwise different, then $\\Psi (E_b)=0$ , therefore both sides of Eq.",
"(REF ) equal zero; If $c_{a}(\\mathbf {i})=r\\ne c_{a+1}(\\mathbf {i})$ then $\\Psi (E_b)(i_{a+1})=0$ and $\\widetilde{K}_a(i_{a+1})=(-1)^{\\bar{i}_{a+1}-\\bar{i}_a}i_{a+1}$ .",
"Then, thanks to Eq.",
"(REF ), both sides of Eq.",
"(REF ) equal; The case $c_{a+1}(\\mathbf {i})=r\\ne c_a(\\mathbf {i})$ can be proved similarly.",
"So $S_a$ commutes with $\\Psi ^{\\otimes n}(E_b)$ for all $E_b\\in {G}$ .",
"In a similar argument, we can show $S_a$ commutes with $\\Psi ^{\\otimes n}(F_b)$ for all $F_b\\in {G}$ .",
"For positive integers $a$ , $b$ , $c$ , we let $\\Pi (a,b;c)=\\left\\lbrace (\\mu ;\\nu )\\left|\\begin{array}{l}\\mu \\vdash s,\\quad \\nu \\vdash t,\\quad s+t=c\\\\ \\ell (\\mu )\\le a,\\ell (\\nu )\\le b, \\mu _a\\ge \\ell (\\nu )\\end{array}\\right.\\right\\rbrace .$ Lemma 4.3 ([4] or [30]) Keeping notations as above, then there is a bijection between $H(a,b;c)$ and $\\Pi (a,b;c)$ given by $\\lambda \\mapsto (\\lambda ^1;\\lambda ^2)$ , where $\\lambda ^1=(\\lambda _1, \\ldots , \\lambda _a)$ and $\\lambda ^2=(\\lambda _1^2,\\ldots , \\lambda _b^2)$ with $\\lambda _i^2=\\max \\lbrace \\lambda _i^*-a,0\\rbrace $ .",
"Recall that the Jucys-Murphy elements of $\\mathcal {H}$ are defined recursively by $J_1\\!",
":=g_0\\quad \\text{ and }\\quad J_{i+1}\\!",
":=g_iJ_ig_i, \\quad i=1, \\cdots , n-1.$ It is known that $J_1$ , $\\ldots , J_n$ generate a maximal commutative subalgebra of $\\mathcal {H}$ .",
"Let $S^{\\lambda }$ be the irreducible $\\mathcal {H}$ -module corresponding to $\\lambda \\in {P}_{m,n}$ .",
"Then $S^{\\lambda }$ has a basis $\\lbrace v_{\\mathfrak {s}}|\\mathfrak {s}\\in \\mathrm {std}(\\lambda )\\rbrace $ satisfying $J_iv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(i)v_{\\mathfrak {s}}, \\quad i=1, \\ldots , n,$ for each $\\mathfrak {s}\\in \\mathrm {std(\\lambda )}$ .",
"Conversely, if $M$ is an $\\mathcal {H}$ -module containing a common eigenvector $v_{\\mathfrak {s}}$ for $J_1, \\ldots , J_n$ satisfying Eq.",
"(REF ) for some $\\mathfrak {s}\\in \\mathrm {std}(\\lambda )$ , then the $\\mathcal {H}$ -submodule $\\mathcal {H}v_{\\mathfrak {s}}$ of $M$ is a sum of copies of $S^{\\lambda }$ .",
"Lemma 4.5 Let $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then $V^{\\otimes n}$ contains an irreducible $\\mathcal {H}$ -module isomorphic to $S^{\\lambda }$ consisting of highest weight vectors of $U_{q}(\\mathfrak {g})$ with highest weight $\\lambda $ .",
"Thanks to [30], [28], we have the following $U_q(\\mathfrak {gl}(k|\\ell ))\\otimes \\mathcal {H}_n(q)$ -module decomposition $V^{\\otimes n}&=&\\bigoplus _{\\lambda \\in H(k,\\ell ;n)}V_{\\lambda }\\otimes S^{\\lambda },$ where $V_{\\lambda }$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_{q}(\\mathfrak {g}(k|\\ell ))$ -module with highest weight $\\lambda $ (resp.",
"the irreducible module of $\\mathcal {H}_n(q)$ corresponding to $\\lambda $ ).",
"Furthermore, the decomposition Eq.",
"(REF ) implies for each $\\lambda \\in H(k,\\ell ;n)$ , the $\\mathcal {H}_n(q)$ -module $S^{\\lambda }$ consisting of highest weight vectors for $V_{\\lambda }$ , that is, if $\\mathfrak {s}$ is a standard $\\lambda $ -tableau then there is a highest weight vector for $V_{\\lambda }$ .",
"Suppose $\\lambda =(\\lambda ^{(1)}; \\ldots ; \\lambda ^{(m)})\\!\\in \\!",
"H(\\mathbf {k}|\\ell ;m,n)$ and let $n_i=\\left|\\lambda ^{(i)}\\right|$ .",
"Thanks to Lemma REF , we may put $\\lambda ^{(i)}\\!=\\!",
"(\\mu ^{(i)};\\nu ^{(i)})\\!\\in \\!\\Pi (k_i,\\ell _i;n_i)$ for $1\\le i\\le m$ .",
"Now we define a standard $\\lambda $ -tableau $\\mathfrak {s}=(\\mathfrak {s}^{(1)},\\ldots , \\mathfrak {s}^{(m)})$ as follows: Set $n_{i,1}=\\left|\\mu ^{(i)}\\right|$ , $n_{i,2}=\\left|\\nu ^{(i)}\\right|$ , $p_{i,1}=n_{i,1}+\\cdots +n_{m,1}$ , $p_{i,2}=n_{i,2}+\\cdots +n_{m,2}$ for $i=1, \\ldots , m$ , and $p_{m+1,1}=p_{m+1,2}=0$ .",
"Define $\\mathfrak {s}^{(i)}$ by filling $n_{i,1}$ numbering $p_{i+1,1}<p_{i+1,1}+2<\\cdots <p_{i,1}$ in the boxes in $\\mu ^{(i)}$ first to all boxes of the first row, and then to all the boxes of the second row, and so on, in increasing order; then by filling $n_{i,2}$ numbering $p_{i+1,2}<p_{i+1,2}+2<\\cdots <p_{i,2}$ in the boxes in $\\nu ^{(i)}$ first to all boxes of the first column, and then to all the boxes of the second column, and so on, in increasing order.",
"It is clearly that $\\mathfrak {s}^{(i)}$ is a standard $\\lambda ^{(i)}$ -tableau.",
"Now consider the action of $U_q(\\mathfrak {gl}(k_i|\\ell _i)\\otimes \\mathcal {H}_{n_i}(q)$ on the ${V^{(i)}}^{n_i}$ and apply Moon's argument in [30], we can obtain a common eigenvector $w_i\\in {V^{(i)}}^{n_i}$ for the Jucys-Murphy elements of $\\mathcal {H}_{n_i}(q)$ with respect to $\\mathrm {res}_{\\mathfrak {s}^{(i)}}$ , which is also a highest weight vector of the irreducible $U_q(\\mathfrak {gl}(k_i|\\ell _i)$ -module $V_{\\lambda ^{(i)}}$ with highest weight $\\lambda ^{(i)}$ .",
"Set $v_{\\mathfrak {s}}=w_1\\otimes w_{2}\\otimes \\cdots \\otimes w_{m}\\in {V^{(1)}}^{\\otimes n_1}\\otimes {V^{(2)}}^{\\otimes n_{2}}\\otimes \\cdots \\otimes {V^{(m)}}^{\\otimes n_m}.$ Then $v_{\\mathfrak {s}}$ is a highest weight vector of $V_{\\lambda }=V_{\\lambda ^{(1)}}\\otimes \\cdots \\otimes V_{\\lambda ^{(m)}}$ .",
"Assume that $a=p_{k+1}<r\\le p_k=b$ .",
"We show that $v_{\\mathfrak {s}}$ is a common eigenvector for Jucys-Murphy elements of $\\mathcal {H}$ , that is, $J_rv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ Clearly $J_rv_{\\mathfrak {s}}$ can be written as $J_rv_{\\mathfrak {s}}=T^{-1}_{r}\\cdots T^{-1}_{n-1}S_{n-1}\\cdots S_1S_0T_1\\cdots T_{r-1}v_{\\mathfrak {s}}.$ First note that $T_a\\cdots T_rv_{\\mathfrak {s}}\\in {V^{(1)}}^{n_1}\\otimes \\cdots \\otimes {V^{(m)}}^{n_m}.$ Let $\\mathbf {i}$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $ T_a\\cdots T_rv_{\\mathfrak {s}}$ .",
"Then $i_a\\in V_r$ , and $i_c>i_a$ for all $c<a$ .",
"Moreover $i_c\\notin V_r$ for $c<a$ , which implies $T_1\\cdots T_{a-1}(\\mathbf {i})=i_a\\otimes i_1\\otimes \\cdots \\otimes i_{a-1}\\otimes i_{a+1}\\otimes \\cdots \\otimes i_n$ and $S_{a-1}\\cdots S_1S_0T_1\\cdots T_{a-1}(\\mathbf {i})=Q_r\\mathbf {i}$ .",
"Thus $J_rv_{\\mathfrak {s}}=Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}.$ On the other hand, we have $S_{b-1}\\cdots S_{a}T_a\\cdots T_{r-1}v_{\\mathfrak {s}}\\in V_{1}^{\\otimes n_1}\\otimes \\cdots \\otimes V_m^{\\otimes n_m}.$ Let $y=y_1\\otimes \\cdots \\otimes y_n$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $S_{b-1}\\cdots S_{a}T_a\\cdots T_{j-1}v_{\\mathfrak {s}}$ .",
"Then $y_b\\in V_{r}$ , $y_i\\notin V_{r}$ for all $i>a$ .",
"Moreover $y_i<y_{b}$ for all $i>b$ .",
"By a similar argument as above, we obtain $T_{b}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_b(y)=y$ .",
"Therefore $J_rv_{\\mathfrak {s}}&=&Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&Q_rT_{r-1}\\cdots T_{a}T_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ We complete the proof.",
"Now we compute the multiplicity $m_{\\lambda }:=[V^{\\otimes n}:V_{\\lambda }]$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Assume that $k_m,\\ell _m>1$ .",
"We let $\\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)$ be the subalgebra of $\\mathfrak {gl}(k_m|\\ell _m)$ corresponding to the basis $\\mathfrak {B}^{(m)}-\\lbrace v^{(m)}_{k_m}\\rbrace $ and let $\\mathfrak {gl}(1,0)$ to be that corresponding to basis $v^{(m)}_{k_m}$ .",
"Put $\\tilde{\\mathfrak {g}}= \\mathfrak {gl}(k_1|\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)\\oplus \\mathfrak {gl}(1|0)$ , which is a subalgebra of $\\mathfrak {g}$ .",
"For $\\lambda =(\\lambda ^{(1)},\\ldots ,\\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ and $\\mu =(\\mu ^{(1)},\\ldots ,\\mu ^{(m)},\\mu ^{(m+1)})\\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , where $\\tilde{\\mathbf {k}}=(k_1, \\ldots ,k_m-1,1)$ and $\\tilde{\\ell }=(\\ell ,0)$ , we write $\\mu \\prec \\lambda $ if $\\lambda ^{(i)}=\\mu ^{(i)}$ for $i=1, \\ldots , m-1$ and $\\lambda ^{(m)}_1\\ge \\mu _{1}^{(m)}\\ge \\cdots \\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge \\mu _{\\ell (\\mu ^{(m)})-1}^{(m)}\\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})},$ where $\\lambda ^{(m)}=(\\lambda ^{(m)}_{1},\\ldots , \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})})$ and $\\mu ^{(m)}=(\\mu ^{(m)}_{1},\\ldots , \\mu ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge 0)$ .",
"Notice that $|\\lambda |=|\\mu |=n$ and $\\ell (\\mu ^{(m+1)})\\le 1$ .",
"Moreover, $\\mu \\prec \\lambda $ implies that $|\\mu ^{(m)}|\\le |\\lambda ^{(m)}|$ .",
"Thus $\\mu ^{(m+1)}$ is determined uniquely whenever $\\mu ^{\\prime }=(\\mu ^{(1)},\\ldots ,\\mu ^{(m)})$ is determined up to $\\prec $ .",
"The following lemma characterize the restriction of $V_{\\lambda }$ as a $U_q(\\tilde{\\mathfrak {g}})$ -modules.",
"Lemma 4.10 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $\\left.V_{\\lambda }\\right|_{U_q(\\tilde{\\mathfrak {g}})}=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }$ .",
"In particular, for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m,n)$ , we have $m_{\\mu }=\\sum _{\\mu \\prec \\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}m_{\\lambda }.$ Note that the lemma is easily reduced to the case $m=1$ , that is, $\\mathfrak {g}=\\mathfrak {gl}(k|\\ell )$ and $\\tilde{\\mathfrak {g}}=\\mathfrak {gl}(k\\!-\\!1|\\ell )\\oplus \\mathfrak {gl}(1,0)$ .",
"Then $U_q(\\mathfrak {gl}(k\\!-\\!1|\\ell ))$ is a subalgebra of $U_q(\\mathfrak {gl}(k|\\ell ))$ .",
"For $\\lambda \\in H(k,\\ell ;n)$ and $\\mu \\in H(k\\!-\\!1,\\ell ;n)$ with $\\mu \\prec \\lambda $ , [5] implies $\\mathrm {Res}_{U_q(\\mathfrak {gl}(k-1|\\ell ))}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda }=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }.$ Note that $U_q(\\mathfrak {g}^{\\prime })=U_q(\\mathfrak {gl}(k-\\!1|\\ell ))\\oplus U_q(\\mathfrak {gl}(1|0))$ .",
"Thus we yield that $\\mathrm {Res}_{U_q(\\tilde{\\mathfrak {g}})}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda } =\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }\\otimes V_{\\nu },$ where $V_{\\nu }$ is an irreducible $U_q(\\mathfrak {gl}(1,0))$ -module with highest weight $\\nu $ which is determined uniquely from $\\mu $ .",
"Since the highest weight $\\mu =(\\mu ,\\nu )$ of $V_{\\mu }=V_{\\mu }\\otimes V_{\\nu }$ satisfies $|\\mu |=n$ and $\\ell (\\nu )\\le 1$ , the condition $\\mu \\prec \\lambda $ is equivalent to $\\mu \\prec \\lambda =\\lambda $ .",
"It completes the proof.",
"Denote by $\\bar{S}^{\\lambda }$ the Specht module of $\\mathbb {C}W_{m,n}$ corresponding to $\\lambda $ .",
"It is well-known that $\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }=\\dim _{\\mathbb {K}}S^{\\lambda }$ .",
"Now we can compute the multiplicity $m_{\\lambda }$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Proposition 4.11 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $m_{\\lambda }=\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }$ .",
"First note that for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , we have $\\dim S^{\\mu }=\\sum _{\\lambda \\succ \\mu }\\dim S^{\\lambda }.$ Indeed we can prove this by a similar argument as Lemma REF .",
"Now we prove the proposition by induction on $\\mathbf {k}|\\ell $ .",
"First assume that $k_i|\\ell _i=1|0$ or $0|1$ for all $i$ and $\\lambda =(\\lambda ^{(1)}, \\ldots , \\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then each $\\lambda ^{(i)}$ can be identified with a non-negative integers, which implies $\\dim V_{\\lambda }=1$ and $m_{\\lambda }=\\dim S^{\\lambda }$ .",
"Assume that there are some $i$ such that $k_i|\\ell _i\\ne 1|0, 0|1$ .",
"Clearly, we may assume that $i=m$ .",
"Given $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ , we choose $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ such that $\\mu \\prec \\lambda $ with $\\mu ^{(m)}_i$ for $i=1, \\ldots , \\ell (\\lambda ^{(m)})-1=d-1$ and $\\mu ^{(m+1)}=\\lambda ^{(m)}_{d}$ .",
"Then for $\\widetilde{\\lambda }=(\\tilde{\\lambda }^{(1)}, \\ldots , \\tilde{\\lambda }^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ , $\\mu \\prec \\widetilde{\\lambda }$ implies that $\\lambda ^{(m)}\\unlhd \\tilde{\\lambda }^{(m)}$ (see §REF ), and $\\lambda ^{(i)}=\\tilde{\\lambda }^{(i)}$ for $i=1, \\ldots , m-1$ .",
"Now by induction argument, we may assume that $m_{\\mu }=\\dim S^{\\mu }$ for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ .",
"Therefore $m_{\\mu } &=&\\sum _{\\widetilde{\\lambda }\\succ \\mu }\\dim S^{\\widetilde{\\lambda }}=\\sum _{\\lambda \\succ \\mu }m_{\\lambda }=\\sum _{\\mu \\prec \\lambda }\\dim S^{\\lambda },$ which implies $m_{\\lambda }=\\dim S^{\\lambda }$ by applying backward induction on the dominant order $\\unlhd $ of weights that $m_{\\widetilde{\\lambda }}=\\dim S^{\\widetilde{\\lambda }}$ for any $\\widetilde{\\lambda }\\ne \\lambda $ .",
"We prove the proposition.",
"Theorem 4.13 Keeping notations as above, then $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))$ and $\\Phi (\\mathcal {H})$ are mutually the fully centralizer algebras of each other, i.e., $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n}), &\\qquad & \\Phi (\\mathcal {H})=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n}).$ More precisely, there is a $U_q(\\mathfrak {g})\\text{-}\\mathcal {H}$ -bimodule isomorphism $V^{\\otimes n}\\cong \\bigoplus _{\\lambda \\in \\Lambda ^+_{\\mathbf {k}|\\ell ,m}(n)}V(\\lambda ) \\otimes S^{\\lambda },$ where $V(\\lambda )$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_q(\\mathfrak {g})$ (resp.",
"$\\mathcal {H}$ )-module indexed by $\\lambda $ .",
"Let ${A}=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n})$ and ${B}=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n})$ .",
"Then $V^{\\otimes n}$ is decomposed as a $U_q(\\lambda )\\otimes {A}$ -module $V^{\\otimes n}=\\bigoplus _{\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}V_{\\lambda }\\otimes \\hat{S}_{\\lambda },$ where $\\hat{S}_{\\lambda }$ is an irreducible ${A}$ -module corresponding to $\\lambda $ .",
"According to Lemma REF , we have $\\Phi (\\mathcal {H})\\subseteq {A}$ .",
"By Lemma REF and Proposition REF , $\\hat{S}_{\\lambda }$ contains $S^{\\lambda }$ as an $\\mathcal {H}$ -submodule and $\\dim \\hat{S}_{\\lambda }=m_{\\lambda }=\\dim S_{\\lambda }$ .",
"Hence $\\hat{S}_{\\lambda }=S_{\\lambda }$ and the decomposition (REF ) follows.",
"It is then clear that ${A}=\\Phi (\\mathcal {H})$ and ${B}=\\Psi (U_q(\\mathfrak {g}))$ .",
"We end this paper with some remarks.",
"Remark 4.15 Combing Remark REF and Theorem REF , we can obtain a Schur-Weyl duality between the universal enveloping superalgebra $U(\\mathfrak {g})$ of $\\mathfrak {g}$ and the group algebra $\\mathbb {C}W_{m,n}$ of $W_{m,n}$ , which is a generalization of the Schur-Sergeev duality in [40], [5].",
"Based on Shoji's work [42] and Mitsuhashi's work [29], we will give a super Frobenius formula for $\\mathcal {H}$ in [46], which is one of our motivation to construct the Schur-Weyl duality between quantum superalgebras and cyclotomic Hecke algebras.",
"In a forthcoming work, we will introduce the cyclotomic $q$ -Schur superalgebras and give an alternative proof of Theorem REF along the line of [20].",
"Inspired by Brundan and Kujawa's work [7] and Du et al's work [13], it would be very interesting to understand the Mullineux involution for Ariki-Koike algebras [22] via the Schur-Weyl duality established in the paper."
],
[
"The sign $q$ -permutation representation",
"This section devotes to introduce an $\\mathcal {H}$ -action on $V^{\\otimes n}$ and prove that it is a (super) representation of $\\mathcal {H}$ by adapting the ideas of [34], [30], [28].",
"3.1.",
"Let $W_{m,n}$ be the complex reflection group of type $G(m,1,n)$ .",
"According to [41], $W_{m,n}$ has a presentation with generators $s_0, s_1, \\dots , s_{n-1}$ where the defining relations are $s_0^m=1, s_1^2=\\cdots =s_{n-1}^2=1$ and the homogeneous relations $&s_0 s_1s_0 s_1=s_1s_0 s_1s_0,&&\\\\& s_is_j=s_js_i,&&\\text{ if } |i-j|>1,\\\\&s_is_{i+1}s_i=s_{i+1}s_{i}s_{i+1},&& \\text{ for }1\\le i\\le n-2.$ It is well-known that $W_{m,n}\\cong (\\mathbb {Z}/m\\mathbb {Z})^{n}\\rtimes \\mathfrak {S}_{n}$ , where $s_1, \\dots , s_{n-1}$ are generators of the symmetric group $\\mathfrak {S}_{n}$ of degree $n$ corresponding to transpositions $(1\\,2)$ , $\\ldots $ , $(n\\!-\\!1\\,n)$ .",
"For $a=1, \\ldots , n-1$ and $\\mathbf {i}=(i_1, \\ldots , i_a,i_{a+1},\\ldots , i_n)$ , we define the following right action $\\mathbf {i}s_a:=(i_1, \\ldots , i_{a-1}, i_{a+1}, i_a, i_{a+2}, \\ldots , i_n).$ Following Sergeev [40] or Berele-Regev [5], there is a right action $\\phi $ of $\\mathbb {C}\\mathfrak {S}_n$ on $V^{\\otimes n}$ defined on generators by $s_a(\\mathbf {i})&:=&\\left\\lbrace \\begin{array}{ll}\\vspace{3.0pt}(-1)^{\\overline{i}_a}\\mathbf {i},& \\text{if }i_a=i_{a+1};\\\\(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,& \\text{if }i_a\\ne i_{a+1}.\\end{array}\\right.$ 3.3.",
"The Ariki-Koike algebra [1] or the cyclotomic Hecke algebra $\\mathcal {H}$ associated to $W_{m,n}$ [21], is the unital associative $\\mathbb {K}$ -algebra generated by $g_0,g_1,\\dots ,g_{n-1}$ and subject to relations $&(g_0-Q_1)\\dots (g_0-Q_m)=0,&&\\\\&g_0g_1g_0g_1=g_1g_0g_1g_0,&&\\\\&g_i^2=(q-q^{-1})g_i+1, &&\\text{ for }1\\le i<n,\\\\&g_ig_j=g_jg_i, &&\\text{ for }|i-j|>2,\\\\&g_ig_{i+1}g_i=g_{i+1}g_{i}g_{i+1}, &&\\text{ for }1\\le i<n-1.$ Let $w\\in \\mathfrak {S}_n$ and let $s_{i_1}s_{i_2}\\cdots s_{i_k}$ be a reduced expression for $w$ .",
"Then $g_{w}:=g_{i_1}g_{i_2}\\cdots g_{i_k}$ is independent of the choice of reduced expression and $\\lbrace g_{w}|w\\in \\mathfrak {S}_n\\rbrace $ is linear basis of the subalgebra $\\mathcal {H}_n(q)$ of $\\mathcal {H}$ generated by $g_1, \\ldots , g_{n-1}$ , that is, $\\mathcal {H}_n(q)$ is the Iwahori-Hecke algebra associated to $\\mathfrak {S}_n$ .",
"For $a=1, \\ldots , n-1$ , we define the endomorphisms $T_a, S_a\\in \\mathrm {End}_K(V^{\\otimes n})$ as follows: $T_a(\\mathbf {i}):=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}(q-q^{-1})\\mathbf {i}+(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}\\frac{(q-q^{-1})+(-1)^{\\overline{i}_a}(q+q^{-1})}{2} \\mathbf {i},& \\text{if }i_a=i_{a+1}; \\\\(-1)^{\\overline{i}_a\\overline{i}_{a+1}}\\mathbf {i}s_a,&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ $S_a(\\mathbf {i})&:=&\\left\\lbrace \\begin{array}{ll}T_a(\\mathbf {i}), & \\hbox{ if } c_a(\\mathbf {i})= c_{a+1}(\\mathbf {i});\\\\s_a(\\mathbf {i}), & \\hbox{ if } c_a(\\mathbf {i})\\ne c_{a+1}(\\mathbf {i}).\\end{array}\\right.$ The following easy verified facts will be used latter.",
"Lemma 3.6 For all $\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)$ and $1\\le a<n$ , we have $T_a(\\mathbf {i})=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}(q-q^{-1})\\mathbf {i}+s_a(\\mathbf {i}),& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}\\frac{q-q^{-1}}{2}\\mathbf {i}+\\frac{q+q^{-1}}{2}s_a(\\mathbf {i}),& \\text{if }i_a=i_{a+1}; \\\\s_a(\\mathbf {i}),&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ $T_a$ is invertible and $T_a^{-1}(\\mathbf {i}):=\\left\\lbrace \\begin{array}{ll}\\vspace{6.0pt}s_a(\\mathbf {i}),& \\text{if }i_a<i_{a+1};\\\\ \\vspace{6.0pt}-\\frac{q-q^{-1}}{2}\\mathbf {i}+\\frac{q+q^{-1}}{2}s_a(\\mathbf {i}),& \\text{if }i_a=i_{a+1}; \\\\(q-q^{-1})\\mathbf {i}+ s_a(\\mathbf {i}),&\\text{if } i_a>i_{a+1}.\\end{array}\\right.$ (i) follows directly by applying Eq.",
"(REF ).",
"Since $T^2_a=(q-q^{-1})T_a+1$ , $T_a^{-1}=T_a-(q-q^{-1})$ .",
"Thus (ii) follows directly by applying (i).",
"Now let $S_0(\\mathbf {i}):=Q_{c_1(\\mathbf {i})}\\mathbf {i}$ and $\\theta =S_{n-1}\\cdots S_{1}$ .",
"We define $T_0\\in \\mathrm {End}_{\\mathbb {K}}(V^{\\otimes n})$ as following $T_0(\\mathbf {i}):&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}\\theta S_0(\\mathbf {i}).$ Thanks to [30], [28], Eq.",
"(REF ) defines a (super) representation of $\\mathcal {H}_n(q)$ .",
"The remainder of this section devotes to show that Eqs.",
"(REF ) and (REF ) define a (super) representation of $\\mathcal {H}$ .",
"Lemma 3.8 For $j,p\\ge 1$ , we denote by $V_{j,p}$ the subspace of $V^{\\otimes n}$ spanned by basis elements $\\mathbf {i}$ such that $c_p(\\mathbf {i})\\ge j$ .",
"If $\\mathbf {i}\\in V_{j,p}$ then $T_{p}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p}(\\mathbf {i})\\in \\mathbf {i}+V_{j+1,p}$ .",
"We use the backward induction on $p$ to prove the claim.",
"Note that for all $p=1, \\ldots , n-1$ , we have $&&T_{p}^{-1}S_{p}(\\mathbf {i})=\\left\\lbrace \\begin{array}{ll}\\mathbf {i}+(q-q^{-1})s_{p}(\\mathbf {i}), & \\hbox{ if }c_{p}(\\mathbf {i})> c_{p+1}(\\mathbf {i});\\\\\\mathbf {i}, & \\hbox{ others}.\\end{array}\\right.$ In particular, the lemma holds for $p=n-1$ .",
"Now assume that for all $p$ and $\\mathbf {i}^{\\prime }\\in V_{j,p}$ , $T_{p}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p}(\\mathbf {i}^{\\prime })\\in \\mathbf {i}^{\\prime }+V_{j+1,p}.$ Thanks to Lemma REF (i), $T_{p-1}(V_{j,p-1})=V_{j,p}$ for all $j\\ge 1$ , which implies $S_{p-1}(V_{j,p-1})\\in V_{j,p}$ due to Eq.",
"(REF ).",
"For any $\\mathbf {i}\\in V_{j,p-1}$ , the induction argument shows $(T_{p-1}^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_{p-1})(\\mathbf {i})&=& T_{p-1}^{-1}(T_p^{-1}\\cdots T_{n-1}^{-1}S_{n-1}\\cdots S_p)(S_{p-1}(\\mathbf {i}))\\\\ &\\in &T_{p-1}^{-1}(S_{p-1}(\\mathbf {i})+V_{j+1,p})\\\\&=&T_{p-1}^{-1}S_{p-1}(\\mathbf {i})+V_{j+1,p}\\\\&\\in & \\mathbf {i}+V_{j+1,p},$ where the last inclusion follows by Eq.",
"(REF ).",
"The lemma is proved.",
"We will need the following facts.",
"Lemma 3.10 For all $j\\ge 2$ , we have the following facts: $S_jS_{j-1}T_j=T_{j-1}S_jS_{j-1},$ $S_jS_{j-1}S_j S_{j-1}^{-1}T_{j-1}=T_jS_jS_{j-1}S_jS_{j-1}^{-1}, \\\\$ $S_jS_{j-1}S_j S_{j-1}T_{j-1}=T_jS_jS_{j-1}S_jS_{j-1}.$ Let $q_+=q+q^{-1}$ and $q_{*}=q-q^{-1}$ .",
"Without loss of generality, we may assume that $j=2$ and $\\mathbf {i}=(i_1, i_2, i_3)$ .",
"Therefore we have the following five cases: If $c_1(\\mathbf {i})=c_2(\\mathbf {i})=c_{3}(\\mathbf {i})$ then $S_1=T_1$ , $S_2=T_2$ .",
"Furthermore Eq.",
"(REF ) follows owing to [30].",
"If $c_1(\\mathbf {i})$ , $c_2(\\mathbf {i})$ and $c_3(\\mathbf {i})$ are pairwise different then $S_1(\\mathbf {i})= s_1(\\mathbf {i})$ , $S_2(\\mathbf {i})=s_2(\\mathbf {i})$ .",
"Thus we only need to consider the following cases: (a) $i_1<i_2<i_3$ ; (b) $i_1<i_2>i_3$ ; (c) $i_1>i_2>i_3$ .",
"Apply Lemma REF (i) and Eq.",
"(REF ), we obtain that $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=(q-q^{-1})s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=(q-q^{-1})s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=(q-q^{-1})s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ $\\begin{array}{lll}&&S_2S_1T_2(\\mathbf {i})=s_2s_1s_2(\\mathbf {i})=s_1s_2s_1(\\mathbf {i})=T_1S_2S_1(\\mathbf {i});\\\\&&S_2S_1S_2 S_1^{-1}T_1(\\mathbf {i})=s_2s_1s_2(\\mathbf {i})=T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});\\\\&&S_2S_1S_2 S_1T_1(\\mathbf {i})=s_1s_2T_1(\\mathbf {i})=T_2s_1s_2(\\mathbf {i})=T_2S_2S_1S_2S_1(\\mathbf {i});\\end{array}$ Therefore, in this case Eq.",
"(REF ) hold.",
"If $c_1(\\mathbf {i})=c_2(\\mathbf {i})\\ne c_3(\\mathbf {i})$ then we only need to check the following six cases: $i_1=i_2<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*S_2S_1(\\mathbf {i})+S_2S_1s_2(\\mathbf {i})\\\\&=&q_*s_2T_1(\\mathbf {i})+T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+q_*s_2s_1(\\mathbf {i})\\!+\\!\\frac{1}{2}q_+s_1s_2s_1(\\mathbf {i})\\!-\\!\\frac{1}{2}q_*^2s_2(\\mathbf {i})\\!-\\!\\frac{1}{2}q_*s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+T_1s_2s_1(\\mathbf {i})-\\frac{1}{2}q_*T_1s_2(\\mathbf {i})\\\\&=&T_1S_2T_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i}); $ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&S_2S_1S_2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+s_2s_1s_2(\\mathbf {i})-\\frac{1}{2}q_*s_1s_2(\\mathbf {i}),$ $T_2S_2S_1S_2S_1^{-\\!1}(\\mathbf {i})&=&T_2S_2S_1S_2T_1^{-1}(\\mathbf {i})\\\\&=&\\frac{1}{2}q_+T_2^2s_1s_2s_1(\\mathbf {i})-\\frac{1}{2}q_*T_2^2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})\\!+\\!\\frac{q_*^2}{2}T_2s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+q_*}{2}T_2s_1s_2s_1(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}); $ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&S_2S_1S_2(\\mathbf {i})+q_*S_2S_1S_2T_1(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_1=i_2>i_3$ : $S_2S_1T_2(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}),$ $T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})&=&T_2^2s_1s_2T_1^{-1}(\\mathbf {i})\\\\&=&q_*T_2s_1s_2T_1^{-1}(\\mathbf {i})+s_1s_2T_1^{-1}(\\mathbf {i})\\\\&=&\\frac{q_*^2}{2}T_2s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+q_*}{2}T_2s_1s_2s_1(\\mathbf {i})\\!+\\!\\frac{q_*}{2}s_1s_2(\\mathbf {i})\\!+\\!\\frac{q_+}{2}s_1s_2s_1(\\mathbf {i})\\\\&=&\\frac{q_+}{2}s_2s_1s_2(\\mathbf {i})-\\frac{q_*}{2}s_1s_2(\\mathbf {i}); $ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_1<i_2<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_2T_1(\\mathbf {i})+T_2s_1s_2(\\mathbf {i})\\\\&=&q_*^2s_2(\\mathbf {i})+q_*(s_1s_2+s_2s_1)(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $T_2S_2S_1S_2S_1^{-1}(\\mathbf {i})&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})\\\\&=&S_2S_1S_2(\\mathbf {i})\\\\&=& S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_3<i_1<i_2$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i}).$ $i_2<i_1<i_3$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2+s_2s_1s_2\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i});$ $i_3<i_2<i_1$ : $S_2S_1T_2(\\mathbf {i})&=&q_*s_1s_2+s_2s_1s_2\\\\&=&q_*T_1s_2(\\mathbf {i})+T_1s_2s_1(\\mathbf {i})\\\\&=&T_1S_2S_1(\\mathbf {i});$ $S_2S_1S_2S_1^{-1}T_1(\\mathbf {i})&=&T_2s_1s_2(\\mathbf {i})\\\\&=&q_*s_1s_2(\\mathbf {i})+s_2s_1s_2(\\mathbf {i})\\\\&=&q_*T_2s_1s_2s_1(\\mathbf {i})+s_1s_2s_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2s_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1^{-1}(\\mathbf {i});$ $S_2S_1S_2S_1T_1(\\mathbf {i})&=&S_2S_1S_2T_1^2(\\mathbf {i})\\\\&=&T_2s_1s_2(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&s_1s_2T_1(\\mathbf {i})+q_*T_2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2^2s_1s_2T_1(\\mathbf {i})\\\\&=&T_2S_2S_1S_2S_1(\\mathbf {i}).$ Therefore Eq.",
"(REF ) hold in this case.",
"The remainder cases $c_1(\\mathbf {i})\\ne c_2(\\mathbf {i})=c_3(\\mathbf {i})$ and $c_1(\\mathbf {i})=c_3(\\mathbf {i})\\ne c_2(\\mathbf {i})$ can be verified by a similar way.",
"As a consequence, we prove the lemma.",
"Now we can prove the main result of this section.",
"Theorem 3.12 Keeping notation as above, then the $\\mathbb {K}$ -linear map $\\Phi : \\mathcal {H}\\rightarrow \\mathrm {End}_{\\mathbb {K}}(V^{\\otimes n})$ defined by $g_a\\mapsto T_a$ ($a=0, \\ldots , n-1$ ) is a (super) representation of $\\mathcal {H}$ .",
"Thanks to [30] or [28], it suffices to show that the following three relations hold: $ &&(T_0-Q_1)\\cdots (T_0-Q_m)=0; \\\\ &&T_0T_1T_0T_1=T_1T_0T_1T_0;\\\\ &&T_0T_i=T_iT_0, \\hbox{ for }i\\ge 2.$ Applying Eq.",
"(REF ) and Lemma REF , $T_0(\\mathbf {i})=Q_{c_1(\\mathbf {i})}\\mathbf {i}+V_{j+1,1}$ for any $\\mathbf {i}\\in V_{j,1}$ .",
"It follows that $(T_0-Q_j)(\\mathbf {i})&=&(Q_{c_1(\\mathbf {i})}-Q_j)\\mathbf {i}+V_{j+1,1}.$ Since $\\mathbf {i}\\in V_{j,1}$ , we have $c_1(\\mathbf {i})\\ge j$ .",
"Therefore $(Q_{c_1(\\mathbf {i})}-Q_j)(\\mathbf {i})=0$ if $c_1(\\mathbf {i})=j$ , and $(Q_{c_1(\\mathbf {i})}-Q_j)(\\mathbf {i})\\in V_{j+1,1}$ if $c_1(\\mathbf {i})>j$ , that is, $(T_0-Q_j)(\\mathbf {i})\\in V_{j+1,1}$ .",
"Finally notice that $V_{1,1}=V^{\\otimes n}$ and $V_{n+1,1}=\\lbrace 0\\rbrace $ .",
"As a consequence, Eq.",
"(REF ) holds.",
"Now note that $S_0$ commutes with $T_2, \\cdots , T_{n-1}$ and $S_iT_j=T_jS_i$ for $|i-j|>2$ .",
"Lemma REF implies $\\theta T_j=T_{j-1}\\theta ,\\qquad j=2, \\ldots , n-1.$ Since $S_iS_j=S_jS_i$ for $|i-j|\\ge 2$ , we have $\\theta ^2T_1 &=& (S_{n-1}\\cdots S_{1})(S_{n-1}\\cdots S_{1})T_1\\\\&=&S_{n-1}(S_{n-2}S_{n-1})\\cdots (S_2S_3)(S_1S_2)S_1T_1\\\\&=&(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})(S_2S_1S_2S_1)T_1\\\\&=&(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})T_2(S_2S_1S_2S_1)\\\\&=&T_{n-1}(S_{n-1}S_{n-2}S_{n-1}S_{n-2}^{-1})(S_{n-2}S_{n-3}S_{n-2}S_{n-3}^{-1})\\cdots (S_3S_2S_3S_{2}^{-1})(S_2S_1S_2S_1)\\\\&=&T_{n-1}\\theta ^2.$ Now we show $S_1^{-1}S_0S_1S_0T_1=T_1S_1^{-1}S_0S_1 S_0$ .",
"To do this, we may assume $\\mathbf {i}=(i_1,i_2)$ .",
"According to Eq.",
"(REF ), $S_1^{-1}S_0S_1S_0T_1(\\mathbf {i}) &=& \\left\\lbrace \\begin{array}{ll}Q_{c_1(\\mathbf {i})}^2T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})=c_2(\\mathbf {i}); \\\\Q_{c_1(\\mathbf {i})}Q_{c_2(\\mathbf {i})}T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})\\ne c_2(\\mathbf {i}).",
"\\end{array} \\right.\\\\T_1S_1^{-1}S_0S_1S_0(\\mathbf {i}) &=& \\left\\lbrace \\begin{array}{ll}Q_{c_1(\\mathbf {i})}^2T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})=c_2(\\mathbf {i}); \\\\Q_{c_1(\\mathbf {i})}Q_{c_2(\\mathbf {i})}T_1(\\mathbf {i}), & \\hbox{ if }c_1(\\mathbf {i})\\ne c_2(\\mathbf {i}).\\end{array}\\right.$ Combing the above two equalities, we get $(\\theta S_0)^2T_1=T_{n-1}(\\theta S_0)^2$ .",
"As a consequence, we yield that $T_0T_1T_0T_1&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0T_1\\\\&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2T_1\\\\&=&(T_{1}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})T_{n-1}(\\theta S_0)^2;\\\\T_1T_0T_1T_0&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0(T_{2}^{-1}\\cdots T_{n-1}^{-1})\\theta S_0\\\\&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2\\\\&=&(T_{2}^{-1}\\cdots T_{n-1}^{-1})(T_{1}^{-1}\\cdots T_{n-2}^{-1})(\\theta S_0)^2.$ Thanks to [2]), we yield that $T_0T_1T_0T_1=T_1T_0T_1T_0$ , i.e., Eq.",
"() holds.",
"Finally, thanks to Eq.",
"(REF ), for all $j\\ge 2$ , we have $T_0T_j &=& T_{1}^{-1}\\cdots T_{n-1}^{-1}\\theta S_0T_j\\\\&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{j-2}^{-1}(T_{j-1}^{-1}T_j^{-1}T_{j-1})T_{j+1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_{1}^{-1}\\cdots T_{j-2}^{-1}(T_{j}T_{j-1}^{-1}T_{j}^{-1})T_{j+1}^{-1}\\cdots T_{n-1}^{-1}T_j\\theta S_0\\\\&=&T_jT_0.$ It completes the proof.",
"Remark 3.15 If $\\ell =0$ then the representation $(\\Phi , V^{\\otimes n})$ reduces to representation defined by Sakamoto and Shoji [34].",
"If $m=1$ then the representation $(\\Phi , V^{\\otimes n})$ reduces to sign $q$ -permutation representation of $\\mathcal {H}_n(q)$ defined by Moon [30] and Mitsuhashi [28].",
"Remark 3.16 It is known that cyclotomic Hecke algebras are cyclotomic quotients of affine Hecke algebras, it would be interesting to define an affine Hecke action on superspaces such that it is stable under the quotient.",
"Moreover, this action would give another way to construct the $\\mathcal {H}$ -action on $V^{\\otimes n}$ .",
"Remark 3.17 Let $q=1$ and $Q_i=\\xi ^i$ , where $\\xi $ is a fixed primitive $m$ -th root of unity.",
"Then $\\mathcal {H}$ reduces to the group algebra $\\mathbb {C}W_{m,n}$ of $W_{m,n}$ and the (super) representation $(\\Phi , V^{\\otimes n})$ of $\\mathcal {H}$ reduces to a (super) representation of $\\mathbb {C}W_{m,n}$ .",
"Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and $\\mathcal {H}$ In this section, we establish the Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and $\\mathcal {H}$ , which can be viewed as a super analogue of the Schur-Weyl reciprocity for cyclotomic Hecke algebras given independently by Sakamoto-Shoji [34] and Hu [20], or as a cyclotomic version of the Schur-Weyl reciprocity between the quantum superalgebra and the Iwahori-Hecke algebra obtained by Moon [30] and Mitsuhashi [28].",
"Lemma 4.1 For any $X\\in \\mathcal {H}$ and $Y\\in U_q(\\mathfrak {g})$ , $\\Phi (X)\\Psi ^{\\otimes n}(Y)=\\Psi ^{\\otimes n}(Y)\\Phi (X)$ .",
"Thanks to [30] or [28], it is enough to show that the $T_0$ -action defined by Eq.",
"(REF ) commutes with the generators of $U_q(\\mathfrak {g})$ , i.e., commutes with those elements listed in the set ${G}$ defined in Eq.",
"(REF ).",
"Clearly $S_0$ commutes with $U_q(\\mathfrak {g})$ .",
"It reduces to show $S_a$ commutes with elements of ${G}$ for all $a\\ge 1$ .",
"It is easy to check that $S_a$ commutes with $K_j^{\\pm 1}$ for all $1\\le j\\le d_m$ .",
"Given $\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)$ , we show that $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)S_a(\\mathbf {i})$ for all $E_b\\in {G}$ .",
"Note that if $c_{a}(\\mathbf {i})=c_{a+1}(\\mathbf {i})$ then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))=c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ .",
"Thus $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=T_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)T_a(\\mathbf {i})$ .",
"If $c_{a}(\\mathbf {i})\\ne c_{a+1}(\\mathbf {i})$ ) then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))\\ne c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ ).",
"In this case, we need to show $s_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)s_a(\\mathbf {i})$ .",
"For $p=0,1,\\ldots , n-1$ , we let $&X_p=\\widetilde{K}_b^{\\otimes p}\\otimes \\Psi (E_b)\\otimes \\mathrm {Id}^{n-1-p}.$ Then $s_a$ commutes with $X_p$ unless $p=a-1,a$ , which implies we only need to show $&&s_a(X_{a-1}+X_{a})(\\mathbf {i})=(X_{a-1}+X_{a})s_a(\\mathbf {i}).$ Since $s_a$ affects only the $a$ and $a\\!+\\!1$ -th factors of $\\mathbf {i}$ and the remaining parts are the same for $X_{a}(\\mathbf {i})$ and $X_{a+1}(\\mathbf {i})$ .",
"We may only consider the $a$ and $a\\!+\\!1$ -th factors, that is, it is enough to verify that $s_a(\\Psi (E_b)\\!\\otimes \\!\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!\\Psi (E_b))(i_{a}\\otimes i_{a\\!+\\!1})\\!=\\!",
"(\\Psi (E_b)\\!\\otimes \\!",
"\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!",
"\\Psi (E_b))s_a(i_{a}\\!\\otimes \\!",
"i_{a\\!+\\!1}).$ Assume that $d_{r-1}<b<d_r$ for some $1\\le r\\le m$ with $d_0=0$ .",
"Thus if $c_{a}(\\mathbf {i})$ , $r$ , $c_{a+1}(\\mathbf {i})$ are pairwise different, then $\\Psi (E_b)=0$ , therefore both sides of Eq.",
"(REF ) equal zero; If $c_{a}(\\mathbf {i})=r\\ne c_{a+1}(\\mathbf {i})$ then $\\Psi (E_b)(i_{a+1})=0$ and $\\widetilde{K}_a(i_{a+1})=(-1)^{\\bar{i}_{a+1}-\\bar{i}_a}i_{a+1}$ .",
"Then, thanks to Eq.",
"(REF ), both sides of Eq.",
"(REF ) equal; The case $c_{a+1}(\\mathbf {i})=r\\ne c_a(\\mathbf {i})$ can be proved similarly.",
"So $S_a$ commutes with $\\Psi ^{\\otimes n}(E_b)$ for all $E_b\\in {G}$ .",
"In a similar argument, we can show $S_a$ commutes with $\\Psi ^{\\otimes n}(F_b)$ for all $F_b\\in {G}$ .",
"For positive integers $a$ , $b$ , $c$ , we let $\\Pi (a,b;c)=\\left\\lbrace (\\mu ;\\nu )\\left|\\begin{array}{l}\\mu \\vdash s,\\quad \\nu \\vdash t,\\quad s+t=c\\\\ \\ell (\\mu )\\le a,\\ell (\\nu )\\le b, \\mu _a\\ge \\ell (\\nu )\\end{array}\\right.\\right\\rbrace .$ Lemma 4.3 ([4] or [30]) Keeping notations as above, then there is a bijection between $H(a,b;c)$ and $\\Pi (a,b;c)$ given by $\\lambda \\mapsto (\\lambda ^1;\\lambda ^2)$ , where $\\lambda ^1=(\\lambda _1, \\ldots , \\lambda _a)$ and $\\lambda ^2=(\\lambda _1^2,\\ldots , \\lambda _b^2)$ with $\\lambda _i^2=\\max \\lbrace \\lambda _i^*-a,0\\rbrace $ .",
"Recall that the Jucys-Murphy elements of $\\mathcal {H}$ are defined recursively by $J_1\\!",
":=g_0\\quad \\text{ and }\\quad J_{i+1}\\!",
":=g_iJ_ig_i, \\quad i=1, \\cdots , n-1.$ It is known that $J_1$ , $\\ldots , J_n$ generate a maximal commutative subalgebra of $\\mathcal {H}$ .",
"Let $S^{\\lambda }$ be the irreducible $\\mathcal {H}$ -module corresponding to $\\lambda \\in {P}_{m,n}$ .",
"Then $S^{\\lambda }$ has a basis $\\lbrace v_{\\mathfrak {s}}|\\mathfrak {s}\\in \\mathrm {std}(\\lambda )\\rbrace $ satisfying $J_iv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(i)v_{\\mathfrak {s}}, \\quad i=1, \\ldots , n,$ for each $\\mathfrak {s}\\in \\mathrm {std(\\lambda )}$ .",
"Conversely, if $M$ is an $\\mathcal {H}$ -module containing a common eigenvector $v_{\\mathfrak {s}}$ for $J_1, \\ldots , J_n$ satisfying Eq.",
"(REF ) for some $\\mathfrak {s}\\in \\mathrm {std}(\\lambda )$ , then the $\\mathcal {H}$ -submodule $\\mathcal {H}v_{\\mathfrak {s}}$ of $M$ is a sum of copies of $S^{\\lambda }$ .",
"Lemma 4.5 Let $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then $V^{\\otimes n}$ contains an irreducible $\\mathcal {H}$ -module isomorphic to $S^{\\lambda }$ consisting of highest weight vectors of $U_{q}(\\mathfrak {g})$ with highest weight $\\lambda $ .",
"Thanks to [30], [28], we have the following $U_q(\\mathfrak {gl}(k|\\ell ))\\otimes \\mathcal {H}_n(q)$ -module decomposition $V^{\\otimes n}&=&\\bigoplus _{\\lambda \\in H(k,\\ell ;n)}V_{\\lambda }\\otimes S^{\\lambda },$ where $V_{\\lambda }$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_{q}(\\mathfrak {g}(k|\\ell ))$ -module with highest weight $\\lambda $ (resp.",
"the irreducible module of $\\mathcal {H}_n(q)$ corresponding to $\\lambda $ ).",
"Furthermore, the decomposition Eq.",
"(REF ) implies for each $\\lambda \\in H(k,\\ell ;n)$ , the $\\mathcal {H}_n(q)$ -module $S^{\\lambda }$ consisting of highest weight vectors for $V_{\\lambda }$ , that is, if $\\mathfrak {s}$ is a standard $\\lambda $ -tableau then there is a highest weight vector for $V_{\\lambda }$ .",
"Suppose $\\lambda =(\\lambda ^{(1)}; \\ldots ; \\lambda ^{(m)})\\!\\in \\!",
"H(\\mathbf {k}|\\ell ;m,n)$ and let $n_i=\\left|\\lambda ^{(i)}\\right|$ .",
"Thanks to Lemma REF , we may put $\\lambda ^{(i)}\\!=\\!",
"(\\mu ^{(i)};\\nu ^{(i)})\\!\\in \\!\\Pi (k_i,\\ell _i;n_i)$ for $1\\le i\\le m$ .",
"Now we define a standard $\\lambda $ -tableau $\\mathfrak {s}=(\\mathfrak {s}^{(1)},\\ldots , \\mathfrak {s}^{(m)})$ as follows: Set $n_{i,1}=\\left|\\mu ^{(i)}\\right|$ , $n_{i,2}=\\left|\\nu ^{(i)}\\right|$ , $p_{i,1}=n_{i,1}+\\cdots +n_{m,1}$ , $p_{i,2}=n_{i,2}+\\cdots +n_{m,2}$ for $i=1, \\ldots , m$ , and $p_{m+1,1}=p_{m+1,2}=0$ .",
"Define $\\mathfrak {s}^{(i)}$ by filling $n_{i,1}$ numbering $p_{i+1,1}<p_{i+1,1}+2<\\cdots <p_{i,1}$ in the boxes in $\\mu ^{(i)}$ first to all boxes of the first row, and then to all the boxes of the second row, and so on, in increasing order; then by filling $n_{i,2}$ numbering $p_{i+1,2}<p_{i+1,2}+2<\\cdots <p_{i,2}$ in the boxes in $\\nu ^{(i)}$ first to all boxes of the first column, and then to all the boxes of the second column, and so on, in increasing order.",
"It is clearly that $\\mathfrak {s}^{(i)}$ is a standard $\\lambda ^{(i)}$ -tableau.",
"Now consider the action of $U_q(\\mathfrak {gl}(k_i|\\ell _i)\\otimes \\mathcal {H}_{n_i}(q)$ on the ${V^{(i)}}^{n_i}$ and apply Moon's argument in [30], we can obtain a common eigenvector $w_i\\in {V^{(i)}}^{n_i}$ for the Jucys-Murphy elements of $\\mathcal {H}_{n_i}(q)$ with respect to $\\mathrm {res}_{\\mathfrak {s}^{(i)}}$ , which is also a highest weight vector of the irreducible $U_q(\\mathfrak {gl}(k_i|\\ell _i)$ -module $V_{\\lambda ^{(i)}}$ with highest weight $\\lambda ^{(i)}$ .",
"Set $v_{\\mathfrak {s}}=w_1\\otimes w_{2}\\otimes \\cdots \\otimes w_{m}\\in {V^{(1)}}^{\\otimes n_1}\\otimes {V^{(2)}}^{\\otimes n_{2}}\\otimes \\cdots \\otimes {V^{(m)}}^{\\otimes n_m}.$ Then $v_{\\mathfrak {s}}$ is a highest weight vector of $V_{\\lambda }=V_{\\lambda ^{(1)}}\\otimes \\cdots \\otimes V_{\\lambda ^{(m)}}$ .",
"Assume that $a=p_{k+1}<r\\le p_k=b$ .",
"We show that $v_{\\mathfrak {s}}$ is a common eigenvector for Jucys-Murphy elements of $\\mathcal {H}$ , that is, $J_rv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ Clearly $J_rv_{\\mathfrak {s}}$ can be written as $J_rv_{\\mathfrak {s}}=T^{-1}_{r}\\cdots T^{-1}_{n-1}S_{n-1}\\cdots S_1S_0T_1\\cdots T_{r-1}v_{\\mathfrak {s}}.$ First note that $T_a\\cdots T_rv_{\\mathfrak {s}}\\in {V^{(1)}}^{n_1}\\otimes \\cdots \\otimes {V^{(m)}}^{n_m}.$ Let $\\mathbf {i}$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $ T_a\\cdots T_rv_{\\mathfrak {s}}$ .",
"Then $i_a\\in V_r$ , and $i_c>i_a$ for all $c<a$ .",
"Moreover $i_c\\notin V_r$ for $c<a$ , which implies $T_1\\cdots T_{a-1}(\\mathbf {i})=i_a\\otimes i_1\\otimes \\cdots \\otimes i_{a-1}\\otimes i_{a+1}\\otimes \\cdots \\otimes i_n$ and $S_{a-1}\\cdots S_1S_0T_1\\cdots T_{a-1}(\\mathbf {i})=Q_r\\mathbf {i}$ .",
"Thus $J_rv_{\\mathfrak {s}}=Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}.$ On the other hand, we have $S_{b-1}\\cdots S_{a}T_a\\cdots T_{r-1}v_{\\mathfrak {s}}\\in V_{1}^{\\otimes n_1}\\otimes \\cdots \\otimes V_m^{\\otimes n_m}.$ Let $y=y_1\\otimes \\cdots \\otimes y_n$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $S_{b-1}\\cdots S_{a}T_a\\cdots T_{j-1}v_{\\mathfrak {s}}$ .",
"Then $y_b\\in V_{r}$ , $y_i\\notin V_{r}$ for all $i>a$ .",
"Moreover $y_i<y_{b}$ for all $i>b$ .",
"By a similar argument as above, we obtain $T_{b}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_b(y)=y$ .",
"Therefore $J_rv_{\\mathfrak {s}}&=&Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&Q_rT_{r-1}\\cdots T_{a}T_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ We complete the proof.",
"Now we compute the multiplicity $m_{\\lambda }:=[V^{\\otimes n}:V_{\\lambda }]$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Assume that $k_m,\\ell _m>1$ .",
"We let $\\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)$ be the subalgebra of $\\mathfrak {gl}(k_m|\\ell _m)$ corresponding to the basis $\\mathfrak {B}^{(m)}-\\lbrace v^{(m)}_{k_m}\\rbrace $ and let $\\mathfrak {gl}(1,0)$ to be that corresponding to basis $v^{(m)}_{k_m}$ .",
"Put $\\tilde{\\mathfrak {g}}= \\mathfrak {gl}(k_1|\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)\\oplus \\mathfrak {gl}(1|0)$ , which is a subalgebra of $\\mathfrak {g}$ .",
"For $\\lambda =(\\lambda ^{(1)},\\ldots ,\\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ and $\\mu =(\\mu ^{(1)},\\ldots ,\\mu ^{(m)},\\mu ^{(m+1)})\\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , where $\\tilde{\\mathbf {k}}=(k_1, \\ldots ,k_m-1,1)$ and $\\tilde{\\ell }=(\\ell ,0)$ , we write $\\mu \\prec \\lambda $ if $\\lambda ^{(i)}=\\mu ^{(i)}$ for $i=1, \\ldots , m-1$ and $\\lambda ^{(m)}_1\\ge \\mu _{1}^{(m)}\\ge \\cdots \\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge \\mu _{\\ell (\\mu ^{(m)})-1}^{(m)}\\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})},$ where $\\lambda ^{(m)}=(\\lambda ^{(m)}_{1},\\ldots , \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})})$ and $\\mu ^{(m)}=(\\mu ^{(m)}_{1},\\ldots , \\mu ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge 0)$ .",
"Notice that $|\\lambda |=|\\mu |=n$ and $\\ell (\\mu ^{(m+1)})\\le 1$ .",
"Moreover, $\\mu \\prec \\lambda $ implies that $|\\mu ^{(m)}|\\le |\\lambda ^{(m)}|$ .",
"Thus $\\mu ^{(m+1)}$ is determined uniquely whenever $\\mu ^{\\prime }=(\\mu ^{(1)},\\ldots ,\\mu ^{(m)})$ is determined up to $\\prec $ .",
"The following lemma characterize the restriction of $V_{\\lambda }$ as a $U_q(\\tilde{\\mathfrak {g}})$ -modules.",
"Lemma 4.10 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $\\left.V_{\\lambda }\\right|_{U_q(\\tilde{\\mathfrak {g}})}=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }$ .",
"In particular, for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m,n)$ , we have $m_{\\mu }=\\sum _{\\mu \\prec \\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}m_{\\lambda }.$ Note that the lemma is easily reduced to the case $m=1$ , that is, $\\mathfrak {g}=\\mathfrak {gl}(k|\\ell )$ and $\\tilde{\\mathfrak {g}}=\\mathfrak {gl}(k\\!-\\!1|\\ell )\\oplus \\mathfrak {gl}(1,0)$ .",
"Then $U_q(\\mathfrak {gl}(k\\!-\\!1|\\ell ))$ is a subalgebra of $U_q(\\mathfrak {gl}(k|\\ell ))$ .",
"For $\\lambda \\in H(k,\\ell ;n)$ and $\\mu \\in H(k\\!-\\!1,\\ell ;n)$ with $\\mu \\prec \\lambda $ , [5] implies $\\mathrm {Res}_{U_q(\\mathfrak {gl}(k-1|\\ell ))}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda }=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }.$ Note that $U_q(\\mathfrak {g}^{\\prime })=U_q(\\mathfrak {gl}(k-\\!1|\\ell ))\\oplus U_q(\\mathfrak {gl}(1|0))$ .",
"Thus we yield that $\\mathrm {Res}_{U_q(\\tilde{\\mathfrak {g}})}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda } =\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }\\otimes V_{\\nu },$ where $V_{\\nu }$ is an irreducible $U_q(\\mathfrak {gl}(1,0))$ -module with highest weight $\\nu $ which is determined uniquely from $\\mu $ .",
"Since the highest weight $\\mu =(\\mu ,\\nu )$ of $V_{\\mu }=V_{\\mu }\\otimes V_{\\nu }$ satisfies $|\\mu |=n$ and $\\ell (\\nu )\\le 1$ , the condition $\\mu \\prec \\lambda $ is equivalent to $\\mu \\prec \\lambda =\\lambda $ .",
"It completes the proof.",
"Denote by $\\bar{S}^{\\lambda }$ the Specht module of $\\mathbb {C}W_{m,n}$ corresponding to $\\lambda $ .",
"It is well-known that $\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }=\\dim _{\\mathbb {K}}S^{\\lambda }$ .",
"Now we can compute the multiplicity $m_{\\lambda }$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Proposition 4.11 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $m_{\\lambda }=\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }$ .",
"First note that for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , we have $\\dim S^{\\mu }=\\sum _{\\lambda \\succ \\mu }\\dim S^{\\lambda }.$ Indeed we can prove this by a similar argument as Lemma REF .",
"Now we prove the proposition by induction on $\\mathbf {k}|\\ell $ .",
"First assume that $k_i|\\ell _i=1|0$ or $0|1$ for all $i$ and $\\lambda =(\\lambda ^{(1)}, \\ldots , \\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then each $\\lambda ^{(i)}$ can be identified with a non-negative integers, which implies $\\dim V_{\\lambda }=1$ and $m_{\\lambda }=\\dim S^{\\lambda }$ .",
"Assume that there are some $i$ such that $k_i|\\ell _i\\ne 1|0, 0|1$ .",
"Clearly, we may assume that $i=m$ .",
"Given $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ , we choose $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ such that $\\mu \\prec \\lambda $ with $\\mu ^{(m)}_i$ for $i=1, \\ldots , \\ell (\\lambda ^{(m)})-1=d-1$ and $\\mu ^{(m+1)}=\\lambda ^{(m)}_{d}$ .",
"Then for $\\widetilde{\\lambda }=(\\tilde{\\lambda }^{(1)}, \\ldots , \\tilde{\\lambda }^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ , $\\mu \\prec \\widetilde{\\lambda }$ implies that $\\lambda ^{(m)}\\unlhd \\tilde{\\lambda }^{(m)}$ (see §REF ), and $\\lambda ^{(i)}=\\tilde{\\lambda }^{(i)}$ for $i=1, \\ldots , m-1$ .",
"Now by induction argument, we may assume that $m_{\\mu }=\\dim S^{\\mu }$ for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ .",
"Therefore $m_{\\mu } &=&\\sum _{\\widetilde{\\lambda }\\succ \\mu }\\dim S^{\\widetilde{\\lambda }}=\\sum _{\\lambda \\succ \\mu }m_{\\lambda }=\\sum _{\\mu \\prec \\lambda }\\dim S^{\\lambda },$ which implies $m_{\\lambda }=\\dim S^{\\lambda }$ by applying backward induction on the dominant order $\\unlhd $ of weights that $m_{\\widetilde{\\lambda }}=\\dim S^{\\widetilde{\\lambda }}$ for any $\\widetilde{\\lambda }\\ne \\lambda $ .",
"We prove the proposition.",
"Theorem 4.13 Keeping notations as above, then $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))$ and $\\Phi (\\mathcal {H})$ are mutually the fully centralizer algebras of each other, i.e., $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n}), &\\qquad & \\Phi (\\mathcal {H})=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n}).$ More precisely, there is a $U_q(\\mathfrak {g})\\text{-}\\mathcal {H}$ -bimodule isomorphism $V^{\\otimes n}\\cong \\bigoplus _{\\lambda \\in \\Lambda ^+_{\\mathbf {k}|\\ell ,m}(n)}V(\\lambda ) \\otimes S^{\\lambda },$ where $V(\\lambda )$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_q(\\mathfrak {g})$ (resp.",
"$\\mathcal {H}$ )-module indexed by $\\lambda $ .",
"Let ${A}=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n})$ and ${B}=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n})$ .",
"Then $V^{\\otimes n}$ is decomposed as a $U_q(\\lambda )\\otimes {A}$ -module $V^{\\otimes n}=\\bigoplus _{\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}V_{\\lambda }\\otimes \\hat{S}_{\\lambda },$ where $\\hat{S}_{\\lambda }$ is an irreducible ${A}$ -module corresponding to $\\lambda $ .",
"According to Lemma REF , we have $\\Phi (\\mathcal {H})\\subseteq {A}$ .",
"By Lemma REF and Proposition REF , $\\hat{S}_{\\lambda }$ contains $S^{\\lambda }$ as an $\\mathcal {H}$ -submodule and $\\dim \\hat{S}_{\\lambda }=m_{\\lambda }=\\dim S_{\\lambda }$ .",
"Hence $\\hat{S}_{\\lambda }=S_{\\lambda }$ and the decomposition (REF ) follows.",
"It is then clear that ${A}=\\Phi (\\mathcal {H})$ and ${B}=\\Psi (U_q(\\mathfrak {g}))$ .",
"We end this paper with some remarks.",
"Remark 4.15 Combing Remark REF and Theorem REF , we can obtain a Schur-Weyl duality between the universal enveloping superalgebra $U(\\mathfrak {g})$ of $\\mathfrak {g}$ and the group algebra $\\mathbb {C}W_{m,n}$ of $W_{m,n}$ , which is a generalization of the Schur-Sergeev duality in [40], [5].",
"Based on Shoji's work [42] and Mitsuhashi's work [29], we will give a super Frobenius formula for $\\mathcal {H}$ in [46], which is one of our motivation to construct the Schur-Weyl duality between quantum superalgebras and cyclotomic Hecke algebras.",
"In a forthcoming work, we will introduce the cyclotomic $q$ -Schur superalgebras and give an alternative proof of Theorem REF along the line of [20].",
"Inspired by Brundan and Kujawa's work [7] and Du et al's work [13], it would be very interesting to understand the Mullineux involution for Ariki-Koike algebras [22] via the Schur-Weyl duality established in the paper."
],
[
"Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and {{formula:cbf7f1c5-0c73-40c0-a5c1-600bde7d6540}}",
"In this section, we establish the Schur-Weyl reciprocity between $U_q(\\mathfrak {g})$ and $\\mathcal {H}$ , which can be viewed as a super analogue of the Schur-Weyl reciprocity for cyclotomic Hecke algebras given independently by Sakamoto-Shoji [34] and Hu [20], or as a cyclotomic version of the Schur-Weyl reciprocity between the quantum superalgebra and the Iwahori-Hecke algebra obtained by Moon [30] and Mitsuhashi [28].",
"Lemma 4.1 For any $X\\in \\mathcal {H}$ and $Y\\in U_q(\\mathfrak {g})$ , $\\Phi (X)\\Psi ^{\\otimes n}(Y)=\\Psi ^{\\otimes n}(Y)\\Phi (X)$ .",
"Thanks to [30] or [28], it is enough to show that the $T_0$ -action defined by Eq.",
"(REF ) commutes with the generators of $U_q(\\mathfrak {g})$ , i.e., commutes with those elements listed in the set ${G}$ defined in Eq.",
"(REF ).",
"Clearly $S_0$ commutes with $U_q(\\mathfrak {g})$ .",
"It reduces to show $S_a$ commutes with elements of ${G}$ for all $a\\ge 1$ .",
"It is easy to check that $S_a$ commutes with $K_j^{\\pm 1}$ for all $1\\le j\\le d_m$ .",
"Given $\\mathbf {i}\\in \\mathcal {I}(k,\\ell ;n)$ , we show that $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)S_a(\\mathbf {i})$ for all $E_b\\in {G}$ .",
"Note that if $c_{a}(\\mathbf {i})=c_{a+1}(\\mathbf {i})$ then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))=c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ .",
"Thus $S_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=T_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)T_a(\\mathbf {i})$ .",
"If $c_{a}(\\mathbf {i})\\ne c_{a+1}(\\mathbf {i})$ ) then $c_{a}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))\\ne c_{a+1}(\\Psi ^{\\otimes n}(E_b)(\\mathbf {i}))$ ).",
"In this case, we need to show $s_a\\Psi ^{\\otimes n}(E_b)(\\mathbf {i})=\\Psi ^{\\otimes n}(E_b)s_a(\\mathbf {i})$ .",
"For $p=0,1,\\ldots , n-1$ , we let $&X_p=\\widetilde{K}_b^{\\otimes p}\\otimes \\Psi (E_b)\\otimes \\mathrm {Id}^{n-1-p}.$ Then $s_a$ commutes with $X_p$ unless $p=a-1,a$ , which implies we only need to show $&&s_a(X_{a-1}+X_{a})(\\mathbf {i})=(X_{a-1}+X_{a})s_a(\\mathbf {i}).$ Since $s_a$ affects only the $a$ and $a\\!+\\!1$ -th factors of $\\mathbf {i}$ and the remaining parts are the same for $X_{a}(\\mathbf {i})$ and $X_{a+1}(\\mathbf {i})$ .",
"We may only consider the $a$ and $a\\!+\\!1$ -th factors, that is, it is enough to verify that $s_a(\\Psi (E_b)\\!\\otimes \\!\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!\\Psi (E_b))(i_{a}\\otimes i_{a\\!+\\!1})\\!=\\!",
"(\\Psi (E_b)\\!\\otimes \\!",
"\\mathrm {Id}\\!+\\!\\widetilde{K}_b\\!\\otimes \\!",
"\\Psi (E_b))s_a(i_{a}\\!\\otimes \\!",
"i_{a\\!+\\!1}).$ Assume that $d_{r-1}<b<d_r$ for some $1\\le r\\le m$ with $d_0=0$ .",
"Thus if $c_{a}(\\mathbf {i})$ , $r$ , $c_{a+1}(\\mathbf {i})$ are pairwise different, then $\\Psi (E_b)=0$ , therefore both sides of Eq.",
"(REF ) equal zero; If $c_{a}(\\mathbf {i})=r\\ne c_{a+1}(\\mathbf {i})$ then $\\Psi (E_b)(i_{a+1})=0$ and $\\widetilde{K}_a(i_{a+1})=(-1)^{\\bar{i}_{a+1}-\\bar{i}_a}i_{a+1}$ .",
"Then, thanks to Eq.",
"(REF ), both sides of Eq.",
"(REF ) equal; The case $c_{a+1}(\\mathbf {i})=r\\ne c_a(\\mathbf {i})$ can be proved similarly.",
"So $S_a$ commutes with $\\Psi ^{\\otimes n}(E_b)$ for all $E_b\\in {G}$ .",
"In a similar argument, we can show $S_a$ commutes with $\\Psi ^{\\otimes n}(F_b)$ for all $F_b\\in {G}$ .",
"For positive integers $a$ , $b$ , $c$ , we let $\\Pi (a,b;c)=\\left\\lbrace (\\mu ;\\nu )\\left|\\begin{array}{l}\\mu \\vdash s,\\quad \\nu \\vdash t,\\quad s+t=c\\\\ \\ell (\\mu )\\le a,\\ell (\\nu )\\le b, \\mu _a\\ge \\ell (\\nu )\\end{array}\\right.\\right\\rbrace .$ Lemma 4.3 ([4] or [30]) Keeping notations as above, then there is a bijection between $H(a,b;c)$ and $\\Pi (a,b;c)$ given by $\\lambda \\mapsto (\\lambda ^1;\\lambda ^2)$ , where $\\lambda ^1=(\\lambda _1, \\ldots , \\lambda _a)$ and $\\lambda ^2=(\\lambda _1^2,\\ldots , \\lambda _b^2)$ with $\\lambda _i^2=\\max \\lbrace \\lambda _i^*-a,0\\rbrace $ .",
"Recall that the Jucys-Murphy elements of $\\mathcal {H}$ are defined recursively by $J_1\\!",
":=g_0\\quad \\text{ and }\\quad J_{i+1}\\!",
":=g_iJ_ig_i, \\quad i=1, \\cdots , n-1.$ It is known that $J_1$ , $\\ldots , J_n$ generate a maximal commutative subalgebra of $\\mathcal {H}$ .",
"Let $S^{\\lambda }$ be the irreducible $\\mathcal {H}$ -module corresponding to $\\lambda \\in {P}_{m,n}$ .",
"Then $S^{\\lambda }$ has a basis $\\lbrace v_{\\mathfrak {s}}|\\mathfrak {s}\\in \\mathrm {std}(\\lambda )\\rbrace $ satisfying $J_iv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(i)v_{\\mathfrak {s}}, \\quad i=1, \\ldots , n,$ for each $\\mathfrak {s}\\in \\mathrm {std(\\lambda )}$ .",
"Conversely, if $M$ is an $\\mathcal {H}$ -module containing a common eigenvector $v_{\\mathfrak {s}}$ for $J_1, \\ldots , J_n$ satisfying Eq.",
"(REF ) for some $\\mathfrak {s}\\in \\mathrm {std}(\\lambda )$ , then the $\\mathcal {H}$ -submodule $\\mathcal {H}v_{\\mathfrak {s}}$ of $M$ is a sum of copies of $S^{\\lambda }$ .",
"Lemma 4.5 Let $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then $V^{\\otimes n}$ contains an irreducible $\\mathcal {H}$ -module isomorphic to $S^{\\lambda }$ consisting of highest weight vectors of $U_{q}(\\mathfrak {g})$ with highest weight $\\lambda $ .",
"Thanks to [30], [28], we have the following $U_q(\\mathfrak {gl}(k|\\ell ))\\otimes \\mathcal {H}_n(q)$ -module decomposition $V^{\\otimes n}&=&\\bigoplus _{\\lambda \\in H(k,\\ell ;n)}V_{\\lambda }\\otimes S^{\\lambda },$ where $V_{\\lambda }$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_{q}(\\mathfrak {g}(k|\\ell ))$ -module with highest weight $\\lambda $ (resp.",
"the irreducible module of $\\mathcal {H}_n(q)$ corresponding to $\\lambda $ ).",
"Furthermore, the decomposition Eq.",
"(REF ) implies for each $\\lambda \\in H(k,\\ell ;n)$ , the $\\mathcal {H}_n(q)$ -module $S^{\\lambda }$ consisting of highest weight vectors for $V_{\\lambda }$ , that is, if $\\mathfrak {s}$ is a standard $\\lambda $ -tableau then there is a highest weight vector for $V_{\\lambda }$ .",
"Suppose $\\lambda =(\\lambda ^{(1)}; \\ldots ; \\lambda ^{(m)})\\!\\in \\!",
"H(\\mathbf {k}|\\ell ;m,n)$ and let $n_i=\\left|\\lambda ^{(i)}\\right|$ .",
"Thanks to Lemma REF , we may put $\\lambda ^{(i)}\\!=\\!",
"(\\mu ^{(i)};\\nu ^{(i)})\\!\\in \\!\\Pi (k_i,\\ell _i;n_i)$ for $1\\le i\\le m$ .",
"Now we define a standard $\\lambda $ -tableau $\\mathfrak {s}=(\\mathfrak {s}^{(1)},\\ldots , \\mathfrak {s}^{(m)})$ as follows: Set $n_{i,1}=\\left|\\mu ^{(i)}\\right|$ , $n_{i,2}=\\left|\\nu ^{(i)}\\right|$ , $p_{i,1}=n_{i,1}+\\cdots +n_{m,1}$ , $p_{i,2}=n_{i,2}+\\cdots +n_{m,2}$ for $i=1, \\ldots , m$ , and $p_{m+1,1}=p_{m+1,2}=0$ .",
"Define $\\mathfrak {s}^{(i)}$ by filling $n_{i,1}$ numbering $p_{i+1,1}<p_{i+1,1}+2<\\cdots <p_{i,1}$ in the boxes in $\\mu ^{(i)}$ first to all boxes of the first row, and then to all the boxes of the second row, and so on, in increasing order; then by filling $n_{i,2}$ numbering $p_{i+1,2}<p_{i+1,2}+2<\\cdots <p_{i,2}$ in the boxes in $\\nu ^{(i)}$ first to all boxes of the first column, and then to all the boxes of the second column, and so on, in increasing order.",
"It is clearly that $\\mathfrak {s}^{(i)}$ is a standard $\\lambda ^{(i)}$ -tableau.",
"Now consider the action of $U_q(\\mathfrak {gl}(k_i|\\ell _i)\\otimes \\mathcal {H}_{n_i}(q)$ on the ${V^{(i)}}^{n_i}$ and apply Moon's argument in [30], we can obtain a common eigenvector $w_i\\in {V^{(i)}}^{n_i}$ for the Jucys-Murphy elements of $\\mathcal {H}_{n_i}(q)$ with respect to $\\mathrm {res}_{\\mathfrak {s}^{(i)}}$ , which is also a highest weight vector of the irreducible $U_q(\\mathfrak {gl}(k_i|\\ell _i)$ -module $V_{\\lambda ^{(i)}}$ with highest weight $\\lambda ^{(i)}$ .",
"Set $v_{\\mathfrak {s}}=w_1\\otimes w_{2}\\otimes \\cdots \\otimes w_{m}\\in {V^{(1)}}^{\\otimes n_1}\\otimes {V^{(2)}}^{\\otimes n_{2}}\\otimes \\cdots \\otimes {V^{(m)}}^{\\otimes n_m}.$ Then $v_{\\mathfrak {s}}$ is a highest weight vector of $V_{\\lambda }=V_{\\lambda ^{(1)}}\\otimes \\cdots \\otimes V_{\\lambda ^{(m)}}$ .",
"Assume that $a=p_{k+1}<r\\le p_k=b$ .",
"We show that $v_{\\mathfrak {s}}$ is a common eigenvector for Jucys-Murphy elements of $\\mathcal {H}$ , that is, $J_rv_{\\mathfrak {s}}=\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ Clearly $J_rv_{\\mathfrak {s}}$ can be written as $J_rv_{\\mathfrak {s}}=T^{-1}_{r}\\cdots T^{-1}_{n-1}S_{n-1}\\cdots S_1S_0T_1\\cdots T_{r-1}v_{\\mathfrak {s}}.$ First note that $T_a\\cdots T_rv_{\\mathfrak {s}}\\in {V^{(1)}}^{n_1}\\otimes \\cdots \\otimes {V^{(m)}}^{n_m}.$ Let $\\mathbf {i}$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $ T_a\\cdots T_rv_{\\mathfrak {s}}$ .",
"Then $i_a\\in V_r$ , and $i_c>i_a$ for all $c<a$ .",
"Moreover $i_c\\notin V_r$ for $c<a$ , which implies $T_1\\cdots T_{a-1}(\\mathbf {i})=i_a\\otimes i_1\\otimes \\cdots \\otimes i_{a-1}\\otimes i_{a+1}\\otimes \\cdots \\otimes i_n$ and $S_{a-1}\\cdots S_1S_0T_1\\cdots T_{a-1}(\\mathbf {i})=Q_r\\mathbf {i}$ .",
"Thus $J_rv_{\\mathfrak {s}}=Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}.$ On the other hand, we have $S_{b-1}\\cdots S_{a}T_a\\cdots T_{r-1}v_{\\mathfrak {s}}\\in V_{1}^{\\otimes n_1}\\otimes \\cdots \\otimes V_m^{\\otimes n_m}.$ Let $y=y_1\\otimes \\cdots \\otimes y_n$ be a basis element of $V^{\\otimes n}$ occurring in the expression of $S_{b-1}\\cdots S_{a}T_a\\cdots T_{j-1}v_{\\mathfrak {s}}$ .",
"Then $y_b\\in V_{r}$ , $y_i\\notin V_{r}$ for all $i>a$ .",
"Moreover $y_i<y_{b}$ for all $i>b$ .",
"By a similar argument as above, we obtain $T_{b}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_b(y)=y$ .",
"Therefore $J_rv_{\\mathfrak {s}}&=&Q_rT_{r}^{-1}\\cdots T_{n-1}S_{n-1}\\cdots S_aT_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&Q_rT_{r-1}\\cdots T_{a}T_{a}\\cdots T_{r-1}v_{\\mathfrak {s}}\\\\&=&\\mathrm {res}_{\\mathfrak {s}}(r)v_{\\mathfrak {s}}.$ We complete the proof.",
"Now we compute the multiplicity $m_{\\lambda }:=[V^{\\otimes n}:V_{\\lambda }]$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Assume that $k_m,\\ell _m>1$ .",
"We let $\\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)$ be the subalgebra of $\\mathfrak {gl}(k_m|\\ell _m)$ corresponding to the basis $\\mathfrak {B}^{(m)}-\\lbrace v^{(m)}_{k_m}\\rbrace $ and let $\\mathfrak {gl}(1,0)$ to be that corresponding to basis $v^{(m)}_{k_m}$ .",
"Put $\\tilde{\\mathfrak {g}}= \\mathfrak {gl}(k_1|\\ell _1)\\oplus \\cdots \\oplus \\mathfrak {gl}(k_m\\!-\\!1|\\ell _m)\\oplus \\mathfrak {gl}(1|0)$ , which is a subalgebra of $\\mathfrak {g}$ .",
"For $\\lambda =(\\lambda ^{(1)},\\ldots ,\\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ and $\\mu =(\\mu ^{(1)},\\ldots ,\\mu ^{(m)},\\mu ^{(m+1)})\\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , where $\\tilde{\\mathbf {k}}=(k_1, \\ldots ,k_m-1,1)$ and $\\tilde{\\ell }=(\\ell ,0)$ , we write $\\mu \\prec \\lambda $ if $\\lambda ^{(i)}=\\mu ^{(i)}$ for $i=1, \\ldots , m-1$ and $\\lambda ^{(m)}_1\\ge \\mu _{1}^{(m)}\\ge \\cdots \\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge \\mu _{\\ell (\\mu ^{(m)})-1}^{(m)}\\ge \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})},$ where $\\lambda ^{(m)}=(\\lambda ^{(m)}_{1},\\ldots , \\lambda ^{(m)}_{\\ell (\\lambda ^{(m)})})$ and $\\mu ^{(m)}=(\\mu ^{(m)}_{1},\\ldots , \\mu ^{(m)}_{\\ell (\\lambda ^{(m)})-1}\\ge 0)$ .",
"Notice that $|\\lambda |=|\\mu |=n$ and $\\ell (\\mu ^{(m+1)})\\le 1$ .",
"Moreover, $\\mu \\prec \\lambda $ implies that $|\\mu ^{(m)}|\\le |\\lambda ^{(m)}|$ .",
"Thus $\\mu ^{(m+1)}$ is determined uniquely whenever $\\mu ^{\\prime }=(\\mu ^{(1)},\\ldots ,\\mu ^{(m)})$ is determined up to $\\prec $ .",
"The following lemma characterize the restriction of $V_{\\lambda }$ as a $U_q(\\tilde{\\mathfrak {g}})$ -modules.",
"Lemma 4.10 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $\\left.V_{\\lambda }\\right|_{U_q(\\tilde{\\mathfrak {g}})}=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }$ .",
"In particular, for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m,n)$ , we have $m_{\\mu }=\\sum _{\\mu \\prec \\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}m_{\\lambda }.$ Note that the lemma is easily reduced to the case $m=1$ , that is, $\\mathfrak {g}=\\mathfrak {gl}(k|\\ell )$ and $\\tilde{\\mathfrak {g}}=\\mathfrak {gl}(k\\!-\\!1|\\ell )\\oplus \\mathfrak {gl}(1,0)$ .",
"Then $U_q(\\mathfrak {gl}(k\\!-\\!1|\\ell ))$ is a subalgebra of $U_q(\\mathfrak {gl}(k|\\ell ))$ .",
"For $\\lambda \\in H(k,\\ell ;n)$ and $\\mu \\in H(k\\!-\\!1,\\ell ;n)$ with $\\mu \\prec \\lambda $ , [5] implies $\\mathrm {Res}_{U_q(\\mathfrak {gl}(k-1|\\ell ))}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda }=\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }.$ Note that $U_q(\\mathfrak {g}^{\\prime })=U_q(\\mathfrak {gl}(k-\\!1|\\ell ))\\oplus U_q(\\mathfrak {gl}(1|0))$ .",
"Thus we yield that $\\mathrm {Res}_{U_q(\\tilde{\\mathfrak {g}})}^{U_q(\\mathfrak {gl}(k|\\ell ))} V_{\\lambda } =\\bigoplus _{\\mu \\prec \\lambda }V_{\\mu }\\otimes V_{\\nu },$ where $V_{\\nu }$ is an irreducible $U_q(\\mathfrak {gl}(1,0))$ -module with highest weight $\\nu $ which is determined uniquely from $\\mu $ .",
"Since the highest weight $\\mu =(\\mu ,\\nu )$ of $V_{\\mu }=V_{\\mu }\\otimes V_{\\nu }$ satisfies $|\\mu |=n$ and $\\ell (\\nu )\\le 1$ , the condition $\\mu \\prec \\lambda $ is equivalent to $\\mu \\prec \\lambda =\\lambda $ .",
"It completes the proof.",
"Denote by $\\bar{S}^{\\lambda }$ the Specht module of $\\mathbb {C}W_{m,n}$ corresponding to $\\lambda $ .",
"It is well-known that $\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }=\\dim _{\\mathbb {K}}S^{\\lambda }$ .",
"Now we can compute the multiplicity $m_{\\lambda }$ of $V_{\\lambda }$ in $V^{\\otimes n}$ .",
"Proposition 4.11 If $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ then $m_{\\lambda }=\\dim _{\\mathbb {C}}\\bar{S}^{\\lambda }$ .",
"First note that for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ , we have $\\dim S^{\\mu }=\\sum _{\\lambda \\succ \\mu }\\dim S^{\\lambda }.$ Indeed we can prove this by a similar argument as Lemma REF .",
"Now we prove the proposition by induction on $\\mathbf {k}|\\ell $ .",
"First assume that $k_i|\\ell _i=1|0$ or $0|1$ for all $i$ and $\\lambda =(\\lambda ^{(1)}, \\ldots , \\lambda ^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ .",
"Then each $\\lambda ^{(i)}$ can be identified with a non-negative integers, which implies $\\dim V_{\\lambda }=1$ and $m_{\\lambda }=\\dim S^{\\lambda }$ .",
"Assume that there are some $i$ such that $k_i|\\ell _i\\ne 1|0, 0|1$ .",
"Clearly, we may assume that $i=m$ .",
"Given $\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)$ , we choose $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ such that $\\mu \\prec \\lambda $ with $\\mu ^{(m)}_i$ for $i=1, \\ldots , \\ell (\\lambda ^{(m)})-1=d-1$ and $\\mu ^{(m+1)}=\\lambda ^{(m)}_{d}$ .",
"Then for $\\widetilde{\\lambda }=(\\tilde{\\lambda }^{(1)}, \\ldots , \\tilde{\\lambda }^{(m)})\\in H(\\mathbf {k}|\\ell ;m,n)$ , $\\mu \\prec \\widetilde{\\lambda }$ implies that $\\lambda ^{(m)}\\unlhd \\tilde{\\lambda }^{(m)}$ (see §REF ), and $\\lambda ^{(i)}=\\tilde{\\lambda }^{(i)}$ for $i=1, \\ldots , m-1$ .",
"Now by induction argument, we may assume that $m_{\\mu }=\\dim S^{\\mu }$ for $\\mu \\in H(\\tilde{\\mathbf {k}}|\\tilde{\\ell };m+1,n)$ .",
"Therefore $m_{\\mu } &=&\\sum _{\\widetilde{\\lambda }\\succ \\mu }\\dim S^{\\widetilde{\\lambda }}=\\sum _{\\lambda \\succ \\mu }m_{\\lambda }=\\sum _{\\mu \\prec \\lambda }\\dim S^{\\lambda },$ which implies $m_{\\lambda }=\\dim S^{\\lambda }$ by applying backward induction on the dominant order $\\unlhd $ of weights that $m_{\\widetilde{\\lambda }}=\\dim S^{\\widetilde{\\lambda }}$ for any $\\widetilde{\\lambda }\\ne \\lambda $ .",
"We prove the proposition.",
"Theorem 4.13 Keeping notations as above, then $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))$ and $\\Phi (\\mathcal {H})$ are mutually the fully centralizer algebras of each other, i.e., $\\Psi ^{\\otimes n}(U_q(\\mathfrak {g}))=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n}), &\\qquad & \\Phi (\\mathcal {H})=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n}).$ More precisely, there is a $U_q(\\mathfrak {g})\\text{-}\\mathcal {H}$ -bimodule isomorphism $V^{\\otimes n}\\cong \\bigoplus _{\\lambda \\in \\Lambda ^+_{\\mathbf {k}|\\ell ,m}(n)}V(\\lambda ) \\otimes S^{\\lambda },$ where $V(\\lambda )$ (resp.",
"$S^{\\lambda }$ ) is the irreducible $U_q(\\mathfrak {g})$ (resp.",
"$\\mathcal {H}$ )-module indexed by $\\lambda $ .",
"Let ${A}=\\mathrm {End}_{U_q(\\mathfrak {g})}(V^{\\otimes n})$ and ${B}=\\mathrm {End}_{\\mathcal {H}}(V^{\\otimes n})$ .",
"Then $V^{\\otimes n}$ is decomposed as a $U_q(\\lambda )\\otimes {A}$ -module $V^{\\otimes n}=\\bigoplus _{\\lambda \\in H(\\mathbf {k}|\\ell ;m,n)}V_{\\lambda }\\otimes \\hat{S}_{\\lambda },$ where $\\hat{S}_{\\lambda }$ is an irreducible ${A}$ -module corresponding to $\\lambda $ .",
"According to Lemma REF , we have $\\Phi (\\mathcal {H})\\subseteq {A}$ .",
"By Lemma REF and Proposition REF , $\\hat{S}_{\\lambda }$ contains $S^{\\lambda }$ as an $\\mathcal {H}$ -submodule and $\\dim \\hat{S}_{\\lambda }=m_{\\lambda }=\\dim S_{\\lambda }$ .",
"Hence $\\hat{S}_{\\lambda }=S_{\\lambda }$ and the decomposition (REF ) follows.",
"It is then clear that ${A}=\\Phi (\\mathcal {H})$ and ${B}=\\Psi (U_q(\\mathfrak {g}))$ .",
"We end this paper with some remarks.",
"Remark 4.15 Combing Remark REF and Theorem REF , we can obtain a Schur-Weyl duality between the universal enveloping superalgebra $U(\\mathfrak {g})$ of $\\mathfrak {g}$ and the group algebra $\\mathbb {C}W_{m,n}$ of $W_{m,n}$ , which is a generalization of the Schur-Sergeev duality in [40], [5].",
"Based on Shoji's work [42] and Mitsuhashi's work [29], we will give a super Frobenius formula for $\\mathcal {H}$ in [46], which is one of our motivation to construct the Schur-Weyl duality between quantum superalgebras and cyclotomic Hecke algebras.",
"In a forthcoming work, we will introduce the cyclotomic $q$ -Schur superalgebras and give an alternative proof of Theorem REF along the line of [20].",
"Inspired by Brundan and Kujawa's work [7] and Du et al's work [13], it would be very interesting to understand the Mullineux involution for Ariki-Koike algebras [22] via the Schur-Weyl duality established in the paper."
]
] | 1808.08484 |
[
[
"Efficiently Processing Workflow Provenance Queries on SPARK"
],
[
"Abstract In this paper, we investigate how we can leverage Spark platform for efficiently processing provenance queries on large volumes of workflow provenance data.",
"We focus on processing provenance queries at attribute-value level which is the finest granularity available.",
"We propose a novel weakly connected component based framework which is carefully engineered to quickly determine a minimal volume of data containing the entire lineage of the queried attribute-value.",
"This minimal volume of data is then processed to figure out the provenance of the queried attribute-value.",
"The proposed framework computes weakly connected components on the workflow provenance graph and further partitions the large components as a collection of weakly connected sets.",
"The framework exploits the workflow dependency graph to effectively partition the large components into a collection of weakly connected sets.",
"We study the effectiveness of the proposed framework through experiments on a provenance trace obtained from a real-life unstructured text curation workflow.",
"On provenance graphs containing upto 500M nodes and edges, we show that the proposed framework answers provenance queries in real-time and easily outperforms the naive approaches."
],
[
"Introduction",
"Many applications are encoded as a workflow which executes a sequence of data manipulation operations on raw input data.",
"Provenance is an important requirement for workflow management systems as it enables various use-cases e.g., data-quality, compliance, problem diagnosis etc.",
"For example, if the value of a data-item is erroneous, we can examine its lineage to investigate which transformation has introduced the error and hence fix this transformation.",
"In this paper, we present efficient Spark algorithms for processing large scale workflow provenance data and answer lineage queries.",
"For a representative example, consider the table Person1.",
"Numbers in bracket represent an id assigned to each attribute-value.",
"Next consider a transformation R1 which filters out persons with age less than 25 and populates the table Person2.",
"Values for attributes Name, City and Age in tuples T5, T6 and T7 are hence derived from values for attributes Name, City and Age in tuples T1, T2 and T3 respectively.",
"Further consider a transformation R2 which works on table Person2 and computes the average age of persons in each city.",
"The resulting output is shown in Table REF .",
"The value for attribute City in tuple T8 is derived from values of attribute City in tuples T5 and T6.",
"Similarly the value for attribute Age in tuple T8 is derived from values of attribute Age in tuples T5 and T6.",
"Values for attributes City and Age in tuple T9 is derived from one value each - value of attribute City and Age in tuple T7.",
"The workflow provenance data captures these lineages among input and output attribute-values across each transformation, as they are executed.",
".",
"Provenance Data Model: We assume that the provenance data is specified as a set of triples $\\langle $$src$ , $dst$ , $op$$\\rangle $ where $src$ and $dst$ represent the ids of the parent and child data-items and $op$ represents the transformation applied along with any metadata (e.g., run-time parameters, timestamp etc).",
"Table REF shows the provenance data associated with the representative example.",
"We also visualize the provenance data as a directed acyclic graph $G(V,E)$ wherein data-items (i.e., $src$ and $dst$ ) in provenance triples form the vertices $V$ and the provenance triples form the edges $E$ (Table REF ).",
"Table: AvgAge Table: Provenance Graph Provenance Query: Given a query data-item $q$ , we want to track its lineage i.e., all its ancestors and the details of all transformations involved.",
"For example, lineage of data-item 23 (i.e., the value of attribute $Age$ of tuple $T8$ in entity $AvgAge$ ) will return that data-item 23 is derived from data-items 15 and 18 via transformation R2 and data-items 15 and 18 are derived from data-items 3 and 6 respectively via transformation R1.",
"Contributions: A naive approach to answer a provenance query is to recursively process the provenance data.",
"We start with the queried data-item $q$ , find those provenance triples which describe its immediate lineage and obtain its parents.",
"We then find the parents of $q$ 's parents and follow this process until we can no longer trace the lineage further.",
"This approach is adopted by many systems e.g., Trio [1], GridDB [2], Titian [3] etc.",
"This obviously takes time as we need to issue many queries.",
"Secondly, as Spark does not support indexing, Spark needs to scan the data to find the parents of a data-item.",
"This hence does not scale for large volumes of data.",
"A second approach is to pre-compute and materialize the transitive closure of the lineage dependencies (i.e., the provenance of each data-item).",
"This allows retrieval of a data-item's lineage using a single query.",
"However this results in a huge increase in the storage cost as the information regarding common ancestors gets replicated multiple times.",
"This approach hence also does not scale.",
"In this paper, we propose a novel approach wherein we first quickly determine a small volume of data which contains the entire provenance output of the queried data-item.",
"We then extract and recursively query this small volume of data.",
"As the recursive querying happens on a small volume of data, we do not incur a large data processing cost.",
"Contributions of this paper are hence as follows.",
"We propose a novel provenance framework wherein we first compute weakly connected components in provenance graph and further partition the large components as a collection of weakly connected sets (section 3).",
"We then effectively navigate the weakly connected components and sets, thus computed, to determine a minimal volume of data containing the entire provenance output of the queried data-item (section 3 and 4).",
"We propose a novel provenance graph partitioning approach wherein we exploit the workflow dependency graph to recursively partition the large components in workflow provenance graph (section 3).",
"Our experiments on provenance graphs obtained from a real-life text curation workflow and containing upto 500M nodes and edges show that the proposed approaches significantly beat the naive approaches (section 5).",
"The performance is realtime if all data can be cached in RAM.",
"Space overheads are (1) storing two set-ids with each provenance triple and (2) storing set-dependencies i.e., how the sets derive each other.",
"The number of set dependencies are upper-bounded by the number of provenance triples and in practice, are only a small fraction of it.",
"The proposed framework hence has a minimal space overhead."
],
[
"Background",
"Apache Spark : Spark uses the resilient distributed data set (RDD) as its basic data type.",
"An RDD partitions the data across the cluster nodes.",
"In this paper, we will be mainly concerned with Spark filter and lookup operations.",
"The filter operation scans each row of an RDD and checks whether the filter conditions are satisfied or not.",
"A lookup is a specific kind of filter where one or more columns are checked for equality.",
"To accelerate lookup operations, we can hash-partition an RDD on one or more columns and, this process moves all rows with same key to one partition.",
"With hash-partitioning enabled, a lookup needs to scan only one partition.",
"Hash-partitioning also accelerates filter performance, if the filter conditions involve checking column equality on hashed columns.",
"RDDs can also be cached and this avoids re-computation of an RDD, each time it is accessed.",
"Weakly Connected Sets and Components: A semipath joining vertices $u_1$ and $u_k$ in a directed graph $G$ =($V$ ,$E$ ) is a sequence of vertices $u_1$ ,$u_2$$\\ldots $ ,$u_k$ s.t.",
"for each $i$ , 1$\\le $$i$$\\le $$k-1$ either there exists an edge $u_i \\rightarrow u_{i+1}$ in $E$ or there exists an edge $u_{i+1} \\rightarrow u_i$ in $E$ .",
"A set of vertices $W$$\\subseteq $$V$ is called weakly connected if there exists a semipath between each vertex pair in $W$ .",
"A maximal weakly connected set of vertices is a weakly connected component in $G$ .",
"Notation: We use the terms “connected component\" and “connected set\" as a shorthand for “weakly connected components\" and “weakly connected sets\", though they are different abstractions in graph theory."
],
[
"Recursive Querying on Spark (RQ)",
"We first discuss the challenges in executing recursive querying (RQ) on Spark.",
"Let us denote the provenance data RDD as provRDD.",
"As discussed, RQ involves executing many queries to trace the entire lineage of a data-item $q$ .",
"The number of queries are equal to the length of the largest provenance path in the lineage of data-item $q$ .",
"Each such query involves finding parents of one or more data-items $\\mathcal {I}$ .",
"As discussed above, if we hash-partition the provRDD on field dst, this moves all provenance triples with the same dst field to one partition and we can hence find the parents of a data-item by scanning one partition of provRDD.",
"To find parents of all data-items in $\\mathcal {I}$ , we need to scan at most $|\\mathcal {I}|$ number of partitions.",
"This is because, some data-items in $|\\mathcal {I}|$ may be in the same partition and the parents of these data-items can hence be obtained by scanning this partition only once.",
"If the lineage size (i.e., number of ancestors) of queried data-item $q$ is $N$ , we hence require scanning a maximum of $|N|$ number of partitions.",
"The overall RQ cost will hence depend upon the number of queries executed, set of lookups made as part of each query, and the distribution of field dst across the provRDD."
],
[
"Connected Components and Provenance",
"We observe that the workflow provenance graph, formed by attribute-values, is a large collection of weakly connected components.",
"This is because many attribute-values do not share any common ancestors.",
"This is best evidenced by looking at Table REF which shows the provenance graph for the representative example.",
"This graph contains 10 weakly connected components.",
"We notice that a data-item and all its ancestors as well as descendants, share the same weakly connected component.",
"This property can be used to speed up the processing of provenance queries.",
"Given a queried data-item $q$ , we first find out its weakly connected component id and then retrieve all provenance triples in this component.",
"We then process the triples in this component recursively to figure out the provenance of data-item $q$ .",
"As the size of a component is much smaller than the whole provenance graph, the recursive querying executes faster.",
"We hence compute weakly connected components on the provenance graph and then append the connected component id with each provenance triple as shown in Table REF .",
"This computation is part of pre-processing and needs to be done only once.",
"[t] Hash-Partitioned Provenance RDD provRDD, data-item $q$ Lineage of data-item $q$ $c$ $\\leftarrow $ Find-Connected-Component(provRDD, $q$ ) $c\\_$provRDD $\\leftarrow $ Find-ProvTriples-In-Component(provRDD, $c$ ) return Recursive-Query($c\\_$provRDD, $q$ ) CCProv Algorithm 1 outlines the algorithm for computing the lineage of a data-item $q$ and it takes the provenance data RDD provRDD, hash-partitioned on column $dst$ as input.",
"We first find out the id of the connected component, the data-item $q$ lies in and let it be $c$ .",
"This can be found by scanning a single partition of provRDD.",
"We then find all provenance triples in component $c$ and let it be $c\\_$provRDD.",
"This is done via a Spark filter operation on provRDD and this preserves the hash-partitioning logic.",
"We then recursively process $c\\_$provRDD to find the lineage of data-item $q$ ."
],
[
"Connected Sets and Provenance",
"Though CCProv provides better performance vis-a-vis RQ, it may not be good-enough when the component size is large as CCProv processes large volume of data (i.e., RDD $c\\_$provRDD).",
"We next discuss CSProv which improves on this aspect.",
"The idea is to pre-process and partition the large components into a collection of weakly connected sets.",
"At query time, we exploit the information regarding how these sets derive each other to quickly find a minimal volume of data containing the entire lineage of the queried data-item.",
"We explain the intuition via a representative example.",
"Table: Set Dependencies[th] Hash-Partitioned Provenance RDD provRDD, Hash-Partitioned Set Dependencies RDD setDepRDD, data-item $q$ Lineage of data-item $q$ $cs$ $\\leftarrow $ Find-Connected-Set(provRDD, $q$ ) $\\mathcal {S}$ $\\leftarrow $ $cs$ $\\cup $ Find-Set-Lineage(setDepRDD, $cs$ ) cs_provRDD $\\leftarrow $ $\\phi $ connected set $s$ in $\\mathcal {S}$ cs_provRDD $\\leftarrow $ cs_provRDD $\\cup $ Find-ProvTriples-With-DerivedItem-In-Set(provRDD,$s$ ) return Recursive-Query(cs_provRDD, $q$ ) CSProv Consider a weakly connected component C as shown in Table REF .",
"Consider, we partition the component C in 4 weakly connected sets - S1, S2, S3 and S4.",
"These sets are formed by data-items {1, 2, 3}, {4, 5, 6}, {7, 8, 9} and {10, 11, 12} respectively.",
"We also maintain the set dependencies - how these sets contribute to the derivation of other sets.",
"The set S1 contributes to the derivation of set S2 as data-items 2 and 3 in set S1 derive data-item 4 in set S2.",
"Set S2 derives set S3 as data-item 5 in set S2 derives data-item 7 in set S3.",
"Similarly set S2 derives set S4.",
"Note that sets S3 and S4 do not contribute to the derivation of any set (Table REF ).",
"Consider that we query the provenance of data-item 8.",
"This belongs to the set S3.",
"From set-dependencies, we find that set S2 derives set S3 and set S1 derives set S2.",
"Hence sets S1 and S2 are relevant to the derivation of set S3.",
"These three sets together contain all ancestors of the data-item 8.",
"We only process those triples whose derived ($dst$ ) data-item is in sets S1, S2 and S3.",
"We do not need to process set S4 triples as the set-dependencies tell us that set S4 neither contributes to the derivation of set S3 nor to the derivation of any ancestor set of set S3.",
"We hence end up processing a smaller volume of data, in this example 3 less provenance triples.",
"CSProv requires the following updates on the provenance data model discussed in section .",
"Provenance Data: Data-items src and dst in a provenance triple may lie in two different weakly connected sets and we hence maintain the set id of both items.",
"We add the columns src_csid and dst_csid in the schema and drop the field ccid from the provenance triple (Table REF ).",
"Set Dependencies: We also maintain how the weakly connected sets are derived from each other (Table REF ).",
"We say a set $s_1$ is derived from $s_2$ if there exists at least one data-item $u$ in $s_1$ and at least one data-item $v$ in $s_2$ s.t.",
"there is a provenance triple where $src$ equals $v$ and $dst$ equals $u$ .",
"There are two columns in the schema - src_csid and dst_csid which denote the set-ids of parent and child connected sets.",
"Algorithm 2 outlines the algorithm CSProv.",
"It takes provenance data provRDD and set dependencies setDepRDD as input, both hash-partitioned on field $dst\\_csid$ .",
"Given queried data-item $q$ , we first find out its connected set $cs$ .",
"We then construct set $\\mathcal {S}$ which includes set $cs$ and its set-lineage i.e., all sets which contribute to the derivation of set $cs$ , directly or indirectly.",
"This is done by executing RQ logic on setDepRDD.",
"RQ on setDepRDD is lightweight due to two reasons.",
"First, the size of setDepRDD is likely to be much smaller vis-a-vis provRDD.",
"Secondly, the size of set-lineage of set $cs$ is likely to be much smaller than the size of lineage of data-item $q$ and hence much smaller number of queries need to be executed.",
"For each set $s$ in $\\mathcal {S}$ , we find the provenance triples s.t., the data-item $dst$ is in connected-set $s$ .",
"As provRDD is hash-partitioned on field $dst\\_csid$ , this requires scanning at most $|\\mathcal {S}|$ number of partitions.",
"As discussed, the size of set $\\mathcal {S}$ is small and this operation is hence light-weight as well.",
"A union of all these provenance triples i.e., $cs\\_$provRDD contains the entire lineage of data-item $q$ .",
"We then recursively process $cs\\_$provRDD to compute the lineage of data-item $q$ .",
"Again, the size of $cs\\_$provRDD is likely to be much smaller that the size of the component, the queried data-item lies in.",
"Recursive querying on $cs\\_$provRDD is hence light-weight as well.",
"Note that when the queried data-item $q$ lies in a small component $c$ , CSProv reduces to CCProv.",
"Small components are not partitioned and each small component is managed as a single weakly connected set (i.e., itself).",
"The set $\\mathcal {S}$ hence only contains the set/component $c$ ."
],
[
"Partitioning Large Components",
"In section REF , we identified the following criteria for algorithm CSProv to work efficiently.",
"C1 - Number of set-dependencies should be small.",
"C2 - The set-lineage of a set should be small.",
"C3 - The size of each connected set should be small.",
"Criteria C1 and C2 imply that CSProv can construct the set-lineage of a set $cs$ cheaply.",
"Criteria C2 and C3 imply that small number of triples (i.e., the size of $cs\\_$provRDD) need to be recursively processed.",
"We next discuss how we partition the large components, so as the resulting sets satisfy these criteria.",
"We exploit the workflow dependency graph for the same.",
"The dependency graph specifies dependencies among the tables and hence an order in which various tables are generated e.g., the dependency graph in Figure REF specifies that the table MTRCS can be generated only after table F10WMTR is generated.",
"We first develop the following notation.",
"Notation: Let $G_{wf}$ represent the workflow dependency graph.",
"Let a split be a sub-set of tables in dependency graph $G_{wf}$ s.t., these tables are weakly-connected in graph $G_{wf}$ .",
"Figure REF shows a partitioning of the dependency graph across three splits - $sp1$ , $sp2$ , $sp3$ .",
"Note that the tables in each split are weakly-connected.",
"Let $V(sp,c)$ be the set of those vertices in provenance graph $G(V,E)$ which belong to component $c$ and belong to a table in split sp.",
"Let $G[V(sp,c)]$ be the subgraph induced by the vertices $V(sp,c)$ .",
"We also call $G[V(sp,c)]$ the provenance subgraph induced by split $sp$ and component $c$ .",
"Let $W(sp,c)$ be the set of weakly connected components in subgraph $G[V(sp,c)]$ .",
"[t] Large Component $c$ , Set of weakly-connected-splits $S$ Set of Weakly-Connected-Sets $W$ $\\leftarrow $ $\\phi $ split $sp$ in $S$ $W(sp,c)$ $\\leftarrow $ Compute-Weakly-Connected-Components($G[V(sp,c)]$ ) component $cn$ in $W(sp,c)$ node-count of $cn$ $\\ge $ $\\theta $ $SS$ $\\leftarrow $ Get-Weakly-Connected-Sub-Splits($sp$ ) $W$ $\\leftarrow $ $W$ $\\cup $ Partition-Large-Component($cn$ , $SS$ ) $W$ $\\leftarrow $ $W$ $\\cup $ cn return $W$ Partition-Large-Component Algorithm 3 outlines the details.",
"We first partition the dependency graph $G_{wf}$ into a set of disjoint splits $S$ .",
"Algorithm Partition-Large-Component takes a large component $c$ and the dependency graph splits $S$ as input, and returns the set of weakly connected sets $W$ as output.",
"For each split $sp$ in $S$ , we first construct the subgraph $G[V(sp,c)]$ and then compute the weakly connected components $W(sp,c)$ in it.",
"The procedure then iterates over each component $cn$ in $W(sp,c)$ .",
"If the number of vertices in component $cn$ is less than a threshold $\\theta $ , it is not processed further and is inserted in the output set $W$ .",
"If not, we further partition split $sp$ into a set of disjoint and weakly connected sub-splits $SS$ and recursively call the procedure Partition-Large-Component with component $cn$ and split-set $SS$ as input.",
"Computing Set Dependencies: After all large components are partitioned, the fields src_csid and dst_csid associated with each provenance triple are populated using the connected sets, thus generated.",
"We then find those provenance triples wherein the columns $src\\_csid$ and $dst\\_csid$ take different values.",
"The set of distinct ($src\\_csid$ , $dst\\_csid$ ) pairs in such triples, form the set dependencies.",
"Discussion: The constraint that all tables in each split are weakly connected, is a key part of the algorithm.",
"Note that for any given large component $c$ and split $sp$ , no two components in $W(sp,c)$ contribute a set-dependency i.e., there is no set-dependency ($cn_1$ , $cn_2$ ) s.t.",
"both $cn_1$ and $cn_2$ are in $W(sp,c)$ .",
"This is because, the set $W(sp,c)$ is obtained by computing weakly connected components on subgraph $G[V(sp,c)]$ and any two components in $W(sp,c)$ are hence, by definition, disconnected.",
"This ensures that the number of set-dependencies are small (criteria C1).",
"Secondly, this increases the likelihood that a data-item's local lineage (i.e., its few immediate ancestors) can be found in the same weakly connected set, this data-item lies in and hence only few sets returned by the procedure are relevant to the lineage of a queried data-item (criteria C2).",
"Finally the condition that the size of each set has to be less than a threshold $\\theta $ , ensures that the size of each set is small (criteria C3).",
"Note that, if we consider each table in dependency graph as a separate split, CSProv reduces to RQ.",
"Each attribute-value is a connected component and provenance triples capture the set dependencies.",
"If we consider all tables in dependency graph as part of one split, CSProv reduces to CCProv."
],
[
"Experimental Evaluation",
"Provenance Data Set: We used a provenance trace obtained from a real-life workflow deployed in our lab for creating financial domain knowledge-bases [4].",
"The workflow parses SEC filing documents [5].",
"Each SEC document contains data pertaining to many thousands of financial metrics and the workflow curates this data.",
"Figure REF shows the workflow dependency graph comprising 25 entities (tables).",
"For each entity, we have only shown its acronym so as to remove any confidential information.",
"The workflow contains various transformations involving entity annotation, extraction and resolution.",
"For each transformation, the lineage relationships among the child and parent attribute-values are captured.",
"The workflow contains many UDFs and the lineage service assumes that each attribute-value in an UDF output is dependent on each attribute-value in the UDF input.",
"The entity FINDocs (marked *) forms the workflow input.",
"This workflow is executed on a set of 532 financial documents.",
"The obtained provenance trace is 1.6GB in size and contains 6.4M triples with 4.6M attribute-values.",
"The provenance graph hence contains 4.6M nodes and 6.4M edges.",
"These attribute-values have widely different derivation patterns.",
"32 attribute-values are being directly derived from more than 100 parent values, with the maximum being 450.",
"3963 values are directly derived from more than 10 parents but less than 100 parents.",
"Rest of the attribute-values have less than 10 parents.",
"Spark Cluster The cluster runs Spark v2.0.2, has 8 nodes with 12 cores each, 2.4GHz processor and 120 GB RAM.",
"Weakly Connected Components: We computed weakly connected components in the provenance graph, using Spark implementation provided at [6] and it took 6 mins to compute them.",
"Three of these components are large containing 1.2M, 0.9M and 0.7M nodes, and 2.7M, 1.4M and 1.2M edges (triples) respectively.",
"We denote these three large components by notations LC1, LC2 and LC3 respectively.",
"132 components contain between 910 and 7453 nodes.",
"Rest of the components have 100 or lesser number of nodes.",
"Table: Weakly Connected Sets StatisticsTable: Class LC-LL Query Times (s)Table: RDDs Cached on Disk, Class LC-LL Query Times (s)Weakly Connected Sets: We next partitioned the three large components using Algorithm 3.",
"We partitioned the workflow dependency graph $G_{wf}$ in three weakly connected splits $sp1$ , $sp2$ , $sp3$ as shown in Figure REF .",
"We set threshold $\\theta $ to 25K nodes.",
"Table REF presents the statistics on the connected sets obtained.",
"For each large component $c$ and for each split $sp$ , we note - (a) number of sets computed i.e., $|W(sp,c)|$ , (b) number of sets in $W(sp,c)$ with $\\ge $ 1000 nodes and (c) number of nodes in the largest set in $W(sp,c)$ (i.e.",
"the set containing maximum nodes).",
"The component LC1 got partitioned in a total of 249595 weakly connected sets with splits $sp1$ , $sp2$ and $sp3$ accounting for 20, 29696 and 219879 sets respectively.",
"Largest sets in $W(sp1,LC1)$ , $W(sp2,LC1)$ and $W(sp3,LC1)$ turned out to contain 490, 21734 and 3291 nodes respectively and hence did not need not further partitioning.",
"The component LC3 got partitioned in 143765 sets with the largest sets in $W(sp1,LC3)$ , $W(sp2,LC3)$ and $W(sp3,LC3)$ containing 313, 2578 and 643 nodes.",
"No set in $W(sp1,LC3)$ , $W(sp2,LC3)$ and $W(sp3,LC3)$ hence required further partitioning.",
"However, the sub-graph $G[V(sp3,LC2)]$ yielded only a single connected component of size 0.9M.",
"Let us denote it as LC2_lc1.",
"This component hence needs to be partitioned further.",
"Split $sp3$ is partitioned in two weakly connected sub-splits $sp4$ and $sp5$ as shown in Figure REF and the procedure Partition-Large-Component is called on component LC2_lc1 and split-set {$sp4$ ,$sp5$ } as input.",
"This time, LC2_lc1 got partitioned into 197336 sets with sub-splits $sp4$ and $sp5$ accounting for 64737 and 132599 sets respectively.",
"None of these sets contained more than $\\theta $ nodes and hence no further partitioning is needed.",
"Overall, the three large components LC1, LC2 and LC3 get partitioned into 590698 sets and these sets involve 645303 set-dependencies.",
"Number of these set-dependencies are hence an order of magnitude smaller than the number of provenance triples and the size on disk is 0.03GB.",
"Scaled Datasets: We replicated the provenance trace by a factor of 9, 24 and 48 and this generated three scaled provenance graphs containing 100M, 250M and 500M nodes and edges respectively.",
"The sizes on disk are 15, 35 and 71GB respectively.",
"As the data is replicated, these scaled datasets contain 27, 72 and 144 large components respectively.",
"These large components are partitioned and the statistics regarding the resulting sets mirror the stats given in Table REF .",
"Number of set dependencies are hence 9, 24 and 48 times vis-a-vis the base dataset and the size on disk are 0.25, 0.67 and 1.3GB respectively.",
"The computation of the connected components and connected sets on these three scaled datasets took 16, 28 and 50 mins respectively.",
"Provenance Queries: We chose three classes of lineage queries to illustrate the effectiveness of proposed approaches.",
"For each class, we chose 10 data-items and queried their lineage using RQ, CCProv and CSProv, on base as well as scaled datasets.",
"The largest provenance path for all LC-LL queries is 10 while it is 7 for all SC-SL and LC-SL queries.",
"SC-SL: We chose data-items in a small component containing 7453 nodes and 8122 edges.",
"Number of ancestors as well as transformations in lineage of these data-items are between 100 and 200.",
"These queries hence track lineage of data-items with small lineage size.",
"LC-SL : We chose data-items in large components LC1, LC2, LC3 s.t., both the number of ancestors and transformations in their lineage are between 100 and 200.",
"These queries track lineage of data-items in large components, but with small lineage size.",
"LC-LL: We chose data-items in large components s.t., both the number of ancestors and transformations in their lineage are between 5000 and 10000.",
"These queries track lineage of data-items in large components, but with considerably larger lineage size vis-a-vis class LC-SL.",
"RDDs Cached in RAM: We first ran experiments with 80 GB executor memory.",
"For all scaled datasets, all RDDs fit in memory with this configuration.",
"The RDDs were hash-partitioned and cached in RAM.",
"All RDDs were loaded with 96 partitions.",
"We executed the lineage queries and measured the average of the time taken by the 10 queries for each class.",
"Tables REF , REF and REF present the results.",
"We note that CSProv performance is real-time, degrades gracefully with datasize and significantly better than RQ and CCProv.",
"RDDs Cached on Disk: A cluster may not have enough RAM to cache all RDDs.",
"We next repeated the experiments but cached the hash-partitioned RDDs on disk.",
"Table REF presents the results.",
"For lack of space, we show results only for class LC-LL.",
"For all steps, RQ, CCProv and CSProv read the data from disk.",
"As the datasize increases, the gap between RQ and CSProv widens.",
"Discussion: We next explain the details of CSProv using a query for each class.",
"One of the 10 data-items queried for LC-SL class, belongs to a connected set in $W(sp3, LC1)$ and this set $cs$ contains 79 nodes and 102 edges.",
"13 sets in $W(sp2, LC1)$ derive the set $cs$ and these 13 sets are found to be derived from one set in $W(sp1, LC1)$ .",
"Set $cs$ and these 14 sets in its set-lineage hence construct the set $\\mathcal {S}$ (Algorithm 2), and these 15 sets are found to contain a total of 1816 nodes and 4177 edges.",
"For all datasets, CSProv hence needs to recursively query only 4177 provenance triples while CCProv needs to query 2.7M triples.",
"This leads to the improved performance of CSProv.",
"A data-item queried for class LC-LL belongs to a connected set $cs$ in $W(sp3, LC1)$ and it contains 3291 nodes and 4403 edges.",
"4 sets in $W(sp2, LC1)$ derive set $cs$ and these 4 sets are found to be derived from 20 sets in $W(sp1, LC1)$ .",
"These 25 sets contain a total of 44196 nodes and 60169 edges.",
"CSProv hence needs to recursively query only 60169 triples while CCProv needs to process 2.7M triples.",
"For SC-SL class, as a small component is not partitioned, both CCProv and CSProv recursively process 8122 triples.",
"GraphX: It is to be noted that GraphX library supports graph-parallel computation on top of Spark but as CCProv/CSProv do not have any graph-parallel computation, we use core Spark RDDs and not GraphX, for our implementations."
],
[
"Related Work",
"Titian [3] is the only major prior work to have looked at provenance data management and querying on Spark.",
"However, Titian focuses on efficiently capturing provenance data in a Spark workflow.",
"Once captured, it uses the recursive querying approach to trace the lineage of a record in an RDD.",
"In comparison, our focus is on leveraging Spark platform for efficiently processing provenance data obtained from a workflow management system and not on capturing provenance data in a Spark workflow.",
"We propose a novel framework for optimizing workflow provenance queries on Spark which exploits the workflow dependency graph to manage the provenance graph as a collection of weakly connected sets.",
"As discussed, this easily beats the recursive querying approach.",
"Few systems focus on capturing minimal volume of lineage data and optimizing the storage using domain properties and the detailed knowledge of transformations applied e.g., SubZero [7], Anand et al.",
"[8] etc.",
"Our paper is domain-agnostic and is targeted towards the black-box lineage scenario wherein the lineage service does not have the details of internals of the transformations/UDFs being applied.",
"Few systems e.g., [9], [10], [11] start with the provenance data representation wherein the transitive closure of the provenance graph (i.e., for each data-item, its full provenance) is materialized and then propose techniques to reduce the storage cost.",
"Our paper focuses on the scenario wherein the provenance data comprises of provenance triples capturing lineages across individual transformations."
],
[
"Conclusions",
"We proposed a provenance framework wherein we manage the workflow provenance graph as a collection of weakly connected sets, by exploiting the workflow dependency graph.",
"The proposed approach is effective and provides significant speed-ups vis-a-vis existing recursive querying based methods."
]
] | 1808.08424 |
[
[
"Alignment Strength and Correlation for Graphs"
],
[
"Abstract When two graphs have a correlated Bernoulli distribution, we prove that the alignment strength of their natural bijection strongly converges to a novel measure of graph correlation $\\rho_T$ that neatly combines intergraph with intragraph distribution parameters.",
"Within broad families of the random graph parameter settings, we illustrate that exact graph matching runtime and also matchability are both functions of $\\rho_T$, with thresholding behavior starkly illustrated in matchability."
],
[
"Overview ",
"Suppose $G$ and $H$ are any two graphs with the same number of vertices.",
"For any positive integer $n$ , define $[n]:=\\lbrace 1,2,3, \\ldots n \\rbrace $ , and let ${[n] \\atopwithdelims ()2}$ denote the set of all 2-element subsets of $[n]$ .",
"For simplicity, suppose that the vertex sets of $G$ and $H$ are both $[n]$ .",
"Let $_n$ denote the set of bijections from $[n]$ to $[n]$ .",
"For each $\\phi \\in _n$ , we define the number of disagreements between $G$ and $H$ under $\\phi $ to be $ d(G,H,\\phi ):= \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\nonumber \\\\\\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}\\mathbb {1}\\Big ( \\ \\mathbb {1}\\big (i \\sim _G j \\big )\\ \\ne \\ \\mathbb {1}\\big (\\phi (i) \\sim _H \\phi (j) \\big ) \\ \\Big ),$ where $\\mathbb {1}(\\cdot )$ denotes the indicator function, and $\\sim _G$ denotes adjacency of vertices in $G$ .",
"For each $\\phi \\in _n$ , we define the alignment strength of $\\phi $ as $ \\mathfrak {str}(G,H,\\phi ):=1-\\frac{d(G,H,\\phi )}{ \\frac{1}{n!",
"}\\sum _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime })}.$ The denominator in this definition of alignment strength serves as a normalizing factor; in particular, if $\\phi $ is an isomorphism between $G$ and $H$ then the alignment strength of $\\phi $ is 1, and if the number of adjacency disagreements for $\\phi $ is merely average among the bijections in $_n$ then the alignment strength of $\\phi $ is 0.",
"(Of course, if $G$ and $H$ are both edgeless or both complete graphs then $\\mathfrak {str}(G,H,\\phi )$ is not defined.)",
"If $\\phi \\in _n$ happens to be a known “natural alignment\" between $G$ and $H$ (for example, if $G$ and $H$ are social networks with the same members, and $\\phi $ maps each member to themselves; e.g.",
"an email network and a Twitter network with the same users) then $\\mathfrak {str}(G,H,\\phi )$ can be viewed as a numerical measure of the structural similarity between $G$ and $H$ .",
"However, if a natural alignment between $G$ and $H$ is not known, then we can use the graph matching problem solution, which is defined as $\\phi _{GM} \\in \\arg \\min _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime })$ ; specifically, $\\mathfrak {str}(G,H,\\phi _{GM})$ can be viewed as a numerical measure of the structural similarity between $G$ and $H$ .",
"Two practical notes regarding computation: Although the denominator $ \\frac{1}{n!",
"}\\sum _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime })$ in the definition of alignment strength (Equation REF ) involves an exponentially sized summation, nonetheless it can be computed efficiently using Equation REF from Section .",
"Also, although the computation of the graph matching problem solution $\\phi _{GM}$ is intractable [[4]], nonetheless there are effective, efficient approximate graph matching algorithms that can be used [[25], [8]], one of which we discuss and use later in this paper.",
"A brief outline of this paper is as follows.",
"In Section we describe a very general random graph setting; $G$ and $H$ are random graphs with a correlated Bernoulli distribution.",
"In particular, $G$ and $H$ share the same vertex set, and the identity bijection ${\\mathcal {I}}\\in _n$ is the natural alignment between $G$ and $H$ .",
"Each pair of vertices is assigned its own probability of adjacency (“Bernoulli parameter\") in $G$ and $H$ , and the indicator Bernoulli random variable for adjacency of the pair in $G$ and the indicator Bernoulli random variable for adjacency of the pair in $H$ have Pearson correlation coefficient $\\varrho _e$ .",
"Inherent to this model is the inter-graph (i.e.",
"between $G$ and $H$ ) statistical correlation $\\varrho _e$ and the intra-graph heterogeneity correlation parameter $\\varrho _h$ , which is a function of the Bernoulli coefficients that measures their variation.",
"Then we define the key parameter $\\varrho _T$ as $1-\\varrho _T:=(1-\\varrho _e)(1-\\varrho _h)$ ; we call $\\varrho _T$ the total correlation.",
"In Section we state and prove our main theoretical result, Theorem REF , which asserts that for $G$ and $H$ with a correlated Bernoulli distribution we have that the alignment strength of the identity bijection $\\mathfrak {str}(G,H,{\\mathcal {I}})$ is asymptotically equal to the total correlation parameter $\\varrho _T$ .",
"This suggests that the total correlation $\\varrho _T$ is a meaningful measure of the structural similarity between the graphs $G$ and $H$ realized from the correlated Bernoulli distribution.",
"Of note is that the total correlation is nicely and cleanly partitioned by the defining formula $1-\\varrho _T=(1-\\varrho _e)(1-\\varrho _h)$ ; this illustrates a symmetry in the affect of (inter-graph parameter) edge correlation $\\varrho _e$ and the affect of (intra-graph parameter) heterogeneity correlation $\\varrho _h$ .",
"The subsequent sections, Section and Section , follow up with empirical illustrations that total correlation $\\varrho _T$ is a meaningful measure.",
"As we vary the edge correlation $\\varrho _e$ together with the heterogeneity correlation $\\varrho _h$ for correlated Bernoulli graphs $G$ and $H$ in broad families of parameter settings, it turns out that the value of $\\varrho _T$ dictates (in Section ) how successful the approximate seeded graph matching algorithm called SGM [[8], [15]] is in recovering the identity bijection (which is the natural alignment here) and (in Section ) $\\varrho _T$ dictates how much time it takes to perform seeded graph matching exactly via binary integer linear programming.",
"The seeded graph matching problem is the graph matching problem wherein we seek to compute $\\phi _{GM} \\in \\arg \\min _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime })$ , except that part of the natural alignment is known; having these “seeds\" can substantially help recover the rest of the natural alignment correctly.",
"In Section , we utilize the SGM Algorithm [[8], [15]] for approximate seeded graph matching on moderately sized graphs, on the order of 1000 vertices, since, unfortunately, exact seeded graph matching can only be done on very small, toy-size graphs (a few tens of non-seed vertices).",
"In Section , where we are interested in comparing runtime, the approximate seeded graph matching algorithms are not appropriate to use, since their run times tend to be monolithic (given the number of vertices) and less sensitive to the parameters of the random graph distribution.",
"So we do exact seeded graph matching, but only on small enough examples."
],
[
"Random graph setting: correlated Bernoulli graphs ",
"In this section we describe the correlated Bernoulli random graph distribution, and three important associated parameters/ functions of parameters; namely $\\varrho _e$ , $\\varrho _h$ , and $\\varrho _T$ .",
"For any positive integer $n$ , the distribution parameters are any given real number $\\varrho _e$ (called the edge correlation) from the interval $[0,1]$ , and any given set of real numbers $\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}$ (called the Bernoulli parameters) from the interval $[0,1]$ such that the Bernoulli parameters are not all equal to 0 and not all equal to 1.",
"Random graphs $G$ and $H$ , each on vertex set $[n]$ , will be called $\\varrho _e$ -correlated random Bernoulli$(\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}})$ graphs if, for each $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ , we have that $\\mathbb {1}(i \\sim _G j)$ is a Bernoulli$(p_{i,j})$ random variable, and $\\mathbb {1}(i \\sim _H j)$ is a Bernoulli$(p_{i,j})$ random variable, and, if $0<p_{i,j}<1$ , then the two random variables $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ have Pearson correlation coefficient $\\varrho _e$ ; other than these specified dependencies, the random variables $\\lbrace \\mathbb {1}(i \\sim _G j)\\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} \\bigcup \\lbrace \\mathbb {1}(i \\sim _H j)\\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} $ are collectively independent.",
"Such $G$ , $H$ can be realized from this distribution as follows.",
"For all $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ independently, first realize $\\mathbb {1}(i \\sim _G j)$ from the Bernoulli$(p_{i,j})$ distribution.",
"Then, conditioned on $\\mathbb {1}(i \\sim _G j)$ , realize $\\mathbb {1}(i \\sim _H j)$ from distribution Bernoulli$(\\varrho _e\\cdot \\mathbb {1}(i \\sim _G j) +(1-\\varrho _e) \\cdot p_{i,j})$ .",
"It is easy to verify that $\\mathbb {1}(i \\sim _H j)$ has a marginal distribution Bernoulli$(p_{i,j})$ and, indeed, the random variables $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ have Pearson correlation $\\varrho _e$ if $0<p_{i,j}<1$ .",
"Moreover, it easy to verify that, for any two Bernoulli$(p_{i,j})$ random variables such that $0<p_{i,j}<1$ , the Pearson correlation coefficient uniquely determines their joint distribution.",
"Also, it is easy to verify that $\\mathbb {P}[i \\sim _G j \\ \\& \\ i \\lnot \\sim _H j ]=(1-\\varrho _e)p_{i,j}(1-p_{i,j})$ .",
"See Appendix A for more of all these details.",
"The identity bijection ${\\mathcal {I}}\\in _n$ is the natural alignment between $G$ and $H$ .",
"When $\\varrho _e=1$ we have that $G,H$ are almost surely isomorphic (via isomorphism ${\\mathcal {I}}$ ), and when $\\varrho _e=0$ we have that $G$ and $H$ are independent (i.e.",
"the indicators for all edges of both graphs are collectively independent).",
"If all Bernoulli parameters $p_{i,j}$ are equal to each other then $G$ and $H$ are Erdos-Renyi random graphs.",
"Associated with the Bernoulli parameters $\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}$ , denote their mean $\\mu :=\\frac{1}{{n \\atopwithdelims ()2}} \\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} p_{i,j}$ and denote their variance $\\sigma ^2:=\\frac{1}{{n \\atopwithdelims ()2}} \\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} (p_{i,j}-\\mu )^2.$ We define the heterogeneity correlation $\\varrho _h$ $\\varrho _h:=\\frac{\\sigma ^2}{\\mu (1-\\mu )} .$ It is simple to show that $0 \\le \\varrho _h\\le 1$ .",
"Furthermore, $\\varrho _h=0$ if and only if all Bernoulli parameters $p_{i,j}$ are equal to each other (i.e.",
"$G$ and $H$ are Erdos-Renyi random graphs), and $\\varrho _h=1$ if and only if all Bernoulli parameters are 0 or 1 (but, recall, the Bernoulli parameters are not all 0 and are not all 1).",
"See Appendix B for more details.",
"Note that $\\varrho _h$ is a measure of heterogeneity within $G$ (and within $H$ ) by virtue of its numerator being the variance (a measure of spread) of the Bernoulli coefficients, although this variance is normalized through division by the denominator of $\\varrho _h$ , where this denominator is a function of the global graph density.",
"(So, among distributions with a common global density, $\\varrho _h$ is just a multiple of the variance $\\sigma ^2$ .)",
"Note that edge correlation $\\varrho _e$ is an inter-graph affect (between $G$ and $H$ ), whereas heterogeneity correlation $\\varrho _h$ is an intra-graph affect.",
"Unlike edge correlation $\\varrho _e$ , heterogeneity correlation $\\varrho _h$ is not a statistical correlation.",
"However, our results will demonstrate that $\\varrho _h$ is interchangeable with edge correlation $\\varrho _e$ with regard to creating alignment strength.",
"We thus take the liberty of calling $\\varrho _h$ “correlation,\" but we do so in a looser, nonstatistical sense, with the meaning that it generates similarity between $G$ and $H$ just like edge correlation does.",
"Finally, define the total correlation $\\varrho _T$ such that $\\varrho _T$ satisfies $1-\\varrho _T:=(1-\\varrho _e)(1-\\varrho _h).$"
],
[
"Alignment strength is total correlation, asymptotically ",
"In this section we state and prove our main theoretical result, Theorem REF , that when $G,H$ have a correlated Bernoulli distribution then the identity bijection ${\\mathcal {I}}\\in _n$ (the natural alignment here) has alignment strength asymptotically equal to the distribution's total correlation $\\varrho _T$ .",
"Let $e_G$ and $e_H$ denote the number of edges in $G$ and $H$ , respectively, and let $\\mathfrak {d}_G:=\\frac{e_G}{{n \\atopwithdelims ()2}}$ and $\\mathfrak {d}_H:=\\frac{e_H}{{n \\atopwithdelims ()2}}$ respectively denote the densities of $G$ and $H$ .",
"Lemma 1 For any graphs $G$ , $H$ on common vertex set $[n]$ , and any $\\phi \\in _n$ , it holds that $\\mathfrak {str}(G,H,\\phi )=1-\\frac{ \\frac{d(G,H,\\phi )}{{n \\atopwithdelims ()2}} }{\\mathfrak {d}_G \\left( 1-\\mathfrak {d}_H \\right)+ \\left( 1-\\mathfrak {d}_G \\right) \\mathfrak {d}_H}.$ Proof: With $G$ and $H$ fixed, consider random $\\varphi \\in _n$ with a discrete-uniform distribution; the expected value of $d(G,H,\\varphi )$ is $ \\frac{1}{n!",
"}\\sum _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime })$ .",
"We next equivalently compute the expected value of $d(G,H,\\varphi )$ using linearity of expectation over the sum of its indicators in Equation REF .",
"Observe that, for any two vertices that form an edge in $G$ , the probability that $\\varphi $ maps them to a nonedge of $H$ is $\\frac{{n \\atopwithdelims ()2}-e_H}{{n \\atopwithdelims ()2}}$ , and, for any two nonadjacent vertices of $G$ , the probability that $\\varphi $ maps them to an edge of $H$ is $\\frac{e_H}{{n \\atopwithdelims ()2}}$ ; the expected value of $d(G,H,\\varphi )$ is thus $& & \\frac{1}{n!",
"}\\sum _{\\phi ^{\\prime } \\in _n}d(G,H,\\phi ^{\\prime }) \\nonumber \\\\&=& e_G \\cdot \\frac{{n \\atopwithdelims ()2}-e_H}{{n \\atopwithdelims ()2}}+ \\left({n \\atopwithdelims ()2}-e_G \\right)\\cdot \\frac{e_H}{{n \\atopwithdelims ()2}} \\nonumber \\\\ &=& {n \\atopwithdelims ()2} \\cdot \\Big [ \\mathfrak {d}_G \\left( 1-\\mathfrak {d}_H \\right)+ \\left( 1-\\mathfrak {d}_G \\right) \\mathfrak {d}_H \\Big ].",
"$ The desired result then follows from substituting Equation REF into the definition of $\\mathfrak {str}(G,H,\\phi )$ in Equation REF .",
"$\\Box $ In the rest of this section we will state and prove limit results for random correlated Bernoulli graphs $G$ , $H$ .",
"This context requires us to consider a sequence of experiments —for each value of $n=1,2,3,\\ldots $ —wherein the chosen edge correlation $\\varrho _e$ is a function of $n$ , and the chosen Bernoulli parameters $\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}$ are also functions of $n$ , and thus $\\varrho _h$ and $\\varrho _T$ are also functions of $n$ .",
"For ease of notation, we do not explicitly write argument $n$ in these functions.",
"However, we will require that there exists a positive lower bound for $\\mu $ over all $n$ , and as well that there exists an upper bound less than 1 for $\\mu $ over all $n$ .",
"(Note that since $\\mu $ is a function of $n$ , we have that the $\\mu $ are a sequence, so the following limit result is expressed as a difference that converges as stated, rather than convergence to $\\mu $ , which would not make technical sense.",
"Similarly for the other results here.)",
"Lemma 2 We have $ \\mathfrak {d}_G -\\mu \\stackrel{a.s.}{\\rightarrow } 0$ and $ \\mathfrak {d}_H - \\mu \\stackrel{a.s.}{\\rightarrow } 0$ .",
"Proof: Clearly $\\mathbb {E}(\\mathfrak {d}_G) =\\mu $ .",
"Also, $e_G$ is the sum of ${n \\atopwithdelims ()2}$ independent Bernoulli random variables, and thus its variance is bounded by ${n \\atopwithdelims ()2}$ , thus the variance of $\\mathfrak {d}_G:=\\frac{e_G}{{n \\atopwithdelims ()2}}$ is of order O$(n^{-2})$ .",
"Next, by Chebyshev's Inequality, for any $\\epsilon >0$ , $\\mathbb {P}\\left[ \\left| \\mathfrak {d}_G- \\mu \\right| \\ge \\epsilon \\right]\\le \\frac{1}{\\epsilon ^2} \\textup {Var} \\left( \\mathfrak {d}_G \\right)$ ; since this probability is O$(n^{-2})$ when $\\epsilon $ is fixed, it has finite sum over $n=1,2,3,\\ldots $ .",
"Thus, since $\\epsilon $ is arbitrary, by the Borel-Cantelli Theorem $ \\mathfrak {d}_G -\\mu \\stackrel{a.s.}{\\rightarrow } 0$ , as desired.",
"$\\Box $ Theorem 3 We have $\\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} - 2 (1-\\varrho _e) \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big ) \\stackrel{a.s.}{\\rightarrow } 0 $ Proof: We begin by taking the expected value of $d(G,H,{\\mathcal {I}})$ ; $& & \\mathbb {E}\\Big [ d(G,H,{\\mathcal {I}}) \\Big ] \\nonumber \\\\ & = & \\mathbb {E}\\left[ \\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} \\mathbb {1}\\Big ( \\ \\mathbb {1}\\big (i \\sim _G j \\big )\\ \\ne \\ \\mathbb {1}\\big (i \\sim _H j \\big ) \\ \\Big ) \\right] \\nonumber \\\\& = & \\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} 2 (1-\\varrho _e) p_{i,j}(1-p_{i,j})\\nonumber \\\\ &=& 2 (1-\\varrho _e) {n \\atopwithdelims ()2} \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big ), $ thus $\\mathbb {E}\\Big [ \\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} \\Big ]= 2 (1-\\varrho _e) \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big )$ .",
"Next, $d(G,H,{\\mathcal {I}})$ is is the sum of ${n \\atopwithdelims ()2}$ independent Bernoulli random variables, and thus its variance is bounded by ${n \\atopwithdelims ()2}$ , thus the variance of $\\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}}$ is of order O$(n^{-2})$ .",
"Next, by Chebyshev's Inequality, for any $\\epsilon >0$ , $\\mathbb {P}\\left[ \\left| \\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} - 2 (1-\\varrho _e) \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big ) \\right| \\ge \\epsilon \\right]\\le \\frac{1}{\\epsilon ^2} \\textup {Var} \\left( \\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} \\right)$ ; since this probability is O$(n^{-2})$ when $\\epsilon $ is fixed, it has finite sum over $n=1,2,3,\\ldots $ .",
"Thus, since $\\epsilon $ is arbitrary, by the Borel-Cantelli Theorem $\\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} - 2 (1-\\varrho _e)\\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big ) \\stackrel{a.s.}{\\rightarrow } 0 $ , as desired.",
"$\\Box $ The following is the main result of this section, and is our main theoretical result.",
"Theorem 4 It holds that $\\mathfrak {str}(G,H, {\\mathcal {I}}) -\\varrho _T\\stackrel{a.s.}{\\rightarrow } 0 $ Proof: By Lemma REF , $ \\mathfrak {d}_G -\\mu \\stackrel{a.s.}{\\rightarrow } 0$ and $ \\mathfrak {d}_H - \\mu \\stackrel{a.s.}{\\rightarrow } 0$ .",
"Because $\\mathfrak {d}_G$ , $\\mathfrak {d}_H$ and $\\mu $ are bounded, we thus have that $\\mathfrak {d}_G \\left( 1-\\mathfrak {d}_H \\right)+ \\left( 1-\\mathfrak {d}_G \\right) \\mathfrak {d}_H - 2 \\mu (1-\\mu ) \\stackrel{a.s.}{\\rightarrow } 0$ .",
"Now, by Theorem REF , we have that $\\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} - 2 (1-\\varrho _e) \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big ) \\stackrel{a.s.}{\\rightarrow } 0$ ; since the relevant sequences are bounded, and $\\mu $ is bounded away from 0 and 1, we have that $\\frac{ \\frac{d(G,H,{\\mathcal {I}})}{{n \\atopwithdelims ()2}} }{\\mathfrak {d}_G \\left( 1-\\mathfrak {d}_H \\right)+ \\left( 1-\\mathfrak {d}_G \\right) \\mathfrak {d}_H } \\ - \\ \\frac{ 2 (1-\\varrho _e) \\Big ( \\mu (1-\\mu ) - \\sigma ^2 \\Big )}{2 \\mu (1-\\mu )}\\stackrel{a.s.}{\\rightarrow } 0.",
"$ Applying Lemma REF and the definitions of $\\varrho _h$ and $\\varrho _T$ we thus have from above that $(1-\\mathfrak {str}(G,H, {\\mathcal {I}}) )- (1 - \\varrho _T) \\stackrel{a.s.}{\\rightarrow } 0 $ , which proves Theorem REF .",
"$\\Box $"
],
[
"Graph matchability and total correlation $\\varrho _T$ ",
"In this section we empirically demonstrate in broad families of parameter settings where $\\varrho _e$ and $\\varrho _h$ vary, that success of an approximate seeded graph matching algorithm is a function of $\\varrho _T$ .",
"Our setting is where $G$ , $H$ are correlated Bernoulli graphs on vertex set $[n]$ .",
"The graph matching problem is to compute $\\phi _{GM} \\in \\arg \\min _{\\phi \\in _n}d(G,H,\\phi )$ .",
"In the seeded graph matching problem, there are $s$ seeds, without loss of generality they are the vertices $1,2,\\ldots ,s$ , and there are $m:=n-s$ ambiguous vertices, which are the other vertices $s+1,s+2, \\ldots , n$ .",
"The meaning of seeded graph matching is that the feasible region $\\phi \\in _n$ of the graph matching problem is restricted to $\\phi \\in _n$ that satisfy $\\phi (i)=i$ for all seeds $i=1,2,\\ldots ,s$ .",
"The graphs $G$ and $H$ are separately observed and the identities of the ambiguous vertices are unobserved for the optimization, so that the natural alignment, which is the identity bijection ${\\mathcal {I}}$ , is only seen for the seeds.",
"If the seeded graph matching solution is ${\\mathcal {I}}$ then we say that $G$ and $H$ are matchable.",
"Even a modest number of seeds can make a very significant increase in the likelihood that $G$ and $H$ are matchable [[15]].",
"Our illustration in this section will be for realistically sized graphs, on the order of a thousand vertices, and we utilize seeds because they will be quite helpful in obtaining reasonable probability of matchability.",
"Unfortunately, exact graph matching –even seeded graph matching– is intractable, only solvable on the smallest, toy examples.",
"So we utilize an approximate seeded graph matching algorithm; the specific one we use is the SGM Algorithm [[8], [15]], which has been demonstrated to have many nice theoretical properties, and it is efficient and quite effective (see [8], [15], [16]).",
"In this section, we will say that $G$ and $H$ are matchable if the SGM-generated approximate seeded graph matching solution is the identity bijection ${\\mathcal {I}}$ .",
"In the experiments that we will perform, we will sample $G$ , $H$ from a correlated Bernoulli distribution for different values of $\\varrho _e$ and $\\varrho _h$ ; the values of the Bernoulli coefficients $\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}$ are selected as follows, in order to obtain specified values of $\\varrho _h$ .",
"Given any real number $p \\in (0,1)$ and real number $\\delta \\in [0, \\min \\lbrace p, 1-p \\rbrace ]$ , we independently randomly sample $\\lbrace p_{i,j} \\rbrace _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}$ from the uniform distribution on the interval $(p-\\delta , p+\\delta )$ .",
"Note that the afore-defined Bernoulli parameter variance $\\sigma ^2$ has expected value $\\frac{\\delta ^2}{3}$ , and $\\sigma ^2$ will be approximately $\\frac{\\delta ^2}{3}$ for large values of $n$ .",
"For a fixed $p$ , as $\\delta $ goes from 0 to $\\min \\lbrace p, 1-p \\rbrace $ , the value of $\\varrho _h=\\frac{\\sigma ^2}{\\mu (1-\\mu )} \\approx \\frac{\\delta ^2}{3p(1-p)} $ monotonically increases from 0 to $\\frac{1}{3} \\cdot \\frac{1-p}{p}$ if $p \\ge \\frac{1}{2}$ and $\\frac{1}{3} \\cdot \\frac{p}{1-p}$ if $p \\le \\frac{1}{2}$ .",
"In this section, when we report values of $\\varrho _e$ and $\\varrho _h$ , we mean that we selected $\\delta $ so that the approximate value of $\\varrho _h$ is as reported.",
"We did three batches of experiments.",
"In the first batch of experiments, for each value of $\\varrho _e= 0, \\frac{1}{120},\\frac{2}{120}, \\frac{3}{120}, \\ldots , \\frac{1}{3}$ and $\\varrho _h= 0, \\frac{1}{120},\\frac{2}{120}, \\frac{3}{120}, \\ldots , \\frac{1}{3}$ , we did 60 replicates of obtaining random graphs $G$ , $H$ with $m=850$ ambiguous vertices and $s=150$ seeds from a correlated Bernoulli distribution with edge correlation $\\varrho _e$ and heterogeneity correlation $\\varrho _h$ based on $p=\\frac{1}{2}$ , and we performed seeded graph matching with the SGM algorithm.",
"If all 60 replicates were matchable then we plotted a green dot in Figure REF at the appropriate coordinates, if between 1 and 5 of the 60 replicates were not matchable then we plotted a yellow dot in the figure, and if more than 5 of the 60 replicates were not matchable then we plotted a red dot.",
"The blue curve in the figure is the set of all pairs of $\\varrho _e$ , $\\varrho _h$ such that $\\varrho _T=\\frac{23}{120}$ .",
"In these experiments and those below, the transition from matchable to anonymized (i.e., not matchable) occurs at a level set of $\\varrho _T$ .",
"We note here that numerous results in the literature have studied this matchability phase transition as a function of edge correlation $\\varrho _e$ (see, for example, [[6], [5], [15]]) and a few papers have considered the impact of network heterogeneity on matchability (see, for example, [[14], [18]]).",
"In the parameterized correlated Bernoulli distribution considered above, these empirical results novelly suggest the form by which matchability is impacted by within and across graph correlation structure.",
"Further understanding this phase transition as a function of $\\varrho _T$ is a necessary next step to understand the dual roles that graph structure ($\\varrho _h$ ) and graph pairedness ($\\varrho _e$ ) play in network alignment problems both theoretical and practical.",
"Figure: Matchability experiment for m=850m=850, s=150s=150, p=1 2p=\\frac{1}{2}.The second batch of experiments differed just in that there were only $s=9$ seeds (with $m=850$ as before), and the range of values of $\\varrho _e$ was $\\frac{1}{3}$ to $\\frac{5}{6}$ in increments of $\\frac{1}{120}$ ; the results are similarly displayed in Figure REF , and the blue curve in the figure is the set of all pairs of $\\varrho _e$ , $\\varrho _h$ such that $\\varrho _T=\\frac{69}{120}$ .",
"In these experiments, we again see the transition in matchability at a level set of $\\varrho _T$ , although the transition is looser due to fewer seeds being considered in this problem setup.",
"Figure: Matchability experiment for m=850m=850, s=9s=9, p=1 2p=\\frac{1}{2}The third batch of experiments differed just in that there were $s=22$ seeds, and now $p=\\frac{1}{3}$ , the range of values of $\\varrho _e$ was $\\frac{1}{4}$ to $\\frac{7}{12}$ in increments of $\\frac{1}{120}$ , and the range of values of $\\varrho _h$ was 0 to $\\frac{1}{6}$ in increments of $\\frac{1}{120}$ ; the results are similarly displayed in Figure REF , and the blue curve in the figure is the set of all pairs of $\\varrho _e$ , $\\varrho _h$ such that $\\varrho _T=\\frac{49}{120}$ .",
"In these experiments, we again see the transition in matchability at a level set of $\\varrho _T$ .",
"Figure: Matchability experiment for m=850m=850, s=22s=22, p=1 3p=\\frac{1}{3}We then repeated the above experiments for each combination of: total number of vertices 300 or 600, number of seeds seeds $5\\%$ or $10\\%$ of the vertices, and values of $p$ being $\\frac{1}{2}$ or $\\frac{1}{3}$ .",
"In all eight such combinations the result of the experiments were like the above; namely, matchability was a function of $\\varrho _T$ .",
"Note that matchability is not universally a function of just $\\varrho _T$ .",
"For example, the number of vertices and the number of seeds have a dramatic affect on matchability.",
"The empirical demonstrations in this section of matchability as a function of $\\varrho _T$ are limited to families of correlated Bernoulli distribution parameterizations of the type that we have used here.",
"New work will be needed to obtain theorems that universally and fully account for matchability.",
"But, nonetheless, we have empirically demonstrated in broad families of parameter settings that the phase transition in matchability occurs at a level set of $\\varrho _T$ , which supports the importance and utility of $\\varrho _T$ as a meaningful measure of graph correlation."
],
[
"Graph matching runtime and total correlation $\\varrho _T$ ",
"Similar to the previous section, in this section we empirically demonstrate, in broad families of parameter settings where $\\varrho _e$ and $\\varrho _h$ vary, that the running time of exact seeded graph matching via binary integer linear programming is a function of $\\varrho _T$ .",
"We consider exact seeded graph matching here because the approximate seeded graph matching algorithms have running times that are relatively monolithic (when the number of vertices are fixed) and not sensitive enough to the parameters in the random graph distribution.",
"Unfortunately, exact graph matching is intractable [[4]], and can only be done for small examples; we will work with graphs that have 20 ambiguous vertices.",
"For this section, the random graphs $G$ ,$H$ have correlated Bernoulli distributions, for various values of $\\varrho _e$ and $\\varrho _h$ .",
"The Bernoulli parameters are chosen in exactly the manner of the previous section, Section ; there is a fixed value $p$ , and then $\\delta $ are selected to attain desired values of $\\varrho _h$ in the manner described in the previous section.",
"We next formulate the binary integer linear program for seeded graph matching.",
"For graphs $G$ and $H$ , say their adjacency matrices are $A$ and $B$ , respectively, and say that there are $s$ seeds and $m$ ambiguous vertices.",
"We partition $A = \\bigl [{\\begin{matrix} A_{11} & A_{12} \\\\ A_{21} & A_{22} \\end{matrix}} \\bigr ]$ and $B = \\bigl [{\\begin{matrix} B_{11} & B_{12} \\\\ B_{21} & B_{22} \\end{matrix}} \\bigr ]$ , where $A_{11},B_{11}~\\in ~\\lbrace 0,1 \\rbrace ^{s \\times s}$ , $A_{12},B_{12}~\\in ~\\lbrace 0,1 \\rbrace ^{s \\times m}$ , $A_{21},B_{21}~\\in ~\\lbrace 0,1 \\rbrace ^{m \\times s}$ , and $A_{22},B_{22} \\in \\lbrace 0,1 \\rbrace ^{m \\times m}$ .",
"(Note that $A_{12}=A_{21}^T$ and $B_{12}=B_{21}^T$ here, since $A$ and $B$ are symmetric, but we do not use this fact in the formulation below so that the formulation is expressed even more generally.)",
"Let $I$ denote the identity matrix (subscripted with its number of rows and columns), let 0 subscripted denote the matrix of zeros of subscripted size, let $\\vec{1}$ denote the column vector of ones with subscripted number of entries, let $\\vec{0}$ denote the column vector of zeros with subscripted number of entries, let $\\otimes $ denote the Kronecker product of matrices, let $\\Vert \\cdot \\Vert _1$ denote the $\\ell _1$ vector norm for matrices (this norm is evaluated by taking the sum of absolute values of the matrix entries), for any matrix $N$ let $\\textup {vec}N$ denote the column vector which is the concatenation of the columns of $N$ (first column of $N$ , then second column of $N$ , etc., then last column of $N$ ), and let ${\\mathcal {P}}_m$ denote the set of $m \\times m$ permutation matrices.",
"Clearly, the seeded graph matching problem is $\\min _{P \\in {\\mathcal {P}}_m}\\Vert A - \\bigl [{\\begin{matrix} I_{s \\times s} & 0_{s \\times m} \\\\ 0_{m \\times s} & P \\end{matrix}} \\bigr ]B \\bigl [{\\begin{matrix} I_{s \\times s} & 0_{s \\times m} \\\\ 0_{m \\times s} & P \\end{matrix}} \\bigr ]^T \\Vert _1$ .",
"By permuting columns of the matrix in the norm, we get an equivalent formulation of the seeded graph matching problem as: $\\min _{P \\in {\\mathcal {P}}_m}\\Vert A \\bigl [{\\begin{matrix} I_{s \\times s} & 0_{s \\times m} \\\\ 0_{m \\times s} & P \\end{matrix}} \\bigr ]- \\bigl [{\\begin{matrix} I_{s \\times s} & 0_{s \\times m} \\\\ 0_{m \\times s} & P \\end{matrix}} \\bigr ] B \\Vert _1.$ Expanding this, we get an equivalent formulation of the seeded graph matching problem as: $ \\hspace{-28.90755pt}\\min _{P \\in {\\mathcal {P}}_m}\\Big ( \\Vert A_{12}P-B_{12} \\Vert _1 + \\Vert A_{21}-PB_{21}\\Vert _1 + \\Vert A_{22}P-PB_{22} \\Vert _1 \\Big ) .$ Now, because of the absolute values in $\\Vert \\cdot \\Vert _1$ , we add artificial variables to obtain simple linearity.",
"For example, (just) minimizing $\\Vert A_{22}P-PB_{22} \\Vert _1$ subject to $P \\in {\\mathcal {P}}_m$ is equivalent to minimizing the sum of the entries of $E+E^{\\prime }$ subject to $A_{22}P-PB_{22}+E-E^{\\prime }=0_{m \\times m}$ , $P \\in {\\mathcal {P}}_m$ , $E,E^{\\prime } \\in \\lbrace 0,1 \\rbrace ^{m \\times m}$ .",
"Of course, there are additional $\\Vert \\cdot \\Vert _1$ terms in the objective function in Equation REF , but the same approach can be used, so that seeded graph matching is equivalent to $\\min \\, \\, &\\bigl [{\\begin{matrix} \\vec{0}_{m^2} \\\\\\vec{1}_{2m^2+4ms} \\end{matrix}} \\bigr ]^Tx\\\\\\text{ s.t.",
"}&[M|E] x=b\\\\&x \\in \\lbrace 0,1 \\rbrace ^{3m^2+4ms}$ where the first $m^2$ entries of $x$ are $\\textup {vec}P$ , and $M$ and $E$ and $b$ are given by: $M=\\left[ \\begin{array}{c}I_{m \\times m} \\otimes A_{22} - B_{22}^T \\otimes I_{m \\times m} \\\\I_{m \\times m} \\otimes A_{12} \\\\ B_{21}^T \\otimes I_{m \\times m} \\\\I_{m \\times m} \\otimes \\vec{1}_m^T \\\\ \\vec{1}_m^T \\otimes I_{m \\times m}\\end{array} \\right]$ $E=\\left[ \\begin{array}{cc}I_{(m^2+2ms) \\times (m^2+2ms)} & -I_{(m^2+2ms) \\times (m^2+2ms)} \\\\0_{ 2m \\times (m^2+2ms)} & 0_{2m \\times (m^2+2ms)}\\end{array} \\right]$ $b=\\left[ \\begin{array}{c} \\vec{0}_{m^2} \\\\ \\textup {vec}B_{12} \\\\ \\textup {vec}A_{21} \\\\ \\vec{1}_m \\\\ \\vec{1}_m \\end{array} \\right]$ We solve the above binary integer linear program exactly using the optimization package GUROBI.",
"The yardstick for runtime that we have chosen to adopt is the number of simplex iterations performed by GUROBI; this measure has the advantage of reducing many sources of platform variability.",
"Figure: Runtime experiment for m=20m=20, s=480s=480, p=1 2p=\\frac{1}{2}.Figure: Runtime experiment for m=20m=20, s=480s=480, p=3 5p=\\frac{3}{5}.Figure: Runtime experiment for m=20m=20, s=480s=480, p=1 3p=\\frac{1}{3}.We performed three batches of experiments.",
"In the first batch of experiments, for each value of $\\varrho _T= \\frac{2}{9},\\frac{3}{9}, \\frac{4}{9}, \\ldots , \\frac{8}{9}$ , we selected various pairs of $\\varrho _e$ , $\\varrho _h$ which have $1-\\varrho _T=(1-\\varrho _e)(1-\\varrho _h)$ for the given value of $\\varrho _T$ ; the values of $\\varrho _h$ are achieved based on $p=\\frac{1}{2}$ , and the chosen pairs $\\varrho _e$ , $\\varrho _h$ are the points plotted with a dot in Figure REF .",
"For each such pair $\\varrho _e$ , $\\varrho _h$ we did 60 replicates of obtaining random graphs $G$ , $H$ with $m=20$ ambiguous vertices and $s=480$ seeds from a correlated Bernoulli distribution with edge correlation $\\varrho _e$ and heterogeneity correlation $\\varrho _h$ , and we solved the seeded graph matching problem for $G$ , $H$ exactly using GUROBI.",
"The average runtimes (measured by the number of simplex iterations performed by GUROBI) are printed above each pair $\\varrho _e$ , $\\varrho _h$ at the appropriate coordinates in Figure REF .",
"The smooth curves on the plot are the level sets of $\\varrho _T$ .",
"These experiments, and those below, suggest that in this parametrized Bernoulli graph model the algorithmic runtimes are approximately constant on the level sets of $\\varrho _T$ .",
"The results in Section suggest that the phase transition of matchability occurs at a level set of $\\varrho _T$ , and these results further reinforce the novel overarching notion: that the theoretic and algorithmic difficulty of matching is a function of $\\varrho _e$ and $\\varrho _h$ only through $\\varrho _T$ .",
"Alone, $\\varrho _e$ and $\\varrho _h$ are insufficient to capture this theoretic and algorithmic difficulty.",
"The second and third batch of experiments are exactly like the first batch, except that for the second batch of experiments the values of $\\varrho _h$ are based on $p=\\frac{3}{5}$ and the results are displayed in Figure REF , and for the third batch of experiments the values of $\\varrho _h$ are based on $p=\\frac{1}{3}$ and the results are displayed in Figure REF .",
"Note that the ranges of $\\varrho _h$ are different in Figures REF , REF , and REF because different values of $p$ put different limitations on $\\delta $ .",
"Just like for matchability in the previous section, it must be pointed out that the runtime of exact seeded graph matching via binary integer linear programming is not universally a function of $\\varrho _T$ .",
"Of course, the number of vertices —particularly the number of ambiguous vertices— has a dominant role in the runtime, and the above experiments show that the graph density likewise plays a very large role.",
"Nonetheless, for families of correlated Bernoulli distributed graphs similar to the ones in the experiments above, we see within a family that the runtime is a function of $\\varrho _T$ ."
],
[
"Discussion and future work ",
"The correlated Bernoulli random graph model considered herein contains many important families of random graph models as subfamilies including stochastic blockmodels [[11], [1]], random dot product graphs [[27], [2]], and more general latent position random graph [[10]].",
"While the edge independent assumption inherent to these models is often not satisfied in real data applications, nonetheless (conditionally) edge-independent random models have shown great utility in capturing statistically relevant structure in a host of real data applications from modeling connectomic structure [[20], [17], [13]], to capturing community and user-level behavior in social networks [[26], [19]].",
"Moreover, these models provide a theoretically tractable environment in which to explore important statistical concepts such as estimation consistency [[3], [21], [22]], consistent hypothesis testing [[24], [23], [12]], and network de-anonymization [[7], [6]], among others.",
"Indeed, it is this appealing mix of theoretical tractability and practical utility that have made these graph models an increasingly popular option in the statistical network inference community.",
"In this paper we prove in a very broad random graph setting —specifically, when $G$ and $H$ have a correlated Bernoulli distribution— that the alignment strength of the natural $G$ , $H$ alignment is asymptotically equal to the total correlation $\\varrho _T$ in the distribution.",
"After this, we empirically demonstrate, for types of families within the distribution, that both matchability and exact-solution-runtime for seeded graph matching of $G$ , $H$ are functions of the total correlation $\\varrho _T$ .",
"Graph matching and seeded graph matching are extremely important in many disciplines; see the surveys [[4]] and [[9]].",
"Unfortunately, these problems are intractable; in their full generality they are NP-hard.",
"Obtaining a function of the distribution parameters that universally predicts matchability via approximate algorithms would be a huge advance in theoretical understanding and in practice.",
"Likewise, it would be a huge advance to predict exact-solution-runtime from a function of the distribution parameters, and it would not just be the number of vertices—the other parameters play a large role.",
"The goals of obtaining these universal functions has not been achieved here; the families we use here are general but not universal.",
"But a universal result will include our families as special cases, thus $\\varrho _T$ will play an important role.",
"There are a number of matchability results already known, see [[6], [5], [14], [15], [16], [18]].",
"However, for the most part these are asymptotic results that do not specify the particular constants involved, and leave gaps in the parameter possibilities where the results are silent.",
"In particular, the empirical matchability demonstrations in this paper are not predictable from the previously known matchability asymptotics.",
"Many of the known matchability results explicitly or implicitly involve edge correlation $\\varrho _e$ .",
"The formulation of $\\varrho _h$ is new to this paper, and $\\varrho _T$ is also new to this paper.",
"Thus we are now opening a fertile new avenue for proof-of-matchability results based on $\\varrho _h$ and $\\varrho _T$ , in the spirit of the existing results for $\\varrho _e$ and also for more powerful types of results."
],
[
"Acknowledgments",
"The authors are grateful to The Maryland Advanced Research Computing Center for use of their supercomputer to conduct the computational experiments.",
"An anonymous contributor made a very useful observation which led to streamlining the main result's proof.",
"The referees' and editor's feedback and remarks greatly strengthened this article, and are very much appreciated.",
"Our research was sponsored by the Air Force Research Laboratory and DARPA, under agreement numbers FA8750-18-2-0035 and FA8750-17-2-0112.",
"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.The views and conclusions contained herein are those of the authors and should not be interpreted as representing official policies or endorsements, expressed or implied, of Air Force Research Laboratory, DARPA, or the U.S. Government.",
"We here provide some details about correlated Bernoulli random graphs.",
"Notation here is as defined in the article.",
"Section A: For any $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ such that $0<p_{i,j}<1$ , suppose that $\\mathbb {1}(i \\sim _G j)$ is a Bernoulli$(p_{i,j})$ random variable and $\\mathbb {1}(i \\sim _H j)$ is a Bernoulli$(p_{i,j})$ random variable, and suppose that the two random variables $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ have Pearson correlation coefficient $\\varrho _e$ ; we derive the joint distribution of $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ as follows: $\\varrho _e& = & \\frac{\\textup {Cov}\\left[ \\mathbb {1}(i \\sim _G j),\\mathbb {1}(i \\sim _H j) \\right]}{\\sqrt{\\textup {Var}[\\mathbb {1}(i \\sim _G j)]},\\sqrt{\\textup {Var}[\\mathbb {1}(i \\sim _H j)]}}\\\\& = & \\frac{ \\mathbb {E}[\\mathbb {1}(i \\sim _G j)\\mathbb {1}(i \\sim _H j)]- \\mathbb {E}[\\mathbb {1}(i \\sim _G j )] \\cdot \\mathbb {E}[\\mathbb {1}(i \\sim _H j)] }{ \\sqrt{p_{i,j}(1-p_{i,j})} \\sqrt{p_{i,j}(1-p_{i,j})} }\\\\& = & \\frac{\\mathbb {P}[ i \\sim _G j \\ \\& \\ i \\sim _H j ] -p_{i,j}^2}{p_{i,j}(1-p_{i,j})},$ from which we obtain $\\mathbb {P}[ i \\sim _G j \\ \\& \\ i \\sim _H j ]= p_{i,j}^2 + \\varrho _ep_{i,j}(1-p_{i,j}) $ .",
"Because $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ are each marginally Bernoulli$(p_{i,j})$ , we obtain that $\\mathbb {P}[ i \\sim _G j \\ \\& \\ i \\lnot \\sim _H j ]=\\mathbb {P}[ i \\lnot \\sim _G j \\ \\& \\ i \\sim _H j ]=p_{i,j}-\\Big ( p_{i,j}^2 + \\varrho _ep_{i,j}(1-p_{i,j}) \\Big ) = (1-\\varrho _e)p_{i,j}(1-p_{i,j})$ , and also that $\\mathbb {P}[ i \\lnot \\sim _G j \\ \\& \\ i \\lnot \\sim _H j ]= (1-p_{i,j})- (1-\\varrho _e)p_{i,j}(1-p_{i,j})=(1-p_{i,j})^2 + \\varrho _ep_{i,j}(1-p_{i,j})$ .",
"Importantly, note that the joint distribution of $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ is uniquely determined by $\\varrho _e$ .",
"Also note that $\\mathbb {P}[ \\mathbb {1}(i \\sim _G j) \\ne \\mathbb {1}(i \\sim _H j) ]=2(1-\\varrho _e)p_{i,j}(1-p_{i,j})$ .",
"Also note that, conditioned on $\\mathbb {1}(i \\sim _G j)$ , the random variable Bernoulli$(\\varrho _e\\cdot \\mathbb {1}(i \\sim _G j) +(1-\\varrho _e) \\cdot p_{i,j})$ results in the joint distribution above, which justifies the method in the article of sampling $\\mathbb {1}(i \\sim _G j)$ and $\\mathbb {1}(i \\sim _H j)$ with marginal Bernoulli$(p_{i,j})$ distribution and Pearson correlation coefficient $\\varrho _e$ .",
"$\\Box $ Section B: We show that $\\varrho _h\\le 1$ , with equality holding if and only if, for all $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ , it holds that $p_{i,j}$ is 0 or 1.",
"Indeed, $& & 1-\\varrho _h\\\\& = & 1-\\frac{\\sigma ^2}{\\mu (1-\\mu )}\\\\& = & \\frac{ \\mu (1-\\mu ) - \\left( \\frac{ \\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}} p_{i,j}^2 }{{n \\atopwithdelims ()2}} -\\mu ^2 \\right) }{\\mu (1-\\mu )}\\\\& = & \\frac{\\sum _{\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}}(p_{i,j}-p_{i,j}^2) }{{n \\atopwithdelims ()2}\\mu (1-\\mu )}$ is clearly nonnegative and equals 0 if and only if, for all $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ it holds that $p_{i,j}=p_{i.j}^2$ , i.e.",
"it holds that $p_{i,j}$ is 0 or 1.",
"Thus $\\varrho _h\\le 1$ with equality holding if and only if, for all $\\lbrace i,j \\rbrace \\in {[n] \\atopwithdelims ()2}$ , it holds that $p_{i,j}$ is 0 or 1.",
"(Except, recall, the Bernoulli parameters are not all 0 and are not all 1, since $\\varrho _h$ would then not be defined.)",
"$\\Box $"
]
] | 1808.08502 |
[
[
"Trustworthy and Privacy-Aware Sensing for Internet of Things"
],
[
"Abstract The Internet of Things (IoT) is considered as the key enabling technology for smart services.",
"Security and privacy are particularly open challenges for IoT applications due to the widespread use of commodity devices.",
"This work introduces two hardware-based lightweight security mechanisms to ensure sensed data trustworthiness (i.e., sensed data protection and sensor node protection) and usage privacy of the sensors (i.e., privacy-aware reporting of the sensed data) for centralized and decentralized IoT applications.",
"Physically unclonable functions (PUF) form the basis of both proposed mechanisms.",
"To demonstrate the feasibility of our PUF-based approach, we have implemented and evaluated PUFs on three platforms (Atmel 8-bit MCU, ARM Cortex M4 32 bit MCU, and Zynq7010 SoC) with varying complexities.",
"We have also implemented our trusted sensing and privacy-aware reporting scheme (for centralized applications) and secure node scheme (for decentralized applications) on a visual sensor node comprising an OV5642 image sensor and a Zynq7010 SoC.",
"Our experimental evaluation shows a low overhead wrt.~latency, storage, hardware, and communication incurred by our security mechanisms."
],
[
"Introduction",
"The future smart world involves a living where people will be automatically and collaboratively served by smart devices and smart spaces interconnected via the Internet of Things (IoT).",
"IoT applications collect data from various data sources.",
"This data is used for intelligence extraction using machine learning models deployed on cloud and/or edge computing infrastructure.",
"The actionable insight obtained from the intelligence extraction is offered as service to end users but also provides resource efficiency, data knowledge and automated decision-making processes to enterprises.",
"Numerous smart services are being conceptualized, researched, prototyped, tested and commercially used today.",
"For instance, smartphones embedded with rich sensing capabilities have enabled navigation [1], m-commerce [2], natural-disaster detection and warning systems [3], environmental monitoring [4], and citizen journalism [5].",
"With wearable health sensors it is now possible to monitor the blood sugar level and heart pace [6], provide assisted living for elderly patients with chronic diseases [7], and document daily sports activities of individuals [8].",
"Smart vehicles have enabled autonomous driving [9], cooperative collision avoidance [10], remote wireless diagnosis of vehicles [11], and traffic flow optimization [12].",
"Likewise, smart homes [13], and safe cities [14] are enabling smart spaces that are intelligent, resource efficient, and secure.",
"Figure: A generic infrastructure for IoT applicationsThe physical infrastructure of today's IoT applications, as depicted in Fig.",
"REF , can be divided into three tiers: data source, edge computation, and cloud computation.",
"The data source tier includes everything that generates data.",
"Sensors are the largest and the most common source of data in IoT applications.",
"Other sources include RFIDs, machine logs, social media feeds and event sources.",
"The edge computing tier comprises host devices and micro data centers, which are responsible for running data processing pipelines, handling network switching, routing, load balancing and security.",
"A host device can either be a commodity device with computing, storage and communication resources such as a smartphone, a computer, a gateway router, a smart vehicle ECU or a processing platform solely dedicated to the attached data sources.",
"Micro data centers host virtualization infrastructure which runs cloud services closer to the data sources.",
"These data centers are distributively located, for example at cellular base-station sites.",
"The cloud computing tier comprises a centralized pool of computing, storage and communication resources, which offers data management, analytics, software or hardware platform or combination of these as-a-Service (aaS).",
"The infrastructure of Fig.",
"REF encompasses three layers of abstraction: technology, middleware, and application layers [15].",
"The technology layer is comprised of sensing, identification, computing and communication resources.",
"The middleware is a software layer or a set of sub-layers, that resides between the technology and the application layer.",
"The middleware hides the technology-level details from the application programmers thereby simplifying the application development process.",
"The application layer is the top most layer exporting all the system's functionality to the end-users by exploiting the functionality of the middleware, standard web interfaces, and protocols."
],
[
"Security and Privacy Threats",
"This work addresses two security threats for IoT applications: sensed data pollution and personal privacy leakage.",
"Sensed data pollution is a major threat posed to IoT applications whereby malicious users or a third-party adversaries contribute manipulated or fabricated sensed data to pollute the application database [16].",
"An adversary can exploit a number of security vulnerabilities present in the infrastructure of Fig.",
"REF to mount this attack.",
"First, the data source layer is mainly comprised of commodity devices embedded with a multitude of resource constrained sensors connected to a host processor.",
"Sensitive data captured by these sensors do not carry any security guarantees.",
"These sensors rely on a resource-rich host device for processing and reporting the sensed data to a micro data center (or a cloud server).",
"Host devices are the commodity devices running a thick, vulnerable software stack [17], [16].",
"As a result, today it is trivial to manipulate or fabricate sensors' readings in these applications by exploiting bugs (e.g., Master Key [18] and Fake ID [19]) in the software stack (e.g., OS) running on the commodity devices.",
"For instance, location readings from a smartphone GPS sensor can be modified to obtain illegitimate access to a location-based service [20], video frames from a surveillance camera can be manipulated to hide or fake an event [21], and patient's blood sugar level measured by wearable sensors can be manipulated to stop the insulin pump [20].",
"Second, the use of public infrastructure for communications (Internet) and storage (public cloud storage servers) further increases the threat surface area of these applications.",
"Third, due to the open and ubiquitous nature of the infrastructure, certain elements such as sensors may not be protected against physical attacks.",
"Physically damaged sensors are another potential source of data pollution attacks.",
"Consequently, any service based on this data lacks trust.",
"The second threat addressed by this work is leakage of personal privacy.",
"The IoT applications collect and process information from almost every aspect of our daily lives, e.g., our private data (photos, medical reports), our routines, habits and preferences (transportation, shopping, political and religious views), our critical infrastructures (energy, emergency systems).",
"By linking individual data points obtained from wearable, personal devices (e.g., smartphones or cars), and private spaces (e.g., home security, assisted living or baby monitoring applications) one can construct the personal profiles of the individuals revealing their sensitive personal information such as home and workplace, contact details, health-status, religious orientation, political affiliations, current and future locations etc."
],
[
"Contributions",
"The goals of this work are twofold: First, we identify and apply security mechanisms to ensure effective and verifiable trustworthiness of the sensed data, collected from vulnerable commodity devices in open networks such as IoT.",
"We adopt the definition of trustworthy sensed data by Liu et al.",
"[20] (i.e., the data carrying integrity, authenticity, and freshness guarantees) and ensure the trustworthiness of sensed data in the IoT applications.",
"Second, certain IoT applications may require these sensors to capture sensitive personal information about the individuals.",
"We incorporate by design personal privacy protection mechanisms which allows individuals to submit sensed data in a privacy-aware manner.",
"The diverse nature of the IoT applications impose varying requirements on the data source tier.",
"We categorize these applications into two groups: (i) applications attributed by centralized processing, i.e., in these applications, raw sensed data is collected at the server side (micro data center or cloud sever) for processing and (ii) applications attributed by distributed processing, i.e., sensed data is processed locally on the sensor nodes.",
"This manuscript extends our preliminary work on trusted sensing [22], [23] and comprehensively introduces the concepts of trusted sensing and secure camera nodes in a holistic IoT setting.",
"In particular, we first expand trusted sensing [22] by addressing the personal privacy leakage caused by the incorporation of trusted sensors into smart devices such as smart phones.",
"Second, we extend secure camera nodes [23] for visual monitoring applications by exploring various PUF sources as root of trust for secure sensor node implementation.",
"Furthermore, both concepts are evaluated using real word application scenarios.",
"Overall, the main contributions of this work can be summarized as follows: First, we present a trusted sensing concept for centralized IoT applications.",
"This concept exploits lightweight security circuits called physically unclonable functions (PUFs) to extract a unique CMOS fingerprint of the sensor.",
"The fingerprint in combination with lightweight security mechanisms ensure non-repudiation (i.e., integrity, authenticity and freshness) on sensor readings.",
"On-chip PUFs assist to detect hardware tampering of the sensor.",
"Second, for the centralized IoT applications, we perform anonymization of the sensed data from trusted sensors on the host device using non-interactive witness indistinguishable proofs to ensure privacy-aware submission of the sensed data to the IoT applications.",
"Third, we present a secure node architecture for applications that require processing of sensed data locally on the sensor nodes.",
"The architecture, implemented as system-on-chip, derives the security keys from sensor's PUF-based CMOS fingerprint.",
"Integrity, authenticity, confidentiality, freshness and access authorization of the sensed data is protected using an encrypt-then-sign technique.",
"Secure boot of the SoC ensures integrity, authenticity and unclonability of the node's firmware.",
"Hardware tampering can be detected due to the tamper evidence property of the on-chip PUF.",
"Fourth, we evaluate both mechanisms using two case studies.",
"The trusted sensing and anonymization of data from trusted sensors are evaluated using a participatory sensing scenario where a secure node approach is evaluated using a private space monitoring scenario.",
"A trusted image sensor and a secure camera node are implemented using Zynq7010 SoC and OV5642 5MP image sensor as platform and latency, hardware, storage and communication overhead incurred by both the approaches is evaluated.",
"To demonstrate the feasibility of PUF-based approach, we also implemented and evaluated PUF on three platforms (Atmel 8-bit MCU, ARM Cortex M4 and Zynq7010 SoC) of varying complexities that are ideally suited as sensing platforms for a broad range of sensors.",
"The remainder of the paper is organized as follows: Section presents the state of the art technologies for ensuring sensed data trustworthiness and sensors' usage privacy in IoT applications.",
"Section provides an overview of the employed approach.",
"We present the details of our trusted sensing and privacy-aware reporting scheme for centralized IoT applications in Section and the secure node scheme for decentralized applications in Section .",
"We evaluate both schemes in Section and discuss relevant security and privacy properties.",
"Section concludes the paper."
],
[
"Related Work",
"This section discusses the relevant available work on sensed data trustworthiness and personal privacy protection mechanisms in IoT scenarios.",
"The threat surface area of the IoT applications necessitates the protection of data closer to the data source(s).",
"Securing sensor nodes in the IoT scenario entails data security, node security and usage privacy [21].",
"Data protection is typically implemented in firmware.",
"Any modification in the underlying hardware can completely bypass the data protection.",
"Therefore, node security is an essential requirement for data protection.",
"Moreover, given the ubiquitous and unprotected nature of IoT infrastructure (especially the sensor nodes), the hardware-, software-, and data-protection mechanisms must consider the possibility of physical access to the nodes.",
"Research on securing the sensed data and the sensing devices has been mainly focused on the integration of trusted platform modules (TPM) and other secure cryptoprocessors into the sensors or host devices.",
"The anonymous attestation feature of TPM is used to attest to the integrity and authenticity of the sensed data closer to the data source.",
"Furthermore, a TPM attests the system state before sensitive information is transmitted.",
"Early work on securing sensor nodes [24] was motivated by participatory sensing.",
"The work made a case for trustworthiness in participatory sensing by content protection.",
"Incorporation of TPM into mobile devices, participatory sensing application servers, and end user devices were proposed.",
"The TPM attests the integrity of the sensed data in the mobile devices for submission to a participatory sensing application server.",
"The proof of concept comprised an add-on circuit board, housing a TPM (TCG v1.2) chip, interfaced to a Nokia N800 phone.",
"Overhead incurred due to the proposed solution amounted to 13 kilobytes of memory (attestation code size), a latency of $1.92$ s (attestation time), verification latency of $0.78$ s. Saroiu and Wolman [16] introduced the concept of trusted sensors and proposed the integration of a TPM functionality into mobile device sensors to ensure integrity of the sensed data within the sensors.",
"The work identified the IoT applications that would benefit from the deployment of trusted sensors.",
"These included participatory sensing, monitoring energy consumption, and documenting evidence of crime scenes.",
"A high-level conceptual design of a trusted sensor, in which a TPM was incorporated into a sensor, was presented.",
"However, the work did not provide any proof of the concept.",
"Dietrich and Winter [25] explored software TPM implementations for embedded systems.",
"Existing CPU extensions like ARM TrustZone were evaluated to implement a software TPM with security guarantees similar to those of dedicated hardware.",
"Aaraj et al.",
"[26] also explored a software TPM solution.",
"In order to achieve a performance improvement, critical functions were implemented on reconfigurable hardware.",
"Our earlier work, TrustCAM [27] and TrustEYE [28], exploited TPM chips for protecting embedded camera nodes.",
"TrustCAM used anonymous attestation and time-stamping features of the TPM to protect the integrity, authenticity and confidentiality of the image data on the host processor.",
"To ensure image data integrity and authenticity, frame-groups were signed using a platform-bound key.",
"Digital signing slowed down the frame rate only by $0.5$ frames per second compared to plain streaming.",
"TrustEYE aimed to protect the captured images closer to the sensor.",
"A TPM chip was integrated into the sensing unit, which has exclusive access to the sensor's data.",
"Integrity, authenticity, confidentiality and freshness of the sensed data were protected at the sensing unit using 2048-bits RSA keys.",
"A cartooning filter was implemented to preserve the privacy of monitored individuals [29].",
"At a resolution of $320\\times 240$ a frame-rate of 9 frames per second was achieved.",
"Table: Classification of the related work on securing sensor nodes and contributions of this workPotkonjak et al.",
"[30] proposed a different approach for the trusted flow of sensed data in remote sensing scenarios.",
"The approach employed public physically unclonable functions (PPUFs).",
"PPUFs are fundamentally different from PUFs in several aspects: First, PPUFs are hardware security circuits which can be modeled by algorithms of high complexity whereas PUFs cannot be modelled.",
"Second, the security of a PPUF relies on the fact that the PPUF hardware output is many orders faster than its software counterpart (i.e., model) whereas the security of a PUF relies on the unclonability of the PUF circuit.",
"The major drawback of the PPUF-based approach lies in the fact that current PPUF designs involve complex circuits that require high measurement accuracy.",
"This slows down the authentication process and therefore the solution is not scalable.",
"Additionally, the solution targets applications where privacy is not a concern.",
"Some recent research efforts have lead to successful identification of PUF behavior on sensors.",
"Sensor PUF is an idea introduced by Rosenfeld et al.",
"[34] whereby the PUF response is determined by the applied challenge as well as the sensor reading.",
"Cao et al.",
"[35] introduced a CMOS image sensor based weak PUF.",
"The PUF response bits are generated by comparing the random fixed pattern noise in selected pixel pairs.",
"Although PUFs are lightweight, hardware security primitives that can be used to offer a scalable solution, identification of PUF behavior on sensors is only a part of the solution.",
"Early work on usage privacy of personal sensing devices was also motivated by participatory sensing.",
"Anonysense [31] is a participatory sensing model that uses a trusted authority to anonymize the sensed data.",
"Instead of submitting the sensed data directly to the application server the mobile devices submit the data to the anonymizing authority.",
"The authority collects the sensed data from the participating mobile devices, anonymizes it, and forwards it to the application server.",
"Mobile devices communicate with the authority via Mix network.",
"The application server assigns sensing tasks to the mobile devices using Tor anonymizing network.",
"Anonysense offers $k$ -anonymity, where $k$ is given by the number of mobile devices contributing sensed data to the application server.",
"Anonysense had a number of limitations: First, observe that in order to guarantee $k$ -anonymity, a Mix network may wait to receive $k$ reports before forwarding them to the application server.",
"This may significantly affect the service offered by the application.",
"Second, anonymization is performed after the data leaves the smartphone, whereas previously bugs [36] have successfully exploited the vulnerabilities in the software stack of the smartphone to leak users' privacy, thereby rendering the entire anonymization process ineffective.",
"PEPSI [32], another participatory sensing framework, used identity based encryption for end-to-end encryption of sensed data reports.",
"Smartphones register with a trusted registration authority and obtain IDs corresponding to the application they intend to participate in.",
"The application server only receives encrypted reports and forwards them to the intended end-user by matching the tags.",
"The solution is only suitable for decentralized applications as the server cannot process the encrypted reports.",
"PEPPeR [33] proposed a protocol for privacy-aware access of the sensed data by the end users (sensed data consumers) in participatory sensing networks.",
"An end user obtains tokens from the application server which reveal nothing about either the identity or its desire to spend the token with a specific sensed data.",
"The token validity, double-spending prevention are incorporated in the protocol using the classic cryptographic techniques.",
"An overview of the discussed related work is summarized in Table REF .",
"To summarize the previous work on sensed data trustworthiness, a TPM-based approach incurs significant hardware overhead on a node which may not be an economical solution for resource constrained sensor nodes.",
"Despite widespread deployment of TPMs in laptops, desktops, and servers for over a decade, TPMs have not yet found their way into resource-constrained embedded devices.",
"Moreover, TPMs do not provide protection against physical attacks.",
"Due to open nature of IoT applications, sensors might be physically accessible to the attackers, which render TPM-based solutions ineffective in the given scenario.",
"Protocols based on complex PPUF primitives are slow, have limited scalability and do not address the privacy protection.",
"In this work, we identify PUF behavior on platforms that can serve as sensing platforms for a broad range of sensors.",
"Furthermore, usage privacy of the sensors was not explicitly considered or addressed by any of the reviewed works.",
"In the related work on sensors' usage privacy, all proposed solutions are based on an online trusted authority.",
"The online nature of the authority significantly increases the risk of keys compromise.",
"Mix network based solutions are slow and may not be ideal for real-time or latency critical applications.",
"Solutions leveraging end-to-end encryption of the sensed data are suitable only for the decentralized applications.",
"By leveraging lightweight cryptographic techniques we propose effective solutions for protecting sensed data, sensor nodes and privacy of data producers, which are hooked into sensor hardware and are therefore harder to bypass.",
"We present protocols for privacy-aware reporting of sensed data in IoT applications for both centralized and decentralized IoT applications.",
"The solution does not uses an offline trusted authority which greatly reduces the risk of compromising keys.",
"The protection of sensed data and privacy locally on the sensing devices, further reduces the risk of collusion and Sybil attacks."
],
[
"Overall Approach for Trustworthy and Privacy-Aware Sensing",
"IoT applications vary significantly in their infrastructure (e.g., cloud vs. edge), sensed data collection mechanisms (e.g., raw data vs. processed information collection), data processing requirements (e.g., processing at data source vs. processing at server), data acceptance criteria and the services.",
"A sensor-centric security solution to ensure sensed data trustworthiness and sensors' usage privacy depends on whether the processing of the sensed data takes place on the sensors or the server side.",
"Therefore, with respect to data processing requirements, we categorize the IoT applications into two classes and propose two schemes tailored for the two classes of applications: The first class of applications is attributed to the collection of raw data from the sensing devices.",
"Processing of the data takes place at a central server.",
"The sensors are (embedded or externally) connected to a host device that reads the sensors and relay the sensed data to the server.",
"Participatory sensing applications are a common example of centralized applications.",
"We propose trusted sensing and privacy-aware reporting for the the centralized applications to ensure (i) trustworthiness of sensed data (i.e., data with integrity, authenticity and freshness guarantees) and (ii) usage privacy (i.e., anonymity of the sensing devices and unlinkability of multiple submissions from a device).",
"The scheme works in two stages.",
"First, the trustworthiness of sensed data is ensured by trusted sensors.",
"Each trusted sensor extracts its unique fingerprint from the sensor hardware using on-chip physically unclonable functions (PUF) and attests to integrity and authenticity of each sensed reading by signing it using an identity-based signature scheme.",
"The signature scheme uses the sensor-bound, unique fingerprint as the signing key.",
"Second, to report the signed readings from the trusted sensors in a privacy preserving manner, all privacy leaking information (e.g., signature with a sensor bound unique key) is anonymized using a non-interactive witness indistinguishable proof system ($P_\\mathrm {NIWI}$ ) [37].",
"Due to resource constraints on the sensors, we offload the privacy protection mechanism to the host processor on the host CPU.",
"Since the host OS is assumed to be untrusted, we leverage a virtualization approach [38], [39] where the user's software environment runs as a guest virtual machine.",
"The root virtual machine, inaccessible to the user, has exclusive access to the trusted sensors and runs the privacy protection mechanism on the sensors' output.",
"The second class of applications leverages the resources on the host devices for processing the sensed data locally on the sensor nodes.",
"Semantic information extracted from the data is delivered to the server.",
"Visual monitoring applications are prominent examples that fall under this class of applications.",
"With the trusted sensing and privacy-aware reporting approach, once sensed data is signed at the sensor, any processing of the data at host devices invalidates the security guarantees, which render the trusted sensing approach unsuitable for the given scenario.",
"Instead, a holistic security solution encompassing the sensor and host device, called secure node, is presented.",
"The secure node approach addresses all layers of sensor node (sensor and host) stack including the applications, middle-ware, OS, and the hardware.",
"We present the details of trusted sensing and privacy-aware reporting and secure node schemes for centralized and decentralized IoT applications in Sections and , respectively."
],
[
"Trusted Sensing and Privacy-Aware Reporting",
"This section presents the trusted sensing and privacy-aware reporting approach for centralized applications.",
"To illustrate our approach, we consider the participatory sensing (PS) scenario of Fig.",
"REF .",
"The individuals interested in contributing sensed data to a PS application register their mobile devices with the PS server (also known as application server).",
"During the registration, a client software is downloaded and installed on the mobile device.",
"In order to contribute sensed data to the PS application, the client software running on the mobile device triggers a system call to the OS to read out the required sensors and return the readings to the client.",
"The client composes them in form of a report and relays them to the server.",
"The mobile devices may use a WiFi or a cellular network Internet service to submit the sensed reports.",
"The PS server collects reports from the contributing mobile devices, archives it for short- or long-term, and performs processing on the collected data.",
"Processing includes filtering the high-quality data, extracting information from the collected data and presenting the information in a format required by the end user.",
"This information is provided to the end users as service.",
"The trusted sensing and privacy-aware reporting aims for two security objectives: (i) trustworthiness of sensed data and (ii) anonymity of sensing devices and unlinkability of multiple submissions from a device.",
"The trustworthiness of sensed data is ensured by trusted sensors that comprise two key components: (i) a PUF framework extracts sensor fingerprint using an on-chip PUF and binds a unique key to the sensor hardware using the fingerprint and (ii) sensed data attestation uses an Identity-based Signature scheme to sign every sensor reading using the sensor-bound key (depicted in Fig.",
"REF ).",
"The scheme uses a trusted authority who securely binds a unique key to each sensor hardware.",
"Figure: A high level infrastructure of participatory sensing (PS) applicationsTo report the signed readings from trusted sensors in a privacy preserving manner we adopt a Non-interactive Witness Indistinguishable Proof System ($P_\\mathrm {NIWI}$ ) [37].",
"Due to resource constraints on the sensors, we offload the privacy protection mechanism to host processor on user device.",
"Since the host OS is assumed to be untrusted, we leverage virtualization approach [38], [39] where the user's software environment runs as a guest virtual machine.",
"The root virtual machine is inaccessible to the user.",
"To ensure anonymity and unlinkability of multiple submissions by a user device, uniquely identifying information in a trusted sensor's output such as a sensor's signature on the reading using the unique sensor-bound key cannot be revealed to the server since it can uniquely identify the sensor, thereby the user device and the user.",
"Instead, the mobile device computes and reports proof of knowledge of the uniquely identifying information.",
"This is done using $P_\\mathrm {NIWI}$ .",
"Given a mobile device embedded with the trusted sensors, the root virtual machine executes the prover algorithm of the $P_\\mathrm {NIWI}$ as follows: (i) read the trusted sensors, (ii) commit to the witness (i.e., uniquely identifying information that we want to anonymize such as sensor identity, the signature, and the certificate) and (iii) generate the proofs of knowledge of the sensor identity,the signature, and the certificate.",
"These commitments and proofs are sent to the server along with the sensed readings.",
"The server executes the verifier algorithm of the $P_\\mathrm {NIWI}$ using received readings, commitments, and the proofs as arguments and verifies that the prover in fact possesses a valid signature-certificate pair for each received reading, thereby verifying the integrity and authenticity of the readings.",
"The witness-indistinguishability of the $P_\\mathrm {NIWI}$ proof system implies that the commitments and proofs do not reveal (to the server) the witness used to construct the commitments and the proofs.",
"Anonymity of the prover is given by the number of possible witnesses.",
"Given $N$ mobile devices equipped with trusted sensors and submitting readings to the PS server, anonymity of each user is given by $N$ .",
"Our scheme aggregates all signatures and certificates in the trusted sensors' output into a single signature and then generates the proof of knowledge of the aggregate signature, as illustrated in Fig.",
"REF .",
"This considerably reduces the communication overhead incurred on the user device.",
"Next, we present the trusted sensing and privacy-aware reporting components of the security scheme in Sections REF and REF , respectively.",
"Figure: Trusted sensing and privacy-aware reporting scheme: Trusted sensors provide integrity and authenticity guarantees on sensed data.",
"Sensor readings are aggregated and anonymized on the root virtual machine (VM) of the host device.",
"The modules of the proposed security scheme are marked in green"
],
[
"Trusted Sensing",
"Trusted sensing is accomplished by the trusted sensors in two steps: fingerprint extraction and sensed data attestation.",
"The former refers to extraction of unique, non-transferable fingerprints from the sensor hardware by a legitimate authority in a secure environment.",
"The later uses a digital signature scheme to attest the integrity and authenticity of sensor readings.",
"The sensor fingerprint serves as the signing key for the sensed data attestation.",
"The receiver (e.g., PS server) can verify the integrity and authenticity of each reading by signature verification.",
"The proposed trusted sensor uses a PUF framework to extract the sensor fingerprint and binds it to the sensor hardware.",
"An Identity Based Signature Scheme (PUF-based Cert-IBS) is used for the sensed data attestation which ensures non-repudiation of the data."
],
[
"PUF Framework",
"Physically unclonable functions (PUF) are special lightweight circuits that use the CMOS manufacturing process variations to generate the fingerprint of the underlying hardware.",
"Typical attributes of a PUF include randomness, uniqueness, physical unclonability, and reliability.",
"A PUF circuit provides a challenge-response mapping that is based on the uncontrollable variations in the physical structure of the integrated circuit (IC) introduced during the manufacturing process.",
"These variations are random and unique for each instance which makes any PUF-enabled electronic hardware uniquely identifiable (uniqueness).",
"Moreover, the chip manufacturer is not able to control or forge these variations (physical unclonability).",
"Reliability implies that a PUF should be able to reproduce the same challenge-response pairs under a range of environmental and operating conditions.",
"However, in practice, multiple responses from a PUF instance obtained under different environmental conditions (e.g., temperature) or operating conditions (e.g., voltage supply) slightly differ from one another.",
"These variations are referred to as PUF noise or error-rate and are measured as intra-Hamming distance ($HD^{intra}$ ).",
"Uniqueness of a PUF mapping is measured in terms of inter-Hamming distance ($HD^{inter}$ ) which is a measure of how different two responses from two PUF instances are.",
"Randomness of a PUF response is measured in terms of Hamming weight ($HW$ ) of the response.",
"Ideally, maximum $HD^{intra} \\approx 0 \\% $ , average $HD^{inter} \\approx 50 \\% $ and average $HW \\approx 50 \\% $ .",
"In order to extract an uniformly distributed random and perfectly reproducible fingerprint from the noisy and biased PUF response, helper data algorithms (HDAs) are used.",
"Our scheme requires the flexibility of masking an externally generated cryptographic key with the device fingerprint; therefore we use the HDA by Tuyls [40].",
"The PUF framework is comprised of two modules: the PUF and the HDA and works in two phases: key binding and key extraction.",
"Key Binding: $W \\leftarrow Gen(r, k)$ It is a one-time protocol carried out by a legitimate authority on the PUF in a secure environment to generate helper data $W$ .",
"A challenge $c$ is applied to the PUF and a response $r$ is obtained.",
"The authority then chooses a random key $k\\in \\lbrace 0,1\\rbrace ^k$ and calculates the corresponding helper-data as $W\\leftarrow r\\mathbin {\\oplus }C_\\mathrm {k}$ , where $C_\\mathrm {k}$ is the nearest code-word chosen from the error-correcting code $\\mathcal {C}$ , with $2^k-1$ code-words.",
"$W$ is integrity protected public information.",
"Key Extraction: $k \\leftarrow Rep(r^{\\prime },W)$ It is performed every time the key extraction from the PUF is desired.",
"The PUF is subjected to the same challenge $c$ and a noisy response $r^{\\prime }$ is obtained.",
"The code-word is then calculated as $C_\\mathrm {k^{\\prime }} \\leftarrow r^{\\prime } \\oplus W$ .",
"If $r^{\\prime }$ corresponds to the same challenge $c$ applied to the same PUF, $k$ is obtained after decoding $C_\\mathrm {k^{\\prime }}$ using $W$ otherwise an invalid code-word is obtained i.e., $k \\leftarrow \\mathtt {Decoding}(C_\\mathrm {k^{\\prime }}),$ if $\\ \\mathtt {Hamming\\ distance}(C_\\mathrm {k},C_\\mathrm {k^{\\prime }}) \\le t$ , where $t$ is error-correction capacity of $\\mathcal {C}$ .",
"This PUF framework offers the following key advantages: (i) it binds a unique key with a PUF-enabled hardware, (ii) it provides secure storage of the key since the key is derived from device properties during start-up and (iii) it offers more cost-effective secure key storage than a secure memory alternative."
],
[
"Identity Based Signature Scheme (PUF-based Cert-IBS)",
"This section explains our Identity based Signature scheme (PUF-based Cert-IBS) that is based on the framework [41] to construct certificate-based identity-based signature scheme (Cert-IBS) from a standard signature (SS) scheme.",
"PUF-based Cert-IBS ensures integrity and authenticity of sensed data in our trusted sensing and privacy-aware reporting and secure node schemes.",
"It uses the PUF framework of Section REF to bind the security key with the sensor fingerprint extracted using PUF.",
"A typical SS comprises three algorithms: key generation $(K)$ , signing $(Sign)$ and verification $(Ver)$ .",
"PUF-based Cert-IBS uses a key generation authority.",
"To setup PUF-based Cert-IBS, the authority generates a master key pair $(msk, mpk)$ using $K$ .",
"We assign an identity $I$ and a PUF instance $PUF$ to each sensor.",
"The identity can be any unique physical identifier of the sensor such as serial number, EPC or a unique bit string written to one-time programmable memory of the sensor.",
"We denote a sensor with identity $I$ and $PUF$ by SEN($I,PUF$ ).",
"Setup.",
"The trusted authority runs the $K$ of $SS$ to generate the master key pair: $(mpk, msk)\\leftarrow K(1^k)$ Enrollment.",
"During the enrollment phase, the authority generates a unique signing key pair $(sk, pk)$ using the key generation algorithm $K$ of SS and binds $sk$ with the on-chip $PUF$ using the key binding algorithm of the PUF framework, i.e., $W_{sk}\\leftarrow Gen(r, sk)$ where $r$ is the $PUF$ response to challenge $c$ selected by the authority and $W_{sk}$ is the helper data corresponding to $sk$ .",
"Further, the authority issues a certificate on the public half of the signing key given by $cert \\leftarrow Sign_{msk}(pk, I)$ .",
"$W_{sk}$ and $cert$ are stored in the sensor's non-volatile memory.",
"Sensed Data Attestation.",
"Sensed data attestation is performed every time the sensor SEN($I, PUF$ ) outputs a new reading.",
"The private key required for signing is reconstructed at the power-up using the key extraction phase of the PUF framework i.e., $sk\\leftarrow Rep(r^{\\prime },W_{sk})$ .",
"PUF-based Cert-IBS signature of SEN($I, PUF$ ) on sensor reading $M$ is given by $(M, I, pk, \\sigma , cert)$ , where $\\sigma \\leftarrow Sign_{sk}(M)$ .",
"PUF-based Cert-IBS verification is successful if $Ver_{pk}(M, \\sigma ) = 1$ and $Ver_{mpk}((I,pk), cert) = 1$ .",
"Successful Cert-IBS verification ensures that reading $M$ is signed by SEN($I, PUF$ ) with its platform-bound private key, assigned and bound to SEN($I,PUF$ ) by the legitimate authority.",
"Given that $SS$ is a uf-cma secure standard signature scheme, theorem 3.5 of [41] proves that the corresponding $\\mathrm {PUF}$ -$\\mathrm {based\\ Cert}$ -$\\mathrm {IBS}$ as per construction of Section REF is a uf-cma secure IBS scheme."
],
[
"Privacy-Aware Reporting of Trustworthy Sensed Data",
"Privacy-aware reporting of sensed data entails anonymity of user devices and unlinkability of multiple submissions from a user device.",
"Given a user device incorporated with the trusted sensors, each element of tuple $(I, pk, \\sigma , cert)$ in a trusted sensor's output uniquely identifies the sensor and therefore cannot be revealed to the data center.",
"For each trusted sensor output $(M, I, pk, \\sigma , cert)$ , the user device computes a proof of knowledge of $(I, pk, \\sigma , cert)$ using the non-interactive witness indistinguishable proof system ($P_\\mathrm {NIWI}$ ) by Groth and Sahai [37].",
"The mobile device (prover) then sends the proof instead of $(I, pk, \\sigma _{M}, cert)$ along with the sensor reading $M$ to the server (verifier) as depicted in Fig.",
"REF .",
"The server verifies the proof.",
"Successful verification ensures that the mobile device knows a witness $(I, pk, \\sigma , cert)$ such that PUF-based Cert-IBS verification equations holds true for the received data $M$ i.e., $Ver_{pk}(M,\\sigma )=Ver_{mpk}((I,pk), cert)=1$ .",
"Given $N$ trusted sensors submitting sensed data to a server, witness indistinguishability of the proof implies that the data center cannot distinguish which witness $\\lbrace (I_i, \\sigma _{i},cert_i)\\rbrace _{i=1}^N$ (i.e., trusted sensor) was used to construct the proof.",
"Therefore, every trusted sensor is $N$ -anonymous with respect to the server.",
"Unlinkability of multiple submissions by the same sensor follows from witness indistinguishability."
],
[
"Primer on Pairings",
"For privacy-aware reporting of sensed data, our scheme uses pairings-based cryptography, therefore we sketch some basics of pairings: Let $G_1$ , $G_2$ and $G_T$ be cyclic groups of the same prime order and $g_1$ and $g_2$ are the generators of $G_1$ and $G_2$ respectively.",
"A pairing is map $e:G_1\\times G_2 \\rightarrow G_T$ that is (i) bilinear, i.e., for all $u\\in G_1$ , $v \\in G_2$ and $a$ ,$b\\in \\mathbb {Z}, e(u^a,v^b) =e(u,v)^{ab}$ , (ii) $e(g_1,g_2)$ generates $G_T$ and (iii) $e$ is efficiently computable.",
"The setting where $G_1 = G_2 = G$ and $g_1 = g_2 = g$ is called symmetric pairing whereas if $G_1 \\ne G_2$ and $g_1 \\ne g_2$ , the pairing is called asymmetric.",
"For simplicity, we explain our scheme for a symmetric setting.",
"For practical implementation, an asymmetric setting is recommended (cp.",
"Section REF )."
],
[
"Non-interactive Witness Indistinguishable Proofs $(P_\\mathrm {NIWI})$",
"A proof system allows a prover who possesses some witness $\\omega $ to convince a verifier that a certain statement $\\chi \\in L$ is true, where $L$ is some language, and $\\omega $ is a witness that attests to this fact.",
"In witness indistinguishable proof systems, the interaction between the prover and the verifier does not reveal information about the witness, even if the verifier behaves maliciously.",
"Furthermore, it is unfeasible for an adversary to decide which of the possible witnesses is used by the prover.",
"In a non-interactive proof system, the prover simply sends the verifier a single message after which the latter verifies correctness of the proof without any further interaction with the prover.",
"Groth and Sahai introduced a non-interactive witness indistinguishable proof system ($P_\\mathrm {NIWI}$ ) for languages involving the satisfiability of equations over bilinear groups, in the common reference string model [37].",
"The main idea underlying $P_\\mathrm {NIWI}$ is as follows: Given groups $A_1, A_2, A_T$ with a bilinear map, $P_\\mathrm {NIWI}$ maps the elements in $A_1, A_2, A_T$ into $B_1, B_2, B_T$ , also equipped with a bilinear map, by using a commitment scheme.",
"The latter groups are larger thereby allowing to hide the elements of $A_1, A_2, A_T$ .",
"Given the equation(s) that we intend to prove, we replace the variables (witness) in the equation(s) with commitments to those variables.",
"Since the commitments are hiding, the equations will no longer be valid.",
"However, we can extract out the additional terms introduced by the randomness of the commitments and provide these terms in the proof to the verifier, who can verify the validity of the equations.",
"Does providing these terms destroy witness indistinguishability?",
"Since there are multiple additional terms introduced by substituting the commitments, the algebraic environment allows us to randomize the terms such that their distribution is uniform over all possible terms satisfying the equations.",
"By definition, $P_\\mathrm {NIWI}$ is a tuple of four probabilistic polynomial time algorithms $(K_{NI}$ , $P_{NI}$ , $V_{NI}$ , $X_{NI})$ , i.e., key generator, prover, verifier, and extractor, respectively.",
"The key generator, $K_{NI}$ , takes the bilinear group description $A_1, A_2, A_T$ as input and outputs a common reference string $crs$ and an extraction key $xk$ .",
"$crs$ comprises the target groups description $B_1, B_2, B_T$ and rules to compute commitments.",
"Given a set of equations (that the prover wants to prove), the prover, $P_{NI}$ , takes $crs$ and a witness $\\omega $ as input and outputs a proof $\\pi $ for each equation.",
"The verifier, $V_{NI}$ , given $crs$ , the set of equations and $\\pi $ outputs 1 if the proof is valid and 0 otherwise.",
"Finally, the extractor, $X_{NI}$ , on a valid proof $\\pi $ may extract $\\omega $ using the extraction key $xk$ .",
"The privacy-aware trusted sensing scheme adopts $P_\\mathrm {NIWI}$ for privacy-aware reporting of sensed data from the trusted sensors.",
"During the setup, the TA runs the key generator algorithm $K_{NI}$ , which takes the group description $(G,G_T, g, e, \\mathrm {p})$ as input and outputs the $csr$ that comprises eight group elements, i.e., $\\in G^8$ and an extraction key $xk$ .",
"The $csr$ is published for the participating mobile devices and the application servers where as $xk$ is kept secret by the TA since the extraction of the witnesses is not required in the proposed privacy-aware trusted sensing scheme.",
"The scheme, however, can be extended with dispute resolution and revocation mechanisms using the extractor algorithm and the $xk$ .",
"A mobile device runs $P_{NI}$ every time it has to submit sensed data using the trusted sensors to an application server.",
"Without a privacy-protection mechanism, a mobile device submits the trusted sensor reading $( M, I, pk,\\sigma , cert)$ as is to the server who verifies the signature and the certificate using the PUF-based Cert-IBS verification equations, $Ver_{mpk}(I,cert)\\stackrel{{\\normalfont \\mbox{?",
"}}}{=}1$ and $Ver_{pk}(M, \\sigma )\\stackrel{{\\normalfont \\mbox{?",
"}}}{=}1$ .",
"In this case, the server requires $(I, pk,\\sigma , cert)$ as input to the PUF-based Cert-IBS verification algorithm.",
"Each element of this tuple uniquely identifies the trusted sensor and the mobile device.",
"With the $P_\\mathrm {NIWI}$ , the root virtual machine running on the mobile device commits to each element of the witness $(I, pk,\\sigma , cert)$ using the $csr$ , $\\mathbf {d}(I),\\ \\mathbf {c}(pk),\\ \\mathbf {c}(\\sigma )$ , and $\\mathbf {c}(cert)$ , where $\\mathbf {c}(.",
")$ denotes a commitment to an element in group $G$ whereas $\\mathbf {d}(.",
")$ denotes a commitment to an element in $\\mathbb {Z}_p$ .",
"The $pk,\\sigma ,\\text{ and } cert \\in G$ each.",
"Although, $I$ is only assumed to be a unique, random value, one can use $I\\mod {p} \\ (\\in \\mathbb {Z}_p)$ instead.",
"Each $\\mathbf {c}(.",
")$ and $\\mathbf {d}(.",
")$ is $\\in G^3$ .",
"It further computes two proofs $\\pi _1$ and $\\pi _2$ by replacing the witness with the commitments in the PUF-based Cert-IBS verification equations.",
"Since we use the symmetric version of the BLS signature scheme [42] as the $SS$ in PUF-based Cert-IBS, the PUF-based Cert-IBS verification equations, denoted as $eq_1$ and $eq_2$ , are given by Eqs.",
"REF and .",
"Each of the $\\pi _1$ and $\\pi _2$ $\\in G^9$ .",
"The proofs and commitments are are then sent to the server along with the sensor reading $M$ .",
"The PS server runs the verification algorithm $V_{NI}$ to verify whether or not the proofs $\\pi _1$ and $\\pi _2$ and the commitments satisfy the following equations $&e(h_{M}, pk)= e(\\sigma , g) \\\\&e(h_{c}, mpk)=e(cert, g)$ where $H(I, pk)$ is denoted by $h_{c}$ and $H(M)$ is denoted by $h_{M}$ .",
"In PS applications a sensed data report comprises typically multiple sensors' readings.",
"For a report comprising $Q$ sensors' readings, the above process is performed $Q$ times.",
"The root virtual machine then provides the tuple $\\lbrace M_i,\\ \\mathbf {d}(I_i),\\mathbf {c}(pk_i),\\mathbf {c}(\\sigma _{i}),\\mathbf {c}(cert_i),\\ \\pi _{1i},\\ \\pi _{2i}\\rbrace _{i=1}^Q$ to the application client running in guest virtual machine on the user device, which sends it to the PS server, who runs the verification algorithm $V_{NI}$ for each sensor reading.",
"For mathematical details and security proofs of the $P_\\mathrm {NIWI}$ construction, the reader is referred to [37].",
"Theorem 17 in [37] proves that $P_\\mathrm {NIWI}$ following the construction of Section REF has perfect completeness, perfect soundness and composable witness indistinguishability for satisfiability of Eqs.",
"REF and in a bilinear group $G$ where DLIN problem is hard.",
"Witness indistinguishability implies that the proofs and the commitments do not reveal what values of the witness $(I,pk, \\sigma , cert)$ were used to generate the commitments and the proofs.",
"Anonymity of the prover $P_{NI}$ with respect to $V_{NI}$ is given by the possible number of values a witnesses can take.",
"Given the PS scenario, anonymity of each trusted sensor with respect to the application server is given by the total number of the trusted sensors contributing sensed data to the server.",
"Assuming the DLIN assumption holds in $G$ , the $P_\\mathrm {NIWI}$ for each sensor reading costs 30 elements of group $G$ , i.e., four commitments $\\mathbf {d}(I),\\ \\mathbf {c}(pk),\\ \\mathbf {c}(\\sigma ),\\ \\mathbf {c}(cert)$ consisting of 3 group elements each and two proofs $\\pi _1$ and $\\pi _2$ of 9 group elements each.",
"A report comprised of $Q$ sensors' readings therefore incurs a communication overhead of $30Q$ elements of $G$ on the mobile device."
],
[
"Communication Overhead Reduction using Signature Aggregation",
"The aggregation property of BLS signatures allows an aggregating party to combine multiple, say $m$ , signatures into a single signature as $\\bar{\\sigma } \\leftarrow \\Pi _i^m\\sigma _i$ , where $\\bar{\\sigma } \\in G$ , thereby reducing the total signatures' size to $1/m$ .",
"For aggregate verification, given $\\bar{\\sigma }$ , the original messages $M_i$ , and public keys $pk_i$ , compute $h_{M_i} \\leftarrow H(M_i)$ and accept if $e(\\bar{\\sigma },g)=\\Pi _i^m e(h_{M_i},pk_i)$ .",
"In our scheme, aggregation is done by the root virtual machine on the mobile device as follows: In a PUF-based trusted sensor output $\\sigma _{i}$ , i.e., $Sign_{sk_i}(M_i)$ , and $cert_i$ , i.e., $Sign_{msk}(I_i,pk_i)$ , are both BLS signatures and can be aggregated.",
"Furthermore, if an application requires every mobile device to submit multiple, say $Q$ sensor readings, the aggregation is done as $\\bar{\\sigma } = \\Pi _{i=1}^Q \\sigma _i \\cdot cert_i$ .",
"A major reduction in communication overhead is achieved since $P_\\mathrm {NIWI}$ is applied on the aggregate of $Q$ readings instead of individual ones.",
"The simultaneous satisfiability of PUF-based Cert-IBS aggregate signature verification is given by Eq.",
"REF .",
"The root virtual machine sets the witness to $(\\lbrace I_i, pk_i\\rbrace _{i=1}^Q,\\bar{\\sigma })$ and equation we want to verify, denoted as $eq$ , to Eq.",
"REF .",
"The prover, using $P_{NI}$ , commits to each element of the witness and generates a proof $\\pi $ by plugging the commitments to the $eq$ .",
"Successful verification using $V_{NI}$ at the server ensures that the prover (mobile device) possesses $Q$ PUF-based Cert-IBS signatures on $Q$ distinct readings $\\lbrace M_i\\rbrace _{i=1}^Q$ such that: $e(\\bar{\\sigma }, g)= \\Pi _{i=1}^Q e(h_{M_i}, pk_i)e(h_{c_i}, mpk)$ The privacy-aware trusted sensing scheme is summarized in Table REF and runs in three phases: setup, enrollment and trusted sensing and privacy-aware reporting.",
"The TA sets up the scheme by generating its master key pair.",
"The bilinear group description and $csr$ are also generated and published for the PS entities during the setup.",
"Enrollment of the privacy-aware trusted sensing scheme is in fact the enrollment of the PUF-based Cert IBS enrollment, performed only once in a secure and trusted environment.",
"The trusted sensing and privacy-aware reporting is performed every time a mobile device contributes sensed data to a PS application: The trusted sensors' output readings with integrity and authenticity guarantees, which are anonymized by the root VM on the host mobile device and sent to the PS server.",
"The PS server verifies the integrity and authenticity of the readings in a privacy-preserving manner using the $P_\\mathrm {NIWI}$ verification algorithm, $V_{NI}$ .",
"$P_\\mathrm {NIWI}$ of $Q$ aggregated readings for satisfiability of $eq$ costs $(6Q+12)$ elements of $G$ compared to $30Q$ elements of $G$ without aggregation.",
"Anonymity of the mobile device with respect to the PS server is given by the total number of mobile devices reporting the PUF-based trusted sensors' readings to the PS server.",
"Table: The Privacy-aware trusted sensing scheme is executed in three phases: Setup, enrollment and trusted sensing and privacy-aware reporting.",
"The TA sets up the scheme by generating its master key pair.",
"The bilinear group description and csrcsr are also generated and published for the PS entities.",
"During enrollment, each sensor is enrolled with the TA using PUF-based Cert IBS enrollment.",
"During the trusted sensing and privacy-aware reporting phase, the trusted sensors output readings with integrity and authenticity guarantees, which are anonymized by the root VM on the host mobile device and sent to the PS server.",
"The PS server verifies the integrity and authenticity of the readings in a privacy-preserving manner using the P NIWI P_\\mathrm {NIWI} verification algorithm V NI V_{NI}"
],
[
"Secure Node",
"This section presents the secure node approach for decentralized IoT applications.",
"Visual sensor networks (VSNs) are becoming increasingly popular in IoT applications that range from surveillance of critical public spaces for law and order maintenance and public safety to private space monitoring such as smart homes, assisted/enhanced living, child monitoring and home security [21].",
"Continuous transmission of visual data requires high bandwidth and memory which is often unfeasible.",
"Therefore, visual monitoring applications typically require processing of visual data locally on camera nodes.",
"A major limitation of the trusted sensing and privacy aware reporting approach of Section is that once data is signed within the sensor, any legitimate modification (processing, compression etc.)",
"of the data at host processor invalidates the security guarantees.",
"With secure node approach, we overcome these limitations by adopting a holistic security solution for the node addressing all layers of a camera stack including the applications, middle-ware, OS, and the hardware.",
"To illustrate our approach, we consider a visual monitoring for assisted living scenario of Fig.",
"REF where one or more cameras monitor the space for events of interest such as fall detection or no movement for long duration.",
"To limit the amount of data transmitted by the camera device, archiving is triggered upon event detection.",
"The event can be triggered internally (e.g., by on board analytics) or externally (e.g., by auxiliary sensors or a request from the caretaker).",
"We assume that integrity and authenticity of an external source is verified before triggering an event.",
"Upon event detection, an alert message is sent to a caretaker and the video data capturing the event is uploaded to a storage server.",
"When the upload has been completed, a push notification is sent by the server to the caretaker who then downloads the data, analysis it and formulates a response.",
"Figure: A high level infrastructure of private space monitoring applications depicting security and privacy requirementsIn order to keep the cost of the camera low, we do not assume availability of permanent storage of videos on the camera node.",
"A public cloud storage server is leveraged for short or long-term video archiving.",
"To limit the amount of data transmitted by the camera device, archiving is triggered upon event detection.",
"The event can be triggered internally (e.g., by onboard analytics) or externally (e.g., by auxiliary sensors or a request from the end user).",
"We assume that integrity and authenticity of an external source is verified before triggering an event.",
"Data security, node security and personal privacy of the monitored individuals are the integral parts of the sensing device.",
"Data security includes integrity, authenticity, confidentiality, and freshness of the visual data and the metadata.",
"Data security is ensured on the sensing device before it is delivered to the server.",
"Since visual data contains identities and behaviors of observed individuals, data confidentiality and access authorization are essential security requirements for personal privacy protection in visual monitoring applications.",
"These security guarantees are valid throughout the entire lifetime of the data.",
"Confidentiality is ensured by encrypting each video frame using AES128 encryption.",
"Integrity and authenticity are ensured by signing the hash-chain of encrypted frames using the PUF-based Cert-IBS scheme of Section REF .",
"PUF-based Cert-IBS and AES128 algorithms use platform-bound security keys.",
"PUF framework of Section REF binds the signing and encryption keys to the camera hardware using on-chip PUF that serves as secure key storage.",
"On event detection, encrypted-hashed-signed footage is uploaded to a public storage server at the edge or cloud tier and an alert message is sent to the end-user who can then download the archived footage from the server on demand.",
"Integrity, authenticity and freshness of data is ensured by verifying PUF-based Cert-IBS signatures.",
"Only the authorized end-user (i.e., having access to the decryption key) can decrypt the frames.",
"Since data security is implemented at application level, in order to ensure effective security guarantees, underlying software and hardware stack of the camera node needs to be protected as well.",
"Node security requirements include integrity, authenticity and unclonability of camera firmware, resistance against hardware tampering and side-channel attacks.",
"Figure: The enrollment, key exchange and monitoring phases of the secure node scheme for visual monitoring for assisted living.",
"The enrollment is performed once in a secure and trusted environment whereby the platform-bound unique cryptographic keys are created.",
"During key exchange, the end user shares verification keys with caretaker's monitoring device using a local interface.",
"Thereafter, the camera is deployed for monitoring"
],
[
"Operational Phases",
"In order to bind the cryptographic keys with the camera platform and limit data access to only legitimate caretakers, the scheme requires two steps namely enrollment and key exchange to be performed before the camera can be deployed for monitoring.",
"Enrollment, key exchange and monitoring phases of the scheme are depicted in Fig.",
"REF .",
"The secure node approach uses a trusted authority (TA).",
"Enrollment is performed by the TA in a secure and trusted environment.",
"During this step, the TA extracts a unique fingerprint of the camera hardware using the PUF framework of Section REF and binds signing and encryption keys with the hardware.",
"During the key exchange, the caretaker securely transfers the signature verification and decryption keys from the camera device to her monitoring device.",
"Afterwards, the camera is deployed for monitoring.",
"To setup our scheme, the TA generates a master key pair $(mpk, msk)\\leftarrow K(1^k)$ .",
"We assign each camera a unique identity $I$ and a PUF instance.",
"We denote a camera with identity $I$ and on-chip PUF instance $PUF$ as CAM($I, PUF$ ) and the trusted authority with master key pair as TA($msk,mpk$ ).",
"Enrollment.",
"During enrollment, TA($msk$ , $mpk$ ) binds a signing key pair ($sk$ , $pk$ ) and an AES-encryption key ($k_E$ ) to the camera node CAM($I$ , $PUF$ ) using the camera fingerprint.",
"The CAM presents TA with its identity $I$ as a request for enrollment.",
"The TA picks two random challenges ($c_1, c_2$ ) and feeds them to the $PUF$ on the camera.",
"The responses from the $PUF$ ($r_1$ , $r_2$ ) are returned to the TA, who then binds a signing and an encryption key with the CAM as follows: First, the TA generates a signing key pair ($sk$ , $pk$ ) using the key generation algorithm of PUF-based Cert IBS.",
"Using the key-binding algorithm of the PUF framework, it binds the private-half of the key pair to the $PUF$ using $r_1$ , i.e., $W_1 \\leftarrow Gen(r_1, sk)$ .",
"Furthermore, the TA issues a certificate consisting of its signature on the CAM's identity and public half of the key pair, i.e., $cert \\leftarrow Sign_{msk}(I, pk)$ .",
"Second, the TA generates a unique, random encryption key $k_E$ and binds it to the $PUF$ on the CAM using $r_2$ , i.e., $W_2 \\leftarrow Gen(r_2, k_E)$ .",
"The tuple ($W_1$ ,$W_2$ , $cert$ ) is stored in non-volatile memory on camera.",
"Key Exchange.",
"Key exchange is performed by the caretaker before deploying the camera for monitoring.",
"During this step, the caretaker transfers the signature verification key $pk$ and the decryption key $k_E$ from the camera device to her monitoring device via a local interface such as NFC.",
"Since the transfer is done in a private space using a local connection, it is assumed that $k_E$ is not leaked to a third party.",
"Securing the keys on mobile devices with vulnerable software stack is out of scope of this work.",
"However, well established techniques such as virtualization [38] (isolates applications requiring trusted infrastructure) and secure vault can be leveraged for this purpose.",
"Monitoring.",
"Once the camera is deployed for monitoring, on every power-up, the signing and encryption keys are generated from noisy $PUF$ responses following the key extraction phase of the PUF framework, i.e., $sk\\leftarrow Rep(r_1^{\\prime },W_1)$ and $k_E\\leftarrow Rep(r_2^{\\prime },W_2)$ .",
"In case of an event of interest, the video footage capturing the event is transferred to the caretaker.",
"Let $N$ be the total number of frames comprising the footage.",
"The value of $N$ can either be a fixed or variable number depending on type of the event.",
"Upon the occurrence of an event, each video frame is encrypted using the AES128 algorithm to ensure confidentiality, i.e., $C_i \\leftarrow Enc_{k_E}(frame[i]) |_{\\ i\\ =\\ 1 \\ldots N}$ .",
"The non-repudiation of the data is ensured by MAC-then-Sign technique.",
"First, each encrypted frame is hashed using the HMAC algorithm to ensure integrity $h_i \\leftarrow HMAC(C_i) |_{\\ i\\ =\\ 1 \\ldots N}$ .",
"This is followed by signing the entire hash-chain of all encrypted frames using the PUF-based Cert-IBS scheme given by $\\sigma \\leftarrow Sign_{sk}(h_1\\parallel h_{2} \\parallel \\cdots \\parallel h_{N} \\parallel \\tau )$ where $\\tau $ is the timestamp given by $SHA256(I\\parallel event\\_count)$ .",
"Signing the entire hash-chain together preserves the frame order.",
"The timestamp is included in the signature to ensure freshness of data and thwart replay attacks.",
"The camera then uploads the encrypted-then-MACed-then-signed footage $\\lbrace C_1,C_2,\\ldots , C_N,\\tau , \\sigma \\rbrace $ to the storage server.",
"An alert message notifies the caretaker about the event and the completion of the upload.",
"The caretaker can then download the footage on-demand and check integrity, authenticity and freshness by verifying the Cert-IBS signature, i.e., $1 \\stackrel{?",
"}{=} Ver_{mpk}(cert, (I, pk))$ and $1 \\stackrel{?",
"}{=} Ver_{pk}(\\sigma , (h_1\\parallel h_{2} \\parallel \\cdots \\parallel h_{N} \\parallel \\tau ))$ .",
"Upon successful verification, the caretaker uses the decryption key to decrypt the frames to obtain footage in raw format, $frame[i] \\leftarrow Dec_{k_E}(C_i)|_{\\ i\\ =\\ 1 \\ldots N}$ ."
],
[
"Secure Camera Architecture & Prototype",
"The key idea underlying the secure camera architecture is to leverage an on-chip PUF to extract the node's fingerprint from the hardware, which serves as basis for on-board data-, node-, and personal privacy protection.",
"Security is rooted in the system hardware making it an intrinsic element of the device and therefore harder to bypass.",
"Video data processing and protection is done inside the SoC and the data leaves the chip with integrity, authenticity, confidentiality, access authorization, and freshness guarantees.",
"The choice of the processing platform is a critical decision in every vision systems design.",
"CPUs perform operations in sequence whereas FPGAs are massively parallel in nature.",
"Typically, FPGA performs vision processing order of magnitude faster than CPUs.",
"However, an FPGA consumes more power and has higher programming complexity as compared to a CPU.",
"An architecture featuring both an FPGA and a CPU presents the best of both worlds and often provides a competitive advantage in terms of performance, cost, and power consumption [43].",
"The secure camera node leverages a system-on-chip (SoC).",
"This offers two advantages: First, the SoC provides a monolithic architecture that allows to architect a security solution tightly integrated with the system logic.",
"Second, the SoC comprises an FPGA and processor part which provides the flexibility to compose a security solution addressing all layers of the node stack.",
"There exist two categories of operating system (OS), that are used for embedded devices: real-time OS (RTOS) and general-purpose OS.",
"An RTOS provides scheduling guarantees to ensure deterministic behaviour and timely response events and interrupts.",
"However, it is inefficient at handling multiple tasks in parallel and lacks board (hardware) support.",
"Embedded Linux and Android dominate the world of general-purpose OS for embedded systems.",
"Android has been widely successful as mobile OS due to its rich support for multimedia, graphics, user interface, and networking.",
"Drawbacks of Android lie in its large memory footprint and extensive CPU resources consumption.",
"Embedded Linux shines when it comes to operating efficiency in terms of memory footprint, power, and computing performance.",
"The secure camera node uses embedded Linux OS.",
"The prototype shown in Fig.",
"REF , realizes the secure camera node architecture.",
"The block diagram of the node showing the core modules of video data path and their mapping on hardware components is depicted in Fig.",
"REF .",
"The hardware and software stack of the camera node are depicted in Fig.",
"REF .",
"The camera hardware is comprised of OV5642—a 5 MP CMOS image sensor array, Zynq7010 SoC, 1 GB SDRAM, and a gigabit Ethernet interface.",
"The SoC houses a dual-core ARM Cortex A9 processor clocked at 666 MHz and FPGA fabric.",
"The ARM-A9 processor runs Embedded Linux that hosts system libraries (OpenSSL, GMP, libjpeg etc.)",
"and user libraries (pbc, motiondetection etc.)",
"to be used by the applications.",
"On top of the OS, a custom application framework is designed, which is responsible for providing the intended (application specific) functionality to the device.",
"The application framework for the secure camera architecture, given by Fig.",
"REF , is divided into the four tasks: sensing, processing, security, and communication.",
"Figure: Application framework of the secure camera node comprises sensing, processing, security and communication tasks.",
"The sensing tasks read image data from the sensor and perform format adaptations.",
"The processing tasks include the application logic (e.g., event trigger based on motion detection).",
"In case of an event detection, data is forwarded to security tasks, where frames are encrypted-MACed-signed.",
"Protected frames are forwarded to communication tasks for uploadSensing tasks include reading the visual data from the image sensor and encoding it the desired format.",
"The OV5642 image sensor is configured to provide data in 640$\\times $ 480 (resolution) 8-bit YUV422 (color-space) format.",
"Processing tasks include the application specific logic; the prototype for private space monitoring performs video compression and event detection by behavioral analysis of video data.",
"Video compression is achieved using the JPEG compression engine on the OV5642 sensing unit.",
"The event is triggered if motion is greater than a predefined threshold.",
"Motion of the monitored individual is computed using the three-frame differencing algorithm by Collins et al. [44].",
"The image difference between frames at time $t$ and $t-1$ and the difference between $t$ and $t-2$ , is performed to determine regions of legitimate motion and to erase ghosting.",
"If an event has been detected, the frames are forwarded to the security tasks.",
"Security tasks entail data security and node security.",
"Data security tasks secure the data on-camera as described in Section REF .",
"After camera power-up, encryption and signing keys are extracted from the ring oscillator (RO) PUF implemented using the reprogrammable fabric and are loaded into the cache.",
"Each frame is encrypted using AES128 and MACed using HMAC-SHA256 to ensure data confidentiality and integrity, respectively.",
"MAC checksums of all frames in the footage are concatenated and signed using the PUF-based Cert-IBS scheme with BLS [45] as underlying standard signature scheme; this ensure authenticity and preserves the frames order.",
"Freshness of data is ensured by including a timestamp $\\tau $ before signing the checksums.",
"For timestamp generation, the camera uses an event counter that increments whenever an event is detected.",
"Given that an event is detected by the motion detection algorithm and the event counter is incremented to $event\\_count$ , then the timestamp is calculated as $\\tau =$ SHA256$(I\\parallel event\\_count)$ .",
"A time-stamp value holds true only for a specific event $event\\_count$ detected by camera device $I$ .",
"Following the event $event\\_count$ , the footage is timstamped with $\\tau $ .",
"The value of the timestamp should not repeat among legitimate footages from different events detected by the same camera or among footages from different cameras.",
"This simple check deters replay attacks.",
"It is important to note that encryption and hashing is performed on frames whereas time-stamping and signing is performed on the complete footage.",
"Node security aims to secure the software and hardware stack of the sensor node; this is achieved by a secure boot of the SoC and the on-chip PUF.",
"The Zynq7010 SoC provides secure boot functionality as part of its boot procedure that verifies authenticity, integrity and unclonability of the camera's software stack based on digital signatures, message authentication code (MAC), and encryption.",
"The boot mechanism is CPU-driven.",
"Other hardware components used in the boot process are the non-volatile memory (NVM), BootROM, on-chip memory (OCM), AES/HMAC module, JTAG, and DDR RAM.",
"The software programs involved in the boot are the BootROM code, the first-stage bootloader (FSBL), U-Boot, the Linux kernel, and user applications.",
"Boot-chain of the camera device is depicted in Fig.",
"REF .",
"The foundation of secure boot is established by placing the BootROM code in a mask ROM, a one-time programmable memory, which implies that the ROM contents cannot be modified.",
"While creating the image for secure boot, each successive component of the boot-chain is signed (RSA), hashed (HMAC-SHA256), and encrypted (AES256).",
"The boot up starts with the BootROM code loading the FSBL, and continues serially with the FSBL loading the FPGA bitstream and the software.",
"During every secure boot up, the chain of trust is established by the successive verification of signature (authentication), MAC checksum (integrity) and decryption (confidentiality) of all software, i.e., FSBL, bitstream, u-boot, OS, and user applications.",
"This procedure prevents an adversary from tampering with software or the FPGA bitstream file.",
"Zynq SoC contains hard IP cores for AES decryption and HMAC computation.",
"As a result, the difference between the boot time of secure and regular boots is negligible [46].",
"The hardware stack is protected by the on-chip PUF.",
"Incorporating a PUF into a chip makes the chip tamper evident [47].",
"Since PUF behavior corresponds to the underlying silicon fabric, any tampering with the fabric modifies the PUF behavior, thereby modifying the camera fingerprint.",
"This leads to generation of incorrect signing and encryption keys thereby incorrect signature and cipher text, which is detected by the verifier.",
"Figure: Chain of trust for secure boot of Zynq7010 SoCData with confidentiality, integrity, authenticity and freshness guarantees is then forwarded to communication module for uploading.",
"The prototype merely demonstrates a proof of the concept and can be extended with wireless communication capabilities such as WiFi."
],
[
"Implementation & Evaluation",
"The section evaluates the trusted sensing and privacy-aware reporting and secure node approaches for IoT-based smart services.",
"The section is divided into five parts: Since both approaches utilize the PUF framework to generate and store sensor bound keys, we present the implementation results of the PUF framework in Section REF .",
"The privacy-aware trusted sensing scheme is evaluated in two parts: First, Section REF presents the trusted image sensor prototype and evaluates the overhead on the sensor with respect to storage, latency and hardware.",
"Second, Section REF evaluates the communication overhead on the mobile device for privacy-aware reporting of the sensed data from trusted sensors.",
"Section REF evaluates the storage, latency, hardware, and communication overhead on the secure camera node (the prototype presented in Section REF ) incurred due to secure node approach.",
"Section REF discusses security properties and limitations of both schemes.",
"The experimentation setup used for the evaluation of the privacy-aware trusted sensing and the secure node approaches is as follows: In order to verify the feasibility of a PUF-based approach for sensors, we implemented the PUF framework on three sensing platforms of different complexities, namely (i) Atmel ATMEGA328P, a lightweight 8-bit MCU running at 8 MHz, (ii) ARM Cortex M4, a 32-bit MCU running at 168 MHz, and (iii) Xilinx Zynq7010 SoC with FPGA and a 32-bit dual core ARM Cortex A9 processor core running at 666 MHz.",
"For objective comparison of the privacy-aware trusted sensing and the secure node approaches, we prototyped a trusted image sensor and a secure camera node using the same platform as shown in Fig.",
"REF .",
"The platform comprises a 5MP OV5642 image sensor module and MicroZed board, which houses Zynq7010 SoC clocked at 666MHz, 1 GB external RAM for frame buffering and a gigabit Ethernet interface to upload video footage.",
"A custom board (mounted below the Micozed board) was designed to interface the image sensor with the MicroZed board and regulate power to both the modules."
],
[
"PUF Framework",
"Various PUF sources are inherent to a typical sensor including SRAM PUF, RO PUF, and sensor-specific PUFs [35], [34], [48].",
"Since we target a broad range of sensors, we seek to identify PUF sources that are commonly available on most sensors such as SRAM and RO PUFs.",
"A PUF is characterized by three quality parameters, i.e., randomness, reliability, and uniqueness.",
"Hamming weight ($HW$ ), the indicator of PUF randomness, measures the deviation of a PUF output from uniform distribution.",
"For an $n$ bit response $r$ obtained from a chip $U$ , $HW$ is given by: $HW(r)=\\frac{1}{n} \\sum \\limits _{i=1}^{n} b_{i}\\cdot 100\\ \\%$ where $b_i$ is the $i^{th}$ binary bit in an $n$ bit PUF response.",
"For a uniformly distributed PUF response, $HW$ should be $50\\ \\%$ .",
"The change in PUF response over varying environmental and operating conditions depicts the (lack of) reliability.",
"The change is referred to as PUF error-rate or noise and is measured in terms of the inter-Hamming distance.",
"An $n$ bit reference response ($r_{ref}$ ) is extracted from the chip $U$ at the room temperature and standard operating conditions.",
"Multiple responses from the same PUF are obtained under different environmental and operating conditions (e.g.",
"varying temperature or supply voltage) and are denoted by $r_i$ .",
"A number of samples of $r_i$ are taken for each combination of the environmental and operating conditions.",
"The PUF error-rate is measured as the average intra-Hamming distance ($HD^{intra}$ ) over $N$ samples obtained under different environmental and operating conditions.",
"For the chip $U$ , it is defined as: $HD^{intra}(r_{ref}, r_{1}, r_{2}, \\cdots , r_{N-1} )=\\frac{1}{N}\\sum \\limits _{i=1}^{N} \\frac{HD^{intra}(r_{ref}, r_{i})}{n} \\cdot 100\\ \\%$ The uniqueness property of a PUF measures how unique are the signatures generated from different chips using the same PUF circuit.",
"The average inter-Hamming distance ($HD^{inter}$ ) of the PUF responses is a commonly used measure of uniqueness.",
"The average $HD^{inter}$ for a group of $M$ chips is defined as the average of all possible pair-wise $HD^{inter}$ among $M$ chips: $HD^{inter}(U_1, U_2, \\cdots ,U_M)= \\frac{2}{M(M-1)}\\sum \\limits _{U_1=1}^{M-1} \\sum \\limits _{U_2=U_1+1}^{M} \\frac{HD^{inter}(r_{U_1}, r_{U_2})}{n} \\cdot 100\\ \\%$ For a truly random PUF output, $HD^{inter}$ should be close to $50\\ \\%$ .",
"We implemented PUFs on three platforms: (i) Atmel ATMEGA328P, a lightweight 8 bit MCU, (ii) ARM Cortex M4, a 32 bit MCU, and (iii) Xilinx Zynq7010 SoC with re-programmable logic and a dual core ARM Cortex A9.",
"These platforms are ideally suited as sensor controllers (see Fig.",
"REF ) for a broad range of sensors.",
"SRAM PUF taps the randomness from the start-up values of the SRAM cells.",
"Once these start-up values are read out, the SRAM can be used as regular memory.",
"As a result, SRAM PUF implementation does not incur any hardware overhead and is therefore preferred for implementation over the RO PUF.",
"However, if SRAM is either not available on board or gets initialized with fixed values during boot up, the RO PUF is implemented.",
"The power-up state of the SRAM cells on the ATMEGA328P and the ARM Cortex M4 show PUF behavior where as the SRAM on the Zynq7010 SoC gets initialized with fixed values during boot up.",
"Therefore, we implemented the SRAM PUF on the ATMEGA328P and the ARM Cortex M4 MCUs and the RO PUF on the Zynq7010 SoC.",
"For the SRAM PUF implementation, 1 kilobytes SRAM on the Atmel ATMEGA328P and 15 kilobytes SRAM on the ARM Cortex M4 were read out and characterized for PUF behavior.",
"The PUF quality parameters were computed from 100 PUF responses (i.e., start-up values of the SRAM cells) obtained at room temperature.",
"Figs.",
"REF and REF depict the error-rate and the uniform distribution of PUF-responses measured as $HD^{intra}$ and $HW$ , respectively.",
"The average and maximum values of $HD^{intra}$ were measured as $3.4\\ \\%$ and $7.2\\ \\%$ for the ATMEGA328P, and $7.66\\ \\%$ and $9.16\\ \\%$ for the Cortex M4.",
"The randomness of the PUF responses for the ATMEGA328P and the Cortex M4 were measured as $63.5\\ \\%$ and $63.96\\ \\%$ , respectively.",
"Furthermore, we implemented the RO PUF, comprised of 1040 3-stage ring oscillators and two 16 bit counters, on the FPGA part of the Zynq7010.",
"1040 ROs can be arranged into 1039 RO independent pairs.",
"To obtain a PUF response, a RO pair is selected.",
"The frequencies generated by the selected ROs, increment the respective counters.",
"The marginal difference between the two frequencies causes one counter to overflow before the other.",
"At the overflow of one counter, the 16 bit value of the other counter is read out.",
"Three bits at positions 8, 9, and 10 in the counter value are read out as the output of the RO pair comparison [49].",
"Therefore, 1039 RO pairs generate a 3117 bit response.",
"We evaluated the RO PUF for $HD^{intra}$ , $HW$ and $HD^{inter}$ .",
"The quality parameters for the Zynq7010 were computed from a total of 800 responses measured over a temperature range of $0-60^{\\circ }\\mathrm {C}$ (i.e., 100 responses at $10^{\\circ }$ intervals, and an additional 100 responses at $25^{\\circ }\\mathrm {C}$ ).",
"The average and maximum $HD^{intra}$ were computed as $3.6\\ \\%$ and $6.97\\ \\%$ , respectively.",
"The average $HW$ was $53.95\\ \\%$ .",
"Furthermore, the same PUF was implemented on 10 Zynq7010 boards, and $HD^{inter}$ was $51.1\\ \\%$ .",
"Instead of designing three separate helper data algorithms (HDAs) for correcting $7.2\\ \\%, 9.16\\ \\%,$ and $6.97\\ \\%$ error-rates of the three PUF, we designed a common HDA that can correct an error-rate of up to $10\\ \\%$ .",
"Guajardo et al.",
"[50] investigated different error correcting codes that are suitable for HDA.",
"The error-correcting code determines the number of required PUF response bits and hence the size of PUF.",
"We evaluated two cases: (i) a simple code using BCH (492,57,171) and (ii) a concatenated code comprising Reed Muller (16,5,8) and Repetition (5,1,5) codes.",
"The failure rate for PUF-based key reconstruction using both these cases is $\\le 10^{-6}$ .",
"Once an error correcting code $(n,k,d)$ is selected, the number of PUF response bits required to generate an $l$ bit key is given by $\\frac{n}{k}\\cdot l$ .",
"According to the PUF framework, the helper data $W$ has the same length as the PUF response.",
"Since, an RO pair generates 3 response bits, $(\\frac{n}{k}\\cdot l)$ bits are generated by $(\\frac{n}{3k}\\cdot l)$ RO pairs or $(\\frac{n}{3k}\\cdot l+1)$ ROs.",
"In our design, each RO is implemented as 3-stage (3 NOT gates), the RO PUF's hardware size is given by $(\\frac{n}{k}\\cdot l+3)$ logic-gates.",
"For a concatenated code $(n_1,k_1,d_1)||(n_2,k_2,d_2)$ , the hardware and helper data sizes can be computed by using the same procedure, with $n=n_1$ and $k=k_2$ .",
"Figure: PUF characterization of (a) 1kB1 kB SRAM on Atmel ATMEGA328P 8-bit MCU, (b) 15kB15 kB SRAM on ARM Cortex M4 32-bit MCU, and (c) RO PUF comprised of 1040 3-stage ring oscillators implemented on re-programmable fabric of Xilinx's Zynq7010 SoC.",
"The PUF quality parameters for (a) and (b) are calculated over 100 PUF responses at room temperature.",
"The mean and maximum values of the intra-Hamming distance ( HD intra )(\\mathrm {HD^{intra}}) representing the PUFs' error-rate for (a) are 3.4%3.4 \\% and 7.2%7.2 \\%, and for (b) 7.66%7.66\\% and 9.16%9.16\\%, respectively.",
"The randomness of PUF responses, measured as the mean Hamming weight ( HW )(\\mathrm {HW}), for (a) and (b) are 63.5%63.5 \\% and 63.96%63.96 \\%, respectively.",
"The quality parameters for (c) are calculated from a total of 800 responses taken over temperature range 0-60 ∘ C0-60^{\\circ }\\mathrm {C} (i.e., 100 responses at 10 ∘ 10^{\\circ } intervals plus a 100 at 25 ∘ C25^{\\circ }\\mathrm {C}) where a response ≈1560\\approx 1560 bits.",
"The mean and maximum values of HD intra \\mathrm {HD^{intra}} are 3.6%3.6 \\% and 6.97%6.97 \\%, respectively.",
"The average HW \\mathrm {HW} is 53.95%53.95 \\%In the trusted sensing and privacy-aware reporting approach, a sensor uses 160 bit $sk$ to sign the sensor readings (sensed data attestation) using PUF-based Cert-IBS with BLS as underlying signature scheme where as the secure node approach uses a 128 bit $k_E$ to encrypt the frames using AES128 algorithm and a 160 bit $sk$ to sign the footage using the PUF-based Cert-IBS with BLS as underlying signature scheme.",
"The PUF framework was implemented on all three platforms.",
"In Table REF we present the overhead incurred by the PUF framework to generate 128 and 160 bit keys, the latency, hardware and memory components.",
"Table: Implementation results of the PUF framework for 128 bit and 160 bit keys generation and storage"
],
[
"Trusted Image Sensor",
"The trusted sensor prototype (Fig.",
"REF ) comprises an $OV5642$ image sensor array (sensing unit) and Zynq7010 SoC (sensor controller) running at 666 MHz.",
"The sensing unit was configured for $640\\times 480$ resolution and $YUV422$ color-space.",
"The evaluation matrix for the trusted image sensor comprises three components of the overhead incurred due to the PUF-based Cert-IBS, i.e., storage, latency, and hardware.",
"During the enrollment phase of the privacy-aware trusted sensing scheme, TA binds a signing key $sk$ to the trusted sensor using the PUF framework and provides a certificate on the public key $pk$ .",
"Using the asymmetric version of the BLS signature scheme as the $SS$ in the PUF-based Cert-IBS, we obtain the sizes of $sk=160$ bits and $cert=480$ bits ($pk=160$ bits, $Sign_{msk}=320$ bits).",
"The RO PUF with BCH error correcting code based framework (summarized in Table REF ) was implemented on the FPGA part of the Zynq7010 SoC.",
"The PUF-based secure key generation and storage framework for a 160 bit $sk$ incurs 1381 bits of memory for the storage of helper data ($W$ ) and 1384 logic-gates of hardware overhead.",
"During the trusted sensing and privacy-aware reporting phase, the trusted sensor performs the sensed data attestation.",
"During this step, a fresh image frame is read from the $OV5642$ image sensor, a MAC checksum is computed over the frame using HMAC-SHA256 algorithm.",
"The checksum is then signed using the PUF-based Cert-IBS with asymmetric version of BLS as the underlying signature scheme.",
"The sensor then outputs the tuple $(M,I,\\sigma , cert)$ .",
"The storage, hardware and latency overhead incurred by the privacy-aware trusted sensing approach on the sensor is summarized as follows: Storage.",
"The trusted sensor stores helper data $W$ and certificate $cert$ .",
"For an $l$ bit key, the helper data $W$ size is given by $\\frac{n}{k}\\cdot l$ , where $n$ and $k$ are parameters of the chosen error correcting code.",
"For a 160 bit key generation framework using $BCH\\ (492,57,171)$ , $l=160$ bits, $n=492$ and $k=57$ , which gives the size of $W=1381$ bits.",
"The $cert$ is given by 480 bits.",
"This amounts to a total of 233 bytes of storage overhead.",
"Latency.",
"Key extraction using the PUF framework is performed only at the start-up and therefore run-time latency overhead is only incurred by the sensed data attestation phase of PUF-based Cert-IBS scheme.",
"During this phase, the sensor controller obtains a new frame from the image sensor, computes a MAC checksum over the frame and signs the checksum using PUF-based Cert-IBS with BLS as underlying signature scheme.",
"Pairings based cryptography library [51] was leveraged for implementation of sensed data attestation.",
"The sensor controller, Zynq7010 SoC, requires $2.5\\ \\mathrm {ms}$ to MAC a frame and $6.27\\ \\mathrm {ms}$ to sign the MAC at $640\\times 480$ resolution and YUV422 color-space, which enables the prototype trusted image sensor to secure 114 frames per second.",
"Hardware.",
"The hardware overhead incurs only in case of an RO PUF implementation and results in 1384 logic-gates of hardware overhead to generate a 160 bit signing key (cp.",
"Table REF )."
],
[
"Privacy-Aware Reporting",
"The evaluation of privacy-aware reporting of the trusted sensors readings aims to compute the communication overhead incurred by the proposed scheme on the user devices.",
"For ease of description, the trusted sensing and privacy-aware reporting scheme was explained using the symmetric pairings settings (Table REF ).",
"However, symmetric pairing can only be realized using supersingular elliptic curves.",
"Supersingular elliptic curves $E$ are defined over the finite field $\\mathbb {F}_q$ , where $G$ and $G_T$ are the groups of elliptic-curve points on $E(\\mathbb {F}_q)$ and $E(\\mathbb {F}_{q^d})$ , respectively and $d$ is called the embedding degree of $E$ .",
"After recent successful attacks on supersingular curves of small characteristic [52], the available supersingular curves, given by $y^2=x^3+x$ over the field $\\mathbb {F}_q$ for some prime $q=3\\ (\\mathrm {mod}\\ 4)$ , have a small embedding degree of 2.",
"This implies that the base field $\\mathbb {F}_q$ must be large enough to obtain sufficient discrete-log security in $\\mathbb {F}_q^2$ .",
"For instance, to obtain 1024 bit discrete-log security in $\\mathbb {F}_{q^2}$ ($\\lceil log_2{q^2}\\rceil \\ge 1024$ ), $q$ must be at least 512 bits ($\\lceil log_2{q}\\rceil \\ge 512$ ).",
"Since $G \\subseteq E(\\mathbb {F}_q)$ , this gives us the size of a group element in $G$ to be 512 bits.",
"Therefore, the communication overhead on a mobile device incurred by our scheme in the symmetric settings is given by $(6Q+12)512$ bits.",
"However, the same level of security can be achieved by more efficient curves using the asymmetric pairings setting.",
"Therefore, we have implemented the proposed scheme using asymmetric pairings setting.",
"First, we briefly outline our scheme in the asymmetric settings to compute the overhead in terms of group elements and then we select the curves to compute the communication overhead on the mobile device.",
"In the asymmetric setting, the PUF-based Cert-IBS uses asymmetric version of BLS signature scheme.",
"Here, the size of a signature, i.e., $Sign_{sk}(M)$ and $Sign_{msk}(I,pk)$ is given by one element in $G_1$ whereas the public half of the signing key, i.e., $pk$ is given by one element in $G_2$ .",
"Proof generation uses the same framework, $P_\\mathrm {NIWI}$ , but follows the asymmetric construction based on the $\\mathrm {SXDH}$ assumption (cp.",
"Section 9 of [37]).",
"Here, commitment to the elements of witness in $G_1$ and $G_2$ costs ${G_1}^2$ and ${G_2}^2$ , respectively.",
"The proof consists of two parts: $\\pi \\in {G_1}^4$ , and $\\theta \\in {G_2}^4$ .",
"Given $Q$ sensors' readings with all signatures aggregated into $\\bar{\\sigma }$ , the prover $P_{NI}$ of $P_{NIWI}$ outputs $(\\lbrace \\mathbf {d}(I_i),\\ \\mathbf {c}(pk_i)\\rbrace _{i=1}^Q,\\ \\mathbf {c}(\\bar{\\sigma }), \\pi , \\theta ))$ which amounts to an overhead of $(6+2Q)G_1 +(4+2Q)G_2$ .",
"Barreto-Naehrig (BN) curves [53] are ideal for an asymmetric setting as they offer much higher security with more efficient curves.",
"An efficient BN curve [51] allows us to represent elements of $G_1$ with 160 bits, $G_2$ with 320 bits and $G_T$ with 1920 bits.",
"This offers 1920 bit of discrete-log security in $G_T$ .",
"In the asymmetric setting, the communication overhead is given as $(6+2Q)160+(4+2Q)320$ bits.",
"For a concrete comparison, we evaluated the communication overhead for two real-word participatory sensing (PS) applications: $\\mathrm {Google\\ Street\\ View}$ [54] and $\\mathrm {Wikicity}$ [55].",
"The former models a typical PS application that requires its participants to submit multiple sensor readings.",
"Moreover, the payload includes multimedia data.",
"The later models the worst-case scenario with respect to overhead, because it requires its participants to submit only a single sensor reading (i.e., scalar value).",
"$\\mathrm {APP\\ 1:\\ Google\\ Street\\ View}$ [54] which requires the participants to capture and upload geotagged images to the Google Maps using their smartphone (user device) cameras and GPS receivers (sensors).",
"Here $Q=2$ , where $M_1$ is image from the camera and $M_2$ location reading from the GPS receiver.",
"Given a sensor configuration of $640\\times 480$ resolution, RGB color-space, 8 bits per color-plane, image size $M_1=900$ kilobytes is given.",
"Typically, the GPS receiver provides location data in NMEA format.",
"The maximum length of an NMEA sentence ($M_2$ ) is 82 bytes.",
"Therefore, the total size of the payload (i.e., $M_1+M_2$ ) is given by 900 kilobytes (approximately).",
"The communication overhead incurred on a smartphone for reporting ($M_1+M_2$ )to $\\mathrm {Google\\ Street\\ View}$ server amounts to 520 bytes that is $0.00056\\ \\%$ of the payload.",
"$\\mathrm {APP\\ 2:\\ Wikicity}$ [55] is an urban planning application that periodically captures the location of the citizens leveraging smartphones and vehicles-embedded location sensors, to monitor their reactions to various events happening in the city.",
"Here, $Q=1$ and $M_1$ is 82 bytes of location data.",
"For $\\mathrm {Wikicity}$ , the communication overhead amounts to 400 bytes.",
"$\\mathrm {APP\\ 1}$ models a typical multimedia report where as $\\mathrm {APP\\ 2}$ determines the minimum communication overhead incurred by the privacy-aware trusted sensing scheme on a user device."
],
[
"Secure Camera Node",
"This section presents the implementation results of the prototype secure camera node as discussed in Section REF .",
"The node uses a 128 bit key, $k_E$ , to encrypt each frame using AES128 and a 160 bit key, $sk$ to sign the footage using BLS signature scheme.",
"A RO PUF and BCH error correcting code based framework (Table REF ) was implemented for the generation of $sk$ and $k_E$ .",
"The total overhead in terms of latency, storage, hardware and communications incurred due to sensing, processing, security and communication tasks of the secure node approach on the camera node is given below.",
"Latency.",
"Since key extraction and secure boot is performed only once at power-up, they do not incur latency during runtime.",
"Time-stamping and signing are performed once per footage.",
"At a resolution of $640\\times 480$ , the runtime latency due to the application framework was measured as 27 ms (i.e., 37 frames per second).",
"However, at the given resolution, the image sensor outputs 30 frames per second, which limits the throughput of the prototype to 30 frames per second.",
"The running times measured for the individual tasks of the camera application framework are given in Table REF .",
"Table: Running times for individual modules of application framework for the Zynq7010 SoC-based secure camera node Storage.",
"Breakdown of the storage overhead incurred by the individual components of the proposed security mechanism is given by the second and third rows of Table REF .",
"The size of the helper data corresponding to 160 bit $sk$ and 128 bit $k_E$ using the BCH codes is given by 1381 bits and 1105 bits, respectively (Table REF ).",
"The node also needs to store $cert$ which requires 480 bits (60 bytes).",
"The Zynq7010 SoC stores three keys, which are used during the secure boot: a private 256 bit key for AES256, a 256 bit private key for HMAC-SHA256, and a 2048 bit modular used by the RSA.",
"The size of a signed and encrypted partition is same as the unencrypted and unsigned one.",
"Therefore, the storage overhead incurred by the secure boot functionality amounts to 320 bytes.",
"The Zynq7010 devices employ RSA for authentication of the boot partitions, which uses large key sizes (2048 bits).",
"This can be replaced by a signature scheme based on elliptic curves such as elliptic curve digital signature algorithm (ECDSA), which provides the same level of security with a 256 bit key and produces much smaller signatures.",
"In comparison to memory consumed by the application logic, the storage overhead of the secure node mechanism is negligible.",
"For example, the size of a double frame buffer (Table REF ) used by the processing and communication tasks (Fig.",
"REF ) incur orders of magnitude greater storage overhead than the secure node mechanism.",
"Table: Memory consumption by the individual modules of the application framework for the Zynq7010 SoC-based secure camera node Hardware.",
"Hardware overhead incurs due to the RO PUF implementation in the FPGA part of the SoC.",
"Table REF gives the total hardware overhead for generating 160 bit $sk$ and 128 bit $k_E$ to 2492 logic-gates, which is negligible as compared to the current state of the art security chip solutions.",
"Communication.",
"Given an assisted living scenario, a camera with the proposed secure node mechanism uploads encrypted-MACed-timestamped-signed footage, i.e., $\\lbrace C_i,\\tau , \\sigma \\rbrace _{i=i..N}$ to the edge or cloud storage server.",
"The encrypted frame $C_i$ has the same size as the original image.",
"A 256 bit timestamp $\\tau $ and a 160 bit signature $\\sigma $ are added to the footage for uploading.",
"For a footage comprised of $N$ frames, the total communication overhead incurred on the secure camera node amounts only to 416 bits, which amounts to $0.008 \\%$ of a single frame size."
],
[
"Security and Privacy Properties",
"This section discusses the security and privacy properties of the two schemes namely trusted sensing and privacy aware reporting and secure node.",
"We also identify the limitations of our approaches and the additional measures that can be taken to over the limitations."
],
[
"Trusted Sensing and Privacy-Aware Reporting",
"The correctness of the trusted sensing and privacy-aware reporting scheme follows from the correctness of the PUF-based Cert-IBS scheme and the $P_\\mathrm {NIWI}$ system.",
"The system parameters $\\Sigma $ , $crs$ and $H$ are generated by the trusted authority.",
"Since, any compromise to the trusted authority nullifies the trust and privacy guarantees, we emphasize that the offline nature of the authority in our scheme greatly reduces the risk of compromise.",
"The security and privacy properties of the privacy-aware trusted sensing scheme are as follows: Sensor-bound Secure Key Storage.",
"The on-chip PUF serves a secure key storage for the sensor.",
"The key is generated from the hardware on sensor power up.",
"During the off state, the key exist in form of unreadable variations introduced in the hardware by the CMOS manufacturing process.",
"In comparison with a secure memory alternative, PUF offers a cost effective and more lightweight secure storage.",
"Moreover, the PUF framework binds the key to the sensor hardware, which never leaves the sensor thereby minimizing the risk of a key compromise.",
"Trusted Sensing.",
"Trusted sensing refers to the sensing techniques that ensure security guarantees about the integrity, authenticity and freshness of the sensed data.",
"The scheme withstands the sensed data corruption attacks due to compromised OS running on mobile device.",
"The OS receives sensors' readings accompanied by the commitments and proofs of $P_\\mathrm {NIWI}$ from the root virtual machine (see Fig.",
"REF ).",
"In order to inject fabricated data at the OS level, an attacker has to produce a valid $P_\\mathrm {NIWI}$ proof of knowledge on a valid PUF-based Cert-IBS signature.",
"A uf-cma secure PUF-based Cert-IBS implies that there is negligible probability that an attacker produces a valid PUF-based Cert-IBS signature.",
"Further, the soundness of $P_\\mathrm {NIWI}$ implies that it is impossible to generate a valid proof without satisfying the PUF-based Cert-IBS verification equations.",
"Further, any manipulation in the values of readings, commitments or proofs results in unsuccessful $P_\\mathrm {NIWI}$ verification.",
"Authenticity of each reading is ensured by signing the data with a unique, sensor-bound private key with-in the sensor.",
"Freshness is ensured by including a time-stamp before signing the reading (sensed data attestation).",
"In the participatory sensing scenario, freshness of sensed data is ensured by including a time-stamp, obtained from the smartphone GPS receiver, before the signature aggregation step.",
"Privacy-Aware Reporting.",
"Anonymity of a user follows from the witness indistinguishability of $P_\\mathrm {NIWI}$ .",
"Given $N$ mobile devices equipped with the privacy-aware trusted sensing scheme, contributing sensed data to an application server, each device is $N$ -anonymous with respect to the server.",
"The scheme provides CPA-anonymity [56] since it does not provide the server oracle access to extract the witness from the proof using the extraction algorithm $X_{NI}$ of $P_\\mathrm {NIWI}$ .",
"Since the server is assumed honest-but-curious threat to privacy, the CPA-anonymity suffices the privacy requirements.",
"Tamper Resistance.",
"Any physical tampering is detected due to the on-chip PUF since the PUF extracts a sensor fingerprint as a function of intrinsic details of the hardware.",
"Any modification in hardware is detected as it results in generation of incorrect key.",
"Identity Management.",
"Every sensor is assigned a unique physical identifier which could be used to track and monitor the state of a node, manage their access privileges in an all open network such as IoT.",
"Next, we enlist some limitations of our scheme and identify some additional measures that may be appended with our scheme to overcome these limitations: Linkability and Location Privacy.",
"In order to ensure effective user privacy at the system level anonymization at the lower layers of the communication stack must be ensured since multiple reports can be trivially linked using the IP address or physical address of a user device (e.g., smartphone).",
"Techniques such as mix networks, onion routing, IP rotation, and MAC address randomization can be leveraged for this purpose.",
"For location privacy, appropriate location blurring technique such as cloaking, perturbation etc.",
"must be used.",
"Event Faking.",
"Event faking attacks refer to capturing of sensed data from the undesirable environment.",
"The proposed mechanism does not address these attacks.",
"However, these can be thwarted by using statistical techniques on the server side."
],
[
"Secure Node",
"Data and node security properties of the secure node scheme are as follows: Secure Key Storage.",
"Secure storage of the keys is ensured by an on-chip PUF.",
"On camera power up, the keys are generated from intrinsic variations of the hardware structure and are loaded into cache.",
"When the camera device is off, the keys exist in form on unreadable variations introduced in the hardware by the CMOS manufacturing process.",
"Compared to secure memory alternatives, PUF offers much cheaper secure storage.",
"Platform-bound Keys.",
"The PUF framework binds signing and encryption keys to the camera platform.",
"The signing key never leaves the platform thereby minimizing the risk of the key compromise.",
"Data Nonrepudiation.",
"All video data leaving the camera carry integrity and authenticity guarantees.",
"Authenticity is ensured by signing the video data using platform bound signing key.",
"PUF-based Cert-IBS with BLS [45] as underlying standard signature scheme is existentially unforgeable under chosen message attack (uf-cma secure), which is the standard security notion for a digital signature algorithm.",
"Any modification or fabrication of data during delivery or archival can be detected at the caretaker monitoring device.",
"Spoofing using offline images can be addressed by using multiple sources for event detection, e.g., on-camera motion detection and sound detection.",
"Data Confidentiality.",
"Privacy of the monitored individual(s) is ensured by end-to-end encryption of each frame using AES128 algorithm.",
"Access Authorization.",
"Secure key exchange and key storage on caretaker monitoring device ensure that only legitimate caretaker can access the decrypted video data.",
"Data Protection Lifetime.",
"Data is protected close to the source (sensor) and the security guarantees on the data remain valid for the entire lifetime of the data (i.e., during transmission, storage on cloud and delivery to monitoring device of caretaker).",
"On the monitoring device, the data is consumed after successful verification of the security guarantees.",
"Camera Firmware Protection.",
"Each component of camera node's boot-chain (i.e., BootROM Code, FSBL, FPGA Bitstream, U-Boot, OS and Apps) is hashed, signed and encrypted using the HMAC-SHA256, RSA, and AES256 algorithms.",
"At every boot-up, the integrity and authenticity of camera firmware is verified by through signature verification of each partition.",
"Furthermore, the encryption of the boot partitions ensures unclonability of the camera firmware, since the decryption key exists within the security perimeter of the camera node.",
"No hardware without the decryption key is capable of decrypting the firmware.",
"Physical Security.",
"First, on-chip PUF offers resistance against hardware tampering of the camera hardware.",
"Since the PUF extracts device fingerprint as a function of intrinsic details of the hardware, hardware tampering is detected as it results in incorrect device fingerprint and keys.",
"Second, in comparison with a TPM-based solution, where data is transferred from host processor to TPM chip (external to the host), where data protection mechanisms are applied.",
"This results in exposed interface with unprotected data which can be tapped to bypass security mechanism.",
"With incorporation of PUF in host SoC, these exposed interfaces are eliminated resulting in better physical security.",
"Next, we enlist some limitations of our scheme and identify some additional countermeasures that may be appended with our scheme to overcome these limitations: Side Channel Attacks.",
"First, although security keys are generated securely inside the SoC, key generation on every power-up opens up electromagnetic and power side channels.",
"Analysis of the side channel information can result in partial recovery of keys at the hands of attackers.",
"Approved effective techniques, masking (e.g, reversible process in which intermediate values of variables are randomized by masked with random numbers) and hiding (e.g., use of dual rail logic to flatten the data dependent leakage) can be leveraged to thwart side channel attacks.",
"Second, data upload is triggered upon every event detection, the transmission pattern of the camera opens up another side-channel that leaks e.g., whether or not someone is at home.",
"Transmitting dummy data at random intervals is a simple countermeasure that can be added to the system to mitigate this threat.",
"Denial of Service.",
"Denial of service attacks by (i) the cloud, such as deleting the archived data, blocking the downloads or (ii) a third party, such as corrupting the video or control data in transit or storage are not addressed by this scheme.",
"Monitoring Device Security.",
"The scheme uses symmetric key encryption (i.e., identical keys for encryption and decryption) since symmetric encryption is orders of magnitude faster and less power hungry than an asymmetric key encryption.",
"However, symmetric key encryption requires secure key exchange between the camera and monitoring device and secure storage of key in the monitoring device.",
"Key exchange is done by the caretaker in a private space using a local interface, so the risk of key is relatively low.",
"On monitoring device with untrusted software stack, virtualization, secure key vault or PUF can be used to securely store the key.",
"To summarize, incorporation of trusted sensing and privacy-aware reporting solution in the mobile devices can protect the participatory sensing applications against data pollution and personal privacy leakage attacks by ensuring three security guarantees: First, the sensed data remotely collected from ubiquitous commodity sensing devices carry integrity (i.e., no malicious manipulation of the data), authenticity (i.e., the data is originated from an authentic hardware sensor), and freshness (i.e., the data is recent and no adversary replayed an old message).",
"Second, trusted sensing and privacy-aware reporting scheme ensures anonymization and unlinkability of multiple submissions from the same device.",
"Anonymization is given by the number of devices registered with or contributing to the application server.",
"This prevents the server from creating personal profiles of the participating individuals.",
"Incorporation of effective privacy protection mechanism would encourage increased user participants in the participatory sensing applications which is currently hindered due to privacy concerns.",
"Third, the scheme improves the physical security of the sensors.",
"On-chip PUF enables resistance against hardware tampering of the sensors.",
"The trusted sensing and privacy-aware reporting approach incurs 233 bytes of storage, $8.77$ ms of latency and 1384 logic-gates of hardware overhead on a sensor.",
"To generalize, the latency overhead depends on the sensor clock and nature of the sensed data whereas the storage and hardware costs are fixed for all sensors.",
"The communication overhead on the mobile device, for submitting $Q$ readings to an application server, is given by $(6+2Q)160+(4+2Q)320$ bits.",
"Participatory sensing application scenario was merely considered to illustrate the approach.",
"However, the solution addresses all IoT applications that collect raw sensed data from commodity sensing devices to centralized server(s) and processing of the data takes place at the application server(s).",
"Implementation of the secure node approach in visual monitoring applications thwarts the illegitimate data access and/or manipulation and illegitimate control (over nodes and/or data) threats in these applications.",
"The mechanism ensures integrity, authenticity, freshness, and confidentiality of the sensed data meanwhile allowing for processing of the data on the nodes.",
"Threats to the personal privacy of the monitored individuals, either from a malicious application server or an unknown adversary, are addressed by ensuring data confidentiality and access authorization.",
"The solution also protects the camera hardware and software against hardware tampering, firmware manipulation, and firmware cloning attacks.",
"This is particularly useful in cases where camera (or other sensor) nodes are deployed in unguarded open spaces.",
"The secure node approach for assisted living scenario incurs 691 bytes of storage, 27 ms of latency, 2492 logic-gates of hardware, and 416 bits of communication overhead on a camera node, configured for $640\\times 480$ YUV422 format.",
"The latency overhead depends on the node clock, type of sensor whereas the storage, hardware, and communication costs are same for any sensor node.",
"The secure node ensures sensed data, personal privacy, and sensor node protection for the IoT applications.",
"The security solution offers more flexibility as compared to trusted sensing and privacy-aware reporting, as it additionally allows for the processing of sensed data on the sensor nodes.",
"The solution is ideally suitable for visual monitoring applications however, it can be leveraged for other IoT applications that require processing of the sensed data locally on the sensor nodes."
],
[
"Conclusion",
"Sensors are the largest and the most common source of data for the IoT-based smart services.",
"Trustworthiness of sensed data and users' privacy are essential security requirements for these services.",
"We presented lightweight and effective approach to protect the sensor nodes, sensed data and users' privacy for applications that require delivery of raw data to the server for processing and applications that require processing of sensed data locally on the sensor nodes.",
"Both solutions ensure integrity, authenticity and freshness of sensed data, integrity of sensor hardware and usage privacy of the sensors.",
"The scheme does not require any additional secure hardware and can be mapped on to the existing sensors' resources.",
"Experimental evaluation of both solutions revealed that the proposed scheme incurs only insignificant latency, storage, hardware and communication overhead on the sensor nodes.",
"Using the proposed sensors can effectively thwart sensed data pollution and privacy leakage attacks.",
"However, there are a number of open challenges that can be addressed as extension of this work.",
"First open research problem is the privacy versus utility trade-off.",
"During the privacy preservation process, the utility of the data diminishes as sensitive information such as the uniquely identifying information is removed, transformed, or distorted to achieve anonymity or confidentiality.",
"Identifying the equilibrium between sensed data privacy and utility in IoT applications is an open challenge.",
"Towards this goal, exploring other nuances of privacy between end-to-end encryption and full access could provide interesting insight.",
"Second, the trusted sensor and secure node prototypes only demonstrate the proof of the concept however they were not optimized for performance.",
"A number of performance improvements can be made in this regard.",
"Considering the resource constraints of the sensors, further lightweight security primitives can be investigated.",
"For instance, digital signature schemes require extensive resources that might not be economically feasible for many resource constrained IoT devices.",
"Identification of lightweight security mechanisms to ensure integrity and authenticity guarantees is another open challenge in this regard."
]
] | 1808.08549 |
[
[
"Twisted cohomology pairings of knots III; triple cup products"
],
[
"Abstract Given a representation of a link group, we introduce a trilinear form, as a topological invariant.",
"We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted) triple cup product and the fundamental relative 3-class.",
"Further, we give some examples of the computation."
],
[
"Introduction",
"This paper examines topological invariants of trilinear forms, while the previous papers [10] in this series discussed of bilinear forms.",
"In general, bilinear form arising from Poincaré duality is a powerful method, as in algebraic surgery theory and classification theorems of some manifolds.",
"In contrast, there are not so many studies of trilinear forms.",
"However, some 3-forms and trilinear cup products appear in 3-dimensional geometry together with topological information (see, e.g., [3], [8], [7], [14], [15]).",
"For example, we mention interesting observations from the Chern-Simons invariant (or $\\phi ^3$ -theory) of the form $ \\frac{k }{2\\pi } \\int _{M} \\mathrm {tr}( A \\wedge d A +\\frac{2}{3}A \\wedge A \\wedge A) .",
"$ We give the definition of trilinear pairing (see (REF )) in a general situation where the coefficients are arbitrary.",
"Let $Y$ be a compact 3-manifold with toroidal boundary with orientation 3-class $[Y, \\partial Y] \\in H_3 ( Y ,\\partial Y ;{Z}) \\cong {Z}.$ Choose a group homomorphism $\\pi _1(Y) \\rightarrow G $ , a right $G$ -module $M$ , and a $G$ -invariant trilinear function $ \\psi : M^3 \\rightarrow A $ over a ring $A$ .",
"Then, we can define the composite map $H^1( Y,\\partial Y ; M )^{ \\otimes 3} \\xrightarrow{} H^{3}( Y ,\\partial Y ; M^{\\otimes 3} )\\xrightarrow{} M^{\\otimes 3 } \\xrightarrow{}A.",
"$ Here $M$ is regarded as the local coefficient of $Y$ via $f$ , and the first map $\\smile $ is the cup product, and the second (resp.",
"third) is defined by the pairing with $[Y, \\partial Y]$ (resp.",
"$\\psi $ ).",
"In contrast to this definition, the 3-form (REF ) is considered to be something uncomputable.",
"Actually, it seems hard to concretely deal with the 3-class $[Y, \\partial Y] $ and the cup products.",
"This paper addresses the link case where $Y$ is the 3-manifold which is obtained from the 3-sphere by removing an open tubular neighborhood of a link $L$ , i.e., $Y=S^3 \\setminus \\nu L.$ In fact, if $L$ is a hyperbolic link, we obtain a diagrammatic method of computing the trilinear pairings.",
"To be precise, in Section REF , starting from a link diagram, we define invariants of trilinear forms, and show (Theorem REF ) that the invariant is equal to (REF ), if $L$ is a hyperbolic link.",
"In addition, we also show a similar theorem in the torus case (see Theorem REF ).",
"The point in the theorem is that, in the computation, we do not need describing $[Y, \\partial Y] $ and cup products; thus, this computation is not so hard.",
"In fact, we give some examples; see §.",
"In addition, as an application (Theorem REF ), when $Y$ is a 3-fold covering space of $S^3$ branched along a hyperbolic link $L$ and $M$ is a trivial coefficient, we give a diagrammatic computation of the trilinear pairing (REF ).",
"This paper is organized as follows.",
"Section 2 formulates the trilinear forms in terms of the quandle cocycle invariants, and states the main theorems.",
"Section 3 discusses a relation to 3-fold branched coverings.",
"Section 4 describes some computations.",
"Section 5 gives the proofs of the theorems.",
"Notation.",
"Every link $L$ is smoothly embedded in the 3-sphere $S^3$ with orientation.",
"We write $E_L$ for the 3-manifold which is obtained from $S^3$ by removing an open neighborhood of $L$ ."
],
[
"Results; diagrammatic formulations of the trilinear forms",
"Our purpose in this section is to give a link invariant of trilinear form (Theorem REF ), and to state the main results in §REF .",
"For this purpose, §REF starts by reviewing colorings, and formulates some link-invariants of linear forms.",
"Thorough this section, we fix a group $G$ and a right $G$ -module $M$ over a ring $A$ ."
],
[
"Preliminary; the formulations of the first cohomology",
"We need some notation from [4], [10] before proceeding.",
"Denote $ M \\times G$ by $X$ .",
"Further, define a binary operation on $X$ by $ \\lhd : (M \\times G) \\times (M \\times G)\\longrightarrow M \\times G, \\ \\ \\ \\ \\ \\ (a,g,b,h) \\longmapsto (\\ (a-b)\\cdot h +b, \\ h^{-1}gh \\ ), $ which was first introduced in [4], and satisfies “the quandle axiom\".",
"Furthermore, we choose a link $L \\subset S^3$ with a group homomorphism $f:\\pi _1(S^3\\setminus L) \\rightarrow G$ .",
"Next, we review colorings.",
"Choose an oriented diagram $D$ of $L.$ Then, it follows from the Wirtinger presentation of $D$ that the homomorphism $f$ is regarded as a map $ \\lbrace \\mbox{arcs of $D$} \\rbrace \\rightarrow G$ .",
"Furthermore, a map $\\mathcal {C}: \\lbrace \\mbox{arcs of $D$} \\rbrace \\rightarrow X$ is an $X$ -coloring if it satisfies $\\mathcal {C}(\\alpha _{\\tau }) \\lhd \\mathcal {C}(\\beta _{\\tau }) = \\mathcal {C}(\\gamma _{\\tau })$ at each crossings of $D$ illustrated as Figure REF .",
"It is worth noticing that the set of all colorings is regarded as a subset of the direct product $X^{\\alpha _D }$ , where $\\alpha _D$ is the number of arcs of $D$ .",
"Let $\\mathrm {Col}_X(D_f) $ denote the set of all $X$ -colorings over $f$ , that is, $ \\mathrm {Col}_X(D_{f}):= \\lbrace \\ \\mathcal {C} \\in (M\\times G)^{\\alpha _D } \\ | \\ \\mathcal {C} \\ \\textrm {is an }X\\textrm {-coloring}, \\ \\ p_G \\circ \\mathcal {C} =f \\ \\rbrace , $ where $p_G$ is the projection $ X = M \\times G \\rightarrow G$ .",
"Then, we can easily verify from the linear operation (REF ) that $\\mathrm {Col}_X(D_{f})$ is made into an abelian subgroup of $M^{\\alpha (D)}$ , and that the diagonal subset $M_{\\rm diag} \\subset M^{\\alpha _D } $ is a direct summand in $\\mathrm {Col}_X(D_{f}) $ .",
"Denoting another summand by $ \\mathrm {Col}^{\\rm red}_X(D_{f}) $ , we have a decomposition $ \\mathrm {Col}_X(D_{f}) \\cong \\mathrm {Col}^{\\rm red}_X(D_{f}) \\oplus M_{\\rm diag} .",
"$ The previous paper [10] gave a topological meaning of the coloring sets as follows: Theorem 2.1 ([10]) Let $E_L$ be a link complement in $S^3$ as in §1.",
"Regard the $G$ -module $M$ as a local system of $E_L$ via $f: \\pi _1(E_L) \\rightarrow G$ .",
"Then, there are isomorphisms $ \\mathrm {Col}_{X } (D_{f}) \\cong H^1(E_L , \\ \\partial E_L ;M ) \\oplus M, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm {Col}_{X }^{\\rm red} (D_{f}) \\cong H^1(E_L , \\ \\partial E_L ;M ) .",
"$ Furthermore, let us review shadow colorings [2], [4].",
"A shadow coloring is a pair of a coloring $\\mathcal {C} $ over $f$ and a map $\\lambda $ from the complementary regions of $D$ to $M$ , satisfying the condition depicted in the right side of Figure REF for every arcs.",
"Let $\\mathrm {SCol}_{X}(D_f )$ denote the set of shadow colorings of $D$ such that the unbounded exterior region is assigned by $0 \\in M$ .",
"Notice that, by the coloring rules, assignments of the other regions are uniquely determined from the unbounded region, and admit, therefore, a shadow coloring; we thus obtain a bijection $ {\\rm Col}_X (D_f) \\simeq \\mathrm {SCol}_{X } (D_f).",
"$ Figure: The coloring conditions at each crossing τ\\tau and around each arcs."
],
[
"Invariants of trilinear forms",
"In addition, we will explain Definition REF below, and show Theorem REF .",
"For this, we need two things: first, we take three $G $ -modules $M_1 ,\\ M_2,\\ M_3 $ and the associated $X_i = M_i \\times G. $ Let $A$ be an abelian group.",
"On the other hand, we prepare a trilinear map $\\psi : M_1 \\times M_2 \\times M_3 \\rightarrow A$ over ${Z}$ satisfying the $G$ -invariance, that is, $\\psi (a_1 \\cdot g ,a_2 \\cdot g ,a_3 \\cdot g ) =\\psi (a_1,a_2,a_3 ) ,$ holds for any $a_i \\in M_i$ and $g \\in G$ .",
"Next, let us consider the map $ X_1 \\times X_2 \\times X_3 \\rightarrow A $ by the formula $ \\bigl ( (b_1,g_1),(b_2,g_2),( b_3,g_3) \\bigr )\\longmapsto \\psi \\bigl ( ( b_1-b_2) \\cdot (1-g_2 ),\\ b_2 -b_3 , \\ b_3 - b_3 \\cdot {g_3}^{-1} \\bigr ), $ for $a_i \\in M_i$ and $g_1,g_2,g_3 \\in G $ .",
"This map was first defined in [9].",
"Furthermore, given three shadow colorings $ \\mathcal {S}_i \\in \\mathrm {SCol}_{X_i }(D_{f})$ with $i \\le 3$ and each crossing $\\tau $ of $D$ , we can find assignments as illustrated in Figure REF .",
"Inspired by the formula (REF ), we define a weight of $\\tau $ to be $ \\mathcal {W}_{\\psi , \\tau }( \\mathcal {S}_1, \\mathcal {S}_2, \\mathcal {S}_3 ):= \\psi \\bigl ( (a_1-b_1)(1-g^{ \\epsilon _{\\tau }}),b_2-c_2 , c_3 -c_3 \\cdot h^{-1} \\bigr ) \\in A,$ where $ \\epsilon _{\\tau } \\in \\lbrace \\pm 1 \\rbrace $ is the sign of $\\tau .$ Figure: Colors around a crossing with respect to three shadow colorings.Definition 2.2 Given a $G$ -invariant trilinear map $\\psi : M_1 \\times M_2 \\times M_3 \\rightarrow A$ , we define a trilinear map $ \\mathcal {T}_{\\psi } : \\prod _{i=1}^3 \\mathrm {SCol}_{X_i }(D_{f}) \\longrightarrow A; \\ \\ \\ \\ \\ \\ \\ \\ \\ ( \\mathcal {S}_1, \\mathcal {S}_2, \\mathcal {S}_3 ) \\longmapsto \\sum _{\\tau } \\mathcal {W}_{\\psi , \\tau }( \\mathcal {S}_1, \\mathcal {S}_2, \\mathcal {S}_3 ), $ where $\\tau $ runs over all the crossings of $D$ .",
"The point is that, given a diagram $D$ , we can diagrammatically deal with the trilinear $ \\mathcal {T}_{\\psi } $ by definitions; see §REF –§REF for examples.",
"Next, we now show the invariance of $\\mathcal {T}_{\\psi }$ up to trilinear equivalence: Theorem 2.3 Let two diagrams $D$ and $D^{\\prime }$ differ by a Reidemeister move.",
"There is a canonical isomorphism $\\mathcal {B}_{i}: \\mathrm {SCol}_{X_i}(D_{f}) \\simeq \\mathrm {SCol}_{X_i}(D_f^{\\prime }) $ , for which the equality $ \\mathcal {T}_{\\psi }= \\mathcal {T}_{\\psi }^{\\prime } \\circ (\\mathcal {B}_1 \\otimes \\mathcal {B}_2 \\otimes \\mathcal {B}_3) $ holds as a map.",
"In particular, the equivalence class of the trilinear map $ \\mathcal {T}_{\\psi }$ depends on only the homomorphism $f:\\pi _1(S^3 \\setminus L) \\rightarrow G $ and the input data $( M_1,M_2, M_3, \\psi )$ .",
"We first focus on Reidemeister move of type III; see Figure REF .",
"Then, considering the correspondence in Figure REF with $x_i,y_i,z_i \\in X_i$ , we have the bijection $\\mathcal {B}_{i}$ .",
"Moreover, we suppose that the left region is colored by $r_i \\in M$ .",
"Thus, it is enough to show the desired equality.",
"For this, take $a_i,b_i,c_i \\in M_i$ and $g,h,k \\in G$ such that $x_i = (a_i,g), \\ y_i = (b_i,h), \\ z_i= (c_i,k) \\in X_i$ .",
"Then, the sum from the left side is, by definition and examining the figure, computed as $ \\psi \\bigl ((r_1 - a_1) (1-g), \\ a_2 - c_2, \\ c_3 (1-k^{-1})\\bigr ) + \\psi \\bigl ((r_1 g- a_1 g+ a_1 -b_1)(1-h), \\ b_2 - c_2, \\ c_3 (1-k^{-1})\\bigr ) $ $ + \\psi \\bigl ((r_1 - a_1) k (1-k^{-1}gk), \\ (a_2 - b_2)k, \\ (b_3 k -c_3k +c_3) (1-k^{-1}h^{-1}k)\\bigr ) .",
"$ On the other hand, the sum from the right side is formulated as $ \\psi \\bigl ((r_1 - a_1) (1-g), \\ a_2 - b_2, \\ b_3 (1-h^{-1})\\bigr ) + \\psi \\bigl ((r_1 - b_1) (1-h), \\ b_2 - c_2, \\ c_3 (1-k^{-1})\\bigr ) $ $ + \\psi \\bigl ((r_1 - a_1) h(1-h^{-1}gh), \\ (b_2 - a_2)h+a_2 -c_2, \\ c_3 (1-k^{-1})\\bigr ) .",
"$ Then, an elementary calculation from (REF ) can show the two sums are equal.",
"However, since the calculation is a little tedious, we omit the detail.",
"Finally, the required equality concerning Reidemeister moves of type I immediately follows from $\\psi (0,y,z)=0$ , and the invariance of type II is clear by a similar discussion.",
"Figure: The 1:1-correspondence associated with a Reidemeister move of type III.Remark 2.4 In this way, the construction for trilinear forms is applicable to not only tame links in $S^3$ , but also handlebody-knots $\\mathcal {H}_g$ in $S^3$ .",
"In fact, as a similar discussion to [4], we can easily check that the trilinear form is invariant with respect to the diagrammatic moves of handlebody-knots; see [4] for the moves."
],
[
"Topological meaning of the trilinear forms",
"As mentioned in the introduction, we will show (Theorems REF and REF ) that the trilinear forms of some links are equal to the trilinear pairings (The proofs of the theorems appear in §).",
"Theorem 2.5 Let $M_1, M_2, M_3$ be $G$ -modules as in Definition REF .",
"Furthermore, choose a fundamental class $ [ E_L ,\\partial E_L ] $ in $ H_3( E_L ,\\partial E_L ;{Z})\\cong {Z}$ .",
"We assume that $L$ is either a hyperbolic link or a prime knot which is neither a cable knot nor a torus knot.",
"Then, via the identification (REF ), the trilinear form $ \\mathcal {T}_{\\psi }$ is equal to the following composite map: $ \\bigotimes _{i: \\ 1\\le i \\le 3 } H^1( E_L ,\\partial E_L ; M_i ) \\xrightarrow{} H^3( E_L ,\\partial E_L ; M_1 \\otimes M_2 \\otimes M_3 ) \\xrightarrow{} A .",
"$ In addition, we mention the torus knot, although we need a condition.",
"More precisely, Theorem 2.6 Let $M_1, M_2, M_3$ , $\\psi $ , and $ [ E_L ,\\partial E_L ] $ be as above.",
"Assume that $L$ is the $(m,n)$ -torus knot.",
"Then, the trilinear form $\\mathcal {T}_{\\psi }$ is equal to the composite (REF ) modulo the integer $nm \\in {Z}$ .",
"As a concluding remark, while the triple cup product of a link often is considered to be speculative and uncomputable, it become computable from only a link diagram without describing $[E_L, \\partial E_L]$ and any triangulation in $S^3 \\setminus L$ .",
"Remark 2.7 Finally, we compare the trilinear forms in Definition REF with the existing results on “the quandle cocycle invariants\", in detail.",
"Briefly speaking, the link invariant in [2] is constructed from a quandle $X$ and a map $\\Phi : X^3 \\rightarrow A $ which satisfy “the quandle cocycle condition\", and is defined to be a certain map $ \\mathcal {J}_{\\Phi } \\ : \\mathrm {SCol}_X(D )\\rightarrow A$ .",
"Then, we note that our trilinear form is a trilinearization of the quandle cocycle invariants with respect to quandles of the form $X= M \\times G $ .",
"To be precise, if $M = M_1=M_2=M_3$ , we can see that the associated invariant $\\mathcal {J}_{\\Phi } : \\mathrm {SCol}_X(D )\\rightarrow A$ is equal to the composite $\\mathcal {T}_{\\psi } \\circ (\\bigtriangleup \\times {\\rm id}) \\circ \\bigtriangleup $ by definitions.",
"In conclusion, the theorems also suggest topological meanings of the quandle cocycle invariants with $X= M \\times G $ ."
],
[
"Relation to 3-fold branched coverings",
"Although we considered relative cohomology, for $n\\in \\mathbb {Z}_{\\ge 0}$ and a closed 3-manifold $N$ , let us consider the triple cup product $ H^1(N;{Z}/n{Z})^{\\otimes 3}\\xrightarrow{} H^3(N;{Z}/n{Z})\\xrightarrow{} {Z}/n ,$ where the coefficient module ${Z}/n$ is trivial.",
"Although there are studies of this map (see, e.g., [14], [3], [15]), there are few examples of the computation.",
"As an application of the theorems above, this section gives a recover of the triple cup products of $N$ , when $N$ is a 3-fold cyclic covering of $S^3 $ branched over a link.",
"To state Theorem REF , we need some terminology.",
"Let $G$ be ${Z}/3= \\langle t | t^3=1 \\rangle $ .",
"Consider the epimorphism $f: \\pi _1(S^3 \\setminus L ) \\rightarrow G$ which sends every meridian to $t$ , and the associated 3-fold cyclic branched covering $\\widetilde{C}_L \\rightarrow S^3 $ .",
"Theorem 3.1 Let $M_1$ , $M_2$ , and $M_3$ be ${Z}[t^{\\pm 1}]/ (n, t^2+t+1)$ .",
"Let $p:{Z}[t^{\\pm 1}]/ (n, t^2 + t+1) \\rightarrow {Z}/n$ be the map which sends $ a+tb$ to $a$ .",
"Set up the map $\\psi _0 : M^3 \\rightarrow {Z}/n$ which takes $ (x,y,z)$ to $ xyz$ .",
"As in Theorem REF , assume that $L$ is either a hyperbolic link or a prime knot which is neither a cable knot nor a torus knot.",
"Then, there is an isomorphism $ \\mathrm {Col}_{X_i}^{\\rm red}(D_f) \\cong H^1(\\widetilde{C}_L;{Z}/n{Z}) $ such that the trilinear map $\\mathcal {T}_{p \\circ \\psi _0}$ is equivalent to (REF ) with $N= \\widetilde{C}_L $ .",
"[Proof of Theorem REF ] We first show the isomorphism $ \\mathrm {Col}_{X_i}^{\\rm red}(D_f) \\cong H^1(\\widetilde{C}_L;{Z}/n{Z}) $ .",
"Let $R$ be the ring $ {Z}[t]/(n, t^2 + t+1)$ .",
"By Theorem REF , we have $\\mathrm {Col}_{X_i}^{\\rm red}(D_f) \\cong H^1(E_L ;\\partial E_L ; M) $ .",
"Notice that $H^i(\\partial E_L ;M) $ is annihilated by $1-t$ .",
"Since $1-t$ and $1 + t+t^2$ are coprime, we have $ H^1(E_L ;\\partial E_L ; M) \\cong H^1(E_L ; M)\\cong \\mathop {\\mathrm {Hom}}\\nolimits _{ R\\textrm {-mod}}( H_1(E_L ,M), R) .$ Let $\\widetilde{E}_L \\rightarrow E_L= S^3 \\setminus L$ be the 3-fold covering.",
"Then, by Shapiro's Lemma (see, e.g.",
"[1]), the canonical inclusion $\\iota : {Z}/n \\rightarrow R$ yields the isomorphisms: $ H^*( \\widetilde{E}_L: {Z}/n ) \\cong H^*( E_L; {Z}[t]/(n, t^3-1)) \\cong H^*( E_L; R) \\oplus H^*( E_L; {Z}[t]/(n,t-1)) .$ Here, the second isomorphism is obtained from the ring isomorphism ${Z}[t]/(n, t^3-1) \\cong R \\oplus {Z}[t]/(n, t-1) $ .",
"Let $ i : \\widetilde{E}_L \\hookrightarrow \\widetilde{C}_L$ be the inclusion.",
"According to [5], the homology $H_1( \\widetilde{C}_L; {Z}) $ is annihilated by $1+t+t^2$ , and the induced map $i_*: H_1( \\widetilde{E}_L; {Z}) \\rightarrow H_1( \\widetilde{C}_L; {Z})$ is a splitting surjection.",
"Thus, dually, the induced map $i^*: H^1( \\widetilde{C}_L; {Z}/n ) \\rightarrow H^1( \\widetilde{E}_L;{Z}/n)$ is injective and the image is isomorphic to $H^1( E_ L; R)$ .",
"In summary, we obtained the desired isomorphism.",
"We will complete the proof.",
"By (REF ), we have a splitting injection $\\mathcal {S}: H^*( \\widetilde{E}_L ,\\partial \\widetilde{E}_L ;R ) \\rightarrow H^*( E_L ,\\partial E_L;{Z}/n ) $ .",
"Take the canonical maps $ j: ( \\widetilde{E}_L ,\\partial \\widetilde{E}_L) \\rightarrow ( \\widetilde{C}_L, \\widetilde{C}_L \\setminus \\widetilde{E}_L ) $ and $ k :( \\widetilde{C}_L, \\emptyset ) \\rightarrow ( \\widetilde{C}_L, \\widetilde{C}_L \\setminus \\widetilde{E}_L) $ .",
"Then, we have the commutative diagrams on the cup products: ${\\normalsize {H^1( E_L, \\partial E_L ; M)^{\\otimes 3}[d]^{\\mathcal {S} } [r]^{\\!\\!\\!\\!\\!",
"\\psi _0 \\circ \\smile } & H^3( E_L, \\partial E_L ; M ) [d]^{\\mathcal {S} }[rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\langle \\bullet , [E_L,\\partial E_L] \\rangle } & & & R \\\\H^1( \\widetilde{E}_L ,\\partial \\widetilde{E}_L ; {Z}/n)^{\\otimes 3} [r]^{\\!\\!\\!\\!\\!\\!\\!\\!",
"\\smile } & H^3 ( \\widetilde{E}_L ,\\partial \\widetilde{E}_L ; {Z}/n ) [rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\langle \\bullet , [\\widetilde{E}_L ,\\partial \\widetilde{E}_L ] \\rangle }& & & {Z}/n @{=}[d] [u]^{\\iota }\\\\H^1( \\widetilde{C}_L, \\widetilde{C}_L \\setminus \\widetilde{E}_L ; {Z}/n)^{\\otimes 3} [r]^{\\!\\!\\!\\!\\!\\!\\!\\!",
"\\smile } [u]^{\\cong }_{j^*}[d]^{k^*} & H^3 ( \\widetilde{C}_L, \\widetilde{C}_L \\setminus \\widetilde{E}_L; {Z}/n) [u]^{\\cong }_{j^*}[d]^{k^*} [rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\langle \\bullet , [ \\widetilde{C}_L, \\widetilde{C}_L \\setminus \\widetilde{E}_L] \\rangle }& & & {Z}/n @{=}[d]\\\\H^1( \\widetilde{C}_L ; {Z}/n)^{\\otimes 3} [r]^{\\!\\!\\!\\!\\!\\!\\!\\!",
"\\smile } & H^3 ( \\widetilde{C}_L; {Z}/n) [rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\langle \\bullet , [ \\widetilde{C}_L] \\rangle }& & & {Z}/n .",
"}}$ Here, the vertical maps $j^*$ are isomorphisms by the excision axiom.",
"Moreover, by the discussion in the above paragraph, the composite $k^* \\circ (j^*)^{-1}\\circ \\mathcal {S}$ is an isomorphism from $ H^1( E_L, \\partial E_L ; M) $ .",
"Hence, since $ p \\circ \\iota :{Z}/n \\rightarrow {Z}/n $ is an isomorphism, the following two composites are equivalent: $ p\\circ \\psi _0 \\circ \\langle \\bullet , [E_L,\\partial E_L] \\rangle \\circ \\smile , \\ \\ \\ \\ \\ \\langle \\bullet , [\\widetilde{C}_L] \\rangle \\circ \\smile .$ By Theorem REF , the left hand side is equal to the trilinear map $\\mathcal {T}_{p\\circ \\psi _0}$ .",
"Hence, $\\mathcal {T}_{p \\circ \\psi _0}$ is equivalent to (REF ) with $N= \\widetilde{C}_L $ as desired."
],
[
"General situation for the trefoil knot and the figure eight knot",
"We will compute the trilinear forms $ \\mathcal {T}_{\\psi }$ associated with some homomorphisms $f : \\pi _L \\rightarrow G$ , where $L$ is either the trefoil knot or the figure eight knot.",
"As a simple example, we will focus on the the trefoil knot $3_1 $ .",
"Let $D$ be the diagram of $K$ as illustrated in Figure REF .",
"Note the Wirtinger presentation $\\pi _L \\cong \\langle \\alpha , \\beta \\ | \\ \\alpha \\beta \\alpha =\\beta \\alpha \\beta \\rangle .",
"$ Then, we can easily see that a correspondence $ \\mathcal {C}: \\lbrace \\alpha , \\beta , \\gamma \\rbrace \\rightarrow X$ with $\\mathcal {C}(\\alpha )=(a_i ,g) , \\ \\ \\ \\ \\mathcal {C}(\\beta )=(b_i ,g), \\ \\ \\ \\ \\mathcal {C}(\\alpha )=(c_i ,g) \\ \\in M_i \\times G $ is an $X$ -coloring $ \\mathcal {C}$ over $f: \\pi _L \\rightarrow G$ , if and only if it satisfies the four equations $ g = f(\\alpha ), \\ \\ h = f(\\beta ), \\ \\ \\ ghg =hgh , $ $ c_i = a_i \\cdot h +b_i \\cdot (1-h ), $ $ (a_i -b_i ) \\cdot (1-g +hg)=(a_i -b_i ) \\cdot (1-h +gh)=0 .",
"$ Furthermore, given a $G$ -invariant linear form $\\psi $ , the sum $\\mathcal {T}_{\\psi }$ is equal to $ \\psi \\bigl ( -a_1 \\cdot (1-g), \\ a_2-b_2, \\ a_3\\cdot (1-h^{-1})\\bigr ) + \\psi \\bigl ( -b_1 \\cdot (1-h), \\ b_2-c_2, \\ c_3 \\cdot (1-h^{-1}g^{-1} h ) \\bigr )$ $ \\ \\ + \\psi \\bigl ( -c_1 \\cdot (1-h^{-1}g h ), \\ c_2-a_2, \\ a_3 \\cdot (1-g^{-1})\\bigr ), $ by definition.",
"Then, by canceling out $c_i$ by using (REF ) and (REF ), we can easily obtain the resulting computation: for $((a_i,g),( b_i,h)) \\in {\\rm SCol}_{X_i} (D_{f})\\subset M_i^2 $ , $ \\mathcal {T}_{\\psi } \\bigl ( (a_1, b_1),( a_2, b_2),( a_3, b_3 ) \\bigr )= \\psi \\bigl ((a_1 -b_1)g^{-1} , \\ (a_2 -b_2)\\cdot h , \\ a_3 -b_3 \\bigr ) \\in A.",
"$ Next, we will compute $\\mathcal {T}_{\\psi } $ of the figure eight knot.",
"However, the computation can be done in a similar way to the trefoil case.",
"So we only describe the outline.",
"Let $D$ be the diagram with arcs as illustrated in Figure REF .",
"Similarly, we can see that a correspondence $ \\mathcal {C}: \\lbrace \\alpha _1, \\alpha _2, \\alpha _3, \\alpha _4\\rbrace \\rightarrow X$ with $\\mathcal {C}(\\alpha _i )=(x_i ,z_i) \\in M_i \\times G $ is an $X$ -coloring $ \\mathcal {C}$ over $f: \\pi _L \\rightarrow G$ , if and only if it satisfies the following equations: $ z_i = f(\\alpha _i ), \\ \\ \\ \\ \\ z_2^{-1} z_1 z_2= z_1^{-1} z_2^{-1}z_1 z_2 z_1^{-1} z_2 z_1 \\in G , $ $ x_3 = (x_1 -x_2 )\\cdot z_2 +x_2 , \\ \\ \\ \\ \\ \\ x_4 = (x_2 -x_1 )\\cdot z_1 +x_1, $ $ (x_1 -x_2) \\cdot (z_1 + z_2 - 1 )=(x_1 -x_2) \\cdot (1-z_2^{-1} ) z_1 z_2 =(x_1 -x_2) \\cdot (1-z_1^{-1} ) z_2 z_1 \\in M.$ Accordingly, it follows from (REF ) that the set $ \\mathrm {Col}_{X } (D_{f}) $ is generated by $x_1, x_2$ .",
"Given a $G$ -invariant trilinear form $\\psi $ , it can be seen that the trilinear form $\\mathcal {T}_{\\psi } \\bigl ( ( x_1, x_2),( x_1^{\\prime }, x_2^{\\prime } ),( x_1^{\\prime \\prime }, x_2^{\\prime \\prime } ) \\bigr )$ is expressed as $ \\psi ((x_1-x_2) \\cdot z_1 z_2^{-1}, \\ x_2^{\\prime }-x_1^{\\prime }, \\ (x_1^{\\prime \\prime }-x_2^{\\prime \\prime }) \\cdot (1-z_2^{-1}))+ \\psi ((x_1-x_2) \\cdot z_2^{-1}z_1 , \\ (x_1^{\\prime }-x_2^{\\prime }) \\cdot (1-z_1), \\ (x_1^{\\prime \\prime }-x_2^{\\prime \\prime }) \\cdot (1-z_2^{-1})z_1).",
"$ Remark 4.1 Here, we should give some examples from concrete $M$ and $\\psi .$ In particular, the author attempted to get non-trivial trilinear form $\\mathcal {T}_{\\psi }$ when $G$ is a Lie group and $M$ is a representation of $G$ .",
"However, even if $G=SL_2(\\mathbb {R})$ or $G=SL_2(\\mathbb {C})$ and $M= \\mathbb {C}^2$ or $\\mathbb {C}^3 $ , the author computed the resulting $\\mathcal {T}_{\\psi }$ equal to zero.",
"Indeed, the author could not find non-trivial examples of $\\mathcal {T}_{\\psi } $ except those in §REF .",
"Thus, it is a problem to find non-trivial examples of $\\mathcal {T}_{\\psi } $ from representations with respect to Lie groups."
],
[
"The $(m,m)$ -torus link {{formula:c9dda0f0-9183-4d4c-b080-1d635358a3d9}}",
"We also calculate the trilinear form $\\mathcal {T}_{\\psi }$ concerning the $(m,m)$ -torus link, following from Definition REF .",
"These calculations will be useful in the paper [12], which suggests invariants of “Hurewitz equivalence classes\".",
"Let $L$ be the $(m,m)$ -torus link $T_{m,m}$ with $m \\ge 2$ , and let $\\alpha _1, \\dots , \\alpha _m$ be the arcs depicted in Figure REF .",
"Furthermore, let us identity $\\alpha _{i+m}$ with $\\alpha _i$ of period $m$ .",
"By Wirtinger presentation, we have a presentation of $ \\pi _L$ as $\\langle \\ a_1, \\dots , a_m \\ | \\ a_1 \\cdots a_m= a_m a_1 a_2 \\cdots a_{m-1}= a_{m-1}a_m a_1 \\cdots a_{m-2}= \\cdots = a_2 \\cdots a_m a_1 \\ \\rangle .",
"$ Given a homomorphism $f:\\pi _L \\rightarrow G$ with $f(\\alpha _i) \\in G $ , let us discuss $X$ -colorings $ \\mathcal {C}$ over $f$ .",
"Then, concerning the coloring condition on the $\\ell $ -th link component, it satisfies the equation $ \\bigl ( \\cdots (\\mathcal {C}(\\alpha _\\ell ) \\lhd \\mathcal {C}(\\alpha _{\\ell +1})) \\lhd \\cdots \\bigr )\\lhd \\mathcal {C}(\\alpha _{\\ell +m-1}) =\\mathcal {C}( \\alpha _\\ell ) ,\\ \\ \\ \\ \\ \\ \\ {\\rm for \\ any \\ }1 \\le \\ell \\le m .",
"$ With notation $ \\mathcal {C} (\\alpha _i):= (x_i,z_i) \\in X$ , this equation (REF ) reduces to a system of linear equations $ ( x_{\\ell -1} -x_{\\ell } ) + \\sum _{ \\ell \\le j \\le \\ell +m-2 }( x_j -x_{j+1} ) \\cdot z_{j+1} z_{j+2} \\cdots z_{m +\\ell } = 0 \\in M , \\ \\ \\ \\ \\ \\mathrm {for \\ any \\ } 1 \\le \\ell \\le m .$ Conversely, we can easily verify that, if a map $\\mathcal {C}: \\lbrace \\mbox{arcs of $D$} \\rbrace \\rightarrow X$ satisfies the equation (REF ), then $\\mathcal {C} $ is an $ X$ -coloring.",
"Denoting the left side in (REF ) by $ \\Gamma _{f,k}(\\vec{x})$ , consider a homomorphism $ \\Gamma _{f}: M^m \\longrightarrow M^m ; \\ \\ \\ (x_1, \\dots , x_m) \\longmapsto (\\Gamma _{f,1}(\\vec{x}) , \\dots , \\Gamma _{f,m}(\\vec{x}) ).",
"$ To conclude, the set $\\mathrm {Col}_X(D_f )$ coincides with the kernel of $\\Gamma _{f} $ .",
"Next, we precisely formulate the resulting trilinear form.",
"Proposition 4.2 Let $f : \\pi _1(S^3 \\setminus T_{m,m})\\rightarrow G$ be as above.",
"Let $\\psi : M^3 \\rightarrow A$ be a $G$ -invariant linear functions.",
"Then, the trilinear form $\\mathcal {T }_{\\psi }: \\mathop {\\mathrm {Ker}}\\nolimits (\\Gamma _{f})^{\\otimes 3} \\rightarrow A$ sends $(w_1, \\dots , w_m) \\otimes (x_1, \\dots , x_m) \\otimes ( y_1, \\dots , y_m)$ to $\\sum _{\\ell =1}^{m } \\sum _{k=1}^{m-1 } \\psi \\bigl ( w_\\ell \\cdot (1 - z_\\ell ) \\hat{z}_{\\ell +1; \\ell +k -1 }, \\ \\sum _{j=1}^{k } (x_{j+\\ell -1}-x_{j +\\ell })\\cdot \\hat{z}_{j+\\ell ;k+ \\ell -1} ,\\ y_{k +\\ell } \\cdot (1 - z_{k + \\ell }^{-1}) \\bigr ) .$ Here, for $s \\le t$ , we use notation $ \\hat{z}_{s;t}:=z_s z_{s+1} \\cdots z_t $ and $\\hat{z}_{s+1; s}:=1 \\in G .$ The formula is obtained by direct calculation and definitions."
],
[
"Examples of Theorem ",
"We will give some examples in Theorem REF .",
"Thus, we should suppose the situation of Theorem REF as follows.",
"Let $G= {Z}/3 = \\langle t | t^3=1 \\rangle $ , and $f : \\pi _1(S^3 \\setminus L) \\rightarrow {Z}/3 $ be the map which sends every meridian to $t $ .",
"Furthermore, take $M_i=A= {Z}[t]/(n,t^2 +t +1) $ for some $n\\in {Z}_{\\ge 0}$ , let $\\psi _0 : M_1 \\times M_2 \\times M_3 \\rightarrow A $ send $(x,y,z)$ to $xyz.$ In this paragraph, we focus on only knots, $K$ , such that $H^1(E_K,\\partial E_K ; A ) \\cong H^1(\\widetilde{B}_K ;{Z}/n) $ is isomorphic to either $A $ or 0.",
"We will write the trilinear map $\\mathcal {T}_{\\psi _0}$ as a cubic polynomial with respect to $a,b,c\\in \\bigl ( H^1(E_K,\\partial E_K ;{Z}/n)\\bigr )^3 $ .",
"Then, we give the resulting computation of $ \\mathcal {T}_{\\psi _0}$ , when $K$ is a prime knot with crossing number $<7$ .",
"The list of the computation is as follows: Table: NO_CAPTION"
],
[
"Proofs of the theorems",
"We will complete the proofs of Theorems REF –REF in §REF .",
"While the statements were described in terms of ordinary cohomology, the proof will be done via the group cohomology.",
"For this purpose, in §REF , we review the relative group homology."
],
[
"Preliminary; Review of relative group cohomology",
"We will spell out the relative group (co)homology in the non-homogeneous terms.",
"Throughout this subsection, we fix a group $ \\Gamma $ and a homomorphism $f : \\Gamma \\rightarrow G$ .",
"Then, we have the action of $\\Gamma $ on the right $G$ -module $M$ via $f$ .",
"Let $ C^{n}_{\\rm gr }(\\Gamma ;M ) $ be $ \\mathrm {Map} ( \\Gamma ^n , M ) $ .",
"For $ \\phi \\in C^{n}_{\\rm gr }(\\Gamma ;M )$ , define the coboundary $\\partial ^n( \\phi ) \\in C_{\\mathrm {gr}}^{n+1} (\\Gamma ;M)$ by the formula $\\partial ^n( \\phi ) (g_1,\\dots , g_{n+1})= $ $\\phi ( g_2, \\dots ,g_{n+1}) +\\!\\sum _{1 \\le i \\le n}\\!\\!",
"(-1)^i \\phi ( g_1, \\dots ,g_{i-1}, g_{i} g_{i+1}, g_{i+2},\\dots , g_{n+1})+(-1)^{n} \\phi ( g_1, \\dots , g_{n}) g_{n+1} .$ Furthermore, we set subgroups $ K_j $ and the inclusions $\\iota _j: K_j \\hookrightarrow \\Gamma $ , where the index $ j$ runs over $ 1 \\le j \\le m $ .",
"Then, we can define the mapping cone of $ \\iota _j$ : More precisely, $C^n( \\Gamma , K_\\mathcal {J}; M ):= \\mathrm {Map} ( \\Gamma ^n , M ) \\oplus \\bigl ( \\bigoplus _{j } \\mathrm {Map }((K_{j})^{n-1} , M ) \\bigr ) .",
"$ For $(h,k_1, \\dots , k_m ) \\in C^n( \\Gamma , K_\\mathcal {J}; M ) $ , let us define $ \\partial ^n(h,k_1, \\dots , k_m )$ in $ C^{n+1}( \\Gamma , K_\\mathcal {J}; M )$ by $ \\partial ^n \\bigl (h,k_1, \\dots , k_m \\bigr )( a, b_1, \\dots , b_m)= \\bigl ( \\partial ^{n} h( a), \\ h (b_1) -\\partial ^{n-1} k_1(b_1), \\dots ,h (b_m) -\\partial ^{n-1} k_m(b_m)\\bigr ) ,$ where $( a, b_1, \\dots , b_m) \\in \\Gamma ^{n+1} \\times K_1^{n} \\times \\cdots \\times K_m^{n} $ .",
"Then we have a complex $ (C^*( \\Gamma , K_\\mathcal {J}; M ), \\partial ^*)$ , and can define the cohomology.",
"We now observe the submodule consisting of 1-cocycles $Z^1( \\Gamma , K_\\mathcal {J}; M ) $ .",
"Let us define the semi-direct product $M \\rtimes G $ by $ (a, g) \\star (a^{\\prime },g^{\\prime }):=( a \\cdot g^{\\prime } + a^{\\prime }, \\ gg^{\\prime }), \\ \\ \\ \\ \\mathrm {for} \\ \\ a,a^{\\prime } \\in M, \\ \\ \\ g,g^{\\prime }\\in G. $ Let $ \\mathop {\\mathrm {Hom}}\\nolimits _f (\\Gamma , M \\rtimes G )$ be the set of group homomorphisms $\\Gamma \\rightarrow M \\rtimes G $ over the homomorphism $f$ .",
"Consider a map $ Z^1( \\Gamma , K_\\mathcal {J}; M ) \\rightarrow \\mathop {\\mathrm {Hom}}\\nolimits _f (\\Gamma , M \\rtimes G ) \\oplus M^m; \\ \\ \\ \\ \\ \\ (h,y_1,\\dots , y_m) \\mapsto (\\gamma \\mapsto ( h(\\gamma ), f(\\gamma )),y_1,\\dots , y_m) .$ Lemma 5.1 ([10]) This map gives an isomorphism between $Z^1( \\Gamma , K_\\mathcal {J}; M )$ and the following set: $ \\bigl \\lbrace \\ (\\widetilde{f} , y_1, \\dots , y_m ) \\in \\mathop {\\mathrm {Hom}}\\nolimits _f ( \\Gamma , M \\rtimes G) \\oplus M^m \\ \\bigl | \\ \\ \\widetilde{f} (h_j) = ( y_j - y_j \\cdot h_j, \\ f_j( h_j) ) , \\ \\ \\mathrm {for \\ any } \\ h_j \\in K_j \\ \\bigr \\rbrace .",
"$ Moreover, the image of $\\partial ^1$ , i.e., $B^1( \\Gamma , K_\\mathcal {J}; M )$ , is equal to the subset $\\lbrace ( \\widetilde{f}_a, a, \\dots , a)\\rbrace _{a \\in M}.",
"$ Here, for $a \\in M,$ the map $ \\widetilde{f}_a : \\Gamma \\rightarrow M \\rtimes G $ is defined as a homomorphism which sends $\\gamma $ to $( a - a \\cdot \\gamma , \\ f (\\gamma ))$ .",
"In particular, if $ K_\\mathcal {J} $ is non-empty, $B^1( \\Gamma , K_\\mathcal {J}; M )$ is a direct summand of $ Z^1( \\Gamma , K_\\mathcal {J}; M )$ .",
"Furthermore, we review the cup product.",
"When $ K_\\mathcal {J} $ is the empty set, the product of $u \\in C^p( \\Gamma ; M )$ and $v \\in C^{q}( \\Gamma ; M^{\\prime } )$ is defined to be $u \\smile v \\in C^{p+q} ( \\Gamma ; M \\otimes M^{\\prime })$ given by $ ( u \\smile v) ( g_1 ,\\dots , g_{p+q}):= (-1)^{pq} \\bigl ( u (g_1 ,\\dots , g_{p} ) g_{p+1} \\cdots g_{p+q} \\bigr ) \\otimes v (g_{p+1} ,\\dots , g_{p+q} ) .",
"$ Furthermore, if $ K_\\mathcal {J} $ is not empty, for two elements $(f,k_1, \\dots , k_m )\\in C^p( \\Gamma , K_\\mathcal {J} ; M )$ and $(f^{\\prime },k^{\\prime }_1, \\dots , k^{\\prime }_m )\\in C^{q}( \\Gamma , K_\\mathcal {J} ; M^{\\prime } )$ , let us define the cup product to be the formula $ ( f \\smile f^{\\prime }, \\ k_1\\smile f^{\\prime }, \\dots , \\ k_m\\smile f^{\\prime }) \\in C^{p+q}( \\Gamma , K_\\mathcal {J} ; M \\otimes M^{\\prime }).",
"$ This formula yields a bilinear map, by passage to cohomology.",
"Finally, we observe another complex.",
"Consider the module of the form $ C^n_{\\rm red}(\\Gamma ):= \\lbrace \\ (c_1, \\dots , c_m ) \\in \\mathrm {Map}( {Z}[\\Gamma ^n], M)^m \\ | \\ c_1 +c_2 + \\cdots +c_m =0\\in \\mathrm {Map}( {Z}[\\Gamma ^n], M)\\ \\rbrace .$ Then, this complex canonically has an inclusion into the direct sum of $ C^n(\\Gamma , K_j)$ : $P_n: C^n_{\\rm red}(\\Gamma ) \\longrightarrow \\bigoplus _{j : \\ 1 \\le j \\le m} C^n(\\Gamma , K_j).",
"$ Then, we define a quotient complex, $D^n( \\Gamma , K_\\mathcal {J}; M )$ , to be the cokernel of $ P_n$ .",
"Then, $C^n( \\Gamma , K_\\mathcal {J}; M )$ is isomorphic to $D^n( \\Gamma , K_\\mathcal {J}; M ) $ , since the kernel of the inclusions $\\oplus _{j=1}^{ m} C^n(\\Gamma , K_j) \\rightarrow C^n(\\Gamma , \\mathcal {K}) $ is the image of $ P_n $ .",
"Remark 5.2 We give a natural relation to usual cohomology.",
"Take the Eilenberg-MacLane spaces of $ \\Gamma $ and of $K_j$ , and consider the map $ (\\iota _j)_* : K(K_j,1 ) \\rightarrow K( \\Gamma ,1 ) $ induced by the inclusions.",
"Then the relative homology $H_n( \\Gamma , K_\\mathcal {J} ; M ) $ is isomorphic to the homology of the mapping cone of $ \\sqcup _j K(K_j,1 )\\rightarrow K(\\Gamma ,1 )$ with local coefficients.",
"Further, the cup product $\\smile $ above coincides with that on the singular cohomology groups.",
"We mention the case where $L$ is either a knot or a hyperbolic link.",
"Then, the complement $S^3 \\setminus L$ is known to be an Eilenberg-MacLane space.",
"Since we only use $\\Gamma $ as $\\pi _1(S^3 \\setminus L)$ in this paper, we may discuss only the relative group cohomology."
],
[
"Review; results of the previous papers {{cite:3cdee1e8397cb23762a8d3c79d4241f28da1af27}} and {{cite:123c86b8f5249c660a47f56283a74addcc142da6}}.",
"Throughout this section, we denote $\\pi _1(S^3 \\setminus L)$ by $\\pi _L$ , and the union of the fundamental groups of the boundaries of $S^3 \\setminus L$ by $\\partial \\pi _L$ , for brevity.",
"Let $m= \\# L $ , and choose a diagram $D.$ Theorem 5.3 ([10]) Let $X$ be $M \\times G$ , as mentioned in (REF ).",
"Let $\\kappa : X \\rightarrow M \\rtimes G $ be a map which sends $(m,g)$ to $ (m-mg , g)$ .",
"Given an $X$ -coloring $\\mathcal {C}$ over $f $ , consider a map $\\lbrace \\mathrm {arcs \\ of \\ }D \\rbrace \\rightarrow M \\rtimes G$ which assigns $ \\alpha $ to $ \\kappa \\bigl ( \\mathcal {C}(\\alpha ) \\bigr ) $ .",
"This assignment yields isomorphisms $ \\mathrm {Col}_X(D_{f}) \\cong Z^1( \\pi _L ,\\partial \\pi _L ; M ), \\ \\ \\ \\ \\mathrm {Col}^{\\rm red}_X(D_{f}) \\cong H^1( \\pi _L ,\\partial \\pi _L ; M ).",
"$ Next, we explain Theorem REF .",
"Choose a relative 1-cocycle $\\tilde{f}: \\pi _L \\rightarrow M \\rtimes \\pi _L$ with $ y_1,\\dots , y_{m}$ .",
"We define the subgroup $K_{\\ell } $ to be $ \\lbrace (y_{\\ell }-y_{\\ell } \\mathfrak {m}_{\\ell }^a \\mathfrak {l}_{\\ell }^b , \\mathfrak {m}_{\\ell }^a \\mathfrak {l}_{\\ell }^b ) \\in M \\rtimes \\pi _L \\ | \\ a,b \\in {Z}^2 \\ \\rbrace .",
"$ Furthermore, given a $G$ -invariant trilinear map $\\psi : M^3 \\rightarrow A$ , consider the map $\\theta _{\\ell }: (M \\rtimes \\pi _1(S^3 \\setminus L) )^3 \\longrightarrow A ; $ $ ((a,g),(b,h),(c,k)) \\longmapsto \\psi ( (a +y_{\\ell } -y_{\\ell } g)\\cdot hk , (b+y_{\\ell } -y_{\\ell } h) \\cdot k, c+y_{\\ell } -y_{\\ell } k) .$ Then, we can easily check that each $\\theta _{\\ell } $ is a 3-cocycle in $C^3( M \\rtimes \\pi _L;A)$ .",
"Then, the collection $\\Psi := (\\theta _{1},\\dots , \\theta _{\\# L} ) $ represents a relative 3-cocycle in $D^3( M \\rtimes \\pi _L,\\mathcal {K} ;A) $ .",
"Proposition 5.4 ([13]) Under the notation above, fix a shadow coloring $ \\mathcal {S}_{\\tilde{f}} $ corresponding the relative 1-cocycle $(\\tilde{f},y_1,\\dots ,y_{\\# L}).$ If $L$ is either a hyperbolic link or a prime knot which is neither a cable knot nor a torus knot, as in Theorem REF , then the diagonal restriction of $ \\mathcal {T}_{ \\psi }$ is equal to the pairing of the 3-class $[E_L,\\partial E_L]$ and the above 3-cocycle $\\Psi $ .",
"To be precise, $ \\mathcal {T}_{ \\psi }( \\mathcal {S}_{\\tilde{f}}, \\mathcal {S}_{\\tilde{f}}, \\mathcal {S}_{\\tilde{f}})=\\psi \\langle \\Psi , \\tilde{f}_*[E_L,\\partial E_L] \\rangle .",
"$ Furthermore, if $L$ is the $(m,n)$ -torus knot, the same equality (REF ) holds modulo $mn.$"
],
[
"Proof of Theorem ",
"[Proof of Theorem REF .]",
"First, we observe (REF ) below.",
"Consider a 0-cochain $\\vec{y}:= (y_1,\\dots , y_{\\# L}) \\in D^0(M \\rtimes \\pi _L, M)$ .",
"Then, $ \\widetilde{f} -\\partial ^{0} \\vec{y}$ is represented by another 1-cocycle $ \\mathcal {C}^{\\prime } :=((\\widetilde{f} -\\bar{y}_1 , \\dots , \\widetilde{f} -\\bar{y}_{\\# L} ),(0,\\dots , 0) ) \\in D^1(M \\rtimes \\pi _L, M),$ where $\\bar{y}_{\\ell }$ means a map $\\pi _L \\rightarrow M $ which takes $g$ to $ y_1 -y_1 g $ .",
"Then, the 3-cocycle $ \\Psi $ explained in (REF ) is equal to the cup product $ \\mathcal {C}^{\\prime } \\smile \\mathcal {C}^{\\prime } \\smile \\mathcal {C}^{\\prime } $ , by definition.",
"Hence, Proposition REF implies $ \\mathcal {T}_{ \\psi }( \\mathcal {S}_{\\tilde{f}}, \\mathcal {S}_{\\tilde{f}}, \\mathcal {S}_{\\tilde{f}})=\\psi \\langle \\mathcal {C}^{\\prime } \\smile \\mathcal {C}^{\\prime }\\smile \\mathcal {C}^{\\prime } , [E_L,\\partial E_L] \\rangle =\\psi \\langle \\mathcal {C} \\smile \\mathcal {C}\\smile \\mathcal {C} , [E_L,\\partial E_L] \\rangle .",
"$ Finally, we will deal with non-diagonal parts, and complete the proof.",
"Here, we define $M$ to be the direct product $M_1 \\times M_2 \\times M_3 $ , and consider the $j$ -th inclusion $ \\iota _j: M_j \\longrightarrow M= M_1 \\times M_2 \\times M_3; \\ \\ \\ \\ x \\longmapsto (\\delta _{1j} x,\\delta _{2j} x,\\delta _{3j} x ).",
"$ Thus, we can decompose $\\mathcal {S}_{\\tilde{f}}$ as $( \\mathcal {S}_{1}, \\mathcal {S}_{2}, \\mathcal {S}_{3}) \\in \\mathrm {Col}_{X_1} (D_f )\\times \\mathrm {Col}_{X_2} (D_f )\\times \\mathrm {Col}_{X_3} (D_f )$ componentwise.",
"In addition, we define a $G$ -invariant trilinear form $ \\overline{\\psi } : M \\times M \\times M \\longrightarrow A; \\ \\ \\ \\ \\bigl ( (a,b,c),(d,e,f),(g,h,i)\\bigr ) \\longmapsto \\psi (a,e,f) .",
"$ Then, the transformation of the coefficients $\\iota _1 \\times \\iota _2 \\times \\iota _3$ yields a diagram ${\\normalsize {\\prod _{i=1}^3 H^1( E_L, \\partial E_L ; M_i) [r]^{\\!\\!\\!\\!\\!",
"\\smile }[d]_{ }^{(\\iota _1 \\times \\iota _2 \\times \\iota _3)_*} & H^3( E_L, \\partial E_L ; M_1 \\times M_2 \\times M_3 ) [rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\psi \\circ \\langle \\bullet , [E_L,\\partial E_L] \\rangle }[d]^{(\\iota _1 \\times \\iota _2 \\times \\iota _3)_*} & & & A @{=}[d] \\\\H^1( E_L ,\\partial E_L; M) [r]^{\\!\\!\\!\\!\\!\\!\\!\\!",
"\\smile _{\\Delta }} & H^3 ( E_L ,\\partial E_L; M \\times M \\times M) [rrr]^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\overline{\\psi } \\circ \\langle \\bullet , [E_L,\\partial E_L] \\rangle }& & & A .",
"}}$ Here, the left bottom $\\smile _{\\Delta } $ is defined by $a \\mapsto a \\smile a \\smile a$ .",
"Then, we can verify the commutativity by definitions.",
"By Proposition REF , the bottom arrow is equal to the left hand side in (REF ).",
"Hence, the pullback to $\\prod _{i=1}^3 H^1( E_L, \\partial E_L ; M_i) $ is equal to the trilinear $\\mathcal {T}_{\\psi }$ as desired.",
"[Proof of Theorem REF .]",
"Let $L$ be the $(m,n)$ -torus knot.",
"According to the latter part in Theorem REF , we need discussions modulo $mn$ .",
"However, the proof runs well in the same manner."
],
[
"Acknowledgments",
"The work is partially supported by JSPS KAKENHI Grant Number 17K05257.",
"DEPARTMENT OF MATHEMATICS TOKYO INSTITUTE OF TECHNOLOGY 2-12-1 OOKAYAMA , MEGURO-KU TOKYO 152-8551 JAPAN E-mail: nosaka@math.titech.ac.jp"
]
] | 1808.08532 |
[
[
"The energy-critical nonlinear wave equation with an inverse-square\n potential"
],
[
"Abstract We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four.",
"In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter.",
"In the focusing case, we prove scattering below the ground state threshold."
],
[
"Introduction",
"We consider the initial-value problem for the energy-critical nonlinear wave equation (NLW) with an inverse-square potential.",
"The underlying linear problem is given in terms of the operator $\\mathcal {L}_a:= -\\Delta + a|x|^{-2}.$ Here we restrict to dimensions $d\\ge 3$ and values $a>-(\\tfrac{d-2}{2})^2$ , and we consider $\\mathcal {L}_a$ as the Friedrichs extension of the quadratic form defined on $C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace 0\\rbrace )$ via $f\\mapsto \\int _{\\mathbb {R}^d} |\\nabla f(x)|^2 + a|x|^{-2}|f(x)|^2\\,dx.$ The lower bound on $a$ guarantees positivity of $\\mathcal {L}_a$ ; in fact, by the sharp Hardy inequality one finds that the standard Sobolev space $\\dot{H}^1$ is equivalent to the Sobolev space $\\dot{H}_a^1$ defined in terms of $\\mathcal {L}_a$ (see Section ).",
"Furthermore, when $a=0$ we recover the standard Laplacian.",
"We consider the following nonlinear wave equation: ${\\left\\lbrace \\begin{array}{ll}\\partial _t^2 u + \\mathcal {L}_a u + \\mu |u|^{\\frac{4}{d-2}}u = 0, \\\\(u,\\partial _t u)|_{t=0} = (u_0,u_1).\\end{array}\\right.",
"}$ Here $u$ is a real-valued function on $\\mathbb {R}^{1+d}$ with $d\\ge 3$ and $\\mu \\in \\lbrace \\pm 1\\rbrace $ corresponds to the defocusing and focusing equations, respectively.",
"This is a Hamiltonian equation, with the conserved energy given by $E_a[\\vec{u}] = \\int \\tfrac{1}{2} |\\nabla u|^2 + \\tfrac{1}{2} |\\partial _t u|^2 + \\tfrac{1}{2} a|x|^{-2}|u|^2 + \\mu \\tfrac{d-2}{2d}|u|^{\\frac{2d}{d-2}}\\,dx,$ where $\\vec{u} = (u,\\partial _t u)$ .",
"The notation $E_a[f]$ should be understood as $E_a[(f,0)]$ .",
"The operator $\\mathcal {L}_a$ arises often in mathematics and physics in scaling limits of more complicated problems, for example in combustion theory, the Dirac equation with Coulomb potential, and the study of perturbations of space-time metrics such as Schwarzschild and Reissner–Nordström [4], [17], [43], [44].",
"One particularly interesting feature of the inverse-square potential is that it has the same scaling as the Laplacian.",
"In particular, one cannot in general treat $\\mathcal {L}_a$ as a perturbation of $-\\Delta $ , which contributes to the mathematical interest of this particular model.",
"An additional consequence is that (REF ) has a scaling symmetry, namely, $\\begin{aligned}u(t,x)&\\mapsto \\lambda ^{\\frac{d}{2}-1} u(\\lambda t,\\lambda x), \\\\\\partial _t u(t,x) &\\mapsto \\lambda ^{\\frac{d}{2}} (\\partial _t u)(\\lambda t,\\lambda x).\\end{aligned}$ This rescaling leaves the energy invariant and identifies the scaling-critical space of initial data to be the energy space $\\dot{H}^1\\times L^2$ .",
"We therefore call (REF ) an energy-critical equation, and indeed when $a=0$ the equation reduces to the standard energy-critical NLW, which has been the center of a great deal of research in recent years.",
"On the other hand, the presence of the inverse-square potential breaks translation symmetry, introducing new challenges into the analysis of (REF ).",
"In particular, our work fits in the context of recent work on dispersive equations in the presence of broken symmetries, which have also attracted a great deal of interest in recent years (see e.g.",
"[9], [10], [11], [12], [13], [14], [15], [20], [24], [25], [28], [29], [35] and in particular [22], [23], [16], [33] for the case of nonlinear Schrödinger equations with an inverse-square potential).",
"We consider the problem of global well-posedness and scattering for (REF ).",
"In the defocusing case, we will prove scattering for arbitrary data in the energy space.",
"In the focusing case, we will prove scattering below the ground state threshold.",
"These results parallel those established for the standard energy-critical NLW (see e.g.",
"[2], [6], [7], [18], [19], [30], [34], [37], [39]), and as in many of those works we will proceed via the concentration-compactness/rigidity approach.",
"Before comparing our work with the existing literature, however, let us state our main results more precisely.",
"Implicit in the statements below is the fact that any initial data in the energy space leads to a unique solution that exists at least locally in time (see Proposition REF ).",
"This local result leads to some restrictions on the parameter $a$ , which we will state in terms of the following constant: $c_d={\\left\\lbrace \\begin{array}{ll} \\tfrac{1}{25} & d=3, \\\\ \\tfrac{1}{9} & d=4.\\end{array}\\right.",
"}$ See Section REF and Section REF for more details.",
"As above, given a solution $u$ to a (linear or nonlinear) wave equation, we write $\\vec{u}=(u,\\partial _t u)$ .",
"We say a solution $u$ to (REF ) scatters if there exist solutions $v_\\pm (t)$ to $(\\partial _t^2+\\mathcal {L}_a)v_\\pm =0$ such that $\\lim _{t\\rightarrow \\pm \\infty } \\Vert \\vec{u}(t) - \\vec{v}_\\pm (t) \\Vert _{\\dot{H}^1\\times L^2} =0.$ Our result in the defocusing case is the following theorem.",
"Theorem 1.1 Let $d\\in \\lbrace 3,4\\rbrace $ , $a>-(\\frac{d-2}{2})^2+c_d$ , and $\\mu =+1$ .",
"For any $(u_0,u_1)\\in \\dot{H}^1\\times L^2$ , the corresponding solution to (REF ) is global and scatters.",
"We next turn to the focusing case.",
"In this case, there exist global nonscattering solutions, and hence we do not expect a scattering result without some size restrictions.",
"Indeed, fixing $a>-(\\frac{d-2}{2})^2$ and defining $\\beta >0$ through the identity $a=(\\tfrac{d-2}{2})^2[\\beta ^2-1]$ , the ground state soliton for (REF ) is the static solution to (REF ) defined by $W_a(x):=[d(d-2)\\beta ^2]^{\\frac{d-2}{4}}\\bigl [\\tfrac{|x|^{\\beta -1}}{1+|x|^{2\\beta }}\\bigr ]^{\\frac{d-2}{2}},$ which arises as an optimizer of a Sobolev embedding inequality (see [22], [38] and Section REF below).",
"Our result in the focusing case is a scattering result below the ground state threshold.",
"In the following, we write $a\\wedge 0=\\min \\lbrace a,0\\rbrace $ and let $\\dot{H}_a^1$ denote the Sobolev space defined in terms of $\\sqrt{\\mathcal {L}_a}$ (see Section REF ).",
"Theorem 1.2 Let $d\\in \\lbrace 3,4\\rbrace ,$ $a>-(\\frac{d-2}{2})^2+c_d$ , and $\\mu =-1$ .",
"Let $(u_0,u_1)\\in \\dot{H}^1\\times L^2$ satisfy $E_a[(u_0,u_1)] < E_{a\\wedge 0}[W_{a\\wedge 0}]\\quad \\text{and}\\quad \\Vert u_0\\Vert _{\\dot{H}_a^1} < \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}^1_{a\\wedge 0}}.$ Then the corresponding solution to (REF ) is global and scatters.",
"As mentioned above, the restriction $a>-(\\tfrac{d-2}{2})^2$ guarantees that the operator $\\mathcal {L}_a$ is positive, while the further restriction $a>-(\\tfrac{d-2}{2})^2+c_d$ arises in the development of the local theory for (REF ).",
"Note that both results still include a range negative values of $a$ , in which case the potential is attractive.",
"This is in contrast to many results for dispersive PDE with potentials, in which case one must consider repulsive (or perturbative) potentials in order to obtain scattering.",
"We restrict to dimensions $d\\in \\lbrace 3,4\\rbrace $ to guarantee that the nonlinearity in (REF ) is algebraic (quintic and cubic, respectively), which is primarily for technical convenience and already includes the most interesting cases.",
"We expect that the results extend to higher dimensions, as well.",
"Finally, let us point out that the results above hold without any radial assumption on the initial data.",
"Theorems REF and REF parallel the existing results for the standard energy-critical NLW without potential (henceforth the free NLW).",
"In fact, as we will see, our results rely in an essential way on these existing results.",
"Comparing with the result of Kenig and Merle on the focusing NLW [19], we find that the scattering threshold for (REF ) is the same as that for the standard NLW in the range $a>0$ .",
"In the free case, one also has a blowup result for solutions below the ground state energy with $\\Vert u_0\\Vert _{\\dot{H}^1}>\\Vert W_0\\Vert _{\\dot{H}^1}$ .",
"The existence ground state soliton then shows that the $E_0[W_0]$ is the correct energy threshold for a simple blowup/scattering dichotomy.",
"The situation is completely analogous when $a\\le 0$ .",
"On the other hand, when $a>0$ the threshold is not given in terms of the soliton $W_a$ ; nonetheless, the condition appearing in Theorem REF is sharp in terms of obtaining uniform space-time bounds.",
"In particular, the proof of Theorem REF will show that the solutions constructed have critical space-time norms controlled by $C(E_{a\\wedge 0}[W_{a\\wedge 0}]-E_a[(u_0,u_1)])$ for some function $C$ ; we will show that this constant diverges as one approaches the threshold.",
"Similar results have been obtained in [29], [23].",
"We summarize the results just mentioned in the following theorem.",
"Theorem 1.3 Let $d\\in \\lbrace 3,4\\rbrace $ , $a>-(\\tfrac{d-2}{2})^2+c_d$ , and $\\mu =-1$ .",
"(i) If $(u_0,u_1)\\in \\dot{H}^1\\times L^2$ satisfy $E_a[(u_0,u_1)]<E_{a\\wedge 0}[W_{a \\wedge 0}] \\quad \\text{and}\\quad \\Vert u_0\\Vert _{\\dot{H}_a^1} > \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1},$ then the corresponding solution to (REF ) blows up in finite time in both time directions.",
"(ii) If $a>0$ , then there exist a sequence of global solutions $u_n$ such that $E_a[\\vec{u}_n]\\nearrow E_0[W_0]\\quad \\text{and}\\quad \\Vert u_n(0)\\Vert _{\\dot{H}_a^1}\\nearrow \\Vert W_0\\Vert _{\\dot{H}^1},$ with $\\lim _{n\\rightarrow \\infty } \\Vert u_n\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\mathbb {R}\\times \\mathbb {R}^d)} = \\infty .$ Our main focus in this paper is on the scattering results, Theorem REF and REF .",
"In the remainder of the introduction, we will primarily discuss the proof of these results.",
"The proof of Theorem REF is relatively straightforward and will be explained in Section .",
"The strategy of proof for Theorems REF and REF is the concentration-compactness approach to induction on energy, often referred to as the Kenig–Merle roadmap.",
"In particular, supposing that either theorem is false, we show that there would exist a special type of solution to (REF ) (constructed as a minimal blowup solution) possessing certain compactness properties.",
"We then preclude the possibility that such solutions can exist.",
"The precise notion of compactness is given by the following: Definition 1.4 (Almost periodic solution) Let $\\vec{u}$ be a nonzero solution to (REF ) on an interval $I$ .",
"We call $\\vec{u}$ almost periodic (modulo symmetries) if there exist a spatial center $x:I\\rightarrow \\mathbb {R}^d$ and frequency scale $N:I\\rightarrow (0,\\infty )$ such that the set $\\bigl \\lbrace \\bigl (N(t)^{\\frac{d}{2}-1}u(t,N(t)[x-x(t)]), N(t)^{\\frac{d}{2}}(\\partial _t u)(t,N(t)[x-x(t)])\\bigr ):t\\in I\\bigr \\rbrace $ is pre-compact in $\\dot{H}^1\\times L^2$ .",
"The first main step of the proof, the reduction to almost periodic solutions, appears as Theorem REF below.",
"The general strategy is well-established, with essentially two key ingredients: (i) a linear profile decomposition adapted to a Strichartz estimate (see Proposition REF ) and (ii) a corresponding `nonlinear profile decomposition'.",
"Both of these steps involve additional difficulties in the setting of equations with broken symmetries.",
"For (i), the new difficulty is related mostly to understanding the convergence of certain linear operators that arise due to the failure of translation symmetry (see Lemma REF and Lemma REF ).",
"For (ii), the key difficulty arises from the construction of (scattering) nonlinear solutions associated to profiles with a translation parameter tending to infinity.",
"Because of the broken translation symmetry, one cannot simply solve (REF ) and then incorporate the translation.",
"Instead, roughly speaking, one constructs a solution to the free NLW, incorporates the translation, and (because the profile lives far from the origin) shows that the result is an approximate solution to (REF ).",
"An application of the stability theory for (REF ) then yields the true solution, as desired.",
"The construction of a suitable (scattering) solution to the free NLW relies on the full strength of [19], [2].",
"For more details, see Proposition REF .",
"With the necessary ingredients in place, the reduction to almost periodic solutions (Theorem REF ) follows along fairly standard lines (see Section REF ).",
"After this, it is useful to make some further reductions to the class of almost periodic solutions that we consider.",
"Recall from Definition REF above that almost periodic solutions are described in terms of a spatial center $x(t)$ and frequency scale $N(t)$ .",
"Because of the result in Proposition REF , we firstly find that we must have $x(t)\\equiv 0$ ; indeed, as described above, profiles with translation parameters tending to infinity correspond to scattering solutions and hence do not arise in the construction of minimal blowup solutions.",
"In Theorem REF , we adapt arguments of [19] to further reduce to two scenarios, namely, the `forward global' scenario and the `self-similar' scenario.",
"The preclusion of both scenarios relies heavily on certain virial/Morawetz identities.",
"For NLW with a general potential $V(x)$ , these identities will involve a term of the form $-\\tfrac{1}{2} x\\cdot \\nabla V$ (see Lemma REF , for example).",
"For repulsive potentials (i.e.",
"those satisfying $x\\cdot \\nabla V\\le 0$ ), this term generally has a good sign and is amenable to deriving monotonicity formulas; this explains in part why scattering results are often restricted to the case of repulsive potentials.",
"For the inverse square potential $V(x)=a|x|^{-2}$ , one has the identity $\\tfrac{1}{2} x\\cdot \\nabla V = -V$ (a consequence of scaling).",
"In particular, while the potential is repulsive only for $a\\ge 0$ , this identity ultimately allows us to prove suitable virial/Morawetz estimates even when $a<0$ .",
"To deal with the forward-global scenario, we use a localized virial estimate (see Section ).",
"The implementation of this argument is fairly straightforward due to the fact that we have $x(t)\\equiv 0$ .",
"This should be contrasted with [19], where the authors must argue (using Lorentz boosts) that minimal blowup solutions have zero momentum, which then allows for sufficient control over $x(t)$ to run the localized virial argument.",
"In the focusing case, we must also rely on the coercivity given by sharp Sobolev embedding and the fact that the minimal blowup solution is below the ground state in the sense of (REF ) (see Lemma REF ).",
"Finally, we rule out the self-similar scenario in Section .",
"In this scenario, the solution $u$ blows up at time $t=0$ and is supported at each $t>0$ in $B_t(0)$ , with $N(t)=t^{-1}$ .",
"To rule out such solutions, we firstly prove a virial/Morawetz estimate (Proposition REF ), which in particular implies the vanishing of the quantity $\\partial _t u + x\\cdot \\nabla u + \\tfrac{d-2}{2}\\tfrac{u}{t}$ along a sequence $t_n\\rightarrow 0$ .",
"This particular estimate is similar to one used to study wave maps (appearing e.g.",
"in [8], [41], [40]), and is particularly closely related to the estimate appearing in [5].",
"Using Proposition REF and almost periodicity, we are able to extract a true self-similar solution to (REF ), that is, a (nonzero) solution of the form $v(t,x)=(1+t)^{-[\\frac{d}{2}-1]}f(\\tfrac{x}{1+t}).$ It follows that $f$ solves a degenerate elliptic PDE in the unit ball and vanishes near the boundary in a certain sense (see Proposition REF ).",
"At this point, we find ourselves essentially in the same position as Kenig and Merle [19]; indeed, for this portion of the argument the difficulty arises only at the boundary of the unit ball, and we may safely include the potential term in the nonlinearity.",
"Following closely the arguments of [19], we employ a change of variables to remove the degeneracy and invoke unique continuation results to deduce that $f\\equiv 0$ , yielding a contradiction and completing the proof of Theorems REF and REF .",
"As just described, our arguments in the self-similar scenario differ from Kenig and Merle [19] essentially only in the extraction of the elliptic solution.",
"In particular, [19] employs a self-similar change of variables, while we utilize a virial/Morawetz estimate (actually closer to the spirit of the arguments of [5]).",
"In fact, one observes that setting $a= 0$ throughout Section leads to a modified proof of the preclusion of self-similar almost periodic solutions for the free NLW.",
"The rest of the paper is organized as follows: In Section we first set up some notation and record some useful lemmas, including the useful virial/Morawetz identity.",
"In Section REF , we collect some harmonic analysis tools adapted to $\\mathcal {L}_a$ , including some results concerning the equivalence of Sobolev spaces as well as Strichartz esimates.",
"In Section REF we record the basic local well-posedness and stability theory for (REF ).",
"Finally, in Section REF , we record some results related to the sharp Sobolev embedding and the ground state solution for (REF ).",
"In Section we develop concentration compactness tools for (REF ).",
"The main result of this section is the linear profile decomposition, Proposition REF .",
"In Section , we prove the existence of minimal blowup solutions under the assumption that Theorem REF or Theorem REF fails (cf.",
"Theorem REF ).",
"We then refine the class of solutions that we need to consider (see Theorem REF ).",
"In Section we preclude the forward global case of Theorem REF .",
"In Section we preclude the self-similar case of Theorem REF , thus completing the proof of Theorems REF and REF .",
"Finally, in Section , we give the proof of Theorem REF ."
],
[
"Acknowledgements",
"C. M. and J. Z.",
"were supported by NSFC Grants 11831004.",
"Part of this work was completed while J. M. was supported by the NSF postdoctoral fellowship DMS-1400706 at UC Berkeley.",
"We are grateful to R. Killip for some useful discussions related to Lemma REF below.",
"We are also grateful to H. Jia for helping us to understand the works [5], [19], which played a key role in developing Section of this paper."
],
[
"Notation and lemmas",
"We will employ some standard geometric notation.",
"We let $g_{\\alpha \\beta }=\\text{diag}(-1,1,\\dots ,1)$ denote the standard Minkowski metric on $\\mathbb {R}^{1+d}$ and denote the inverse metric by $g^{\\alpha \\beta }$ .",
"Greek indices take values in $\\lbrace 0,1,\\dots ,d\\rbrace $ while Roman indices take values in $\\lbrace 1,\\dots ,d\\rbrace $ .",
"We employ Einstein summation convention, and we raise and lower indices with respect to the metric: $\\partial ^\\alpha =g^{\\alpha \\beta }\\partial _\\beta $ .",
"For example, (REF ) may be written $\\partial ^\\alpha \\partial _\\alpha u = a|x|^{-2}u + \\mu |u|^{\\frac{4}{d-2}}u.$ We write $(x^{\\alpha })$ for space-time coordinates, with $x^0=t$ .",
"We use $\\nabla $ for the gradient in the spatial variables only; the space-time gradient will be denoted by $\\nabla _{t,x}$ .",
"We use the standard Lebesgue spaces $L^p(\\mathbb {R}^d)$ , as well as the mixed space-time norms $L_t^q L_x^r(\\mathbb {R}^{1+d})$ , defined by $\\Vert u\\Vert _{L_t^q L_x^r(\\mathbb {R}^{1+d})} = \\bigl \\Vert \\,\\Vert u(t)\\Vert _{L_x^r(\\mathbb {R}^d)}\\, \\Vert _{L_t^q(\\mathbb {R})}.$ We write $q^{\\prime }\\in [1,\\infty ]$ for the dual exponent of $q\\in [1,\\infty ]$ , i.e.",
"the solution to $\\tfrac{1}{q}+\\tfrac{1}{q^{\\prime }}=1$ .",
"We write $A\\lesssim B$ to denote $A\\le CB$ for some $C>0$ .",
"We can similarly define $A\\gtrsim B$ .",
"We write $a\\pm $ to denote a quantity of the form $a\\pm \\varepsilon $ for some small $\\varepsilon >0$ .",
"We introduce a mapping $T_a$ that takes a pair of real-valued functions $(\\phi ,\\psi )$ and returns a single complex-valued function defined by $T_a(\\phi ,\\psi ):=\\phi +i\\mathcal {L}_a^{-\\frac{1}{2}}\\psi .$ We apply this mapping to $\\vec{u} = (u,\\partial _t u)$ , where $u$ solves (REF ), that is, $T_a\\vec{u}= u + i\\mathcal {L}_a^{-\\frac{1}{2}}\\partial _t u.$ Thus $u=\\operatornamewithlimits{Re}T_a\\vec{u}$ , and $u$ solves (REF ) with data in $\\dot{H}_a^1\\times L^2$ if and only if the complex-valued function $v:=T_a\\vec{u}$ solves $i\\partial _t v - \\mathcal {L}_a^{\\frac{1}{2}}v - \\mu \\mathcal {L}_a^{-\\frac{1}{2}}(|\\operatornamewithlimits{Re}v|^{\\frac{4}{d-2}}\\operatornamewithlimits{Re}v)=0$ with data in $\\dot{H}_a^1$ .",
"In these variables we have $E_a[\\vec{u}] = \\tilde{E}_a[T_a\\vec{u}]:= \\int \\tfrac{1}{2} |\\mathcal {L}_a^{\\frac{1}{2}}T_a\\vec{u}|^2 + \\mu \\tfrac{d-2}{2d}|\\operatornamewithlimits{Re}T_a\\vec{u}|^{\\frac{2d}{d-2}}\\,dx.$ We will need the following refinement of Fatou's lemma in Section .",
"Lemma 2.1 (Refined Fatou, [3]) Let $1\\le p<\\infty $ and let $\\lbrace f_n\\rbrace $ be bounded in $L^p$ .",
"If $f_n\\rightarrow f$ almost everywhere, then $\\lim _{n\\rightarrow \\infty } \\int \\bigl | |f_n|^p - |f_n-f|^p - |f|^p \\bigr |\\,dx =0.$ Finally, we record the following virial identity that will be used on a few occasions below.",
"The proof follows from direct computation and integration by parts.",
"Lemma 2.2 (Virial identity) Fix a weight $w:\\mathbb {R}^d\\rightarrow \\mathbb {R}$ and a solution $u:\\mathbb {R}^{1+d}\\rightarrow \\mathbb {R}$ to the wave equation $\\partial _\\alpha \\partial ^\\alpha u = Vu+G^{\\prime }(u).$ Then $\\partial _t \\int -\\partial _tu[\\nabla u\\cdot \\nabla w + \\tfrac{1}{2} u\\Delta w]\\,dx & = \\int \\nabla u \\cdot \\nabla ^2 w\\nabla u + \\Delta w[\\tfrac{1}{2} uG^{\\prime }(u) - G(u)] \\\\& \\quad \\quad -\\tfrac{1}{2} \\nabla w\\cdot \\nabla V u^2 - \\tfrac{1}{4} \\Delta \\Delta w(u^2)\\,dx.$"
],
[
"Harmonic analysis tools",
"A harmonic analysis toolkit adapted to $\\mathcal {L}_a$ was developed in [21].",
"In this section, we will import several relevant results.",
"We will also record some Strichartz estimates adapted to the linear wave equation with inverse-square potential, which were established in [4].",
"For $r\\in (1,\\infty )$ we write $\\dot{H}_a^{1,r}$ and $H_a^{1,r}$ for the homogeneous and inhomogeneous Sobolev spaces defined in terms of $\\mathcal {L}_a$ ; these have norms $\\Vert f\\Vert _{\\dot{H}_a^{1,r}} = \\Vert \\sqrt{\\mathcal {L}_a} f\\Vert _{L^r},\\quad \\Vert f\\Vert _{H_a^{1,r}} = \\Vert \\sqrt{1+\\mathcal {L}_a} f\\Vert _{L^r}.$ When $r=2$ we write $\\dot{H}_a^{1,2}=\\dot{H}_a^{1}$ .",
"Let us introduce the parameter $\\sigma = \\tfrac{d-2}{2}-\\bigl [(\\tfrac{d-2}{2})^2+a\\bigr ]^{\\frac{1}{2}}.$ One of the main results in [21] is the following result concerning the equivalence of Sobolev spaces.",
"Lemma 2.3 (Equivalence of Sobolev spaces, [21]) Let $d\\ge 3$ , $a>-(\\tfrac{d-2}{2})^2$ , and $s\\in (0,2)$ .",
"If $p\\in (1,\\infty )$ satisfies $\\tfrac{s+\\sigma }{d}<\\tfrac{1}{p}<\\min \\lbrace 1,\\tfrac{d-\\sigma }{d}\\rbrace ,$ then $\\Vert |\\nabla |^s f\\Vert _{L^p} \\lesssim \\Vert \\mathcal {L}_a^{\\frac{s}{2}}f\\Vert _{L^p}.$ If $p\\in (1,\\infty )$ satisfies $\\max \\lbrace \\tfrac{s}{d},\\tfrac{\\sigma }{d}\\rbrace <\\tfrac{1}{p}<\\min \\lbrace 1,\\tfrac{d-\\sigma }{d}\\rbrace ,$ then $\\Vert \\mathcal {L}_a^{\\frac{s}{2}}f\\Vert _{L^p} \\lesssim \\Vert |\\nabla |^s f\\Vert _{L^p}.$ We will use Littlewood–Paley projections defined through the heat kernel, i.e.",
"$P_N^a = e^{-\\mathcal {L}_a/N^2}-e^{-4\\mathcal {L}_a/N^2},$ where $N\\in 2^{\\mathbb {Z}}$ .",
"As was shown in [31], [32], the heat kernel $e^{-t\\mathcal {L}_a}(x,y)$ has upper and lower bounds of the form $C(1\\wedge \\tfrac{\\sqrt{t}}{|x|})^{\\sigma }(1\\wedge \\tfrac{\\sqrt{t}}{|y|})^\\sigma e^{-|x-y|^2/ct}.$ To state results, it will be useful to define the exponent $q_0={\\left\\lbrace \\begin{array}{ll} \\infty & \\text{if }a\\ge 0, \\\\ \\tfrac{d}{\\sigma } & \\text{if} -(\\tfrac{d-2}{2})^2<a<0.\\end{array}\\right.",
"}$ We write $q_0^{\\prime }$ for the dual exponent in both cases.",
"We record the harmonic analysis tools we need in the following proposition.",
"Proposition 2.4 (Harmonic analysis tools, [21]) Let $q_0^{\\prime }<q\\le r<q_0$ .",
"We have the following expansion: $f=\\sum _{N\\in 2^{\\mathbb {Z}}} P_N^a f \\quad \\text{as elements of}\\quad L^r.$ We have the following Bernstein estimates: The operators $P_N^a$ are bounded on $L^r$ .",
"The operators $P_N^a$ are bounded from $L^q$ to $L^r$ with norm bounded by $N^{\\frac{d}{q}-\\frac{d}{r}}$ .",
"For any $s\\in \\mathbb {R}$ , $N^s\\Vert P_N^a f\\Vert _{L^r} \\sim \\Vert \\mathcal {L}_a^{\\frac{s}{2}} P_N^a f\\Vert _{L^r}.$ We have the square function estimate: $\\biggl \\Vert \\biggl ( \\sum _{N\\in 2^{\\mathbb {Z}}} |P_N^a f|^2\\biggr )^{\\frac{1}{2}}\\biggr \\Vert _{L^r} \\sim \\Vert f\\Vert _{L^r}.$ We next turn to Strichartz estimates for the linear wave equation with inverse-square potential.",
"Here we import results of Burq, Planchon, Stalker, and Tahvildar-Zadeh [4], specifically Theorem 5 and Theorem 9 therein.",
"We state the estimates in terms of the operators $e^{\\pm it\\sqrt{\\mathcal {L}_a}}$ and specialize to dimensions $d\\in \\lbrace 3,4\\rbrace $ .",
"Proposition 2.5 (Strichartz) Let $q,r\\ge 2$ satisfy the wave admissibility condition $\\tfrac{1}{q}+\\tfrac{d-1}{2r}\\le \\tfrac{d-1}{4},$ where in $d=3$ we additionally require $q,\\tilde{q}>2$ .",
"Define $\\gamma $ via the scaling relation $\\tfrac{1}{q}+\\tfrac{d}{r}=\\tfrac{d}{2}-\\gamma .$ For any time interval $I$ we have $\\Vert e^{\\pm it\\sqrt{\\mathcal {L}_a}} f\\Vert _{L_t^q L_x^r(I\\times \\mathbb {R}^d)} \\lesssim \\Vert f\\Vert _{\\dot{H}^{\\gamma }(\\mathbb {R}^d)}$ provided the following conditions hold: If $d=3$ , then we require $-\\min \\lbrace 1,\\sqrt{a+\\tfrac{9}{4}}-\\tfrac{1}{2},\\sqrt{a+\\tfrac{1}{4}}+1\\rbrace <\\gamma <\\min \\lbrace 2,\\sqrt{a+\\tfrac{9}{4}}+\\tfrac{1}{2},\\sqrt{a+\\tfrac{1}{4}}+1-\\tfrac{1}{q}\\rbrace .$ If $d=4$ , then we require $-\\min \\lbrace \\tfrac{5}{6},\\sqrt{a+4}-\\tfrac{7}{6},\\sqrt{a+1}+1\\rbrace <\\gamma <\\min \\lbrace \\tfrac{5}{2},\\sqrt{a+4}+\\tfrac{1}{2},\\sqrt{a+1}+1-\\tfrac{1}{q}\\rbrace .$ We will also need an inhomogeneous estimate.",
"In particular, using Proposition REF , Lemma REF , and the Christ–Kiselev lemma, we have the following: Corollary 2.6 (Strichartz) Let $q,r,\\gamma $ be as in Proposition REF and let $\\tilde{q},\\tilde{r},\\tilde{\\gamma }$ be defined similarly.",
"Suppose $q,\\tilde{q}>2$ .",
"Then for any time interval $I\\ni t_0$ , we have $\\biggl \\Vert \\int _{t_0}^t e^{i(t-s)\\sqrt{\\mathcal {L}_a}}F(s)\\,ds\\biggr \\Vert _{L_t^q L_x^r(I\\times \\mathbb {R}^d)} \\lesssim \\Vert |\\nabla |^{\\gamma +\\tilde{\\gamma }} F\\Vert _{L_t^{\\tilde{q}^{\\prime }} L_x^{\\tilde{r}^{\\prime }}(I\\times \\mathbb {R}^d)}.$"
],
[
"Local well-posedness and stability",
"We next develop the local theory for (REF ), including a stability result.",
"As the arguments are rather standard, we will be rather brief.",
"As usual, the results rely primarily on Strichartz estimates, which were recorded in the previous section.",
"We will construct solutions that lie locally in $L_t^\\infty (\\dot{H}^1\\times L^2)$ as well as the Strichartz space $S(I):=L_{t,x}^{\\frac{2(d+1)}{d-2}}(I\\times \\mathbb {R}^d) \\cap L_t^{\\frac{d+2}{d-2}}L_x^{2(\\frac{d+2}{d-2})}(I\\times \\mathbb {R}^d).$ Note that the scaling of $S$ corresponds to $\\gamma = 1$ in the Strichartz estimates appearing above, and that we are able to use these spaces provided we choose $a>{\\left\\lbrace \\begin{array}{ll} -\\tfrac{1}{4} + \\tfrac{1}{25} & d=3 \\\\ -1 + \\tfrac{1}{9} & d=4.\\end{array}\\right.",
"}$ This is the origin of the constant $c_d$ defined in (REF ) and appearing in the statements of the main results, Theorem REF and Theorem REF .",
"Writing the Duhamel formulation of (REF ), namely, $u(t)=\\cos (t\\sqrt{\\mathcal {L}_a})u_0+\\frac{\\sin (t\\sqrt{\\mathcal {L}_a})}{\\sqrt{\\mathcal {L}_a}}u_1 -\\mu \\int _0^t \\frac{\\sin ((t-s)\\sqrt{\\mathcal {L}_a})}{\\sqrt{\\mathcal {L}_a}}(|u|^{\\frac{4}{d-2}}u)(s)\\,ds,$ we can run a contraction mapping in the space $L_t^\\infty (\\dot{H}^1\\times L^2)\\cap S$ by relying on the nonlinear estimate $\\biggl \\Vert \\int _0^t \\frac{\\sin ((t-s)\\sqrt{\\mathcal {L}_a})}{\\sqrt{\\mathcal {L}_a}}(|u|^{\\frac{4}{d-2}}u)(s)\\,ds\\biggr \\Vert _{L_t^\\infty \\dot{H}^1\\cap S} \\lesssim \\Vert |u|^{\\frac{4}{d-2}}u\\Vert _{L_t^1 L_x^2} \\lesssim \\Vert u\\Vert _{S}^{\\frac{d+2}{d-2}}.$ The conclusion is the following local result.",
"Proposition 2.7 (Local well-posedness) Let $(u_0,u_1)\\in \\dot{H}^1\\times L^2$ , $d\\in \\lbrace 3,4\\rbrace $ , and $a>-(\\frac{d-2}{2})^2+c_d$ .",
"There exists $\\eta _0$ such that if $\\Vert \\cos (t\\sqrt{\\mathcal {L}_a})u_0+\\frac{\\sin (t\\sqrt{\\mathcal {L}_a})}{\\sqrt{\\mathcal {L}_a}}u_1\\Vert _{S(I)}< \\eta $ for some $0<\\eta <\\eta _0$ , then there exists a solution to (REF ) on $I$ satisfying $\\Vert u\\Vert _{S(I)}\\lesssim \\eta $ .",
"In particular, data in $\\dot{H}^1\\times L^2$ leads to local-in-time solutions.",
"The solution may be extended as long as the $S$ -norm remains finite, and if the solution is global with $\\Vert u\\Vert _{S(\\mathbb {R})}<\\infty $ then the solution scatters in both time directions.",
"Finally, given a final state $(u_0^+,u_1^+)\\in \\dot{H}^1\\times L^2$ , we may construct a solution on an interval $(T,\\infty )$ that scatters to $(u_0^+,u_1^+)$ as $t\\rightarrow \\infty $ .",
"A similar result holds backward in time.",
"Standard arguments relying primarily on the same Strichartz estimates as above yield the following stability result for (REF ).",
"Proposition 2.8 (Stability) Let $d,a$ be as in Proposition REF .",
"$I$ be a time interval and let $\\tilde{u}$ satisfy $\\partial _t^2\\tilde{u} + \\mathcal {L}_a\\tilde{u} + \\mu |\\tilde{u}|^{\\frac{4}{d-2}}\\tilde{u} + e=0$ for some function $e:I\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ .",
"Suppose that $\\Vert \\tilde{u}\\Vert _{S(I)}+\\Vert (\\tilde{u}(t_0),\\partial _t \\tilde{u}(t_0))\\Vert _{\\dot{H}^1\\times L^2} \\le L$ for some $t_0\\in I$ and $L>0$ .",
"There exists $\\varepsilon _0=\\varepsilon _0(L)$ such that for $0<\\varepsilon <\\varepsilon _0$ we have the following: if $\\Vert \\vec{u}_0-(\\tilde{u}(t_0),\\partial _t \\tilde{u}(t_0))\\Vert _{\\dot{H}^1\\times L^2} + \\Vert e\\Vert _{L_t^1 L_x^2(I\\times \\mathbb {R}^d)} <\\varepsilon ,$ then there exists a solution $u$ to (REF ) on $I$ with $\\vec{u}(t_0)=\\vec{u}_0$ satisfying $\\Vert u-\\tilde{u}\\Vert _{S(I)} \\lesssim _L \\varepsilon \\quad \\text{and}\\quad \\Vert \\vec{u}\\Vert _{L_t^\\infty (I;\\dot{H}^1\\times L^2)}+\\Vert u\\Vert _{S(I)}\\lesssim 1.$ For an introduction to these types of results, we refer the reader to [26].",
"We remark that by applying the transformation $T_a$ introduced above, one has equivalent local well-posedness and stability results stated in terms of the equation (REF ) with initial data in $\\dot{H}^1$ .",
"We will use both versions of these results below."
],
[
"Variational analysis",
"In this section we record results related to the sharp Sobolev embedding $\\Vert f\\Vert _{L^{\\frac{2d}{d-2}}(\\mathbb {R}^d)} \\le C_a \\Vert f\\Vert _{\\dot{H}_a^1(\\mathbb {R}^d)},$ where $C_a$ denotes the sharp constant.",
"Much of the analysis that we need was carried out in [22]; see also [38].",
"For $a>-(\\frac{d-2}{2})^2$ , we define $\\beta >0$ by $a=(\\tfrac{d-2}{2})^2[\\beta ^2-1]$ .",
"We may also write $\\sigma =\\tfrac{d-2}{2}(1-\\beta )$ .",
"The ground state soliton is defined by $W_a(x) = [d(d-2)\\beta ^2]^{\\frac{d-2}{4}}\\bigl [ \\tfrac{|x|^{\\beta -1}}{1+|x|^{2\\beta }}\\bigr ]^{\\frac{d-2}{2}}.$ We have that $W_a$ solves $\\mathcal {L}_aW_a - |W_a|^{\\frac{4}{d-2}}W_a = 0,$ and $\\Vert W_a\\Vert _{\\dot{H}_a^1}^2 = \\Vert W_a\\Vert _{L^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}=\\tfrac{\\pi d(d-2)}{4}\\bigl [\\tfrac{2\\sqrt{\\pi }\\beta ^{d-1}}{\\Gamma (\\frac{d+1}{2})}\\bigr ]^{\\frac{2}{d}}.$ Note that the first identity above and [22] imply $C_a=\\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}^{-\\frac{2}{d}}.$ In Section REF , we will need to construct scattering nonlinear solutions to (REF ) that are parametrized by a spatial center approaching infinity.",
"To do this, we need to approximate by solutions to the nonlinear wave equation without potential; in particular, we need to rely on the scattering result of [19].",
"Consequently, in the focusing case we need to be sure that initial data lying below the threshold stated in Theorem REF also lie below the appropriate threshold for the equation without potential.",
"This fact is guaranteed by the following corollary.",
"Corollary 2.9 (Comparison of thresholds) Let $a>-(\\tfrac{d-2}{2})^2$ .",
"Then $E_{a\\wedge 0}[W_{a\\wedge 0}] \\le E_{0}[W_{0}]\\quad \\text{and}\\quad \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1} \\le \\Vert W_{0}\\Vert _{\\dot{H}_{0}^1}.$ There is nothing to prove when $a\\ge 0$ , so let us fix $a<0$ .",
"We begin by observing that $\\Vert f\\Vert _{\\dot{H}_a^1}<\\Vert f\\Vert _{\\dot{H}_0^1},$ which implies $C_0\\le C_a$ ; indeed, $\\Vert f\\Vert _{L^{\\frac{2d}{d-2}}} \\le C_a \\Vert f\\Vert _{\\dot{H}_a^1} <C_a \\Vert f\\Vert _{\\dot{H}_0^1}.$ In light of (REF ), we have $\\Vert W_a\\Vert _{\\dot{H}_a^1} \\le \\Vert W_0\\Vert _{\\dot{H}_0^1},$ which is one of the desired inequalities.",
"For the remaining inequality, we again call on (REF ) and use the inequality just established to observe that $E_a(W_a)=\\tfrac{1}{d}\\Vert W_a\\Vert _{\\dot{H}_a^1}^2 \\le \\tfrac{1}{d}\\Vert W_0\\Vert _{\\dot{H}_0^1}^2 = E_0(W_0).$ This completes the proof.",
"Finally, we record the following lemma, which is almost identical to [22] and in particular follows from the same proof appearing there.",
"Lemma 2.10 (Coercivity) Let $d\\ge 3$ and $a>-(\\frac{d-2}{2})^2$ .",
"Suppose $u:I\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ is a solution to (REF ) with $\\mu =-1$ and initial data $\\vec{u}_0\\in \\dot{H}^1\\times L^2$ satisfying $E_a[\\vec{u}_0]\\le (1-\\delta )E_{a\\wedge 0}[W_{a\\wedge 0}]$ for some $\\delta >0$ .",
"If $\\Vert u_0\\Vert _{\\dot{H}_a^1} \\le \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}^1_{a\\wedge 0}}$ , then for all $t\\in I$ : $\\Vert u(t)\\Vert _{\\dot{H}_a^1}\\le (1-\\delta ^{\\prime })\\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}^1_{a\\wedge 0}}$ $\\int |\\mathcal {L}_a u(t,x)|^2 - |u(t,x)|^{\\frac{2d}{d-2}}\\,dx \\gtrsim _\\delta \\Vert u(t)\\Vert _{\\dot{H}_a^1}^2$ , $E_a(\\vec{u})\\sim _\\delta \\Vert \\vec{u}(t)\\Vert _{\\dot{H}^1\\times L^2}^2$ , for some $\\delta ^{\\prime }>0$ depending on $\\delta $ ."
],
[
"Concentration compactness",
"A key step in the proofs of Theorem REF and Theorem REF is to prove that if the result is false, then we can construct minimal blowup solutions with good compactness properties.",
"This will be carried out in Section (see Theorem REF and Theorem REF below).",
"In the present section, we will develop a key technical ingredient needed for this step, namely, a linear profile decomposition adapted to Strichartz estimates for $e^{it\\sqrt{\\mathcal {L}_a}}$ (see Proposition REF ).",
"We introduce the following notation, which helps keep track of the lack of translation symmetry in $\\mathcal {L}_a$ .",
"Definition 3.1 Given a sequence $\\lbrace y_n\\rbrace \\subset \\mathbb {R}^d$ , we define $\\mathcal {L}_a^n:=-\\Delta + \\tfrac{a}{|x+y_n|^2} \\quad \\text{and}\\quad \\mathcal {L}_a^\\infty :={\\left\\lbrace \\begin{array}{ll} -\\Delta +\\frac{a}{|x+y_\\infty |^2} \\quad &\\text{if}\\quad y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d,\\\\-\\Delta &\\text{if}\\quad |y_n|\\rightarrow \\infty .\\end{array}\\right.",
"}$ We therefore have $\\mathcal {L}_a[\\phi (\\cdot -y_n)]=[\\mathcal {L}_a^n\\phi ](\\cdot -y_n)]$ .",
"Proposition 3.2 (Linear profile decomposition) Let $f_n$ be a bounded sequence in $\\dot{H}_a^1$ .",
"Passing to a subsequence, there exist $J^*\\in \\lbrace 0,1,\\dots ,\\infty \\rbrace $ , profiles $\\lbrace \\phi ^j\\rbrace _{j=1}^{J^*}\\subset \\dot{H}_a^1$ , scales $\\lbrace \\lambda _n^j\\rbrace _{j=1}^{J^*}\\subset (0,\\infty )$ , and space-time positions $\\lbrace (t_n^j,x_n^j)\\rbrace \\subset \\mathbb {R}^{1+d}$ such that for any finite $0\\le J\\le J^*$ we have the decomposition $f_n=\\sum _{j=1}^J \\phi _n^j + r_n^J,\\quad \\text{where}\\quad \\phi _n^j(x) =(\\lambda _n^j)^{-(\\frac{d}{2}-1)}\\bigl [e^{it_n^j\\sqrt{\\mathcal {L}_a^{n_j}}}\\phi ^j\\bigr ](\\tfrac{x-x_n^j}{\\lambda _n^j}),$ where $\\mathcal {L}_a^{n_j}$ is as in Definition REF with $y_n^j=\\frac{x_n^j}{\\lambda _n^j}$ .",
"This decomposition satisfies the following properties for any finite $1\\le J\\le J^*$ : The remainder term satisfies $\\lim _{J\\rightarrow J^*} \\limsup _{n\\rightarrow \\infty } \\Vert e^{it\\sqrt{\\mathcal {L}_a}} r_n^J\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\mathbb {R}\\times \\mathbb {R}^d)}&=0, \\\\\\lim _{n\\rightarrow \\infty } (\\lambda _n^J)^{\\frac{d}{2}-1}[e^{-it_n^J \\sqrt{\\mathcal {L}_a}}r_n^J](\\lambda _n^J x + x_n^J)&\\rightharpoonup 0 \\quad \\text{weakly in}\\quad \\dot{H}^1.",
"$ For $j\\ne k$ we have the orthogonality condition $\\lim _{n\\rightarrow \\infty } \\big |\\log \\tfrac{\\lambda _n^j}{\\lambda _n^k} \\big |+\\tfrac{|x_n^j-x_n^k|^2}{\\lambda _n^j\\lambda _n^k} +\\tfrac{|t_n^j-t_n^k|^2}{\\lambda _n^j\\lambda _n^k}=\\infty .$ We also have the decouplings for each finite $0\\le J\\le J^*$ : $\\lim _{n\\rightarrow \\infty }\\Bigl \\lbrace \\Vert f_n\\Vert _{\\dot{H}^1_a}^2-\\sum _{j=1}^J\\Vert \\phi _n^j\\Vert _{\\dot{H}_a^1}^2 -\\Vert r_n^J\\Vert _{\\dot{H}^1_a}^2\\Bigr \\rbrace =0,\\\\\\lim _{n\\rightarrow \\infty }\\big \\lbrace \\Vert f_n\\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}-\\sum _{j=1}^J\\Vert \\phi _n^j\\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}-\\Vert r_n^J\\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}\\big \\rbrace =0.$ Finally, we may assume that for each $j$ either $t_n^j\\equiv 0$ or $t_n^j\\rightarrow \\pm \\infty $ and either $x_n^j\\equiv 0$ or $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty $ .",
"The strategy for proving Proposition REF is well-established: we remove one bubble at a time until the Strichartz norm is depleted.",
"The key to isolating an individual bubble is to first identify a scale for concentration, which can be done via a refinement of the usual Strichartz estimate.",
"One then finds a space-time position for concentration via Hölder's inequality.",
"The broken space translation symmetry introduces some additional technical difficulties, related the manner in which we have convergence of the operators $\\mathcal {L}_a^{n_j}$ to the limiting operator $\\mathcal {L}_a^\\infty $ .",
"We begin by collecting a few lemmas related to this latter point.",
"The first follows from the arguments of [22].",
"Lemma 3.3 (Convergence of operators, [22]) Let $a>-(\\frac{d-2}{2})^2+c_d$ .",
"Suppose $t_n\\rightarrow t\\in \\mathbb {R}$ and $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"Let $\\mathcal {L}_a^n$ and $\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"Then the following hold: $\\lim _{n\\rightarrow \\infty }\\big \\Vert \\bigl [\\sqrt{\\mathcal {L}_a^n} - \\sqrt{\\mathcal {L}_a^\\infty } \\,\\bigr ] \\psi \\big \\Vert _{L^2}&=0 \\quad \\text{for all}\\quad \\psi \\in \\dot{H}^1, \\\\\\lim _{n\\rightarrow \\infty }\\big \\Vert \\bigl [\\sqrt{\\mathcal {L}_a^n} - \\sqrt{\\mathcal {L}_a^\\infty } \\,\\bigr ] \\psi \\big \\Vert _{\\dot{H}^{-1}}&=0 \\quad \\text{for all}\\quad \\psi \\in L^2.$ Furthermore, if $y_\\infty \\ne 0$ , then $\\lim _{n\\rightarrow \\infty }\\big \\Vert \\big [e^{-\\mathcal {L}_a^n}-e^{-\\mathcal {L}_a^\\infty }\\big ] \\delta _0\\big \\Vert _{\\dot{H}^{-1}(\\mathbb {R}^d)}=0.",
"$ For the next result, an analogous statement appears in [22] for the case of the Schrödinger propagator; however, the proof relies on (endpoint) Strichartz estimates and hence we need a new argument in our case.",
"Lemma 3.4 (Convergence of operators) Let $a>-(\\frac{d-2}{2})^2$ .",
"Suppose $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"Let $\\mathcal {L}_a^n$ and $\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"Then $\\lim _{n\\rightarrow \\infty } \\Vert (e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }})\\psi \\Vert _{L_t^\\infty L_x^2(\\mathbb {R}\\times \\mathbb {R}^d)} = 0 \\quad \\text{for all}\\quad \\psi \\in L^2.$ By approximation, it suffices to consider $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace 0\\rbrace )$ .",
"It also suffices to consider the case $|y_n|\\rightarrow \\infty $ or $y_n\\rightarrow 0$ (by applying a fixed translation).",
"Case 1.",
"Suppose $y_n\\rightarrow 0$ .",
"Then $\\mathcal {L}_a^\\infty = \\mathcal {L}_a$ .",
"Let us define $u_n(t,x) = [e^{it\\sqrt{\\mathcal {L}_a}}\\psi (\\cdot -y_n)](x).$ Then by the triangle inequality, we have $\\Vert (e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{it\\sqrt{\\mathcal {L}_a}})\\psi \\Vert _{L_t^\\infty L_x^2} & \\le \\Vert u_n(t,x+y_n)-u_n(t,x)\\Vert _{L_t^\\infty L_x^2} \\\\& \\quad + \\Vert e^{it\\sqrt{\\mathcal {L}_a}}[\\psi (\\cdot -y_n)] - e^{it\\sqrt{\\mathcal {L}_a}}\\psi \\Vert _{L_t^\\infty L_x^2}.",
"$ For (), we observe $(\\ref {SC2}) = \\Vert \\psi (\\cdot -y_n)-\\psi \\Vert _{L_x^2} = o(1)\\quad \\text{as}\\quad n\\rightarrow \\infty $ by continuity of translation in $L^2$ .",
"For (REF ), we use the fundamental theorem of calculus and equivalence of Sobolev spaces to bound $\\Vert u_n(t,x+y_n)-u_n(t,x)\\Vert _{L^2} \\lesssim |y_n| \\Vert \\nabla u_n(t)\\Vert _{L^2} \\lesssim |y_n| \\Vert \\nabla \\psi \\Vert _{L^2}$ uniformly in $t$ .",
"Thus (REF ) is $o(1)$ as well and the desired result follows.",
"Case 2.",
"Suppose $|y_n|\\rightarrow \\infty $ .",
"Then $\\mathcal {L}_a^\\infty =-\\Delta $ .",
"Define the propagator $S_n(t)(f,g)=\\cos (t\\sqrt{\\mathcal {L}_a^n})f+\\tfrac{\\sin (t\\sqrt{\\mathcal {L}_a^n})}{\\sqrt{\\mathcal {L}_a^n}}g,$ which generates solutions to $(\\partial _t^2+\\mathcal {L}_a^n)u=0$ .",
"Define $S_\\infty (t)$ similarly.",
"We may write $e^{it\\sqrt{\\mathcal {L}_a^n}}\\psi =S_n(t)(\\psi ,0)+iS_n(t)(0,\\sqrt{\\mathcal {L}_a^n}\\psi ),$ and similarly for $e^{it\\sqrt{\\mathcal {L}_a^\\infty }}\\psi $ .",
"Thus we have $(e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }})\\psi & = iS_n(t)(0,(\\sqrt{\\mathcal {L}_a^n}-\\sqrt{\\mathcal {L}_a^\\infty }\\psi ) \\\\& \\quad + S_n(t)(\\psi ,0)-S_\\infty (t)(\\psi ,0) \\\\& \\quad + i[S_n(t)(0,\\sqrt{\\mathcal {L}_a^\\infty }\\psi )-S_\\infty (t)(0,\\sqrt{\\mathcal {L}_a^\\infty }\\psi )].",
"$ Using (REF ), we first see $\\Vert (\\ref {E:coo1})\\Vert _{L_t^\\infty L_x^2}\\rightarrow 0$ .",
"We turn to (); the treatment of () is similar.",
"Define $V_n(x) = \\mathcal {L}_a^\\infty - \\mathcal {L}_a^n = \\tfrac{a}{|x+y_n|^2}$ and set $U_n(t) = S_n(t)(\\psi ,0)-S_\\infty (t)(\\psi ,0),$ so that $(\\partial _t^2+\\mathcal {L}_a^n)U_n = V_n(x)S_\\infty (t)(\\psi ,0)$ with zero initial data.",
"In particular, we may also write $U_n(t) = \\int _0^t \\tfrac{\\sin ((t-s)\\sqrt{\\mathcal {L}_a^n})}{\\sqrt{\\mathcal {L}_a^n}}V_n(x)S_\\infty (s)(\\psi ,0)\\,ds.$ Using (REF ), we can firstly observe that $u_n$ is bounded in $L_t^\\infty \\dot{H}_x^{0-}$ .",
"Thus, it suffices to prove that $u_n$ tends to zero in $L_t^\\infty \\dot{H}_x^{\\frac{1}{2}}$ .",
"For this, we use the Duhamel formulation and use Strichartz to estimate $\\Vert U_n\\Vert _{L_t^\\infty \\dot{H}_x^{\\frac{1}{2}}} \\lesssim \\Vert V_n S_\\infty (\\psi ,0)\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d+3}}}.$ Recalling that $\\psi \\in C_c^\\infty $ and that $u(t,x):=S_\\infty (\\psi ,0)$ solves the (free) linear wave equation, we are left to prove that $\\lim _{n\\rightarrow \\infty } \\Vert |x+y_n|^{-2} u(t,x)\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d+3}}} =0$ on $[0,\\infty )\\times \\mathbb {R}^d$ (say), where $u$ satisfies the following: $u(t)$ is supported in the ball of radius $t+C$ for some $C>0$ , $u(t)$ is uniformly bounded in $L_x^2$ , $u(t)$ decays like $t^{-\\frac{d-1}{2}}$ in $L_x^\\infty $ .",
"Let $0<\\varepsilon \\ll 1$ .",
"We first consider the contribution of $0<t<\\varepsilon |y_n|$ .",
"By the support properties of $u$ , we have $|x+y_n|^{-2}\\lesssim |y_n|^{-2}$ in this region.",
"Thus $\\Vert |x+y_n|^{-2}&u\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d+3}}(\\lbrace |t|<\\varepsilon |y_n|\\rbrace \\times \\mathbb {R}^d)} \\\\& \\lesssim |y_n|^{-2} \\Vert \\langle t\\rangle ^{\\frac{d}{d+1}}\\Vert _{L_t^{\\frac{2(d+1)}{d+3}}(\\lbrace |t|<\\varepsilon |y_n|\\rbrace )} \\Vert u\\Vert _{L_t^\\infty L_x^2} \\lesssim |y_n|^{-\\frac{1}{2}},$ which is acceptable.",
"We next consider the contribution of $t>\\varepsilon |y_n|$ .",
"We split the spatial integral into the regions where $|x+y_n|\\ge 1$ and $|x+y_n|<1$ , respectively.",
"The estimates on each region are similar, so let us consider the first case.",
"Using the decay properties of $u(t)$ , we have $\\Vert |x+y_n|^{-2}&u\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d+3}}(\\lbrace |t|>\\varepsilon |y_n|\\rbrace \\times \\lbrace |x+y_n|\\ge 1\\rbrace )} \\\\& \\lesssim \\bigl \\Vert \\Vert |x|^{-2}\\Vert _{L_x^{\\frac{d}{2}+}(\\lbrace |x|>1\\rbrace )} \\Vert u(t)\\Vert _{L_x^{\\frac{2d(d+1)}{d^2-d-4}-}} \\bigr \\Vert _{L_t^{\\frac{2(d+1)}{d+3}}(\\lbrace |t|>\\varepsilon |y_n|\\rbrace )} \\\\& \\lesssim \\bigl \\Vert |t|^{-\\frac{(d-1)(d+2)}{d(d+1)}+} \\bigr \\Vert _{L_t^{\\frac{2(d+1)}{d+3}}(\\lbrace |t|>\\varepsilon |y_n|\\rbrace )} \\\\& \\lesssim (\\varepsilon |y_n|)^{-\\frac{(d-1)(d+2)}{d(d+1)}+\\frac{d+3}{2(d+1)}+},$ which is acceptable.",
"As the modifications necessary to treat $|x+y_n|<1$ are straightforward (e.g.",
"we put $|x|^{-2}$ in $L^{\\frac{d}{2}-}$ ), this completes the proof.",
"We now record a few corollaries that will be of use below.",
"We first have the following.",
"Corollary 3.5 Let $a>-(\\tfrac{d-2}{2})^2+c_d$ .",
"Suppose $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ and let $\\mathcal {L}_a^n,\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"Then $\\lim _{n\\rightarrow \\infty } \\Vert [e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{S(\\mathbb {R})}=0 \\quad \\text{for all}\\quad \\psi \\in \\dot{H}^1,$ where $S(\\cdot )$ is as in (REF ).",
"Let us show the proof for the $L_{t,x}^{\\frac{2(d+1)}{d-2}}$ component of the $S$ -norm.",
"By Strichartz, the quantity in question is finite for $\\psi \\in \\dot{H}^1$ .",
"Thus, we can reduce to the case of $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace y_\\infty \\rbrace )$ (if $y_n\\rightarrow y_\\infty $ ) or $C_c^\\infty (\\mathbb {R}^d)$ (if $|y_n|\\rightarrow \\infty $ ).",
"For such $\\psi $ , we use Hölder to estimate $\\Vert [e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}} & \\lesssim \\Vert [e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{L_t^\\infty L_x^2}^\\theta \\\\& \\quad \\times \\Vert [e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{L_t^q L_x^r}^{1-\\theta },$ where $\\theta \\in (0,1)$ and $\\tfrac{\\theta }{2}+\\tfrac{1-\\theta }{r}=\\tfrac{d-2}{2(d+1)},\\quad \\tfrac{1-\\theta }{q}=\\tfrac{d-2}{2(d+1)}.$ Choosing $\\theta $ sufficiently small so that we may apply Stirchartz and the equivalence of Sobolev spaces holds, we can estimate $\\Vert [e^{it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{L_t^q L_x^r} \\lesssim \\Vert |\\nabla |^{\\frac{d}{2}-\\frac{1}{q}-\\frac{d}{r}}\\psi \\Vert _{L^2} \\lesssim 1,$ so that the result follows from Lemma REF .",
"Next, we have the following, which is completely analogous to equation (3.4) in [22].",
"Corollary 3.6 Let $a>-(\\frac{d-2}{2})^2+c_d$ .",
"Suppose $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"Let $\\mathcal {L}_a^n$ and $\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"If $t_n\\rightarrow t\\in \\mathbb {R}$ , then $\\lim _{n\\rightarrow \\infty }\\bigl \\Vert [e^{-it_n\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{\\dot{H}^{-1}} = 0 \\quad \\text{for all}\\quad \\psi \\in \\dot{H}^{-1}.$ By the equivalence of Sobolev spaces, we may write $\\psi \\in \\dot{H}^{-1}$ in the form $\\sqrt{\\mathcal {L}_a^\\infty }\\phi $ for some $\\phi \\in L^2$ .",
"We write $[e^{it_n\\sqrt{\\mathcal {L}_a^n}}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }}]\\sqrt{\\mathcal {L}_a^\\infty }\\phi & = e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\bigl [\\sqrt{\\mathcal {L}_a^\\infty }-\\sqrt{\\mathcal {L}_a^n}]\\phi \\\\& \\quad + \\sqrt{\\mathcal {L}_a^n}[e^{it_n\\sqrt{\\mathcal {L}_a^n}}-e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}]\\phi \\\\& \\quad + \\sqrt{\\mathcal {L}_a^n}[e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }}]\\phi \\\\& \\quad + \\bigl [\\sqrt{\\mathcal {L}_a^n}-\\sqrt{\\mathcal {L}_a^\\infty }\\bigr ]e^{it\\sqrt{\\mathcal {L}_a^\\infty }}\\phi .$ Applying () to the first and last terms and applying Lemma REF to the second term, we find that $\\limsup _{n\\rightarrow \\infty } \\Vert [e^{it_n\\sqrt{\\mathcal {L}_a^n}}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }}]\\psi \\Vert _{\\dot{H}^{-1}} \\le \\limsup _{n\\rightarrow \\infty }\\Vert [e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}-e^{it\\sqrt{\\mathcal {L}_a^\\infty }}]\\phi \\Vert _{L^2},$ which vanishes by the spectral theorem.",
"This completes the proof.",
"The next result will be important proving energy decoupling in the profile decomposition.",
"Corollary 3.7 Let $a>-(\\frac{d-2}{2})^2+c_d$ and $\\psi \\in \\dot{H}^1$ .",
"Given a sequence $t_n\\rightarrow \\pm \\infty $ and any sequence $\\lbrace y_n\\rbrace \\subset \\mathbb {R}^d$ , we have $\\lim _{n\\rightarrow \\infty } \\Vert e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi \\Vert _{L_x^\\frac{2d}{d-2}}=0,$ where $\\mathcal {L}_a^n$ is as in Definition REF .",
"Without loss of generality, we assume $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"We let $\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"We begin by using Sobolev embedding to estimate $\\Vert e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi \\Vert _{L_x^{\\frac{2d}{d-2}}} & \\lesssim \\Vert \\sqrt{\\mathcal {L}_a^\\infty }\\bigl [e^{it_n\\sqrt{\\mathcal {L}_a^n}}-e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}\\bigr ]\\psi \\Vert _{L_x^2} \\\\& \\quad + \\Vert e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}\\psi \\Vert _{L_x^{\\frac{2d}{d-2}}}.$ To estimate (REF ), we first use the triangle inequality and find $\\Vert \\sqrt{\\mathcal {L}_a^\\infty }\\bigl [e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi -e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}\\psi \\bigr ]\\Vert _{L_x^2} &\\lesssim \\Vert \\bigl [\\sqrt{\\mathcal {L}_a^\\infty }-\\sqrt{\\mathcal {L}_a^n}\\bigr ]e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi \\Vert _{L_x^2} \\\\& \\quad + \\Vert e^{it_n\\sqrt{\\mathcal {L}_a^n}}[\\sqrt{\\mathcal {L}_a^n}-\\sqrt{\\mathcal {L}_a^\\infty }]\\psi \\Vert _{L_x^2} \\\\& \\quad + \\Vert [e^{it_n\\sqrt{\\mathcal {L}_a^n}} - e^{it_n\\sqrt{\\mathcal {L}_a^\\infty }}]\\sqrt{\\mathcal {L}_a^\\infty }\\psi \\Vert _{L_x^2}.$ The second term is $o(1)$ as $n\\rightarrow \\infty $ by (REF ).",
"The third term is $o(1)$ as $n\\rightarrow \\infty $ by Lemma REF (bounding an individual $t_n$ by the $L^\\infty $ norm in time).",
"Thus we need to show that the first term is $o(1)$ as $n\\rightarrow \\infty $ , as well.",
"To see this, we use duality to write $\\Vert [\\sqrt{\\mathcal {L}_a^\\infty }-\\sqrt{\\mathcal {L}_a^n}]e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi \\Vert _{L_x^2}&=\\sup \\bigl | \\bigl \\langle e^{it_n\\sqrt{\\mathcal {L}_a^n}}\\psi ,[\\sqrt{\\mathcal {L}_a^\\infty }-\\sqrt{\\mathcal {L}_a^n}]g\\rangle \\bigl |\\\\& \\lesssim \\Vert [ \\sqrt{\\mathcal {L}_a^\\infty }-\\sqrt{\\mathcal {L}_a^n}]g\\Vert _{\\dot{H}_x^{-1}}\\Vert \\psi \\Vert _{\\dot{H}_x^1},$ where the supremum is over $g\\in L^2$ with $\\Vert g\\Vert _{L^2}=1$ .",
"The claim now follows from ().",
"It remains to estimate ().",
"By density, we may assume $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace y_\\infty \\rbrace )$ if $y_n\\rightarrow y_\\infty $ and $\\psi \\in C_c^\\infty $ if $|y_n|\\rightarrow \\infty $ .",
"Writing $F(t)=\\Vert e^{it\\sqrt{\\mathcal {L}_a^\\infty }}\\psi \\Vert _{L_x^{\\frac{2d}{d-2}}},$ we have by Strichartz that $F\\in L_t^{q}(\\mathbb {R})$ for sufficiently large $q<\\infty $ .",
"Furthermore, $F$ is Lipschitz; indeed, by Sobolev embedding $|\\partial _t F(t)| \\le \\Vert \\partial _t e^{it\\sqrt{\\mathcal {L}_a^\\infty }}\\psi \\Vert _{L_x^{\\frac{2d}{d-2}}} \\lesssim \\Vert \\sqrt{\\mathcal {L}_a^\\infty } \\psi \\Vert _{\\dot{H}_x^1} \\lesssim 1.$ Thus $F(t_n)\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"This completes the proof.",
"We turn now to the linear profile decomposition, Proposition REF .",
"As mentioned above, the starting point is a refined Strichartz estimate for identifying a scale at which concentration occurs.",
"We have the following: Lemma 3.8 (Refined Strichartz estimate) There exists $\\theta \\in (0,1)$ so that $\\Vert e^{-it\\sqrt{\\mathcal {L}_a}} f\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}} \\lesssim \\Vert f\\Vert _{\\dot{H}_x^1}^{1-\\theta }\\sup _{N\\in 2^{\\mathbb {Z}}} \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}P_N^a f\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}}^{\\theta }.$ Denote $f_N=P_N^af$ , $u(t)=e^{-it\\sqrt{\\mathcal {L}_a}}f$ , $u_N=P_N^a u$ , and so on.",
"Let us also write $r=\\frac{2(d+1)}{d-2}$ .",
"By the square function estimate and Bernstein (see Proposition REF ), as well as Strichartz, we have $\\Vert u\\Vert _{L_{t,x}^r}^r & \\lesssim \\iint \\biggl (\\sum _N |u_N|^2\\biggr )^{\\frac{r}{2}}\\,dx\\,dt \\\\& \\lesssim \\Vert \\bigl (\\sum _N |u_N|^2\\bigr )^{\\frac{1}{2}}\\Vert _{L_{t,x}^r}^{r-4}\\sum _{N_1\\le N_2} \\Vert u_{N_1}\\Vert _{L_t^r L_x^{r+}}\\Vert u_{N_1}\\Vert _{L_{t,x}^r} \\Vert u_{N_2}\\Vert _{L_{t,x}^r}\\Vert u_{N_2}\\Vert _{L_t^r L_x^{r-}} \\\\& \\lesssim \\Vert u\\Vert _{L_{t,x}^r}^{r-4}\\bigl [\\sup _N \\Vert u_N\\Vert _{L_{t,x}^r}\\bigr ]^2\\sum _{N_1\\le N_2}N_1^{0+}\\Vert u_{N_1}\\Vert _{L_{t,x}^r}\\Vert f_{N_2}\\Vert _{\\dot{H}_x^{1-}} \\\\& \\lesssim \\Vert f\\Vert _{\\dot{H}^1}^{r-4}\\bigl [\\sup _N \\Vert u_N\\Vert _{L_{t,x}^r}\\bigr ]^2 \\sum _{N_1\\le N_2}\\bigl (\\tfrac{N_1}{N_2}\\bigr )^{0+}\\Vert f_{N_1}\\Vert _{\\dot{H}_x^1}\\Vert f_{N_2}\\Vert _{\\dot{H}_x^1}.$ Applying Cauchy–Schwarz, the result now follows with $\\theta =\\tfrac{2}{r}$ .",
"The next ingredient for the linear profile decomposition is the following inverse Strichartz estimate, which demonstrates how to remove each bubble of concentration.",
"Proposition 3.9 (Inverse Strichartz) Let $a>-(\\frac{d-2}{2})^2 +c_d$ and suppose $f_n\\in \\dot{H}^1$ satisfy $\\lim _{n\\rightarrow \\infty }\\Vert f_n\\Vert _{\\dot{H}^1}=A<\\infty \\quad \\text{and}\\quad \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}f_n\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}} =\\varepsilon >0.$ Passing to a subsequence, there exist $\\phi \\in \\dot{H}^1$ , $N_n\\in 2^{\\mathbb {Z}}$ , and $(t_n,x_n)\\in \\mathbb {R}^{1+d}$ such that $& g_n(x) = N_n^{-(\\frac{d}{2}-1)}[e^{-it_n\\sqrt{\\mathcal {L}_a}} f_n]\\big (\\tfrac{x}{N_n}+x_n\\big ) \\rightharpoonup \\phi (\\cdot ) \\quad \\text{weakly in}\\quad \\dot{H}_x^1, \\\\& \\Vert \\phi \\Vert _{\\dot{H}_a^1} \\gtrsim \\varepsilon (\\tfrac{\\varepsilon }{A})^{c}.$ Furthermore, defining $\\phi _n(x) = N_n^{\\frac{d}{2}-1}e^{it_n\\sqrt{\\mathcal {L}_a}}[\\phi (N(x-x_n))]=N_n^{\\frac{d}{2}-1}[e^{i N_n t_n\\sqrt{\\mathcal {L}_a^n}}\\phi ](N_n(x-x_n)),$ where $\\mathcal {L}_a^n$ is as in Definition REF with $y_n=N_n x_n$ , we have $& \\lim _{n\\rightarrow \\infty } \\bigl \\lbrace \\Vert f_n\\Vert _{\\dot{H}^1_a}^2 - \\Vert f_n-\\phi _n\\Vert _{\\dot{H}^1_a}^2 - \\Vert \\phi _n\\Vert _{\\dot{H}^1_a}^2 \\bigr \\rbrace =0, $ and $\\lim _{n\\rightarrow \\infty }\\Bigl \\lbrace \\Vert f_n\\Vert _{L^{\\frac{2d}{d-2}}_x}^{\\frac{2d}{d-2}}-\\Vert f_n-\\phi _n\\Vert _{L^{\\frac{2d}{d-2}}_x}^{\\frac{2d}{d-2}}-\\Vert \\phi _n\\Vert _{L^{\\frac{2d}{d-2}}_x}^{\\frac{2d}{d-2}}\\Bigr \\rbrace =0,$ Finally, we may assume that either $N_nt_n\\rightarrow \\pm \\infty $ or $t_n\\equiv 0$ , and that either $N_n|x_n|\\rightarrow \\infty $ or $x_n\\equiv 0$ .",
"Let $r=\\frac{2(d+1)}{d-2}$ .",
"We use $c$ to denote a positive constant that may change throughout the proof.",
"Using Lemma REF , there exists $N_n$ such that $\\Vert e^{-it\\mathcal {L}_a}P_{N_n}^a f_n\\Vert _{L_{t,x}^r}\\gtrsim \\varepsilon (\\tfrac{\\varepsilon }{A})^c.$ Using Hölder followed by Bernstein, we have $\\Vert P_N^a F\\Vert _{L_x^r(|x|\\le C N^{-1})} \\lesssim C^{0+}\\Vert P_N^a F\\Vert _{L_x^r},$ and hence for $C$ sufficiently small we have $\\Vert e^{-it\\mathcal {L}_a}P_{N_n}^a f_n\\Vert _{L_{t,x}^r(\\mathbb {R}\\times \\lbrace |x|>CN_n^{-1}\\rbrace )}\\gtrsim \\varepsilon (\\tfrac{\\varepsilon }{A})^c.$ Thus, applying Hölder, Strichartz, and Bernstein, we deduce $\\varepsilon (\\tfrac{\\varepsilon }{A})^c & \\lesssim \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}P_{N_n}^af_n\\Vert _{L_{t,x}^{(1-\\theta )r}}^{1-\\theta } \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}P_{N_n}^a f_n\\Vert _{L_{t,x}^\\infty }^{\\theta } \\\\& \\lesssim N_n^{-\\theta [\\frac{d}{2}-1]}A^{1-\\theta }\\Vert e^{-it\\sqrt{\\mathcal {L}_a}}P_{N_n}^a f_n\\Vert _{L_{t,x}^\\infty }^\\theta $ for small $\\theta >0$ .",
"It follows that there exist $(\\tau _n,x_n)$ with $|x_n|N_n\\ge C$ and $N_n^{-(\\frac{d}{2}-1)}\\bigl | (e^{-i\\tau _n\\sqrt{\\mathcal {L}_a}}P_{N_n}^a f_n)(x_n)\\bigr | \\gtrsim \\varepsilon (\\tfrac{\\varepsilon }{A})^c.$ Passing to a subsequence, we may assume $N_n\\tau _n\\rightarrow \\tau _\\infty \\in [-\\infty ,\\infty ]$ .",
"If $\\tau _\\infty $ is finite, define $t_n\\equiv 0$ ; otherwise, let $t_n=\\tau _n$ .",
"We now let $g_n(x):= N_n^{-(\\frac{d}{2}-1)}[e^{it_n\\sqrt{\\mathcal {L}_a}}f_n](\\tfrac{x}{N_n}+x_n).$ Note that $\\Vert g_n\\Vert _{\\dot{H}^1} \\lesssim \\Vert f_n\\Vert _{\\dot{H}^1} \\lesssim A.$ Therefore there exists $\\phi \\in \\dot{H}^1$ so that $g_n\\rightharpoonup \\phi $ weakly in $\\dot{H}^1$ , yielding (REF ).",
"Expanding inner products and appealing to Lemma REF , we can also deduce (REF ).",
"We turn to ().",
"We now wish to define $h_n$ so that $|\\langle g_n,h_n\\rangle | = N_n^{-(\\frac{d}{2}-1)}|(e^{-i\\tau _n\\sqrt{\\mathcal {L}_a}}P_{N_n}^a f_n)(x_n)| \\gtrsim \\varepsilon (\\tfrac{\\varepsilon }{A})^c.$ A computation shows that we should take $h_n = e^{iN_n(\\tau _n-t_n)\\sqrt{\\mathcal {L}_a^n}} P_1^n \\delta _0$ where $ P_1^n = e^{-\\mathcal {L}_a^n}-e^{-4\\mathcal {L}_a^n}$ and $\\mathcal {L}_a^n$ is as in Definition REF with $y_n=N_n x_n$ .",
"Using (REF ) and (REF ), we find that $h_n \\rightarrow h_\\infty :={\\left\\lbrace \\begin{array}{ll} P_1^\\infty \\delta _0 & \\tau _\\infty \\in \\lbrace \\pm \\infty \\rbrace , \\\\ e^{-i\\tau _\\infty \\sqrt{\\mathcal {L}_a^\\infty }} P_1^\\infty \\delta _0& \\tau _\\infty \\in \\mathbb {R}\\end{array}\\right.",
"}$ strongly in $\\dot{H}^{-1}$ , where $P_1^\\infty = e^{-\\mathcal {L}_a^\\infty }-e^{-4\\mathcal {L}_a^\\infty }$ .",
"Therefore, by strong convergence of $h_n$ and weak convergence of $g_n$ , we can conclude that $\\varepsilon (\\tfrac{\\varepsilon }{A})^c \\lesssim \\Vert \\phi \\Vert _{\\dot{H}^1}\\Vert h_\\infty \\Vert _{\\dot{H}^{-1}}.$ Using the heat kernel estimates (cf.",
"(REF )), the embedding $L^{\\frac{2d}{d+2}}\\hookrightarrow \\dot{H}^{-1}$ , and $N_n|x_n|\\gtrsim c$ , we can show that $\\Vert h_\\infty \\Vert _{\\dot{H}^{-1}} \\lesssim 1,$ and hence () holds.",
"Finally, we turn to (REF ).",
"If $t_n\\equiv 0$ , then using Rellich–Kondrashov (to get $g_n\\rightarrow \\phi $ a.e.)",
"and Lemma REF we get $\\lim _{n\\rightarrow \\infty } \\biggl [\\Vert g_n\\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}-\\Vert g_n-\\phi \\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}-\\Vert \\phi \\Vert _{L_x^{\\frac{2d}{d-2}}}^{\\frac{2d}{d-2}}\\biggr ]=0,$ which yields (REF ) after a change of variables.",
"In the case that $t_n=\\tau _n$ , the result follows from the fact that $\\phi _n\\rightarrow 0$ in $L^{\\frac{2d}{d-2}}$ (by Corollary REF ).",
"Finally, by passing to a further subsequence, we can assume that either $N_n|x_n|\\rightarrow \\infty $ or $N_n x_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ .",
"In the latter case, we may take $x_n\\equiv 0$ by replacing $\\phi $ with $\\phi (\\cdot -y_\\infty )$ .",
"We now turn to the proof of the linear profile decomposition Proposition REF .",
"As the proof follows along well-established lines, we will be somewhat brief.",
"The decompositon (REF ) and the decouplings (REF ) and () follow by induction.",
"One sets $r_n^0=f_n$ and applies Proposition REF to the sequence $r_n^0$ to find $\\phi _n^1$ (and we set $\\lambda _n^1=[N_n^1]^{-1}$ ); one then applies Proposition REF to the sequences $r_n^J:=r_n^{J-1}-\\phi _n^J$ .",
"The process terminates at a finite $J^*$ if $\\Vert e^{-it\\sqrt{\\mathcal {L}_a}}r_n^{J^*}\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}}=0$ .",
"Defining $\\varepsilon _J = \\lim _{n\\rightarrow \\infty } \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}r_n^J\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}}\\quad \\text{and}\\quad A_J = \\lim _{n\\rightarrow \\infty } \\Vert r_n^J\\Vert _{\\dot{H}^1},$ we have that $\\varepsilon _J\\rightarrow 0$ as a consequence of (REF ), (), and $A_J\\le A_0$ ; in fact, $\\sum _{j=1}^J \\varepsilon _{j-1}^2\\bigl (\\tfrac{\\varepsilon _{j-1}}{A_0}\\bigr )^{c} \\lesssim \\sum _{j=1}^J\\Vert \\phi _n^j\\Vert _{\\dot{H}_a^1}^2 \\lesssim A_0^2.$ By construction and (REF ), we have $(\\lambda _n^J)^{\\frac{d}{2}-1}[e^{-it_n^J\\sqrt{\\mathcal {L}_a}}r_n^{J-1}](\\lambda _n^J x+x_n^J\\bigr )\\rightharpoonup \\phi ^J\\quad \\text{for each finite}\\quad J\\ge 1.$ Recalling that $r_n^J=r_n^{J-1}-\\phi ^J$ , we deduce ().",
"This will also play an important role in proving the orthogonality condition (REF ), to which we now turn.",
"Putting $(j,k)$ in lexicographical order, we suppose toward a contradiction that (REF ) fails for the first time at some $(j,k)$ with $j<k$ .",
"Thus $\\tfrac{\\lambda _n^j}{\\lambda _n^k}\\rightarrow \\lambda _0,\\quad \\tfrac{x_n^j-x_n^k}{\\sqrt{\\lambda _n^j \\lambda _n^k}}\\rightarrow y_0,\\quad \\text{and}\\quad \\tfrac{t_n^j-t_n^k}{\\sqrt{\\lambda _n^j \\lambda _n^k}}\\rightarrow t_0,$ but (REF ) holds for every pair $(j,\\ell )$ with $j<\\ell <k$ .",
"Now, using (REF ) to get an expression for both $r_n^j$ and $r_n^{k-1}$ , we have $r_n^j -\\sum _{\\ell =j+1}^{k-1}\\phi _n^\\ell = r_n^{k-1}.$ Therefore, using (REF ), we have $(\\lambda _n^k)^{\\frac{d}{2}-1}[e^{-it_n^k\\sqrt{\\mathcal {L}_a}}r_n^j](\\lambda _n^kx+x_n^k)-\\sum _{\\ell =j+1}^{k-1}(\\lambda _n^k)^{\\frac{d}{2}-1}[e^{-it_n^k\\sqrt{\\mathcal {L}_a}}\\phi _n^\\ell ](\\lambda _n^kx+x_n^k)\\rightharpoonup \\phi ^k(x).$ To get a contradiction, we will show that both of the terms above converge weakly to zero, contradicting that $\\phi ^k$ is nontrivial.",
"For the first term in (REF ), we will use (REF ).",
"Let us introduce the notation $(g_n^j)^{-1}f(x) = (\\lambda _n^j)^{\\frac{d}{2}-1}f(\\lambda _n^j x + x_n^k).$ We rewrite the first term in (REF ) as $(g_n^k)^{-1}e^{i(t_n^j-t_n^k)\\sqrt{\\mathcal {L}_a}}g_n^j\\bigl [(g_n^j)^{-1}e^{-it_n^j\\sqrt{\\mathcal {L}_a}}r_n^j\\bigr ],$ and observe that the term inside the square brackets converges weakly to zero.",
"The operator preceding the square brackets can be rewritten $(g_n^k)^{-1}g_n^j e^{i(\\lambda _n^j)^{-1}(t_n^j-t_n^k)\\sqrt{\\mathcal {L}_a^{n_j}}},$ where $\\mathcal {L}_a^{n_j}$ is as in Definition REF with the sequence $y_n=(\\lambda _n^j)^{-1}x_n^j$ .",
"Noting that (REF ) implies that the adjoint of $(g_n^k)^{-1}g_n^j$ converges strongly and that the sequence $(\\lambda _n^j)^{-1}(t_n^j-t_n^k)$ converges to some finite real number, the claim now reduces to the following lemma.",
"This lemma (and its proof) is completely analogous to [22] Lemma 3.10 Suppose $f_n\\in \\dot{H}^1$ converges to zero weakly in $\\dot{H}^1$ and $t_n\\rightarrow t_\\infty \\in \\mathbb {R}$ .",
"Then for any $y_n\\in \\mathbb {R}^d$ , $e^{-it_n\\sqrt{\\mathcal {L}_a^n}}f_n\\rightharpoonup 0 \\quad \\text{weakly in}\\quad \\dot{H}^1,$ where $\\mathcal {L}_a^n$ is as in Definition REF with the sequence $y_n$ .",
"Without loss of generality, we assume $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"We let $\\mathcal {L}_a^\\infty $ be as in Definition REF .",
"We claim that it suffices to prove $e^{-it_\\infty \\sqrt{\\mathcal {L}_a^n}} f_n\\rightharpoonup 0\\quad \\text{weakly in}\\quad \\dot{H}^1.$ To see this, given $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace y_\\infty \\rbrace )$ (if $y_n\\rightarrow y_\\infty $ ) or $\\psi \\in C_c^\\infty (\\mathbb {R}^d)$ (if $|y_n|\\rightarrow \\infty $ ), we estimate $\\bigl |\\bigl \\langle [e^{-it_n\\sqrt{\\mathcal {L}_a^n}}-e^{-it_\\infty \\sqrt{\\mathcal {L}_a^n}}]f_n, \\psi \\bigr \\rangle _{\\dot{H}^1_x}\\bigr |&\\lesssim \\big \\Vert [e^{-it_n\\sqrt{\\mathcal {L}_a^n}}-e^{-it_\\infty \\sqrt{\\mathcal {L}_a^n}}]f_n\\big \\Vert _{L^2_x} \\Vert \\Delta \\psi \\Vert _{L^2_x}\\\\&\\lesssim |t_n-t_\\infty | \\Vert \\sqrt{\\mathcal {L}_a^n}f_n\\Vert _{L^2_x}\\Vert \\Delta \\psi \\Vert _{L^2_x},$ where we have used the spectral theorem and the simple inequality $|e^{-it_n\\sqrt{\\lambda }}-e^{-it_\\infty \\sqrt{\\lambda }}| \\lesssim |t_n-t_\\infty |\\sqrt{\\lambda }$ for $\\lambda \\ge 0$ .",
"Thus the claim follows.",
"To prove (REF ), we take $\\psi $ as above and begin by estimating $|\\langle e^{-it_\\infty \\sqrt{\\mathcal {L}_a^n}}f_n,\\psi \\rangle _{\\dot{H}^1}| & \\lesssim |\\langle f_n, [e^{it_\\infty \\sqrt{\\mathcal {L}_a^n}}-e^{it_\\infty \\sqrt{\\mathcal {L}_a^\\infty }}](-\\Delta \\psi )\\rangle _{L^2}| \\\\& \\quad + |\\langle f_n, e^{it_\\infty \\sqrt{\\mathcal {L}_a^\\infty }}(-\\Delta \\psi )\\rangle _{L^2}|.$ The first term on the right-hand side converges to zero by (REF ), using the fact that $\\Delta \\psi \\in \\dot{H}^{-1}$ , while the second term converges to zero due to the weak convergence of $f_n$ .",
"This completes the proof.",
"We turn now to the second term in (REF ).",
"This time we take a similar approach, relying on the fact that (REF ) holds for each pair $(j,\\ell )$ with $j<\\ell <k$ .",
"Omitting some of the details, the claim boils down to the following lemma.",
"This lemma (and its proof) is again completely analogous to [22].",
"Lemma 3.11 Let $f\\in \\dot{H}^1$ and let $(t_n,x_n)\\in \\mathbb {R}^{1+d}$ and $y_n\\in \\mathbb {R}^d$ .",
"Writing $\\mathcal {L}_a^n$ as in Definition REF with the sequence $y_n$ , we have $[e^{-it_n\\sqrt{\\mathcal {L}_a^n}}f](x+x_n)\\rightharpoonup 0 \\quad \\text{weakly in}\\quad \\dot{H}^1$ whenever $|t_n|\\rightarrow \\infty $ or $|x_n|\\rightarrow \\infty $ .",
"Without loss of generality, assume $y_n\\rightarrow y_\\infty \\in \\mathbb {R}^d$ or $|y_n|\\rightarrow \\infty $ .",
"Take $\\mathcal {L}_a^\\infty $ as in Definition REF .",
"Suppose $t_n\\rightarrow \\infty $ ; the case $t_n\\rightarrow -\\infty $ is similar.",
"We let $\\psi \\in C_c^\\infty (\\mathbb {R}^d\\backslash \\lbrace y_\\infty \\rbrace )$ (if $y_n\\rightarrow y_\\infty $ ) or $C_c^\\infty (\\mathbb {R}^d)$ (if $|y_n|\\rightarrow \\infty $ ).",
"Define $F_n(t) = \\langle e^{-it\\sqrt{\\mathcal {L}_a^n}}f](x+x_n),\\psi \\rangle _{\\dot{H}^1}.$ We need to show that $F_n(t_n)\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"To this end, we first compute the time derivative and observe that $|\\partial _t F_n|\\lesssim 1$ uniformly in $n$ .",
"Thus, letting $r=\\tfrac{2(d+1)}{d-2}$ , we have by the fundamental theorem of calculus that $|F_n(t_n)|^{r+1} \\lesssim |F_n(t_0)|^{r+1} + \\Vert F_n\\Vert _{L_t^{r}(t_n,t_0)}^r\\quad \\text{for any}\\quad t_0>t_n.$ In particular, it suffices to show that each $F_n \\in L_t^r$ (which yields $F_n\\rightarrow 0$ as $t\\rightarrow \\infty $ for each fixed $n$ ) and that $\\lim _{n\\rightarrow \\infty } \\Vert F_n\\Vert _{L_t^r(t_n,\\infty )}=0.$ That $F_n \\in L_t^r$ follows from Hölder's inequality and Strichartz.",
"For the second point, we estimate by Hölder's inequality $\\Vert F_n\\Vert _{L_t^r([t_n,\\infty ])} & \\lesssim \\Vert [e^{-it\\sqrt{\\mathcal {L}_a^n}}-e^{-it\\sqrt{\\mathcal {L}_a^\\infty }}]f\\Vert _{L_{t,x}^r([t_n,\\infty ]\\times \\mathbb {R}^d)} \\\\& \\quad + \\Vert e^{-it\\sqrt{\\mathcal {L}_a^\\infty }} f\\Vert _{L_{t,x}^r([t_n,\\infty )\\times \\mathbb {R}^d)}.$ The first term converges to zero by Corollary REF , while the second term tends to zero as $n\\rightarrow \\infty $ by Strichartz and the monotone convergence.",
"This completes the proof in the case $t_n\\rightarrow \\infty $ .",
"Finally, suppose $t_n$ is bounded (and $t_n\\rightarrow t_\\infty $ , say) but $|x_n|\\rightarrow \\infty $ .",
"In this case, we can move the translation inside appeal to Lemma REF , cf.",
"$[e^{-it_n\\sqrt{\\mathcal {L}_a^n}}f](\\cdot +x_n) = e^{-it\\sqrt{\\tilde{\\mathcal {L}}_a^n}}[f(\\cdot + x_n)]$ where $\\tilde{\\mathcal {L}}_a^n$ is as in Definition REF with the sequence $x_n+y_n$ .",
"This completes the proof.",
"With Lemma REF and Lemma REF in place, we complete the proof of (REF ) and hence the proof of Proposition REF ."
],
[
"Existence of minimal blowup solutions",
"In this section, we prove that if Theorem REF or Theorem REF fails, then we can construct minimal blowup solutions.",
"We then prove the existence of scattering nonlinear profiles associated to linear profiles with translation parameters tending to infinity (Proposition REF ).",
"With these two ingredients in place, we can then follow fairly standard arguments to deduce the existence of minimal blowup solutions (see Theorem REF ).",
"Finally, arguments from [19] will allow us to further reduce the class of solutions under consideration (see Theorem REF ).",
"We recall the mapping $T_a$ introduced in (REF ), which takes a pair of real-valued functions and returns a single complex-valued function through $T_a(f,g)=f+i\\mathcal {L}_a^{-\\frac{1}{2}}g.$ We also recall the notation $\\tilde{E}_a$ from (REF ).",
"Note that $T_0(f,g)=f+i|\\nabla |^{-1}g.$"
],
[
"Construction of nonlinear profiles",
"We will construct nonlinear profiles via approximation by solutions to the free nonlinear wave equation.",
"To construct scattering solutions to the free NLW, we rely on the result of [19].",
"Theorem 4.1 (Scattering for the free NLW, [19], [2]) Let $(w_0,w_1)\\in \\dot{H}^1\\times L^2$ and $\\mu \\in \\lbrace \\pm 1\\rbrace $ .",
"If $\\mu =-1$ , assume further that $E_0[(w_0,w_1)]<E_0[W_0] \\quad \\text{and}\\quad \\Vert w_0\\Vert _{\\dot{H}^1} < \\Vert W_0\\Vert _{\\dot{H}^1}.$ There exists a unique global solution $w$ to $\\partial _t^2 w - \\Delta w + \\mu |w|^{\\frac{4}{d-2}} w =0$ that scatters in both time directions and obeys global $L_{t,x}^{\\frac{2(d+1)}{d-2}}$ space-time bounds.",
"Furthermore, given $(w_0,w_1)\\in \\dot{H}^1\\times \\dot{L}^2$ satisfying $\\tfrac{1}{2} \\Vert w_0\\Vert _{\\dot{H}_x^1}^2+\\tfrac{1}{2}\\Vert w_1\\Vert _{L_x^2}^2 < E_0[W_0]\\quad \\text{and}\\quad \\Vert w_0\\Vert _{\\dot{H}_x^1} <\\Vert W_0\\Vert _{\\dot{H}^1}$ in the case $\\mu =-1$ , there exists a unique global solution to (REF ) that scatters to $(w_0,w_1)$ as $t\\rightarrow \\infty $ (or as $t\\rightarrow -\\infty $ ).",
"We turn to the main result of this section.",
"We assume $d\\in \\lbrace 3,4\\rbrace $ and $a>-(\\tfrac{d-2}{2})^2+c_d$ , as usual.",
"In light of the application below, we will state the following result in terms of constructing scattering solutions to (REF ), rather than the original equation (REF ).",
"Proposition 4.2 (Construction of nonlinear profiles) Suppose $t_n\\in \\mathbb {R}$ satisfy $t_n\\equiv 0$ or $t_n\\rightarrow \\pm \\infty $ , and suppose $x_n\\in \\mathbb {R}^d$ satisfy $|x_n|\\lambda _n^{-1}\\rightarrow \\infty $ .",
"Let $\\phi \\in \\dot{H}^1$ and define $\\phi _n(x) = \\lambda _n^{-(\\frac{d}{2}-1)}[e^{-it_n\\sqrt{\\mathcal {L}_a^n}}\\phi ](\\tfrac{x-x_n}{\\lambda _n}),$ where $\\mathcal {L}_a^n$ is as in Definition REF with the sequence $y_n=\\lambda _n^{-1} x_n$ .",
"If $\\mu =+1$ (the defocusing case), then for $n$ sufficiently large there exists a global solution $v_n$ to (REF ) with $v_n(0)=\\phi _n$ satisfying $\\Vert v_n\\Vert _{\\dot{S}^1(\\mathbb {R})} \\lesssim 1,$ where the implicit constant depends on ${\\Vert \\phi \\Vert _{\\dot{H}^1}}$ .",
"If $\\mu =-1$ (the focusing case), the same result holds provided $E_0[T_0^{-1}\\phi ]< E_{ 0}[W_{0}] \\quad \\text{and}\\quad \\Vert \\operatornamewithlimits{Re}\\phi \\Vert _{\\dot{H}_x^1}< \\Vert W_{0}\\Vert _{\\dot{H}_{0}^1},$ if $t_n\\equiv 0$ , and $\\tfrac{1}{2}\\Vert \\phi \\Vert _{\\dot{H}_x^1}^2 < E_0[W_0]\\quad \\text{and}\\quad \\Vert \\operatornamewithlimits{Re}\\phi \\Vert _{\\dot{H}_x^1}< \\Vert W_{0}\\Vert _{\\dot{H}_{0}^1},$ if $t_n\\rightarrow \\pm \\infty $ .",
"Furthermore, for every $\\eta >0$ there exists $N_\\eta $ and $\\psi _\\eta \\in c^\\infty (\\mathbb {R}^{1+d})$ such that for $n\\ge N_\\eta $ , $\\Vert v_n(t-\\lambda _n t_n,x+x_n)-\\lambda _n^{-[\\frac{d}{2}-1]}\\psi _\\eta (\\lambda _n^{-1} t,\\lambda _n^{-1}x)\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\mathbb {R}^{1+d})} < \\eta .$ Our ultimate goal is to construct solutions $v_n$ to (REF ) with data $v_n(0)=\\phi _n$ .",
"Equivalently, we need to construct solutions $u_n$ to (REF ) with data $\\vec{u}_n(0)=T_a^{-1}\\phi _n$ .",
"The starting point is to appeal to Theorem REF to construct a solution $u$ associated to the initial data $(\\operatornamewithlimits{Re}\\phi , |\\nabla |\\operatornamewithlimits{Im}\\phi ).$ The assumptions (REF ) and (REF ) guarantee that we are in a position to apply Theorem REF .",
"If $t_n\\equiv 0$ , we take $u$ to be the solution to (REF ) with initial data $\\vec{\\psi }:=(\\operatornamewithlimits{Re}\\phi ,|\\nabla |\\operatornamewithlimits{Im}\\phi ).$ If $t_n\\rightarrow \\pm \\infty $ , we instead of $u$ be the solution to (REF ) with $\\lim _{t\\rightarrow \\pm \\infty }\\Vert \\vec{u}(t) - (S_0(t)\\vec{\\psi }, \\partial _t S_0(t)\\vec{\\psi })\\Vert _{\\dot{H}^1\\times L^2}=0,$ where $S_0(t)(f,g)=\\cos (t|\\nabla |)f + |\\nabla |^{-1}\\sin (t|\\nabla |)g$ is the free linear wave propagator.",
"In both cases, we have that $u$ obeys global space-time bounds.",
"We will now use $u$ to construct approximate solutions to (REF ).",
"For each $n$ , we let $\\chi _n$ be a smooth function such that $\\chi _n(x) ={\\left\\lbrace \\begin{array}{ll} 0 & |x_n+\\lambda _n x|\\le \\tfrac{1}{4} |x_n|, \\\\ 1 & |x_n+\\lambda _n x|\\ge \\tfrac{1}{2}|x_n|.\\end{array}\\right.",
"}$ In particular, $\\chi _n(x)\\rightarrow 1$ as $n\\rightarrow \\infty $ for each $x$ .",
"We further impose that $\\chi _n$ obey the symbol bounds $\\sup _x |\\partial ^\\alpha \\chi _n(x)| \\lesssim [\\lambda _n^{-1} |x_n|]^{-|\\alpha |}$ for all multi-indices $\\alpha $ .",
"For $\\tau >0$ , we now let $u_{n,\\tau }(t,x)={\\left\\lbrace \\begin{array}{ll} \\lambda _n^{-[\\frac{d}{2}-1]}(\\chi _n u)(\\lambda _n^{-1}t,\\lambda _n^{-1}(x-x_n)) & |t|\\le \\lambda _n\\tau , \\\\[S(t-\\tau \\lambda _n)\\vec{u}_{n,\\tau }(\\lambda _n\\tau )](x) & t>\\lambda _n \\tau , \\\\[S(t+\\tau \\lambda _n)\\vec{u}_{n,\\tau }(-\\lambda _n\\tau )](x) & t<-\\lambda _n\\tau ,\\end{array}\\right.",
"}$ where $S(t)(f,g)=\\cos (t\\sqrt{\\mathcal {L}_a})f+\\mathcal {L}_a^{-\\frac{1}{2}}\\sin (t\\sqrt{\\mathcal {L}_a})g.$ We claim that the $u_{n,\\tau }$ are approximate solutions to (REF ) that asymptotically agree with $T_a^{-1}\\phi _n$ , so that we may appeal to the stability result (Proposition REF ) to construct true solutions to (REF ) with initial data $T_a^{-1}\\phi _n$ .",
"To do this requires that we verify the following: $&\\limsup _{\\tau \\rightarrow \\infty }\\limsup _{n\\rightarrow \\infty }\\bigl \\lbrace \\Vert \\vec{u}_{n,\\tau }\\Vert _{L_t^\\infty (\\dot{H}^1\\times L^2)} + \\Vert u_{n,\\tau }\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}}\\bigr \\rbrace \\lesssim 1, \\\\&\\limsup _{\\tau \\rightarrow \\infty }\\limsup _{n\\rightarrow \\infty }\\Vert \\vec{u}_{n,\\tau }(\\lambda _nt_n)-T_a^{-1}\\phi _n\\Vert _{\\dot{H}^1\\times L^2} = 0, \\\\&\\limsup _{\\tau \\rightarrow \\infty }\\limsup _{n\\rightarrow \\infty }\\Vert (\\partial _t^2+\\mathcal {L}_a)u_{n,\\tau } + F(u_{n,\\tau })\\Vert _{L_t^1 L_x^2} = 0, $ where we have denoted $F(z)=\\mu |z|^{\\frac{4}{d-2}}z$ and $\\vec{u}_{n,\\tau }=(u_{n,\\tau },\\partial _t u_{n,\\tau })$ .",
"We begin by estimating $\\Vert \\vec{u}_{n,\\tau }\\Vert _{L_t^\\infty (\\dot{H}^1\\times L^2)} \\lesssim \\Vert \\chi _n \\vec{u}\\Vert _{\\dot{H}^1\\times L^2} \\lesssim 1.$ The space-time bound in (REF ) then follows from Strichartz and the corresponding bounds for $u$ .",
"We turn to ().",
"We begin with the case $t_n\\equiv 0$ .",
"By construction and a change of variables, we estimate the $\\dot{H}^1$ component by $\\Vert (1-\\chi _n)\\operatornamewithlimits{Re}\\phi \\Vert _{\\dot{H}^1} =o(1)\\quad \\text{as}\\quad n\\rightarrow \\infty .$ We turn to the $L^2$ component.",
"Again, by construction and a change of variables, we have $\\Vert \\chi _n |\\nabla |\\operatornamewithlimits{Im}\\phi - (\\mathcal {L}_a^n)^{\\frac{1}{2}}\\operatornamewithlimits{Im}\\phi \\Vert _{L^2}=o(1)\\quad \\text{as}\\quad n\\rightarrow \\infty ,$ where we have also made use of (REF ).",
"We turn to the case $t_n\\rightarrow \\infty $ , with the case $t_n\\rightarrow -\\infty $ being similar.",
"Note that $t_n>\\tau $ for $n$ sufficiently large.",
"It is enough to prove $\\limsup _{\\tau \\rightarrow \\infty }\\limsup _{n\\rightarrow \\infty }\\big \\Vert T_a\\vec{u}_{n,\\tau }(\\lambda _nt_n)-\\phi _n\\big \\Vert _{\\dot{H}^1}=0.$ To this end, set $u(t,x):=S(t)(f,g).$ Then $T_a\\vec{u}(t)=e^{-it\\sqrt{\\mathcal {L}_a}}T_a(f,g),$ which implies $T_a\\vec{u}_{n,\\tau }(\\lambda _nt_n)=e^{-i(\\lambda _nt_n-\\lambda _n\\tau )\\sqrt{\\mathcal {L}_a}}T_a\\vec{u}_{n,\\tau }(\\lambda _n\\tau ).$ Thus, performing a change of variables, we have $\\big \\Vert T_a\\vec{u}_{n,\\tau }(\\lambda _nt_n)-\\phi _n\\big \\Vert _{\\dot{H}^1}&= \\big \\Vert e^{-i(\\lambda _nt_n-\\lambda _n\\tau )\\sqrt{\\mathcal {L}_a}}T_a\\vec{u}_{n,\\tau }(\\lambda _n\\tau )-e^{-it_n\\lambda _n\\sqrt{\\mathcal {L}_a}}g_n\\phi \\big \\Vert _{\\dot{H}^1}\\\\&\\lesssim \\big \\Vert T_a\\vec{u}_{n,\\tau }(\\lambda _n\\tau )-e^{-i\\lambda _n\\tau \\sqrt{\\mathcal {L}_a}}g_n\\phi \\big \\Vert _{\\dot{H}^1}\\\\&\\lesssim \\big \\Vert \\chi _nT_a^n\\vec{u}(\\tau )-e^{-i\\tau \\sqrt{\\mathcal {L}_a^n}}g_n\\phi \\big \\Vert _{\\dot{H}^1}\\\\&\\lesssim \\big \\Vert T_a^n\\vec{u}(\\tau )-e^{-i\\tau \\sqrt{\\mathcal {L}_a^n}}\\phi \\big \\Vert _{\\dot{H}^1}+o_n(1)$ as $n\\rightarrow \\infty $ , where $T_a^n(f,g):=f+i(\\mathcal {L}_a^n)^{-\\frac{1}{2}}g.$ Furthermore, using (REF ), Corollary REF and (REF ), we derive that $\\big \\Vert &T_a\\vec{u}_{n,\\tau }(\\lambda _nt_n)-\\phi _n\\big \\Vert _{\\dot{H}^1}\\\\ &\\lesssim \\big \\Vert T_a^n\\vec{u}(\\tau )-T_0\\vec{u}(\\tau )\\big \\Vert _{\\dot{H}^1} +\\big \\Vert e^{-i\\tau \\sqrt{-\\Delta }}\\phi -e^{-i\\tau \\sqrt{\\mathcal {L}_a^n}}\\phi \\big \\Vert _{\\dot{H}^1}\\\\& \\quad +\\big \\Vert T_0\\vec{u}(\\tau )-e^{-i\\tau \\sqrt{-\\Delta }}\\phi \\big \\Vert _{\\dot{H}^1}+o_n(1)\\\\& \\lesssim \\big \\Vert \\partial _tu(\\tau )-\\sqrt{\\mathcal {L}_a^n}|\\nabla |^{-\\frac{1}{2}}\\partial _tu(\\tau )\\big \\Vert _{L^2}+o_n(1)\\\\&\\lesssim o_n(1)$ as $n\\rightarrow \\infty .$ Turning to (), we define the errors $e_{n,\\tau } = (\\partial _t^2 + \\mathcal {L}_a)u_{n,\\tau } + F(u_{n,\\tau }),\\quad F(z)=\\mu |z|^{\\frac{4}{d-2}}z,$ which we need to estimate in $L_t^1 L_x^2(\\mathbb {R}^{1+d})$ .",
"We first consider the contribution of times $t>\\lambda _n\\tau $ , with the case $t<-\\lambda _n\\tau $ being analogous.",
"In this regime, we have $e_{n,\\tau }=F(u_{n,\\tau }).$ Thus, by construction and a change of variables, we have $\\Vert e_{n,\\tau }\\Vert _{L_t^1 L_x^2(\\lbrace t>\\lambda _n\\tau \\rbrace \\times \\mathbb {R}^d)} & \\lesssim \\Vert u_{n,\\tau }\\Vert _{L_t^{\\frac{d+2}{d-2}}L_x^{\\frac{2(d+2)}{d-2}}(\\lbrace t>\\lambda _n\\tau \\rbrace \\times \\mathbb {R}^d)}^{\\frac{d+2}{d-2}} \\\\& \\lesssim \\Vert S_n(t)[\\chi _n \\vec{u}(\\tau )]\\Vert _{L_t^{\\frac{d+2}{d-2}}L_x^{\\frac{2(d+2)}{d-2}}((0,\\infty )\\times \\mathbb {R}^d)}^{\\frac{d+2}{d-2}},$ where $S_n(t)(f,g)=\\cos (t\\sqrt{\\mathcal {L}_a^n})f+\\tfrac{\\sin (t\\sqrt{\\mathcal {L}_a^n})}{\\sqrt{\\mathcal {L}_a^n}}g.$ We claim that this term tends to zero as $n,\\tau \\rightarrow \\infty $ , which will yield () in the region $|t|>\\lambda _n\\tau $ .",
"Recalling the notation $\\vec{\\psi }$ from (REF ) and observing that we can replace $\\chi _n$ with 1 up to errors that are $o(1)$ as $n\\rightarrow \\infty $ , we are led to estimate $\\Vert S_n(t)\\vec{u}(\\tau )\\Vert _{S(0,\\infty )} & \\lesssim \\Vert S_0(t)\\vec{\\psi }\\Vert _{S(\\tau ,\\infty )} \\\\& \\quad +\\Vert [S_n(t)-S_0(t)]\\vec{u}(\\tau )\\Vert _{S(0,\\infty )} \\\\& \\quad + \\big \\Vert S_0(t)\\big [\\vec{u}(\\tau )-\\big (S_0(\\tau )\\vec{\\psi },\\partial _tS_0(\\tau )\\vec{\\psi }\\big )\\big ]\\big \\Vert _{S(0,\\infty )}.$ Now (REF ) is $o(1)$ as $\\tau \\rightarrow \\infty $ by Strichartz and monotone convergence.",
"Next, () is $o(1)$ for each $\\tau $ by Corollary REF .",
"Finally, () is $o(1)$ as $\\tau \\rightarrow \\infty $ by Strichartz and (REF ).",
"This completes the proof of () in the region $|t|>\\lambda _n\\tau $ .",
"Finally, we turn to () in the region $|t|\\le \\lambda _n\\tau $ .",
"Recalling that $u$ is a solution to (REF ), we compute that in this region $e_{n,\\tau } & = \\lambda _n^{-(\\frac{d}{2}+1)}\\mu [(\\chi _n-\\chi _n^{\\frac{d+2}{d-2}})F(u)](\\lambda _n^{-1} t,\\lambda _n^{-1}(x-x_n)) \\\\& \\quad + 2\\lambda _n^{-(\\frac{d}{2}+1)}[\\nabla \\chi _n\\cdot \\nabla u](\\lambda _n^{-1} t,\\lambda _n^{-1}(x-x_n)) \\\\& \\quad + \\lambda _n^{-(\\frac{d}{2}+1)}[\\Delta \\chi _n u](\\lambda _n^{-1} t,\\lambda _n^{-1}(x-x_n)) \\\\& \\quad + \\lambda _n^{-(\\frac{d}{2}-1)}a|x|^{-2}[\\chi _n u](\\lambda _n^{-1} t,\\lambda _n^{-1}(x-x_n)).",
"$ Changing variables, we estimate the contribution of () and () by $\\tau &\\bigl \\lbrace \\Vert \\nabla \\chi _n\\Vert _{L^\\infty }\\Vert \\nabla u\\Vert _{L^2} + \\Vert \\Delta \\chi _n\\Vert _{L^d}\\Vert u\\Vert _{L^{\\frac{2d}{d-2}}}\\bigr \\rbrace \\lesssim \\tau \\tfrac{\\lambda _n}{|x_n|} = o(1)$ as $n\\rightarrow \\infty $ .",
"For (REF ), we change variables and observe that $F(u)\\in L_t^1 L_x^2$ (since $u$ obeys $L_t^{\\frac{d+2}{d-2}}L_x^{\\frac{2(d+2)}{d-2}}$ bounds); thus the contribution of this term is $o(1)$ as $n\\rightarrow \\infty $ by the dominated convergence theorem.",
"Finally, for () we will use Hardy's inequality and a change of variables.",
"Recalling the notation $g_n$ from above, first observe that in the support of $g_n\\chi _n$ , we have $|x|\\gtrsim |x_n|$ .",
"Thus $\\Vert & |x|^{-2}g_n[\\chi _n u(\\lambda _n^{-1}t)]\\Vert _{L_t^1 L_x^2(\\lbrace |t|\\le \\lambda _n \\tau \\rbrace \\times \\mathbb {R}^d)}\\\\& \\lesssim \\tfrac{\\lambda _n}{|x_n|} \\Vert |x|^{-1} g_n[\\chi _n u(\\lambda _n^{-1}t)]\\Vert _{L_t^\\infty L_x^2} \\\\& \\lesssim \\tfrac{\\lambda _n}{|x_n|}\\Vert \\nabla g_n(\\chi _n u(\\lambda _n^{-1}t))\\Vert _{L_t^\\infty L_x^2} \\\\& \\lesssim \\tfrac{\\lambda _n}{|x_n|}\\Vert \\lambda _n^{-\\frac{d}{2}}\\nabla [\\chi _n u](\\lambda _n^{-1} t, \\lambda _n^{-1}(x-x_n))\\Vert _{L_t^\\infty L_x^2} \\\\& \\lesssim \\tfrac{\\lambda _n}{|x_n|}\\Vert \\nabla [\\chi _n u]\\Vert _{L_t^\\infty L_x^2} = o(1)$ as $n\\rightarrow \\infty $ .",
"This completes the proof of ().",
"Applying Proposition REF , we deduce that for $n$ sufficiently large exist true solutions $\\tilde{u}_n$ to (REF ) with initial data $T_a^{-1}\\phi _n$ .",
"Furthermore, this solution obeys global space-time bounds.",
"We now define $v_n=T_a\\vec{\\tilde{}}u_n$ to obtain the desired solutions to (REF ).",
"Finally, the approximation result follows from the same argument in [22]."
],
[
"Reduction to almost periodic solutions",
"In this section we prove the following theorem.",
"Theorem 4.3 Suppose Theorem REF or Theorem REF fails.",
"Then there exists a maximal-lifespan solution $u:I_{\\max }\\times \\mathbb {R}^4\\rightarrow \\mathbb {R}$ to (REF ) that blows up in both time directions and is almost periodic modulo symmetries with $x(t)\\equiv 0$ .",
"In the focusing case, we have $E_a[\\vec{u}(0)] < E_{a\\wedge 0}[W_{a\\wedge 0}] \\quad \\text{and}\\quad \\Vert u(0)\\Vert _{\\dot{H}_a^1} < \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}.$ We define $L(\\mathcal {E}) =\\sup \\bigl \\lbrace \\Vert u\\Vert _{L_{t,x}^{\\frac{2(d-2)}{d+1}}(I\\times \\mathbb {R}^d)}\\bigr \\rbrace ,$ where the supremum is taken over all maximal-lifespan solutions $u:I\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ to (REF ) such that $E_a[\\vec{u}]\\le \\mathcal {E}$ .",
"In the focusing case, we also restrict to solutions satisfying $\\Vert u(t)\\Vert _{\\dot{H}_a^1}\\le \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}$ for some $t\\in I$ .",
"By the small-data theory, we have that $L(\\mathcal {E})<\\infty $ for $\\mathcal {E}$ small enough.",
"Therefore, if Theorem REF or Theorem REF fails, there exists a critical $\\mathcal {E}_c\\in (0,\\infty )$ (in the defocusing case) or $\\mathcal {E}_c\\in (0,E_{a\\wedge 0}[W_{a\\wedge 0}])$ (in the focusing case) such that $L(\\mathcal {E})<\\infty \\quad \\text{for}\\quad \\mathcal {E}<\\mathcal {E}_c\\quad \\text{and}\\quad L(\\mathcal {E})=\\infty \\quad \\text{for}\\quad \\mathcal {E}>\\mathcal {E}_c.$ The key to establishing Theorem REF is the following convergence result.",
"Proposition 4.4 Suppose $u_n:I_n\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ is a sequence of solutions to (REF ) with $E_a[\\vec{u}_n]\\rightarrow \\mathcal {E}_c,$ and suppose $t_n\\in I_n$ are such that $\\lim _{n\\rightarrow \\infty }\\Vert u_n\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\lbrace t>t_n\\rbrace \\times \\mathbb {R}^d)} = \\lim _{n\\rightarrow \\infty } \\Vert u_n\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\lbrace t<t_n\\rbrace \\times \\mathbb {R}^d)}=\\infty .$ In the focusing case, assume additionally that $\\Vert u_n(t_n)\\Vert _{\\dot{H}_a^1} \\le \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}.$ Then, passing to a subsequence, the sequence $\\lbrace u(t_n),\\partial _tu(t_n)\\rbrace $ converges in $\\dot{H}^1\\times L^2$ modulo scaling.",
"Assuming Proposition REF , the proof of Theorem REF is straightforward.",
"If Theorem 1.1 or Theorem 1.2 fails, one can find a sequence of solutions and times satisfying the hypotheses of Proposition REF .",
"Therefore, one can extract (after rescaling) a subsequential limit.",
"The solution $v$ to (REF ) with this initial data satisfies the conclusions of Theorem REF .",
"To check the compactness, for example, one applies Proposition REF with $u_n\\equiv v$ for any sequence $t_n$ in the orbit of $v$ .",
"Thus, it remains to establish Proposition REF .",
"By time-translation symmetry, we may assume $t_n\\equiv 0$ .",
"As we developed the requisite concentration-compactness tools for the operator $e^{-it\\sqrt{\\mathcal {L}_a}}$ , we will generally apply the mapping $T_a$ introduced in (REF ) and work with solutions to (REF ).",
"We apply the linear profile decomposition (Proposition REF ) to the sequence $T_a\\vec{u}_n(0)=u_n(0)+i\\mathcal {L}_a^{-\\frac{1}{2}}\\partial _t u_n(0)$ to write $T_a\\vec{u}_n(0)=\\sum _{j=1}^J \\phi _n^j + r_n^J$ with all of the properties stated in Proposition REF .",
"We need to prove that $J^*=1$ , $r_n^1\\rightarrow 0$ in $\\dot{H}^1$ , $x_n^1\\equiv 0$ , and $t_n^1\\equiv 0$ .",
"Note that by the decouplings (REF ) and (), we have $\\lim _{n\\rightarrow \\infty }\\biggl \\lbrace E_a[\\vec{u}_n] - \\sum _{j=1}^J \\tilde{E}_a[\\phi _n^j]-\\tilde{E}_a[r_n^J]\\biggr \\rbrace =0,$ where we recall the notation from (REF ).",
"Let us first show that $\\liminf _{n\\rightarrow \\infty } \\tilde{E}_a[\\phi _n^j]>0\\quad \\text{for each}\\quad j.$ To see this, first observe that $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty \\Rightarrow \\Vert \\phi _n^j\\Vert _{\\dot{H}_a^1}\\rightarrow \\Vert \\phi ^j\\Vert _{\\dot{H}^1}>0,$ which is a consequence of (REF ).",
"Thus, the claim follows from (REF ), (REF ), and Lemma REF .",
"Similarly, we deduce $\\liminf _{n\\rightarrow \\infty } \\tilde{E}_a[r_n^J]\\ge 0$ for each $J$ .",
"There are now two possible cases.",
"Case 1.",
"Suppose $\\sup _j \\limsup _{n\\rightarrow \\infty } \\tilde{E}_a[\\phi _n^j]=\\mathcal {E}_c$ .",
"In this case, the energy decoupling and (REF ) imply that $J^*=1$ , and hence we can write $T_a\\vec{u}_n(0) = \\phi _n + r_n,$ and in fact we can deduce that $r_n\\rightarrow 0$ in $\\dot{H}^1$ .",
"It therefore remains to preclude $\\lambda _n^{-1}|x_n|\\rightarrow \\infty $ and $t_n\\rightarrow \\pm \\infty $ .",
"To this end, first suppose $\\lambda _n^{-1}|x_n|\\rightarrow \\infty $ .",
"We will apply Proposition REF to the profile $\\phi _n$ .",
"If $t_n\\equiv 0$ , then the hypotheses of Proposition REF follow from (REF ), the fact that $r_n\\rightarrow 0$ in $\\dot{H}^1$ , Lemma REF and Corollary REF .",
"If instead $t_n\\rightarrow \\pm \\infty $ then we utilize Corollary REF , as well.",
"Thus, by Proposition REF , for $n$ large there exists a global solution $v_n$ to (REF ) with $(v_n(0),\\partial _t v_n(0))=\\phi _n$ satisfying global space-time bounds.",
"Then $T_a^{-1}v_n$ is a global solution to (REF ) with global space-time bounds.",
"However, noting that $\\Vert (u_n(0),\\partial _t u_n(0))-T_a^{-1}\\phi _n\\Vert _{\\dot{H}^1\\times L^2} \\lesssim \\Vert T_a\\vec{u}_n(0)-\\phi _n\\Vert _{\\dot{H}^1}\\rightarrow 0,$ we can therefore apply the stability result (Proposition REF ) to deduce that the $u_n$ obey global spacetime bounds, contradicting (REF ).",
"We conclude that $x_n\\equiv 0$ .",
"Next, if $t_n\\rightarrow \\infty $ , we observe that by Strichartz, monotone convergence, $r_n\\rightarrow 0$ in $\\dot{H}^1$ , and $x_n\\equiv 0$ , we have $\\Vert e^{-it\\sqrt{\\mathcal {L}_a}}T_a\\vec{u}_n(0)\\Vert _{L_{t,x}^r(\\lbrace t>0\\rbrace \\times \\mathbb {R}^d)} & \\lesssim \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}r_n\\Vert _{L_{t,x}^r} + \\Vert e^{-it\\sqrt{\\mathcal {L}_a}}\\phi \\Vert _{L_{t,x}^r((t_n,\\infty )\\times \\mathbb {R}^d)} \\\\& \\rightarrow 0 \\quad \\text{as}\\quad n\\rightarrow \\infty ,$ where $r=\\frac{2(d+1)}{d-2}$ .",
"By the small-data theory, this again implies global space-time bounds for the $u_n$ , yielding a contradiction.",
"A similar argument precludes the possibility that $t_n\\rightarrow -\\infty $ .",
"It therefore remains to preclude the following case: Case 2.",
"Suppose towards a contradiction that $\\sup _j \\limsup _{n\\rightarrow \\infty } \\tilde{E}_a[\\phi _n^j]<\\mathcal {E}_c - 3\\delta \\quad \\text{for some}\\quad \\delta >0.$ In this case, for each finite $J\\le J^*$ , we have $\\tilde{E}_a[\\phi _n^j] \\le \\mathcal {E}_c-2\\delta \\quad \\text{for}\\quad 1\\le j\\le J\\quad \\text{and}\\quad n\\quad \\text{large}.$ Recalling (REF ), (REF ), and Lemma REF , we also have $\\Vert \\operatornamewithlimits{Re}\\phi _n^j\\Vert _{\\dot{H}^1_a} <(1-\\delta ^{\\prime }) \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}^1_{a\\wedge 0}}\\quad \\text{for}\\quad 1\\le j\\le J\\quad \\text{and}\\quad n\\quad \\text{large}.$ We now introduce nonlinear solutions to (REF ) associated to each $\\phi _n^j$ as follows: If $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty $ then, arguing as above, the hypotheses of Proposition REF hold for $\\phi ^j$ and hence we obtain a global solution $v_n^j$ to (REF ) with $v_n^j(0)=\\phi _n^j$ .",
"If $x_n^j\\equiv 0$ and $t_n^j\\equiv 0$ , then we let $v^j$ be the maximal-lifespan solution to (REF ) with $v^j(0)=\\phi ^j$ .",
"If $x_n^j\\equiv 0$ and $t_n^j\\rightarrow \\pm \\infty $ , we use Proposition REF to find the maximal lifespan solution $v^j$ to (REF ) that scatters to $e^{-it\\sqrt{\\mathcal {L}_a}}\\phi ^j$ in $\\dot{H}^1$ as $t\\rightarrow \\pm \\infty $ .",
"In the latter two cases, we define $v_n^j(t,x) = (\\lambda _n^j)^{-(\\frac{d}{2}-1)}v^j(\\tfrac{t}{\\lambda _n^j}+t_n^j,\\tfrac{x}{\\lambda _n^j}).$ In particular, $v_n^j$ is also a solution to (REF ) with 0 in the maximal-lifespan for large enough $n$ and satisfying $\\lim _{n\\rightarrow \\infty }\\Vert v_n^j(0)-\\phi _n^j\\Vert _{\\dot{H}^1} =0.$ In particular, it follows that $\\tilde{E}_a[v_n^j]\\le \\mathcal {E}_c-\\delta $ for $1\\le j\\le J$ and $n$ large enough.",
"By the definition of $\\mathcal {E}_c$ , (REF ), and Proposition REF (for those $j$ for which $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty $ ), we have that each $v_n^j$ is global in time with uniform space-time bounds; moreover, (again using Proposition REF if $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty $ ) for any $\\eta >0$ we may find $\\psi _\\eta ^j\\in C_c^\\infty (\\mathbb {R}^{1+d})$ such that $\\Vert v_n(t-\\lambda _n t_n,x+x_n)-\\lambda _n^{-(\\frac{d}{2}-1)}\\psi _\\eta (\\lambda _n^{-1} t,\\lambda _n^{-1}x)\\Vert _{L_{t,x}^{\\frac{2(d+1)}{d-2}}(\\mathbb {R}^{1+d})} <\\eta $ for $n$ sufficiently large.",
"We will now construct approximate solutions to (REF ) that asymptotically match $T_a\\vec{u}_n(0)$ , but which have uniform space-time bounds.",
"Using the stability result (Proposition REF ), this will lead to a contraction to (REF ).",
"We define $w_n^J = \\sum _{j=1}^J v_n^j(t)+ e^{-it\\sqrt{\\mathcal {L}_a}}r_n^J,$ which we immediately observe satisfies $\\lim _{n\\rightarrow \\infty } \\Vert w_n^J(0) - T_a\\vec{u}_n(0)\\Vert _{\\dot{H}^1} = 0 \\quad \\text{for all}\\quad J.$ We claim that it remains to prove the following lemma.",
"Lemma 4.5 (Approximate solutions) The functions $w_n^J$ satisfy $\\limsup _{n\\rightarrow \\infty } \\bigl \\lbrace \\Vert w_n^J(0)\\Vert _{\\dot{H}^1} + \\Vert w_n^J\\Vert _{S(\\mathbb {R})} \\bigr \\rbrace \\lesssim 1 \\quad \\text{uniformly in}\\quad J, $ and $\\lim _{J\\rightarrow J^*}\\limsup _{n\\rightarrow \\infty } \\Vert \\mathcal {L}_a^{\\frac{1}{2}}\\bigl [(i\\partial _t-\\mathcal {L}_a^{\\frac{1}{2}})w_n^J -\\mu \\mathcal {L}_a^{-\\frac{1}{2}}|\\operatornamewithlimits{Re}w_n^J|^{\\frac{4}{d-2}}\\operatornamewithlimits{Re}w_n^J\\bigr ]\\Vert _{L_t^1 L_x^2} = 0.",
"$ Indeed, with Lemma REF in place, we can use Proposition REF and (REF ) to deduce that the solutions $T_a\\vec{u}_n$ to (REF ) inherit the uniform space-time bounds of the $u_n^J$ for large $n$ , contradicting (REF ).",
"The proof of Lemma REF follows along standard lines, so we will be somewhat brief.",
"One essential ingredient is the orthogonality of parameters given in (REF ).",
"In particular, (REF ) and approximation by functions that are $C_c^\\infty $ in space-time imply the following: Lemma 4.6 (Orthogonality) For $j\\ne k$ , we have $\\lim _{n\\rightarrow \\infty } \\Vert v_n^j v_n^k\\Vert _{L_{t,x}^{\\frac{d+1}{d-2}}} + \\Vert v_n^j v_n^k\\Vert _{L_t^{\\frac{d+2}{2(d-2)}}L_x^{\\frac{d+2}{d-2}}} = 0.$ We turn to Lemma REF .",
"The $\\dot{H}^1$ bound in (REF ) is straightforward.",
"Using this and decoupling, we deduce $\\limsup _{n\\rightarrow \\infty }\\sum _{j=1}^J \\Vert \\phi _n^j\\Vert _{\\dot{H}^1}^2 \\lesssim 1$ uniformly in $J$ .",
"Utilizing (REF ) for those $j$ with $(\\lambda _n^j)^{-1}|x_n^j|\\rightarrow \\infty $ , this implies $\\sum _{j=1}^\\infty \\Vert \\phi ^j\\Vert _{\\dot{H}^1}^2 \\lesssim 1.$ Thus for $J_0$ sufficiently large (depending on the small-data threshold), we can use the small-data theory to deduce $\\sup _J \\limsup _{n\\rightarrow \\infty } \\sum _{j=J_0}^J \\Vert v_n^j\\Vert _{S(\\mathbb {R})}^2 \\lesssim \\sum _{j\\ge J_0}\\Vert \\phi ^j\\Vert _{\\dot{H}^1}^2 \\ll 1,$ from which we then get $\\limsup _{n\\rightarrow \\infty }\\sum _{j=1}^J \\Vert v_n^j\\Vert _{S(\\mathbb {R})}^2 \\lesssim 1\\quad \\text{uniformly in}\\quad J.$ Writing $r=\\tfrac{2(d+1)}{d-2}$ , we use Lemma REF to estimate $\\biggl | \\biggl \\Vert \\sum _{j=1}^J v_n^j \\biggr \\Vert _{L_{t,x}^r}^r - \\sum _{j=1}^J \\Vert v_n^j\\Vert _{L_{t,x}^r}^r\\biggr | \\lesssim _J \\sum _{j\\ne k} \\Vert v_n^j\\Vert _{L_{t,x}^r}^{r-2} \\Vert v_n^j v_n^k\\Vert _{L_{t,x}^{\\frac{r}{2}}} \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"As the remainder term $r_n^J$ is controlled in $L_{t,x}^r$ uniformly, we deduce the $L_{t,x}^r$ bound appearing in (REF ).",
"We turn to (REF ) and set $F(z)=\\mu \\bigl [|\\operatornamewithlimits{Re}z|^{\\frac{4}{d-2}}\\operatornamewithlimits{Re}z\\bigr ],$ so that (using that each $v_n^j$ solves (REF )) $\\mathcal {L}_a^{\\frac{1}{2}}(i\\partial _t - \\mathcal {L}_a^{\\frac{1}{2}})w_n^J - F(w_n^J) & = \\sum _{j=1}^J F(v_n^j) - F(\\sum _{j=1}^J v_n^j) \\\\& \\quad + F(w_n^J - e^{-it\\sqrt{\\mathcal {L}_a}}r_n^J) - F(w_n^J).",
"$ In particular, we need to estimate (REF ) and () in $L_t^1 L_x^2$ .",
"First, by Lemma REF , $\\lim _{n\\rightarrow \\infty }\\Vert (\\ref {enj1})\\Vert _{L_t^1 L_x^2} & \\lesssim _J \\lim _{n\\rightarrow \\infty }\\sum _{j\\ne k} \\Vert v_n^j v_n^k\\Vert _{L_t^{\\frac{d+2}{2(d-2)}}L_x^{\\frac{d+2}{d-2}}}\\Vert v_n^k\\Vert _{L_t^{\\frac{d+2}{d-2}} L_x^{\\frac{2(d+2)}{d-2}}}^{\\frac{6-d}{d-2}} =0$ for all $J$ .",
"Thus $\\lim _{J\\rightarrow J^*}\\limsup _{n\\rightarrow \\infty } \\Vert (\\ref {enj1})\\Vert _{L_t^1 L_x^2} = 0,$ as desired.",
"We turn to ().",
"Employing the vanishing condition (REF ) (and interpolation), we find $\\lim _{J\\rightarrow J^*}&\\lim _{n\\rightarrow \\infty }\\Vert (\\ref {enj2})\\Vert _{L_t^1 L_x^2} \\\\& \\lesssim \\lim _{J\\rightarrow J^*}\\lim _{n\\rightarrow \\infty }\\Vert e^{-it\\sqrt{\\mathcal {L}_a}}r_n^J\\Vert _{L_t^{\\frac{d+2}{d-2}} L_x^{\\frac{2(d+2)}{d-2}}}\\bigl [\\Vert w_n^J\\Vert _{L_t^{\\frac{d+2}{d-2}} L_x^{\\frac{2(d+2)}{d-2}}}+\\Vert r_n^J\\Vert _{\\dot{H}^1}\\bigr ]^{\\frac{4}{d-2}} =0,$ as desired.",
"This completes the proof of Lemma REF .",
"With Lemma REF in place, we complete the proof of Proposition REF (and hence the proof of Theorem REF ."
],
[
"Further reductions",
"In this section, we perform some further reductions to the class of solutions constructed in Theorem REF .",
"We begin with the observation that the frequency scale of an almost periodic solution obeys a local constancy property, namely, $N(t)\\sim N(t^{\\prime })$ whenever $|t-t^{\\prime }|\\ll N(t)^{-1}$ .",
"This is essentially a consequence of the local theory (cf.",
"[26], for example).",
"Using this, we may always divide the lifespan of an almost periodic solution into characteristic subintervals $J_k$ on which $N(t)$ is equal to some constant $N_k$ , with $|J_k|\\sim N_k^{-1}$ .",
"We next record a `non-triviality' condition for almost periodic solutions.",
"Note that while the $\\dot{H}^1 \\times L^2$ -norm of $\\vec{u}(t)$ is bounded away from zero, each component individually may spend some time near zero.",
"Nonetheless, by adapting the arguments of [27] one readily observes that almost periodicity implies that for any $\\delta >0$ , we have $|\\lbrace t\\in [t_0,t_0+\\delta N(t_0)^{-1}]:\\Vert u(t)\\Vert _{\\dot{H}^1}\\ge \\varepsilon \\rbrace | \\ge \\varepsilon N(t_0)^{-1}$ for some small $\\varepsilon =\\varepsilon (\\delta ,u)>0$ (uniformly in $t_0$ ).",
"We will proceed in a similar fashion to [19] prove the following.",
"Theorem 4.7 Suppose there exist almost periodic solutions to (REF ) as in Theorem REF .",
"Then we may find an almost periodic solution $u:I_{\\max }\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ to (REF ) conforming to one of the following two scenarios.",
"(i) Let $I_0=[0,\\infty )$ .",
"Then $I_{\\max }\\supset I_0$ , $x(t)\\equiv 0$ , and $\\inf _{t\\in I_0}N(t)\\ge 1$ .",
"(ii) Let $I_0=(0,1]$ .",
"Then $I_{\\max }\\supset I_0$ with $\\inf I_{\\max }=0$ , $x(t)\\equiv 0$ , and $N(t) = t^{-1}$ .",
"Furthermore, for each $t\\in I_0$ , $(u,\\partial _t u)$ is supported in $B_t(0)$ .",
"In the focusing case, we have $E_a(u,\\partial _tu) < E_{a\\wedge 0}(W_{a\\wedge 0},0) \\quad \\text{and}\\quad \\sup _{t\\in I_0}\\Vert u(t) \\Vert _{\\dot{H}_a^1} < \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}.$ We call scenario (i) the forward-global case and scenario (ii) the self-similar case.",
"As mentioned above, we follow the arguments in [19].",
"In fact, the proof is simplified by the fact that the solutions in Theorem REF have $x(t)\\equiv 0$ .",
"Take $u:I_{\\max }\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ as in Theorem REF .",
"A standard rescaling argument shows that we may assume $N(t)\\ge 1$ on half of the maximal lifespan of $u$ , say $[0,T_{max})$ .",
"We then split into two cases, namely $T_{\\max }=\\infty $ or $T_{\\max }<\\infty $ .",
"If $T_{\\max }=\\infty $ , then we are in scenario (i).",
"Thus it remains to show that if $T_{\\max }<\\infty $ , then we may extract a solution conforming to scenario (ii).",
"Suppose $T_{\\max }<\\infty $ .",
"By time reversal and scaling, we may assume that $I_{\\max }\\supset I_0=(0,1]$ with $\\inf I_{\\max }=0$ .",
"A standard rescaling argument relying on almost periodicity and local well-posedness shows that we must have $N(t)\\gtrsim _u t^{-1}$ .",
"We begin by showing that for each $t\\in I_0$ , $(u(t),\\partial _tu(t))$ is supported in $B_t(0)$ .",
"Using the fact that $x(t)\\equiv 0$ and $N(t)\\rightarrow \\infty $ as $t\\rightarrow 0$ , we first deduce $\\lim _{t\\rightarrow 0^+}\\int _{|x|>R} |\\nabla u(t,x)|^2 + |\\partial _tu(t,x)|^2 \\,dx = 0 \\quad \\text{for any}\\quad R>0.$ Using the small-data theory and finite speed of propagation, this implies $\\lim _{t\\rightarrow 0^+} \\int _{|x|\\ge \\frac{3}{2}R + |t-s|} |\\nabla u(s,x)|^2 + |\\partial _su(s,x)|^2\\,dx = 0 \\quad \\text{for any}\\quad R>0,\\ s\\in [0,1).$ Now fix $\\eta >0$ and $R$ so that $\\tfrac{3}{2} R < \\eta $ and let $s\\in (0,1]$ .",
"Choosing $t_n\\rightarrow 0^+$ , we have for $n$ sufficiently large that $t_n<s$ and $\\lbrace |x|\\ge s+\\eta \\rbrace \\subset \\lbrace |x|\\ge \\tfrac{3}{2} R + s-t_n\\rbrace .$ Thus, sending $n\\rightarrow \\infty $ , we get $\\int _{|x|\\ge s+\\eta } |\\nabla u(s,x)|^2 + |\\partial _su(s,x)|^2\\,dx = 0.$ As $\\eta ,s$ were arbitrary, the claim follows.",
"We next wish to show that $N(t)\\lesssim _u t^{-1}$ .",
"Combining this with the upper bound, we will then be able to modify the compactness modulus by a uniformly bounded function and take $N(t)=t^{-1}$ .",
"To this end, we will apply the virial identity Lemma REF with the weight $w(x)=\\tfrac{1}{2} |x|^2$ .",
"We write $M(t) = \\int -\\partial _t u[x\\cdot \\nabla u + \\tfrac{d}{2} u]\\,dx.$ Because of the support properties of $(u,\\partial _tu)$ , we do not need to truncate the weight $w$ .",
"In fact, using Hölder's inequality and Sobolev embedding, $|M(t)|\\lesssim t \\rightarrow 0$ as $t\\rightarrow 0^+$ .",
"With $G(u) = \\mu \\tfrac{d-2}{2d}|u|^{\\frac{2d}{d-2}}$ and $V(x) = \\tfrac{a}{|x|^2}$ , we have $\\tfrac{1}{2} u G^{\\prime }(u) - G(u) = \\tfrac{\\mu }{d}|u|^{\\frac{2d}{d-2}}\\quad \\text{and}\\quad -\\tfrac{1}{2} x\\cdot \\nabla V = V.$ Thus the virial identity becomes $M^{\\prime }(t) = \\int |\\mathcal {L}_a u|^2 + \\mu |u|^{\\frac{2d}{d-2}} \\,dx.$ In particular, using Lemma REF in the focusing case, we deduce $M^{\\prime }(t)\\gtrsim \\Vert u(t)\\Vert _{\\dot{H}^1}^2.$ Using the fundamental theorem of calculus (cf.",
"$M(t)\\rightarrow 0$ as $t\\rightarrow 0+$ ), breaking into characteristic subintervals, and employing (REF ), this further implies $M(t)\\gtrsim t$ .",
"Now suppose toward a contradiction that there exists $t_n\\rightarrow 0^+$ so that $N(t_n)t_n\\rightarrow \\infty $ .",
"We will show that ${M(t_n)}=o(t_n)$ , contradicting the fact that $M(t_n)\\gtrsim _u t_n$ .",
"To see this, we fix $\\eta >0$ and note that $C(\\eta )<N(t_n)t_n$ for $n$ large, where $C(\\cdot )$ is the compactness modulus of $u$ .",
"We then write $|M(t_n)| \\le \\biggl | \\int _{|x|\\le \\frac{C(\\eta )}{N(t_n)}} \\partial _tu[x\\cdot \\nabla u + \\tfrac{d}{2} u] \\,dx \\biggr | + \\biggl | \\int _{\\frac{C(\\eta )}{N(t_n)}\\le | x|\\le t_n} \\partial _tu[x\\cdot \\nabla u + \\tfrac{d}{2} u] \\,dx\\biggr |.$ By almost periodicity, Hölder's inequality, and Sobolev embedding, the second term is controlled by $\\eta \\cdot t_n$ .",
"The first term is controlled by $\\tfrac{C(\\eta )}{N(t_n)}=o(t_n)$ .",
"As $\\eta $ was arbitrary, we conclude $M(t_n)=o(t_n)$ , as desired.",
"To complete the proof of our main results, Theorem REF and Theorem REF , it therefore suffices to rule out the possibility of solutions to (REF ) as in scenarios (i) and (ii) of Theorem REF ."
],
[
"Preclusion of the forward global case",
"In this section we suppose that $u$ is an almost periodic solution to (REF ) conforming to scenario (i) in Theorem REF and derive a contradiction.",
"In particular, we have $I_{max}\\supset [0,\\infty )$ , $N(t)\\ge 1$ , and $x(t)\\equiv 0$ .",
"Moreover, in the focusing case, $u$ is below the ground state threshold.",
"We will apply the virial identity Lemma REF with $w(x) = R^2 \\phi (\\tfrac{x}{R})$ , where $\\phi $ is a smooth function satisfying $\\phi (x) = {\\left\\lbrace \\begin{array}{ll} \\tfrac{1}{2}|x|^2 & |x|\\le 1 \\\\ 2 & |x|>3.\\end{array}\\right.",
"}$ We recall that with $G(u) = \\mu \\tfrac{d-2}{2d}|u|^{\\frac{2d}{d-2}}$ and $V(x) = \\tfrac{a}{|x|^2}$ , we have $\\tfrac{1}{2} u G^{\\prime }(u) - G(u) = \\tfrac{\\mu }{d}|u|^{\\frac{2d}{d-2}}\\quad \\text{and}\\quad -\\tfrac{1}{2} x\\cdot \\nabla V = V.$ Applying Lemma REF with $w$ as above and employing the fundamental theorem of calculus, Hölder's inequality, and Sobolev embedding, we deduce that $\\int _{t_1}^{t_2}&\\int _{\\mathbb {R}^d}|\\mathcal {L}_a u|^2+ \\mu |u|^{\\frac{2d}{d-2}}\\,dx\\,dt \\\\&\\lesssim \\sup _{t\\in [t_1,t_2]} R\\Vert \\nabla _{t,x} u\\Vert _{L_x^2}^2 + \\mathcal {O}\\biggl (\\int _{t_1}^{t_2}\\int _{R<|x|<3R} |\\nabla u|^2 + R^{-2} |u|^2+ |u|^{\\frac{2d}{d-2}}\\,dx\\,dt\\biggr ) \\\\& \\quad + \\biggl |\\int _{t_1}^{t_2} \\int _{|x|>R} |\\nabla u|^2 + \\tfrac{a}{|x|^2}|u|^2 + |u|^{\\frac{2d}{d-2}} \\,dx\\,dt \\biggr |$ for any $0<t_1<t_2<\\infty $ .",
"We will seek lower bounds for the left-hand side and upper bounds for the right-hand hand side that together will yield a contradiction.",
"We begin with the left-hand side.",
"Using Lemma REF in the focusing case, we firstly observe that $\\int _{\\mathbb {R}^d} |\\mathcal {L}_a u|^2 + \\mu |u|^{\\frac{2d}{d-2}}\\,dx\\gtrsim \\Vert u(t)\\Vert _{\\dot{H}^1}^2.$ Thus, utilizing (REF ) and breaking into characteristic subintervals, we deduce that $\\int _0^T\\int _{\\mathbb {R}^d} |\\mathcal {L}_a u|^2 + \\mu |u|^{\\frac{2d}{d-2}}\\,dx\\gtrsim T\\delta ,$ uniformly in $T$ for some small $\\delta =\\delta (u)>0$ .",
"We now let $\\eta >0$ .",
"By almost periodicity and the fact that $\\inf _{t\\in [0,\\infty )}N(t)\\ge 1$ , we may choose $R=R(\\eta )$ large enough that $\\sup _{t\\in [0,\\infty )}\\int _{|x|>R} |\\nabla u(t,x)|^2 + \\tfrac{a}{|x|^2}|u(t,x)|^2 + |u|^{\\frac{2d}{d-2}} \\,dx \\,dt < \\eta .$ Using Hölder's inequality as well, we can take $R$ possibly even larger to guarantee that $\\sup _{t\\in [0,\\infty )}\\int _{R<|x|<3R} |\\nabla u|^2 + R^{-2} |u|^2 + |u|^\\frac{2d}{d-2} \\,dx < \\eta .$ Combining the estimates above on an interval fo the form $[0,T]$ , we deduce that $T\\delta \\lesssim _uR + T\\eta $ for any $T>0$ .",
"However, choosing $\\eta =\\eta (u,\\delta )$ sufficiently small and then $T=T(\\eta )$ sufficiently large, this leads to a contradiction.",
"We conclude that there are no solutions to (REF ) as in scenario (i) of Theorem REF ."
],
[
"Preclusion of the self-similar case",
"In this section we preclude the possibility of self-similar almost periodic solutions as in Theorem REF .",
"Recall that a self-similar almost periodic solution satisfies $x(t)\\equiv 0$ and $N(t)=t^{-1}$ .",
"In particular such solutions blow up at $t=0$ ; furthermore, at each $t>0$ they are supported in $B_t(0)$ .",
"We recall the notation $x^\\beta =(t,x)$ .",
"Proposition 6.1 (Virial/Morawetz estimate) Suppose $u:(0,1)\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ is a self-similar almost periodic solution to (REF ) as in Theorem REF .",
"For any $t_2>t_1>0$ , we have $\\int _{t_1}^{t_2} \\int _{|x|<t} \\bigl [\\tfrac{x^\\beta }{t} \\partial _\\beta u +\\tfrac{d-2}{2}\\tfrac{u}{t}\\bigr ]^2\\,dx \\,\\tfrac{dt}{t} \\lesssim \\log (\\tfrac{t_2}{t_1})^{\\frac{3}{4}}.$ We write the equation in the form $\\partial ^\\alpha \\partial _\\alpha u = Vu + G^{\\prime }(u),$ where $V(x) = a|x|^{-2}\\quad \\text{and}\\quad G(u)=\\tfrac{d-2}{2d} |u|^{\\frac{2d}{d-2}}.$ We next introduce the function $\\rho =\\rho (t,x) = \\bigl [(1+\\varepsilon ^2)t^2 - |x|^2\\bigr ]^{-\\frac{1}{2}},$ which satisfies $x^\\beta \\partial _\\beta \\rho = - \\rho .$ We now define the space-time region $S=\\bigcup _{t_1<t<t_2} \\lbrace (t,x):|x|<t\\rbrace \\subset \\mathbb {R}^{1+d}.$ Let $n_\\beta $ denote the outward-pointing unit normal vector at $(t,x)\\in \\partial S$ .",
"We may write $\\partial S = \\Sigma _1 \\cup \\Sigma _2,$ where $\\Sigma _1&=\\bigcup _{t_1<t<t_2}\\lbrace (t,x):|x|=t\\rbrace ,\\\\\\Sigma _2 &= \\lbrace (t_1,x):|x|<t_1\\rbrace \\cup \\lbrace (t_2,x):|x|<t_2\\rbrace .$ Note that $u\\equiv 0$ on $\\Sigma _1$ .",
"We have $n_\\beta =\\tfrac{1}{\\sqrt{2}}(-1,\\tfrac{x}{t})&\\quad \\text{and}\\quad x^\\beta n_\\beta = 0,\\quad (t,x)\\in \\Sigma _1, \\\\n_\\beta =(\\pm 1,0)&\\quad \\text{and}\\quad x^\\beta n_\\beta = \\pm t,\\quad (t,x)\\in \\Sigma _2.\\nonumber $ We multiply the equation (REF ) by $\\rho [x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2}u]$ and integrate over $S$ .",
"This yields $0 &= \\int _S \\rho [x^\\beta \\partial _\\beta u+\\tfrac{d-2}{2}u][\\partial ^\\alpha \\partial _\\alpha u - Vu - G^{\\prime }(u)] \\nonumber \\\\& = \\int _S \\rho [x^\\beta \\partial ^\\alpha (\\partial _\\alpha u \\partial _\\beta u) - \\tfrac{1}{2} x^\\beta \\partial _\\beta (\\partial _\\alpha u \\partial ^\\alpha u)] \\\\& \\quad + \\int _S \\tfrac{d-2}{2}\\rho [\\partial ^\\alpha (u\\partial _\\alpha u) - \\partial ^\\alpha u \\partial _\\alpha u ] \\\\& \\quad - \\int _S \\rho [\\tfrac{1}{2} x^\\beta V\\partial _\\beta (u^2) + \\tfrac{d-2}{2} Vu^2] \\\\& \\quad - \\int _S \\rho [x^\\beta \\partial _\\beta G(u) + \\tfrac{d-2}{2}G^{\\prime }(u) u].",
"$ Integration by parts (using (REF ), (REF ), $\\partial ^\\alpha x^\\beta = g^{\\alpha \\beta }$ , and $\\partial _\\beta x^\\beta = d+1$ ) yields $(\\ref {12283})+(\\ref {12284}) & =\\int _S -\\partial ^\\alpha \\rho \\partial _\\alpha u\\bigl [x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2}u\\bigr ] \\\\& \\quad + \\int _{\\Sigma _2} \\rho (\\partial _\\alpha u)( x^\\beta \\partial _\\beta u) g^{\\alpha \\gamma } n_\\gamma \\\\& \\quad - \\int _{\\Sigma _2} \\tfrac{1}{2}\\rho (\\partial _\\alpha u)( \\partial ^\\alpha u) x^\\beta n_\\beta \\\\& \\quad + \\int _{\\Sigma _2} \\tfrac{d-2}{2} \\rho u(\\partial _\\alpha u) g^{\\alpha \\gamma }n_\\gamma .",
"$ Further integration by parts (using $\\tfrac{1}{2} x^\\beta \\partial _\\beta V = -V$ and (REF )) yields $(\\ref {12285}) = -\\int _{\\partial S}\\tfrac{1}{2} \\rho x^\\beta n_\\beta Vu^2 & = -\\int _{\\Sigma _2}\\tfrac{1}{2} \\rho x^\\beta n_\\beta Vu^2, \\\\(\\ref {12286}) = -\\int _{\\partial S} \\rho x^\\beta n_\\beta G(u) & = -\\int _{\\Sigma _2} \\rho x^\\beta n_\\beta G(u).$ Collecting the identities above now yields $\\int _S \\partial ^\\alpha \\rho \\partial _\\alpha u\\bigl [x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2}u\\bigr ] & = \\int _{\\Sigma _2} \\rho (\\partial _\\alpha u)( x^\\beta \\partial _\\beta u) g^{\\alpha \\gamma } n_\\gamma \\\\& \\quad +\\int _{\\Sigma _2} \\tfrac{d-2}{2} \\rho u(\\partial _\\alpha u) g^{\\alpha \\gamma }n_\\gamma \\\\& \\quad - \\int _{\\Sigma _2} \\tfrac{1}{2}\\rho (\\partial _\\alpha u)(\\partial ^\\alpha u)x^\\beta n_\\beta \\\\& \\quad - \\int _{\\Sigma _2} \\tfrac{1}{2} \\rho x^\\beta n_\\beta [V u^2+G(u)].$ To estimate (REF )–(), we use $\\rho \\le (\\varepsilon t)^{-1}$ on $\\Sigma _2$ and $x^\\beta n_\\beta = \\pm t$ on $\\Sigma _2$ .",
"Then, since $t^{-1}\\le |x|^{-1}$ on $\\Sigma _2$ , we have by $\\dot{H}^1\\times L^2$ bounds (and Hardy's inequality) that $(\\ref {6e2})+(\\ref {6e4})+(\\ref {6e5})+(\\ref {6e6}) \\lesssim \\varepsilon ^{-1}.$ We now turn to the left-hand side.",
"We wish to exhibit a coercive term and control the remaining error terms.",
"To this end, note that $\\partial ^\\alpha \\rho \\partial _\\alpha u = \\rho ^3[x^\\alpha \\partial _\\alpha u + \\varepsilon ^2 t\\partial _t u ],$ so that the left-hand side of (REF ) is given by $\\partial ^\\alpha \\rho \\partial _\\alpha u[x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2}u] & = \\rho ^3(x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2} u)^2 \\\\& \\quad + \\rho ^3(x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2} u)(\\varepsilon ^2 t \\partial _t u - \\tfrac{d-2}{2}u).",
"$ We further expand () to write $(\\ref {ss-error1}) & = -\\tfrac{d-2}{2} \\rho ^3 x^\\beta \\partial _\\beta ( \\tfrac{1}{2} u^2 )- \\tfrac{(d-2)^2}{4}\\rho ^3 u^2 \\\\& \\quad + \\varepsilon ^2\\rho ^3 (x^\\beta \\partial _\\beta u+\\tfrac{d-2}{2}u)t\\partial _t u.$ An integration by parts shows $\\int _S (\\ref {ss-error12}) & = \\int _S \\tfrac{(d+1)(d-2)}{4} \\rho ^3 u^2 + \\tfrac{3(d-2)}{4}\\rho ^2 (x^\\beta \\partial _\\beta \\rho ) u^2 - \\tfrac{(d-2)^2}{4}\\rho ^3 u^2 \\\\& \\quad - \\int _{\\Sigma _2} \\rho ^3\\tfrac{d-2}{2}x^\\beta n_\\beta \\tfrac{1}{2} u^2.$ The first term on the right-hand side is zero, and hence using $\\rho ^3\\lesssim \\varepsilon ^{-3}t^{-3} \\lesssim \\varepsilon ^{-3}t^{-1}|x|^{-2}\\quad \\text{and}\\quad |x^\\beta n_\\beta |=t\\quad \\text{on}\\quad \\Sigma _2,$ we have by Hardy's inequality $\\biggl | \\int _S (\\ref {ss-error12})\\biggr | \\lesssim \\varepsilon ^{-3}\\int _{\\Sigma _2} \\tfrac{u^2}{|x|^2} \\lesssim \\varepsilon ^{-3}.$ Collecting our estimates, we have so far established $\\int _S \\rho ^3(x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2} u)^2 + \\varepsilon ^2\\rho ^3 (x^\\beta \\partial _\\beta u+\\tfrac{d-2}{2}u)t\\partial _t u \\lesssim \\varepsilon ^{-1}+\\varepsilon ^{-3}.$ We next use $\\varepsilon ^2\\rho ^3(x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2}u)t\\partial _t u \\le \\tfrac{1}{2}\\rho ^3(x^\\beta \\partial _\\beta u+\\tfrac{d-2}{2}u)^2 + \\varepsilon ^4 \\rho ^3 \\tfrac{1}{2} (t\\partial _t u)^2,$ along with the fact that $\\varepsilon ^4 \\int _{t_1}^{t_2} \\int _{|x|<t} \\rho ^3 t^2 (\\partial _t u)^2 \\,dx\\,dt \\lesssim \\varepsilon \\log (\\tfrac{t_2}{t_1})$ (cf.",
"$\\rho \\le \\varepsilon ^{-1}t^{-1}$ ) to deduce $\\int _S \\rho ^3(x^\\beta \\partial _\\beta u + \\tfrac{d-2}{2} u)^2 \\lesssim \\varepsilon ^{-1}+\\varepsilon ^{-3}+\\varepsilon \\log (\\tfrac{t_2}{t_1}).$ Noting that that $\\rho ^3\\gtrsim t^{-3}$ for $(t,x)\\in S$ , we finally conclude $\\int _{t_1}^{t_2} \\int _{|x|<t} \\bigl [\\tfrac{x^\\beta }{t} \\partial _\\beta u +\\tfrac{d-2}{2}\\tfrac{u}{t}\\bigr ]^2\\,dx \\,\\tfrac{dt}{t} \\lesssim \\varepsilon ^{-1}+\\varepsilon ^{-3}+\\varepsilon \\log (\\tfrac{t_2}{t_1}).$ Optimizing in $\\varepsilon $ yields $\\int _{t_1}^{t_2} \\int _{|x|<t} \\bigl [\\tfrac{x^\\beta }{t} \\partial _\\beta u +\\tfrac{d-2}{2}\\tfrac{u}{t}\\bigr ]^2\\,dx \\,\\tfrac{dt}{t} \\lesssim \\log (\\tfrac{t_2}{t_1})^{\\frac{3}{4}},$ which completes the proof.",
"Using Proposition REF , we will now extract a nontrivial solution to a (degenerate) elliptic equation satisfying some integrability properties.",
"Below we will use a unique continuation result to conclude that such a solution cannot exist, thereby reaching a contradiction to the existence of self-similar almost periodic solutions to (REF ), as desired.",
"We let $B$ denote the unit ball centered at the origin.",
"Proposition 6.2 Suppose there exists a self-similar almost periodic solution to (REF ) as in Theorem REF .",
"Then there exists a nonzero $H^1$ solution $f:B\\rightarrow \\mathbb {R}$ to $\\Delta f - x\\cdot \\nabla ^2 f x - dx\\cdot \\nabla f = \\tfrac{d(d-2)}{4}f + a|x|^{-2}f +\\mu |f|^{\\frac{4}{d-2}} f$ satisfying $f|_{\\partial B}=0$ and $\\int _B \\frac{|f|^{\\frac{2d}{d-2}}}{(1-|x|)^{\\frac{1}{2}}}\\,dx + \\int _{B} \\frac{|{\\nabla } f|^2}{(1-|x|)^{\\frac{1}{2}}} \\,dx \\lesssim 1,$ where ${\\nabla }$ denotes the angular derivative.",
"Suppose $u$ is a self-similar almost periodic solution.",
"Let us first extract the stationary solution $f$ .",
"We first claim that Proposition REF yields a sequence $t_n\\downarrow 0$ such that $\\int _{t_n}^{2t_n}\\int _{|x|<t} \\bigl [\\tfrac{x^\\beta }{t}\\partial _\\beta u + \\tfrac{d-2}{2}\\tfrac{u}{t}\\bigr ]^2\\,dx\\tfrac{dt}{t} \\rightarrow 0.$ To see this, we argue as in [5].",
"We apply Proposition REF with $t_2=2^{-J}$ and $t_1=4^{-J}$ for some large $J>0$ .",
"Then Proposition REF implies that there exists $j=j(J)$ such that $\\int _{2^j 4^{-J}}^{2^{j+1}4^{-J}} \\int _{|x|<t} \\bigl [\\tfrac{x^\\beta }{t}\\partial _\\beta u + \\tfrac{d-2}{2}\\tfrac{u}{t}\\bigr ]^2\\,dx\\tfrac{dt}{t} \\lesssim J^{-\\frac{1}{4}}.$ Now choose a sequence $J_n\\rightarrow \\infty $ such that $J_n\\ge 2J_{n-1}$ and take $t_n = 2^{j(J_n)}4^{-J_n}$ .",
"Then $t_n\\downarrow 0$ and (REF ) holds along this sequence.",
"By almost periodicity (and the fact that $N(t)=t^{-1}$ ), we have (passing to a further subsequence) $(\\tilde{u}_n(0), \\partial _t \\tilde{u}_n(0)):= (t_n^{\\frac{d}{2}-1} u(t_n,t_n\\cdot ),t_n^{\\frac{d}{2}}(\\partial _t u)(t_n,t_n \\cdot )) \\rightarrow (v_0,v_1)$ strongly in $\\dot{H}^1\\times L^2$ for some $(v_0,v_1)\\in \\dot{H}^1\\times L^2$ .",
"Note that $(v_0,v_1)$ are supported in the unit ball.",
"We let $v:[0,\\delta )\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ be the solution to (REF ) with initial data $(v_0,v_1)$ .",
"Now observe that by scaling symmetry, $\\tilde{u}_n(t,x)=t_n^{\\frac{d}{2}-1}u(t_n(1+t),t_nx)$ is the solution to (REF ) with initial data $(\\tilde{u}_n(0),\\partial _t \\tilde{u}_n(0))$ .",
"By a change of variables, (REF ) implies $\\lim _{n\\rightarrow \\infty }\\int _0^1 \\int _{|y|<1}\\bigl |\\partial _t \\tilde{u}_n(s,y) + \\tfrac{y}{s+1}\\cdot \\nabla \\tilde{u}_n(s,y)+\\tfrac{d-2}{2}\\tfrac{\\tilde{u}_n(s,y)}{s+1}\\bigr |^2 \\,dx \\,dt = 0,$ from which we deduce $\\partial _t v + \\tfrac{x}{t+1}\\nabla v + \\tfrac{d-2}{2} \\tfrac{v}{t+1}\\equiv 0$ in $[0,\\delta ]\\times \\lbrace |x|<1\\rbrace $ .",
"By the method of characteristics, this implies that $v(t,x)=(t+1)^{-[\\frac{d}{2}-1]} f(\\tfrac{x}{1+t})$ for some $f\\in H^1$ supported in the unit ball.",
"Combining this form with the fact that $v$ solves (REF ), we immediately deduce that $f$ solves (REF ).",
"We turn to establishing (REF ).",
"To begin, we collect a few properties about the solution $f$ .",
"First, because the solution $v$ belongs to $L_{t,x}^{\\frac{2(d+1)}{d-2}}$ locally in time, a change of variables yields $f\\in L_x^{\\frac{2(d+1)}{d-2}}.$ Next, we observe that $\\int \\frac{|f|^2}{(1-|x|)^s}\\,dx \\lesssim 1 \\quad \\text{for all}\\quad s\\le 2.$ The case $s=0$ is clear, while the case $s=2$ can be deduced as a consequence of Hardy's inequality.",
"Using (REF ) and Hölder's inequality, we also observe that $\\int \\frac{|f|^{\\frac{2d}{d-2}}}{(1-|x|)^{\\frac{1}{2}}}\\,dx \\lesssim \\biggl (\\int \\frac{|f|^2}{(1-|x|)^{2}} \\biggr )^{\\frac{1}{4}} \\biggl (\\int |f|^{\\frac{2(3d+2)}{3(d-2)}}\\,dx \\biggr )^{\\frac{3}{4}}\\lesssim 1,$ where we use $\\tfrac{2(3d+2)}{3(d-2)}<\\tfrac{2(d+1)}{d-2}.$ This gives the first bound in (REF ).",
"We turn to the second estimate in (REF ).",
"Let us define the weight $w(x)=(1-|x|^2)^{-\\frac{1}{2}}$ .",
"We multiply both sides of (REF ) by $fw$ and integrate by parts.",
"This yields $-\\int [|\\nabla f|^2&-(x\\cdot \\nabla f)^2]w + \\int (x\\cdot \\nabla f)fw +f\\lbrace (x\\cdot \\nabla f)(x\\cdot \\nabla w)-\\nabla f\\cdot \\nabla w\\rbrace \\,dx \\\\& = \\int \\tfrac{d(d-2)}{4}|f|^2 w + a|x|^{-2}|f|^2 w + \\mu |f|^{\\frac{2d}{d-2}}w\\,dx.$ Noting that $\\nabla w = xw^3$ , we find that the second term on the left-hand side vanishes.",
"Using (REF ) and (REF ), we deduce that $\\int \\frac{\\bigl | |\\nabla f|^2 - (x\\cdot \\nabla f)^2\\bigr |}{(1-|x|)^{\\frac{1}{2}}}\\,dx\\lesssim 1.$ As $|\\nabla f|^2 - (x\\cdot \\nabla f)^2 = (1-|x|^2)|\\nabla f|^2+|x|^2|{\\nabla }f|^2,$ we deduce that (REF ) holds.",
"To rule out the self-similar scenario of Theorem REF , it therefore suffices to preclude the possibility of a solution to (REF ) as in Proposition REF .",
"For this we will rely on unique continuation results for elliptic PDE.",
"In fact, at this point we are in almost an identical situation to [19].",
"Indeed the remaining issues to address are all related to the degeneracy of (REF ) at $|x|=1$ ; in particular, the presence of the potential term $a|x|^{-2}f$ plays essentially no role.",
"For the sake of completeness, however, let us not simply quote [19] and conclude the proof.",
"Instead let us briefly go through the argument (parallel to that in [19]) to preclude the existence of a solution as in Proposition REF .",
"It remains to prove the following: Proposition 6.3 Suppose $f$ is a solution to (REF ) as in Proposition REF .",
"Then $f\\equiv 0$ .",
"As just mentioned, the PDE (REF ) is a degenerate elliptic PDE, with the degeneracy occurring as $|x|\\rightarrow 1$ .",
"In particular, by standard unique continuation results (see [1]), the result will follow if we can prove $f\\equiv 0$ on $\\lbrace 1-\\delta <|x|<1\\rbrace $ for some small $\\delta >0$ .",
"The idea is to introduce a change of variables that removes this degeneracy and to study the resulting PDE.",
"We begin by changing to polar coordinates $f=f(r,\\omega )$ and rewriting the left-hand side of the PDE (REF ) as $(1-r^2)\\partial _r^2 f + (\\tfrac{d-1}{r}-dr)\\partial _r f + \\tfrac{1}{r^2}{\\Delta } f,$ where ${\\Delta }$ denotes the spherical Laplacian.",
"We now introduce $g(s,\\omega )=f(r(s),\\omega )$ for a function $r(s)$ to be defined shortly.",
"The left-hand side of (REF ) becomes $\\tfrac{1-r^2}{(r^{\\prime })^2}\\partial _s^2g + \\bigl \\lbrace -\\tfrac{(1-r^2)r^{\\prime \\prime }}{(r^{\\prime })^3}+\\tfrac{d-1}{rr^{\\prime }}-\\tfrac{dr}{r^{\\prime }}\\bigr \\rbrace \\partial _s g - \\tfrac{1}{r^2}{\\Delta } g,$ where $r=r(s)$ .",
"If we choose $r(s) = 1-\\tfrac{1}{4}(1-s)^2$ (as in [19]), then the expression above becomes $(1+r)\\partial _s^2 g + (1-r)^{-\\frac{1}{2}}\\bigl \\lbrace \\tfrac{d-1}{r}-(d-\\tfrac{1}{2})r+\\tfrac{1}{2}\\rbrace \\partial _s g-\\tfrac{1}{r^2}{\\Delta }g.$ Furthermore, the domain $\\lbrace 1-\\delta <r<1\\rbrace $ corresponds to $\\Omega :=\\lbrace 1-\\delta ^{\\prime }<s<1\\rbrace $ , where $\\delta ^{\\prime }=2\\sqrt{\\delta }$ .",
"In particular, the PDE for $g$ , namely $(1+r)\\partial _s^2 g +(1-r)^{-\\frac{1}{2}}\\lbrace \\tfrac{d-1}{2}-(d-\\tfrac{1}{2})r+\\tfrac{1}{2}\\rbrace \\partial _s g - \\tfrac{1}{r^2}{\\Delta } g = \\mathcal {N}(g),$ where $\\mathcal {N}(g)=\\tfrac{d(d-2)}{4}g+ar^{-2}g+\\mu |g|^{\\frac{4}{d-2}}g,$ is nondegenerate and hence amenable to unique continuation results.",
"From this point on, the strategy is as follows: (i) Collect bounds on $g$ that show, in particular, that it is a standard solution to (REF ).",
"(ii) Define $\\tilde{g}(s,\\omega )=\\chi (s)g(s,\\omega )$ , where $\\chi $ is the characteristic function of $(0,1)$ , and show that $\\tilde{g}$ is a weak solution to (REF ) on $\\lbrace 1-\\delta ^{\\prime }<s<2\\rbrace $ .",
"With (i) and (ii) in place, we can (as in [19]) invoke unique continuation (cf.",
"[1]) to deduce that $g\\equiv 0$ on $\\Omega $ , and hence complete the proof.",
"(i) First, a change of variables in the (REF ) yields $\\int _{\\Omega } |g|^{\\frac{2d}{d-2}} + |{\\nabla }g|^2 \\lesssim 1.$ Similarly, $\\int _{\\Omega } \\frac{|\\partial _s g|^2}{1-s}\\lesssim \\int |\\partial _r f|^2 \\lesssim 1\\quad \\text{and}\\quad \\int _{\\Omega } \\frac{|g|^2}{(1-s)^3}\\lesssim \\int \\frac{|f|^2}{(1-r)^2} \\lesssim 1,$ where we use (REF ) for the last bound.",
"(ii) We turn to (ii) define $\\tilde{g}$ as above.",
"In order to write down the weak formulation of (REF ), it is useful to rewrite (REF ) as $\\partial _s(1+r)^{\\frac{1}{2}}\\partial _s g +\\tfrac{d-1}{r}(1-r)^{\\frac{1}{2}}\\partial _sg + (1+r)^{-\\frac{1}{2}}r^{-2}{\\Delta }g=(1+r)^{-\\frac{1}{2}}\\mathcal {N}(g).$ Now recall that $g$ solves (REF ) on $\\Omega $ .",
"Thus (letting $\\phi $ be a test function and integrating by parts), we find that to prove that $\\tilde{g}$ is a weak solution reduces to proving $\\int \\partial _s\\phi (1+r)^{\\frac{1}{2}}\\partial _s(\\chi g)\\,ds\\,d\\omega = \\int \\partial _s g(1+r)^{\\frac{1}{2}}\\partial _s(\\chi \\phi )\\,ds\\,d\\omega $ and $\\int g\\chi \\partial _s\\lbrace \\tfrac{d-1}{r}(1-r)^{\\frac{1}{2}}\\phi \\rbrace \\,ds\\,d\\omega = -\\int \\lbrace \\tfrac{d-1}{r}(1-r^2)^{\\frac{1}{2}}\\partial _s g\\rbrace \\chi \\phi \\,ds\\,d\\omega .$ Letting $\\chi _\\varepsilon $ be smooth approximations to $\\chi $ , the problem therefore reduces to proving $&\\lim _{\\varepsilon \\rightarrow 0} \\int \\lbrace |g\\partial _s\\phi |+|\\partial _sg \\phi |\\rbrace (1+r)^{\\frac{1}{2}}\\partial _s\\chi _\\varepsilon \\,ds\\,d\\omega = 0, \\\\&\\lim _{\\varepsilon \\rightarrow 0}\\int g\\lbrace \\tfrac{d-1}{r}(1-r)^{\\frac{1}{2}}\\phi \\rbrace \\partial _s\\chi _\\varepsilon \\,ds\\,d\\omega = 0.$ The bounds established in (i) are well-suited for proving (REF ) and ().",
"Consider for example, the second term in (REF ).",
"Assuming $\\partial _s\\chi _\\varepsilon $ is of size $\\varepsilon ^{-1}$ supported in an interval $I_\\varepsilon =(1-2\\varepsilon ,1-\\varepsilon )$ , we get the bound $\\varepsilon ^{-1}\\int _{s\\in I_\\varepsilon } |\\phi |\\,|\\partial _s g|\\,ds\\,d\\omega \\lesssim \\int _{s\\in I_\\varepsilon }\\tfrac{|\\partial _s g|}{|1-s|}\\,ds \\lesssim \\biggl (\\int _{s\\in I_\\varepsilon }\\frac{|\\partial _s g|^2}{|1-s|}\\biggr )^{\\frac{1}{2}}[\\log (\\tfrac{1-\\varepsilon }{1-2\\varepsilon })]^{\\frac{1}{2}},$ which tends to zero as $\\varepsilon \\rightarrow 0$ .",
"As the other terms can be treated similarly, this completes the proof."
],
[
"Proof of Theorem ",
"In this section we give the proof of Theorem REF , which contains two statements: (i) a blowup result below the ground state energy, and (ii) the failure of uniform space-time bounds as one approaches the ground state threshold in the case $a>0$ .",
"The proof of (i) is similar to the blowup result appearing in the work of [19], while the proof of (ii) follows a similar strategy as [23], [29].",
"Consequently, our presentation will be rather brief.",
"Suppose $(u_0,u_1)\\in \\dot{H}^1\\times L^2$ satisfies $E_a[(u_0,u_1)]<E_{a\\wedge 0}[W_{a\\wedge 0}]\\quad \\text{and}\\quad \\Vert u_0\\Vert _{\\dot{H}_a^1} > \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}$ and $u$ is the corresponding solution to (REF ).",
"We will show that $u$ blows up in finite time.",
"To this end we introduce the function $y_R(t) := \\int |u(t,x)|^2 \\varphi (\\tfrac{x}{R})\\,dx,$ where $R>0$ and $\\varphi $ is a smooth function satisfying $\\varphi \\equiv 1$ for $|x|\\le 1$ , $\\varphi \\equiv 0$ for $|x|>2$ and $0\\le \\varphi \\le 1$ for $1<|x|<2$ .",
"Direct computation using (REF ) yields $y_R^{\\prime }(t) = 2\\int u\\partial _t u \\varphi (\\tfrac{x}{R})\\,dx$ and $y_R^{\\prime \\prime }(t) = 2\\int |\\partial _t u|^2 - |\\nabla u|^2 - \\tfrac{a}{|x|^2}|u|^2 + |u|^{\\frac{2d}{d-2}}\\,dx + r(R),$ where $r(R)& :=2\\int [1-\\varphi (\\tfrac{x}{R})]\\cdot [|\\partial _t u|^2-|\\nabla u|^2 - \\tfrac{a}{|x|^2}|u|^2 + |u|^{\\frac{2d}{d-2}}] \\,dx \\\\& \\quad - \\tfrac{2}{R}\\int u\\nabla u \\cdot [\\nabla \\varphi ](\\tfrac{x}{R})\\,dx.$ Now, using $E_{a}[\\vec{u}] < E_{a\\wedge 0}[W_{a\\wedge 0}]$ , we can show that $2\\int |u|^{\\frac{2d}{d-2}}\\,dx \\ge \\tfrac{2d}{d-2}\\int \\bigl [|\\partial _t u|^2 + |\\nabla u|^2+ \\tfrac{a}{|x|^2}|u|^2\\bigr ]\\,dx - \\tfrac{4}{d-2}\\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}^2 + \\delta $ for some $\\delta >0$ .",
"Combining this with (REF ), we get $y_R^{\\prime \\prime }(t)& \\ge \\tfrac{4(d-1)}{d-2}\\int |\\partial _t u|^2\\,dx + \\tfrac{4}{d-2}\\bigl (\\Vert u\\Vert _{\\dot{H}_a^1}^2 - \\Vert W_{a\\wedge 0}\\Vert _{\\dot{H}_{a\\wedge 0}^1}^2\\bigr ) + \\delta + r(R) \\\\& \\ge \\tfrac{4(d-1)}{d-2}\\int |\\partial _t u|^2 \\,dx + \\delta + r(R).$ Now observe that if we additionally assume $u_0\\in L^2$ , then the formulas above make sense even as $R\\rightarrow \\infty $ (in which case $r(R)$ becomes identically zero).",
"In this case an application of Cauchy–Schwarz leads to the lower bound $y(t)y^{\\prime \\prime }(t)\\ge \\tfrac{d-1}{d-2}[y^{\\prime }(t)]^2,$ from which an ODE argument yields finite time blowup.",
"In the general case, we need to estimate the term $r(R)$ .",
"To this end note that by finite speed of propagation, for any $\\varepsilon >0$ we may find $M=M(\\varepsilon )$ such that $\\int _{|x|>M+t} |\\partial _t u|^2 + |\\nabla u|^2 + \\tfrac{|a|}{|x|^2}|u|^2\\,dx <\\varepsilon .$ Choosing $\\varepsilon \\ll \\tfrac{1}{2}\\delta $ and $R>2M$ , we deduce $|r(R)| \\ll \\tfrac{1}{2}\\delta \\quad \\text{uniformly for}\\quad t\\in (0,\\tfrac{1}{2}R).$ Thus we have $y_R^{\\prime \\prime }(t)\\ge \\tfrac{4(d-1)}{d-2}\\int |\\partial _t u|^2\\varphi (\\tfrac{x}{R})\\,dx\\quad \\text{for}\\quad t\\in (0,\\tfrac{1}{2}R),$ which in particular yields an estimate like (REF ) for $y_R$ on the interval $(0,\\tfrac{1}{2}R)$ .",
"In particular, a similar ODE type argument (with a careful choice of parameters) once again yields finite-time blowup.",
"As the complete details appear in [19] (cf.",
"Theorem 3.7 and Theorem 7.1(ii) therein) and apply equally well in our case, we omit the details here.",
"Finally, we turn to the proof of Theorem REF (ii).",
"Recall we are in the setting of $a>0$ and $\\mu =-1$ (the focusing case).",
"We define $\\phi _n(x):=(1-\\varepsilon _n)W_0(x-x_n),$ where $\\varepsilon _n\\rightarrow 0$ and $|x_n|\\rightarrow \\infty .$ We can show that $E_a[\\phi _n]\\nearrow E_0[W_0],\\quad \\text{and}\\quad \\Vert \\phi _n\\Vert _{\\dot{H}^1_a}\\nearrow \\Vert W_0\\Vert _{\\dot{H}^1},$ and hence (by Theorem REF ) there exist global scattering solutions $u_n$ to (REF ) with data $(\\phi _n,0)$ .",
"Our goal is to show that the Strichartz norm of these solutions diverges as $n\\rightarrow \\infty $ .",
"To this end, we define $\\tilde{u}_n(t,x)=(1-\\varepsilon _n)[\\chi _nW_0](x-x_n),$ where $\\chi _n$ is as in Proposition REF , that is, a smooth function satisfying $\\chi _n(x)={\\left\\lbrace \\begin{array}{ll}0, & \\mbox{if } |x+x_n|\\le \\frac{1}{4}|x_n| \\\\1, & \\mbox{if } |x+x_n|\\ge \\frac{1}{2}|x_n|\\end{array}\\right.",
"}\\quad \\text{with}\\quad \\sup _x\\big |\\partial ^\\alpha \\chi _n(x)|\\lesssim |x_n|^{-|\\alpha |}$ for all multi-indices $\\alpha .$ One can verify that $\\Vert \\tilde{u}_n(0)-u_n(0)\\Vert _{\\dot{H}^1}=\\big \\Vert \\big [(1-\\epsilon _n)\\chi _n-1\\big ]W_0\\big \\Vert _{\\dot{H}^1}\\rightarrow 0$ as $n\\rightarrow \\infty $ , with $\\Vert \\tilde{u}_n\\Vert _{L_{t,x}^\\frac{2(d+1)}{d-2}([-T,T]\\times \\mathbb {R}^d)}\\gtrsim T^\\frac{d-2}{2(d+1)}\\quad \\text{for}\\quad T>0.$ Using the equation $-\\Delta W_0=|W_0|^\\frac{4}{d-2}W_0,$ we find $\\nonumber e_n&:=(\\partial _t^2+\\mathcal {L}_a)\\tilde{u}_n-|\\tilde{u}_n|^\\frac{4}{d-2}\\tilde{u}_n\\\\&= \\big [(1-\\epsilon _n)\\chi _n(x-x_n)-(1-\\epsilon _n)^\\frac{d+2}{d-2} \\chi _n(x-x_n)^\\frac{d+2}{d-2}\\big ]|W_0|^\\frac{4}{d-2}W_0 \\\\&\\quad +(1-\\epsilon )\\big [W_0\\Delta \\chi _n+2\\nabla \\chi _n\\cdot \\nabla W_0\\big ](x-x_n)\\\\&\\quad -\\tfrac{a}{|x|^2}(1-\\epsilon _n)[\\chi _nW_0](x-x_n).$ We now claim that for any fixed $T>0$ , $\\Vert e_n\\Vert _{L_t^1L_x^2([-T,T]\\times \\mathbb {R}^d)}\\rightarrow 0\\quad \\text{as}\\quad n\\rightarrow \\infty .$ We begin with the estimate of (REF ).",
"As $W_0\\in L^p$ for any $p>\\tfrac{d}{d-2}$ , we have $\\Vert &(\\ref {equ:erren1})\\Vert _{L_t^1L_x^2([-T,T]\\times \\mathbb {R}^d)}\\\\& \\lesssim T\\big \\Vert \\big [(1-\\epsilon _n)\\chi _n(x-x_n)-(1-\\epsilon _n)^\\frac{d+2}{d-2} \\chi _n(x-x_n)^\\frac{d+2}{d-2}\\big ]|W_0|^\\frac{4}{d-2}W_0\\big \\Vert _{L_t^\\infty L_x^2} \\\\& \\lesssim T\\big \\Vert \\big [(1-\\epsilon _n)\\chi _n(x-x_n)-(1-\\epsilon _n)^\\frac{d+2}{d-2}\\chi _n(x-x_n)^\\frac{d+2}{d-2}\\big ]W_0\\big \\Vert _{L_t^\\infty L_x^{2d}} \\\\& \\quad \\times \\Vert W_0\\Vert _{L_x^\\frac{8d}{(d-1)(d-2)}}^\\frac{4}{d-2}\\\\&\\rightarrow 0\\quad \\text{as}\\quad n\\rightarrow \\infty .$ Next, we estimate of ().",
"We have $\\Vert (\\ref {equ:erren2})\\Vert _{L_t^1L_x^2([-T,T]\\times \\mathbb {R}^d)}&\\lesssim T \\bigl (\\Vert \\Delta \\chi _n\\Vert _{L_x^{\\frac{2d}{4-d}-}}\\Vert W_0\\Vert _{L_x^{\\frac{d}{d-2}+}}+\\Vert \\nabla \\chi _n\\Vert _{L_x^\\infty }\\Vert W_0\\Vert _{\\dot{H}^1}\\bigr )\\\\&\\lesssim T(|x_n|^{-\\frac{d}{2}+}+|x_n|^{-1})\\rightarrow 0\\quad \\text{as}\\quad n\\rightarrow \\infty .$ Finally, we have $\\Vert (\\ref {equ:erren3})\\Vert _{L_t^1L_x^2([-T,T]\\times \\mathbb {R}^d)}&\\lesssim T \\big \\Vert \\tfrac{\\chi _n}{|\\cdot +x_n|^2}\\big \\Vert _{L_x^{\\frac{2d}{4-d}-}}\\Vert W_0\\Vert _{L_x^{\\frac{d}{d-2}+}}\\\\& \\lesssim T|x_n|^{-\\frac{d}{2}+}\\rightarrow 0\\quad \\text{as}\\quad n\\rightarrow \\infty .$ Applying the stability result (Proposition REF ), we deduce $\\Vert u_n\\Vert _{L_{t,x}^\\frac{2(d+1)}{d-2}([-T,T]\\times \\mathbb {R}^d)}\\gtrsim T.$ As $T>0$ was arbitrary, this implies the result."
]
] | 1808.08571 |
[
[
"Commensurability growth of branch groups"
],
[
"Abstract Fixing a subgroup $\\Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\\Delta$ of $G$ with $[\\Gamma: \\Gamma \\cap \\Delta][\\Delta : \\Gamma \\cap \\Delta] = n$.",
"For pairs $\\Gamma \\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $\\Gamma$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals.",
"For almost all known branch groups $\\Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.)",
"acting on $p$-regular trees, this function is precisely $\\aleph_0$ for any $n = p^k$."
],
[
"Introduction",
"Two subgroups $\\Delta _1$ and $\\Delta _2$ of a group $G$ are commensurable if their commensurability index $\\operatorname{c}(\\Delta _1, \\Delta _2) := [\\Delta _1 : \\Delta _1 \\cap \\Delta _2][\\Delta _2 : \\Delta _1 \\cap \\Delta _2]$ is finite.",
"For a pair of groups $\\Gamma \\le G$ , the commensurability growth function $\\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ assigns to each $n \\in \\mathbb {N}$ the cardinality ${\\bf c}_n(\\Gamma ,G) := | \\lbrace \\Delta \\le G : \\operatorname{c}\\left(\\Gamma , \\Delta \\right) = n \\rbrace |.$ This function was first systematically studied in [2], where it was used to give regularity results on the structure of arithmetic lattices in a unipotent algebraic group.",
"Here, we continue this study to pairs of groups naturally arising from the class of finitely generated residually finite groups.",
"This extends the study of commensurability growth beyond the class of linear groups.",
"Associated to any residually finite group $\\Gamma $ are many rooted finite-valent trees $T$ where $\\Gamma \\le \\operatorname{Aut}(T)$ , the automorphism group of $T$ .",
"Such pairs are particularly beautiful and useful when the rooted tree is $d$ -regular, denoted $T_d$ , and the subgroup is branch.",
"For instance, the first Grigorchuk group, a branch subgroup of $\\operatorname{Aut}(T_2)$ , has intermediate growth [8], is commensurable with its direct product, and is a counter-example to the Burnside Problem [6].",
"To what extent does the sequence $\\lbrace {\\bf c}_n(\\Gamma , \\operatorname{Aut}(T_d))\\rbrace _{n=1}^\\infty $ distinguish branch subgroups of $\\operatorname{Aut}(T_d)$ among the collection of subgroups of $\\operatorname{Aut}(T_d)$ ?",
"A simple example of a non-branch subgroup of $\\operatorname{Aut}(T_d)$ is the embedding of $\\mathbb {Z}$ into $\\operatorname{Aut}(T_2)$ known as the binary adding machine.",
"See § for the definition.",
"Proposition A Let $\\mathcal {A}$ be the binary adding machine subgroup of $\\operatorname{Aut}(T_2)$ .",
"Then for every natural number $k$ , ${\\bf c}_{2^k}(\\mathcal {A}, \\operatorname{Aut}(T_2)) = \\aleph _1$ .",
"On the other hand, there exists an infinite dihedral group $H \\le \\operatorname{Aut}(T_2)$ , containing $\\mathcal {A}$ as a subgroup of index two such that ${\\bf c}_2(H, \\operatorname{Aut}(T_2)) = 3$ .",
"Our proof of Proposition REF , given in §, uses results from [3].",
"For every natural number $n$ , the group $\\mathcal {A}$ acts transitively on vertices of distance $n$ from the root in $T_2$ .",
"Thus, while $\\mathcal {A}$ in some sense fills up $\\operatorname{Aut}(T_2)$ , there are many subgroups of finite commensurability index with $\\mathcal {A}$ .",
"In contrast to this behavior, our main result shows that most well-studied examples of branch groups sitting inside the Sylow pro-$p$ subgroup of $\\operatorname{Aut}(T_p)$ , where $p$ is a prime, have the same commensurability growth values.",
"These examples include the first Grigorchuk group, the twisted twin of the Grigorchuk group, the Pervova groups, the Gupta-Sidki $p$ -groups, the Fabrykowski-Gupta group and an infinite family of generalizations of the Fabrykowski-Gupta group, and GGS groups with non-constant accompanying vector.",
"Important to our proof is that all these examples satisfy the rigid congruence subgroup property, a weakening of the usual congruence subgroup property.",
"See § for definitions of these groups and their properties.",
"Theorem B Let $\\Gamma $ be a finitely generated, self-similar, regular branch group over a branching subgroup $K$ in $\\operatorname{Aut}(T_p)$ .",
"Suppose $\\Gamma $ is contained in the Sylow pro-$p$ subgroup of $\\operatorname{Aut}(T_p)$ and satisfies the rigid congruence subgroup property.",
"Then ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p)) = \\aleph _0$ for all $k$ .",
"Röver's theorem [15] on abstract commensurators of branch groups is key to the proof of the upper bound ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p)) \\le \\aleph _0$ (see §).",
"Commensurability growth is a generalization of subgroup growth to pairs of groups [13].",
"While the subgroup growth function of a finitely generated group is always finite, this paper gives the first naturally occurring pairs where the commensurability growth function has infinite values.",
"We are grateful to Benson Farb, Andrew Putman, Benjamin Steinberg, and Slobodan Tanushevski for their conversations and support.",
"We thank Benson Farb for helpful comments on an earlier draft."
],
[
"Preliminaries",
"The groups we shall consider will all be subgroups of the group $\\operatorname{Aut}(T_d)$ of automorphisms of a $d$ -regular rooted tree $T_d$ .",
"We will always consider our trees $T_d$ to arise from the following construction.",
"Let $X$ be a finite alphabet with $|X|=d\\ge 2$ and given a fixed total ordering.",
"The vertex set of the tree $T_X$ is the set of finite sequences over $X$ ; two sequences are connected by an edge when one can be obtained from the other by right-adjunction of a letter in $X$ .",
"The root is the empty sequence $\\emptyset $ , and the children of $v$ are all $v x$ for $x\\in X$ .",
"The length of a sequence $v$ is denoted by $|v|$ .",
"The set $X^n\\subset T_X$ , of all sequences of length $n$ , is called the $n$ th level of the tree $T_X$ ."
],
[
"Automorphisms",
"Let $g$ be an automorphism of the rooted tree $T_X$ .",
"For a vertex $v\\in T_X$ , consider the rooted subtrees $vT_X=\\lbrace vw\\mid w\\in T_X\\rbrace $ and $g(v)T_X=\\lbrace g(v)w\\mid w\\in T_X\\rbrace $ with roots $v$ and $g(v)$ respectively.",
"Notice that the map $vT_X\\rightarrow g(v)T_X$ , given by $vw \\mapsto g(v)w,$ is a morphism of rooted trees.",
"Moreover, the subtrees $vT_X$ and $g(v)T_X$ are naturally isomorphic to $T_X$ .",
"Identifying $vT_X$ and $g(v)T_X$ with $T_X$ we get an automorphism $g|_v\\colon T_X\\rightarrow T_X$ uniquely defined by the condition $g(vw)=g(v)g|_v(w)$ for all $w\\in T_X$ .",
"We call the automorphism $g|_v$ the section of $g$ at $v$ .",
"Observe the following obvious properties of the sections: $g|_{v_1 v_2}&=g|_{v_1}|_{v_2}\\\\(g_1\\cdot g_2)|_v&=g_1|_{g_2(v)}\\cdot g_2|_v.$ It follows that the action of the automorphism $g \\in \\operatorname{Aut}(T_X)$ can be written as $g = (g_1,\\dots , g_{|X|}) \\pi _g$ , where $\\pi _g \\in \\operatorname{Sym}(X)$ is the permutation defined by the action of $g$ on the first level of the tree, and $g_1,\\dots , g_{|X|} \\in \\operatorname{Aut}(T_X)$ are the sections of $g$ at the vertices of the first level of $T_X$ .",
"This gives an isomorphism $\\operatorname{Aut}(T_X)\\cong \\operatorname{Aut}(T_X)\\wr \\operatorname{Sym}(X)$ .",
"For $H\\le \\operatorname{Aut}(T_X)$ , we write $X\\star H$ to indicate the group of automorphisms $g$ with $\\pi _g=1$ and $g_i\\in H$ for all $1\\le i \\le |X|$ .",
"Similarly, let $X^n\\star H$ indicate the group of elements $g\\in \\operatorname{Aut}(T_X)$ with $\\pi _{g|_v}=1$ for all $v$ on level less than $n$ and $g|_u \\in H$ for all $u$ on the $n$ th level."
],
[
"Self-similar and branch groups",
"A subgroup $G$ of $\\operatorname{Aut}(T_X)$ is self-similar if for every $g\\in G$ and every $v\\in T_X$ the section $g|_v \\in G$ .",
"For example, the full automorphism group $\\operatorname{Aut}(T_X)$ is itself self-similar.",
"Let $G\\le \\operatorname{Aut}(T_X)$ be a group of automorphisms of the rooted tree $T_X$ .",
"For a vertex $v\\in T_X$ the vertex stabilizer is the subgroup consisting of the automorphisms that fix the sequence $v$ : $\\operatorname{stab}_G(v)=\\lbrace g\\in G\\mid g(v)=v\\rbrace .$ The $n$ th level stabilizer (also called the $n$ th principal congruence subgroup) is the subgroup $\\operatorname{stab}_G(n)$ consisting of the automorphisms that fix all vertices of the $n$ th level: $\\operatorname{stab}_G(n)=\\cap _{v\\in X^n} \\operatorname{stab}_G(v).$ Stabilizer subgroups $\\operatorname{stab}_G(n)$ with $n\\ge 0$ are normal in $G$ .",
"Notice that any $g\\in \\operatorname{stab}_G(n)$ can be identified in a natural way with the sequence of sections at vertices in $X^n$ $(g_1, \\dots , g_{|X|^n})$ taken in the lexicographical ordering on $X^n$ .",
"We say that $g$ is of level $n$ if $g\\in \\operatorname{stab}_G(n)\\setminus \\operatorname{stab}_G(n+1)$ .",
"The rigid stabilizer $\\operatorname{rist}_G(v)$ of a vertex $v\\in T_X$ is the subgroup of $G$ of all automorphisms acting non-trivially only on the vertices of the form $vu$ with $u\\in T_X$ : $\\operatorname{rist}_G(v)=\\lbrace g\\in G\\mid g(w)=w \\text{ for all } w\\notin vT_X\\rbrace $ The $n$ th level rigid stabilizer $\\operatorname{rist}_G(n)=\\langle \\operatorname{rist}_G(v)\\mid v\\in X^n\\rangle $ is the subgroup generated by the union of the rigid stabilizers of the vertices of the $n$ th level.",
"We say that a subgroup $G \\le \\operatorname{Aut}(T_X)$ is level-transitive if $G$ acts transitively on each level of $T_X$ .",
"An automorphism $g$ is level transitive if $\\langle g \\rangle $ is level-transitive.",
"A level-transitive subgroup $G\\le Aut(T_X)$ is branch if $\\operatorname{rist}_G(n)$ is of finite index in $G$ for all $n \\ge 1$ .",
"In this article we will restrict ourselves to the particularly important type of branch groups introduced by the following definition.",
"Definition 1.1 A level-transitive group $G\\le \\operatorname{Aut}(T_X)$ is regular branch if there exists a finite-index subgroup $K$ of $G$ such that $K$ contains $X\\star K$ of finite index.",
"In this case, $K$ is called a branching subgroup for $G$ .",
"Call $G$ layered if $G$ itself is a branching subgroup for $G$ .",
"A subgroup $G$ of $\\operatorname{Aut}(T_X)$ is said to satisfy the congruence subgroup property if any finite index subgroup $H$ of $G$ contains a principal congruence subgroup $\\operatorname{stab}_G(n)$ for some $n\\ge 1$ .",
"Definition 1.2 A subgroup $G$ of $\\operatorname{Aut}(T_X)$ has the rigid congruence subgroup property if every level rigid stabilizer of $G$ contains a level stabilizer of $G$ ."
],
[
"The Sylow pro-$p$ subgroup",
"$\\operatorname{Aut}(T_d)$ is a profinite group; it is canonically isomorphic to $\\underset{n \\ge 1}{\\varprojlim } \\operatorname{Aut}(T_d(n))$ where $T_d(n)$ is the finite subtree of $T_d$ consisting of vertices of level less than or equal to $n$ .",
"In the case that $d = p$ for a prime $p$ , fix a cyclic permutation $\\sigma \\in \\operatorname{Sym}(X)$ of order $p$ .",
"The Sylow pro-$p$ subgroup $\\operatorname{Aut}_p(T_p) \\le \\operatorname{Aut}(T_p)$ consists of automorphisms $g \\in \\operatorname{Aut}(T_p)$ such that at every vertex $v\\in X^*$ the section $g|_v$ acts on $X$ as $\\sigma ^i$ for some $0\\le i \\le p-1$ (see [9] pages 133-134).",
"For a self-similar group $G \\le \\operatorname{Aut}_p(T_p)$ we have the containment $G\\le G\\wr \\langle \\sigma \\rangle $ under the isomorphism $\\operatorname{Aut}_p(T_p) \\cong \\operatorname{Aut}_p(T_p)\\wr \\left\\langle \\sigma \\right\\rangle $ .",
"If $G$ is layered, there is an inclusion $X \\star G \\le G$ .",
"Since a layered group is level-transitive, it follows that a self-similar and layered subgroup $G\\le \\operatorname{Aut}_p(T_p)$ satisfies $G\\cong G\\wr \\left\\langle \\sigma \\right\\rangle $ ."
],
[
"Examples",
"The following examples are self-similar regular branch groups with the rigid congruence subgroup property.",
"The First Grigorchuk group: Let $X=\\lbrace 1,2\\rbrace $ .",
"Define automorphisms of $T_X$ inductively by $a = \\sigma ,\\; b = (a,c),\\; c = (a,d), \\text{ and } d = (1,b),$ where $\\sigma $ is the transposition $(1,2)\\in Sym(X)$ .",
"The first Grigorchuk group is $\\Gamma :=\\langle a, b, c, d \\rangle $ .",
"Clearly, $\\Gamma $ is self-similar.",
"Moreover, it is regular branch [8] over the subgroup $ K=\\langle (ab)^2, (bada)^2, (abad)^2\\rangle .$ It also has the congruence subgroup property [9].",
"The Twisted Twin: Let $X=\\lbrace 1,2\\rbrace $ .",
"Define automorphisms of $T_X$ inductively by $a=\\sigma , \\; \\beta =(\\gamma , a), \\; \\gamma =(a, \\delta ), \\text{ and } \\delta =(1, \\beta ).$ The Twisted Twin of the Grigorchuk group is $G := \\langle a, \\beta , \\gamma , \\delta \\rangle $ .",
"It is a self-similar regular branch group [4] with branching subgroup $K=\\langle \\langle [a,\\beta ], [\\beta , \\gamma ], [\\beta , \\delta ], [\\gamma , \\delta ], \\beta \\delta \\gamma \\rangle \\rangle ^G.$ It does not have the congruence subgroup property but does have the rigid congruence subgroup property [5].",
"Gupta-Sidki groups: Let $X=\\lbrace 1,\\dots , p\\rbrace $ where $p$ is odd prime.",
"Define automorphisms $x$ and $y$ of $T_X$ inductively by: $x= \\sigma \\text{ and } y =(x,x^{-1}, 1,\\dots , 1,y ),$ where $\\sigma $ is the cyclic permutation $(1,2,\\dots p)$ on $X$ .",
"The Gupta-Sidki $p$ -group is $G_p := \\langle x, y \\rangle $ .",
"Clearly, $G_p$ is self-similar.",
"It is regular branch over its commutator subgroup [10], [11].",
"Moreover, $G_p$ has the congruence subgroup property (see [12]).",
"Gupta-Sidki variations: There are various modifications of the Gupta-Sidki group which are self-similar, regular branch groups having the congruence subgroup property.",
"Here is an example of such a modification.",
"Let $G$ be the subgroup of automorphisms on the rooted $p$ -regular tree for $p\\ge 7$ generated by $x=(1,2,\\dots , p)$ and $y=(x^{i_1}, x^{i_2}, \\dots , x^{i_p-3},1,1,1,y)$ for $0\\le i_j\\le p-1$ and $i_1\\ne 0$ .",
"The group $G$ is regular branch over its commutator subgroup (see [9]).",
"Fabrykowski-Gupta group: Let $X = \\lbrace 1, 2, 3\\rbrace $ .",
"Define automorphisms of $T_X$ inductively by $a=(1,2,3) \\text{ and } b=(a,1,b).$ The Fabrykowski-Gupta group is $\\mathcal {G} := \\langle a,b \\rangle $ .",
"Then $\\mathcal {G}$ is a regular branch group with the congruence subgroup property (see [1]).",
"A natural generalization of the Fabrykowski-Gupta group is a group $\\mathcal {G}_p$ generated by automorphisms $a=(1,2,\\dots , p)$ and $b=(a,1,\\dots , 1, b)$ of a $p$ -regular tree.",
"For every prime $p\\ge 5$ , the Fabrykowski-Gupta group $\\mathcal {G}_p$ is regular branch with the congruence subgroup property [9].",
"EGS groups: Let $X = \\lbrace 1, \\cdots , p\\rbrace $ and let $\\bar{\\iota }= (i_1, i_2, \\dots , i_{p-1})$ be a non-symmetric vector of integers between 0 and $p-1$ , so that $i_j \\ne i_{p-j}$ for some $j$ .",
"Define automorphisms of $T_X$ inductively by $a=\\sigma , \\; b=(a^{i_1}, a^{i_2}, \\dots , a^{i_{p-1}}, b), \\text{ and } c=(c, a^{i_1}, a^{i_2}, \\dots , a^{i_{p-1}})$ where $\\sigma $ is the permutation $(1,2, \\dots , p)$ .",
"The extended Gupta-Sidki (EGS) group is $\\Gamma _{\\bar{\\iota }} := \\langle a,b,c \\rangle $ .",
"Pervova constructed the EGS groups as the first examples of branch groups failing to have the congruence subgroup property [14].",
"It was shown in [5] that these groups nevertheless do satisfy the rigid congruence subgroup property.",
"These groups are clearly self-similar and moreover are regular branch groups having their commutator subgroup as a branching subgroup [14]."
],
[
"The adding machine: Proof of Proposition ",
"The binary adding machine $\\mathcal {A}\\le \\operatorname{Aut}(T_2)$ is the infinite cyclic subgroup generated by $\\tau := (1, \\tau )\\sigma $ .",
"Proposition 2.1 For every natural number $k$ , ${\\bf c}_{2^k}(\\mathcal {A}, \\operatorname{Aut}(T_2)) = \\aleph _1$ .",
"Note that it suffices to show that ${\\bf c}_2(\\mathcal {A}, \\operatorname{Aut}(T_2)) \\ge \\aleph _1$ , since the cardinal of the collection of all finitely generated subgroups of $\\operatorname{Aut}(T_2)$ is $\\aleph _1$ .",
"In Theorem 4.13 from [3], it is shown that $\\langle \\tau \\rangle $ is normalized by elements of the form $\\tau ^x u_y$ where $y$ is an odd integer, $x$ is a 2-adic integer, and $u_y := (u_y, u_y \\tau ^{(y-1)/2}).$ Notice that for any fixed $x\\in \\mathbb {Z}_2$ , the element $\\tau ^x u_{-1}$ has order two and normalizes $\\mathcal {A}$ , and hence $\\langle \\tau ^x u_{-1}, \\mathcal {A}\\rangle $ contains $\\mathcal {A}$ as a subgroup of index two.",
"Moreover, since $\\tau ^x u_{-1}$ has order two and $u_{-1} \\tau u_{-1} = \\tau ^{-1}$ , we have set equalities $\\langle \\tau ^x u_{-1}, \\mathcal {A}\\rangle = \\langle \\tau ^x u_{-1}, \\tau \\rangle =\\lbrace \\tau ^{x+n} u_{-1} : n \\in \\mathbb {Z}\\rbrace \\sqcup \\mathcal {A}.$ Since canonically, $\\lbrace \\tau ^x : x \\in \\mathbb {Z}_2\\rbrace \\cong \\mathbb {Z}_2$ , it follows that the cardinality of all such sets $\\langle \\tau ^x u_{-1}, \\mathcal {A}\\rangle $ as $x$ varies over $\\mathbb {Z}_2^*$ is equal to the cardinality of $\\mathbb {Z}_2^*/\\mathbb {Z}$ , which is $\\aleph _1$ , as desired.",
"Now, any two level-transitive automorphisms in $\\operatorname{Aut}(T_d)$ are conjugate in $\\operatorname{Aut}(T_d)$ (see [7] Corollary 4.1).",
"Since $\\mathcal {A}$ is clearly level-transitive, we get the following immediate corollary.",
"Corollary 2.2 Let $g\\in \\operatorname{Aut}(T_2)$ be a level-transitive automorphism.",
"Then for every natural number $k$ , ${\\bf c}_{2^k}(\\langle g \\rangle , \\operatorname{Aut}(T_2))=\\aleph _1$ .",
"We now prove the second half of Proposition REF .",
"Fix the element $\\delta = (\\delta , \\delta ) \\sigma $ and set $H = \\langle \\delta , \\tau \\rangle $ .",
"We use the following theorem from [3].",
"Theorem 2.3 (Theorem 4.12 [3]) The group $H$ is infinite dihedral.",
"Moreover, $H$ is its own normalizer in $\\operatorname{Aut}(T_2)$ .",
"Proposition 2.4 The group $H$ satsifies ${\\bf c}_2(H, \\operatorname{Aut}(T_2)) = 3$ .",
"Notice that if $H_0 \\le \\operatorname{Aut}(T_2)$ contains $H$ with $[H_0 : H] = 2$ , then $H \\lhd H_0$ and so $H_0$ is contained in the normalizer of $H$ .",
"Hence, there does not exist a supergroup $H_0$ containing $H$ as a subgroup of index two.",
"Moreover, since $H$ is infinite dihedral, there are only three subgroups of $H$ of index two, and so ${\\bf c}_2(H, \\operatorname{Aut}(T_2) = 3$ as desired."
],
[
"Branch groups: Proof of Theorem ",
"Lemma 3.1 Let $\\Gamma \\le Aut_p(T_p)$ be self-similar and finitely generated.",
"Then $\\Gamma $ is not layered.",
"Suppose that $\\Gamma $ is layered.",
"We will show that it can not be finitely generated.",
"Indeed, let $C_p$ be a cyclic group of order $p$ and for each $i$ define a homomorphism $\\psi _i:\\Gamma \\rightarrow C_p$ by $\\prod _{|v|=i} \\pi _{g|_v}$ .",
"Now for each $n$ let $\\Psi _n = \\prod _{i=0}^n \\psi _i: \\Gamma \\rightarrow \\bigoplus _{i=0}^n C_p$ .",
"Since $\\Gamma $ is layered, the remarks of §REF give an isomorphism $\\Gamma \\cong \\Gamma \\wr \\langle \\sigma \\rangle $ under the isomorphism $\\operatorname{Aut}_p(T_p)\\cong \\operatorname{Aut}_p(T_p)\\wr \\left\\langle \\sigma \\right\\rangle $ .",
"This implies that $\\Psi _n$ is surjective for each $n$ .",
"Since the groups $\\bigoplus _{i=0}^n C_p$ require arbitrarily many generators as $n$ tends to infinity, the group $\\Gamma $ is not finitely generated.",
"We now discuss some consequences of the rigid congruence subgroup property.",
"For a regular branch group $\\Gamma $ with maximal branching subgroup $K$ , Corollary 1.6 in [5] says $K$ contains a level rigid stabilizer.",
"Consequently, if $\\Gamma $ has the rigid congruence subgroup property, $K$ also contains a level stabilizer.",
"In particular, there exists an $m$ with $\\operatorname{stab}_{\\Gamma }(m)\\le K$ .",
"Lemma 3.2 Let $\\Gamma $ be a self-similar regular branch group with maximal branching subgroup $K$ and with the rigid congruence subgroup property.",
"Then for all $n\\ge 0$ , $\\operatorname{stab}_\\Gamma (m+n)=X^n\\star \\operatorname{stab}_\\Gamma (m)$ where $m$ is such that $\\operatorname{stab}_\\Gamma (m)\\le K$ .",
"Let $\\Gamma $ be a self-similar regular branch group with the rigid congruence subgroup property and let $K$ be the maximal branching subgroup for $\\Gamma $ .",
"Since $\\Gamma $ is self-similar, $\\operatorname{stab}_\\Gamma (m+n)\\le X^n\\star \\operatorname{stab}_\\Gamma (m)$ for all $m$ and $n$ .",
"Now let $m$ be such that $\\operatorname{stab}_\\Gamma (m)\\le K$ .",
"As $K$ is a branching subgroup, for all $n\\ge 0$ , $X^n\\star K\\le K$ and so we get the following set of inclusions: $X^n\\star \\operatorname{stab}_{\\Gamma }(m)\\le X^n\\star K \\le K \\le \\Gamma $ and so $X^n\\star \\operatorname{stab}_{\\Gamma }(m)$ is contained in $\\Gamma $ and stabilizes level $(m+n)$ .",
"Thus we conclude $\\operatorname{stab}_\\Gamma (m+n)= X^n\\star \\operatorname{stab}_\\Gamma (m)$ as desired.",
"Let $Q_n=\\lbrace g\\in \\operatorname{Aut}_p(T_p) \\mid g|_v=1 \\text{ for all $v$ with } |v| \\ge n \\rbrace $ .",
"Observe that the group $(X^n\\star \\Gamma )\\cap Q_n=\\lbrace 1\\rbrace $ and so $\\langle X^n\\star \\Gamma , Q_n \\rangle = (X^n \\star \\Gamma )\\rtimes Q_n$ .",
"Lemma 3.3 Let $\\Gamma \\in \\operatorname{Aut}_p(T_p)$ be a self-similar regular branch group with maximal branching subgroup $K$ and the rigid congruence subgroup property.",
"Let $n\\ge 0$ .",
"Then for all sufficiently large $k$ , $\\operatorname{stab}_{\\Gamma }(k)=\\operatorname{stab}_{(X^n\\star \\Gamma )\\rtimes Q_n}(k)$ .",
"Note that it suffices to show equality for a fixed $k$ , as the stabilizer of the $k+1$ level is precisely the set of elements in the stabilizer of level $k$ which also stabilize level $k+1$ .",
"Let $k=n+m$ where $m$ is such that $\\operatorname{stab}_{\\Gamma }(m)\\le K$ .",
"As $\\Gamma $ is self-similar, $\\Gamma \\le (X^n\\star \\Gamma )\\rtimes Q_n$ and so similarly $\\operatorname{stab}_{\\Gamma }(k)\\le \\operatorname{stab}_{(X^n\\star \\Gamma )\\rtimes Q_n}(k)$ .",
"For the other inclusion, note that only the identity element in $Q_n$ stabilizes level $n+m$ and so $\\operatorname{stab}_{(X^n\\star \\Gamma )\\rtimes Q_n}(n+m)=\\operatorname{stab}_{(X^n\\star \\Gamma )}(n+m)$ .",
"Moreover, we have $\\operatorname{stab}_{(X^n\\star \\Gamma )}(n+m)=X^n\\star \\operatorname{stab}_{\\Gamma }(m)$ , which by Lemma REF is precisely equal to $\\operatorname{stab}_{\\Gamma }(n+m)$ .",
"We now establish an upper bound on the commensurabilty growth.",
"Our proof uses the abstract and relative commensurators.",
"The relative commensurator of a subgroup $H$ in a group $G$ is $\\operatorname{Comm}_G(H) := \\left\\lbrace g\\in G \\mid \\operatorname{c}(g H g^{-1}, H) < \\infty \\right\\rbrace .$ The abstract commensurator of $G$ is the set of equivalence classes of isomorphisms $\\phi : H_1 \\rightarrow H_2$ for finite-index subgroups $H_1,H_2\\le G$ , where two isomorphisms are equivalent if they are both defined and equal on a common finite-index subgroup of $G$ .",
"Proposition 3.4 Let $\\Gamma $ be a finitely generated, self-similar, regular branch group over a branching subgroup $K$ in $\\operatorname{Aut}(T_p)$ .",
"Suppose $\\Gamma $ is contained in the Sylow pro-$p$ subgroup of $\\operatorname{Aut}(T_p)$ and satisfies the rigid congruence subgroup property.",
"Then ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p)) \\le \\aleph _0.$ Consider the map $\\Phi : \\operatorname{Aut}(T_p) \\rightarrow \\operatorname{Homeo}(\\partial T_p)$ that sends an element to its induced action on the boundary.",
"Note that $\\Phi $ is injective.",
"If $\\Delta \\le \\operatorname{Aut}(T_p)$ satisfies $\\operatorname{c}(\\Gamma , \\Delta ) < \\infty $ then $\\Phi (\\Delta ) \\le \\operatorname{Comm}_{\\operatorname{Homeo}(\\partial T_p)}(\\Gamma )$ .",
"Therefore, the map $\\Phi $ faithfully maps the collection of subgroups of $\\operatorname{Aut}(T_p)$ commensurable with $\\Gamma $ into the collection of finitely generated subgroups of $\\operatorname{Comm}_{\\operatorname{Homeo}(\\partial T_p)}(\\Gamma )$ .",
"Röver [15] has shown that $\\operatorname{Comm}_{\\operatorname{Homeo}(\\partial T_p)}(\\Gamma )$ is isomorphic to $\\operatorname{Comm}(\\Gamma )$ , the abstract commensurator of $\\Gamma $ .",
"Because $\\Gamma $ is finitely generated, $\\operatorname{Comm}(\\Gamma )$ is countable.",
"Therefore there are countably many finitely generated subgroups of $\\operatorname{Comm}_{\\operatorname{Homeo}(\\partial T_p)}(\\Gamma )$ , and so there are countably many subgroups $\\Delta \\le \\operatorname{Aut}(T_p)$ commensurable with $\\Gamma $ .",
"We finish the proof by supplying the $\\aleph _0$ lower bound: Theorem 3.5 Let $\\Gamma $ be a finitely generated, self-similar, regular branch group over a branching subgroup $K$ in $\\operatorname{Aut}(T_p)$ .",
"Suppose $\\Gamma $ is contained in the Sylow pro-$p$ subgroup of $\\operatorname{Aut}(T_p)$ and satisfies the rigid congruence subgroup property.",
"Then ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p)) = \\aleph _0$ for all $k$ .",
"Fix $k\\ge 1$ .",
"Proposition REF provides an upper bound ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p))\\le \\aleph _0$ .",
"To prove the lower bound, fix a subgroup $H\\le \\Gamma $ of index $p^{k-1}$ containing $\\operatorname{stab}_{\\Gamma }(N)$ for some $N \\in \\mathbb {N}$ .",
"We will construct infinitely many index $p$ extensions of $H$ not contained in $\\Gamma $ .",
"To find these extensions, we will inductively construct an infinite sequence of pairs $(\\tilde{H}_i, n_i)_{i=1}^\\infty $ such that $H \\le \\tilde{H}_i$ with $[\\tilde{H}_i: H]=p$ and $H/ \\operatorname{stab}_{H}(n_{i-1})= \\tilde{H}_i/\\operatorname{stab}_{\\tilde{H}_i}(n_{i-1})$ , but $H/ \\operatorname{stab}_{H}(n_i)\\ne \\tilde{H}_i/\\operatorname{stab}_{\\tilde{H}_i}(n_i)$ .",
"It is immediate from the latter condition that the $\\tilde{H}_i$ 's are pairwise distinct.",
"In the case that $k=1$ , in which $H = \\Gamma $ , this completes the proof.",
"See below for the end of the argument in the case $k\\ge 2$ .",
"For the base case of the induction, choose $1\\ne \\gamma _1 \\in (X\\star \\Gamma )\\setminus \\Gamma $ .",
"Such a $\\gamma _1$ exists because $\\Gamma $ is finitely generated and thus is not layered by Lemma REF .",
"Let $\\Gamma _1=\\langle \\Gamma , \\gamma _1 \\rangle $ .",
"Then $[\\Gamma _1: \\Gamma ]=p^{k_1}$ for some $k_1\\ge 1$ .",
"To see this, let $Q_1=\\langle \\sigma \\rangle $ be as defined above and recall that Lemma REF gives an inclusion $\\operatorname{stab}_{(X\\star \\Gamma )\\rtimes Q_1}(n)\\le \\Gamma $ for sufficiently large $n$ .",
"Therefore there is a chain of subgroups $\\operatorname{stab}_{(X\\star \\Gamma )\\rtimes Q_1}(n)\\le \\Gamma \\le \\Gamma _1 \\le (X\\star \\Gamma )\\rtimes Q_1$ and the index of $\\operatorname{stab}_{(X\\star \\Gamma )\\rtimes Q_1}(n)$ in $(X\\star \\Gamma )\\rtimes Q_1$ is a power of $p$ as $(X\\star \\Gamma )\\rtimes Q_1 \\le \\operatorname{Aut}_p(T_p)$ .",
"Now select $\\tilde{H}_1 \\le \\Gamma _1$ such that $H \\le \\tilde{H}_1$ and $[\\tilde{H}_1:H] = p$ where $\\tilde{H}_1 = \\langle H, \\tilde{h}_1 \\rangle $ for some $\\tilde{h}_1$ .",
"Now, since $\\Gamma \\le \\Gamma _1 \\le (X\\star \\Gamma )\\rtimes Q_1$ , by Lemma REF there exists $n_1 > N$ with $\\operatorname{stab}_{\\Gamma _1}(n_1)=\\operatorname{stab}_{\\Gamma }(n_1)\\le \\Gamma $ , and therefore $\\operatorname{stab}_{\\tilde{H}_1}(n_1) \\le \\operatorname{stab}_{\\Gamma }(n_1)$ .",
"On the other hand, since $\\operatorname{stab}_{\\Gamma }(N) \\le H \\le \\tilde{H}_1$ , we clearly have $\\operatorname{stab}_{\\Gamma }(n_1) \\le \\operatorname{stab}_{\\tilde{H}_1}(n_1)$ .",
"Therefore $\\operatorname{stab}_{\\tilde{H}_1}(n_1) = \\operatorname{stab}_{\\Gamma }(n_1)$ .",
"Letting $\\pi _{n_1} : \\tilde{H}_1 \\rightarrow \\tilde{H}_1 / \\operatorname{stab}_{\\tilde{H}_1}(n_1)$ , we see that $\\pi _{n_1}(\\tilde{h}_1) \\notin \\pi _{n_1}(H).$ This completes the base case of the induction.",
"Now assume for some $j$ , we have built a sequence of pairs $(\\tilde{H}_i, n_i)_{i=1}^{j}$ as described above.",
"Choose $1\\ne \\gamma _{j+1}\\in (X^{n_j}\\star \\Gamma ) \\setminus \\Gamma $ .",
"Then, as in the argument in the base case, $\\Gamma _{j+1}=\\langle \\Gamma , \\gamma _{j+1}\\rangle $ contains $\\Gamma $ as a subgroup of index $p^{k_{j+1}}$ for some $k_{j+1}$ .",
"There exists $\\tilde{h}_{j+1}$ such that $\\tilde{H}_{j+1}=\\langle \\tilde{h}_{j+1}, H\\rangle \\le \\Gamma _{j+1}$ contains $H$ as a subgroup of index $p$ .",
"Clearly, $\\tilde{H}_{j+1}/\\operatorname{stab}_{\\tilde{H}_{j+1}}(n_j)=H/\\operatorname{stab}_{H}(n_j)$ as $\\tilde{h}_{j+1}$ stabilizes level $n_j$ .",
"Moreover, again as in the argument in the base case, there exists an $n_{j+1} > N$ such that $\\operatorname{stab}_{\\tilde{H}_{j+1}}(n_{j+1})=\\operatorname{stab}_{\\Gamma }(n_{j+1})$ .",
"Hence, taking $\\pi _{n_{j+1}}: \\tilde{H}_{n_{j+1}} \\rightarrow \\tilde{H}_{n_{j+1}}/\\operatorname{stab}_{\\tilde{H}_{n_{j+1}}}(n_{j+1}),$ we see that $\\pi _{n_{j+1}}(\\tilde{h}_{n_{j+1}})\\notin \\pi _{n_{j+1}}(H)=H/\\operatorname{stab}_{H}(n_{j+1})$ .",
"The induction is complete.",
"The proof is complete in the case $k=1$ , so consider now the case $k\\ge 2$ .",
"Each $H_i$ constructed satisfies either $\\tilde{H}_i\\cap \\Gamma = H$ or $\\tilde{H}_i \\le \\Gamma $ with $[\\Gamma : \\tilde{H}_i] = p^{k-2}$ .",
"Since there are finitely many subgroups of $\\Gamma $ of index $p^{k-2}$ and there are infinitely many distinct $\\tilde{H}_i$ with $[\\tilde{H}_i :H] = p$ , we know that there exists an index set $S$ of cardinality $\\aleph _0$ such that $\\tilde{H}_i \\cap \\Gamma = H$ for every $i \\in S$ .",
"For every $i \\in S$ , we then have $\\operatorname{c}(\\Gamma , \\tilde{H}_i)=[\\Gamma :H][\\tilde{H}_i:H]=p^{k-1}p=p^k$ , giving us the desired lower bound on ${\\bf c}_{p^k}(\\Gamma , \\operatorname{Aut}(T_p))$ ."
]
] | 1808.08660 |
[
[
"Non-expansive bijections, uniformities and polyhedral faces"
],
[
"Abstract We extend the result of B. Cascales at al.",
"about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets.",
"We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting."
],
[
"Introduction",
"Let $E_1, E_2$ be metric spaces.",
"A map $F \\colon E_1 \\rightarrow E_2$ is called non-expansive (resp.",
"non-contractive) if $d(F(x),F(y)) \\leqslant d(x, y)$ (resp.",
"$d(F(x),F(y)) \\geqslant d(x, y)$ ) for all $x,y \\in E_1$ .",
"A metric space $E$ is called expand-contract plastic (or simply, an EC-space) if every non-expansive bijection from $E$ onto itself is an isometry.",
"[5] or [10] imply that every totally bounded metric space is an EC-space, but there are examples of EC-spaces that are not totally bounded (and even unbounded).",
"In general bounded closed subsets of infinite-dimensional Banach spaces are not EC-spaces, see [2].",
"It is not known whether it is true that for every Banach space $X$ its unit ball $B_X$ is an EC-space.",
"There is no known counterexample and there are some known partial positive results: finite-dimesional spaces (the unit ball is compact), strictly convex Banach spaces (see [2]) or $\\ell _1$ -sum of strictly convex Banach spaces (see [7]).",
"A more general problem is studied in [12]: Problem 1.1 Let $X$ and $Y$ be Banach spaces and $F:B_X\\rightarrow B_Y$ be a bijective non-expansive map.",
"Is $F$ an isometry?",
"There are positive answers when $Y$ is $\\ell _1$ , a finite-dimensional Banach space or a strictly convex Banach space (see [12]).",
"The unit sphere of a strictly convex space consists of its extreme points.",
"The main result of our paper is Theorem REF , in which we substitute extreme points by finite-dimensional polyhedral extreme subsets.",
"Namely, we demonstrate that if $X$ , $Y$ are Banach spaces, $F: B_X \\rightarrow B_Y$ is a bijective non-expansive map and $S_Y$ is the union of all its finite-dimensional polyhedral extreme subsets, then $F$ is an isometry.",
"Let us briefly explain the structure of the paper.",
"In Section we extend the results about EC-spaces in totally bounded metric spaces to totally bounded uniform spaces, see Theorem REF and Lemma REF .",
"In Section we recollect some known results about bijective non-expansive maps between unit balls that we will need in the sequel.",
"The goal of Section is to demonstrate the main result.",
"On the way we collect as much as possible information about preimages under a bijective non-expansive map of finite-dimensional faces of the unit ball.",
"Using this information we obtain positive answers for the Problem REF for the case when $X$ is strictly convex (Theorem REF ) and for the case when $S_Y$ is the union of all its finite-dimensional polyhedral extreme subsets (Theorem REF ).",
"We dedicate this paper to the memory of Bernardo Cascales, who passed away in April, 2018.",
"From the very beginning our activity related to EC-spaces was motivated by Bernardo's interest to the subject.",
"It was his idea to search for a definition of EC-spaces that could be applicable to topological vector spaces and to uniform spaces.",
"It was Bernardo who communicated to us the Ellis' result [4] and the way how this result was used by Isaac Namioka [11] in his elegant demonstration of EC-plasticity of compact metric spaces.",
"We were planning to start a joint project, but..."
],
[
"Non-expansive maps and uniformities",
"The aim of this section is to extend the results about EC-spaces in metric spaces to uniform spaces.",
"We denote by $(E,{{\\mathcal {U}}})$ a uniform space and by ${{\\mathcal {B}}}$ a basis of the uniformity.",
"For $A\\subset E$ , $U,V \\subset E\\times E$ , and $u,v,w \\in E$ we denote: $A^{-1} = \\lbrace (u,v) \\colon (v, u) \\in A \\rbrace , \\,\\, \\,\\, (u,v)\\circ (v,w)=(u,w);$ $U\\circ V=\\lbrace (u,v) \\colon \\text{ there is }w\\in E\\text{ such that }(u,w)\\in U\\text{ and } (w,v)\\in V \\rbrace ;$ $U[A]=\\lbrace u\\in E\\text{ such that there is }v\\in A\\text{ with }(u,v)\\in U\\rbrace .$ The uniform space $(E,{{\\mathcal {U}}})$ is called totally bounded if for every $U\\in {{\\mathcal {B}}}$ there is a finite subset $\\widetilde{E} \\subset E$ such that $E = U[\\widetilde{E}]$ .",
"Let us recall the following definitions that were introduced in [3] and extend the concepts of non-expansive, non-contractive and isometric maps to uniform spaces.",
"Definition 2.1 Let $(E,{{\\mathcal {U}}})$ be a uniform space, ${{\\mathcal {B}}}$ a basis of the uniformity and $F:E\\rightarrow E$ a map.",
"We say that $F$ is non-contractive for the basis ${{\\mathcal {B}}}$ if for every $V \\in {{\\mathcal {B}}}$ $(F(x),F(y))\\in V \\Rightarrow (x,y)\\in V$ We say that $F$ is non-expansive for the basis ${{\\mathcal {B}}}$ if for every $V \\in {{\\mathcal {B}}}$ $(x,y)\\in V \\Rightarrow (F(x),F(y))\\in V.$ We say that $F$ is an isobasism for the basis ${{\\mathcal {B}}}$ if for every $V \\in {{\\mathcal {B}}}$ $(F(x),F(y))\\in V \\Leftrightarrow (x,y)\\in V.$ For unexplained standard definitions and terminology we refer to [8].",
"The next proposition will be used in the proof of Theorem REF .",
"Proposition 2.2 (Ellis [4]) Let $K$ be a compact space, let $S\\subset C(K,K)$ be a semigroup for the composition, and let $\\Sigma :=\\overline{S}\\subset K^K$ .",
"The following are equivalent: each member of $\\Sigma $ is onto, each member of $\\Sigma $ is one to one, $\\Sigma $ is a group and $id:K\\rightarrow K$ is the identity element of the group.",
"Theorem 2.3 Let $K$ be a compact Hausdorff uniform space, ${{\\mathcal {B}}}$ a basis for the uniformity made of open sets in $K\\times K$ .",
"If $F:K\\rightarrow K$ is a non-contractive bijection for the basis ${{\\mathcal {B}}}$ , then $F$ is an isobasism for the basis ${{\\mathcal {B}}}$ .",
"The demonstration follows the idea of Namioka's unpublished proof of EC-plasticity of compact metric spaces [11], and is presented here with his kind permission.",
"Observe that since $F$ is non-contractive, then $F^{-1}$ is non-expansive and then $(x,y)\\in V \\Rightarrow (F^{-1}(x),F^{-1}(y))\\in V,$ so $F^{-1}$ is a continuous bijection between compact spaces and then $F$ is a continuous function.",
"Consider the semigroup $S=\\lbrace F^n:n\\in {\\mathbb {N}}\\rbrace \\subset C(K,K)$ and let $G\\in \\Sigma =\\overline{S}$ be the pointwise closure of $S$ .",
"Choose a net $(G_i)_{i\\in I}$ in $S$ that converges to $G$ .",
"Let $x,y\\in K$ and $V\\in {{\\mathcal {B}}}$ be such that $(G(x),G(y))\\in V$ .",
"There is $j\\in I$ such that $(G_j(x),G_j(y))\\in V$ .",
"Let $n\\in {\\mathbb {N}}$ be such that $G_j=F^n$ .",
"Then since $F$ is non-contractive we have that $(F^n(x),F^n(y))\\in V\\Rightarrow (F^{n-1}(x),F^{n-1}(y))\\in V\\Rightarrow \\cdots \\Rightarrow (x,y)\\in V.$ We have proved that $(G(x),G(y))\\in V \\Rightarrow (x,y)\\in V$ for every $G\\in \\Sigma $ , $x,y\\in K$ and $V\\in {{\\mathcal {B}}}$ .",
"Then we have that $G$ is one to one.",
"By Proposition REF , $\\Sigma $ is a group so $F^{-1}\\in \\Sigma $ and then by (REF ) we have that $(F^{-1}(x),F^{-1}(y))\\in V \\Rightarrow (x,y)\\in V$ for every $x,y\\in K$ and $V\\in {{\\mathcal {B}}}$ , so $F^{-1}$ is non-contractive and then $F$ is an isobasism for the basis ${{\\mathcal {B}}}$ .",
"We know that every totally bounded metric space is an EC-space.",
"The above theorem generalizes this result for uniformities when the space is compact.",
"We can use some ideas of [10] to get the following results for uniformities in totally bounded spaces.",
"Lemma 2.4 Let $(E,{{\\mathcal {U}}})$ be a totally bounded uniform space, ${{\\mathcal {B}}}$ a basis for the uniformity in $E\\times E$ and $F:E\\rightarrow E$ a non-contractive bijection for ${{\\mathcal {B}}}$ .",
"Then $F$ satisfies that for every $V\\in {{\\mathcal {B}}}$ $\\begin{split}(x,y)\\in V\\Rightarrow &\\text{ for each }W\\in {{\\mathcal {U}}}\\text{ there is }k\\in {\\mathbb {N}}\\text{ such that }\\\\&(F^k(x),F^k(y))\\in W\\circ V \\circ W.\\end{split}$ Choose $x,y\\in E$ and $V\\in {{\\mathcal {B}}}$ with $(x,y)\\in V$ .",
"Choose $W\\in {{\\mathcal {U}}}$ , $W^{\\prime }\\in {{\\mathcal {B}}}$ a subset of $W$ , $Z\\in {{\\mathcal {B}}}$ such that $Z\\circ Z\\subset W^{\\prime }$ and $U\\in {{\\mathcal {B}}}$ such that $U\\subset Z\\cap Z^{-1}$ .",
"Since $E$ is totally bounded there is a finite set $\\widetilde{E} \\subset E$ such that $E = U[\\widetilde{E}]$ .",
"Then there is a infinite set $M\\subset {\\mathbb {N}}$ and $z_1,z_2\\in \\widetilde{E}$ such that $\\lbrace F^n(x):n\\in M\\rbrace \\subset U[z_1]$ and $\\lbrace F^n(y):n\\in M\\rbrace \\subset U[z_2]$ .",
"Pick $n,m\\in M$ with $m>n$ and let $k=m-n$ .",
"Then $(F^m(x),F^n(x))=(F^m(x),z_1)\\circ (z_1,F^n(x))\\in U\\circ U^{-1}\\subset Z\\circ Z\\subset W^{\\prime }.$ Then by (REF ) we have that $(F^{k}(x),x)\\in W^{\\prime }\\subset W$ .",
"Analogously $(y,F^{k}(y))\\in W$ .",
"Then $(F^k(x),F^k(y))=(F^k(x),x)\\circ (x,y)\\circ (y,F^k(y))\\in W\\circ V\\circ W.$ Corollary 2.5 Let $(E,{{\\mathcal {U}}})$ be a totally bounded uniform space, ${{\\mathcal {B}}}$ a basis for the uniformity and $F \\colon E \\rightarrow E$ a non-contractive bijection for ${{\\mathcal {B}}}$ .",
"Then $F$ satisfies that $(x,y)\\in V\\Rightarrow (F(x),F(y))\\in \\overline{V}$ for every $V\\in {{\\mathcal {B}}}$ .",
"Choose $x,y\\in E$ and $V\\in {{\\mathcal {B}}}$ with $(x,y)\\in V$ .",
"By Lemma REF we have that for each $W\\in {{\\mathcal {B}}}$ there is $k\\in {\\mathbb {N}}$ such that $(F^k(x),F^k(y))\\in W\\circ V\\circ W$ .",
"Then since $F^k$ is a bijection, we can choose $w,z\\in E$ such that $(F^k(x),F^k(w))\\in W$ , $(F^k(w),F^k(z))\\in V$ and $(F^k(z),F^k(y))\\in W$ .",
"Since $F$ is a non-contractive map we have that $(F(x),F(w))\\in W$ , $(F(w),F(z))\\in V$ and $(F(z),F(y))\\in W$ so $(F(x),F(y))\\in W\\circ V\\circ W$ and then $(F(x),F(y))\\in \\bigcap _{W\\in {{\\mathcal {B}}}}W\\circ V\\circ W=\\overline{V}.$ Let $(E,d)$ be a metric space, if we denote $U_\\varepsilon =\\lbrace (x,y):d(x,y)<\\varepsilon \\rbrace $ then ${{\\mathcal {B}}}=\\lbrace U_\\varepsilon :\\varepsilon >0\\rbrace $ is a basis of the uniformity and $F:E\\rightarrow E$ is non-expansive, non-contractive or an isometry for the metric $d$ if and only if $F$ is non-expansive, non-contractive or an isobasism for the basis of the uniformity ${{\\mathcal {B}}}$ .",
"Then Corollary REF implies the following result: Corollary 2.6 ([5]) Let $(E,d)$ be a totally bounded metric space and $F:E\\rightarrow E$ a bijective non-contractive (or non-expansive) map.",
"Then $F$ is an isometry.",
"Corollary 2.7 Let $X$ be a topological vector space, $A\\subset X$ a totally bounded set and ${{\\mathcal {B}}}$ a basis of closed neighborhoods of 0.",
"Let $F: A\\rightarrow A$ be a bijection such that for every $x, y \\in A$ and $V\\in {{\\mathcal {B}}}$ $F(x)-F(y)\\in V \\Rightarrow x-y\\in V.$ Then $f$ satisfies that for every $x, y \\in A$ and $V\\in {{\\mathcal {B}}}$ $x-y \\in V \\Rightarrow F(x)-F(y) \\in V.$ This result follows from Corollary REF applied to the set $A$ and the basis for a uniformity $\\lbrace U_V:V\\in {{\\mathcal {B}}}\\rbrace $ where $U_V=\\lbrace (x,y):x-y\\in V\\rbrace $ for each $V\\in {{\\mathcal {B}}}$ .",
"The following result is a reformulation of the last corollary: Corollary 2.8 Let $X$ be a topological vector space, $A\\subset X$ a totally bounded set and ${{\\mathcal {B}}}$ a basis of closed neighborhoods of 0.",
"Let $F: A\\rightarrow A$ be a bijection.",
"If there is $x,y\\in A$ and $V\\in {{\\mathcal {B}}}$ such that $x-y\\in V\\text{ and }F(x)-F(y)\\notin V,$ then there is $z,w\\in A$ and $W\\in {{\\mathcal {B}}}$ such that $F(z)-F(w)\\in W \\text{ and } z-w\\notin W.$"
],
[
"Notation and auxiliary statements for Banach spaces",
"In this short section we fix the necessary notation and recollect some known results that we will need in the sequel.",
"Below the letters $X$ and $Y$ always stand for real Banach spaces.",
"We denote by $S_X$ and $B_X$ the unit sphere and the closed unit ball of $X$ respectively.",
"For a convex set $A \\subset X$ denote by $\\mathop {\\rm ext}\\nolimits (A)$ the set of extreme points of $A$ ; that is, $x \\in \\mathop {\\rm ext}\\nolimits (A)$ if $x \\in A$ and for every $y \\in X\\setminus \\lbrace 0\\rbrace $ either $x + y \\notin A$ or $x - y \\notin A$ .",
"Recall that $X$ is called strictly convex if all elements of $S_X$ are extreme points of $B_X$ , or in other words, $S_X$ does not contain non-trivial line segments.",
"Strict convexity of $X$ is equivalent to the strict triangle inequality $\\Vert x +y\\Vert < \\Vert x\\Vert + \\Vert y\\Vert $ holding for all pairs of vectors $x, y \\in X$ that do not have the same direction.",
"For subsets $A, B \\subset X$ we use the standard notation $A+B = \\lbrace x + y{:}\\ x \\in A, y \\in B\\rbrace $ and $a A = \\lbrace ax {:}\\ x \\in A\\rbrace $ .",
"Proposition 3.1 (P. Mankiewicz's [9]) If $A\\subset X$ and $B \\subset Y$ are convex subsets with non-empty interior, then every bijective isometry $F : A \\rightarrow B$ can be extended to a bijective affine isometry $\\widetilde{F} : X \\rightarrow Y$ .",
"Taking into account that in the case of $A$ , $B$ being the unit balls every isometry maps 0 to 0, this result implies that every bijective isometry $F : B_X \\rightarrow B_Y$ is the restriction of a linear isometry from $X$ onto $Y$ .",
"Proposition 3.2 (Brower's invariance of domain principle [1]) Let $U$ be an open subset of ${\\mathbb {R}}^n$ and $f : U \\rightarrow {\\mathbb {R}}^n$ be an injective continuous map, then $f(U)$ is open in ${\\mathbb {R}}^n$ .",
"Proposition 3.3 ([6]) Let $X$ be a finite-dimensional normed space and $V$ be a subset of $B_X$ with the following two properties: $V$ is homeomorphic to $B_X$ and $V \\supset S_X$ .",
"Then $V=B_X$ .",
"The remaining results of this section listed below appeared first in [2] for the particular case of $X = Y$ .",
"The generalizations to the case of two different spaces were made in [12] and [7].",
"The following theorem appears in [12] and it can be demonstrated repeating the proof of [2] almost word to word (see [13] for details).",
"Theorem 3.4 Let $F: B_X \\rightarrow B_Y$ be a bijective non-expansive (briefly, a BnE) map.",
"In the above notations the following hold.",
"$F(0) = 0$ .",
"$F^{-1}(S_Y) \\subset S_X$ .",
"If $F(x)$ is an extreme point of $B_Y$ , then $x$ is also an extreme point of $B_X$ , $F(ax) = aF(x)$ for all $a \\in [-1,1]$ .",
"Moreover, if $Y$ is strictly convex, then $F$ maps $S_X$ bijectively onto $S_Y$ ; $F(ax) = a F(x)$ for all $x \\in S_X$ and $a \\in [-1, 1]$ .",
"Lemma 3.5 ([12]) Let $F: B_X \\rightarrow B_Y$ be a BnE map such that $F(S_X) = S_Y$ .",
"Let $V \\subset S_X$ be the subset of all those $v \\in S_X$ that $F(av) = a F(v)$ for all $a \\in [-1,1]$ .",
"Denote $A = \\lbrace tx: x \\in V, t \\in [-1,1] \\rbrace $ , then $F|_A$ is a bijective isometry between $A$ and $F(A)$ .",
"Lemma 3.6 ([7]) Let $F: B_X \\rightarrow B_Y$ be a BnE map such that for every $v \\in F^{-1}(S_Y)$ and every $t \\in [-1,1]$ the condition $F(tv) = t F(v)$ holds true.",
"Then $F$ is an isometry.",
"Proposition 3.7 ([12]) Let $F: B_X \\rightarrow B_Y$ be a BnE map.",
"If $Y$ is strictly convex, then $F$ is an isometry.",
"Let us list some more definitions.",
"An extreme subset of a set $B\\subset X$ is a subset $C \\subset B$ with the property $\\forall _{y_1, y_2 \\in B} \\textit { } \\forall _{\\alpha \\in (0, 1) } \\left(\\alpha y_1 + (1-\\alpha )y_2 \\in C\\right) \\Longrightarrow (y_1, y_2 \\in C).$ The generating subspace of a convex set $C$ is $\\mathop {\\rm span}\\nolimits (C - C)$ .",
"The dimension of a convex set $C$ is the dimension of its generating subspace.",
"For a convex set $B\\subset X$ we will say that a point $x \\in B$ is $n$ -extreme if for any $(n+1)$ -dimensional subspace $E \\subset X$ and any $\\varepsilon > 0$ there is an element $e \\in S_E$ , such that $x + \\varepsilon e \\notin B$ .",
"For $n \\in {\\mathbb {N}}$ a point $x$ of the convex set $B$ is called sharp $n$ -extreme in $B$ if it is $n$ -extreme and is not $(n-1)$ -extreme.",
"Remark, that in the definition we do not demand the convexity of extreme subsets.",
"This is done in order to enjoy the following easy to verify property: the union of any collection of extreme subsets of $B$ is an extreme subset of $B$ .",
"Nevertheless, we mostly deal with convex sets and convex extreme subsets.",
"Observe also that being 0-extreme point and being extreme point of $B$ in the usual sense are equivalent.",
"Every $n$ -extreme point of $B$ is also $(n+1)$ -extreme point of $B$ .",
"Every $n$ -dimensional convex extreme subset $C$ of a convex set $B$ consists of $n$ -extreme points of $B$ and contains a sharp $n$ -extreme point.",
"If $E$ is the generating subspace of the $n$ -dimensional convex extreme subset $C \\subset B$ , then $x \\in C$ is a sharp $n$ -extreme point of $B$ if and only if $x$ belongs to the relative interior of $C$ in the affine subspace $x + E = C + E$ .",
"For a convex set $C\\subset X$ with generating subspase $E$ by $\\partial C$ we denote the relative boundary of $C$ in $C+E$ .",
"Evidently, in a normed space for collinear vectors $x, y$ looking in the same direction (codirected vectors) we have $ \\Vert x+y\\Vert =\\Vert x\\Vert +\\Vert y\\Vert .$ In spaces that are not strictly convex the converse statement is not true, which motivates the following definition.",
"Definition 3.8 Elements $x,y \\in X$ are said to be quasi-codirected, if they satisfy (REF ).",
"By the triangle inequality, in order to verify (REF ) it is sufficient to check $\\Vert x+y\\Vert \\geqslant \\Vert x\\Vert +\\Vert y\\Vert $ .",
"The next lemma is well-known, but this is the example of a fact which is much easier to demonstrate than to find out when and who observed it first Lemma 3.9 If $x,y \\in X$ are quasi-codirected, then for every $a,b > 0$ the elements $ax$ and $by$ are quasi-codirected as well.",
"By symmetry we may assume $a \\geqslant b$ .",
"Then, $\\Vert ax+by\\Vert = \\Vert a(x+y) - (a - b)y\\Vert $ $\\geqslant a\\Vert x+y\\Vert - (a - b)\\Vert y\\Vert = a\\Vert x\\Vert +b\\Vert y\\Vert $ .",
"Geometrically speaking $x,y \\in S_X$ are quasi-codirected, if the whole segment $[x, y] := \\lbrace tx + (1-t)y: t \\in [0,1]\\rbrace $ lies on the unit sphere.",
"If $C \\subset S_X$ is convex, then every two elements of $C$ are quasi-codirected."
],
[
"Non-expansive maps and finite-dimensional faces",
"The aim of this section is, in the setting of Section and using some similar ideas, to obtain as much as possible information about preimages of finite-dimensional faces of the unit ball.",
"The main result is Theorem REF that gives a positive answer for the Problem REF when $S_Y$ is the union of all its finite-dimensional polyhedral extreme subsets.",
"Let us start with a very simple observation.",
"Lemma 4.1 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, and $y_1, y_2 \\in S_Y$ be quasi-codirected.",
"Then, $F^{-1}(y_1)$ is quasi-codirected with $-F^{-1}(-y_2)$ , so if $F^{-1}(-y_2) = - F^{-1}(y_2)$ , then $F^{-1}(y_1)$ is quasi-codirected with $F^{-1}(y_2)$ .",
"In particular if $y_2$ is an extreme point of $B_Y$ , then $F^{-1}(y_1)$ is quasi-codirected with $F^{-1}(y_2)$ .",
"$\\left\\Vert F^{-1}(y_1) + \\left(-F^{-1}(-y_2)\\right) \\right\\Vert &= \\left\\Vert F^{-1}(y_1) - F^{-1}(-y_2) \\right\\Vert \\\\&\\geqslant \\left\\Vert y_1 - (-y_2) \\right\\Vert = \\left\\Vert y_1 + y_2 \\right\\Vert = 2.$ The above lemma readily implies the following natural counterpart to Proposition REF .",
"Theorem 4.2 Let $X$ , $Y$ be Banach spaces, $X$ be strictly convex and $F: B_X \\rightarrow B_Y$ be a BnE map.",
"Then $F$ is an isometry.",
"According to Proposition REF it is sufficient to demonstrate that $Y$ is strictly convex.",
"Assume to the contrary that $S_Y$ contains a non-void segment $[y_0, y_1] := \\lbrace ty_1 + (1-t)y_0: t \\in [0,1]\\rbrace $ .",
"Since $X$ is strictly convex, the only element of $S_X$ quasi-codirected with $F^{-1}(y_1)$ is $F^{-1}(y_1)$ itself.",
"But, according to (1) of Lemma REF all elements $-F^{-1}(-y_t)$ , where $y_t := ty_1 + (1-t)y_0$ , $t \\in [0,1]$ , are quasi-codirected with $F^{-1}(y_1)$ .",
"This contradiction completes the proof.",
"Let $Y$ be a Banach space, $y_1, y_2 \\in S_Y$ be quasi-codirected.",
"Denote $\\nonumber D_1(y_1, y_2) &:=& (y_1 + B_Y) \\cap (- y_2 + B_Y) \\\\&=& \\left\\lbrace y \\in Y \\colon \\Vert y_1 - y\\Vert \\leqslant 1 \\mathrm {\\, and \\, } \\ \\Vert y_2 + y\\Vert \\leqslant 1 \\right\\rbrace \\\\\\nonumber &=& \\bigl \\lbrace y \\in Y \\colon \\Vert y_1 - y\\Vert = \\Vert y_2 + y\\Vert = 1 \\bigr \\rbrace .$ Some evident properties of $D_1(y_1, y_2)$ are listed below without proof.",
"Lemma 4.3 Let $Y$ be a Banach space, $y_1, y_2 \\in S_Y$ be quasi-codirected.",
"Then $D_1(y_1, y_2)$ is a convex closed subset of $Y$ .",
"$0 \\in D_1(y_1, y_2)$ .",
"$t D_1(y_1, y_2) \\subset D_1(y_1, y_2)$ for every $t \\in [0,1]$ .",
"$D_1(y_1, y_2) \\subset 2 B_Y$ , consequently $\\frac{1}{2} D_1(y_1, y_2) \\subset D_1(y_1, y_2) \\cap B_Y$ .",
"Lemma 4.4 Let $Y$ be a Banach space, $y_1, y_2 \\in S_Y$ be quasi-codirected, and $h \\in Y$ be such that $y_1 \\pm h \\in S_Y$ .",
"Then $ \\Bigl \\lbrace \\frac{1}{2}(y_1- y_2) \\pm \\frac{1}{2} h\\Bigr \\rbrace \\subset D_1(y_1, y_2).$ In particular, substituting $y_2 = y_1$ we obtain $\\pm \\frac{1}{2} h \\in D_1(y_1, y_1).$ Substituting $h = 0$ we obtain $\\frac{1}{2}(y_1 - y_2) \\in D_1(y_1, y_2),$ which implies that for all $t \\in [0, 1/2]$ $ t (y_1 - y_2) \\in D_1(y_1, y_2).$ We have to verify two inequalities: $\\left\\Vert \\frac{1}{2}(y_1- y_2) \\pm \\frac{1}{2} h + y_2 \\right\\Vert \\leqslant 1 \\mathrm {\\, and \\, } \\ \\left\\Vert \\frac{1}{2}(y_1- y_2) \\pm \\frac{1}{2} h - y_1 \\right\\Vert \\leqslant 1.$ Each of them reduces to the same inequality $\\left\\Vert \\frac{1}{2}(y_1 + y_2) \\pm \\frac{1}{2} h \\right\\Vert \\leqslant 1.$ Let us demonstrate this: $\\left\\Vert (y_1 + y_2) \\pm h \\right\\Vert = \\left\\Vert y_2 + (y_1 \\pm h) \\right\\Vert \\leqslant \\left\\Vert y_2 \\right\\Vert + \\left\\Vert y_1 \\pm h\\right\\Vert = 2$ .",
"Lemma 4.5 Let $Y$ be a Banach space, $C \\subset S_Y$ be a convex extreme subset, and $E$ be the generating subspace of $C$ .",
"Then $D_1(y_1, y_2) \\subset E$ for every $y_1, y_2 \\in C$ .",
"Let $y \\in D_1(y_1, y_2)$ .",
"Then, $y_1 - y, y_2 + y \\in B_Y$ and $\\frac{1}{2}\\bigl ((y_1 - y) +( y_2 + y)\\bigr ) = \\frac{1}{2}(y_1 + y_2) \\in C.$ Consequently, by the definition of extreme subset, $y_2 + y \\in C$ , so $y = ( y_2 + y) - y_2 \\in C-C \\subset E$ .",
"Lemma 4.6 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, $y_1, y_2 \\in S_Y$ be quasi-codirected, $x_1 = F^{-1}(y_1) \\in S_X$ , $x_2 = -F^{-1}(-y_2) \\in S_X$ .",
"Then $F\\left(D_1(x_1, x_2) \\cap B_X\\right) \\subset D_1(y_1, y_2) \\cap B_Y.$ In particular, $F\\left(\\frac{1}{2} D_1(x_1, x_2) \\right) \\subset D_1(y_1, y_2) \\cap B_Y$ .",
"According to (1) of Lemma REF , $x_1$ and $x_2$ are quasi-codirected, so the set $D_1(x_1, x_2) $ is well-defined.",
"Consider arbitrary $x \\in D_1(x_1, x_2) \\cap B_X$ .",
"We have $\\Vert x_1 - x\\Vert \\leqslant 1$ and $\\Vert (-x_2) - x\\Vert = \\Vert x_2 + x\\Vert \\leqslant 1$ , so $\\Vert F(x_1) - F(x)\\Vert \\leqslant 1$ and $\\Vert F(-x_2) - F(x)\\Vert \\leqslant 1$ .",
"In other words, $\\Vert y_1 - F(x)\\Vert \\leqslant 1$ and $\\Vert (-y_2) - F(x)\\Vert = \\Vert y_2 + F(x)\\Vert \\leqslant 1$ , which means that $ F(x) \\in D_1(y_1, y_2)$ .",
"Lemma 4.7 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, $n\\in {\\mathbb {N}}$ , and $C \\subset S_Y$ be an $n$ -dimensional convex extreme subset.",
"Then for every $y_1 \\in C$ its preimage $x_1 = F^{-1}(y_1) \\in S_X$ is an $n$ -extreme point of $B_X$ .",
"Denote $x_2 = -F^{-1}(-y_1) \\in S_X$ .",
"Assume that $x_1$ is not $n$ -extreme point of $B_X$ .",
"Then, according to the definition, there exist an $(n+1)$ -dimensional subspace $E \\subset X$ and an $\\varepsilon > 0$ such that $x_1 + \\varepsilon B_E \\subset S_X$ .",
"According to Lemma REF $\\frac{1}{2}(x_1- x_2) + \\varepsilon B_E \\subset D_1(x_1, x_2).$ The above inclusion implies that $\\frac{1}{2} D_1(x_1, x_2)$ contains an $(n+1)$ -dimensional ball.",
"Then Lemma REF implies that $D_1(y_1, y_1)$ contains a homeomorphic copy of $(n+1)$ -dimensional ball, which is impossible by Lemma REF .",
"Note, that under conditions of the previous lemma $x$ may be also $m$ -extreme point for some $m < n$ .",
"Now we are coming to the most important and at the same time most difficult result of the paper.",
"Theorem 4.8 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, then for every $n\\in {\\mathbb {N}}$ the preimage of any $n$ -dimensional convex polyhedral extreme subset $C \\subset S_Y$ is an $n$ -dimensional convex polyhedral extreme subset of $S_X$ .",
"Moreover, the equality $- F^{-1}(-C) = F^{-1}(C)$ holds true.",
"We will use the induction in $n$ .",
"The initial case of $n = 0$ (i.e., of extreme points) is covered by the assertion (3) of Theorem REF .",
"Let us assume that the theorem is demonstrated for extreme subsets of dimension smaller than $n$ , and let us demonstrate it for a given $n$ -dimensional polyhedral extreme subset $C \\subset S_Y$ .",
"Denote $E$ the generating subspace of $C$ , $\\dim E = n$ .",
"The boundary $\\partial C$ of polyhedron $C$ consists of finite union of its convex $(n-1)$ -dimensional polyhedral extreme subsets, so, by the inductive hypothesis, $A:= F^{-1}(\\partial C)$ also consists of finite union of some convex $(n-1)$ -dimensional polyhedral extreme subsets of $S_X$ .",
"Consequently, $A$ is an extreme subset of $B_X$ .",
"Also, $A$ is compact, and $F|_A$ performs a homeomorphism between $A$ and $F(A) = \\partial C$ .",
"Let $y_1 \\in C \\setminus \\partial C$ be an arbitrary point.",
"Denote $x_1 = F^{-1}(y_1)$ .",
"Since $y_1$ is quasi-codirected with every point $y_2 \\in \\partial C$ , $x_1$ is quasi-codirected with the corresponding $x_2 = -F^{-1}(-y_2) \\in S_X$ .",
"By the inclusion (REF ) and Lemmas REF and REF $F\\left(t (x_1 - x_2)\\right) \\in F\\left(\\frac{1}{2} D_1(x_1, x_2) \\right) \\subset D_1(y_1, y_2) \\subset E$ for all $t \\in \\left[0, \\frac{1}{4}\\right]$ .",
"By the inductive hypothesis, when $y_2$ runs through $\\partial C$ the corresponding $x_2$ runs through the whole $A$ .",
"So, denoting $\\widetilde{A} = \\left[0, \\frac{1}{4}\\right] (x_1 - A) = \\left\\lbrace t (x_1 - x_2) \\colon t \\in \\left[0, \\frac{1}{4}\\right], x_2 \\in A \\right\\rbrace $ we obtain $ F\\bigl (\\widetilde{A}\\bigr ) \\subset E.$ Let us demonstrate that the segments $\\left(0, \\frac{1}{4}\\right] (x_1 - x_2)$ with different $x_2 \\in A$ are pairwise disjoint.",
"We will argue by contradiction.",
"Let two segments of the form $\\left(0, \\frac{1}{4}\\right] (x_1 - \\widehat{x}_2)$ , $\\left(0, \\frac{1}{4}\\right] (x_1 - \\widetilde{x}_2)$ with $\\widehat{x}_2,\\widetilde{x}_2 \\in A$ , $\\widehat{x}_2 \\ne \\widetilde{x}_2$ intersect at some point $y$ .",
"Then the corresponding closed segments $\\left[0, \\frac{1}{4}\\right] (x_1 - \\widehat{x}_2)$ , $\\left[0, \\frac{1}{4}\\right] (x_1 - \\widetilde{x}_2)$ intersect in two points (0 and $y$ ), so either they coincide or one segment contains the other one.",
"That is, $(x_1 - \\widehat{x}_2)$ and $(x_1 - \\widetilde{x}_2)$ are codirected.",
"There are two cases:$(x_1 - \\widehat{x}_2) = \\lambda (x_1 - \\widetilde{x}_2)$ or $(x_1 - \\widetilde{x}_2) = \\lambda (x_1 - \\widehat{x}_2)$ with some $0<\\lambda <1$ .",
"We will discuss the first one, the second one is analogous.",
"We get $\\widehat{x}_2=\\lambda \\widetilde{x}_2+(1-\\lambda )x_1$ , so these three points are on the same segment and $\\widehat{x}_2$ lies between $x_1$ and $\\widetilde{x}_2$ .",
"Since $A$ is extreme subset, we get $x_1\\in A$ , which contradicts the fact $y_1\\notin \\partial C$ .",
"The set $(x_1 - A)$ is homeomorphic to the unit sphere of ${\\mathbb {R}}^n$ .",
"Let us show, that $\\widetilde{A}$ is homeomorphic to the unit ball of ${\\mathbb {R}}^n$ , with 0 mapped to 0.",
"Let $S^n$ and $B^n$ denote the unit sphere and the unit ball of ${\\mathbb {R}}^n$ respectively, and $h \\colon S^n \\rightarrow (x_1 - A)$ be a homeomorphism.",
"One may define the mapping $H\\colon B^n \\rightarrow \\widetilde{A}$ as $H(x)={\\left\\lbrace \\begin{array}{ll}0, & \\mbox{when } x=0 \\\\\\frac{1}{4}\\Vert x\\Vert h\\left(\\frac{x}{\\Vert x\\Vert }\\right), & \\mbox{when } x\\in B^n\\backslash \\lbrace 0\\rbrace .\\end{array}\\right.",
"}$ Obviously, this mapping is bijective and continuous at 0.",
"We are going to show that $H$ is continuous at all points.",
"Let us consider some sequence $\\lbrace x_n\\rbrace _{n=1}^{\\infty }$ in $B^n$ converging to an $x \\in B^n \\setminus \\lbrace 0\\rbrace $ , that is $\\lim _{n\\rightarrow \\infty }x_n = x \\ne 0.$ Then $\\lim _{n\\rightarrow \\infty }H(x_n)&=\\lim _{n\\rightarrow \\infty }\\frac{1}{4}\\Vert x_n\\Vert h\\left(\\frac{{x}_n}{\\Vert x_n\\Vert }\\right)=\\frac{1}{4}\\lim _{n\\rightarrow \\infty }\\Vert x_n\\Vert \\lim _{n\\rightarrow \\infty }h\\left(\\frac{{x}_n}{\\Vert x_n\\Vert }\\right)\\\\&=\\frac{1}{4}\\Vert x\\Vert h\\left(\\lim _{n\\rightarrow \\infty }\\frac{x_n}{\\Vert x_n\\Vert }\\right)=\\frac{1}{4}\\Vert x\\Vert h\\left(\\frac{{x}}{\\Vert x\\Vert }\\right) = H(x).$ So, $H$ is a bijective continuous map from compact $B^n$ to Hausdorf space, thus $H$ is a homeomorphism.",
"Consequently, $F\\bigl (\\widetilde{A}\\bigr ) \\subset E$ is homeomorphic to the unit ball of ${\\mathbb {R}}^n$ , with 0 being a relative (in $E$ ) interior point of $F\\bigl (\\widetilde{A}\\bigr )$ .",
"Consider now any point $\\widetilde{y}_2 \\in C \\setminus \\partial C$ , $\\widetilde{y}_2 \\ne y_1$ , such that the corresponding $\\widetilde{x}_2 = -F^{-1}\\bigl (-\\widetilde{y}_2\\bigr )$ is not equal to $x_1$ .",
"By the same reason as before, the segment $F\\left(\\left[0, \\frac{1}{4}\\right](x_1 - \\widetilde{x}_2)\\right) \\subset D_1(y_1, \\widetilde{y}_2) \\subset E$ .",
"The set $F\\left(\\left[0, \\frac{1}{4}\\right] (x_1 - \\widetilde{x}_2)\\right)$ is a continuous curve in $E$ connecting $F(\\frac{1}{4}(x_1 - \\widetilde{x}_2))$ with 0, which is an interior point of $F\\bigl (\\widetilde{A}\\bigr )$ .",
"So there is a $t_0 \\in \\left(0, \\frac{1}{4}\\right]$ such that $F\\bigl (t_0 (x_1 - \\widetilde{x}_2)\\bigr ) \\in F\\bigl (\\widetilde{A}\\bigr )$ , that is $t_0 (x_1 - \\widetilde{x}_2) \\in \\widetilde{A}$ .",
"This means that for some $t_1 \\in \\left(0, \\frac{1}{4}\\right]$ and some $x_2 \\in A$ we have $t_0 (x_1 - \\widetilde{x}_2) = t_1 (x_1 - x_2)$ .",
"In other words, there is an $\\alpha > 0$ such that $ x_1 - \\widetilde{x}_2 = \\alpha (x_1 - x_2).$ Let us demonstrate that $\\alpha < 1$ .",
"Indeed, if $\\alpha \\geqslant 1$ , the above formula would give the representation $x_2 = \\left(1 - \\frac{1}{\\alpha }\\right)x_1 + \\frac{1}{\\alpha }\\widetilde{x}_2$ of $x_2 \\in A$ as a convex combination of $x_1, \\widetilde{x}_2 \\in S_X \\setminus A$ , which contradicts the fact that $A$ is extreme in $S_X$ .",
"Since $\\alpha < 1$ , the formula (REF ) gives the representation $\\widetilde{x}_2 = (1 - \\alpha ) x_1 + \\alpha x_2$ of $\\widetilde{x}_2$ as a convex combination of $x_1$ and some $x_2 \\in A$ .",
"If we consider the BnE mapping $G: B_X \\rightarrow B_Y$ defined as $G(x) = -F(-x)$ , all the above reasoning is applicable for $G$ as well, because by the inductive hypothesis $G^{-1}(\\partial C) = F^{-1}(\\partial C) = A$ .",
"Since $\\widetilde{x}_2 = G^{-1}\\bigl (\\widetilde{y}_2\\bigr )$ and $x_1 = - G^{-1}(- y_1)$ the roles of these elements for $G$ interchange, and we deduce that also $x_1$ is a convex combination of $\\widetilde{x}_2$ and some $x_3 \\in A$ .",
"So, we obtain the following properties of sets $F^{-1}(C \\setminus \\partial C)$ and $G^{-1}(C \\setminus \\partial C)$ : Properties.",
"(i) For every $u \\in F^{-1}(C \\setminus \\partial C)$ $G^{-1}(C \\setminus \\partial C) \\subset \\left\\lbrace t x + (1-t)u \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace .$ (ii) For every $v \\in G^{-1}(C \\setminus \\partial C)$ $F^{-1}(C \\setminus \\partial C) \\subset \\left\\lbrace t x + (1-t)v \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace .$ (iii) For every $u \\in F^{-1}(C \\setminus \\partial C)$ and every $v \\in G^{-1}(C \\setminus \\partial C)$ , $u \\ne v$ there are (unique) elements $w, z \\in A$ such that $[u, v] \\subset [w, z]$ .",
"Properties (i) and (ii) imply that $F^{-1}(C)$ and $G^{-1}(C)$ lie in some finite-dimensional subspace of $X$ .",
"Since both these sets are bounded and closed, they are compacts.",
"Continuous mappings $F$ and $G$ map corresponding compacts $F^{-1}(C)$ and $G^{-1}(C)$ to $C$ bijectively, so both $F^{-1}(C)$ and $G^{-1}(C)$ are homeomorphic to $C$ , i.e.",
"homeomorphic to the unit ball of ${\\mathbb {R}}^n$ .",
"Since the set $\\left\\lbrace t x + (1-t)u \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace $ for a fixed $u$ is also homeomorphic to the unit ball of ${\\mathbb {R}}^n$ and $A$ corresponds to the unit sphere and belongs to both $\\left\\lbrace t x + (1-t)u \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace $ and $G^{-1}(C)$ , the inclusion (i) and Proposition REF imply that (i)' for every $u \\in F^{-1}(C \\setminus \\partial C)$ $G^{-1}(C) = \\left\\lbrace t x + (1-t)u \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace ,$ and by the same reason (ii)' For every $v \\in G^{-1}(C \\setminus \\partial C)$ $F^{-1}(C) = \\left\\lbrace t x + (1-t)v \\colon t \\in \\left[0, 1\\right], x \\in A \\right\\rbrace .$ In particular, from (i)' it follows that every $u \\in F^{-1}(C \\setminus \\partial C)$ belongs to $G^{-1}(C)$ , so $F^{-1}(C) \\subset G^{-1}(C)$ , and (ii)' implies the inverse inclusion $G^{-1}(C) \\subset F^{-1}(C)$ , so $G^{-1}(C) = F^{-1}(C).$ Coming back to the already used inclusion (REF ) and Lemmas REF and REF we obtain that for all $x_1, x_2 \\in F^{-1}(C)$ $F\\left(\\frac{1}{4}(x_1 - x_2)\\right) \\in E,$ in other words $F\\left(\\frac{1}{4}( F^{-1}(C) - F^{-1}(C))\\right) \\subset E.$ Recall, that by the inductive hypothesis, $A= F^{-1}(\\partial C)$ consists of finite union of some convex $(n-1)$ -dimensional polyhedral extreme subsets $\\widetilde{W}_i$ , $i = 1, \\ldots , N$ which are preimages of corresponding parts of $\\partial C$ .",
"Let us fix some $v\\in F^{-1}(C \\setminus \\partial C)$ .",
"Denote $W_i = \\left\\lbrace t x + (1-t)v \\colon t \\in \\left[0, 1\\right], x \\in \\widetilde{W}_i \\right\\rbrace .$ These $W_i$ are $n$ -dimensional convex polyhedrons, and, according to (ii)', $F^{-1}(C) = \\bigcup _{i=1}^N W_i.$ We state that all polyhedrons $W_i$ (and also their union $ F^{-1}(C)$ ) are situated in one and the same $n$ -dimensional affine subspace $\\widetilde{E}$ .",
"To this end, consider the generating subspaces $Z_i = \\mathop {\\rm span}\\nolimits (W_i - W_i)$ of $W_i$ and let us demonstrate that all of $Z_i$ are equal one to another, i.e.",
"all of them are equal to some $n$ -dimensional linear subspace $Z$ .",
"Then $\\widetilde{E} = v + Z$ will be the $n$ -dimensional affine subspace $\\widetilde{E}$ we are looking for.",
"Let us argue “ad absurdum”.",
"Assume that $Z_i \\ne Z_j$ for some $i \\ne j$ .",
"Then $Z_i + Z_j$ has dimension strictly greater than $n$ , and $\\dim (W_i-W_j) = \\dim (\\mathop {\\rm span}\\nolimits ((W_i-W_j) - (W_i-W_j))) = \\dim (Z_i + Z_j) > n.$ Taking into account that $W_i-W_j \\subset (F^{-1}(C) - F^{-1}(C))$ the dimension of $F^{-1}(C) - F^{-1}(C)$ is strictly greater than $n$ , which makes the inclusion (REF ) impossible.",
"It remains to demonstrate that $ F^{-1}(C)$ is convex and is an extreme subset.",
"For the convexity let us show that $ F^{-1}(C) = B_X \\cap \\widetilde{E}$ .",
"We have already known, that $ F^{-1}(C)\\subset B_X \\cap \\widetilde{E}$ .",
"Let us show the inverse inclusion.",
"Again we will argue by contradiction.",
"Suppose there is a point $z\\in (B_X \\cap \\widetilde{E})\\setminus F^{-1}(C)$ .",
"We may fix some $v\\in F^{-1}(C \\setminus \\partial C)$ and consider the segment $[z,v]$ .",
"As we already remarked, $F^{-1}(C)$ is homeomorphic to $C$ and hence to $B^n$ , that is, $v$ lies in the relative interior of $F^{-1}(C)$ in $\\widetilde{E}$ .",
"So, the segment $[z,v]$ must intersect $A = F^{-1}(\\partial C)$ in some point.",
"In other words, there is $\\lambda \\in (0,1)$ such that $\\lambda z+ (1-\\lambda )v\\in A$ , which contradicts the fact, that $A$ is an extreme subset in $B_X$ .",
"Theorem 4.9 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, then for every $n$ -dimensional convex polyhedral extreme subset $C \\subset S_Y$ the following equality holds true: $F({\\mathop {\\rm conv}\\nolimits }(0,F^{-1}(C)))= {\\mathop {\\rm conv}\\nolimits }(0,C)$ .",
"We will carry out the proof by induction in $n$ .",
"For $n = 0$ (i.e., when C is extreme point) the required equality may be obtained from the assertion (3) of Theorem REF .",
"Suppose our theorem is proved for all extreme subsets of dimension smaller than $n$ , and let us show the same for a given $n$ -dimensional polyhedral extreme subset $C \\subset S_Y$ .",
"Consider $x\\in F^{-1}(C\\setminus \\partial C)$ and $\\alpha \\in (0,1)$ .",
"Since $F$ is non-expansive we have $\\Vert F(\\alpha x)\\Vert \\leqslant \\Vert \\alpha x\\Vert , \\textrm { and } \\Vert F(x)-F(\\alpha x)\\Vert \\leqslant \\Vert x-\\alpha x\\Vert .$ Also $\\nonumber 1=\\Vert F(x)\\Vert &\\leqslant \\Vert F(\\alpha x)\\Vert +\\Vert F(x)-F(\\alpha x)\\Vert \\\\&\\leqslant \\Vert \\alpha x\\Vert +\\Vert x-\\alpha x\\Vert =1.$ That is why $\\Vert F(\\alpha x)\\Vert +\\Vert F(x)-F(\\alpha x)\\Vert = 1.$ So one may write $F(x)$ as a convex combination $F(x) = \\Vert F(\\alpha x)\\Vert \\frac{F(\\alpha x)}{\\Vert F(\\alpha x)\\Vert }+\\Vert F(x)-F(\\alpha x)\\Vert \\frac{F(x)-F(\\alpha x)}{\\Vert F(x)-F(\\alpha x)\\Vert }.$ Since $F(x)\\in C$ and $C$ is extreme subset in $B_X$ we get $\\frac{F(\\alpha x)}{\\Vert F(\\alpha x)\\Vert }\\in C$ and $\\frac{F(x)-F(\\alpha x)}{\\Vert F(x)-F(\\alpha x)\\Vert }\\in C$ .",
"So, $F(\\alpha x) = \\Vert F(\\alpha x)\\Vert \\frac{F(\\alpha x)}{\\Vert F(\\alpha x)\\Vert }\\in {\\mathop {\\rm conv}\\nolimits }(\\frac{F(\\alpha x)}{\\Vert F(\\alpha x)\\Vert },0) \\subset {\\mathop {\\rm conv}\\nolimits }(0,C)$ and thus $F({\\mathop {\\rm conv}\\nolimits }(0,F^{-1}(C)))\\subset {\\mathop {\\rm conv}\\nolimits }(0,C)$ .",
"By the inductive hypothesis $F({\\mathop {\\rm conv}\\nolimits }(0,A))= {\\mathop {\\rm conv}\\nolimits }(0,\\partial C)$ and $\\partial {\\mathop {\\rm conv}\\nolimits }(0,C) \\subset F({\\mathop {\\rm conv}\\nolimits }(0,F^{-1}(C)))$ .",
"Besides, ${\\mathop {\\rm conv}\\nolimits }(0,F^{-1}(C))$ is homeomorphic to $B^{n+1}$ and $\\partial {\\mathop {\\rm conv}\\nolimits }(0,C)$ is homeomorphic to $S^{n+1}$ .",
"In this way Proposition REF implies the statement of the theorem.",
"Lemma 4.10 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map, then $\\Vert F(\\alpha x)\\Vert =\\Vert \\alpha x\\Vert =\\alpha $ for all $x\\in F^{-1}(S_Y)$ , $\\alpha \\in [0,1]$ .",
"Since $F$ is non-expansive, we may use inequalities (REF ) and (REF ).",
"The inequality (REF ) implies $\\Vert F(\\alpha x)\\Vert +\\Vert F(x)-F(\\alpha x)\\Vert =\\Vert \\alpha x\\Vert +\\Vert x-\\alpha x\\Vert ,$ and application of (REF ) concludes the proof.",
"Theorem 4.11 Let $X$ , $Y$ be Banach spaces, $F: B_X \\rightarrow B_Y$ be a BnE map and $S_Y$ be the union of all its finite-dimensional polyhedral extreme subsets.",
"Then $F$ is an isometry.",
"Let us first show, that $F( S_X) = S_Y$ .",
"Since $S_Y=\\bigcup _{i\\in I}^{}C_i,$ where $C_i$ are finite-dimensional polyhedral extreme subsets of $S_Y$ and $I$ is some index set, one may deduce $B_Y=\\bigcup _{i\\in I}{\\mathop {\\rm conv}\\nolimits }(0,C_i).$ Due to bijectivity of $F$ , theorem REF implies $B_X=\\bigcup _{i\\in I}{\\mathop {\\rm conv}\\nolimits }(0,F^{-1}(C_i)).$ Consequently, there is no other norm-one points in $B_X$ except for points from $F^{-1}(C_i)$ , and we get $S_X=\\bigcup _{i\\in I}F^{-1}(C_i)=F^{-1}(S_Y).$ To prove that $F$ is an isometry we will use lemmas REF and REF .",
"We are going to show for the set $V$ from lemma REF that $F^{-1}(C)\\subset V$ for every $n$ -dimensional polyhedral extreme subset $C$ of $S_Y$ .",
"To do that, we will use induction by dimension.",
"For 0-dimensional sets, i.e.",
"extreme points, the statement we need follows from item (3) of theorem REF .",
"Now suppose that the inclusion is proved for all $(n-1)$ -dimensional polyhedral extreme subsets and let us prove it for dimension $n$ .",
"Consider some $n$ -dimensional extreme subset $C$ in $S_Y$ .",
"For every pair $x,y\\in F^{-1}(C)$ there are $u,v\\in F^{-1}(\\partial C)$ such that $x=\\lambda u + (1-\\lambda )v $ and $y=\\mu u + (1-\\mu )v, \\lambda ,\\mu \\in (0,1)$ .",
"Without loss of generality one may account $\\lambda >\\mu $ .",
"Since $\\partial C$ consists of $(n-1)$ -dimensional polyhedral extreme subsets, the inductive hypothesis and lemma REF give that $\\Vert u-v\\Vert =\\Vert F(u)-F(v)\\Vert $ .",
"Since $F$ is non-expansive, $\\Vert u-v\\Vert &=\\Vert F(u)-F(v)\\Vert \\leqslant \\Vert F(u)-F(x)\\Vert +\\Vert F(x)-F(y)\\Vert \\\\&+\\Vert F(y)-F(v)\\Vert \\leqslant \\Vert u-x\\Vert +\\Vert x-y\\Vert +\\Vert y-v\\Vert \\\\&= (1-\\lambda )\\Vert u-v\\Vert +(\\lambda -\\mu )\\Vert u-v\\Vert +\\mu \\Vert u-v\\Vert =\\Vert u-v\\Vert .$ So we get $\\Vert F(u)-F(x)\\Vert =\\Vert u-x\\Vert $ , $\\Vert F(y)-F(v)\\Vert =\\Vert y-v\\Vert $ , $\\Vert F(x)-F(y)\\Vert = \\Vert x-y\\Vert $ .",
"Thus, $F$ is bijective isometry between $F^{-1}(C)$ and $C$ and Proposition REF implies that $F$ is affine on $F^{-1}(C)$ .",
"Lemma REF together with Theorem REF give the equality $F(\\alpha F^{-1}(C)) = \\alpha C$ for $\\alpha \\in [0, 1]$ , and application of the “moreover\" part of theorem REF extends this to $\\alpha \\in [-1,1]$ .",
"The same way as before, the inductive hypothesis and lemma REF imply that $F$ is bijective isometry between $\\alpha F^{-1}(C)$ and $\\alpha C$ , so $F$ is affine on $\\alpha F^{-1}(C)$ .",
"We are going to show that $F(\\alpha x)=\\alpha F(x)$ for all $x \\in F^{-1}(C)$ , $\\alpha \\in [-1,1]$ .",
"Every $x \\in F^{-1}(C)$ is of the form $x=\\lambda u+(1-\\lambda )v$ , where $u,v\\in F^{-1}(\\partial C)$ and $\\lambda \\in (0,1)$ .",
"We obtain $F(\\alpha x) = F( \\lambda \\alpha u+(1-\\lambda )\\alpha v) = \\lambda F(\\alpha u)+(1-\\lambda )F(\\alpha v),$ because $F$ is affine on $\\alpha F^{-1}(C)$ .",
"By the inductive hypothesis $F(\\alpha u) =\\alpha F(u)$ , $F(\\alpha v) = \\alpha F(v)$ , so $F(\\alpha x) = \\lambda \\alpha F(u) + (1-\\lambda ) \\alpha F(v) = \\alpha (\\lambda F(u) + (1-\\lambda )F(v)).$ It remains to use the fact that $F$ is affine on $F^{-1}(C)$ to conclude that $F(\\alpha x) = \\alpha F(\\lambda u + (1-\\lambda )v) = \\alpha F(x).$ So, the required inclusion (REF ) is demonstrated.",
"At last, (REF ) and the written above imply that for every $v \\in F^{-1}(S_Y)$ and every $t \\in [-1,1]$ $F(tv) = t F(v)$ .",
"So, the application of lemma REF completes the proof of the theorem."
]
] | 1808.08534 |
[
[
"Energy Distribution of Radial Solutions to Energy Subcritical Wave\n Equation with an Application on Scattering Theory"
],
[
"Abstract The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\\partial_t^2 u - \\Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3\\leq p<5$) whose initial data are radial and come with a finite energy.",
"We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: \"scattering energy\" which concentrates around the light cone $|x|=|t|$ and moves to infinity at the light speed and \"retarded energy\" which is at a distance of at least $|t|^\\beta$ behind when $|t|$ is large.",
"Here $\\beta$ is an arbitrary constant smaller than $\\beta_0(p) = \\frac{2(p-2)}{p+1}$.",
"A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data $(u_0,u_1)$ than previously known results.",
"More precisely, we assume \\[ \\int_{{\\mathbb R}^3} (|x|^\\kappa+1)\\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx < +\\infty.",
"\\] Here $\\kappa>\\kappa_0(p) =1-\\beta_0(p) = \\frac{5-p}{p+1}$ is a constant."
],
[
"Background",
"In this work we consider the Cauchy problem of the defocusing semi-linear wave equation $\\left\\lbrace \\begin{array}{ll} \\partial _t^2 u - \\Delta u = - |u|^{p-1}u, & (x,t) \\in {\\mathbb {R}}^3 \\times {\\mathbb {R}}; \\\\u(\\cdot , 0) = u_0; & \\\\u_t (\\cdot ,0) = u_1.",
"& \\end{array}\\right.\\quad (CP1)$ If $u$ is a solution as above and $\\lambda $ is a positive constant, then the function $u_\\lambda = \\lambda ^{-2/(p-1)} u(x/\\lambda ,t/\\lambda )$ is another solution to (CP1) with initial data $& u_\\lambda (\\cdot ,0) = \\lambda ^{-\\frac{2}{p-1}} u_0(\\cdot /\\lambda );& &\\partial _t u_\\lambda (\\cdot ,0) = \\lambda ^{-\\frac{2}{p-1}-1} u_1(\\cdot /\\lambda ).&$ Both pairs of initial data share the same $\\dot{H}^{s_p}\\times \\dot{H}^{s_p-1}({\\mathbb {R}}^3)$ norm if we choose $s_p = 3/2-2/(p-1)$ .",
"As a result, this Sobolev space is called the critical Sobolev space of this equation.",
"It has been proved that this problem is locally well-posed for any initial data in this critical Sobolev space.",
"Please see [8], for instance, for more details.",
"There is also an energy conservation law for suitable solutions: $E(u, u_t) = \\int _{{\\mathbb {R}}^3} \\left(\\frac{1}{2}|\\nabla u(\\cdot , t)|^2 +\\frac{1}{2}|u_t(\\cdot , t)|^2 + \\frac{1}{p+1}|u(\\cdot ,t)|^{p+1}\\right)\\,dx = \\hbox{Const}.$ The question about global behaviour of solutions is more difficult.",
"In early 1990's M. Grillakis [4] gave a satisfying answer in the energy critical case $p=5$ : Any solution with initial data in the critical space $\\dot{H}^1 \\times L^2({\\mathbb {R}}^3)$ must scatter in both two time directions.",
"In other words, the asymptotic behaviour of any solution mentioned above resembles that of a free wave.",
"We expect that a similar result holds for other exponent $p$ as well.",
"Conjecture 1.1 Any solution to (CP1) with initial data $(u_0,u_1) \\in \\dot{H}^{s_p} \\times \\dot{H}^{s_p-1}$ must exist for all time $t \\in {\\mathbb {R}}$ and scatter in both two time directions.",
"This is still an open problem, although we do have progress in two different aspects:"
],
[
"Scattering result with a priori estimates",
"It has been proved that if a radial solution $u$ with a maximal lifespan $I$ satisfies an a priori estimate $\\sup _{t \\in I} \\left\\Vert (u(\\cdot ,t), u_t(\\cdot , t))\\right\\Vert _{\\dot{H}^{s_p} \\times \\dot{H}^{s_p-1} ({\\mathbb {R}}^3)} < +\\infty , $ then $u$ is defined for all time $t$ and scatters.",
"The proof uses a compactness-rigidity argument.",
"The compactness part is nowadays a standard procedure in the study of wave and Schödinger equations; while the rigidity part does depend on specific situations.",
"In fact, different methods were used for different range of $p$ 's.",
"The details can be found in Kenig-Merle [6] for $p>5$ , Shen [10] for $3<p<5$ and Dodson-Lawrie [1] for $1+\\sqrt{2}<p\\le 3$ .",
"The author would also like to mention that the same result still holds in the non-radial case in the energy supercritical case $p>5$ , as shown in the paper [7].",
"Finally please pay attention that (REF ) is automatically true in the energy critical case $p=5$ , as long as initial data are contained in the critical Sobolev space $\\dot{H}^1 \\times L^2$ , thanks to the energy conservation law."
],
[
"Strong Assumptions on Initial Data",
"There is also multiple scattering results if we assume that the initial data satisfy stronger regularity and/or decay conditions.",
"These results are usually proved via a suitable global space-time integral estimate.",
"In the energy sub-critical case $3\\le p < 5$ , the solutions always scatter if initial data satisfy an additional regularity-decay condition $ \\int _{{\\mathbb {R}}^3} \\left[(|x|^2+1) (|\\nabla u_0 (x)|^2 + |u_1(x)|^2) + |u_0(x)|^2 \\right] dx < \\infty .$ The main tool is the following conformal conservation law $\\frac{d}{dt} Q(t, u, u_t) = \\frac{4(3-p)t}{p+1} \\int _{{\\mathbb {R}}^3} \\left|u(x,t)\\right|^{p+1} dx.$ Here $Q(t,\\varphi ,\\psi ) = Q_0(t, \\varphi , \\psi ) + Q_1(t, \\varphi )$ is called the conformal charge with $Q_0(t,\\varphi ,\\psi ) &= \\left\\Vert x\\psi + t \\nabla \\varphi \\right\\Vert _{L^2({\\mathbb {R}}^3)}^2 + \\left\\Vert (t\\psi +2\\varphi )\\frac{x}{|x|} +|x|\\nabla \\varphi \\right\\Vert _{L^2({\\mathbb {R}}^3)}^2;\\\\Q_1(t, \\varphi ) &= \\frac{2}{p+1}\\int _{{\\mathbb {R}}^3} (|x|^2+t^2)|\\varphi (x,t)|^{p+1} dx.$ The assumption (REF ) guarantees the finiteness of conformal charge $Q(t,u,u_t)$ when $t=0$ .",
"It immediately gives a global space-time integral estimate $\\int _{|t|>1} \\int _{{\\mathbb {R}}^3} |u(x,t)|^{p+1}\\,dxdt \\lesssim _p \\sup _{t\\in {\\mathbb {R}}} Q_1(t, u(\\cdot ,t)) \\le \\sup _{t\\in {\\mathbb {R}}} Q(t,u,u_t) = Q(0,u_0,u_1)<+\\infty ,$ which then implies the scattering of solutions.",
"For more details please see [3], [5].",
"The author's previous work [11] proved the scattering result for $3 \\le p<5$ if initial data $(u_0, u_1) \\in \\dot{H}^1 \\times L^2$ are radial and satisfy $\\int _{{\\mathbb {R}}^3} (|x|+1)^{1+2\\varepsilon }\\left(|\\nabla u_0|^2 + |u_1|^2\\right) dx < \\infty $ for an arbitrary constant $\\varepsilon >0$ , by introducing a conformal transformation: If $u$ is a solution as assumed, then for any $t_0\\in {\\mathbb {R}}$ , the function $v(y, \\tau ) = \\frac{\\sinh |y|}{|y|} e^\\tau u \\left( e^\\tau \\frac{\\sinh |y|}{|y|}\\cdot y, t_0 + e^\\tau \\cosh |y|\\right), \\quad (y,\\tau ) \\in {\\mathbb {R}}^3 \\times {\\mathbb {R}}$ solves another wave equation $v_{\\tau \\tau } - \\Delta _y v = - \\left(\\frac{|y|}{\\sinh |y|}\\right)^{p-1} e^{-(p-3)\\tau } |v|^{p-1}v.$ We then apply a Morawetz-type estimate on the solutions $v$ of the second equation and rewrite it in the form of original solutions $u$ .",
"This helps to give a global space-time integral $\\Vert u\\Vert _{L^{2(p-1)} L^{2(p-1)}({\\mathbb {R}}\\times {\\mathbb {R}}^3)} < +\\infty $ and finishes the proof.",
"In the author's recent work [12] we proved the same scattering result for radial solutions under a weaker assumption on initial data $\\int _{{\\mathbb {R}}^3} (|x|^\\kappa +1)\\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2 + \\frac{1}{p+1}|u_0|^{p+1} \\right) < +\\infty .$ Here $\\kappa >\\kappa _1(p) = \\frac{3(5-p)}{p+3}$ is a constant.",
"The proof uses a detailed version of the classic Morawetz estimate (see Section REF below) to give a decay rate of the space-time integral $\\int _{-\\infty }^{+\\infty } \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt \\lesssim R^{-\\kappa }.$ This then gives the same estimate $\\Vert u\\Vert _{L^{2(p-1)} L^{2(p-1)}({\\mathbb {R}}\\times {\\mathbb {R}}^3)} < +\\infty $ and implies the scattering."
],
[
"Main Results",
"In this paper we always consider radial solutions to (CP1) with a finite energy.",
"Energy-subcriticality guarantees the global existence of the solutions.The goal of this work is two-fold.",
"We want to understand the spatial distribution of the energy as $t$ goes to infinity.",
"This gives plentiful information about the global behaviour of solutions.",
"If the energy of initial data satisfies an additional decay assumption, we prove the scattering results.",
"In this subsection we give two main theorems.",
"Theorem 1.2 Assume $3\\le p<5$ .",
"Let $u$ be a radial solution to (CP1) with a finite energy $E$ .",
"Then there exist a three-dimensional free wave $v^-(x,t)$ , with an energy $\\tilde{E}_-\\le E$ , so that We have scattering outside any backward light cone ($R \\in {\\mathbb {R}}$ ) $\\lim _{t \\rightarrow - \\infty } \\left\\Vert \\left(\\nabla u(\\cdot ,t), u_t(\\cdot ,t)\\right)- \\left(\\nabla v^- (\\cdot ,t), v_t^- (\\cdot ,t)\\right)\\right\\Vert _{L^2(\\lbrace x\\in {\\mathbb {R}}^3:|x|>R+|t|\\rbrace )} = 0.$ If we have $\\tilde{E}_- = E$ , then the scattering happens in the whole space in the negative time direction.",
"$\\lim _{t \\rightarrow - \\infty } \\left\\Vert \\left(u(\\cdot ,t), u_t(\\cdot ,t)\\right)- \\left(v^- (\\cdot ,t), v_t^- (\\cdot ,t)\\right)\\right\\Vert _{\\dot{H}^1\\times L^2({\\mathbb {R}}^3)} = 0.$ If $\\tilde{E}_- < E$ , the remaining energy (also called “retarded energy”) can be located: for any constants $c \\in (0,1)$ and $\\beta <\\frac{2(p-2)}{p+1}$ we have $\\lim _{t \\rightarrow - \\infty } \\int _{c|t|<|x|<|t|-|t|^\\beta } \\left(\\frac{1}{2}|\\nabla u(x,t)|^2 + \\frac{1}{2}|u_t(x,t)|^2 + \\frac{1}{p+1}|u(x,t)|^{p+1}\\right) dx = E - \\tilde{E}_-.$ The asymptotic behaviour in the positive time direction is similar.",
"Remark 1.3 If $\\tilde{E}_-<E$ , then the energy distribution is illustrated in figure REF .",
"The “gap” between scattering energy, which travels at the light speed, and “retarded energy”, which travels slightly slower, becomes wider and wider as time goes to infinity.",
"Figure: Illustration of travelling energyTheorem 1.4 Assume $3\\le p<5$ .",
"Let $\\kappa >\\kappa _0(p)=\\frac{5-p}{p+1}$ be a constant.",
"If $u$ is a radial solution to (CP1) with initial data $(u_0,u_1)$ so that $\\int _{{\\mathbb {R}}^3} (|x|^\\kappa +1)\\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2+\\frac{1}{p+1}|u_0|^{p+1}\\right) dx < +\\infty ,$ then $u$ must scatter in both two time directions.",
"More precisely, there exists $(v_0^\\pm ,v_1^\\pm ) \\in \\dot{H}^1 \\times L^2({\\mathbb {R}}^3)$ , so that $\\lim _{t \\rightarrow \\pm \\infty } \\left\\Vert \\begin{pmatrix} u(\\cdot ,t)\\\\ \\partial _t u(\\cdot ,t)\\end{pmatrix} -\\mathbf {S}_L (t)\\begin{pmatrix}u_0^\\pm \\\\ u_1^\\pm \\end{pmatrix}\\right\\Vert _{\\dot{H}^1 \\times L^2({\\mathbb {R}}^3)} = 0.$ Here $\\mathbf {S}_L (t)$ is the linear wave propagation operator.",
"Remark 1.5 The assumptions in our main theorems can not guarantee that $(u_0,u_1) \\in \\dot{H}^{s_p}\\times \\dot{H}^{s_p-1}$ .",
"For example, we can choose a radial function $u_0 \\in C^\\infty ({\\mathbb {R}}^3)$ with decay $&u_0(x) \\simeq |x|^{-\\frac{2(p+4)}{(p+1)^2}-\\varepsilon }; & &|\\nabla u_0(x)| \\simeq |x|^{-\\frac{2(p+4)}{(p+1)^2}-1-\\varepsilon };& &|x|\\gg 1.&$ Here $\\varepsilon $ is an sufficiently small positive constant.",
"One can check that $(u_0,0)$ satisfies all the assumptions on initial data in two main theorems but $u_0 \\notin L^{3(p-1)/2}({\\mathbb {R}}^3)$ .",
"The latter implies that $u_0 \\notin \\dot{H}^{s_p}({\\mathbb {R}}^3)$ since we have the Sobolev embedding $\\dot{H}^{s_p}({\\mathbb {R}}^3) \\hookrightarrow L^{3(p-1)/2}({\\mathbb {R}}^3)$ .",
"As a result we have $(u(\\cdot ,t),u_t(\\cdot ,t)) \\notin \\dot{H}^{s_p} \\times \\dot{H}^{s_p-1}({\\mathbb {R}}^3)$ for any time $t$ .",
"This is the reason why in Theorem REF we measure the distance of $u$ and free waves by $\\dot{H}^1 \\times L^2$ norm instead.",
"This is a phenomenon which has not been covered by previous results mentioned above.",
"All the solutions discussed in those results come with initial data $(u_0,u_1) \\in \\dot{H}^{s_p} \\times \\dot{H}^{s_p-1}({\\mathbb {R}}^3)$ ."
],
[
"The idea",
"In this subsection we give the main idea of this paper and outline the proof of main theorems.",
"The details can be found in later sections."
],
[
"Transformation to 1D",
"In order to take full advantage of our radial assumption, we use the following transformation: if $u$ is a radial solution to (CP1), then $w(r,t) = r u(x,t)$ , where $|x|=r$ , is a solution to one-dimensional wave equation $w_{tt} - w_{rr} = - \\frac{|w|^{p-1}w}{r^{p-1}}.$ A basic calculation shows that $&2\\pi \\int _a^b (|w_r(r,t)|^2+|w_t(r,t)|^2) dr \\nonumber \\\\&\\qquad = 2\\pi \\left[\\int _a^b \\left(r^2|u_r(r,t)|^2 + r^2|u_t(r,t)|^2\\right)dr +b|u(b,t)|^2 - a|u(a,t)|^2\\right].",
"$ Since for any radial $\\dot{H}^1({\\mathbb {R}}^3)$ function $f(r)$ , we have $\\lim _{r\\rightarrow 0^+} r|f(r)|^2 = \\lim _{r\\rightarrow +\\infty } r|f(r)|^2 = 0.$ It immediately follows that $2\\pi \\int _0^\\infty (|w_r(r,t)|^2+|w_t(r,t)|^2) dr = \\int _{{\\mathbb {R}}^3} \\left(\\frac{1}{2}|\\nabla u|^2 + \\frac{1}{2} |u_t|^2 \\right) dx.",
"$ The new solution $w$ also satisfies an energy conservation law $E(w,w_t) \\doteq 2\\pi \\int _0^\\infty (|w_r(r,t)|^2+|w_t(r,t)|^2+\\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{r-1}}) dr = E(u,u_t) = \\hbox{Const}.$ The main tool to understand spatial energy distribution is the energy flux formula for inward and outward energy."
],
[
"Inward and Outward Energy",
"Let us first define Definition 1.6 Let $w$ be a solution as above.",
"We define the inward and outward energy $E_- (t) & = \\pi \\int _{0}^{\\infty } \\left(\\left|w_r(r,t)+w_t(r,t)\\right|^2+\\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dr\\\\E_+ (t) & = \\pi \\int _{0}^{\\infty } \\left(\\left|w_r(r,t)-w_t(r,t)\\right|^2+\\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dr$ We also need to consider their truncated versions $E_- (t;r_1,r_2) & = \\pi \\int _{r_1}^{r_2} \\left(\\left|w_r(r,t)+w_t(r,t)\\right|^2+\\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dr\\\\E_+ (t;r_1,r_2) & = \\pi \\int _{r_1}^{r_2} \\left(\\left|w_r(r,t)-w_t(r,t)\\right|^2+\\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dr$ Remark 1.7 The inward energy travels toward the origin as time $t$ increases; the outward energy travels in the opposite direction, as indicated by their names."
],
[
"Energy Flux",
"We consider flux of inward and outward energies through either characteristic lines $t+r = \\hbox{Const}$ , $t-r = \\hbox{Const}$ or the $t$ -axis.",
"This helps to give the following results (a) Almost all energy is outward energy as $t\\rightarrow +\\infty $ .",
"$&\\lim _{t\\rightarrow +\\infty } E_-(t) = 0;& &\\lim _{t\\rightarrow +\\infty } E_+(t) = E.&$ (b) Almost all energy is inward energy as $t\\rightarrow -\\infty $ .",
"$&\\lim _{t\\rightarrow -\\infty } E_+(t) = 0;& &\\lim _{t\\rightarrow +\\infty } E_-(t) = E.&$ (c) The energy flux through characteristic lines are bounded from the above.",
"In particular, the following inequalities hold for any $s, \\tau \\in {\\mathbb {R}}$ .",
"$&\\int _{-\\infty }^s \\frac{|w(s-t,t)|^{p+1}}{(s-t)^{p-1}} dt \\lesssim _p E;& &\\int _\\tau ^\\infty \\frac{|w(t-\\tau ,t)|^{p+1}}{(t-\\tau )^{p-1}} dt \\lesssim _p E.&$"
],
[
"Asymptotic Behaviour",
"The characteristic line method gives $&\\frac{\\partial }{\\partial t}\\left[(w_r+w_t)(s-t,t)\\right] = -\\frac{|w|^{p-1}w(s-t,t)}{(s-t)^{p-1}};\\\\&\\frac{\\partial }{\\partial t}\\left[(w_r-w_t)(t-\\tau ,t)\\right] = \\frac{|w|^{p-1}w(t-\\tau ,t)}{(t-\\tau )^{p-1}}.$ Combining these two identities with part (c) above, we have the following convergence holds uniformly in any bounded interval.",
"$&\\lim _{t \\rightarrow -\\infty } (w_r+w_t)(s-t,t) = g_-(s);& &\\lim _{t \\rightarrow +\\infty } (w_r-w_t)(t-\\tau ,t) = g_+(\\tau ).&$ We also have the following $L^2$ convergence for any $s_0,\\tau _0\\in {\\mathbb {R}}$ .",
"$& \\left\\Vert w_r(t-\\tau ,t) - \\frac{1}{2}g_+(\\tau )\\right\\Vert _{L^2((-\\infty ,\\tau _0])} \\rightarrow 0,& &\\left\\Vert w_t(t-\\tau ,t) +\\frac{1}{2}g_+(\\tau )\\right\\Vert _{L^2((-\\infty ,\\tau _0])}\\rightarrow 0,& &\\hbox{as} \\; t\\rightarrow +\\infty ;&\\\\& \\left\\Vert w_r(s-t,t) - \\frac{1}{2}g_-(s)\\right\\Vert _{L^2([s_0,\\infty ))} \\rightarrow 0,& &\\left\\Vert w_t(s-t,t) -\\frac{1}{2}g_-(s)\\right\\Vert _{L^2([s_0,\\infty ))}\\rightarrow 0,& &\\hbox{as} \\; t\\rightarrow -\\infty .&$ This gives us the free waves $v^-(x,t)$ and $v^+(x,t)$ in the conclusion part (a) of theorem REF .",
"$&v^-(x,t) = \\frac{1}{2|x|}\\int _{t-|x|}^{t+|x|} g_-(s) ds;& &v^+(x,t) = \\frac{1}{2|x|}\\int _{t-|x|}^{t+|x|} g_+(\\tau ) d\\tau .&$ We can then prove Part (b) and (c) by considering the energy located in different regions via the energy flux formula.",
"More details are given in Section ."
],
[
"Morawetz Estimates",
"Another important ingredient of Theorem REF 's proof is a more detailed version of the classic Morawetz estimate as given below, which plays an key role in the author's recent work [12] as well.",
"This is a little different from the original inequality given by Perthame and Vega in the work [9], but can be deduced from the original one without difficulty.",
"Please see Subsection REF for more details.",
"$& \\frac{1}{2R}\\int _{-\\infty }^{+\\infty } \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt + \\frac{1}{4R^2} \\int _{-\\infty }^{+\\infty } \\int _{|x|=R} |u|^2 d\\sigma _R dt \\\\& \\qquad + \\frac{p-3}{2(p+1)R} \\int _{-\\infty }^{+\\infty } \\int _{|x|<R} |u|^{p+1} dx dt +\\frac{p-1}{2(p+1)} \\int _{-\\infty }^{+\\infty } \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt \\le E.$ If we pick up the last term in the left hand side and make $R\\rightarrow 0^+$ , we obtain the most frequently used Morawetz estimate: $\\int _{-\\infty }^{+\\infty } \\int _{{\\mathbb {R}}^3} \\frac{|u|^{p+1}}{|x|} dx dt \\le CE.$ In this work, however, we choose large radius $R$ in the long inequality above and observe an important fact: The first term in the left hand side itself is almost equal to $E$ .",
"This is because for almost all time $t \\in [-R,R]$ , as long as $t$ is not too closed to $R$ or $-R$ , almost all energy concentrates in the ball of radius $R$ , thanks to finite speed of propagation.",
"As a result, we discard all other terms in the left hand and focus on the first term: $\\int _{-\\infty }^{+\\infty } \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt \\le 2RE.$ We can substitute the right hand side by $\\int _{-R}^R \\int _{{\\mathbb {R}}^3} \\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt$ , and rewrite the inequality in another form $&\\int _{|t|>R} \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt\\\\& \\qquad \\le \\int _{-R}^{+R} \\int _{|x|>R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt.$ In physics this means that the total contribution of “retarded energy” when $t>|R|$ is always smaller or equal to the total contribution of energy which escapes the ball of radius $R$ in the time interval $[-R,R]$ .",
"This enables us to prove Theorem REF by a contradiction.",
"On one hand, our theory on energy distribution of solutions gives a lower bound of the contribution of “retarded energy”, unless the solution scatters.",
"On the other hand, if we assume that the initial data satisfy a suitable decay condition, we can find an upper bound of the escaping energy.",
"Thus we can simply make $R\\rightarrow +\\infty $ and compare the upper and lower bounds to finish the proof."
],
[
"The Structure of This Paper",
"This paper is organized as follows.",
"In section 2 we collect notations, recall the classic Morawetz estimates and give a few preliminary results.",
"Next in Section 3 we give a general formula for inward and outward energy flux.",
"This helps to prove the energy distribution properties of the solutions in Section 4.",
"Finally we prove the scattering of the solution $u$ under an additional decay assumption in the last section."
],
[
"The $\\lesssim $ symbol",
"We use the notation $A \\lesssim B$ if there exists a constant $c$ , so that the inequality $A \\le c B$ always holds.",
"In addition, a subscript of the symbol $\\lesssim $ indicates that the constant $c$ is determined by the parameter(s) mentioned in the subscript but nothing else.",
"In particular, $\\lesssim _1$ means that the constant $c$ is an absolute constant."
],
[
"Radial functions",
"Let $u(x,t)$ be a spatially radial function.",
"By convention $u(r,t)$ represents the value of $u(x,t)$ when $|x| = r$ ."
],
[
"Uniform Pointwise Estimates",
"In this subsection we first recall Lemma 2.1 (Please see Lemma 3.2 of Kenig and Merle's work [6]) If $u$ is a radial $\\dot{H}^1({\\mathbb {R}}^3)$ function, then $|u(r)| \\lesssim _1 r^{-1/2} \\Vert u\\Vert _{\\dot{H}^1({\\mathbb {R}}^3)}.$ Therefore we always have $|w(r,t)| \\lesssim _1 E^{1/2} r^{1/2}$ .",
"Because $u$ is not only a $\\dot{H}^1({\\mathbb {R}}^3)$ function but also an $L^{p+1}({\\mathbb {R}}^3)$ function, this can be further improved if $r$ is large.",
"Lemma 2.2 If $w: [0,\\infty ) \\rightarrow {\\mathbb {R}}$ satisfies $2\\pi \\int _{0}^\\infty \\left(|w_r(r)|^2 + \\frac{2}{p+1}\\frac{|w(r)|^{p+1}}{r^{p-1}}\\right)dr \\le E,$ then we have $|w(r)| \\lesssim _{p} E^{2/(p+3)} r^{(p-1)/(p+3)} $ .",
"First of all, we observe $|w(r)-w(r_0)| = \\left|\\int _{r_0}^r w_r(s) ds \\right| \\le |r-r_0|^{1/2} \\left(\\int _{r_0}^r |w_r(s)|^2 ds\\right)^{1/2} \\le (E/2\\pi )^{1/2} |r-r_0|^{1/2}.$ Thus $w(r)$ converges as $r\\rightarrow 0^+$ .",
"Since the singular integral of $|w(r)|^{p+1}/r^{p-1}$ near zero converges, it is clear that $w(0)=0$ .",
"Plugging $r_0=0$ in the estimate above, we re-discover the pointwise estimate $|w(r)|<(Er)^{1/2}$ .",
"Assume $|w(r_0)| = S \\le (Er_0)^{1/2}$ .",
"By the inequality above we have $|w(r)-w(r_0)|<S/2$ thus $|w(r)|>S/2$ for $r \\in [r_0,r_0+S^2/E]$ .",
"As a result $E \\ge \\int _{r_0}^{r_0+S^2/E} \\frac{|w(r)|^{p+1}}{r^{p-1}} \\ge \\frac{(S/2)^{p+1}}{(r_0+S^2/E)^{p-1}}\\cdot S^2/E \\ge \\frac{(S/2)^{p+1}}{(2r_0)^{p-1}}\\cdot S^2/E.$ This means $E \\gtrsim _p S^{p+3}/(r_0^{p-1}E)$ .",
"Thus we have $S \\lesssim _p E^{2/(p+3)} r_0^{(p-1)/(p+3)}$ and finish the proof."
],
[
"Morawetz Estimates",
"Theorem 2.3 (Please see Perthame and Vega's work [9], we use the 3-dimensional case) Let $u$ be a solution to (CP1) defined in a time interval $[0,T]$ with a finite energy $E$ .",
"Then we have the following inequality for any $R>0$ $& \\frac{1}{2R}\\int _0^T \\!\\!\\int _{|x|<R}(|\\nabla u|^2+|u_t|^2) dx dt + \\frac{1}{2R^2} \\int _0^T \\!\\!\\int _{|x|=R} |u|^2 d\\sigma _R dt + \\frac{p-2}{(p+1)R} \\int _0^T \\!\\!\\int _{|x|<R} |u|^{p+1} dx dt \\nonumber \\\\& \\qquad + \\frac{p-1}{p+1} \\int _0^T \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt + \\frac{1}{R^2} \\int _{|x|<R} |u(x,T)|^2 dx \\le 2E.",
"$ Remark 2.4 The notations $p$ and $E$ represent slightly different constants in the original paper [9] and this current paper.",
"Here we rewrite the inequality in the setting of the current work.",
"The coefficient before the integral $\\int _{B(0,R)} |u(T)|^2 dx$ was $\\frac{d^2-1}{4R^2}$ (in the 3-dimensional case $\\frac{2}{R^2}$ ) in the original paper.",
"But the author believes that this is a minor typing mistake.",
"It should have been $\\frac{d^2-1}{8R^2}$ instead.",
"Remark 2.5 The upper bound of time interval $T$ does not appear in any of the coefficients above.",
"We also have an energy conservation law.",
"As a result, we can substitute the time interval $[0,T]$ by any bounded time interval $[T_1,T_2]$ or even $(-\\infty ,T]$ .",
"If we ignore the final term on the left hand side, we can all use the time interval $(-\\infty ,\\infty )$ .",
"$& \\frac{1}{2R}\\int _{-\\infty }^{\\infty } \\!\\int _{|x|<R}(|\\nabla u|^2+|u_t|^2) dx dt + \\frac{1}{2R^2} \\int _{-\\infty }^{\\infty } \\!\\int _{|x|=R} |u|^2 d\\sigma _R dt\\nonumber \\\\& \\qquad + \\frac{p-2}{(p+1)R} \\int _{-\\infty }^{\\infty } \\!\\int _{|x|<R} |u|^{p+1} dx dt + \\frac{p-1}{p+1} \\int _{-\\infty }^{\\infty } \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt \\le 2E.",
"$ This Morawetz estimate plays two different roles in this work.",
"On one hand, it gives a few global integral estimates, which our theory of energy distribution depends on.",
"One of the most popular ones is $\\int _{-\\infty }^\\infty \\int _{{\\mathbb {R}}^3} \\frac{|u(x,t)|^{p+1}}{|x|} dx dt \\lesssim E$ .",
"On the other hand, this Morawetz estimate also gives direct information on energy distribution, as we mentioned in the introduction part.",
"Let us discuss these aspects one-by-one."
],
[
"Global Estimates",
"If we pick the second term in the left hand side of (REF ) and use the radial assumption, we obtain $\\sup _{r>0} \\int _{-\\infty }^{+\\infty } |u(r,t)|^2 dt \\le \\frac{E}{\\pi }.$ We recall $w = ru$ , use the Morawetz estimate (REF ) again and obtain $& \\int _{-\\infty }^{+\\infty } \\int _0^R (|w_r(r,t)|^2+|w_t(r,t)|^2) dr dt \\\\& \\qquad = \\int _{-\\infty }^{+\\infty } \\int _0^R r^2 (|u_r(r,t)|^2 +|u_t(r,t)|^2) dr dt + \\int _{-\\infty }^{+\\infty } R|u(R,t)|^2 dt \\le \\frac{RE}{\\pi }$ We can also pick the third term of (REF ) and rewrite it in term of $w$ $\\int _{-\\infty }^{+\\infty } \\int _0^R \\frac{|w|^{p+1}}{r^{p-1}} dr dt \\le \\frac{(p+1)}{2(p-2)\\pi }RE.$ Finally we pick the forth term, make $R\\rightarrow 0^+$ and rewrite it in term of $w$ $\\int _{-\\infty }^{+\\infty } \\int _{{\\mathbb {R}}^3} \\frac{|u|^{p+1}}{|x|} dx dt \\le \\frac{2(p+1)}{p-1}E\\Rightarrow \\int _{-\\infty }^{+\\infty } \\int _0^\\infty \\frac{|w|^{p+1}}{r^p} dr dt \\le \\frac{p+1}{2(p-1)\\pi } E.$ In summary, we have (The second line immediately follows the first line) Corollary 2.6 Let $u$ be a radial solution to (CP1) with a finite energy $E$ .",
"Then $u$ and $w = ru$ satisfy $\\int _{-\\infty }^{+\\infty } \\int _0^R \\left(|w_r(r,t)|^2+|w_t(r,t)|^2+\\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dr dt & \\lesssim _{p} RE;\\\\\\liminf _{r\\rightarrow 0^+} \\int _{-\\infty }^{+\\infty } \\left(|w_r(r,t)|^2+|w_t(r,t)|^2+\\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dt & \\lesssim _p E;\\\\\\int _{-\\infty }^{+\\infty } \\int _0^\\infty \\frac{|w(r,t)|^{p+1}}{r^p} dr dt & \\lesssim _{p} E.\\\\\\sup _{r>0} \\int _{-\\infty }^{+\\infty } |u(r,t)|^2 dt & \\lesssim _p E.$"
],
[
"Energy Distribution Information",
"We can combine part of the third term of (REF ) with the first term, divide both sides by 2 and obtain $& \\frac{1}{2R}\\int _{-\\infty }^{+\\infty } \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt + \\frac{1}{4R^2} \\int _{-\\infty }^{+\\infty } \\int _{|x|=R} |u|^2 d\\sigma _R dt\\nonumber \\\\& \\qquad + \\frac{p-3}{2(p+1)R} \\int _{-\\infty }^{+\\infty } \\int _{|x|<R} |u|^{p+1} dx dt +\\frac{p-1}{2(p+1)} \\int _{-\\infty }^{+\\infty } \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt \\le E. $ This helps to give a decay rate $\\int _{-\\infty }^\\infty \\int _{|x|>R} \\frac{|u(x,t)|^{p+1}}{|x|} dx dt \\lesssim R^{-\\kappa }$ in the author's recent work [12].",
"In this work, we only need a weaker version of this inequality.",
"Since every term in the left hand side is nonnegative, we can focus on the first term and obtain $\\int _{-\\infty }^{+\\infty } \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt \\le 2RE.$ We substitute the right hand side by $\\int _{-R}^R \\int _{{\\mathbb {R}}^3} \\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt$ , split the integral in the left hand side into two parts, with time $t\\in [-R,R]$ and $|t|>R$ respectively, combine the first part with the right hand side and finally obtain Proposition 2.7 For any $R>0$ , we have $&\\int _{|t|>R} \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt\\\\& \\qquad \\le \\int _{-R}^{+R} \\int _{|x|>R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt.$ The following will not be used in the argument of this paper, instead it is a corollary of our theory on energy distribution.",
"Remark 2.8 A careful review of Perthame and Vega's calculation shows that for a radial solution $u$ , we actually have an identity for any $R>0$ and $T_1<T_2$ $& \\frac{1}{2R}\\int _{T_1}^{T_2} \\int _{|x|<R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt + \\frac{1}{4R^2} \\int _{T_1}^{T_2} \\int _{|x|=R} |u|^2 d\\sigma _R dt\\nonumber \\\\& \\qquad + \\frac{p-3}{2(p+1)R} \\int _{T_1}^{T_2} \\int _{|x|<R} |u|^{p+1} dx dt +\\frac{p-1}{2(p+1)} \\int _{T_1}^{T_2} \\int _{|x|>R} \\frac{|u|^{p+1}}{|x|} dx dt \\\\= & 2\\pi \\left[\\int _0^R \\frac{r}{R} w_t(r,T_1) w_r(r,T_1) dr + \\int _R^\\infty w_t(r,T_1)w_r(r,T_1) dr \\right]\\\\& \\qquad -2\\pi \\left[\\int _0^R \\frac{r}{R} w_t(r,T_2) w_r(r,T_2) dr + \\int _R^\\infty w_t(r,T_2)w_r(r,T_2) dr \\right]$ By the identity $w_t w_r = \\frac{1}{4}\\left[(w_r+w_t)^2 - (w_r-w_t)^2\\right]$ , Proposition REF , Corollary REF and Lemma REF , we know that if we make $T_1\\rightarrow -\\infty $ , $T_2 \\rightarrow +\\infty $ , then the limit of the right hand side is exactly $E$ .",
"Therefore the inequality (REF ) is actually an identity.",
"Please note that the radial assumption is essential because we discard a term in the form of $\\int _{T_1}^{T_2} \\int _{|x|>R} {\\mathbf {D}} u \\cdot {\\mathbf {D}}^2 \\Phi \\cdot {\\mathbf {D}} u \\,dx dt$ in the left hand side, where ${\\mathbf {D}}^2 \\Phi (x)$ is a positive semidefinite matrix whose eigenvalue 0 has a single eigenvector $x$ .",
"This term vanishes only for radial solutions."
],
[
"Energy Flux for Inward and Outward Energies",
"In this section we consider the inward and outward energies given in Definition REF and give energy flux formula of them."
],
[
"General Energy Flux Formula",
"The following is a simple application of Green's Theorem, but it is indeed a general version of energy flux formula.",
"Proposition 3.1 (General Energy Flux) Let $\\Omega $ be a closed region in the right half $(0,\\infty )\\times {\\mathbb {R}}$ of $r-t$ space.",
"Its boundary $\\Gamma $ consists of finite line segments, which are paralleled to either $t$ -axis, $r$ -axis or characteristic lines $t\\pm r = 0$ , and is oriented counterclockwise.",
"Then we have an identity $\\pi \\int _{\\Gamma } \\left(|w_r+w_t|^2 + \\frac{2}{p+1}\\cdot \\frac{|w|^{p+1}}{r^{p-1}}\\right) dr & + \\left(|w_r+w_t|^2 - \\frac{2}{p+1}\\cdot \\frac{|w|^{p+1}}{r^{p-1}}\\right) dt \\nonumber \\\\& \\qquad -\\frac{2\\pi (p-1)}{p+1} \\iint _{\\Omega } \\frac{|w|^{p+1}}{r^p} dr dt = 0.",
"\\\\\\pi \\int _{\\Gamma } \\left(|w_r-w_t|^2 + \\frac{2}{p+1}\\cdot \\frac{|w|^{p+1}}{r^{p-1}}\\right) dr & + \\left(-|w_r-w_t|^2 + \\frac{2}{p+1}\\cdot \\frac{|w|^{p+1}}{r^{p-1}}\\right) dt \\nonumber \\\\&\\qquad + \\frac{2\\pi (p-1)}{p+1} \\iint _{\\Omega } \\frac{|w|^{p+1}}{r^p} dr dt = 0 .$ Furthermore, there exists a finite, nonnegative, continuousContinuity means the function $\\mu ((-\\infty ,t])$ is a continuous function of $t$ .",
"measure $\\mu $ with $\\mu ({\\mathbb {R}}) \\lesssim _p E$ , which is solely determined by $u$ and independent of $\\Omega $ , so that the identities above also hold for regions $\\Omega $ with part of its boundary on the $t$ -axis.",
"In this case the line integral from the point $(0,t_2)$ downward to $(0,t_1)$ along $t$ -axis is understood as $- \\pi \\int _{t_1}^{t_2} 1 \\,d\\mu (t)$ in identity (REF ) or $\\pi \\int _{t_1}^{t_2} 1 \\,d\\mu (t)$ in identity ()"
],
[
"Line integrals",
"Before we give the outline of the proof, let us first have a look at what the line integrals look like for different types of boundary line segments.",
"In table REF we always assume $r_1<r_2$ , $t_1<t_2$ .",
"For example, when we say a line segment $r=r_0$ goes downward, it starts at time $t_2$ and ends at time $t_1$ .",
"Table: Line integrals in energy flux"
],
[
"Physical Meaning",
"The physical meaning of the flux across characteristic lines, which corresponds to light cone in ${\\mathbb {R}}^3$ , and other related terms, are given as The terms in the form of $\\int _{t_1}^{t_2} |w_r\\pm w_t|^2 dt$ are the amount of energy which moves across characteristic lines due to liner wave propagation.",
"The terms in the form of $\\int _{t_1}^{t_2} \\frac{|w|^{p+1}}{r^{p-1}} dt$ are the amount of energy which moves across characteristic lines due to nonlinear effect.",
"The terms in the form of $\\int _{t_1}^{t_2} 1 d\\mu (t)$ are the amount of energy which moves across the line $r=0$ , thus transforms from inward energy to outward energy during the given period of time.",
"The double integral in the identities is the amount of energy which transform from inward energy to outward energy due to nonlinear effect in the given space-time region.",
"Now let us give a outline of proof for Proposition REF Without loss of generality let us prove the first identity in the proposition.",
"The second one can be prove in the same way.",
"If the region is away from the $t$ -axis, then we only need to apply Green's formula on the line integrals of the given vector fields and use the equation $w_{tt}-w_{rr} = - \\frac{|w|^{p-1}w}{r^{p-1}}$ .",
"In the process of calculation the second derivatives are involved, thus we apply smooth approximation techniques when necessary.",
"Now let us consider the case when part of the boundary is on the $t$ -axis.",
"In this case a limiting process $r \\rightarrow 0^+$ is required.",
"It suffices to show there exists a nondecreasing continuous function $P(t)$ with $P(+\\infty )-P(-\\infty ) \\lesssim _p E$ so that for all $t_1<t_2$ , we have $ \\lim _{r\\rightarrow 0^+} \\int _{t_1}^{t_2} \\left(|w_r(r,t)+w_t(r,t)|^2 - \\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dt = P(t_2)-P(t_1).$ We first choose $\\Omega = [r,1]\\times [t_1,t_2]$ , which is away from the $t$ -axis, thus we can use formula (REF ) and write $- \\int _{t_1}^{t_2} \\left(|w_r(r,t)+w_t(r,t)|^2 - \\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right) dt + E_-(t_1;r,1) & \\nonumber \\\\+\\int _{t_1}^{t_2} \\left(|w_r(1,t)+w_t(1,t)|^2 -\\frac{2}{p+1}\\cdot |w(1,t)|^{p+1}\\right) dt - E_-(t_2;r,1)& \\\\-\\frac{2\\pi (p-1)}{p+1} \\int _{t_1}^{t_2} \\int _r^1 \\frac{|w(r^{\\prime },t)|^{p+1}}{(r^{\\prime })^p} dr^{\\prime } dt & = 0 \\nonumber $ All the other four terms in the identity above converge as $r \\rightarrow 0^+$ , therefore we know the limit in (REF ) converges for any time $t_1<t_2$ .",
"Let us define $P(t) = \\lim _{r\\rightarrow 0^+} \\int _{0}^{t} \\left(|w_r(r,t^{\\prime })+w_t(r,t^{\\prime })|^2 - \\frac{2}{p+1}\\cdot \\frac{|w(r,t^{\\prime })|^{p+1}}{r^{p-1}}\\right) dt^{\\prime }.$ This guarantees that identity (REF ) always holds.",
"By (REF ) we also have $P(t) = & E_-(0;0,1) - E_-(t;0,1) + \\int _{0}^{t} \\left(|w_r(1,t^{\\prime })+w_t(1,t^{\\prime })|^2 -\\frac{2}{p+1}\\cdot |w(1,t^{\\prime })|^{p+1}\\right) dt^{\\prime } \\nonumber \\\\& -\\frac{2\\pi (p-1)}{p+1} \\int _{0}^{t} \\int _0^1 \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{(r^{\\prime })^p} dr^{\\prime } dt^{\\prime }.",
"$ We claim that $E_-(t;0,1) = \\pi \\int _0^1 \\left(|w_r(r,t)+w_t(r,t)|^2 + \\frac{2}{p+1}\\cdot \\frac{|w(r,t)|^{p+1}}{r^{p-1}}\\right)dr$ is a continuous function of $t$ by the following facts: $(u(\\cdot ,t),u_t(\\cdot ,t)) \\in C({\\mathbb {R}}_t; \\dot{H}^1 \\times L^2({\\mathbb {R}}^3)) \\Rightarrow (w_r(\\cdot ,t),w_t(\\cdot ,t)) \\in C({\\mathbb {R}}_t; L^2 \\times L^2)$ ; $4\\pi \\int _0^1 \\frac{|w(r,t)|^{p+1}}{r^{p-1}} dr = \\int _{B(\\mathbf {0},1)} |u(x,t)|^{p+1} dx$ ; The Sobolev embedding $\\dot{H}^1({\\mathbb {R}}^3) \\hookrightarrow L^{p+1}(B({\\mathbf {0}},1))$ ; Each term in the right hand side of (REF ) is a continuous function of $t$ , thus $P(t)$ is also a continuous function.",
"Furthermore, we can combine the inequality $w(r,t) \\lesssim _E r^{1/2}$ and Corollary REF to obtain $\\int _{-\\infty }^{\\infty } \\frac{|w(r,t)|^{p+1}}{r^{p-1}} dt = \\int _{-\\infty }^{\\infty } \\frac{|w(r,t)|^{p-1} r^2 |u(r,t)|^2}{r^{p-1}} dt\\lesssim _{p,E} r^{\\frac{5-p}{2}} \\int _{-\\infty }^\\infty |u(r,t)|^2 dt \\lesssim _{p,E} r^{\\frac{5-p}{2}}.",
"$ Thus we can redefine $P(t) = \\lim _{r\\rightarrow 0^+} \\int _{0}^{t} \\left(|w_r(r,t^{\\prime })+w_t(r,t^{\\prime })|^2 \\right) dt^{\\prime }.$ This means $P(t)$ is also a non-decreasing function with $P(0)=0$ .",
"By the monotonicity and continuity of $P(t)$ we can always define a Borel measure $\\mu $ by $\\mu ((a,b))=P(b)-P(a)$ .",
"This enable us to rewrite (REF ) into $\\lim _{r\\rightarrow 0^+} \\int _{t_1}^{t_2} \\left(|w_r(r,t)+w_t(r,t)|^2 - \\frac{2}{p+1}\\cdot \\frac{|w(r,t^{\\prime })|^{p+1}}{r^{p-1}}\\right) dt = \\int _{t_1}^{t_2} 1 d\\mu (t).$ Please note that it is unnecessary to specify whether the integral $\\int _{t_1}^{t_2} 1d\\mu (t)$ include the endpoints or not.",
"Because the continuity of $P(t)$ implies $\\mu (\\lbrace t_0\\rbrace )=0$ for any single point set $\\lbrace t_0\\rbrace $ .",
"Finally we use the second inequality of Corollary REF to deduce $\\mu ({\\mathbb {R}}) \\lesssim _p E$ .",
"Remark 3.2 If the radial solution $u(x,t)$ is sufficiently smooth at the origin, then a simple calculation shows that $d\\mu (t) = |u(0,t)|^2 dt$ .",
"In fact, we have $ \\lim _{r\\rightarrow 0^+} |w_r(r,t)+w_t(r,t)|^2 = \\lim _{r\\rightarrow 0^+} |w_r(r,t)-w_t(r,t)|^2 = |u(0,t)|^2$ If we follow the same procedure as above to prove the energy flux formula for outward energy, then the limiting process will give the same measure $\\mu $ because of (REF ) and smooth approximation techniques.",
"Another way to prove the energy flux formula for outward energy is to write $E_+ = E-E_-$ .",
"Remark 3.3 We can also prove $d\\mu (t) = |\\xi (t)|^2 dt$ .",
"Here $\\xi (t) \\in L^2({\\mathbb {R}})$ .",
"We prefer $|\\xi (t)|^2$ to a single $L^1$ function, because when $u$ is sufficiently smooth, we have $d\\mu (t) = |u(0,t)|^2 dt$ , as mentioned above.",
"Since the proof uses elements from later sections, we put the proof in Appendix for convenience."
],
[
"Notation for Flux",
"For convenience we write the energy flux across a characteristic line $t\\pm r = \\hbox{Const}$ , which corresponds to a light cone in ${\\mathbb {R}}^3$ in the following way.",
"Definition 3.4 (Notations of Energy flux) Given $s,\\tau \\in {\\mathbb {R}}$ , we define full energy flux $Q_-^- (s) & = \\frac{4\\pi }{p+1} \\int _{-\\infty }^{s} \\frac{|w(s-t,t)|^{p+1}}{(s-t)^{p-1}} dt;\\\\Q_+^-(s) & = 2\\pi \\int _{-\\infty }^{s} |w_r(s-t,t)-w_t(s-t,t)|^2 dt;\\\\Q_-^+(\\tau ) & = 2\\pi \\int _{\\tau }^{+\\infty } |w_r(t-\\tau ,t)+w_t(t-\\tau ,t)|^2 dt;\\\\Q_+^+(\\tau ) & = \\frac{4\\pi }{p+1} \\int _{\\tau }^{+\\infty } \\frac{|w(t-\\tau ,t)|^{p+1}}{(t-\\tau )^{p-1}} dt.$ A negative sign upper index means that this is energy flux across the characteristic line $t+r = s$ ; otherwise this is energy flux across the characteristic line $t-r = \\tau $ .",
"The lower index indicates whether this is energy flux of inward energy ($-$ ) or outward energy ($+$ ).",
"We can also consider their truncated version, which is the energy flux across a line segment of a characteristic line.",
"$Q_-^- (s; t_1,t_2) & = \\frac{4\\pi }{p+1} \\int _{t_1}^{t_2} \\frac{|w(s-t,t)|^{p+1}}{(s-t)^{p-1}} dt,& &t_1<t_2\\le s;\\\\Q_+^-(s; t_1,t_2) & = 2\\pi \\int _{t_1}^{t_2} |w_r(s-t,t)-w_t(s-t,t)|^2 dt,& &t_1<t_2\\le s; \\\\Q_-^+(\\tau ; t_1,t_2) & = 2\\pi \\int _{t_1}^{t_2} |w_r(t-\\tau ,t)+w_t(t-\\tau ,t)|^2 dt,& &\\tau \\le t_1<t_2;\\\\Q_+^+(\\tau ; t_1,t_2) & = \\frac{4\\pi }{p+1} \\int _{t_1}^{t_2} \\frac{|w(t-\\tau ,t)|^{p+1}}{(t-\\tau )^{p-1}} dt,& &\\tau \\le t_1<t_2.$"
],
[
"Energy Flux Formula for Triangle",
"We can apply our general energy flux formula for a triangle region.",
"This will be frequently used in Section .",
"Proposition 3.5 (Triangle Law) Given any $s_0>t_0$ , we can define $\\Omega = \\lbrace (r,t):t>t_0, r>0, r+t<s_0\\rbrace $ and write $E_-(t_0;0,s_0-t_0) = \\pi \\int _{t_0}^{s_0} 1\\mu (t) + Q_-^-(s_0;t_0,s_0) + \\frac{2\\pi (p-1)}{p+1} \\iint _{\\Omega } \\frac{|w(r,t)|^{p+1}}{{r}^p} dr dt.$ We can substitute $s_0$ by $t_0+r_0$ with $r_0>0$ and rewrite this into the form $E_-(t_0;0,r_0) = \\pi \\int _{t_0}^{t_0+r_0} 1\\mu (t) + Q_-^-(t_0+r_0;t_0,t_0+r_0) + \\frac{2\\pi (p-1)}{p+1} \\iint _{\\Omega } \\frac{|w(r,t)|^{p+1}}{{r}^p} dr dt.$"
],
[
"Limits of Inward and Outward Energies",
"All inward and outward energies are clearly bounded above by the full energy $E$ , since $E= E_-+E_+$ .",
"This immediately gives Lemma 4.1 (Boundedness of Energy Flux) We have a uniform upper bound for all $s, \\tau \\in {\\mathbb {R}}$ , $Q_{-}^{-}(s), Q_+^+ (\\tau ), \\pi \\int _{-\\infty }^{+\\infty } 1 d\\mu (t) \\le E.$ We apply triangle law (Proposition REF ) on the region $\\Omega (s,t_0) = \\lbrace (r,t): r+t\\le s, r>0, t>t_0\\rbrace $ for any $t_0<s$ and obtain $E_-(t_0;0,s-t_0) = \\pi \\int _{t_0}^s 1 d\\mu (t) + Q_-^-(s;t_0,s) + \\frac{2(p-1)\\pi }{p+1} \\int _{\\Omega (s,t_0)} \\frac{|w(r,t)|^{p+1}}{r^p} dr dt,$ as shown in figure REF .",
"Making $t_0 \\rightarrow -\\infty $ we have $\\pi \\int _{-\\infty }^s 1 d\\mu (t) + Q_-^-(s) + \\frac{2(p-1)\\pi }{p+1} \\iint _{r>0, r+t\\le s} \\frac{|w(r,t)|^{p+1}}{r^p} dr dt \\le E.$ Thus we have $Q_-^-(s) \\le E$ for any $s \\in {\\mathbb {R}}$ .",
"Finally we let $s \\rightarrow +\\infty $ in the inequality above to obtain $\\pi \\int _{-\\infty }^{\\infty } 1 d\\mu (t) \\le E$ .",
"The outward energy flux $Q_+^+$ can be dealt with in the same way.",
"Figure: Illustration for proof of Lemma Before we consider the monotonicity and asymptotic behaviour of inward and outward energy as $t \\rightarrow \\pm \\infty $ , let us first give a technical lemma.",
"Lemma 4.2 Given any $t_0\\in {\\mathbb {R}}$ , we have $\\liminf _{r\\rightarrow +\\infty } Q_-^-(t_0+r,t_0,t_0+r) = 0.$ Fix $r_0>0$ .",
"We consider the following integral and apply the change of variable $\\bar{r} = t_0+r-t$ $\\int _{r_0}^{2r_0} Q_{-}^{-} (t_0+r,t_0,t_0+r) dr & = \\frac{4\\pi }{p+1} \\int _{r_0}^{2r_0} \\int _{t_0}^{t_0+r} \\frac{|w(t_0+r-t,t)|^{p+1}}{(t_0+r-t)^{p-1}} dt dr \\\\& = \\frac{4\\pi }{p+1} \\iint _{\\Omega (r_0)} \\frac{|w(\\bar{r},t)|^{p+1}}{\\bar{r}^{p-1}} d\\bar{r} dt\\\\& \\le \\frac{8\\pi r_0}{p+1} \\iint _{\\Omega (r_0)} \\frac{|w(\\bar{r},t)|^{p+1}}{\\bar{r}^p} d\\bar{r} dt$ Here $\\Omega (r_0) = \\lbrace (\\bar{r},t):\\bar{r}>0,t>t_0,t_0+r_0<t+\\bar{r}<t_0+2r_0\\rbrace \\subset [0,2r_0]\\times {\\mathbb {R}}$ , as shown in figure REF .",
"Figure: Illustration of integral regionNow we are able to apply the mean value theorem to conclude there exists a number $r \\in [r_0,2r_0]$ so that $Q_{-}^{-} (t_0+r,t_0,t_0+r) \\le \\frac{8\\pi }{p+1} \\iint _{\\Omega (r_0)} \\frac{|w(\\bar{r},t)|^{p+1}}{\\bar{r}^p} d\\bar{r} dt.$ If we make $r_0 \\rightarrow +\\infty $ in the argument above and observe the fact $\\lim _{r_0 \\rightarrow + \\infty } \\iint _{\\Omega (r_0)} \\frac{|w(\\bar{r},t)|^{p+1}}{\\bar{r}^p} d\\bar{r} dt = 0,$ we obtain the lower limit as desired.",
"Now we have Proposition 4.3 (Monotonicity of Inward and Outward Energies) The inward energy $E_-(t)$ is a decreasing function of $t$ , while the outward energy $E_+(t)$ is an increasing function of $t$ .",
"In addition $&\\lim _{t\\rightarrow +\\infty } E_-(t) = 0; & &\\lim _{t\\rightarrow -\\infty } E_+(t) = 0.&$ Let us prove the monotonicity and limit of inward energy.",
"The outward energy can be dealt with in the same way.",
"First of all, we apply inward energy flux formula on the region $\\Omega = \\lbrace (r,t): r>0,t_1<t<t_2, r+t<s\\rbrace $ for $s\\ge t_2>t_1$ , as shown in the upper half of figure REF .",
"$& E_-(t_2; 0, s-t_2) - E_-(t_1;0,s-t_1) \\nonumber \\\\& \\quad = -\\pi \\int _{t_1}^{t_2} 1 d\\mu (t) - Q_-^-(s;t_1,t_2) - \\frac{2\\pi (p-1)}{p+1} \\iint _{\\Omega } \\frac{|w(r,t)|^{p+1}}{r^p} dr dt.",
"$ Now let us recall the inequality $|w(r,t)| \\lesssim _{p,E} r^{(p-1)/(p+3)}$ given by Lemma REF .",
"This implies $Q_-^-(s;t_1,t_2) \\rightarrow 0$ as $s\\rightarrow \\infty $ .",
"Therefore we can make $s\\rightarrow \\infty $ in the identity above and obtain $E_-(t_2) - E_-(t_1) = -\\pi \\int _{t_1}^{t_2} 1 d\\mu (t) - \\frac{2\\pi (p-1)}{p+1} \\int _{t_1}^{t_2} \\int _0^\\infty \\frac{|w(r,t)|^{p+1}}{r^p} dr dt < 0.$ This gives the monotonicity.",
"Next we apply triangle law with $t_0\\in {\\mathbb {R}}$ and $r_0>0$ , as shown in the lower part of figure REF $E_-(t_0;0,r_0) & = \\pi \\int _{t_0}^{t_0+r_0} 1 d\\mu (t) + Q_-^-(t_0+r_0; t_0, t_0+r_0) \\\\& \\qquad + \\frac{2\\pi (p-1)}{p+1} \\int _{t_0}^{t_0+r_0} \\int _0^{t_0+r_0-t} \\frac{|w(r,t)|^{p+1}}{r^p} dr dt.$ According to Lemma REF , we can take a limit $r_0 \\rightarrow +\\infty $ and write $E_-(t_0) \\le \\pi \\int _{t_0}^{\\infty } 1 d\\mu (t) + \\frac{2\\pi (p-1)}{p+1} \\int _{t_0}^{+\\infty } \\int _0^{+\\infty } \\frac{|w(r,t)|^{p+1}}{r^p} dr dt.",
"$ Finally we can make $t_0\\rightarrow +\\infty $ and finish the proof.",
"Figure: Illustration for proof of Proposition This immediately gives Corollary 4.4 We have the limits $&\\lim _{t\\rightarrow -\\infty } E_-(t) = E; & && &\\lim _{t\\rightarrow +\\infty } E_+(t) = E.&\\\\&\\lim _{t\\rightarrow \\pm \\infty } \\int _{0}^\\infty \\frac{|w(r,t)|^{p+1}}{r^{p-1}} dr = 0& &\\Leftrightarrow &&\\lim _{t\\rightarrow \\pm \\infty } \\int _{{\\mathbb {R}}^3} |u(x,t)|^{p+1} dx = 0.&$ We also have asymptotic behaviour of energy flux Proposition 4.5 We have the limits $&\\lim _{s \\rightarrow +\\infty } Q_-^-(s) = 0;& &\\lim _{\\tau \\rightarrow -\\infty } Q_+^+(\\tau ) = 0.&$ Again we only give the proof for inward energy flux.",
"An application of inward energy flux on the parallelogram $\\Omega = \\lbrace (r,t): t_0<t<s, s<r+t<s^{\\prime }\\rbrace $ with $t_0<s<s^{\\prime }$ gives (Please see figure REF ) $& E_-(s; 0, s^{\\prime }-s) - E_-(t_0;s-t_0,s^{\\prime }-t_0) \\nonumber \\\\&\\qquad = Q_-^-(s;t_0,s)- Q_-^-(s^{\\prime };t_0,s) - \\frac{2\\pi (p-1)}{p+1} \\int _{t_0}^s \\int _{s-t}^{s^{\\prime }-t} \\frac{|w(r,t)|^{p+1}}{r^p} dr dt.",
"\\nonumber $ Making $s^{\\prime }\\rightarrow \\infty $ we can discard the term $Q_-^-(s^{\\prime };t_0,s)$ as in the proof of Proposition REF and rewrite the identity above into $E_-(s) + \\frac{2\\pi (p-1)}{p+1} \\int _{t_0}^s \\int _{s-t}^{\\infty } \\frac{|w(r,t)|^{p+1}}{r^p} dr dt = E_-(t_0; s-t_0,\\infty ) + Q_-^-(s;t_0,s).$ Next we can take a limit as $t_0\\rightarrow - \\infty $ .",
"$E_-(s) + \\frac{2\\pi (p-1)}{p+1} \\int _{-\\infty }^s \\int _{s-t}^{\\infty } \\frac{|w(r,t)|^{p+1}}{r^p} dr dt = \\lim _{t\\rightarrow -\\infty } E_-(t;s-t,\\infty ) + Q_-^-(s).$ Finally we observe that both terms in left hand converges to zero as $s \\rightarrow +\\infty $ and finish the proof.",
"Figure: Illustration for proof of Proposition Remark 4.6 This asymptotic behaviour of $Q_-^-$ means that the inequality (REF ) is actually an identity as below, since now we have the limit of $Q_-^-$ vanishes rather than a lower limit.",
"$E_-(t) = \\pi \\int _{t}^{\\infty } 1 d\\mu (t^{\\prime }) + \\frac{2\\pi (p-1)}{p+1} \\int _{t}^{+\\infty } \\int _0^{+\\infty } \\frac{|w(r,t^{\\prime })|^{p+1}}{r^p} dr dt^{\\prime }.$ Making $t \\rightarrow -\\infty $ we have an identity $\\pi \\int _{-\\infty }^{\\infty } 1 d\\mu (t) + \\frac{2\\pi (p-1)}{p+1} \\int _{-\\infty }^{+\\infty } \\int _0^{+\\infty } \\frac{|w(r,t)|^{p+1}}{r^p} dr dt = E.$ This means that all the energy eventually transform from inward energy to outward energy by either passing through the origin or nonlinear effect when $t$ moves from $-\\infty $ to $+\\infty $ ."
],
[
"Asymptotic Behaviour of $w_r\\pm w_t$",
"We can rewrite the equation $w_{tt}-w_{rr} = -\\frac{|w|^{p-1}w}{r^{p-1}}$ into $\\left(\\partial _t -\\partial _r\\right)(\\partial _t + \\partial _r) w = - \\frac{|w|^{p-1}w}{r^{p-1}}.$ Therefore we have Proposition 4.7 Let $\\tau < t_1<t_2<s$ $(w_r+w_t)(s-t_2,t_2) - (w_r+w_t)(s-t_1,t_1) & = -\\int _{t_1}^{t_2} \\frac{|w|^{p-1}w(s-t,t)}{(s-t)^{p-1}} dt.\\\\(w_r-w_t)(t_2-\\tau ,t_2) - (w_r-w_t)(t_1-\\tau ,t_1) & = +\\int _{t_1}^{t_2} \\frac{|w|^{p-1}w(t-\\tau ,t)}{(t-\\tau )^{p-1}} dt.$ As a result, we can use the boundedness of energy flux $Q$ and obtain an estimate Proposition 4.8 Let $\\tau <t_1<t_2<s$ $\\left|(w_r+w_t)(s-t_2,t_2) - (w_r+w_t)(s-t_1,t_1)\\right| & \\lesssim _{p,E} (s-t_2)^{-\\frac{p-2}{p+1}}\\\\\\left|(w_r-w_t)(t_2-\\tau ,t_2) - (w_r-w_t)(t_1-\\tau ,t_1)\\right| & \\lesssim _{p,E} (t_1-\\tau )^{-\\frac{p-2}{p+1}}.$ Both estimates are proved in the same way.",
"Let us prove the first one.",
"According to Proposition REF , it suffices to show $I = \\left|\\int _{t_1}^{t_2} \\frac{|w|^{p-1}w(s-t,t)}{(s-t)^{p-1}} dt\\right| \\lesssim _{p,E} (s-t_2)^{-\\frac{p-2}{p+1}}$ This follows a straightforward calculation.",
"$I & \\le \\int _{t_1}^{t_2} \\frac{|w(s-t,t)|^{p}}{(s-t)^{p-1}} dt\\\\& \\le \\left(\\int _{t_1}^{t_2} \\left(\\frac{|w(s-t,t)|^p}{(s-t)^{\\frac{(p-1)p}{p+1}}}\\right)^{\\frac{p+1}{p}}dt\\right)^{\\frac{p}{p+1}}\\left(\\int _{t_1}^{t_2}\\left(\\frac{1}{(s-t)^{\\frac{p-1}{p+1}}}\\right)^{p+1} dt\\right)^{\\frac{1}{p+1}}\\\\& \\lesssim _p \\left(Q_-^-(s;t_1,t_2)\\right)^{\\frac{p}{p+1}}\\left[(s-t_2)^{2-p}-(s-t_1)^{2-p}\\right]^{\\frac{1}{p+1}}\\\\& \\lesssim _{p,E} (s-t_2)^{-\\frac{p-2}{p+1}}.$"
],
[
"The limit of $w_r \\pm w_t$",
"By Proposition REF , we immediately have pointwise limits $&\\lim _{t \\rightarrow -\\infty } (w_r+w_t)(s-t,t) = g_-(s);& &\\lim _{t \\rightarrow +\\infty } (w_r-w_t)(t-\\tau ,t) = g_+(\\tau ).&$ By Fatou's Lemma, we have $\\Vert g_-\\Vert _{L^2({\\mathbb {R}})}, \\Vert g_+\\Vert _{L^2({\\mathbb {R}})} \\le E/\\pi $ .",
"Let us define $&\\tilde{E}_- = \\pi \\int _{\\mathbb {R}}|g_-(s)|^2 ds;& &\\tilde{E}_+ = \\pi \\int _{\\mathbb {R}}|g_+(\\tau )|^2 d\\tau .&$ Energy flux of full energy gives us $& E(t;s-t,+\\infty ) \\le E(0;s,\\infty ), t<0,s>0& &\\Rightarrow & &\\lim _{s\\rightarrow +\\infty } \\sup _{t<0} \\int _{s-t}^\\infty |w_r(r,t)+w_t(r,t)|^2 dt = 0;&\\\\& E(t;t-\\tau ,+\\infty ) \\le E(0;-\\tau ,\\infty ),t>0, \\tau <0& &\\Rightarrow & &\\lim _{\\tau \\rightarrow -\\infty } \\sup _{t>0} \\int _{t-\\tau }^\\infty |w_r(r,t)-w_t(r,t)|^2 dt = 0.&$ As a result, we have Proposition 4.9 There exist functions $g_-(s), g_+(\\tau )$ with $\\Vert g_-\\Vert _{L^2({\\mathbb {R}})}^2, \\Vert g_+\\Vert _{L^2({\\mathbb {R}})}^2 \\le E/\\pi $ .",
"so that $&\\left|(w_r+w_t)(s-t,t) - g_-(s)\\right| \\lesssim _{p,E} (s-t)^{-\\frac{p-2}{p+1}},& &t<s;&\\\\&\\left|(w_r-w_t)(t-\\tau ,t) - g_+(\\tau )\\right| \\lesssim _{p,E} (t-\\tau )^{-\\frac{p-2}{p+1}},& &t>\\tau .&$ Furthermore we have the following $L^2$ convergence for any $s_0,\\tau _0 \\in {\\mathbb {R}}$ $&\\lim _{t \\rightarrow -\\infty } \\Vert (w_r+w_t)(s-t,t) - g_-(s)\\Vert _{L_s^2([s_0,\\infty ))} = 0; \\\\&\\lim _{t \\rightarrow +\\infty } \\Vert (w_r-w_t)(t-\\tau ,t) - g_+(\\tau )\\Vert _{L_\\tau ^2((-\\infty , \\tau _0])} = 0.$"
],
[
"The scattering target",
"Now we can define $&V^-(r,t) = \\frac{1}{2}\\int _{t-r}^{t+r} g_-(s) ds;& &V^+(r,t) = \\frac{1}{2}\\int _{t-r}^{t+r} g_+(\\tau ) d\\tau .&$ One can check $(V^\\pm ,V_t^\\pm ) \\in C({\\mathbb {R}}_t; \\dot{H}^1\\times L^2)$ are one-dimensional free waves.",
"We can also define $v^\\pm (x,t) = V^\\pm (|x|,t)/|x|$ as three-dimensional free waves with energy $\\tilde{E}_\\pm $ .",
"By Proposition REF and the fact $\\displaystyle \\lim _{t\\rightarrow \\pm \\infty } E_\\mp (t) = 0$ (given by Corollary REF ) we can conduct a simple calculation and conclude for $s_0,\\tau _0 \\in {\\mathbb {R}}$ $& \\lim _{t\\rightarrow -\\infty } \\left\\Vert \\begin{pmatrix} w_r(\\cdot ,t)\\\\ w_t(\\cdot ,t) \\end{pmatrix} -\\begin{pmatrix} V_r^-(\\cdot ,t)\\\\ V_t^-(\\cdot ,t) \\end{pmatrix} \\right\\Vert _{L^2([s_0-t,+\\infty ))} = 0.",
"\\\\& \\lim _{t\\rightarrow +\\infty } \\left\\Vert \\begin{pmatrix} w_r(\\cdot ,t)\\\\ w_t(\\cdot ,t) \\end{pmatrix} -\\begin{pmatrix} V_r^+(\\cdot ,t)\\\\ V_t^+(\\cdot ,t) \\end{pmatrix} \\right\\Vert _{L^2([t-\\tau _0,+\\infty ))} = 0.",
"\\nonumber $ Rewriting this in term of the original solution $u$ and $v^\\pm $ via identity (REF ), we obtain Part (a) of Theorem REF .",
"Next we prove Part (b), i.e.",
"an equivalent condition for scattering.",
"Proposition 4.10 If $\\tilde{E}_- = E$ , then the solution $u$ scatters in the negative time direction $\\lim _{t\\rightarrow -\\infty } \\left\\Vert \\begin{pmatrix} u(\\cdot ,t)\\\\ u_t(\\cdot ,t) \\end{pmatrix} -\\begin{pmatrix} v^-(\\cdot ,t)\\\\ v_t^-(\\cdot ,t) \\end{pmatrix} \\right\\Vert _{\\dot{H}^1\\times L^2({\\mathbb {R}}^3)} = 0.$ Given $\\varepsilon >0$ , there exists a number $s_0 \\in {\\mathbb {R}}$ , so that $\\pi \\int _{-\\infty }^{s_0} |g_-(s)|^2 ds < \\varepsilon $ .",
"According to Proposition REF , for sufficiently large negative time $t<t_1$ , we have $E(t;s_0-t,\\infty ) \\ge \\pi \\int _{s_0-t}^\\infty \\left|(w_r+w_t)(r,t)\\right|^2 dr > E-\\varepsilon \\Rightarrow E(t;0,s_0-t)<\\varepsilon .",
"$ By the definition of $V^-$ and our assumption on $s_0$ we also have $& \\pi \\int _{0}^{s_0-t} \\left|(V_r^- +V_t^-)(r,t)\\right|^2 dr = \\pi \\int _{t}^{s_0} |g_-(s)|^2 ds < \\varepsilon ,& &\\hbox{for}\\; t<s_0;&\\\\& \\pi \\int _{0}^{s_0-t} \\left|(V_r^- -V_t^-)(r,t)\\right|^2 dr = \\pi \\int _{2t-s_0}^{t} |g_-(s)|^2 ds \\rightarrow 0,& &\\hbox{as}\\; t \\rightarrow -\\infty .&$ Therefore we have $2\\pi \\int _0^{s_0-t} \\left(|V_r^-|^2 + |V_t^-|^2\\right) dr < \\varepsilon $ for sufficiently small $t<t_2$ .",
"Combining this with (REF ) we have the following inequality for any $t<\\min \\lbrace t_1,t_2\\rbrace $ : $2\\pi \\int _0^{s_0-t} \\left(|V_r^- -w_r|^2 + |V_t^- -w_r|^2\\right) dr < 4\\varepsilon .$ Combining this with (REF ), we know for sufficiently large negative $t$ $2\\pi \\int _0^{+\\infty } \\left(|V_r^- -w_r|^2 + |V_t^- -w_r|^2\\right) dr < 4\\varepsilon .$ We can rewrite this in terms of $u$ and $v^-$ by (REF ) and finish the proof."
],
[
"Remaining Energy",
"Proposition REF implies that if we did not have the scattering in the negative time direction, then the difference $E-\\tilde{E}_-$ would be positive.",
"Let us try to locate the remaining energy $E-\\tilde{E}_-$ .",
"Proposition REF tells us that the location is inside the ball $B({\\mathbf {0}}, s_0-t)$ when $|t|$ is sufficiently large for any given $s_0$ .",
"In other words, this part of energy eventually enters and stays in any backward light cone as $t\\rightarrow -\\infty $ .",
"It travels slower than the light.",
"The following lemma, however, shows that its out-going speed is closed to the light speed.",
"Lemma 4.11 Given $c\\in (0,1)$ , we have the limit $\\lim _{t \\rightarrow \\pm \\infty } E_{\\pm }(t;0,c|t|) = 0.$ Again we only need to give a proof for inward energy.",
"First of all, we can apply the triangle law on the triangle region $\\lbrace (r^{\\prime },t^{\\prime }): t^{\\prime }>t, r^{\\prime }>0, t^{\\prime }+r^{\\prime }<t+r\\rbrace $ with $c|t|<r<\\frac{c+1}{2}|t|$ .",
"$E_-(t;0,r) & = \\pi \\int _{t}^{t+r} 1 d\\mu (t^{\\prime }) + Q_-^-(t+r; t, t+r) \\\\& \\qquad + \\frac{2\\pi (p-1)}{p+1} \\int _{t}^{t+r} \\int _0^{t+r-t^{\\prime }} \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{r^{\\prime p}} dr^{\\prime } dt^{\\prime }.$ By the upper bound of $r$ we have $E_-(t;0,r) \\le P(t) + Q_-^-(t+r; t, t+r)$ where $P(t) = \\pi \\int _{t}^{\\frac{1-c}{2}t} 1 d\\mu (t^{\\prime }) + \\frac{2\\pi (p-1)}{p+1} \\int _{t}^{\\frac{1-c}{2}t} \\int _0^{\\frac{1-c}{2}t-t^{\\prime }} \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{r^{\\prime p}} dr^{\\prime } dt^{\\prime }$ satisfies $\\displaystyle \\lim _{t\\rightarrow -\\infty } P(t) = 0$ .",
"We can integrate the inequality above for $r \\in \\left(c|t|,\\frac{c+1}{2}|t|\\right)$ and obtain $\\frac{1-c}{2}|t| \\cdot E_{-}(t,0,c|t|) & \\le \\int _{c|t|}^{\\frac{c+1}{2}|t|} E_-(t,0,r) dr \\nonumber \\\\& \\le \\frac{1-c}{2}|t|P(t) + \\int _{c|t|}^{\\frac{c+1}{2}|t|} Q_-^-(t+r;t,t+r) dr. $ Following the same argument in Lemma REF , as shown in figure REF , we have $\\int _{c|t|}^{\\frac{c+1}{2}|t|} Q_-^-(t+r;t,t+r) dr & = \\frac{4}{p+1} \\iint _{\\Omega (t)} \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{r^{\\prime p-1}} dr^{\\prime } dt^{\\prime }\\\\& \\le \\frac{2(1+c)|t|}{p+1} \\iint _{\\Omega (t)} \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{r^{\\prime p}} dr^{\\prime } dt^{\\prime }.$ Here $\\Omega (t) = \\left\\lbrace (r^{\\prime },t^{\\prime }): t^{\\prime }>t,r^{\\prime }>0,(1-c)t<t^{\\prime }+r^{\\prime }<\\frac{1-c}{2}t\\right\\rbrace \\subset \\lbrace (r^{\\prime },t^{\\prime }): r^{\\prime }\\le \\frac{c+1}{2}|t|\\rbrace $ .",
"Plugging this upper bound in (REF ) and dividing both sides by $\\frac{1-c}{2}|t|$ , we obtain $E_{-}(t,0,c|t|) \\le P(t) + \\frac{4(1+c)}{(p+1)(1-c)} \\iint _{\\Omega (t)} \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{r^{\\prime p}} dr^{\\prime } dt^{\\prime }.$ Now we can take the limit $t\\rightarrow -\\infty $ on both sides and finish the proof.",
"Figure: Illustration for proof of Proposition This shows that the remaining energy is in the sphere shell $\\lbrace x: c|t|<|x|<|t|\\rbrace $ as $|t|$ is large.",
"The upper bound of $|x|$ can be further improved, so that we can locate the remaining energy in a region further and further away from the light cone $|x|=|t|$ as $t$ goes to $-\\infty $ .",
"This is the reason why we give this remaining energy another name “retarded energy”.",
"Lemma 4.12 Given any $\\beta < \\frac{2(p-2)}{p+1}$ , we have the limit $\\lim _{t \\rightarrow - \\infty } E_{-}(t;|t|-|t|^\\beta ,+\\infty ) = \\tilde{E}_-.$ Our conclusion is a combination of the following two limits.",
"Here we can choose $s_0=0$ in identity ().",
"$\\lim _{t \\rightarrow - \\infty } E_{-}(t;|t|-|t|^\\beta ,|t|) &= \\pi \\int _{-\\infty }^0 |g_-(s)|^2 ds;& \\\\\\lim _{t \\rightarrow - \\infty } E_{-}(t;s_0-t,+\\infty ) &= \\pi \\int _{s_0}^\\infty |g_-(s)|^2 ds, \\quad s_0 \\in {\\mathbb {R}}.& $ We start by proving the first one.",
"Let $I = \\left(\\int _{-\\infty }^0 |g_-(s)|^2 ds\\right)^{1/2}$ .",
"By Corollary REF we only need to show $\\lim _{t \\rightarrow -\\infty } \\left\\Vert (w_r+w_t)(r,t)\\right\\Vert _{L^2([|t|-|t|^\\beta ,|t|])} = \\lim _{t \\rightarrow -\\infty } \\left\\Vert (w_r+w_t)(s-t,t)\\right\\Vert _{L_s^2([-|t|^\\beta ,0])} = I.$ Since we have $\\displaystyle \\lim _{t \\rightarrow -\\infty } \\Vert g_-\\Vert _{L^2([-|t|^\\beta ,0])} = I$ , it suffices to show $\\lim _{t\\rightarrow -\\infty } \\left\\Vert (w_r+w_t)(s-t,t) - g_-(s)\\right\\Vert _{L_s^2([-|t|^\\beta ,0])} = 0.$ This immediately follows the pointwise estimate given in Proposition REF .",
"The same argument as above with the $L^2$ convergence part of Proposition REF instead proves the second limit ().",
"Combining Lemma REF , Lemma REF , Proposition REF and Corollary REF , we are able to prove Part (c) of Theorem REF , i.e.",
"we have the following limits for any $c\\in (0,1)$ and $0<\\beta <\\frac{2(p-2)}{p+1}$ : $\\lim _{t \\rightarrow - \\infty } E_{-}(t;c|t|,|t|-|t|^\\beta ) = \\lim _{t \\rightarrow -\\infty } E(t;c|t|,|t|-|t|^\\beta ) = E - \\tilde{E}_-.$ Remark 4.13 A combination of identity () and Corollary REF gives ($s_0 \\in {\\mathbb {R}}$ ) $\\lim _{t \\rightarrow -\\infty } E_-(t;0,s_0-t) = E-\\pi \\int _{s_0}^\\infty |g_-(s)|^2 ds.$ Remark 4.14 If we apply triangle law on the triangle region $\\Omega (s,t_0) = \\lbrace (r,t): r>0,t>t_0,r+t<s\\rbrace $ , make $t_0\\rightarrow -\\infty $ with Remark REF in mind and finally consider the limit as $s \\rightarrow -\\infty $ , we obtain another expression of the retarded energy $E - \\tilde{E}_- = \\lim _{s \\rightarrow -\\infty } Q_-^-(s).$"
],
[
"Scattering with Additional Decay on Initial Data",
"In this section we prove Theorem REF , i.e.",
"the solution to (CP1) scatters in both time directions if the initial data satisfy additional decay assumptions.",
"The proof is by a contradiction.",
"If the solution failed to scatter in the negative direction, we would have $\\tilde{E}_-<E$ ."
],
[
"Additional Contribution by Retarded Energy",
"According to Part (c) of Theorem REF , given any $\\beta < \\frac{2(p-2)}{p+1}$ , there exists a negative time $t_1$ , so that the inequality $\\int _{|x|<|t|-|t|^\\beta } \\left(\\frac{1}{2}|\\nabla u(x,t)|^2 + \\frac{1}{2}|u_t(x,t)|^2 + \\frac{1}{p+1}|u(x,t)|^{p+1}\\right) dx > \\frac{E-\\tilde{E}_-}{2}.$ holds for any time $t<t_1$ .",
"If $R$ is a large number $R>|t_1|$ and $t \\in (-R-R^\\beta ,-R)$ , we have $R<|t|<R+R^\\beta \\Rightarrow |t|^\\beta >R^\\beta $ .",
"Thus $|t|-|t|^\\beta < (R+R^\\beta ) - R^\\beta = R$ .",
"This means $\\int _{|x|<R} \\left(\\frac{1}{2}|\\nabla u(x,t)|^2 + \\frac{1}{2}|u_t(x,t)|^2 + \\frac{1}{p+1}|u(x,t)|^{p+1}\\right) dx > \\frac{E-\\tilde{E}_-}{2}.$ As a result we have for sufficiently large $R$ : $ \\int _{-R-R^\\beta }^{-R} \\int _{|x|<R} \\left(\\frac{1}{2}|\\nabla u(x,t)|^2 + \\frac{1}{2}|u_t(x,t)|^2 + \\frac{1}{p+1}|u(x,t)|^{p+1}\\right) dx dt > \\frac{E-\\tilde{E}_-}{2}\\cdot R^\\beta .$ This gives a lower bound of the left hand side of the inequality in Proposition REF ."
],
[
"Upper Bound on Energy Leak",
"Now let us give an upper bound on the amount of energy escaping the ball $B(0,R) = \\lbrace x \\in {\\mathbb {R}}^3 :|x|<R\\rbrace $ for time $t \\in [-R,R]$ under our decay assumption on the energy.",
"In fact we have Proposition 5.1 Let $u$ be a solution to (CP1) with a finite energy and satisfy $I = \\int _{{\\mathbb {R}}^3} |x|^{\\kappa } \\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2 + \\frac{1}{p+1}|u_0|^{p+1}\\right) dx < \\infty .$ Then we have the function $I(t) = \\int _{|x|>|t|} (|x|-|t|)^{\\kappa } \\left(\\frac{1}{2}|\\nabla u|^2 + \\frac{1}{2}|u_t|^2 + \\frac{1}{p+1}|w|^{p+1}\\right) dx \\le I, \\quad t\\in {\\mathbb {R}}.$ Since the wave equation is time-invertible, it suffices to prove this inequality for $t>0$ .",
"A basic calculation then shows $I^{\\prime }(t)\\le 0$ for $t>0$ .",
"Please see [12] for more details."
],
[
"Escaping Energy",
"Given any $t \\in (-R,R)$ , we have $& \\int _{|x|>R}\\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx\\\\& \\qquad \\le (R-|t|)^{-\\kappa } \\int _{|x|>R} (|x|-|t|)^{\\kappa } \\left(\\frac{1}{2}|\\nabla u|^2 + \\frac{1}{2}|u_t|^2 + \\frac{1}{p+1}|w|^{p+1}\\right) dx\\\\& \\qquad \\le (R-|t|)^{-\\kappa } I(t) \\le (R-|t|)^{-\\kappa } I.$ We integrate $t$ from $-R$ to $R$ and obtain $ \\int _{-R}^R \\int _{|x|>R} \\left(\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}|u_t|^2+\\frac{1}{p+1}|u|^{p+1}\\right) dx dt \\le \\frac{2}{1-\\kappa } R^{1-\\kappa } I.$ This is an upper bound of the right hand side of the inequality in Proposition REF ."
],
[
"Completion of the Proof",
"Plugging both the lower bound (REF ) and the upper bound (REF ) in the inequality given by Proposition REF , we obtain the following inequality for any given $\\beta <\\frac{2(p-2)}{p+1}$ and sufficiently large $R>R(u,\\beta )$ .",
"$\\frac{E-\\tilde{E}_-}{2}\\cdot R^\\beta \\le \\frac{2}{1-\\kappa } R^{1-\\kappa } I.$ If $\\kappa >\\kappa _0(p) = 1-\\frac{2(p-2)}{p+1}$ , we can always choose $\\beta <\\frac{2(p-2)}{p+1}$ so that $\\beta >1-\\kappa $ .",
"This immediately gives a contradiction for sufficiently large $R$ 's thus finishes the proof."
],
[
"Appendix",
"In this final section we prove the measure $\\mu $ in the energy flux formula satisfies $d\\mu = |\\xi (t)|^2 dt$ .",
"Here $\\xi (t)$ satisfies $\\pi \\Vert \\xi (t)\\Vert _{L^2({\\mathbb {R}})}^2 \\le E$ .",
"Therefore we can substitution $\\pi \\int _{t_1}^{t_2} 1d\\mu (t)$ by $\\pi \\int _{t_1}^{t_2} |\\xi (t)|^2 dt$ in the energy flux formula.",
"It suffices to show the function $P(t) = \\mu ((\\infty ,t])$ is an absolutely continuous function.",
"The idea is to find a relation between the measure $\\mu $ and the function $g_+(\\tau ) \\in L^2$ introduced in Proposition REF .",
"In fact $\\mu $ and $g_+(\\tau )$ are limits of $w_r(r,r+\\tau )-w_t(r,r+\\tau )$ as $r\\rightarrow 0^+$ and $r\\rightarrow +\\infty $ , respectively.",
"We start by Lemma 6.1 Given any interval $I = (\\tau _1,\\tau _2)$ , we always have $\\mu (I) \\le 2 \\int _{\\tau _1}^{\\tau _2} |g_+(\\tau )|^2 d \\tau + 2 \\int _{\\tau _1}^{\\tau _2} \\left(\\int _\\tau ^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt\\right)^2 d\\tau .$ First of all, we apply energy flux formula of the outward energy on the region $\\lbrace (r,t): 0<r<r^{\\prime }, \\tau _1<t-r<\\tau _2\\rbrace $ , then let $r^{\\prime } \\rightarrow 0^+$ and obtain $\\pi \\mu (I) = \\pi \\lim _{r^{\\prime }\\rightarrow 0^+} \\int _{\\tau _1+r^{\\prime }}^{\\tau _2+r^{\\prime }} \\left(|w_r(r^{\\prime },t^{\\prime })-w_t(r^{\\prime },t^{\\prime })|^2 - \\frac{2}{p+1}\\cdot \\frac{|w(r^{\\prime },t^{\\prime })|^{p+1}}{{r^{\\prime }}^{p-1}}\\right) dt^{\\prime }.$ We recall the estimate (REF ), apply change of variable $\\tau = t^{\\prime }-r^{\\prime }$ $\\mu (I) = \\lim _{r^{\\prime }\\rightarrow 0^+} \\int _{\\tau _1+r^{\\prime }}^{\\tau _2+r^{\\prime }} |w_r(r^{\\prime },t^{\\prime })-w_t(r^{\\prime },t^{\\prime })|^2 dt^{\\prime } = \\lim _{r^{\\prime }\\rightarrow 0^+} \\int _{\\tau _1}^{\\tau _2} |w_r(r^{\\prime },\\tau +r^{\\prime })-w_t(r^{\\prime },\\tau +r^{\\prime })|^2 d\\tau .$ By Proposition REF and Proposition REF we have the following upper bound $|w_r(r^{\\prime },\\tau +r^{\\prime })-w_t(r^{\\prime },\\tau +r^{\\prime })| \\le |g_+(\\tau )| + \\int _{\\tau +r^{\\prime }}^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt.$ We put this upper bound into the expression of $\\mu (I)$ above, $\\mu (I) & \\le \\limsup _{r^{\\prime }\\rightarrow 0^+} \\left(2\\int _{\\tau _1}^{\\tau _2} |g_+(\\tau )|^2 d\\tau + 2\\int _{\\tau _1}^{\\tau _2} \\left(\\int _{\\tau +r^{\\prime }}^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt\\right)^2 d\\tau \\right)\\\\& \\le 2\\int _{\\tau _1}^{\\tau _2} |g_+(\\tau )|^2 d\\tau + 2\\int _{\\tau _1}^{\\tau _2} \\left(\\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt\\right)^2 d\\tau .$ Let us have a look at the upper bound given by Lemma REF .",
"The first term is the integral of an integrable function over the same interval $I$ .",
"This will not make any double.",
"Now let us deal with the second term.",
"We first introduce a new notation Definition 6.2 Given $\\tau \\in {\\mathbb {R}}$ , we define $M(\\tau ) = \\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^{p+1}}{(t-\\tau )^p} dt.$ We claim $M \\in L^1({\\mathbb {R}})$ .",
"Because we can apply the change of variable $\\tau = t-r$ and obtain $\\int _{-\\infty }^{\\infty } M(\\tau ) d\\tau = \\int _{-\\infty }^{\\infty } \\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^{p+1}}{(t-\\tau )^p} dt d\\tau = \\int _{-\\infty }^\\infty \\int _0^\\infty \\frac{|w(r,t)|^{p+1}}{r^p} dr dt \\lesssim _p E.$ Lemma 6.3 For all $\\tau \\in {\\mathbb {R}}$ we have $\\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt \\lesssim _p Q_+^+(\\tau )^{\\frac{2}{p+1}}M(\\tau )^\\frac{p-2}{p+1}.$ We split the integral into two parts $\\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt = \\int _{\\tau }^{T} \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt + \\int _{T}^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt$ Following the same argument as in the proof of Proposition REF , we can find an upper bound of the second term above $\\int _{T}^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt \\lesssim _p (T-\\tau )^{-\\frac{p-2}{p+1}}Q_+^+(\\tau )^{\\frac{p}{p+1}}.$ Next we find an upper bound of the first term using $M(\\tau )$ $\\int _{\\tau }^{T} \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt & \\le \\left(\\int _{\\tau }^T \\left(\\frac{|w(t-\\tau ,t)|^{p}}{(t-\\tau )^{\\frac{p^2}{p+1}}}\\right)^{\\frac{p+1}{p}}dt\\right)^{\\frac{p}{p+1}} \\left(\\int _\\tau ^T \\left((t-\\tau )^{\\frac{1}{p+1}}\\right)^{p+1}dt \\right)^{\\frac{1}{p+1}}\\\\& \\le (T-\\tau )^\\frac{2}{p+1} M(\\tau )^{\\frac{p}{p+1}}.$ In summary, we have for any $T>\\tau $ $\\int _{\\tau }^\\infty \\frac{|w(t-\\tau ,t)|^p}{(t-\\tau )^{p-1}} dt \\lesssim _p (T-\\tau )^\\frac{2}{p+1} M(\\tau )^{\\frac{p}{p+1}} +(T-\\tau )^{-\\frac{p-2}{p+1}}Q_+^+(\\tau )^{\\frac{p}{p+1}}$ Finally we choose $T=\\tau +Q_+^+(\\tau )/M(\\tau )$ and finish the proof.",
"Now we are able to prove Proposition 6.4 The function $P(t) = \\mu ((-\\infty ,t])$ is an absolutely continuous function.",
"Given any interval $I=(\\tau _1,\\tau _2)$ , we combine Lemma REF and Lemma REF to obtain $\\mu (I) \\lesssim _p \\int _{\\tau _1}^{\\tau _2} |g_+(\\tau )|^2 d\\tau + E^\\frac{4}{p+1} \\int _{\\tau _1}^{\\tau _2} M(\\tau )^{\\frac{2(p-2)}{p+1}} d\\tau .$ As a result, if $\\displaystyle A = \\bigcup _{k=1}^n (a_k,b_k)$ is a union of finite many intervals so that $\\displaystyle \\sum _{k=1}^n (b_k-a_k)<\\delta $ , then we have $\\sum _{k=1}^n |P(b_k)-P(a_k)| = \\mu (A) & \\lesssim _p \\int _A |g_+(\\tau )|^2 d\\tau + E^\\frac{4}{p+1} \\int _A M(\\tau )^{\\frac{2(p-2)}{p+1}} d\\tau \\\\& \\lesssim _p \\int _A |g_+(\\tau )|^2 d\\tau + E^\\frac{2p}{p+1}\\delta ^{\\frac{5-p}{p+1}}.$ The right hand side above converges to zero, as long as $\\delta \\rightarrow 0^+$ , regardless of the choice of $n$ , $a_k$ and $b_k$ 's.",
"Therefore $P(t)$ is absolutely continuous by definition."
],
[
"The $L^2$ function {{formula:ecf48e80-360f-4b9e-94fc-8d4b112f66fd}}",
"Now for any interval $I = (t_1,t_2)$ we are able to write $\\mu (I) = P(t_2) -P(t_1) = \\int _{t_1}^{t_2} P^{\\prime }(t) dt\\quad \\Rightarrow \\quad d\\mu (t) = P^{\\prime }(t) dt.$ Here $P^{\\prime }(t)$ is a nonnegative, locally integrable function.",
"In addition, the finiteness of $\\mu $ implies that $P^{\\prime }(t) \\in L^1({\\mathbb {R}})$ .",
"Finally we only need to rewrite $P^{\\prime }(t) = |\\xi (t)|^2$ , with $\\xi (t) \\in L^2({\\mathbb {R}})$ and finish the proof."
]
] | 1808.08656 |
[
[
"On a reverse H\\\"older inequality for Schr\\\"odinger operators"
],
[
"Abstract We obtain a reverse H\\\"older inequality for the eigenfuctions of the Schr\\\"odinger operator with slowly decaying potentials.",
"The class of potentials includes singular potentials which decay like $|x|^{-\\alpha}$ with $0<\\alpha<2$, in particular the Coulomb potential."
],
[
"Introduction",
"In this paper we are concerned with a reverse Hölder inequality for the eigenfunctions of the Schrödinger operator $-\\Delta +V(x)$ in $L^2(\\mathbb {R}^n)$ .",
"More generally, we consider second-order elliptic operators of the form $Lu=- \\sum _{i,j=1}^n \\frac{\\partial }{\\partial x_j} (a_{ij}(x)u_{x_i}) + V(x)u$ where $a_{ij}(x)$ is a measurable and real-valued function, and the matrix $(a_{ij}(x))_{n\\times n}$ is uniformly elliptic.",
"Namely, there exists a positive constant $\\Lambda $ such that $\\sum _{i,j=1}^n a_{ij}(x) \\xi _i\\xi _j \\ge \\Lambda |\\xi |^2$ for $x,\\xi \\in \\mathbb {R}^n $ .",
"Particularly when $a_{ij}=\\delta _{ij}$ (Kronecker delta function), the operator $L$ becomes equivalent to the Schrödinger operator.",
"In this regard, we shall call a real-valued function $V(x)$ the potential.",
"The reverse Hölder inequalities for solutions to the following Dirichlet boundary problem have been studied for a long time: ${\\left\\lbrace \\begin{array}{ll}\\begin{split}Lu=- \\sum _{i,j=1}^n \\frac{\\partial }{\\partial x_j} (a_{ij}(x)u_{x_i}) + V(x)u &= \\lambda u \\,\\,\\quad \\text{in} \\,\\,\\,\\,\\Omega \\\\u &=0 \\qquad \\text{on} \\,\\,\\,\\partial \\Omega \\end{split}\\end{array}\\right.",
"}$ where $\\Omega $ is a bounded region in $ \\mathbb {R}^{n}$ .",
"When $n=2$ , Payne and Rayner [6] showed that if $\\lambda $ is the first eigenvalue and $u$ is the corresponding eigenfunction of the problem (REF ) with $a_{ij}=\\delta _{ij}$ and $V(x)\\equiv 0$ , ${\\left\\lbrace \\begin{array}{ll}\\begin{split}- \\Delta u &= \\lambda u \\,\\,\\quad \\text{in} \\,\\,\\,\\,\\Omega \\\\u &=0 \\qquad \\text{on} \\,\\,\\,\\partial \\Omega ,\\end{split}\\end{array}\\right.",
"}$ then the following reverse Schwarz inequality holds: $\\Vert u\\Vert _{L^2(\\Omega )} \\le \\sqrt{\\frac{\\lambda }{4 \\pi }} \\Vert u\\Vert _{L^1(\\Omega )},$ which was extended to higher dimensions by Kohler-Jobin [5] (see also [7]).",
"In the general setting (REF ), the reverse Hölder inequalities, $\\Vert u\\Vert _{L^q(\\Omega )} \\le C_{p,q,\\lambda ,n} \\Vert u\\Vert _{L^p(\\Omega )},\\quad q\\ge p>0,$ were obtained later by Talenti [8] for $q=2$ and $p=1$ , and by Chiti [1] for all $q\\ge p>0$ , but with a nonnegative potential $V\\ge 0$ and with symmetric coefficients $a_{ij}=a_{ji}$ .",
"Our aim in this paper is to remove these restrictions.",
"Namely, we obtain a reverse Hölder inequality for solutions of (REF ) where $V$ is allowed to be negative and we do not need to assume the symmetry, $a_{ij}=a_{ji}$ .",
"Before stating our results, we introduce the Morrey-Campanato class ${\\mathcal {L}}^{\\alpha ,r}$ of potentials $V$ , which is defined for $\\alpha >0$ and $1\\le r\\le n/\\alpha $ by $V \\in {\\mathcal {L}}^{\\alpha ,r} \\quad \\Leftrightarrow \\quad \\sup _{x \\in \\mathbb {R}^n, \\rho >0}\\rho ^{\\alpha -n/r} \\left( \\int _{B_\\rho (x)} |V(y)|^r dy \\right)^{1/r} < \\infty ,$ where $B_\\rho (x)$ is the ball centered at $x$ with radius $\\rho $ .",
"In particular, ${\\mathcal {L}}^{\\alpha ,n/\\alpha } = L^{n/\\alpha }$ and $1/|x|^\\alpha \\in L^{n/\\alpha , \\infty } \\subset {\\mathcal {L}}^{\\alpha ,r}$ if $1 \\le r < n/\\alpha $ .",
"Let us next make precise what we mean by a weak solution of the problem (REF ).",
"We say that a function $u \\in H_0^1(\\Omega )$ is a weak solution if $\\int _{\\Omega } \\sum _{i,j=1}^{n} a_{ij}(x) u_{x_i} \\phi _{x_j} dx+ \\int _{\\Omega }V(x) u(x) \\phi (x) dx = \\int _{\\Omega }\\lambda u(x) \\phi (x) dx$ for every $\\phi \\in H_0^1(\\Omega )$ .",
"Our result is then the following theorem.",
"Theorem 1.1 Let $n\\ge 3$ .",
"Assume that $ u \\in H_{0}^1 (\\Omega ) $ is a weak solution of the problem (REF ) with $\\lambda \\in \\mathbb {R}$ and $V\\in \\mathcal {L}^{\\alpha ,r}$ for $\\alpha <2$ and $r>2/\\alpha $ .",
"Then we have $\\Vert u\\Vert _{L^{q} (\\Omega )}\\le CC_\\alpha ^{\\frac{n}{2p}} \\max \\lbrace p,2\\rbrace ^{\\frac{n}{p(2-\\alpha )}} \\Big (\\frac{n}{n-2}\\Big )^{\\frac{n(n-2)}{p(2-\\alpha )}}\\Vert u\\Vert _{L^p (\\Omega )}$ for all $q\\ge p>0$ .",
"Here, $C$ is a constant depending on $\\Lambda , \\lambda , p, q, n$ and $\\Omega $ , and $C_\\alpha =1+\\alpha ^{\\frac{\\alpha }{2-\\alpha }}\\Big (\\frac{2C_n}{\\Lambda }\\Vert V\\Vert _{\\mathcal {L}^{\\alpha , r}}\\Big )^{2/(2-\\alpha )}$ with a constant $C_n$ depending on $n$ and arising from the Fefferman-Phong inequality (REF ).",
"Remark 1.2 The class $\\mathcal {L}^{\\alpha ,r}$ , $\\alpha <2$ , of potentials in the theorem includes the positive homogeneous potentials $a|x|^{-\\alpha }$ with $a>0$ and $0<\\alpha <2$ in three and higher dimensions, especially the Coulomb potential.",
"The rest of this paper is organized as follows: In Section we prove Theorem REF .",
"Compared with the previous results [5], [1] based on rearrangements of functions, our approach works also for negative potentials and for non-symmetric coefficients $a_{ij}$ .",
"It is completely different approach and is based on a combination between the Fefferman-Phong inequality and the classical Moser's iteration technique.",
"Throughout this paper, we denote $A \\lesssim B$ to mean $A \\le CB$ with unspecified constant $C>0$ which may be different at each occurrence."
],
[
"Proof of Theorem ",
"In this section we prove the reverse Hölder inequality (REF ).",
"Since a complex-valued solution $u$ satisfies (REF ) for every complex $\\phi \\in H_0^1(\\Omega )$ , one can easily see that real and imaginary parts of the solution also satisfy (REF ) for every real $\\phi \\in H_0^1(\\Omega )$ .",
"On the other hand, once we prove the inequality for the real and imaginary parts, we get the same inequality for $u$ .",
"Indeed, using the inequality $(a+b)^s \\le C(a^s + b^s)$ for $a,b>0$ and $s>0$ , one can see $\\Vert u\\Vert _{L^q (\\Omega )}^q &= \\int _{\\Omega } (|\\textrm {Re}\\,u+i\\textrm {Im}\\,u|^2)^{q/2} dx \\\\&= \\int _{\\Omega } (|\\textrm {Re}\\,u|^2 + |\\textrm {Im}\\,u|^2 )^{q/2} dx \\\\&\\le C \\int _{\\Omega } |\\textrm {Re}\\,u|^q + |\\textrm {Im}\\,u|^q dx \\\\&\\le C(\\Vert \\textrm {Re}\\,u\\Vert _{L^q (\\Omega )}^q + \\Vert \\textrm {Im}\\,u\\Vert _{L^q (\\Omega )}^q) \\\\&\\le C(\\Vert \\textrm {Re}\\,u\\Vert _{L^p (\\Omega )}^q + \\Vert \\textrm {Im}\\,u\\Vert _{L^p (\\Omega )}^q) \\\\&\\le C\\Vert u\\Vert _{L^p (\\Omega )}^q.$ Hence we may assume that the solution $u$ is a real-valued function.",
"Now we decompose $u$ into two parts, $f= \\max \\lbrace u,0 \\rbrace $ and $g=-\\min \\lbrace u,0\\rbrace $ .",
"Then it is enough to prove that (REF ) holds for $f$ and $g$ .",
"Indeed, $\\Vert u\\Vert _{L^{q}(\\Omega )} &= \\Vert f - g\\Vert _{L^{q}(\\Omega )} \\\\&\\le \\Vert f\\Vert _{L^{q}(\\Omega )} + \\Vert g\\Vert _{L^{q}(\\Omega )} \\\\&\\le C(\\Vert f\\Vert _{L^{p}(\\Omega )} + \\Vert g\\Vert _{L^{p}(\\Omega )}) \\\\&\\le C\\Vert u\\Vert _{L^{p}(\\Omega )}.$ We only consider $f$ because the proof for $g$ follows obviously from the same argument.",
"To prove (REF ) for $f$ , we now divide cases into two parts, $p\\ge 2$ and $p<2$ ."
],
[
"The case $p \\ge 2$",
"In this case we will show that for all $\\tau \\ge 2$ $\\Vert f\\Vert _{L^{\\tau \\omega } (\\Omega )} \\lesssim C_\\alpha ^{1/\\tau } \\tau ^{\\frac{2}{\\tau (2-\\alpha )}} \\Vert f\\Vert _{L^{\\tau } (\\Omega )}$ with $\\omega = n/(n-2)$ .",
"Beginning with $\\tau = p$ , we then iterate as $\\tau = p, p\\omega , p\\omega ^2 , \\cdots ,$ to obtain (REF ).",
"Indeed, first put $\\tau _i = p \\omega ^i$ for $i=0,1,2,\\cdots $ .",
"Since $\\tau _i = \\tau _{i-1} \\omega $ , we then get for $i=1,2,\\cdots ,$ $\\Vert f\\Vert _{L^{\\tau _i} (\\Omega )} &\\lesssim C_\\alpha ^{\\frac{1}{\\tau _{i-1}}} {\\tau _{i-1}}^{\\frac{2}{(2-\\alpha )\\tau _{i-1}}} \\Vert f\\Vert _{L^{\\tau _{i-1}}(\\Omega )} \\\\&= C_\\alpha ^{\\frac{1}{p\\omega ^{i-1}}} \\left( p\\omega ^{i-1} \\right)^{\\frac{2}{p(2-\\alpha )\\omega ^{i-1}}} \\Vert f\\Vert _{L^{\\tau _{i-1}}(\\Omega )},$ which implies by iteration that $\\Vert f\\Vert _{L^{\\tau _i}(\\Omega )}\\lesssim \\big ( C_\\alpha p^{{2}/(2-\\alpha )} \\big )^{\\sum _{k=1}^i (p\\omega ^{k-1})^{-1}}\\big (\\omega ^{2/(2-\\alpha )}\\big )^{\\sum _{k=1}^{i} (k-1)(p\\omega ^{k-1})^{-1}} \\Vert f\\Vert _{L^p(\\Omega )}.$ Since $\\omega = n/(n-2) >1$ , by letting $i \\rightarrow \\infty $ , this implies $\\Vert f\\Vert _{L^q (\\Omega )}\\lesssim \\Vert f\\Vert _{L^{\\infty } (\\Omega )}\\lesssim C_\\alpha ^{\\frac{n}{2p}} p^{\\frac{n}{p(2-\\alpha )}} \\Big (\\frac{n}{n-2} \\Big )^{\\frac{n(n-2)}{p(2-\\alpha )}}\\Vert f\\Vert _{L^p (\\Omega )}$ as desired.",
"It remains to show (REF ).",
"Since $f \\in H_0^1 (\\Omega )$ is a positive part of the weak solution $u$ , it follows that $ \\int _{\\Omega } \\sum _{i,j=1}^{n} a_{ij}(x) f_{x_i} \\phi _{x_j} dx+ \\int _{\\Omega }V(x) f(x) \\phi (x) dx = \\int _{\\Omega }\\lambda f(x) \\phi (x) dx$ for every real $\\phi \\in H_0^1(\\Omega )$ supported on $\\lbrace x\\in \\mathbb {R}^n:u(x)>0\\rbrace $ .",
"For $l>0$ and $m>0$ , we set $\\tilde{f}=f + l$ , and let $\\tilde{f}_m = {\\left\\lbrace \\begin{array}{ll} l+m \\quad \\text{if} \\quad f \\ge m,\\\\ \\tilde{f} \\quad \\text{if}\\quad f<m.\\end{array}\\right.",
"}$ We now consider the following test function $\\phi = \\tilde{f}_m^{\\beta } \\tilde{f} - l^{\\beta +1} \\in H_{0}^1 (\\Omega )$ for $\\beta \\ge 0$ .",
"We then compute $\\phi _{x_j} &= \\beta {\\tilde{f}_m}^{\\beta -1} (\\tilde{f}_m)_{x_j}\\tilde{f} + {\\tilde{f}_m}^\\beta \\tilde{f}_{x_j}\\\\&=\\beta {\\tilde{f}_m}^{\\beta } (\\tilde{f}_m)_{x_j} + {\\tilde{f}_m}^\\beta \\tilde{f}_{x_j}$ using the fact that $(\\tilde{f}_m)_{x_j}=0\\,\\,\\text{ in}\\,\\, \\lbrace x: f(x) \\ge m\\rbrace \\quad \\text{and}\\quad \\tilde{f}_m = \\tilde{f}\\,\\,\\text{ in}\\,\\, \\lbrace x:f(x)<m\\rbrace .$ Substituting $\\phi $ into (REF ) and using (REF ) together with the trivial fact $f_{x_i} = \\tilde{f}_{x_i}$ , the first term on the left-hand side of (REF ) is written as $\\int _{\\Omega } \\sum _{i,j=1}^{n} & a_{ij}(x) f_{x_i} \\phi _{x_j} dx \\\\&= \\beta \\int _{\\Omega } \\tilde{f}_m^{\\beta } \\sum _{i,j=1}^n a_{ij}(x) (\\tilde{f}_m)_{x_i} (\\tilde{f}_m)_{x_j} dx + \\int _{\\Omega } {\\tilde{f}_m}^\\beta \\sum _{i,j=1}^n a_{ij}(x) \\tilde{f}_{x_i} \\tilde{f}_{x_j} dx.$ Then it follows from the ellipticity (REF ) that $\\int _{\\Omega } \\sum _{i,j=1}^{n} a_{ij}(x) f_{x_i} \\phi _{x_j} dx \\ge \\Lambda \\beta \\int _{\\Omega } \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}_m|^2 dx + \\Lambda \\int _{\\Omega } \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}|^2 dx.$ Combining (REF ) and (REF ), we conclude that $\\Lambda \\beta \\int _{\\Omega } \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}_m|^2 dx &+ \\Lambda \\int _{\\Omega } \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}|^2 dx \\\\&\\le \\int _{\\Omega } -V f (\\tilde{f}_m^{\\beta } \\tilde{f} - l^{\\beta +1}) dx + \\int _{\\Omega } \\lambda f (\\tilde{f}_m^{\\beta } \\tilde{f} -l^{\\beta +1}) dx.$ Note here that $|\\nabla (\\tilde{f}_m^{\\beta /2}\\tilde{f})|^2 \\le 2 (1+\\beta ) \\Big (\\beta \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}_m|^2 + \\tilde{f}_m^{\\beta } |\\nabla \\tilde{f}|^2 \\Big )$ which follows by a direct computation together with (REF ).",
"We therefore get $\\nonumber \\int _{\\Omega } |\\nabla &(\\tilde{f}_m^{\\beta /2} \\tilde{f})|^2 dx \\\\\\nonumber &\\le \\frac{2(1+\\beta )}{\\Lambda }\\int _{\\Omega } -Vf (\\tilde{f}_m^{\\beta }\\tilde{f}-l^{\\beta +1})dx + \\frac{2\\lambda (1+\\beta )}{\\Lambda } \\int _{\\Omega } f(\\tilde{f}_m^{\\beta }\\tilde{f}-l^{\\beta +1}) dx\\\\&\\le \\frac{2(1+\\beta )}{\\Lambda }\\int _{\\Omega } |V| \\tilde{f}_m^{\\beta }\\tilde{f}^2dx + \\frac{2|\\lambda |(1+\\beta )}{\\Lambda } \\int _{\\Omega } \\tilde{f}_m^{\\beta }\\tilde{f}^2 dx.$ To control the term involving the potential in (REF ), we now use the so-called Fefferman-Phong inequality ([4]), $\\int _{\\mathbb {R}^n}|g|^2 v(x)dx\\le C_n\\Vert v\\Vert _{\\mathcal {L}^{2,r}} \\int _{\\mathbb {R}^n}|\\nabla g|^2dx,$ where $C_n$ is a constant depending on $n$ , and $1<r\\le n/2$ .",
"(It is not valid for $r=1$ as remarked in [2].)",
"Applying this inequality along with Hölder's inequality, the first integral on the right-hand side of (REF ) is bounded as $\\int _{\\Omega } |V| \\tilde{f}_m^{\\beta } \\tilde{f}^2 dx &\\le \\bigg ( \\int _{\\Omega } |V|^{\\frac{2}{\\alpha }} \\tilde{f}_m^{\\beta } \\tilde{f}^2 dx \\bigg )^{\\frac{\\alpha }{2}} \\bigg ( \\int _{\\Omega } \\tilde{f}_m^{\\beta }\\tilde{f}^2 dx \\bigg )^{\\frac{2-\\alpha }{2}} \\\\&\\le C_n\\Vert |V|^{\\frac{2}{\\alpha }}\\Vert _{\\mathcal {L}^{2,\\tilde{r}}}^{\\frac{\\alpha }{2}} \\bigg ( \\int _{\\Omega } |\\nabla (\\tilde{f}_m^{\\beta /2} \\tilde{f})|^2 dx \\bigg )^{\\frac{\\alpha }{2}} \\bigg ( \\int _{\\Omega }\\tilde{f}_m^{\\beta } \\tilde{f}^2 dx \\bigg )^{\\frac{2-\\alpha }{2}}$ for all $1<\\tilde{r}\\le n/2$ .",
"We note here that $\\Vert |V|^{\\frac{2}{\\alpha }}\\Vert _{\\mathcal {L}^{2,\\tilde{r}}}^{\\frac{\\alpha }{2}}=\\Vert V\\Vert _{\\mathcal {L}^{\\alpha ,2\\tilde{r}/\\alpha }}$ and apply Young's inequality $ab \\le \\frac{\\alpha }{2} (\\varepsilon a)^{2/\\alpha } + \\frac{2-\\alpha }{2} (\\varepsilon ^{-1} b)^{2/(2-\\alpha )}$ with $0<\\alpha <2$ and $\\varepsilon >0$ to obtain $ \\nonumber \\int _{\\Omega }& |V| \\tilde{f}_m^{\\beta } \\tilde{f}^2 dx \\\\&\\le C_n\\Vert V\\Vert _{\\mathcal {L}^{\\alpha ,2\\tilde{r}/\\alpha }} \\bigg ( \\frac{\\alpha }{2} \\varepsilon ^{2/\\alpha } \\int _{\\Omega } |\\nabla (\\tilde{f}_m^{\\beta /2}\\tilde{f})|^2 dx + \\frac{2-\\alpha }{2} \\varepsilon ^{{-2}/(2-\\alpha )} \\int _{\\Omega }\\tilde{f}_m^\\beta \\tilde{f}^2 dx \\bigg ).$ By settingSince $\\widetilde{r}>1$ , setting $r= 2\\tilde{r}/\\alpha $ determines the condition $r>2/\\alpha $ in the theorem.",
"$r= 2\\tilde{r}/\\alpha $ and taking $\\varepsilon ^{2/\\alpha } =c(1+\\beta )^{-1}$ with $c=\\frac{\\Lambda }{2\\alpha C_n \\Vert V\\Vert _{\\mathcal {L}^{\\alpha ,r}}}$ so that $C_n\\Vert V\\Vert _{\\mathcal {L}^{\\alpha ,r}}\\frac{\\alpha }{2} \\varepsilon ^{2/\\alpha }\\frac{2(1+\\beta )}{\\Lambda }=1/2,$ the gradient term in (REF ) can be absorbed into the left-hand side of (REF ), as follows: $\\nonumber \\int _{\\Omega } |\\nabla (\\tilde{f}_m^{\\beta /2} \\tilde{f})|^2 dx &\\le \\,\\,c^{-\\frac{2}{2-\\alpha }} \\bigg ( \\frac{2-\\alpha }{\\alpha } \\bigg ) (\\beta +1)^{\\frac{2}{2-\\alpha }} \\int _{\\Omega } \\tilde{f}_m^{\\beta }\\tilde{f}^2 dx \\\\& + \\frac{2|\\lambda | (1+\\beta )}{\\Lambda } \\int _{\\Omega } \\tilde{f}_m^{\\beta } \\tilde{f}^2 dx.$ Finally, applying the Gagliardo-Nirenberg-Sobolev inequality ([3]) to the left-hand side of (REF ), we see $\\bigg (\\int _{\\Omega }|\\tilde{f}_m^{\\beta /2}\\tilde{f}|^{2\\omega }dx\\bigg )^{1/\\omega }\\lesssim \\int _{\\Omega } |\\nabla (\\tilde{f}_m^{\\beta /2} \\tilde{f})|^2 dx$ with $\\omega =n/(n-2)$ .",
"Using the fact that $\\tilde{f}_m \\le \\tilde{f}$ and setting $\\beta +2=\\tau $ , we therefore get $\\nonumber \\bigg (\\int _{\\Omega }\\tilde{f}_m^{\\tau \\omega } dx\\bigg )^{1/\\omega } \\le \\,\\,c^{-\\frac{2}{2-\\alpha }}\\bigg ( \\frac{2-\\alpha }{\\alpha } \\bigg ) (\\tau -1)^{\\frac{2}{2-\\alpha }} \\int _{\\Omega } \\tilde{f}^{\\tau } dx + \\frac{2|\\lambda | (\\tau -1)}{\\Lambda } \\int _{\\Omega } \\tilde{f}^{\\tau } dx.$ which implies the desired estimate $\\bigg ( \\int _{\\Omega } {f}^{\\tau \\omega } dx \\bigg )^{1/\\omega } \\lesssim \\Big ( 1+\\alpha ^{\\frac{\\alpha }{2-\\alpha }}\\Big (\\frac{2C_n}{\\Lambda }\\Vert V\\Vert _{\\mathcal {L}^{\\alpha , r}}\\Big )^{\\frac{2}{2-\\alpha }} \\Big ) \\tau ^{\\frac{2}{2-\\alpha }} \\int _{\\Omega } f^{\\tau } dx.$ by letting $m \\rightarrow \\infty $ and $l \\rightarrow 0$ ."
],
[
"The case $p<2$",
"From the case $p=2$ , we have $\\Vert f\\Vert _{L^{\\infty }(\\Omega )} < \\infty $ and $\\Vert f\\Vert _{L^\\infty (\\Omega )} &\\lesssim C_\\alpha ^{\\frac{n}{4}} 2^{\\frac{n}{2(2-\\alpha )}}\\Big ( \\frac{n}{n-2}\\Big )^{\\frac{n(n-2)}{2(2-\\alpha )}} \\Vert f\\Vert _{L^2(\\Omega )} \\nonumber \\\\&\\le C_\\alpha ^{\\frac{n}{4}} 2^{\\frac{n}{2(2-\\alpha )}} \\Big ( \\frac{n}{n-2}\\Big )^{\\frac{n(n-2)}{2(2-\\alpha )}} \\Vert f\\Vert _{L^\\infty (\\Omega )}^{(2-p)/2} \\Vert f\\Vert _{L^p(\\Omega )}^{p/2} \\nonumber \\\\&\\le \\frac{1}{2} \\Vert f\\Vert _{L^\\infty (\\Omega )} + C_\\alpha ^{\\frac{n}{2p}}\\frac{p}{2}\\Big (\\frac{1}{2-p}\\Big )^{1-\\frac{2}{p}} {2}^{\\frac{n}{p (2-\\alpha )}}\\Big ( \\frac{n}{n-2}\\Big )^{\\frac{n(n-2)}{p(2-\\alpha )}} \\Vert f\\Vert _{L^{p}(\\Omega )}.$ For the third inequality, we used here Young's inequality, $ab \\le \\Big ( 1- \\frac{p}{2}\\Big ) (\\epsilon a)^{\\frac{2}{2-p}} + \\frac{p}{2} (\\epsilon ^{-1} b)^{\\frac{2}{p}}$ with $\\epsilon = ( \\frac{1}{2-p})^{(2-p)/2}$ , $a=\\Vert f\\Vert _{L^{\\infty }(\\Omega )}^{(2-p)/2} \\quad \\textrm {and} \\quad b= C_\\alpha ^{\\frac{n}{4}} 2^{\\frac{n}{2(2-\\alpha )}}\\Big ( \\frac{n}{n-2}\\Big )^{\\frac{n(n-2)}{2(2-\\alpha )}} \\Vert f\\Vert _{L^p(\\Omega )}^{p/2}.$ By absorbing the first term on the right-hand side of (REF ) into the left-hand side, we conclude that $\\Vert u\\Vert _{L^q (\\Omega )}\\lesssim \\Vert u\\Vert _{L^{\\infty } (\\Omega )}\\lesssim C_\\alpha ^{\\frac{n}{2p}} 2^{\\frac{n}{p(2-\\alpha )}} \\Big (\\frac{n}{n-2} \\Big )^{\\frac{n(n-2)}{p(2-\\alpha )}}\\Vert u\\Vert _{L^p (\\Omega )}$ as desired."
]
] | 1808.08516 |
[
[
"Deep Convolutional Neural Network with Mixup for Environmental Sound\n Classification"
],
[
"Abstract Environmental sound classification (ESC) is an important and challenging problem.",
"In contrast to speech, sound events have noise-like nature and may be produced by a wide variety of sources.",
"In this paper, we propose to use a novel deep convolutional neural network for ESC tasks.",
"Our network architecture uses stacked convolutional and pooling layers to extract high-level feature representations from spectrogram-like features.",
"Furthermore, we apply mixup to ESC tasks and explore its impacts on classification performance and feature distribution.",
"Experiments were conducted on UrbanSound8K, ESC-50 and ESC-10 datasets.",
"Our experimental results demonstrated that our ESC system has achieved the state-of-the-art performance (83.7%) on UrbanSound8K and competitive performance on ESC-50 and ESC-10."
],
[
"Introduction",
"Sound recognition is a front and center topic in today's pattern recognition theories, which covers a rich variety of fields.",
"Some of sound recognition topics have made remarkable research progress, such as automatic speech recognition (ASR)[10], [9] and music information retrieval (MIR)[4], [31].",
"Environmental sound classification (ESC) is an another important branch of sound recognition and is widely applied in surveillance[21], home automation[33], scene analysis[3] and machine hearing[14].",
"However, unlike speech and music, sound events are more diverse with a wide range of frequencies and often less well defined, which make ESC tasks more difficult than ASR and MIR.",
"Hence, ESC still faces critical design issues in performance and accuracy improvement.",
"Traditional ASR techniques such as MFCC, LPC, PLP are applied directly to ESC fields in previous works[7], [28], [13], [16].",
"However, state-of-the-art performance has been achieved when using more discriminative representations such as Mel filterbank features[5], Gammatone features[34] and wavelet-based features[8].",
"These features were modeled with some typical machine learning algorithms such as SVM[32], GMM[17] and KNN[20] for ESC tasks.",
"However, the performance gain introduced by these approaches is still unsatisfying.",
"One main reason is that traditional classifiers do not have feature extraction ability.",
"Over the past few years, deep neural networks (DNNs) have made great success in ASR and MIR[10], [25].",
"For audio signals, DNNs have ability to extract features from raw data or hand-draft feature.",
"Therefore, some DNN-based ESC systems[15], [12] were proposed and performed much better than SVM-based ESC system.",
"However, deep fully-connected architecture of DNNs is not robust for transformative features[22].",
"Some new researchs find convolutional neural networks (CNNs) have strong abilities to explore inherit and hidden patterns through huge amount of training data.",
"Several attempts that apply CNN to ESC have received performance boosts by learning spectrogram-like features from environment sounds[35], [19], [23].",
"However, the existing networks for ESC mostly use shallow architecture, such as 2 convolutional layers[19], [35] and 3 convolutional layers[23].",
"Getting a more discriminative and powerful information usually requests a deeper model.",
"Therefore in this paper, we propose an enhanced CNN architecture with a deeper network based on VGG Net[26].",
"The main contributions of this paper includes We propose a novel CNN network based on VGG Net.",
"We find that simply using stacked convolutional layers with 3x3 convolution filters is unsatisfying in our tasks.",
"So we redesign a novel CNN architecture in our ESC system.",
"Instead of 3x3 convolution filters, We use 1-D convolution filters to learn local patterns across frequency and time, respectively.",
"And our method performs better than CNN using 3x3 convolution filters with same depth of network.",
"Mixup is applied in our ESC system for ESC tasks.",
"Every training sample is created by mixing two examples randomly selected from original training dataset when using mixup.",
"And the training target is also changed to the mix ratio.",
"The effectiveness of mixup on classification performance and feature distribution is then explored further.",
"Experiments were conducted on UrbanSound8K, ESC-50 and ESC-10 datasets, the result of which demonstrated that our ESC system has achieved the state-of-the-art performance (83.7$\\%$ ) on UrbanSound8K and competitive performance on ESC-50 and ESC-10.",
"The rest of this paper is organized as follows.",
"Recent related works of ESC are introduced in Section .",
"Section provides detailed introduction of our methods.",
"Section presents the experiments settings on ESC-10, ESC-50 and UrbanSound8K datasets, and Section gives both experimental results and detailed discussions of our results.",
"Finally, Section concludes the paper."
],
[
"Related Work",
"In this section, we introduce the recent deep learning methods for environmental sound classification.",
"Piczak[20] proposed to apply CNNs to the log mel spectrogram which is calculated for each frame of audio and represents the squared magnitude of each frequency area.",
"Piczak created a two-channel feature by applying log mel spectrogram and its delta information as the input of his CNN model and gave a 20.5$\\%$ improvement over Random Forest method on ESC-50 dataset.",
"Takahashi et al.",
"[27] also used log mel spectrogram and their delta and delta-delta information as a three-channel input in a manner similar to the RGB inputs of the image.",
"Dharmesh et al.",
"[1] used gammatone spectrogram and a similar CNN architecture as Piczak [18] and claimed that they achieved 79.1$\\%$ and 85.34$\\%$ accuracy on ESC-50 and UrbanSound8K dataset, respectively.",
"However, since their results were not reproducible, we contacted with the author and realized that the results achieved by them didn't follow the official cross validation methods, which means they used different training data and validation data than main published papers and not comparable.",
"So we will not compare our results with the results from [1].",
"Some researchers also proposed to train model directly from raw waveforms.",
"Dai et al.",
"[6] proposed a deep CNN architecture (up to 34 layers) with 1-D convolutional layers using 1-D raw data as input and they showed competitive accuracy with CNN using log mel spectrogram inputs[20].",
"Tokozume et al.",
"[29] proposed a end-to-end network named EnvNet using raw data as inputs and reported EnvNet could extract a discriminative feature that complements the log mel features.",
"In [30], they constructed a deeper recognition network based on EnvNet, referred as EnvNet-v2, and achieved better performance.",
"In addition, some researchers proposed to use external data for sound recognition.",
"Mun et al.",
"[18] proposed a DNN based transfer learning method for ESC.",
"They first trained a DNN model using merged different web accessible environmental sound datasets.",
"Then, they transferred the parameters of the pre-trained model and adapted the sound recognition system for target domain task using additional layers.",
"Aytar et al.",
"[2] proposed to learn rich sound representations from large amounts of unlabeled sound and videos dataset.",
"They transferred the knowledge of pre-trained visual recognition network into the sound recognition network.",
"Then, they used a linear-SVM classifier to classify the feature which is the output of the hidden layer of the sound recognition network to the target task."
],
[
"Convolutional Neural Network",
"CNN is a stack of multi-layer neural networks including a group of convolutional layers, pooling layers and a limited number of fully connected layers.",
"In this section, we propose a novel CNN as our ESC system model inspired by VGG Net[26], the architecture of which is presented in Table 1.",
"The proposed CNN architecture is comprised of eight convolutional layers and two fully connected layers.",
"We first use 2 convolutional layers with large filter kernals as a basic feature extractor.",
"Then, we learn local patterns across frequency and time using 3x1 and 1x5 convolution filters, respectively.",
"Next, we use small convolution filters (3x3) to learn joint time-frequency patterns.",
"Batch normalization[11] is applied to the output of convolutional layers to speed up training.",
"We use the Rectified Linear Units (ReLU) to model the non-linearly for the output of each layer.",
"After every two convolutional layers, a pooling layer is used to reduce the dimensions of the convolutional features maps, where maximum pooling is chosen in our network.",
"To reduce the risks of overfitting, the dropout technique is applied after the first fully connected layers, with the probability of $0.5$ .",
"L2-regularization is applied to the weights of each layer with the coefficient $0.0001$ .",
"In the output layer, softmax function is used as the activation function which outputs probabilities of all classes.",
"Table: Configuration of proposed CNN.",
"Out shape represents the dimension in (frequency, time, channel).",
"Batch Normalization is applied for each convolutional layer."
],
[
"Mixup",
"Mixup is an simple but effective method to generate training data[36].",
"Fig REF shows the pipeline of mixup.",
"Different from traditional augmentation approaches, mixup constructs virtual training samples by mixing training samples.",
"Normally, a model is optimized by using a mini-batch optimization method, such as mini-batch SGD, and each mini-batch data is selected from the whole original training data.",
"In mixup, however, each data and label of a mini-batch is generated by mixing two training samples, which are determined by $\\left\\lbrace \\begin{aligned}\\hat{\\mathbf {x}}= {\\lambda }x_i + (1-\\lambda )x_j\\\\\\hat{\\mathbf {y}}= {\\lambda }y_i + (1-\\lambda )y_j\\end{aligned}\\right.$ where $x_i$ and $x_j$ are two samples randomly selected from training data, and $y_i$ and $y_j$ are their one-hot labels.",
"The mix factor $\\lambda $ is decided by a hyper-parameter $\\alpha $ and $\\lambda $ $\\sim $ Beta($\\alpha $ , $\\alpha $ ).",
"Therefore, mixup extends the training data distribution by mixing various training data within or without the same class by a linear way, leading to a linear interpolation of the associated targets.",
"Note that we do not use mixup for testing phase."
],
[
"Dataset",
"Three publicly available datasets are used for model training and performance evaluation of the proposed approach, including ESC-10, ESC-50[20] and UrbanSound8K[24], the detailed information of which is shown in Table REF .",
"The ESC-50 dataset consists of 2000 short environmental records which are divided into 50 classes in 5 major categories, including animals, natural soundscapes and water sounds, human non-speech sounds, interior/domestic sounds, and exterior/urban noises.",
"All audio samples are 5 seconds with 44.1 kHz sampling frequency.",
"The ESC-10 dataset is a subset of 10 classes (400 samples) selected from the ESC-50 dataset (dog bark, rain, sea waves, baby cry, clock tick, person sneeze, helicopter, chainsaw, rooster, fire crackling).",
"The UrbanSound8K dataset is a collection of 8732 short (up to 4 seconds) audio clips of urban sound areas.",
"And the audio clips are prearranged into 10 folds.",
"The dataset is divided into 10 classes: air conditioner, car horn, children playing, dog bark, drilling, engine idling, gun shot, jackhammer, siren, and street music.",
"Table: Information of datasets."
],
[
"Preprocessing",
"We use a 44.1kHz sampling rate for ESC-10, ESC-50, UrbanSound8K datasets.",
"All audio samples are normalized into a range from $-1$ to 1.",
"In order to avoid overfitting and to effectively utilize the limited data, we use Time Stretch[23] and Pitch Shift[23] deformation methods to generate new audio samples.",
"We use two spectrogram-like representations, log mel spectrogram (Mels) and gammatone spectrogram (GTs).",
"Both features are extracted from all recordings with hamming window size of 1024, hop length of 512 and 128 bands.",
"Then, the resulting spectrograms are converted into logarithmic scale.",
"In our experiments, we use a simple energy-based silence drop algorithm to drop silence regions.",
"Finally, the spectrograms are split into 128 frames (approximately $1.5s$ ) length with 50$\\%$ overlap.",
"The delta information of the original spectrogram is calculated, which is the first temporal derivative of the spectrogram feature.",
"Then, we use the segments with their deltas as a two-channel input to the network."
],
[
"Training settings",
"All models are trained using mini-batch stochastic gradient descent (SGD) with Nesterov momentum of 0.9.",
"We used a learning rate decrease schedule with a initial learning rate of 0.1, and then divided the learning rate by 10 every 80 epoch for UrbanSound8K and 100 epoch for ESC-10 and ESC-50.",
"Every batch consists of 200 samples randomly selected from training set without repetition.",
"The models are trained for 200 epochs for UrbanSound8K and 300 epochs for ESC-50 and ESC-10.",
"We initialize all the weights to zero mean Gaussian noise with a standard deviation of 0.05.",
"We use cross entropy as the loss function, which is typically used for multi classification task.",
"In the test stage, feature extraction and audio cropping patterns are the same as those used in the training stage.",
"Prediction probability of a test audio sample is the average of predicted class probability of each segment.",
"The predicted label of the test audio sample is the class with the highest posterior possibility.",
"The classification performance of the methods is evaluated by the $K$ -fold cross-validation.",
"For the ESC-50 and ESC-10 dataset, $K$ is set to 5, while for the UrbanSound8K dataset, $K$ is set to 10.",
"All models are trained using Keras library with TensorFlow backend on an Nvidia P100 GPU with a 12GB memory."
],
[
"Results and Analysis",
"The classification accuracy of the proposed method compared with recent related works is shown in Table REF .",
"It can be observed that our method achieved the state-of-the-art performance (83.7$\\%$ ) on UrbanSound8K dataset and competitive performance (91.7$\\%$ , 83.9$\\%$ ) on ESC-10 and ESC-50.",
"The average classification accuracy of our methods with Mels outperformed PiczakCNN[19] (baseline) by 10.8$\\%$ , 17.6$\\%$ , 9.9$\\%$ on ESC-10, ESC-50 and UrbanSound8K datasets, respectively.",
"Data augmentation is an important technique for increasing performance for limited dataset, which gave an improvement of 1.1$\\%$ , 3.3$\\%$ and 5.3$\\%$ on ESC-10, ESC-50 and UrbanSound8K, respectively.",
"In addition, GTs improved by 0.4$\\%$ , 1.4$\\%$ and 1.1$\\%$ over Mels on ESC-10, ESC-50 and UrbanSound8K, respectively.",
"We can see that classification accuracy with GTs is always better than accuracy with Mels on on ESC-10, ESC-50 and UrbanSound8K datasets, which indicates that feature representation is a critical factor for classification performance.",
"What's more, mixup is a powerful way to improve performance which can always perform better results than that without mixup.",
"In our experiments, Mixup gave an improvement of 1.5$\\%$ , 2.4$\\%$ and 2.6$\\%$ with Mels on ESC-10, ESC-50, UrbanSound8k datasets, respectively.",
"As mentioned in Section , mixup trains a network using a linear combination of training examples and their labels and leads to a regularization for neural network and generalization for unseen data.",
"For the effect of mixup, we do a further exploration in the following parts.",
"Table: Classification accuracy (%\\%) of different ESC systems.",
"In our ESC system, we compare two different features with augmentation and without augmentation.",
"'aug' stands for augmentation, including Pitch Shift, Time Stretch.",
"Note that we will not compare with the results of Dharmesh which was discussed in Section .Figure: Training curves of our proposed CNN on (a) ESC-50 and (b) UrbanSound8K datasets."
],
[
"Comparison of network architecture",
"We compare our proposed CNN with a VGG network architecture with same depth of network.",
"This VGG network has same network parameters with our proposed CNN except for replacing to use 3x3 convolution filters and 2x2 stride pooling and we refer to this architecture as VGG10.",
"In Table REF , we provide classification accuracy of proposedCNN and VGG10 on ESC-10, ESC-50 and UrbanSound8K datasets.",
"The results shows that our proposed CNN always performs better than VGG10 on three datasets.",
"Table: Comparison between proposed CNN and VGG10 Net (%\\%)."
],
[
"Effects of Mixup",
"Analysis.",
"The confusion matrix by the proposed CNN with Mels and mixup for the UrbanSound8K dataset is given in Fig.REF (a).",
"We can observe that the most misrecognition happened between two noise-like classes, such as jackhammer and drilling, engine idling and jackhammer, and air conditioner and engine idling.",
"In Fig.REF (b), we provide the difference of the confusion for the proposed CNN method with and without mixup.",
"We see that mixup gives an improvement for most classes, especially for air conditioner, drilling, jackhammer and siren.",
"However, mixup also has a slightly harmful effect on the accuracy for some classes and increases confusion between some specific pairs classes.",
"For example, although mixup reduces the confusion between jackhammer and engine idling, it increases the confusion between jackhammer and siren.",
"Figure: Visualization of the feature distribution at the output of FC1 using PCA (a) without mixup and (b) with mixup.To gain further insights to the effect of mixup, we visualized the feature distributions for UrbanSound8K with mixup and without mixup using PCA in Fig.REF .",
"The feature dots represent the high-level feature vectors obtained at the output of the first fully connected layer (FC1).",
"We can observe that it is quite different between feature distributions with and without mixup.",
"Fig.REF (a) shows the feature distributions of different classes with mixup.",
"Some classes have a large within-class variance of the feature distribution, while some have a small within-class variance.",
"In addition, the between-class distances of different pairs of classes are also varied, which may make models more sensitive to some classes.",
"However, features of most classes distribute within a small space with a relative smaller within-class variance and the boundary of most classes is clear as shown in Fig.REF (b).",
"Hyper-parameter $\\alpha $ selected.",
"In order to achieve a better performance for our system on ESC, the effect of mixup hyper-parameter $\\alpha $ is further explored.",
"Fig.REF shows the change of accuracy with different $\\alpha $ ranging from $[0.1, 0.5]$ .",
"We see that when $\\alpha = 0.2$ , the best accuracy is achieved on all three datasets.",
"Figure: Curves of an accuracy with different α\\alpha for ESC-10, ESC-50, UrbanSound8K"
],
[
"Conclusion",
"In this paper, we proposed a novel deep convolutional neural network architecture for environmental sound classification.",
"We compared our proposed CNN with VGG10 and results showed that our proposed CNN always performed better.",
"To further improve the classification accuracy, mixup was applied in our ESC system.",
"As a result, the proposed ESC system achieved state-of-the-art performance on UrbanSound8K dataset and competitive performance on ESC-10 and ESC-50 dataset.",
"Furthermore, we explored the impacts of mixup on the classification accuracy and feature space distribution of different classes on UrbanSound8K dataset.",
"The results showed that mixup is a powerful method to improves classification accuracy.",
"Our future work will focus on the network design and exploration for using mixup method for specific classes."
]
] | 1808.08405 |
[
[
"Phonon-induced linewidths of graphene electronic states"
],
[
"Abstract The linewidths of the electronic bands originating from the electron-phonon coupling in graphene are analyzed based on model tight-binding calculations and experimental angle-resolved photoemission spectroscopy (ARPES) data.",
"Our calculations confirm the prediction that the high-energy optical phonons provide the most essential contribution to the phonon-induced linewidth of the two upper occupied $\\sigma$ bands near the $\\bar{\\Gamma}$-point.",
"For larger binding energies of these bands, as well as for the $\\pi$ band, we find evidence for a substantial lifetime broadening from interband scattering $\\pi \\rightarrow \\sigma$ and $\\sigma \\rightarrow \\pi$, respectively, driven by the out-of-plane ZA acoustic phonons.",
"The essential features of the calculated $\\sigma$ band linewidths are in agreement with recent published ARPES data [F. Mazzola et al., Phys.~Rev.~B.",
"95, 075430 (2017)] and of the $\\pi$ band linewidth with ARPES data presented here."
],
[
"Introduction",
"Numerous experimental and theoretical studies of graphene have been presented during the last decade.",
"[1], [2] These investigations have revealed remarkable mechanical [3], electronic [4], optical [5], and thermal [6] properties.",
"However, graphene is not considered to be a good BCS superconductor [7] because of very weak electron-phonon coupling (EPC).",
"[8] This is probably true when considering the $\\pi $ bands crossing the Fermi level: for neutral graphene, two nearly-linearly dispersing bands touch at the Fermi level at the charge neutrality point, also called Dirac point.",
"In this case, or even for extreme ranges of doping levels or electrostatic gating conditions, the density of states (DOS) is low, as well as the electron-phonon matrix element.",
"On the other hand, the situation can be considerably different in other parts of the electronic spectrum.",
"Recently, Mazzola et al.",
"[9], [10] reported evidence of strong EPC in the $\\sigma $ -band; which was revealed in angle-resolved photoemission spectroscopy (ARPES) measurements with a substantial lifetime broadening and a pronounced kink in the dispersion.",
"The aim of the present combined theory and experimental study has been to get insight to the scattering process determining the EPC induced linewidths of the occupied $\\sigma $ bands and $\\pi $ band.",
"Of particular interest was to investigate the relative importance of the intraband and interband scattering as well as which dominant phonon modes drive the scattering.",
"To the best of our knowledge, linewidth analysis of the $\\sigma $ bands is still missing in the literature.",
"The phonon induced linewidth of the $\\pi $ band has been studied by Park et al.",
"[11] in the binding energy range 0-2.5 eV.",
"In the lower part of this energy range our results agrees reasonably while in the upper part, our linewidths are about twice as large.",
"The intraband scattering, which is found to be driven by the high energy in-plane optical phonons, is an important scattering channel for both the $\\sigma $ and the $\\pi $ band.",
"For the two occupied uppermost $\\sigma $ bands this channel dominates near the EPC induced “kink”, about 200 meV below the top of these bands.",
"However, the interband $\\pi \\rightarrow \\sigma $ and $\\sigma \\rightarrow \\pi $ scattering can be mediated by the existence of out-of-plane vibrational modes.",
"Our calculations reveal a substantial contribution from these scattering channels, driven by in particular the out-of-plane acoustic ZA mode, at higher binding energies.",
"The paper is organized as follows.",
"In next section, Sec.",
"II we introduce the theoretical formulation of the EPC linewidth and outline the calculation of the electron and phonon band structure.",
"In addition we give some details about the approximations used when constructing the deformation potential.",
"In Sec.",
"III we present the results of the linewidth calculations for $\\sigma $ bands and the $\\pi $ band and compare with experimental data.",
"Our summary and conclusions and some perspectives for future research are presented in Sec.",
"IV."
],
[
"EPC-induced linewidth",
"Our calculations are based on the traditional theoretical framework where the distortion of the electronic Hamiltonian caused by lattice vibrations can be considered to be of first order.",
"In the low-temperature limit, which is the relevant case in the experiment reported by Mazzola et al.",
"[9], the thermal energy ($\\approx 6$ meV) is less than the typical phonon energy ($\\approx 170$ meV).",
"In this case phonon emission dominates, while phonon absorption is suppressed.",
"The EPC contribution to the linewidth of a particular electron band $n$ and wave vector $\\mathbf {k}$ is calculated applying first order time dependent perturbation theory, the Fermi Golden Rule $&&\\Gamma _{n \\mathbf {k}} = \\\\&& 2 \\pi \\sum _{n^{\\prime } \\nu \\mathbf {q}}|\\langle n \\mathbf {k} |\\delta V^{\\nu }_{\\mathbf {q}}|n^{\\prime } \\mathbf {k} + \\mathbf {q}\\rangle |^2 \\delta (\\varepsilon _{n^{\\prime }\\mathbf {k} + \\mathbf {q}}-\\varepsilon _{n \\mathbf {k} } - \\hbar \\omega _{\\nu \\mathbf {q}}) \\ , \\nonumber $ where $\\varepsilon _{n \\mathbf {k} }$ and $\\omega _{\\nu \\mathbf {q}}$ represent the electron band energy and phonon frequency, respectively.",
"The phonons are described by band index $\\nu $ and wave vector $\\mathbf {q}$ .",
"In the harmonic approximation the deformation potential is written $&&\\delta V_{\\mathbf {q}}^{\\nu }(\\mathbf {r}) = \\\\ && \\sqrt{\\frac{\\hbar }{2M\\omega _{\\nu \\mathbf {q}}}}\\sum _{\\mathbf {R}} \\mathbf {e}^{\\nu }(\\mathbf {q})\\cdot \\mathbf {V}^{\\prime }(\\mathbf {R} + \\mathbf {r}_s; \\mathbf {r}) e^{-i\\mathbf {q} \\cdot (\\mathbf {R} + \\mathbf {r}_s)} \\nonumber \\ ,$ where $\\mathbf {R}$ denotes the center position of the unit cells and $\\mathbf {r_s}$ the positions of the A and B atoms within the unit cell.",
"$\\mathbf {e}^{\\nu }(\\mathbf {q})$ is a six dimensional polarization vector with components $e^{\\nu }_{si}(\\mathbf {q})$ , where $s$ =(A,B) and index $i$ refers to the three Cartesian coordinates of the displacement vector $ \\mathbf {X}=(X,Y,Z) $ .",
"The derivative of the one-electron potential $\\mathbf {V}^{\\prime }$ has six components $V^{\\prime }_{si}= {\\partial V_s}/{\\partial X_i} $ .",
"A calculation of the EPS linewidth apparently requires information about the electron structure – band structure and wave functions, and phonon structure – band structure and polarizations fields.",
"The electron structure is achieved from a tight-binding (TB) calculations and the phonon structure from a force constant model (FCM)."
],
[
"Electron structure",
"In the TB approximation the wave functions $\\psi _{n\\mathbf {k}}$ are written $\\psi _{n\\mathbf {k}}(\\mathbf {r}) = \\sum _{js} c_{nsj}(\\mathbf {k}) \\Psi _{sj}(\\mathbf {k},\\mathbf {r}) \\ ,$ where the Bloch orbitals are given by $\\Psi _{sj}(\\mathbf {k},\\mathbf {r}) = \\frac{1}{\\sqrt{N}} \\sum _{\\mathbf {R}} \\phi _{j}(\\mathbf {r} - (\\mathbf {R} + \\mathbf {r}_s )) e^{i\\mathbf {k}\\cdot (\\mathbf {R}+\\mathbf {r}_s)} ,$ where $N$ denotes the number of unit cells to be summed over, $\\phi _j$ the basis $\\lbrace \\phi _{2s},\\phi _{2p_x},\\phi _{2p_y},\\phi _{2p_z}\\rbrace $ .",
"The electronic bands $\\varepsilon _{n\\mathbf {k}}$ and coefficients $c_{nsj}(\\mathbf {k})$ are obtained by solving the generalized eigenvalues problem: $\\sum _{J^{\\prime }} [H_{JJ^{\\prime }}(\\mathbf {k}) - \\varepsilon _{n\\mathbf {k}} S_{JJ^{\\prime }}(\\mathbf {k})]c_{nJ^{\\prime }}(\\mathbf {k}) = 0 \\ ,$ where we use the short hand index notation $J=js$ .",
"$S_{JJ^{\\prime }}$ denotes the overlap matrix elements.",
"We apply the TB parameter-set shown in Table.",
"REF .",
"Table: Tight binding parameters.",
"Direct terms ε 2s \\varepsilon _{2s} and ε 2p \\varepsilon _{2p} and hopping parameters V ssσ V_{ss\\sigma }, V spσ V_{sp\\sigma }, V ppσ V_{pp\\sigma } and V ppπ V_{pp\\pi } are all given in units of eV while the overlap parameters S ssσ S_{ss\\sigma }, S spσ S_{sp\\sigma }, S ppσ S_{pp\\sigma } and S ppπ S_{pp\\pi } are dimensionless.",
"Based on published values of these parameters, , , we have made some slight adjustments to fit our own previously published DFT band structure calculation.",
"The calculated band structure is shown in Fig.",
"REF and agrees well with our Density functional theory (DFT) based calculation published recently.",
"[10] Figure: (Color online) Occupied part of the electron band structure.",
"The σ\\sigma bands in red and the π\\pi band in blue."
],
[
"Phonon structure",
"All six phonon modes are considered, three optical and three acoustic.",
"The optical phonon modes are: longitudinal optical (LO), transversal optical (TO) and out-of-plane optical (ZO).",
"The acoustic phonon modes are: longitudinal acoustic (LA), transversal acoustic (TA) and the out-of-plane acoustic (ZA).",
"Applying a FCM the dynamical matrix $D$ is calculated including up to third order nearest neighbor interactions.",
"The force constants $\\Phi _{ll^{\\prime }}^{As}$ are defined by $D_{ll^{\\prime }}^{As}(\\mathbf {q}) = \\sum _{\\mathbf {R}_s} \\Phi _{ll^{\\prime }}^{As}(\\mathbf {R}_s)e^{-i\\mathbf {q}\\cdot \\mathbf {R}_s} ,$ where index $l$ denotes the components of a complex vector $(\\xi ,\\eta )$ where $\\xi =X+iY$ and $\\eta =X-iY$ and $\\mathbf {X}_{\\parallel }=X\\hat{x}+Y\\hat{y}$ being the atomic in-plane displacement vector.",
"$\\mathbf {R}_s$ labels the vectors from a center A atom to the three nearest B atom, the six next-nearest A atoms and the three next-next-nearest B atoms.",
"The in-plane force constants, in the $(\\xi ,\\eta )$ representation, are parametrized according to Falkovsky.",
"[15] To achieve the dynamical matrix elements in the $(X,Y)$ representation we have to transform the force constants in the $(\\xi ,\\eta )$ representation to the $(X,Y)$ representation.",
"We then derive $D^{As^{\\prime }}_{XX}(\\mathbf {q}) &=& 2D^{As^{\\prime }}_{\\xi \\eta }(\\mathbf {q}) + D^{As^{\\prime }}_{\\xi \\xi }(\\mathbf {q}) + D^{As^{\\prime }}_{\\eta \\eta }(\\mathbf {q}) \\nonumber \\\\D^{As^{\\prime }}_{YY}(\\mathbf {q}) &=& 2D^{As^{\\prime }}_{\\xi \\eta }(\\mathbf {q}) - D^{As^{\\prime }}_{\\xi \\xi }(\\mathbf {q}) - D^{As^{\\prime }}_{\\eta \\eta }(\\mathbf {q}) \\nonumber \\\\D^{As^{\\prime }}_{XY}(\\mathbf {q}) &=& i[D^{As^{\\prime }}_{\\xi \\xi }(\\mathbf {q}) - D^{As^{\\prime }}_{\\eta \\eta }(\\mathbf {q})] \\ .$ Fourier transforming the equation of motion we then get the eigenvalue problem: $\\sum _{s^{\\prime }i^{\\prime }} [D_{ii^{\\prime }}^{ss^{\\prime }}(\\mathbf {q}) - \\omega ^2_{\\nu }(\\mathbf {q}) \\delta _{ss^{\\prime }}\\delta _{ii^{\\prime }}]e^{\\nu }_{s^{\\prime }i^{\\prime }}(\\mathbf {q}) = 0 \\ ,$ Table: Force constants in units of 10 5 ^5 cm -2 ^{-2} for nearest-neighbors (nn), next-nearest-neighbors (nnn) and next-next-nearest-neighbors (nnnn) interaction.",
"The force constants are given in the complex representation (ξ,η)(\\xi ,\\eta ), where ξ=X+iY\\xi =X+iY and η=X-iY\\eta = X - iY, where (X,Y)(X,Y) is the Cartesian coordinate representation.All force constants are taken from Falkovsky , except Φ zz SUB \\Phi ^{SUB}_{zz} which is the force constant representing a spring connecting the carbon atoms to a rigid substrate.",
"The value 0.38 for this force constant is set to reproduce the finite energy of the ZA mode at Γ ¯\\bar{\\Gamma } of 24 meV in a recent DFT based calculation of graphene on SiC.where the subscript $i$ labels the three components $X$ , $Y$ and $Z$ of the Cartesian displacement vector $\\mathbf {X}=X\\hat{x}+Y\\hat{y}+Z\\hat{z}$ , where $Z$ denotes the out-of-plane displacement.",
"In addition to the parameters of Falkovsky [15], we add a spring between all carbon atoms of the graphene layer and a rigid substrate in order to take into account the influence of the substrate in a first order approximation.",
"The force constant of this spring is adjusted to fit the out-of-plane phonon mode dispersion from a $\\it {first}$ $\\it {principles}$ calculation of the phonon band structure of graphene on SiC.",
"[16] The complete set of force constants are shown in Table.",
"REF and the phonon dispersion relation, solving Eq.",
"(REF ), is shown in Fig.",
"REF .",
"The solid lines represent the phonon bands of graphene on SiC and the dashed lines the dispersion of the out of plane modes of unsupported graphene.",
"The phonon band dispersion of unsupported graphene agrees well with our published DFT based calculation.",
"[10] Figure: (Color online) Phonon band structure of unsupported graphene (dashed lines) and graphene supported on a rigid substrate (solid lines).",
"Red indicates out-of-plane modes and blue indicates in-plane (transverse and longitudinal) modes."
],
[
"Deformation potential",
"The deformation potential in the EPC matrix element $g^{\\nu }(n\\mathbf {k},n^{\\prime }\\mathbf {k^{\\prime }})$ is calculated according to the Rigid Ion Approximation (RIA), displacing a spherically symmetric screened one-electron atomic model potential $V(r) = -V_0 e^{-(r/r_o)^2}$ .",
"Then the basis orbital EPC matrix elements take the form $\\langle \\phi _i^I | \\partial V/\\partial X^{I^{\\prime \\prime }}_j|\\phi _{i^{\\prime }}^{I^{\\prime }}\\rangle =- \\langle \\phi _i^I | \\partial V/\\partial x^{I^{\\prime \\prime }}_j|\\phi _{i^{\\prime }}^{I^{\\prime }}\\rangle &=& \\nonumber \\\\\\frac{2 V_o}{r_o^2}\\langle \\phi _i^I |x^{I^{\\prime \\prime }}_j e^{-(r^{I^{\\prime \\prime }}/r_o)^2}|\\phi _{i^{\\prime }}^{I^{\\prime }}\\rangle \\ .$ $X_j^I$ and $x_j^I$ denotes the Cartesian atomic displacement coordinates and electron coordinates relative the equilibrium atomic position $\\mathbf {R}_I$ , respectively.",
"The parameters, strength $V_0$ and the screening length $r_o$ are set to fit both an experimentally observed linewidth and a $first$ $principles$ calculation of EPC matrix elements.",
"The experimental linewidth refers to the measured linewidth of the $\\sigma _o$ band 200 meV below the top of the $\\sigma $ bands [10], and to the $first$ $principles$ calculation of the quantity $\\sum _{\\nu =LO,TO}|\\langle \\sigma _o,\\mathbf {k}|\\delta V^\\nu _{\\mathbf {q} = - \\mathbf {k}}|\\sigma _o, \\mathbf {o} \\rangle |^2 ,$ which varies weakly over the square area: - 0.1 au $\\le $ k$_x$ ,k$_y$ $\\le $ +0.1 au.",
"[10] ARPES is a powerful tool to investigate the many-body nature of solid-state systems [17], [18].",
"Indeed, it gives a direct measure of the spectral function of a material, which intrinsically contains information on the real and imaginary parts of a self energy $\\Sigma $ .",
"$\\Sigma $ describes the many-body interactions, among which the most significant contributions typically come from electron phonon coupling (EPC), electron impurity scattering (EIS) and electron electron scattering (EES).",
"For these contributions we can write $\\Sigma =\\Sigma ^{EPC}+\\Sigma ^{EIS}+\\Sigma ^{EES}$ .",
"In addition to this, the linewidth of the ARPES spectra is closely related to the imaginary part of $\\Sigma $ and it is therefore necessarily affected by all these contributions [19].",
"Whilst the ARPES linewidth intrinsically contains contributions from all relevant many-body interactions, EPC is commonly responsible for abrupt changes in the linewidth.",
"Furthermore, such abrupt changes will occur on an energy-scale corresponding to the energy of the relevant phonon mode(s).",
"These factors generally allow the EPC contribution to the linewidth to be disentangled from EIS and EES [19], [18].",
"In this section we compare the linewidth extracted form ARPES measurements with our corresponding tight-binding calculated linewidths, due to EPC.",
"We will focus on the linewidth of the sigma bands $\\sigma _o$ and $\\sigma _i$ and the $\\pi $ band in the high symmetry directions of the Brillouin zone.",
"We aim at understanding which phonon modes are most important in assisting the electron scattering and to judge the relative importance of interband and intraband scattering."
],
[
"$\\sigma $ bands",
"We analyze the origin of the observed kink in the $\\sigma $ bands about 200 meV below the top of the $\\sigma $ bands, referring to recently presented ARPES data.",
"[9], [10] In Fig.",
"REF we show the calculated linewidth of the inner and outer $\\sigma $ bands ($\\sigma _i$ and $\\sigma _o$ ) in the two high symmetry directions $\\bar{\\Gamma } \\rightarrow $ K̄ and $\\bar{\\Gamma } \\rightarrow $ M̄.",
"The binding energy range of about 1 eV below the top of the occupied $\\sigma _o$ and $\\sigma _i$ bands corresponds to a region close to the $\\bar{\\Gamma }$ -point.",
"The sudden increase of the calculated total linewidth ($\\sigma _o$ : sum of contributions from $\\sigma _o\\rightarrow \\sigma _o$ , $\\sigma _i\\rightarrow \\sigma _o$ and $\\pi \\rightarrow \\sigma _o$ and $\\sigma _i$ : sum of contributions from $\\sigma _o\\rightarrow \\sigma _i$ , $\\sigma _i\\rightarrow \\sigma _i$ and $\\pi \\rightarrow \\sigma _i$ ) is found in both symmetry directions at about 200 meV below the top of these sigma bands.",
"Figure: (Color online) Calculated linewidth of the sigma bands σ o \\sigma _o and σ i \\sigma _i versus the binding energy of the σ\\sigma band maximum at the Γ ¯\\bar{\\Gamma } point, E σ _\\sigma .",
"The black solid line represents the full linewidth and the contributions from different assisting phonon modes are shown.",
"Results are shown for the two high symmetry directions Γ ¯→\\bar{\\Gamma } \\rightarrow K̄ and Γ ¯→\\bar{\\Gamma } \\rightarrow M̄.",
"The inset shows the color notation for the contribution to the linewidth from the different phonon modes.The main contributions originate from $\\sigma $ inter- and intraband scattering assisted by the two high energy optical phonon modes LO and TO.",
"Analysing the linewidth of the $\\sigma _o$ band in Fig.",
"REF , panels (a) and (b), it is interesting to note that in the $\\bar{\\Gamma } \\rightarrow $ K̄ direction there is an increasing contribution from the interband scattering $\\pi \\rightarrow \\sigma _o$ assisted by the out-plane acoustic ZA mode for increasing binding energies.",
"In the direction $\\bar{\\Gamma } \\rightarrow $ M̄ the out-of-plane ZA mode driven interband scattering $ \\pi \\rightarrow \\sigma _o $ is of minor importance.",
"The linewidth of the inner $\\sigma $ band $\\sigma _i$ is shown in panels (c) and (d) in Fig.",
"REF .",
"The result is reversed.",
"The ZA mode driven $\\pi \\rightarrow \\sigma _i$ scattering is in this case more important in the $\\bar{\\Gamma } \\rightarrow $ M̄ direction.",
"The reason for this is to be found in the EPC matrix element.",
"In the direction $\\bar{\\Gamma } \\rightarrow $ K̄ then $\\langle \\sigma _o|\\delta V_{ZA}|\\pi \\rangle $ is nearly totally symmetric while in the direction $\\bar{\\Gamma } \\rightarrow $ M̄, $\\langle \\sigma _i|\\delta V_{ZA}|\\pi \\rangle $ is nearly totally symmetric.",
"We conclude that the sudden increase of the calculated full linewidth of the $\\sigma _o$ and $\\sigma _i$ bands at about 200 meV below the top of these bands is clear and in good agreement with experimental findings.",
"[10] The sudden increase of the linewidth, $\\Gamma $ = 2 Im$\\Sigma ^{EPC}$ , connects to a sudden change - a kink - in the observed band energy, $\\varepsilon _o$ = $\\varepsilon _o^0$ + Re$\\Sigma ^{EPC}$ , where $\\Sigma ^{EPC}$ represents the EPC self energy.",
"For the $\\sigma _o$ band the origin stems from the $\\sigma _i \\rightarrow \\sigma _o$ and $\\sigma _o \\rightarrow \\sigma _o$ scattering and in the case of the $\\sigma _i$ band from $\\sigma _i \\rightarrow \\sigma _i$ and $\\sigma _o \\rightarrow \\sigma _i$ scattering.",
"The calculations show that the main contributions originate from assisting TO and LO phonons.",
"Furthermore, we find that the linewidth of $\\sigma _o$ and $\\sigma _i$ bands are anisotropic in the surface Brillouin zone in the energy region investigated.",
"The increasing contribution from the interband $\\pi \\rightarrow \\sigma $ scattering, assisted by the ZA phonon mode, indicates that this anisotropy will be even more pronounced at greater binding energies.",
"Figure: (Color online) Calculated and measured linewidth of the π\\pi band of graphene supported by SiC.",
"(a) Calculated linewidth in the K̄-Γ ¯\\bar{\\Gamma } direction and (b) in the M̄-Γ ¯{\\bar{\\Gamma }} direction.",
"Calculations are shown for four different vibrational energy cut-offs; 50 meV (green), 100 meV (blue), 150 meV (red) and 200 meV (black).",
"The dashed line is the total contribution for the unsupported graphene.",
"(c) and (d) ARPES measurements and corresponding linewidth values.",
"(c) ARPES data (gold-black shading) for the π\\pi band of monolayer graphene on SiC acquired along the Γ ¯{\\bar{\\Gamma }}- K̄ direction of the Brillouin zone, together with the extracted linewidth (blue curve) and spectra.",
"To help the data visualization a baseline has been overlaid to the data.",
"(d) ARPES data along the M̄- Γ ¯\\bar{\\Gamma } direction (gold-black shading) and the extracted linewidth.",
"In all pictures, the yellow, purple and blue areas represent the energy ranges where the intra and inter band transitions manifest.",
"The experimentally determined values of the σ\\sigma band maximum and VH-singularity as indicated.",
"Green shading indicates the contributions to the linewidth given by the replica bands ."
],
[
"$\\pi $ band",
"The linewidth of the $\\pi $ band has also been investigated applying our tight-binding model based calculations.",
"In Fig.",
"REF we show the result of the calculation, along the high symmetry directions K̄ $\\rightarrow \\bar{\\Gamma }$ and K̄ $\\rightarrow $ M̄.",
"The results for the linewidth for the four different phonon frequency cut-offs, 50, 100, 150 and 200 meV show that the optical high frequency modes dominates the intra $\\pi $ band scattering from the $\\bar{M}$ point down to the top of the $\\sigma $ band.",
"The peak at E$_B$ $\\approx $ 3 eV arises because of the increased electronic density of states due to the flat $\\pi $ band in region of the $\\bar{M}$ point (a.k.a.",
"the van Hove singularity) and is discussed further below.",
"Below the $\\sigma $ band maximum, E$_B$ $\\approx $ 4 eV, the interband scattering $\\sigma _i \\rightarrow \\pi $ and $\\sigma _o \\rightarrow \\pi $ becomes increasingly more important.",
"The green line in Fig.",
"REF , corresponding to a phonon energy maximum of 50 meV clearly indicates that it is only the acoustic out-of-plane ZA mode which is in operation in the $\\sigma _i \\rightarrow \\pi $ and $\\sigma _o \\rightarrow \\pi $ scattering.",
"As the bottom of the $\\pi $ is approached, the contribution from these scattering channels dominate completely.",
"This is explained by the increase of the phase space of the initial electron states, referring to the $\\sigma _i$ and $\\sigma _o$ bands and in addition also to the reduced slope of the $\\pi $ band as the $\\bar{\\Gamma }$ point is reached.",
"The peak in the linewidth of the $\\pi $ band of unsupported graphene (dashed lines in Fig.",
"REF (a) an (b)) close to the bottom of the $\\pi $ band is due to a large contribution near the crossing of the $\\pi $ band and the $\\sigma _o$ band and the $\\sigma _i$ band in the direction K̄ $\\rightarrow \\bar{\\Gamma }$ ( Fig.",
"REF (a)) and M̄ $\\rightarrow \\bar{\\Gamma }$ ( Fig.",
"REF (b)), respectively.",
"This peak signals the instability of unsupported graphene due to the $\\sim q^2$ dispersion of the ZA mode.",
"This singularity is lifted when graphene is supported on a substrate.",
"[21] The finite frequency, $\\sim $ 24 meV, at the $\\bar{\\Gamma }$ -point (see Fig.",
"REF ) stabilizes the crystal structure of the graphene layer.",
"The reason why the peaks appear at different binding energies is again, just as for the $\\sigma $ bands, to be found in the EPC matrix element.",
"Considering the unit cell including the $A$ and the $B$ atom, we have that in the $\\bar{\\Gamma } \\rightarrow $ K̄ direction $|\\sigma _o\\rangle \\approx \\frac{1}{\\sqrt{2}}(|2p^A_x\\rangle -|2p^B_x\\rangle )$ (with the $x$ -axis in the $A$ to $B$ direction) while in the $\\bar{\\Gamma } \\rightarrow $ M̄ direction $|\\sigma _i\\rangle $ has the same form.",
"Thus $\\langle \\pi |\\delta V_{ZA}|\\sigma _o\\rangle $ will be totally symmetric in the $\\bar{\\Gamma } \\rightarrow $ K̄ direction, while in the $\\bar{\\Gamma } \\rightarrow $ M̄ the $\\langle \\pi |\\delta V_{ZA}|\\sigma _i\\rangle $ will be totally symmetric.",
"ARPES measurements on monolayer graphene on SiC and their corresponding spectral linewidth are shown in Fig.",
"REF (c) and (d) for the $\\pi $ band of graphene acquired along the $\\bar{\\Gamma }$ -K̄ and $\\bar{\\Gamma }$ -M̄ directions, respectively.",
"Each cut at constant energy in Fig.",
"REF (c,d), a so-called momentum distribution curve (MDC), has been fitted by Lorentzian curves with inclusion of a cubic polynomial background and from the fit results, the linewidth as a function of energy is extracted.",
"As can be seen in the figure, the linewidth shows several sudden changes occurring at E$_{B} \\approx 2$ ; $3.3$ ; $4.5$ and $5.5$ eV.",
"Unlike our calculation (which only includes EPC contributions to the linewidth), the ARPES measurement intrinsically includes all relevant interactions.",
"It is therefore necessary to discuss the origin of the experimentally observed linewidth changes.",
"At E$_{B} \\approx 2$ eV, (green area in Fig.",
"REF (b)) a change in the spectral linewidth is observed.",
"At such an energy the graphene/SiC electronic dispersion is known to be affected by replica-bands.",
"These bands originate from the interaction between the graphene and the substrate on which it is grown [20].",
"These bands have a weak intensity and in our experimental data are difficult to see, however the lineshape of the ARPES spectra in this region indicates the presence of additional components, and we conclude that they are responsible for the linewidth change observed experimentally at this $E_B$ value.",
"The linewidth changes at E$_{B}\\approx 3.3$ eV and E$_{B}\\approx 4.5$ eV cannot be explained by replica bands or substrate interactions (Fig.",
"REF (c-d), orange and purple areas, respectively): At $E_{B}\\approx 3.3$ eV the e-DOS suddenly increases, due to the electron accumulation at the Van Hove (VH) singularity [22].",
"The VH-singularity constitutes a local maximum of the $\\pi $ -band at the M point of the BZ, as indicated in Fig.",
"REF .",
"Therefore, the VH-singularity creates an increase in the e-DOS, and hence the probability of phonon mediated refilling of the photo-hole is dramatically increased, in good agreement with our calculations (for example, Fig.",
"REF (a).",
"At $E_{B}\\approx 4.5$ eV, a similar change in the measured linewidth is also seen.",
"This also occurs at an energy where the e-DOS is dramatically increased, but in this case it is because of the maximum of the $\\sigma $ -band.",
"Again, because the e-DOS shows a strong increase, the probability of phonon-mediated refilling dramatically increased and hence the lifetime of a photo-hole is reduced.",
"In agreement with our tight-binding calculation, this is observed as an increase in the linewidth due to EPC.",
"At $E_{B}\\approx 5.5$ eV, the ARPES linewidth has become very broad, and also appears to show a peak.",
"It is difficult to unambiguously disentangle the EPC contribution to the linewidth since EES may also play a significant role here.",
"Also, the dramatically increased ARPES linewidth hinders accurate analysis.",
"However, it is interesting to note that the tight-binding calculation (which only includes EPC contributions to the linewidth) predicts that the linewidth will dramatically increase in this $E_B$ range, hence it seems feasible that the large measured linewidth is at least partially due to increased EPC.",
"Unlike the previous cases where the increase in EPC was primarily due to an increase in the e-DOS, in this case, it is the crossing of the $\\pi $ -band with $\\sigma _i$ and $\\sigma _o$ which dramatically increases the efficiency of EPC by allowing phonon modes with little energy and momentum (i.e.",
"acoustic modes) to make a large contribution.",
"It should also be noted that the experimentally reported changes in the spectral linewidth are small and, purely from the experiment, we cannot exclude a priori that there might be other contributions to such linewidth changes; however, the good agreement between the experiment and the tight-binding calculation (which predicts the same linewidth changes) significantly strengthens the validity of our interpretations."
],
[
"Summary and Conclusions",
"We present a theoretical investigation of the electron-phonon interaction in pristine graphene and compare with experimental ARPES data.",
"The interaction is found to be considerably stronger in the $\\sigma $ band than in the $\\pi $ band.",
"The theoretical linewidth analysis of the two uppermost occupied $\\sigma $ bands in the region of the $\\bar{\\Gamma }$ -point supports the picture that the scattering is primarily driven by the high energy optical phonon modes LO and TO.",
"The calculations also reveal a strong anisotropy of these $\\sigma $ bands in the surface Brillouin zone.",
"In the $\\bar{\\Gamma } \\rightarrow $ K̄ direction, the interband scattering $\\pi \\rightarrow \\sigma $ , driven by the out-of-plane phonon mode ZA, dominates in most of the energy region where the $\\sigma $ and $\\pi $ bands overlap.",
"The calculated linewidth of the $\\pi $ band is compared in detail along the K̄ $\\rightarrow $ $\\bar{\\Gamma }$ and M̄ $\\rightarrow $ $\\bar{\\Gamma }$ directions with ARPES data.",
"The main features are reproduced by the calculations.",
"In the energy regions where the $\\sigma $ and $\\pi $ overlap the linewidth is found to be nearly isotropic in the surface Brillouin zone.",
"Also for the $\\pi $ band, the interband scattering, now $\\sigma \\rightarrow \\pi $ , dominates and the acoustic ZA mode is most important.",
"We show that in order to understand the variation of the linewidth it is not enough to only consider the density of state effects (for example the van Hove singularities) - it is also important to consider the symmetry of the EPC matrix element.",
"The latter is of central importance in some regions of the BZ (for example, at the $\\sigma $ -band maximum).",
"We also demonstrate that when taking the graphene-substrate coupling into account, the lattice instability of unsupported graphene caused by the acoustic ZA vibrational mode is removed and the sharply peaked $\\pi $ -band linewidth increase is reduced such that it is in better agreement with the experimental data.",
"This work was partly supported by the Research Council of Norway through its Centres of Excellence funding scheme, project number 262633, “QuSpin”, and through the Fripro program, project number 250985 “FunTopoMat”.",
"The linewidth calculations were performed on resources at Chalmers Center for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).",
"TF acknowledges Grant.",
"FIS2017-83780-P from the Spanish Ministerio de Economía y Competitividad."
]
] | 1808.08620 |
[
[
"StreamChain: Rethinking Blockchain for Datacenters"
],
[
"Abstract Permissioned blockchains promise secure decentralized data management in business-to-business use-cases.",
"In contrast to Bitcoin and similar public blockchains which rely on Proof-of-Work for consensus and are deployed on thousands of geo-distributed nodes, business-to-business use-cases (such as supply chain management and banking) require significantly fewer nodes, cheaper consensus, and are often deployed in datacenter-like environments with fast networking.",
"However, permissioned blockchains often follow the architectural thinkining behind their WAN-oriented public relatives, which results in end-to-end latencies several orders of magnitude higher than necessary.",
"In this work, we propose StreamChain, a permissioned blockchain design that eliminates blocks in favor of processing transactions in a streaming fashion.",
"This results in a drastically lower latency without reducing throughput or forfeiting reliability and security guarantees.",
"To demonstrate the wide applicability of our design, we prototype StreamChain based on the Hyperledger Fabric, and show that it delivers latency two orders of magnitude lower than Fabric, while sustaining similar throughput.",
"This performance makes StreamChain a potential alternative to traditional databases and, thanks to its streaming paradigm, enables further research around reducing latency through relying on modern hardware in datacenters."
],
[
"Introduction",
"Blockchains (distributed ledgers) offer strong data integrity and reliability properties in untrusted environments that make them worth considering for distributed data management use cases beyond crypto-currencies.",
"Their adoption, in particular that of permissioned blockchains (in which membership is vetted, e.g., [9]) is expected to bring benefits to enterprise consortia thanks to the logical centralization of datasets that previously resided at separate sites and were interconnected through various protocols.",
"Prominent examples include financial use cases [22], supply-chain management [17] and provenance [26], to name but a few.",
"As most enterprise software, permissioned blockchains are deployed predominantly in cloud environments, with all leading cloud providers such as Amazon [2], IBM [3], Microsoft [7] and Oracle [4] offering hosted enterprise blockchain infrastructure and services.",
"Even though this introduces the cloud provider as a centralization point, which is at odds with blockchain decentralization, in the long run, cloud providers are likely to offer ways of interconnecting blockchain nodes across multiple clouds while retaining high bandwidth and low latency communication.",
"Such multi-cloud blockchain offerings, that span multiple cloud providers, are in fact already starting to appear [6].",
"Figure: The latency of Fabric is dominated by block-based batching and disk access.",
"By removing batching and applying further optimizations, latency can be reduced by two orders of magnitude.Even in cross-cloud enterprise blockchain deployments, geo-distribution is often neither required nor beneficial.",
"For instance, in performance sensitive blockchain applications, such as stock exchanges [5], [1], datacenter-like deployments, characterized by geographical proximity, are a better fit.",
"Fault-diversity may be ensured by running nodes across different datacenters belonging to different providers, while geographical proximity allows network latencies to remain low and bandwidth high.",
"Unfortunately, when deployed in datacenter-like environments, all current permissioned distributed ledgers exhibit poor performance.",
"For instance, they will take hundreds of milliseconds to commit transactions [9], even if the networking latencies are in the order of microseconds (see Fig.",
"REF ).",
"This is because the design of most permissioned blockchains is based on that of public blockhcains and, as a result, they are often optimized for wide area networks.",
"For instance, the batching of transactions into blocks is useful for amortizing the high computational cost of Proof-of-Work (PoW) consensus, as well as networking overheads; however, in permissioned blockchains, blocks are a source of added latency and should be removed.",
"In the light of emerging datacenter-like deployments, we introduce latency as the first-class blockchain performance metric to complement throughput, which is generally perceived as the main metric of interest [30].",
"We revisit the design decisions of permissioned blockchains from the perspective of reducing latency and, as a concrete instantiation of our design recommendations, we prototype StreamChain, our streaming variant of Hyperledger Fabric [9].",
"In a nutshell, StreamChain departs from block-based system design and chains transactions directly, in a streaming fashion, drastically reducing latency, but without jeopardizing security, reliability or throughput guarantees.",
"The contributions of this work are as follows: Inroducing a latency-efficient permissioned distributed ledger for datacenter-like environments: By not batching transactions into blocks and by exploiting the parallelism of modern multicores, StreamChain can achieve end-to-end latencies close to 1.5 ms without compromising throughput.",
"As shown in Figure REF , its latency is two orders of magnitude smaller than vanilla Fabric running on a local area network, and one order of magnitude smaller than Fabric naively configured to form single transaction blocks.",
"Overall, StreamChain provides latencies that are comparable to traditional databases, thereby making it a more realistic alternative to consider.",
"Showcasing integration with modern datacenter accelerators: Even though consensus can be adapted for fast networks [19], [24], current permissioned ledgers do not benefit significantly from such optimizations due to the heavy use of batching.",
"Conversely, latency improvements of consensus make a difference in StreamChain and, as proof-of-concept, we implemented an FPGA-based consensus service that makes it possible to reach commit latencies (i.e., latency excluding smart contract execution) below a millisecond.",
"Platform for future research: StreamChain is an open-sourceSource code available at: https://gitlab.software.imdea.org/zistvan-public/streamchain and https://gitlab.software.imdea.org/zistvan-public/streamchain-benchmarks platform to explore throughput and latency improvements in permissioned ledgers.",
"StreamChain builds on a long-term support (LTS) release of Fabric (v.1.4) which allows future platform improvements to remain compatible with currently existing application ecosystems.",
"This paper is organized as follows: we provide an overview of permissioned blockchains and Hyperledger Fabric in Section .",
"The design and implementation of StreamChain is described in Section .",
"We evaluate the system in Section with a YCSB microbenchmark and a Supply Chain Management application.",
"We talk about the next steps in Section and conclude in Section .",
"Public ledgers do not authenticate nodes and instead they rely on PoW and similar consensus protocols to make Sybil attacks impractical.",
"Permissioned (private) blockchains, on the other hand, authenticate all nodes and therefore allow the use of much cheaper consensus protocols.",
"Nonetheless, permissioned ledgers inherit their design from public ones with many systems evolving from crypto-currency to more general use through “smart contracts” (e.g.",
"Ethereum [31]).",
"Since the execution of the smart contracts has to be serial, many of these systems are severely limited in throughput when running complex contracts.",
"As a result, Ethereum, for instance, adds a “complexity” fee to smart contracts in the form of “gas”, thereby limiting usability in enterprise scenarios.",
"Hyperledger Fabric [9] is an open-source permissioned ledger that allows smart contracts (“chaincode”) to be developed in general purpose programming languages.",
"It also removes the execution bottleneck by using an Execute-Order-Validate (EOV) approach (see Figure REF ).",
"As opposed to the Order-Execute (OE) model used, for instance, in Ethereum, where smart contracts are ordered and then shipped to all nodes for execution, in EOV only a subset of nodes execute them.",
"This is done in isolation on top of the “materialized view” of the ledger (Fabric's State DB) to determine their read/write sets and values (TX descriptors).",
"The ordering nodes run a consensus protocol and establish the global order of these read/write sets which are then validated on all peers on their current State DB and committed to the ledger.",
"The EOV model saves compute resources at peers and allows the execution of feature-rich smart contracts.",
"The drawback of the EOV model is that, since execution and committing happen concurrently on the nodes, it is possible that in the time it takes to order transactions, the underlying state of the variables in the read/write set have changed.",
"This will result in a “failed” transaction whose results are not applied to the ledger state, even though its execution is recorded.",
"This means that in case multiple clients execute chaincode that shares state, the successful transaction throughput (goodput) can be significantly lower than the raw throughput of the system.",
"Figure: Fabric processes transactions in three steps, relying on two types of nodes: peers for the first and third and orderers for the second."
],
[
"Improvements and Alternative Designs",
"Most related work sets out to increase the throughput of permissioned ledgers but end-to-end latencies of single transactions are typically not addressed.",
"However, we believe that this is an important feature to improve, especially in use-cases where network latencies are less than in a wide area deployment.",
"The authors of FastFabric [16] propose various optimizations to Fabric which result in an unprecedented 20,000 ops/s throughput.",
"They achieve this through a combination of multi-threaded batch processing, changes to the ordering service and caching deserialized data structures in the validation phase.",
"We have adopted the latter technique in StreamChain and reduced, as a result, the cost of the validation step by 10% (see details in Section REF ).",
"Sharma et al.",
"[27] focus on increasing goodput in the EOV model and demonstrate that, by relying on database techniques for concurrency control, it is possible to reorder transactions within a batch inside the ordering service in a way that minimizes the number of failing transactions.",
"This idea is applicable to our case as well but it changes an assumption that Fabric makes about the ordering service, namely, that the contents of the transactions are opaque.",
"Depending on the trust model, this raises concerns related to malicious reordering.",
"CAPER [8] demonstrates how multiple applications that share the same distributed ledger can leverage the fact that they operate on disjoint parts of the dataset in order to increase the overall throughput.",
"This is achieved by separating the ordering of operations inside an application from that of ordering across applications.",
"The authors propose a supply chain management workload for benchmarking, similar to the one we use for evaluation.",
"There is an emerging class of distributed ledgers that do not incorporate the notion of blocks at all and operate on a per-transaction basis.",
"Two well known examples are Corda [11] and IOTA [25], which were designed with a very specific financial services use-case in mind.",
"Instead of storing data in a single chain, a directed acyclic graph (DAG) was used.",
"This allows executing transactions between various subsets of the peers without the burden of global ordering, but also results in systems with different properties than traditional permissioned ledgers."
],
[
"Preliminary Version of This Work",
"In a preliminary version of this work [20] we demonstrated that the main idea behind StreamChain is viable by using a mock implementation that consisted of running Fabric v1.0 configured with 1TX per block and to run in main-memory (i.e., without persistence).",
"In this work we provide a full implementation and further optimizations by: re-implementing StreamChain based on Hyperledger Fabric 1.4 LTS in a way that produces a transaction log that remains compatible with vanilla Fabric; introducing additional pipelining in the validation stage to increase throughput; providing a tunable disk write batching mechanism that ensures persistence while hiding I/O cost; and showing that the streaming execution model allows us to further optimize the ordering service, for instance, by including specialized hardware based on FPGAs.",
"Even though the throughput of StreamChain could be further increased, it is already a feasible alternative to traditional databases in emerging use-cases, even for contention-heavy workloads.",
"Furthermore, StreamChain opens up novel opportunities for incorporating specialized hardware, such as FPGAs or SmartNICs, at various levels since the time spent within each processing step is directly exposed."
],
[
"Which Scenarios is It For?",
"StreamChain targets datacenter-like deployments that are closer to a single datacenter than to geo-distributed environements and offer high bandwidth and low latency networking across nodes.",
"Given that, at the moment, virtually all hosted permissioned ledgers are single-cloud, in the current prototype StreamChain tolerates Byzantine peers but uses crash fault tolerant (CFT) ordering, just like Fabric v1.4 LTS.",
"This means that the participants have to trust ordering nodes not to be malicious/Byzantine.",
"In the long run, with the emergence of multi-cloud deployments, it will be necessary to integrate a Byzantine fault tolerant (BFT) ordering service (e.g., [28]) with our streaming optimizations.",
"We discuss the path to achieving this in Section ."
],
[
"Insights and System Design",
"StreamChain builds on the following insight: in permissioned ledgers that have fast consensus, blocks have a negative effect on response times in low latency networks.",
"Removing blocks, however, does not change the total amount of computation in the system.",
"It still requires careful parallelization and pipelining of cryptographic operations and disk accesses in order to achieve reasonable throughput.",
"An important side-effect of reducing latency in EOV systems is that it results in higher goodput: lower commit latencies mean less stale data for endorsement, which in turn yields less failed transactions in validation."
],
[
"Batching and Caching",
"In Fabric, both the ordering and the validation (commit) stages are writing to disk for each block that is processed in the system, and all transactions, even those that only read from the state, have to be persisted in the ledger.",
"If we would naively set the size of blocks to a single transaction then latencies would be dominated by the disk flushing cost (see Figure REF ) and throughput would be reduced to the IOPS of the underlying device.",
"In StreamChain, we replace the code-path that flushes the ledger to disk with a local batcher that waits until enough data has accumulated or a time-out is reached (e.g., 100ms) before flushing to disk in an asynchronous manner.",
"To avoid inconsistencies, all buffers are flushed immediately if the code on top performs a read of the underlying data.",
"The same approach applies to all data structures in the peers that are persisted to disk (ledger state, index structures, etc.",
"), with the exception of the materialized view of the ledger that is stored in the State DB.",
"In Fabric, LevelDB is used by default for the State DB and it persists its own data.",
"The main reason for writing to disk in LevelDB is to speed up recovery in case a node fails and restarts, but we argue that it would be enough to checkpoint the key-value store off the critical path.",
"Hence, we configure LevelDB to run in main-memory.",
"Since the ledger is still persisted to disk, durability is not compromised.",
"Fabric uses gRPC for the communication between participants of the network and it uses Protocol Buffers to marshal and unmarshal messages.",
"This can be an expensive operation with the block data structure that the orderers send to the peers.",
"Unmarshalling happens selectively and at different levels of granularity; since parts of the block are used several times during the validation phase this leads to inefficiencies.",
"The data structure we use to wrap transactions in StreamChain are very similar to blocks with less overall metadata, and therefore this inefficiency applies to StreamChain as well.",
"As a solution, we implement an idea from FastFabric [16], namely, a cache that stores unmarshalled block contents to avoid repeated work.",
"Since transactions are received on a single thread and never modified inside the peer, the cache doesn't need to be protected by locks against concurrency.",
"Beyond the above optimizations, we also modified the way in which the peer reads in its configuration parameters, caching them for subsequent accesses.",
"Furthermore, we disabled an auxiliary data structure, called the history DB that exists to support data provenance queries.",
"Nonetheless, the same batching techniques apply to this data structure as to the main ledger structures."
],
[
"Using Available Parallelism",
"The peers receive transactions (or blocks in the case of Fabric) from the ordering nodes over the network.",
"These transactions are passed to the Validation logic that performs signature and read/write set checks before it records them in the ledger.",
"In StreamChain we divided the Validation logic into 3 pipeline stages to take advantage of multi-core machines.",
"This layout extends Fabric's two-stage pipeline by one stage and, whereas in Fabric parallel signature checks are only carried out within a block, in StreamChain they work on incoming transactions directly.",
"The pipeline is organized as follows: Message authentication and endorsement signature check – this step is the most expensive one because it requires checking various signatures to decide whether each transaction has been properly endorsed and ordered.",
"Since there are no data dependencies across transactions, we rely on several cores to parallelize this check and collect the results in a FIFO order.",
"In principle all idle cores of the CPU could be used for this purpose, but we found that in our experimental setup 6-8 were enough to result in pipeline stages with almost equal processing times.",
"Read/write set check and ledger commit – this step iteratively checks the transactions' read/write sets against the current ledger state (materialized in the State DB) and if no conflicts are found, it commits them to the ledger and updates the State DB.",
"This operation needs to be performed sequentially for transactions so it runs in a single thread.",
"Fabric uses a single reader/writer lock around the State DB and limits parallelism in our pipelined version between endorsement and validation.",
"Therefore, we replace the single lock with a series of locks that allow more fine-grained access.",
"Additional housekeeping – after updating the ledger state, various additional operations are performed on auxiliary data structures used, for instance, in the gossip protocol.",
"Even though we disable gossip in our deployment (as it's not needed in deployments with plenty of networking bandwidth between ordering nodes and peers), it is important that StreamChain doesn't remove Fabric features.",
"Therefore, these data structure updates are still performed but in their own pipeline stage, which moves them off the critical path."
],
[
"Exploring New Opportunities",
"Since StreamChain exposes the cost of each execution step, it acts as a platform for future exploration and lets us evaluate the usefulness of various emerging hardware features in the datacenter.",
"In this work we focus on the ordering step and show that by incorporating Field Programmable Gate Arrays (FPGAs), which are increasingly available in datacenters, into the ordering service we are able to reduce its latency further and allow it to scale to higher throughput in the future.",
"We target a scenario where the permissioned ledger is hosted by a service provider that has existing infrastructure (e.g.",
"Microsoft Catapult) that can be used to improve the overall performance of the blockchain."
],
[
"FPGAs in the Datacenter",
"Field programmable gate arrays (FPGAs) are hardware chips that are composed of a collection of small look-up tables (LUTs), on-chip memory (BRAM) and specialized digital signal processing units (DSPs), which can be configured and interconnected to implement any hardware circuit.",
"In comparison to traditional processors, FPGAs allow for fine-grained dataflow parallelism due to the fact that all “code” is executing in parallel in the device.",
"The energy footprint of FPGAs is an order of magnitude lower than that of server-grade CPUs (and even though it is higher than that of ASICs, FPGAs can be reprogrammed freely, whereas ASICs have fixed functionality).",
"For a more detailed description of FPGA internals and a summary of their strengths in data processing scenarios, we direct the reader to the book by Teubner and Woods [29].",
"FPGAs are being explored in cloud context both for compute intensive tasks related to machine learning [12] and as low latency key-value store solutions [32], [15], [18] mainly due to their ability to provide predictable, line-rate behavior even when performing near-data processing tasks.",
"Importantly, they are already being used in production for infrastructure acceleration, for instance in Project Catapult [13], to offload networking tasks in Azure virtual machines.",
"With the general availability of FPGAs in the cloud (the number Catapult-enabled Azure servers in 2018 was reported to be more than a million [13]), they could be used to provide functionality packaged as a service with a lower energy footprint, hence lower cost, and more predictable performance than software-based solutions.",
"Figure: We integrate a crash fault tolerant ordering service built with FPGAs into StreamChain by using a software node to act as facade between the service and the blockchain peers."
],
[
"FPGA-based Ordering",
"Fabric is designed such that the ordering service can easily be replaced with custom implementations.",
"In version 1.4, it has a service built with Raft [23] as its main way of ordering blocks/transactions.",
"As an alternative, an Apache Kafka based one is available but this introduces higher latencies and more complexity to the system.",
"Fabric also offers a “solo” orderer comprising of a single node (hence no fault tolerance) that is useful for development and testing purposes.",
"For the FPGA-based ordering service, we build upon our earlier work on CFT consensus on stand-alone FPGAs [19].",
"That work implements Zookeeper's Atomic Broadcast protocol to replicate write operations in a key-value store running on the same nodes.",
"One of the main benefits of using FPGAs in this context is that they can achieve 10Gbps network-bound performance similar to RDMA-based systems without specializing the network protocol, relying instead on commodity networks and TCP sockets for communication across the consensus nodes and the clients.",
"For this work, we modified the interface of the nodes to expose the following three operations: 1) order, that takes a transaction (a BLOB), replicates it across nodes and returns the sequence number assigned to the transaction, 2) get, to retrieve the transaction ordered with a specific sequence number and 3) getLast, to retrieve the latest ordered transaction and its sequence number.",
"To achieve this functionality, we modified the key-value store implementation on the FPGA to expose the sequence numbers of the consensus algorithm that were hidden before.",
"The design does not rely on blocks and computes the SHA256 hash of each individual transaction inside the FPGA.",
"This hash value is appended to the transactions before they are stored and is returned with gets, thereby forming the “links” of the chain.",
"Our prototyping boards do not have persistent storage, therefore we explored the feasibility of using NVDIMMs for providing durability in the future.",
"We added a module in front of the memory controller on the FPGA to simulate the timings and bandwidth of Intel's Optane NVDIMMs [21].",
"Given the throughput levels of StreamChain, using lower bandwidth NVMe Flash could also be an option.",
"For integration with the rest of the nodes in StreamChain we rely on the code from the “solo” orderer to act as a facade between peers and FPGAs (Figure REF ).",
"It deals with aspects such as TLS flow termination, management messages, etc., and unwraps the transactions and submits them to the FPGA nodes for ordering.",
"For our prototype we used a single facade, but since it is stateless, it could be deployed on multiple nodes to help with fast failover."
],
[
"Benchmarking Applications",
"In the Evaluation of StreamChain, we focus on answering the following questions: Can low latency be achieved simply by using blocks of a single transaction?",
"What is the relative cost of the different steps and pipeline stages?",
"What is the throughput of StreamChain and Fabric with more complex chaincodes?",
"Do failing transactions reduce the useful throughput (goodput) of the system significantly?",
"Could StreamChain compete with a database?",
"To answer these questions, we designed two different benchmarks (YCSB and SCM), as described in the following.",
"As a micro-benchmark we relied on the YCSB suite to generate 1KB key-value operations and encoded these as invocations of chaincodes that insert/update/read a value belonging to a key.",
"Experiments were driven by a peer executable that connects to endorsers and invokes the chaincode directly, avoiding an additional RPC from the original Java-based YCSB client.",
"We noticed that, since in the original Fabric version most time is spent on disk writes to update the ledger state, the actual type of operation (set or get) didn't play an important role for performance.",
"We used two workloads: 90% inserts/10% reads (YCSB-90) and equal read/update ratios (YCSB-50).",
"Figure: The SCM example implemented in MySQL consists of tables describing relationships between vendors and keeps track of their orders and current inventories.",
"Both transactional and analytical queries are implemented as stored procedures."
],
[
"Supply Chain Management Scenario",
"Since supply chain management (SCM) is an important use-case for distributed ledgers, we designed a benchmark targeting this scenario: vendors order products from each other and update their inventories, provided that they have put a contract in place (n.b.",
"this is not a smart contract, but a condition that allows ordering a specific type of product).",
"The corresponding schema is shown in Figure REF .",
"In addition to transactional queries, such as create contract, place order, update order, etc., two analytical queries are provided: one to compute the days of supply (local analytics) and one to compute the bullwhip coefficient (global analytics).",
"The former determines the number of days for which a vendor can supply a product based on current demand and inventory; the latter calculates a number that represents the overall variation between demand and supply across the supply-chain [14].",
"We implemented all operations as stored procedures in MySQL 8.0 and ran the benchmark from a multi-threaded Java application issuing requests in a closed loop over JDBC.",
"For the distributed ledger version we implemented the queries as chaincode written in Go and organized the key-space by the indexes and primary keys defined in the SQL schema.",
"Data is encoded as JSON within the ledger's key-value pairs.",
"For running the benchmark on Fabric and StreamChain, we used the Java application to export chaincode invocations and we used the same peer executable as for the microbenchmark.",
"In the supply chain management scenario we focused on the case where contention happens with high probability and, as a result, failing transactions could reduce goodput.",
"To this end, we ran the benchmark with 50 vendors selling 500 products with 6000 contracts between them.",
"Each order issued as part of the benchmark was based on one of these contracts.",
"We varied the portion of transactional queries between 95% (SCM-95) and 99% percent (SCM-99).",
"We ran the experiments on a local 10Gbps cluster of eight servers with Intel Xeon E-2186G CPUs (6x3.80GHz), 32GB of RAM and regular HDDs.",
"The blockchain network was composed of 5 peers and 3 ordering service nodes for Raft, respectively 1 facade node for the FPGA ordering version with three Xilinx VCU1525 FPGA boards connected to the same switch.",
"We ran MySQL Version 8.0.15 on one of the servers.",
"For benchmarking we used a single client machine with the two workloads described in the previous section and issued 10000 operations not counting data loading, warm-up and cool-down phases (first and last 10% of operations).",
"For benchmarking MySQL, the client was situated on the same machine.",
"Unless otherwise stated we used Fabric with the default block batching of 10 transactions as a baseline."
],
[
"Latency and Throughput",
"Can low latency be achieved simply by using blocks of a single transaction?",
"To demonstrate that naively setting the block size of Fabric to one does not result in great improvements, and hence the optimizations in StreamChain are needed, we show latency as a function of throughput in Figure REF when using the YCSB-90 workload (we also compare to Fabric in [9], batching 500TXs on average, that uses a comparable micro benchmark).",
"If we set the block size to one (1TX), disk access will dominate both latency and throughput.",
"If, in addition, we “remove” the disk overhead by running on a RamDisk (as we did in our preliminary work [20]), latency can be lowered significantly but the system becomes non-persistent.",
"In StreamChain we can maintain the low latency behavior and keep persistence, while achieving higher throughput.",
"What is the relative cost of the different steps and pipeline stages?",
"The latencies inside StreamChain are reduced and balanced across the three stages of the EOV model.",
"Figure REF shows the breakdown of latency as well as its evolution with increasing load.",
"When using FPGAs to run the ordering service instead of Raft, half of its latency (0.3 ms) can be saved.",
"A further reduction of latency would be possible by replacing the facade node with in-FPGA TLS termination.",
"Overall, commit latencies (O+V) are below 1 ms and end-to-end latencies (E+O+V) under 1.5 ms up to 2000TX/s.",
"Figure: Fabric with 1TX/Block lowers response times at the cost of throughput.",
"StreamChain offers as low a response time as Fabric run on top of a RamDisk, but 2x better throughput.Figure: The main components of the StreamChain pipeline have all predictable latency behavior.",
"The FPGA-based ordering service cuts latency in half when compared to the Raft-based one."
],
[
"SCM Benchmark",
"What is the throughput with more complex chaincodes?",
"The SCM workload allows us to measure the latency of the systems with more complex operations.",
"Table REF shows the end-to-end latencies of a single client.",
"In the case of Fabric and StreamChain, the latency decreases when the fraction of transactional queries is increased from 95% to 99%.",
"This is expected because analytical queries take longer than transactional ones and often keep a lock on the entire key-space (e.g., in the case of the Bullwhip coefficient) while being endorsed, thereby slowing all validations down.",
"The database exhibits the opposite behavior, whereby each transactional query incurs several write disk accesses, but analytics can be performed on the buffered pages.",
"While there certainly are databases that offer better performance than MySQL, overall, StreamChain delivers competitive latencies.",
"How much do failing transactions reduce the useful throughput (goodput)?",
"We explore the effect of small working set sizes (more read-after-write contention) on the goodput of StreamChain using the YCSB-50 workload with equal updates and reads and pick the working set as 100%, respectively 10%, of the 10k key space.",
"As shown in Figure REF , Fabric has a negligible number of failing transactions until it reaches 87% of its maximum throughput, where the percentage of failing transactions quickly rises to 40%.",
"Failing transactions are not reissued in this benchmark, but if they were, depending on the backoff policy it could happen that the system is not able to make meaningful progress at this point.",
"Thanks to the reduced data staleness, StreamChain shows a more graceful degradation that happens much later (beyond 97% of its maximum throughput).",
"Even though it cannot eliminate the problem entirely, StreamChain can operate close to saturation with virtually no failing transactions as soon as workloads don't have very small “hotspots”.",
"Figure: When running the YCSB-50 microbenchmark, transactions start failing sooner in Fabric than in StreamChain, which retains almost 100% goodput close to maximum throughput (1700TX/s, resp.",
"3200 TX/s).",
"The difference in behavior is even more visible with a 1000 key working set.Table: The SCM benchmark provides a real-world estimate of end-to-end latencies since each query/chaincode performs several reads and writes.We also measure goodput with the SCM benchmark.",
"The results, in Figure REF , show two trends: First, by increasing the percentage of transactional queries (from 95% to 99%), the blockchain solutions become faster.",
"Second, both Fabric and StreamChain have a drop in goodput sooner than with the YCSB benchmark.",
"Both trends are to be expected.",
"The endorsment of the analytical operations takes longer and requires locking a large portion of the key-space, thereby slowing down validation.",
"Furthermore, concurrent updates to the inventory of vendors result in failing validations, thereby lowering goodput.",
"StreamChain, however, consistently outperforms Fabric.",
"Could StreamChain compete with a database?",
"when executing the SCM benchmark, MySQL delivers somewhat lower throughput than StreamChain, but of course no failing transactions (in Figure REF we artificially extend the lines to the right once MySQL reaches maximum throughput).",
"The database is limited by high lock contention and disk flushes for the transactional queries.",
"If we run MySQL on a RamDisk, removing the disk overhead, its throughput increases significantly, to 3000TX/s for SCM-95 and 7000TX/s for SCM-99.",
"These numbers can be regarded as an upper bound for traditional database throughput.",
"Overall, StreamChain can deliver throughput within the range defined by these lower and upper-bounds on a single machine even though it executes in a distributed manner.",
"Figure: StreamChain delivers higher goodput than Fabric for the SCM use-case, even though failed transactions become more common with increased load and a higher percentage of analytical queries."
],
[
"BFT Ordering",
"In order to reduce the necessary trust in a service provider and to ensure that permissioned distributed ledgers can be run in multi-cloud deployments, it is important to have a Byzantine fault tolerant (BFT) ordering service implementation.",
"Using a BFT consensus protocol would also make the system more resilient to arbitrary failures.",
"Even though there is already work providing BFT ordering in Fabric [28], integration with StreamChain will require modifications to the BFT implementation.",
"This is because batching, used as a default in such protocols [10], [28], [33], has to be eliminated without reducing throughput to impractical levels.",
"An additional challenge is that, since each peer has to be connected to a majority of the BFT ordering nodes to accept transactions as ordered, the jitter across ordering nodes has to be minimized without reducing their ability to sustain high network bandwidth.",
"There are no BFT implementations that fulfill all of the above requirements at the same time because traditionally they have been optimized for geo-distributed operation.",
"Nonetheless, even though challenging, we believe that it is possible to build such an ordering service if we rely on low latency networks and using hardware accelerators both for cryptographic operations (signing, verifying, etc.)",
"and data movement operations (marshaling and unmarshaling) withing the protocols."
],
[
"Concurrent State Access",
"Currently, endorsement and validation have to concurrently access the State DB inside endorsing peers.",
"This limits throughput and can increase latencies when locking large portions of the key-space.",
"For this reason, concurrency-enabling optimizations of the State DB will yield the biggest immediate benefits in StreamChain.",
"Of course, fundamentally, the problem of failed transactions cannot be eliminated by changing the State DB.",
"Instead, research is required into a hybrid between the OE and EOV execution models that allows more flexibility in post-order validation."
],
[
"Conclusions",
"In this work we make the case that the design of permissioned blockchains should be revisited for datacenter-like environments and that latency should be considered as a first-class performance metric.",
"With StreamChain, we propose a streaming design and show that it achieves low latency while maintaining high throughput.",
"We demonstrate that StreamChain could be already practical in real-world deployments with a supply chain management benchmark.",
"Our approach is complementary to ideas for further increasing throughput and, thanks to the elimination of batching overhead, novel research directions open up for hardware-accelerated consensus and cryptography operations in the context of permissioned ledgers.",
"StreamChain is able to take advantage of low latency networking and modern datacenter hardware, including emerging platforms such as FPGAs, which, until now, have not been beneficial in such systems."
],
[
"Acknowledgments",
"This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No.",
"842956."
]
] | 1808.08406 |
[
[
"Deep Emotion: A Computational Model of Emotion Using Deep Neural\n Networks"
],
[
"Abstract Emotions are very important for human intelligence.",
"For example, emotions are closely related to the appraisal of the internal bodily state and external stimuli.",
"This helps us to respond quickly to the environment.",
"Another important perspective in human intelligence is the role of emotions in decision-making.",
"Moreover, the social aspect of emotions is also very important.",
"Therefore, if the mechanism of emotions were elucidated, we could advance toward the essential understanding of our natural intelligence.",
"In this study, a model of emotions is proposed to elucidate the mechanism of emotions through the computational model.",
"Furthermore, from the viewpoint of partner robots, the model of emotions may help us to build robots that can have empathy for humans.",
"To understand and sympathize with people's feelings, the robots need to have their own emotions.",
"This may allow robots to be accepted in human society.",
"The proposed model is implemented using deep neural networks consisting of three modules, which interact with each other.",
"Simulation results reveal that the proposed model exhibits reasonable behavior as the basic mechanism of emotion."
],
[
"Emotions are very important for human intelligence.",
"For example, emotions are closely related to the appraisal of the internal bodily state and external stimuli.",
"This helps us to respond quickly to the environment.",
"Another important perspective in human intelligence is the role of emotions in decision-making.",
"Moreover, the social aspect of emotions is also very important.",
"Therefore, if the mechanism of emotions were elucidated, we could advance toward the essential understanding of our natural intelligence.",
"In this study, a model of emotions is proposed to elucidate the mechanism of emotions through the computational model.",
"Furthermore, from the viewpoint of partner robots, the model of emotions may help us to build robots that can have empathy for humans.",
"To understand and sympathize with people's feelings, the robots need to have their own emotions.",
"This may allow robots to be accepted in human society.",
"The proposed model is implemented using deep neural networks consisting of three modules, which interact with each other.",
"Simulation results reveal that the proposed model exhibits reasonable behavior as the basic mechanism of emotion."
],
[
"Keywords:",
"Emotion Model, Human–Robot Interaction, Empathic Communication, Machine Learning, Recurrent Attention Model, Convolutional Long Short-Term Memory, Deep Deterministic Policy Gradient"
],
[
"Introduction",
"The development of artificial intelligence (AI) in recent years has been remarkable.",
"In certain tasks such as object recognition, it is said that AI has surpassed human capabilities.",
"However, one might think that emotion separates human intelligence from AI.",
"Is this true?",
"If the human mind is created as a result of the calculations of the brain, then emotions could be simulated by a computer.",
"Would this imply the possibility that a robot could have emotions?",
"To answer this question, we should start thinking of the basic mechanism of emotion.",
"If the mechanism of emotion were elucidated, we could get closer to the essential understanding of what a human being is.",
"Because emotions are very important to human beings, many studies on emotions have been carried out in the past.",
"Here, the conventional studies on emotions are organized from the viewpoint of their research approach.",
"Many psychological studies, among others, have tried to capture emotional phenomena.",
"Emotional facial expression studies, for instance, are based on the idea of basic emotions theory.",
"It is well known that Ekman insisted that there are six basic emotions, regardless of culture [14].",
"Plutchik and Izard respectively assumed eight and ten basic emotions [54], [55], [23].",
"Because the basic emotions theory is based on the evolutionary point of view, emotional expressions are defined at the nerve level, and the categories of expression and recognition of basic facial expressions are universal.",
"The dimensional model of emotions is another well known approach for emotions [56], [59].",
"This expresses emotions in approximately two to three cognitive dimensions based on a factor analysis of judgment on emotional stimuli such as expressive photographs or emotional expression words.",
"Although many emotion-related studies are based on dimensional models, the mechanism behind emotional phenomena cannot be revealed.",
"Various emotional models have been proposed in the literature from the physiological viewpoint [24], [8].",
"The central idea in the James–Lange theory is represented in the quote “We don't laugh because we're happy, we're happy because we laugh” by James.",
"However, the Cannon–Bard theory contradicts it.",
"The question of which of these theories is correct has long been controversial.",
"Schachter, in contrast, advocated a two-factor theory and developed an emotional theory that included these two competitive theories [58].",
"Cognitive theory is also famous for incorporating cognitive activities in the form of judgments, evaluations, or thoughts [1], [31], [51].",
"These models give important implications for emotion; however, because they do not model the entire mechanism of emotions, they do not necessarily clarify what emotions are.",
"Moreover, they are not computational models.",
"In other words, there is also a problem that the models cannot be directly implemented on a computer.",
"Neuroscience has revealed neural circuits, such as the Papez circuit [52] and Yakovlev circuit [65], that are relevant to emotions.",
"LeDoux discussed the function of the brain in emotions in detail based on the anatomical point of view.",
"He proposed the dual pathway theory, which claims that there are two types of emotional processing paths: automatic and rapid processing by the limbic system, and complicated and higher cognitive processing from the neocortex to the amygdala [32], [33], [34].",
"More recently, the quartet theory of emotions, which claims four important systems for emotions in the brain, was proposed [27].",
"These are the brainstem-centered, diencephalon-centered, hippocampus-centered, and orbitofrontal-centered systems.",
"As a matter of course, the authors argue that the limbic/paralimbic structure, i.e., the basal ganglia, amygdala, insular cortex, and cingulate cortex, are also of importance for affective processes.",
"As shown in computational neuroscience studies, the cortical-basal ganglia loop can be considered as a reinforcement learning module.",
"In particular, the striatum plays a very important role in sensorimotor, attentional, and emotional processes.",
"These neuroscientific findings are not only important for concretely considering emotion models, but also have direct implications on computational models.",
"From the viewpoint of human–robot interaction, emotion is one of the most important factors for partner robots.",
"The intuition that the difference between humans and AI (robots) is in emotion implies that, in other words, the realization of the emotion model may be the key to realize robots and/or AI with high affinity for human beings.",
"Picard proposed the idea of affective computing, in which the emotion recognition in humans has been studied extensively, mainly by examining facial expressions [53].",
"The success of deep learning in recent years has accelerated this line of research.",
"Of course, the classification of a person's inner state based on facial expressions is very useful for the robot to communicate with us, because it can select its response according to the recognized result.",
"However, it is fair to say that the recognition of facial expressions is different from a “true understanding” of the emotional states of others, even though a highly accurate facial expression recognition method is available thanks to deep learning technologies.",
"Robots need to understand, sympathize, and act according to their partners' complex emotional states in order to become accepted members of human society.",
"Toward this goal, many efforts on designing emotional expressions for social robots have been made [5].",
"However, almost all these emotions have been designed manually.",
"High-level complex social emotions for robots are difficult to preprogram manually.",
"In fact, conventional studies have only been able to accomplish simple basic emotions such as happiness and sadness [42], [63].",
"An emotion model for robots based on the difference equation was proposed [44]; however, the system was too simple to generate complex higher-level emotions.",
"The basic idea underlying this study is that the problem of emotions should be formulated as “understanding by a generative process of emotions” rather than “classification.” If we abandon the manual design of emotions, emotional differentiation [6], [36] must be the right path to follow in order to achieve this ultimate goal.",
"This idea shares the same goal as the affective developmental robotics proposed by Asada [2].",
"Accordingly, we propose a computational model of emotions, which is based on certain neurological and psychological findings in the literature.",
"The purpose of this paper is first to present a general meta-level framework for the mechanism behind emotions.",
"The literature on emotions in the past as discussed above motivates us to propose a three-layer model to cover emotions.",
"The first layer corresponds to the appraisal module, which is responsible for quick evaluation of the external world and internal body.",
"Interoception in particular, which is sensitivity to stimuli originating inside the body, is a very important factor.",
"The second layer has an emotional memory to adjust the innate appraisal module in the first layer to the surrounding environment, which the agent is facing.",
"The third layer includes reinforcement learning and sequence learning modules that correspond to the cortical-basal ganglia loop.",
"This is because the important aspect of emotions is their role in decision-making [47].",
"The dual path theory by LeDoux is one of the important theories of emotions, which forms the basis of our proposed three-layer model.",
"Moreover, this three-layer model roughly matches the recent neurobiological emotion model [27].",
"We also attempt to implement the three-layer model of emotion using deep neural networks.",
"Our proposed implementation relies on a combination of the recurrent attention model (RAM) [45] for the first layer, as well as the convolutional long short-term memory (LSTM) [64] and the reinforcement learning module using the deep deterministic policy gradient (DDPG) [39] for the third layer.",
"The second layer is realized based on a mechanism of nonlinear smoothing, which makes the whole emotion system adaptable to the surrounding environment.",
"Then, the implemented computational model of emotion is tested by employing certain tasks simulating mother-–infant interaction to evaluate the plausibility of the model.",
"Some promising results are obtained in the experiment.",
"For example, we found that the policy network represents emotional states and exhibits emotion differentiation in the proposed three-layer model.",
"We believe this constructive approach toward emotions may yield a clue to the elucidation of human emotions.",
"Moreover, the generative model of emotion is also important for achieving empathic communication between humans and robots.",
"The contributions of this study are threefold.",
"First, this study investigates a meta-level model of emotion as a whole.",
"Second, an implementation of the emotion model using deep leaning modules is provided.",
"Third, we design a simulation task mimicking mother–infant interaction in order to evaluate the model, which reveals that the proposed model is indeed able to show emotion differentiation.",
"It should be noted that our previous studies showed some preliminary ideas and examinations of the proposed emotion model [21], [19], [20].",
"Although the basic idea of this work is shared with the previous studies, this paper provides a detailed explanation of the model and full implementation as an entire emotion network, which were not given in the previous works.",
"Moreover, we repeated the entire experiments and the results presented in this paper are completely new.",
"The remainder of this paper is organized as follows.",
"In the next section, the literature on emotions is discussed and then a model of emotions is proposed.",
"Section 3 provides an implementation of the proposed emotion model using deep neural networks.",
"Each module of the network is explained in detail.",
"The experiments are presented in Section 4, which indicate plausibility of the proposed deep emotion model.",
"Finally, this paper is summarized in Section 5."
],
[
"Model of Emotion",
"Here, we present an overview of the basic idea of our proposed emotion model.",
"Some important findings for our proposal are reviewed first, followed by the proposed model of emotions."
],
[
"Emotions in literature",
"What is emotion?",
"In order to propose a model of emotion, we should start from this important question.",
"Moreover, we need to clarify the definition of emotion.",
"To the best of our knowledge, there is no universal consensus on the definition of emotion; however, recent research reveals the importance of the body in emotion.",
"This is what William James claimed long ago, also called the peripheral origin theory of emotion [24].",
"Recent studies in cognitive neuroscience have revealed that interoception, which is a perception of the internal bodily state, is a key for the subjective experience of emotion [61].",
"In the quartet theory of emotions, the brainstem-centered system corresponds to this type of emotion system [27].",
"The brainstem is the oldest brain structure and the reticular formation plays an important role in this system.",
"Another important aspect of embodiment in emotion is Damasio's somatic marker hypothesis, which hypothesized that emotions evaluate external stimuli efficiently through our own body [10].",
"This motivates us to consider both internal and external appraisals simultaneously.",
"In any case, the physical body is an origin of emotions and is indispensable.",
"In this paper, we consider the body and interoception as one subsystem, i.e., the first layer in the proposed model.",
"In fact, despite the differentiating property of emotions, some basic emotions such as anger, joy, disgust, fear, sorrow, and surprise exist regardless of culture [14].",
"This is possible because we as human beings share similar physical bodies and environments.",
"This result supports the fact that emotions are based on our physical bodily states.",
"The idea of active inference is also related to this embodied system, e.g., visual attention is relevant to the appraisal of visual stimuli [16], [60].",
"Another important aspect of emotion is related to decision-making [35] and inference on causal attribution.",
"For example, a misattribution of arousal, which is also known generally as the suspension-bridge effect, has been found to happen to someone who experiences the effects of fear of physical danger while meeting someone, and who mistakenly believes that the other person is the cause of their physical responses [13].",
"This higher-level cognitive process seems to be deeply related to the orbitofrontal-centered system.",
"The reinforcement learning module, which originates from the cortical-basal ganglia loop, is also related.",
"The relationship between active inference and reinforcement learning has been discussed [15], which implies this system is also related to the active inference.",
"In this paper, we consider the decision-making as another subsystem, which we will call the third-layer in the proposed model.",
"The discussion so far implies that one's emotional system is divided into two systems: 1) the hardwired innate system and 2) the learning system, which is related to the decision-making.",
"Now, we define emotions (emotional states) and feelings (emotional feelings).",
"Emotions are defined as a set of physical reactions, state changes of visceral and skeletal muscles, and changes in internal conditions.",
"These changes are evoked by the above systems 1) and 2).",
"In contrast, feelings are defined as perceptions of the emotional states.",
"These definitions are based on Damasio's definition [11].",
"It should be noted that the term “emotion” corresponds to “affect” in the area of psychology: this paper uses the term “emotion,” which is generally acceptable.",
"Figure: Illustration of “anger” and “fear,” which highlights the difference: (a) emotional feeling of anger, and (b) emotional feeling of fear.In order to clarify the definition of emotions/feelings used in this paper, Fig.",
"REF illustrates concrete examples.",
"In the figure, there are a stimulus A and a bodily state that evoke the “Fight” action, whereas a stimulus B and a bodily state activate the “Flight” action.",
"In this case, the emotional state that stimulus A and the bodily sate cause is labeled as “anger,” and the emotional state caused by the stimulus B and the bodily sate is labeled as “fear.” This definition directly connects emotions to the somatic marker hypothesis, which means that the emotion should be generated by considering internal appraisal, external appraisal, and decision-making mechanisms.",
"Regarding the learning system, a memory-based system is an important candidate as a building block of the emotion model.",
"In the quartet theory, the hippocampus-centered system corresponds to the memory-based system, in which the hippocampus and amygdala are mainly involved [27].",
"The activity of the amygdala in emotion is particularly important and has been studied for a long time.",
"Yakovlev's circuit is one of the well known limbic systems and the amygdala is involved in the circuit [65].",
"Papez's circuit is another well known limbic circuit, which includes the hippocampus [52].",
"Although these are independent as circuits, they have mutual interaction and are closely related each other through the cortex, basal ganglia, and diencephalon [43].",
"In this paper, we consider the memory-based system as another subsystem, which we call the second layer in the proposed model.",
"This subsystem gives flexibility to the innate-appraisal system, i.e., first layer, in order to adapt the whole system to the environment.",
"Eventually, emotions cannot be viewed locally, and need to be thought of as a network.",
"Therefore, the abovementioned subsystems should be connected as a network to generate emotions.",
"Furthermore, the important aspect of the model is its ability to explain various phenomena known in the art.",
"Among others, emotion differentiation is an important phenomenon, because it is a key to implementing emotions for robots, as mentioned earlier.",
"Bridges claimed that excitement, which is the origin of emotion, can be divided into several emotional categories based on observations of infants [6].",
"More recently, it has been reported that emotions such as pleasure, interest, surprise, sadness, anger, and fear were recognized one year after birth; pride, shame, guilty feelings, etc.",
"emerged from approximately two and a half years; and not all but great majority of emotions appear by the age of three [37].",
"Our proposed model of emotion is discussed in the next subsection.",
"Then, the model is implemented using deep neural networks and tested to determine whether it develops the emotional categories."
],
[
"Proposed model of emotion",
"The proposed emotion model is illustrated in Fig.",
"REF .",
"The emotion model is divided into three layers: the first layer that reacts bodily to stimuli very fast, the second layer that accesses memories such that stimuli can be evaluated through experiences, and the third layer that makes future predictions and actions.",
"These are derived from the abovementioned implications.",
"Figure: Schematic diagram of our proposed three-layer model of emotion.The first layer reacts to stimuli very quickly using the body, which is called external appraisal.",
"Moreover, this part reflects the situation of the body itself, i.e., internal appraisal, regardless of external perception.",
"This layer is the reason why emotions depend on the physical body.",
"Because the reactions are preprogrammed innately, they usually contain errors, which cause overreactions to stimuli.",
"To alleviate this problem, the second layer accesses memories such that stimuli can be evaluated through experiences.",
"This second layer makes it possible to suppress unnecessary reactions and, at the same time, react quickly to important problems.",
"Of course, this is a trade-off between processing cost and accuracy of response to stimuli.",
"Hence, the output of the first layer, which is modulated by the second layer to be precise, can be considered as the perception of dimensionally reduced evaluated results of the external and internal worlds, i.e., internal representation.",
"Therefore, the perception of the output of the first layer can be regarded as interoception.",
"In the third layer, the output of the first layer is used together with the input stimuli for causal inference and prediction, as shown in Fig.",
"REF .",
"Subsequently to the prediction, decision-making is carried out using the input stimuli and the results of the prediction.",
"The most important part of the third layer is reinforcement learning, which is responsible for the learning of optimal decision-making.",
"One of the most important aspects of the reinforcement learning is the definition of a reward.",
"In the model of emotion, the idea of “homeostasis,” which is a regulatory mechanism of the agent's internal state, should be adopted.",
"This is based on the drive reduction theory, which is the basic theory of motivation [49].",
"It interesting that homeostasis is closely related to the diencephalon, which is one of the emotion systems in the quartet theory of emotions [27].",
"Hence, a reward is provided when the output of the first layer, i.e., interoception, remains constant.",
"This constant is not a completely constant value.",
"It takes the average value of emotional state over a time window with a certain length.",
"In other words, homeostasis is set not to keep the emotional state completely constant, but to discourage rapid changes.",
"In our model, the average value that gradually changes in time is defined as “mood.” Thereafter, the neural patterns of the policy in the striatum, i.e., emotional states, are consciously recognized as emotional feelings.",
"After the decision-making process, the prediction error is calculated followed by updating of the model in the third layer.",
"Experiences are stored in the hippocampus as episodic memories, with emotional evaluation in the second layer.",
"It is worth noting that the learning process exists only in the second and third layers.",
"Figure: Our proposed emotion model for implementation, which is a redrawn version of Fig.",
".Figure REF is a redrawn version of the proposed model in Fig.",
"REF , in order to make it comprehensive for implementation.",
"This figure directly claims some important points of our proposal.",
"First, the internal and external appraisals, i.e., embodiment, are the sources of emotions.",
"It is fair to say that without the physical body, there should be no emotions.",
"Second, prediction is indispensable in the proposed model.",
"Third, another key point in our model is the decision-making part, which is relevant to the somatic marker hypothesis.",
"These viewpoints remind us to note the close relationship between our proposed emotion model and the embodied predictive interoception coding (EPIC) model, which was proposed recently [3].",
"The idea of the EPIC model is based on predictive coding and active inference [17].",
"Although we developed our proposed model independently of the EPIC model, some important ideas are shared between the two models.",
"The main difference between the EPIC model and the model proposed in this study is that we propose the actual implementation of the proposed model by combining several deep learning modules, which are described in the next section.",
"On the contrary, the EPIC model is a conceptual model and sticks firmly to the predictive coding.",
"Another important aspect for the model is the design of artificial emotional systems, which Cañamero contends [7].",
"She claimed that emotions must be grounded in an internal value system that is meaningful for the robot's physical and social niche.",
"The model should establish a link between emotions, motivation, behavior, perception, and various aspects of “cognition,” and the link must be rooted in the body of the agent.",
"As already discussed, our proposed emotion model has the potential to fulfill these requirements."
],
[
"Implementation",
"This section proposes an implementation of the emotion model described in the previous section.",
"The proposed implementation consists of a combination of deep neural networks, such as RAM, LSTM, and DDPG, except for the second layer.",
"The second layer is realized by a simple smoothing mechanism to make the learning system tractable.",
"In the following, we will look at the implementation of each module in turn."
],
[
"Appraisal module (1st layer)",
"As we discussed earlier, the first layer is responsible for generating interoception based on both internal and external appraisals.",
"The problem here is the generation of a physical response to stimuli.",
"The model is required to generate a “human-like” response in order to replicate human emotions.",
"In this study, we attempt to generate suitable affect values, i.e., a pair of valence and arousal values, from input visual stimuli using a neural network instead of generating physical body reactions [21].",
"To replicate human-like innate reactions, we utilize several databases such as the international affective picture system (IAPS) database [29], [30], the open affective standardized image set (OASIS) [28], the Nencki affective picture system (NAPS) [41], and the Geneva affective picture database (GAPED) [12], to train the network in order to generate two-dimensional valence and arousal values for a given visual stimulus.",
"Therefore this is a regression problem.",
"To take the active interoceptive inference into consideration, we propose the use of the RAM [45].",
"This is because visual attention is a very important factor for estimating arousal and valence values, and the RAM makes it possible to learn the visual attention and affect values simultaneously, as shown in Fig.",
"REF .",
"Please refer to the appendix for details on the RAM.",
"It should be noted that the RAM improved the performance of regression compared with the convolutional neural network (CNN), which is directly trained using pairs of images and ground truth.",
"Figure: Overview of the first layer implemented by the recurrent attention model: (a) block diagram of the first layer and image examples from IAPS , and (b) network architecture of the RAM ."
],
[
"Emotional memory module (2nd layer)",
"The second layer shown in Fig.",
"REF (a) can be regarded as an adaptation using data in the actual environment for the innate and fixed system, i.e., the first layer.",
"The memory-based learning increases the accuracy of the prediction by using the past accumulated information experienced by the agent.",
"Here, we formulate this as a problem of calculating the expected value $E\\left[x(t)|{z}_0^T \\right]$ ($0 \\le t \\le T $ ), where $x(t)$ and ${z}_0^T$ represent the target value to be estimated and the stored data, respectively.",
"This is a type of smoothing problem and the second layer is realized by a simple nonlinear smoothing technique, as shown in Fig.",
"REF (b).",
"More specifically, a time series including affect values and stimuli during a certain period of time is stored in the memory.",
"The idea here is that the output of the RAM is modified by the compensation term ${L}(\\cdot )$ in the second layer as follows: ${a}^{\\prime }(t) & =& RAM(I_{t}^{k}) + {L}(k),\\\\{a}(t) &= &{a}^{\\prime }(t) + IA(t),$ where ${a}^{\\prime }(t)$ is an external appraisal at time $t$ , $RAM(I_{t}^{k})$ represents the output of the first layer for the input image $I^k_t$ at time $t$ , $k$ indicates the category of the input image, $IA(t)$ is an internal appraisal at time $t$ , which will be described later, and ${L}(k)$ represents the output of the second layer (compensation term) for the image $I_t^k$ .",
"${L}(k)$ , which modifies the first layer output $RAM(\\cdot )$ , is updated using the stored data as follows: ${L}(k) \\leftarrow {L}(k) + \\gamma \\frac{1}{|\\phi _k|} \\sum _{i \\in \\phi _k}\\left\\lbrace {a}(i+1)-{a}(i)\\right\\rbrace = {L}(k)+\\gamma \\frac{1}{|\\phi _k|} \\sum _{i \\in \\phi _k} {\\Delta }_{i \\in \\phi _k},$ where $\\gamma $ is the learning rate and is set to 0.1 in the later experiment.",
"$\\phi _k$ is a collection of time indices $t$ with the same image category $k$ and $|\\phi _k|$ represents the number of images belonging to $\\phi _k$ .",
"As shown in Fig.",
"REF (b), the smoothing process can capture temporal information.",
"For example, when the next affect values for a particular image category $k=1$ increase frequently, the term ${L}(1)$ compensates the affect value of the corresponding image input in the upward direction.",
"However, the next affect values for $k=2$ vary and the sum of ${\\Delta }_{i \\in \\phi _1}$ cancels out.",
"With this smoothing process, we can expect the effect of lowering the load on the body, and it improves the prediction performance of the next layer.",
"Long-term potentiation (LTP) is a well known mechanism for connecting memory and learning.",
"The hippocampus and amygdala are closely related to the LTP mechanism.",
"The smoothing mechanism in this layer is assumed to mimic LTP in the functional level.",
"Moreover, the amygdala is involved in the classical conditioning based on LTP.",
"Thus, the process of this layer may replicate the classical conditioning.",
"Fig.",
"REF (b) also explains this mechanism in a simple way.",
"The second layer learns that the image category $k=1$ is the trigger of high valence and arousal values.",
"The output of RAM and second layer represents external appraisal.",
"According to our definition, interoception is a combination of external appraisal, i.e., the output of RAM modulated by the second layer, and internal appraisal representing internal energy, as shown in Fig.",
"REF .",
"The internal energy increases or decreases according to the selected action.",
"For example, moving the body forcefully consumes energy, consequently the internal energy decreases (the internal appraisal increases).",
"Because the internal appraisal depends on the definition of the agent to be assumed, we explain certain details on the implemented internal appraisal module in the next subsection."
],
[
"Internal appraisal",
"In this study, the internal appraisal module is implemented in a rule-based manner.",
"Essentially, the internal appraisal increases when the agent acts as the internal energy is decreased.",
"When the agent shows sadness or closes his eyelids, the internal appraisal decreases.",
"This is because we assume that showing sadness leads to getting milk, and closing his eyelids corresponds to sleeping, which restores physical strength.",
"The internal appraisal implies a physical strength bias in general, and the interoception is expressed by applying the physical strength bias to the external appraisal, as shown in Fig.",
"REF .",
"These assumptions are made because of a mother–infant interaction scenario in our later experiment.",
"The agent has four facial parts to move according to external stimuli.",
"Each facial part can be continuously controlled by the agent at the cost of corresponding power consumption.",
"Thus, the agent has to learn (in the third layer) suitable facial expressions according to the external and internal worlds.",
"More precisely, the internal appraisal $IA(t)$ can be rewritten as the following formula: $IA(t) &=& \\sum _{n=0}^{3} \\left( 1-exp \\left\\lbrace -\\frac{A_{n}(t)}{\\tau } \\right\\rbrace \\right), \\\\A_{n}(t) &=&\\left\\lbrace \\begin{array}{l}|A_{n}(t-1)-d| ~:for~closing~eyelids~actions~or~showing~sadness\\\\A_{n}(t-1)+a^c_{t-1}+\\eta ~: otherwise\\end{array},\\right.",
"$ where $a^c_{t-1}$ is an action cost at time $t-1$ , and $n=0, \\cdots , 3$ represents facial parts.",
"$\\eta $ represents constant physical fatigue and is set to $0.01$ in the later experiments.",
"As four facial parts are assumed, $IA(t)$ has a value in the range from 0 to 4.",
"Eq.",
"(REF ) denotes a basic curve of physical strength with a time constant $\\tau $ ($\\tau =50$ in the later experiments).",
"Eq.",
"() represents change in the parameter of the basic curve.",
"It is natural that the parameter $A_{n}(t)$ increases as the action is taken by the agent.",
"When the agent closes his eyelids, the parameter is set such that the physical strength recovers.",
"Additionally, even when the agent expresses a sad expression, the parameter is set for restoring physical strength.",
"As we mentioned earlier, these settings are based on the assumption of mother-–infant interaction.",
"In this study, $d$ is set to 50 for closing eyelids and 75 for showing sadness.",
"If these two values are the same, the two types of actions become meaningless.",
"Therefore, they are set to different values, such that each action becomes meaningful in the reinforcement learning module.",
"It should be noted that it is possible to design other rule-based internal appraisal modules according to the physical body of the agent and the scenario of the world in which the agent exists."
],
[
"Decision-making module (3rd layer)",
"As shown in Fig.",
"REF , the decision-making module is implemented using convolutional LSTM and DDPG.",
"In a previous study, we have implemented using LSTM–DQN [19].",
"The LSTM–DQN has a drawback that continuous actions cannot be dealt with.",
"That is why the convolutional LSTM–DDPG is employed in this study.",
"Please refer to the appendix for details on the convolutional LSTM and DDPG.",
"To train the network (reinforcement learning), combinations of an input image such as Fig.",
"REF (a), and the result of subtraction between an output of the RAM and an internal appraisal, i.e., interoception, are used.",
"Another important part of reinforcement learning in general is the actions.",
"This means that the implementation of the proposed emotion model requires actions, because the reinforcement learning is employed.",
"Here, we discuss the actions used in this study.",
"To consider the actions in the reinforcement learning, we need to assume the robot/agent to be used, because the actions to take vary depending on the body of the robot/agent.",
"Without loss of generality, we assume the agent that is used in our later experiment in this study.",
"The agent has action commands of its own facial expressions for given visual stimuli and interoception in the first layer, i.e., valence and arousal values.",
"The convolutional LSTM is responsible for predicting an image and interoception values at the next time-step from the input image and current interoception values.",
"The DDPG module generates an action command by taking the input image, interoception values, and predicted results by the convolutional LSTM, as an input.",
"Figure REF illustrates the overall processing of the decision-making module (third layer).",
"As discussed in the meta-level model, the idea of homeostasis is used for calculating the reward as follows: $R(t)& =& C - {\\left\\Vert {m}(t) - {a}(t) \\right\\Vert }^2_2 , \\\\{m}(t)& =& \\frac{1}{2}\\left( \\bar{{a}} + \\frac{1}{N}\\sum _{i=1}^{N} {a}(t-i) \\right), $ where $R(t)$ and ${a}(t)$ represent, respectively, the reward value and the vector consisting of valence and arousal values, i.e., interoception values, at time $t$ .",
"${m}(t)$ represents the mood of the agent at that moment and is calculated as a mean vector of $\\bar{{a}}$ and the average of the past $N$ frames.",
"$\\bar{{a}}$ , $N$ , and $C$ represent a vector consisting of intermediate values between maximum and minimum interoception values, number of averaging frames, and a constant value, which translates the differential value to a reward value.",
"Eq.",
"(REF ) is intended to represent a mood, which is less likely to be provoked by a particular stimulus and is determined by the average of the last $N$ interoception values."
],
[
"Learning of the model",
"Because our proposed model consists of several learning modules, several patterns can be considered as the timing of these updates.",
"This study takes a simple idea of updating the LSTM and the second layer at each timing based on DDPG update loop.",
"The entire learning algorithm of the proposed model is shown in Algorithm REF .",
"In the algorithm, we set two parameters empirically as $T_{LSTM}=100$ and $T_{L2}=1000$ .",
"Figure REF shows the whole network architecture of the proposed model.",
"One can see the detailed parameters, such as number of input/output nodes, in the figure.",
"Deep emotion learning algorithm Train the recurrent attention model $RAM(\\cdot )$ (offline) Initialize the mood of the agent ${m}(0)$ Initialize the second layer ${L}(k)$ Randomly initialize critic network $Q(s,a|\\theta ^{Q})$ and actor $\\mu (s|\\theta ^{\\mu })$ with weights $\\theta ^{Q}$ and $\\theta ^{\\mu }$ Initialize target network $Q^{\\prime }$ and $\\mu ^{\\prime }$ with weights $\\theta ^{Q^{\\prime }} \\leftarrow \\theta ^{Q}, \\theta ^{\\mu ^{\\prime }} \\leftarrow \\theta ^{\\mu }$ Initialize replay buffer $B$ Initialize a random process $\\mathcal {N}$ for action exploration Receive an initial input image $I_{0}^{k}$ Calculate interoception ${a}(0)$ using Eq.",
"(REF ) Predict next image $\\bar{I}_{0}^{k}$ and interoception $\\bar{{a}}(0)$ by LSTM module Set $s_1 = \\left\\lbrace I_{0}^{k}, {a}(0), \\bar{I}_{0}^{k}, \\bar{{a}}(0) \\right\\rbrace $ $e$ = 1, $M$ Select action $a_{e} = \\mu (s_{e}|\\theta ^{\\mu }) + \\mathcal {N}_{t} $ according to the current policy and exploration noise Execute action $a_{e}$ and observe reward $R(e)$ Receive an input image $I_{e}^{k}$ Calculate interoception ${a}(e)$ using Eq.",
"(REF ) Predict next image $\\bar{I}_{e}^{k}$ and interoception $\\bar{{a}}(e) $ by LSTM module Set $s_{e+1} = \\left\\lbrace I_{e}^{k}, {a}(e), \\bar{I}_{e}^{k}, \\bar{{a}}(e) \\right\\rbrace $ Store transition ($s_{e}, a_{e}, R_{e}, s_{e+1}$ ) in $B$ Sample a random minibatch of $N_B$ transitions ($s_{i}, a_{i}, R_{i}, s_{i+1}$ ) from $B$ Set $y_{i}=R_{i}+\\gamma Q^{\\prime } \\left( s_{i+1},\\mu ^{\\prime }(s_{i+1}|\\theta ^{\\mu ^{\\prime }})|\\theta ^{Q^{\\prime }} \\right) $ Update critic by minimizing the loss: $L=\\frac{1}{N_B}\\sum _{i} \\left\\lbrace y_{i}-Q(s_{i},a_{i}|\\theta ^{Q}) \\right\\rbrace ^{2}$ Update the actor policy using the sampled policy gradient: $\\nabla _{\\theta ^{\\mu }}J \\approx \\frac{1}{N_B}\\sum _{i}\\nabla _{a}Q(s,a|\\theta ^{Q})|_{s=s_{i},a=\\mu (s_{i})}\\nabla _{\\theta ~{\\mu }} \\mu (s|\\theta ^{\\mu })_{s_{i}}$ Update the target networks: $\\theta ^{Q^{\\prime }} \\leftarrow \\zeta \\theta ^{Q}+(1-\\zeta )\\theta ^{Q^{\\prime }}$ $\\theta ^{\\mu ^{\\prime }} \\leftarrow \\zeta \\theta ^{\\mu }+(1-\\zeta )\\theta ^{\\mu ^{\\prime }}$ Store the loss of LSTM Store interoception value and image for the second layer and the mood $e$ is divisible by $T_{LSTM}$ Update LSTM module $e$ is divisible by $T_{L2}$ Update the mood of the agent ${m}(t)$ Update the second layer ${L}(k)$ 's Figure: Whole network architecture of the proposed deep emotion."
],
[
"Experiments",
"We explain the experiment in this section.",
"The experiment is roughly divided into three parts.",
"In the first experiment, we verify the performance of the RAM (first layer).",
"Because the first layer is assumed to return innate responses to stimuli, it is qualitatively evaluated through our subjective sense and children's tendencies.",
"In the second experiment, we combined the RAM (first layer) and convolutional LSTM–DDPG (third layer), and observe the agent's behavior and internal representation of the emotional state.",
"Because the second layer is responsible for the adaptation of the system to the environment, we focused on the implementation of first and third layers in this experiment.",
"In the third experiment, we combined the first, second, and third layers, implemented the whole emotion model, and verified its behavior.",
"Then, by comparing with the second experiment, the significance of the second layer is examined."
],
[
"Experiments on RAM (1st layer)",
"In order to test the performance of the RAM, we conducted the following experiment.",
"As explained in REF , the RAM was trained using a set of images from IAPS, OASIS, NAPS, and GAPED.",
"We used in total 24,270 images (4,045 original images $\\times $ 6 types of deformation such as rotations, flipping, and affine transformations) for training, and 100 images for testing (randomly selected from IAPS).",
"After the training, the RAM was evaluated using the evaluation data.",
"To qualitatively examine the property of the model, we also input single-color images to the RAM and observed the results.",
"This evaluation is expected to provide an insight on the color preference of the trained network.",
"Moreover, we also input face images with a certain facial expression to the RAM.",
"The Japanese female facial expression (JAFFE) database [9] was used in this experiment.",
"The JAFFE database contains 213 images of seven facial expressions (pleasure, sad, angry, fearful, surprised, disgusted, and neutral).",
"We visualized outputs from the RAM, i.e., valence and arousal values, for these input images."
],
[
"Results",
"Fig.",
"REF (a) represents the ground truth, i.e., values from the IAPS database, and the results output by the RAM.",
"The mean absolute errors for 100 test images are 0.48 for arousal and 0.46 for valence.",
"These errors are sufficiently small as compared with the variation of human evaluation [29], [30].",
"Fig.",
"REF (b) represents the results of visual attention for two different test images.",
"One can see that the system successfully paid attention to visually important locations and estimated reasonable arousal and valence values in both cases.",
"Fig.",
"REF (c) shows the results of inputting single-color images; high values are observed around 45 degrees of hue, which corresponds to the color yellow.",
"Additionally, high values are seen in the center, which corresponds to the color white.",
"On the other hand, low values are observed around 270 and 100 degrees of hue, which correspond to purple and green, respectively.",
"According to Yamawaki, for infants six months of age, warm colors, such as yellow, white, and pink, have high preference and cold colors, such as blue, green, and violet, have low preference [66].",
"This result implies that the output of the RAM shows reasonable reactions compared with an infant.",
"In the result of the facial image input, the output, i.e., valance and arousal values, tends to coincide with the category of the facial expression.",
"The results are given in Fig.",
"REF (d).",
"For instance, when the facial images with pleasure expression are input to the RAM, the output of the RAM tends to have a high valence value.",
"However, anger facial expressions tend to draw low valence and high arousal values.",
"These results indicate that a response called emotional contagion [18], [4] is observed in the trained RAM network.",
"Figure: Results of the first layer: (a) comparison between the output of the RAM and ground truth, (b) examples of the locations paid attention by the RAM (the red rectangle in each image represents the location of attention and a part of the facial image was blurred to make it impossible to identify individuals), (c) visualization of the RAM's output for input single-color images, and (d) heat map of arousal/valence frequency for facial images."
],
[
"Experimental setup",
"This experiment intends to show the performance of the decision-making mechanism in the proposed emotion model.",
"Therefore, we connected the RAM and the third layer, which is implemented by the convolutional LSTM–DDPG.",
"The second layer is not included in this experiment, because the whole emotion model is used in the next experiment and the results are compared to examine the importance of the second layer.",
"The virtual agent (we use a free software package called “MakeHuman” for the modeling of 3-dimensional agent http://www.makehumancommunity.org/wiki/MakeHuman_resources), which can change its facial expressions by moving eyelids, eyebrows, mouth, and corners of the mouth, is used as the body and the RAM and the convolutional LSTM–DDPG are implemented in the virtual agent.",
"Then, we designed a “facial expression” task based on the mother–infant interaction scenario.",
"In this task, the interaction partner, which is also a computer agent (mother agent), recognizes the agent's facial expressions as one of four categories, and expresses back the corresponding facial expressions in the same category as that of the virtual agent (infant agent).",
"The facial expression recognition of the infant agent by the mother agent is based on the following rules: 1) pleasure (when the corner of the mouth is raised), 2) anger (when the corner of the mouth falls, eyebrows are knitted, and eyes are more than half open), 3) sadness (when the corner of the mouth falls, eyebrows are knitted and eyes are more than half closed), and 4) neutral (otherwise).",
"This experimental design is based on a known phenomenon called “mirroring,” in which the mother intuitively imitates the infant's expression on a daily basis [62].",
"This is said to be important for young infants to learn emotional adjustment and social response [48].",
"Especially for smiling, interactive smile games between infants and their caregivers are known as an important milestone in infant social development and build the foundation for later forms of social competence [26].",
"Ruvolo and colleagues revealed that there exists a strategy for the timing when the child smiles and the relationship with his/her mother [57].",
"Thus, the purpose of this experiment is to observe the behavior learned by the infant agent and the change in interoception, emotional state, and emotion due to learning of a facial expression strategy.",
"In this experiment, we have two different conditions: “face-only condition” and “face+natural condition.” These conditions are set to compare the ideal condition of seeing only the face of the mother and the case where environmental factors exist.",
"In the “face-only” condition, the infant agent always receives a facial image according to the infant agent expression (mirroring).",
"The top row of Fig.",
"REF represents information on facial images, which are selected from JAFFE database [9].",
"There are two different facial images for each emotional category.",
"One of these two images is selected randomly to present to the infant agent.",
"Although the actual images used in this experiment cannot be shown, one can check the images by downloading the database from http://www.kasrl.org/jaffe.html.",
"“JAFEE ID” corresponds to the filename of each image.",
"On the contrary, in the “face+natural” condition, the infant agent randomly receives one of the facial images in the top row of Fig.",
"REF or one of the IAPS images in the bottom row of Fig.",
"REF as a visual stimulus.",
"The natural images from IAPS mimic environmental stimuli.",
"It should be noted that the facial images are stimuli that the infant agent can manipulate, because the facial images are selected according to the infant agent action.",
"The IAPS images are, however, stimuli that cannot be manipulated, as they are randomly chosen.",
"In other words, it is expected that the infant agent learns a policy to acquire intended stimulus according to a given facial image, and learns countermeasures, e.g., closing eyes, when an undesirable stimulus is presented from the IAPS images.",
"In both cases, the image of the closed eyes portion is displayed as a black image when the agent closes his eyes.",
"We performed 100000 epochs of this training using the proposed emotion model and the abovementioned scenario.",
"Each time learning progresses, we visualize the middle layer of the policy network in Fig.",
"REF using principal component analysis (PCA) in order to observe the state space constructed by the infant agent through the mother–infant interaction.",
"If our emotions were correctly defined and properly implemented, then this state space could be divided into emotional categories by actions.",
"Figure: Images used in the experiment: (a) facial images selected according to the infant agent's facial expression (JAFFE IDs are shown instead of actual images because of personality rights), and (b) natural images selected randomly."
],
[
"Results",
"Figures REF (a) and (b) show the learning curves of this experiment for the face-only and face+natural conditions, respectively.",
"On the top row, the learning curves of the LSTM are shown.",
"From these figures, one can see that the LSTM learns to predict the next stimuli and interoception values.",
"The training loss rapidly decreases within 5000 epochs.",
"By comparing the face-only condition and face+natural condition, it is natural that the prediction error is smaller in the face-only condition.",
"For the reward in the bottom row of Figs.",
"REF (a) and (b), similar properties can be seen.",
"In fact, the reward rapidly increases for less than 5000 epochs.",
"The reward does not converge to a constant value.",
"This fluctuation occurs because the reward is based on homeostasis, which is a difference between a current interoception value and the past averaged interoception values.",
"If there is a sudden change such as recovery of strength, the reward tends to change suddenly.",
"In spite of this fluctuation of the reward, it can be clearly seen that the face-only condition achieved the higher reward in total.",
"This is because the prediction in face-only condition works better than the face+natural case.",
"In other words, the infant agent can better control the external environment as the mother agent always shows the facial expression in response to the infant agent.",
"Now, we examine the change in internal representation, i.e., emotional state, behind this reinforcement learning.",
"The results are shown in Fig.",
"REF .",
"Figures REF (a) and (b) show plots of the external appraisal and interoception values, respectively.",
"The visualization of the middle layer of the policy network using PCA is shown in Fig.",
"REF (c).",
"These results correspond to (a), (b), and (c) in Fig.",
"REF .",
"Each color represents a category of facial expression recognized by the mother agent.",
"Specifically, green, yellow, blue, and red represent neutral, pleasure, sadness, and anger, respectively.",
"The top rows of Figs.",
"REF (a)–(c) show the results of the face-only condition, and the bottom rows show the results of the face+natural condition.",
"As mentioned previously, the face-only condition indicates that only stimuli that can be controlled by the infant agent are provided, whereas in the face+natural condition, half of the stimuli can be controlled by the infant agent and the other half cannot.",
"From the results, one can see that the colors are mixed all over in Figs.",
"REF (a) and (b).",
"This implies that the external appraisal and interoception do not explicitly provide emotion differentiation functionality.",
"Moreover, it can be seen that the space does not expand in Figs.",
"REF (a) and (b) as the learning progresses.",
"However, the state space expands and is divided for each color as learning progresses in Fig.",
"REF (c).",
"We hypothesize that this is the basic mechanism of emotion differentiation, which is observed in the middle layer of the policy network.",
"Because the interoception and external appraisal did not show differentiation, these results indicate the plausibility of the proposed emotion model.",
"We stop the learning process at certain epochs and run the infant agent using the learned model at each epoch to observe the behavior of the infant agent.",
"From these observations, we found that the agent had the following behavioral changes (One can download the demo video of the running agent using learned models from https://youtu.be/DHOIbe4qEEY).",
"In 20000 epochs model, the infant agent often opens his eyes.",
"In the model of 40000 epochs, he often closes his eyes.",
"He changes facial expressions by stimulation in the model of 60000 epochs.",
"He closes his eyes at the times when the internal appraisal increases in the model of 80000 epochs.",
"Finally, after 100000 epochs, he shows various facial expressions and has succeeded in stabilizing emotions.",
"Essentially, the agent smiles very often and makes the other person smile.",
"This behavior also seems to be altruistic, such that the agent is trying to make the partner smile.",
"This behavior seems to be consistent with the findings in [57].",
"Actually, it is interesting that the infant agent is just smiling for the desired stimulus, that is, the agent learned a selective smile.",
"Figure: Learning curves of the LSTM and the DDPG: (a) face-only condition in experiment 4.2, (b) face+natural condition in experiment 4.2, and (c) face+natural condition in experiment 4.3.Figure: Visualization of the internal representations in experiment 4.2 (first + third layers): (a) external appraisal during each period of epochs, (b) interoception values during each period of epochs, (c) PCA visualization of the middle layer of the policy network during each period of epochs.",
"It should be noted that the top and bottom rows represent the results of the face-only and face+natural conditions, respectively."
],
[
"Experimental setup",
"In Section REF , we conducted an experiment using the first and third layers.",
"This is because the integration of the first and third layers is the core part of the proposed emotion model.",
"We are interested in the core mechanism of the emotion model and evaluated the model without using the second layer in the previous section.",
"The whole system, including the second layer, is the focus of our interest in this section.",
"Moreover, we can determine the importance of the second layer by comparing the results to the previous ones.",
"We use exactly the same experimental protocol as in Section REF ; however, only the face+natural condition is adopted as it is obvious that the face-only condition gives better performance in terms of prediction."
],
[
"Results",
"The learning curve of the whole system is given in Fig.",
"REF (c).",
"The upper graph represents the LSTM loss versus the number of epochs.",
"This graph shows a similar tendency to the previous experiment, which means that the LSTM learns to predict the next image and interoception values.",
"The lower graph shows the reward with respect to the number of epochs.",
"This also shows the same tendency as the previous experiment.",
"It is interesting to compare the results between the previous and current experiments in terms of average errors in the LSTM and average reward that the agent obtained.",
"For the LSTM, the averaged losses, which are represented by $\\ell _a$ (face-only condition without second layer), $\\ell _b$ (face+natural condition without second layer), and $\\ell _c$ (face+natural condition with second layer) are expected to be in the order $\\ell _{a}$ $<$ $\\ell _{c}$ $<$ $\\ell _{b}$ .",
"In fact, the averaged losses are $\\ell _{a}= 1.26$ , $\\ell _{b}=1.89$ , and $\\ell _{c}=1.66$ ; and the order of these values is as expected.",
"Exactly the same observations can be made with respect to the reward (larger is better in this case).",
"The averaged reward values are $\\bar{R}_{a}= 38.72$ , $\\bar{R}_{b}=32.35$ , and $\\bar{R}_{c}=35.24$ ($\\bar{R}_a > \\bar{R}_c > \\bar{R}_b$ ).",
"These results are obtained because the face-only condition is the easiest setting for the infant agent.",
"Moreover, the second layer improves the prediction of the next situation, which leads to $\\bar{R}_{c} > \\bar{R}_{b}$ .",
"In order to show that the second layer actually works, the learned models (both with and without the second layer) were run for 3000 epochs and the interoception values were collected.",
"Then, the mean absolute differences (MAD) $|{a}_t -{a}_{t+1}|$ of both models were compared.",
"For the valence, the MAD of the previous experiment (face+natural condition without second layer) was $0.21$ .",
"The MAD of the current experiment (face+natural condition with second layer) was $0.16$ .",
"For the arousal, the MAD of the previous experiment and the current experiment were, respectively, $0.20$ and $0.15$ .",
"The t-test was performed on the MAD of both models and revealed that the MAD was significantly smaller in the case with the second layer ($p < 0.01$ ).",
"This indicates that the second layer works as we expected, and it improves the learning performance.",
"Here, discuss in detail the representation inside the network to find the basis of emotions.",
"Figure REF shows plots of the external appraisal, interoception values, and visualization of the middle layer of the policy network using PCA.",
"These results correspond to (a), (b), and (c) in Fig.",
"REF .",
"Each color represents a category of facial expression recognized by the mother agent, as mentioned in the previous section.",
"From these figures, one can see that the representation in the policy network divides the emotional category very clearly compared with the external appraisal and the interoception.",
"Moreover, it is also clear that the policy network represents categories far better compared with Fig.",
"REF , which does not include the second layer.",
"In this experiment, we also run the agent using the learned model.",
"Figure REF shows some typical facial expressions by the infant agent for each model at specific epochs.",
"From the observations of the infant agent's behavior, we found that the agent had the following behavioral changes (One can download the demo video of the running agent using learned models from https://youtu.be/Phjn58kJ2ns): –20000 epochs: The agent often closes his eyes.",
"–40000 epochs: The agent often closes his eyes.",
"–60000 epochs: The agent often opens his eyes and he changes expressions by stimulation.",
"–80000 epochs: The agent closes his eyes at the times when the internal appraisal increases.",
"–100000 epoch: The agent shows various facial expressions (surprise, anger, etc.)",
"and has succeeded in stabilizing affects.",
"Figure: Visualization of the internal representations in experiment 4.3 (the whole model): (a) external appraisal, (b) interoception, and (c) PCA visualization of the middle layer of the policy network during each period of epochs.Figure: Examples of facial expressions by the infant agent using the learned model with the second layer.",
"Please note that the facial input image on top right is blurred for personality rights.Figure: Frequency ratio of facial expressions for each condition."
],
[
"Discussion",
"In the first experiment, the RAM was evaluated.",
"The results of this experiment show that the RAM has an ability to replicate the innate reactions of a human against specific stimulation.",
"It is interesting that although the network does not learn the reactions directly, it can learn general human reactions.",
"For example, when an image of a pleasure facial expression is input to the RAM, the arousal and valence values corresponding to pleasure are generated.",
"Moreover, similar responses of infants to color are learned by the RAM.",
"These facts indicate the existence of an innate and general response of humans to visual stimuli, and the RAM can extract such visual features.",
"For the implementation of emotional robots, emotions are usually designed manually by the robot designer and the above results may free the robot designer from this difficult design task.",
"In the second experiment, we evaluated the proposed emotion models.",
"According to the PCA results with the face-only condition, pleasure occupies half, and the remaining half seems to consist of a mix of neutral and sadness, and anger can be seen to a lesser degree.",
"In an environment that the agent can always control, anger is not necessary to be output; that is, the agent learned to deal with stimuli by pleasure or otherwise.",
"However, according to the result of the face+natural condition, although pleasure is predominant, anger increases, and neutral and sadness are separated as compared with the face-only condition.",
"This is due to the necessity of selecting actions by classifying stimuli in more detail because of uncontrollable stimuli.",
"Therefore, it can be surmised that not only the controllable stimuli but also the uncontrollable stimuli create our human-like rich emotions.",
"The uncontrollable stimuli also give a very important meaning to learning to predict the future; that is, if the world is simple enough to predict perfectly, then the learning does not mean anything.",
"In the third experiment, the whole emotion model including the second layer was evaluated.",
"By having the second layer in the emotion model, the state space, i.e., the middle layer of the policy network, has the representations of basic emotions such as anger, sadness, pleasure, and neutral.",
"More interestingly, these emotional categories are located as assumed in the dimensional model; that is, neutral is located at the center, and pleasure, sad, and anger are located surrounding the neutral.",
"Pleasure does not occupy the PCA space anymore, and it seems to be relatively evenly divided.",
"In particular, the frequency of anger increases as shown in Fig.",
"REF .",
"Because the second layer works as a smoothing function, the interoception values of temporally adjacent stimuli are made closer, and sudden changes are reduced.",
"As a result, prediction in the LSTM improves, and categorization of the stimulus is promoted.",
"It is thought that these effects result in relatively uniform and distinct differentiation of the boundary surface of emotional categories.",
"Now, let us consider the behavioral output of the infant agent with the whole emotion model.",
"In the early stage of learning, the agent closed his eyes well and the eyes are opened well in the second half of the early stage.",
"This is similar to the development of infants.",
"In general, infants initially almost always have their eyes closed (sleeping), and the time with their eyes open increases gradually.",
"This process may be mainly dependent on the developmental process of the physical bodies of young infants.",
"However, in the course of action selection, infants may have a stage to learn that the best policy is to close the eyes at the beginning, and gradually shift toward the policy of keeping their eyes open.",
"This is only a speculation, which should be verified in the future.",
"Additionally, the 100000 epoch result in Fig.",
"REF shows that the infant agent looks surprised by the snake.",
"In the PCA space of the middle layer of the policy network, i.e., internal representation of emotional states, it is not clear whether the surprise category was generated, because the actions were classified with only four emotional categories.",
"However, there is a possibility that a richer emotional space emerged as the internal representation of the proposed emotion model.",
"This point still needs further analysis.",
"Here, the limitations of our proposed model are discussed.",
"Because the IAPS used adult human subjects to label the arousal and valence values, there must be an issue of the RAM using the IAPS in the first place.",
"However, we think that the averaging process of the labeled values reduced the individuality of the data and innate reactions were extracted.",
"The results of the first experiment using the RAM implies that this is in fact true.",
"Currently, biosignals from a real human body instead of the IAPS database are prepared to use for training the RAM as another direction of this research.",
"Another issue to be addressed is the reward for reinforcement learning, which is currently based solely on the idea of “homeostasis.” The idea of intrinsic motivation that appeared as a series of counterarguments to drive reduction theory cannot be ignored [25].",
"More complex tasks should be considered in the future, because the current facial expression task is too simple to examine the full functionality of the emotion model.",
"We also consider using a real robot to examine more complex internal appraisals.",
"From the viewpoint of empathic communication, “other” should appear in Fig.REF .",
"Moreover, self/other discrimination must be considered in the model for generating higher-level social emotions.",
"Language is another important aspect of the emotion model [38].",
"We are currently working on the “emotional symbol grounding problem” using the idea of language acquisition by robots [50].",
"In addition, it is necessary to consider empathy.",
"For example, Lim et al.",
"proposed multimodal emotional intelligence [40].",
"Their model was inspired by the mirror neuron system, which is a mechanism underlying human cognition [22].",
"In considering empathy, the work on mirror neurons cannot be ignored."
],
[
"Conclusions",
"In this study, a computational model of emotion, which consists of three layers was proposed.",
"As the first layer, we examined a method for generating valence and arousal values by given visual stimuli using the RAM.",
"Some promising results were obtained, which verified that the first layer is plausible for generating human-like quick reactions against specific stimuli.",
"Next, we examined a decision-making mechanism, which is the third layer, by employing a convolutional LSTM and DDPG.",
"As a result, the agent learned a selective smile and emotion differentiation was observed.",
"Finally, the whole model including the second layer was integrated and its performance was studied.",
"The results obtained in this experiment show that the second layer provided far better results compared with the model without the second layer.",
"For future work, we will evaluate the proposed model using more complex tasks.",
"The implementation on a real physical robot is also left for future work."
],
[
"Author Contributions",
"CH and TN conceived of the presented idea.",
"CH developed the theory and implemented the system.",
"CH and TN analyzed the results, and all authors discussed the results.",
"CH wrote the manuscript with support from TH and TN.",
"This research was subsidized by JSPS Science Research Fund JP 16 J 04930, JST CREST (JPMJCR15E3), and Grant-in-Aid for Scientific Research on Innovative Areas (26118001)."
],
[
"Recurrent attention model (RAM)",
"The RAM is a recurrent neural network (RNN) with visual attention proposed by Mnih et al.",
"[45].",
"In general, humans focus attention selectively on parts of the visual space instead of processing whole scene at once.",
"Human visual perception acquires information when and where it is needed, and combine information from different fixations over time.",
"This is how we build up an internal representation of the scene and we use the representation for decision making.",
"Based on this idea the RAM, which is a novel framework for attention-based task-driven visual processing with neural networks, has been developed.",
"As shown in Fig.",
"REF (b), images with multiple resolutions are acquired from the original image $x_{t}$ at the center point $l_{t-1}$ .",
"Then, each point and multiple images are input to the linear layer as $g_{t} = f_{g}(x_{t},l_{t-1};\\theta _{g})$ .",
"$f_{h}(\\theta _{h})$ is the core network and takes $h_{t-1}$ , which is a previous internal representation, as an input.",
"The action network $f_ {a}(\\theta _{a})$ and the location network $f_{l}(\\theta _{l})$ take $h_t$ to calculate the valence/arousal values and location of the next step, respectively.",
"The parameters of RAM are defined as $\\theta = \\lbrace \\theta _{g}, \\theta _{h}, \\theta _{a} \\rbrace $ , and $\\theta $ is optimized such that the total reward the agent can obtain when interacting with the environment is maximized.",
"More specifically, the policy of the agent induces a distribution over possible interaction sequences $s_{1:N}$ and the reward is maximize under this distribution: $J(\\theta )={E}_{p(s_{1:T};\\theta )}[\\sum _{t=1}^{T}r_{t}]={E}_{p(s_{1:T};\\theta )}[R],$ where $p(s_{1:T};\\theta )$ depends on the policy.",
"Although it is difficult to maximize $J$ exactly, we can apply some techniques form the reinforcement learning by viewing the problem as a partially observable Markov decision process.",
"In this case, the gradient can be expressed as $\\nabla _{\\theta }J=\\sum _{t=1}^{T} {E}_{p(s_{1:T};\\theta )}[\\nabla _{\\theta }log\\pi (u_{t} | s_{1:t};\\theta )R] \\approx \\frac{1}{M}\\sum _{i=1}^{M}\\sum _{t=1}^{T}\\nabla _{\\theta }log\\pi (u_{t}^{i} | s_{1:t}^{i};\\theta )R^{i},$ where $s^{i}$ are interaction sequences obtained by running the current agent $\\pi _{\\theta }$ for $i = 1 \\cdots M$ episodes.",
"The learning rule is also known as the REINFORCE rule.",
"It involves running the agent with its current policy to obtain samples of interaction sequences $s_{1:T}$ .",
"Then, the parameters $\\theta $ of the agent are adjusted such that the log-probability of the chosen actions that have led to high cumulative reward is increased, while that of actions having produced low reward is decreased.",
"Eq.",
"(REF ) requires us to compute $\\nabla _{\\theta }log\\pi (u_{t} | s_{1:t};\\theta )$ ; however, this is the gradient of the RNN that defines the agent evaluated at time step $t$ and can be computed by standard backpropagation.",
"In our scenario, the RAM must output the arousal/valence values for the input image as the final action.",
"For the training images, these values are known and the policy, that outputs the correct values associated with a training image at the end of an observation sequence, can be directly optimized.",
"This can be achieved by maximizing the conditional probability of the true values given the observations from the image, i.e., by maximizing $log\\pi (a_{T}^{*}|s_{1:T};\\theta )$ , where $a_{T}^{*}$ corresponds to the ground-truth associated with the image from which observations $s_{1:T}$ were obtained.",
"The original RAM follows this approach for classification problems, where it optimizes the cross-entropy loss to train the action network $f_{a}$ and the gradients are backpropagated through the core and glimpse networks.",
"The location network $f_{l}$ is always trained with REINFORCE, which provides the parameter $\\theta _l$ ."
],
[
"Convolutional long short-term memory (LSTM)",
"Convolutional LSTM is a method combining CNN, which captures the features of images, and LSTM, which can handle long-term time series information, proposed by Xingjian et al.",
"[64].",
"Specifically, it is a network in which multiplication by the weight of LSTM is convolution, and the constituent element is composed of a memory cell $C_{t}$ , input gate $i_{t}$ , forget gate $f_{t}$ , and output gate $o_{t}$ .",
"$i_{t}&=&\\sigma (W_{xi}*X_{t}+W_{hi}*H_{t-1}+W_{ci}\\circ C_{t-1}+b_{i}), \\\\f_{t}&=&\\sigma (W_{xf}*X_{t}+W_{hf}*H_{t-1}+W_{cf}\\circ C_{t-1}+b_{f}), \\\\C_{t}&=&f_{t}\\circ C_{t-1}+i_{t}\\circ tanh(W_{xc}*X_{t}+W_{hc}*H_{t-1}+b_{c}), \\\\o_{t}&=&\\sigma (W_{xo}*X_{t}+W_{ho}*H_{t-1}+W_{co}\\circ C_{t}+b_{o}), \\\\H_{t}&=&o_{t}\\circ tanh(C_{t}),$ where $X_{t}$ are inputs, $H_{t}$ are hidden states, the $W$ terms denote weight matrices, the $b$ terms denote bias vectors, $*$ denotes the convolution operator, and $\\circ $ denotes the Hadamard product.",
"The memory cells are responsible for storing past states.",
"The input gate has a role of adjusting the value added to the memory cell.",
"It is possible to prevent the important information possessed by the memory cell from being lost due to the influence of the most unrelated information that is most recent, owing to the existence of this gate.",
"The forget gate has a role of adjusting how much the value of the memory cell is held at the next time.",
"The output gate serves to adjust how much the value of the memory cell affects the next layer.",
"The existence of this gate can prevent the entire network from being disturbed by short-term memory and interruption of long-term memory.",
"In this study, we use two layers of convolutional LSTM; the filter is $5 \\times 5 \\times 5$ and the error is calculated by the mean square error.",
"The learning rate is adaptive moment estimation (Adam) ($\\alpha =0.001,\\beta _{1}=0.9,\\beta _{2}=0.999,\\epsilon =10^{-8}$ )."
],
[
"Deep deterministic policy gradient (DDPG)",
"DDPG is a reinforcement learning method using deep learning proposed by Lillicrap et al.",
"[39].",
"As recently reported, “Deep Q Network” (DQN) algorithm [46] is capable of human-level performance on many Atari video games using unprocessed pixels for input.",
"Whereas DQN solves problems with high-dimensional observation spaces, it can only handle discrete and low-dimensional action spaces.",
"Then, they presented a model-free, off-policy actor-critic algorithm (DDPG) using deep function approximators that can learn policies in high-dimensional, continuous action spaces.",
"The DDPG algorithm is shown in Algorithm .",
"The learning rate is Adam (actor network: $\\alpha =10^{-4},\\beta _{1}=0.9,\\beta _{2}=0.999,\\epsilon =10^{-8}$ , critic network: $\\alpha =10^{-3},\\beta _{1}=0.9,\\beta _{2}=0.999,\\epsilon =10^{-8}$ ).",
"$\\mathcal {N}$ is the Ornstein–Uhlenbeck process.",
"$N_B$ is 200.",
"The size of $B$ is 500.",
"When new data comes in, old data is discarded.",
"We used batch normalization.",
"DDPG algorithm Randomly initialize critic network $Q(s,a|\\theta ^{Q})$ and actor $\\mu (s|\\theta ^{\\mu })$ with weights $\\theta ^{Q}$ and $\\theta ^{\\mu }$ .",
"Initialize target network $Q^{\\prime }$ and $\\mu ^{\\prime }$ with weights $\\theta ^{Q^{\\prime }} \\leftarrow \\theta ^{Q}, \\theta ^{\\mu ^{\\prime }} \\leftarrow \\theta ^{\\mu }$ Initialize replay buffer $B$ episode = 1, M Initialize a random process $\\mathcal {N}$ for action exploration Receive initial observation state $s_{1}$ t = 1, T Select action $a_{t} = \\mu (s_{t}|\\theta ^{\\mu }) + \\mathcal {N}_{t} $ according to the current policy and exploration noise Execute action $a_{t}$ and observe reward $r_{t}$ and observe new state $s_{t+1}$ Store transition ($s_{t}, a_{t}, r_{t}, s_{t+1}$ ) in $B$ Sample a random minibatch of $N_B$ transitions ($s_{i}, a_{i}, r_{i}, s_{i+1}$ ) from $B$ Set $y_{i}=r_{i}+\\gamma Q^{\\prime }\\left(s_{i+1},\\mu ^{\\prime }(s_{i+1}|\\theta ^{\\mu ^{\\prime }})|\\theta ^{Q^{\\prime }} \\right) $ Update critic by minimizing the loss: $L=\\frac{1}{N_B}\\sum _{i} \\left\\lbrace y_{i}-Q(s_{i},a_{i}|\\theta ^{Q}) \\right\\rbrace ^{2}$ Update the actor policy using the sampled policy gradient: $\\nabla _{\\theta ^{\\mu }}J \\approx \\frac{1}{N_B}\\sum _{i}\\nabla _{a}Q(s,a|\\theta ^{Q})|_{s=s_{i},a=\\mu (s_{i})}\\nabla _{\\theta ~{\\mu }} \\mu (s|\\theta ^{\\mu })_{s_{i}}$ Update the target networks: $\\theta ^{Q^{\\prime }} \\leftarrow \\eta \\theta ^{Q}+(1-\\eta )\\theta ^{Q^{\\prime }}$ $\\theta ^{\\mu ^{\\prime }} \\leftarrow \\eta \\theta ^{\\mu }+(1-\\eta )\\theta ^{\\mu ^{\\prime }}$"
]
] | 1808.08447 |
[
[
"Aperiodic Array Synthesis for Multi-User MIMO Applications"
],
[
"Abstract This paper demonstrates the advantages of aperiodic arrays in multi-user multiple-input multiple-output systems for future mobile communication applications.",
"We propose a novel aperiodic array synthesis method which account for the statistics of the propagation channel and the adaptive beamforming algorithm.",
"Clear performance gains in line-of-sight dominated propagation environments are achieved in terms of the signal-to-interference-plus-noise ratio, the sum rate capacity, as well as the spread of the amplifier output power as compared to their regular counterparts.",
"We also show that the performance is not sacrificed in rich scattering environments.",
"Hence, aperiodic array layouts can provide performance gains in millimeter-wave applications with a dominating line-of-sight component."
],
[
"Introduction",
"The continuously growing need for higher capacity and user data rates in wireless communications calls for new multi-antenna concepts, such as the massive multiple-input multiple-output (MIMO) concept.",
"The practical implementation of such complex antenna systems is very challenging, particularly if power-efficient and cost-effective solutions are to be realized.",
"Typical systems require hundreds up to thousands of active antenna elements, each of which is equipped with a signal digitizing circuit [1].",
"Research on massive MIMO solutions has mainly focused on classical uniform array layouts.",
"However, aperiodic array layouts are potentially advantageous in suppressing spatially distributed interference through minimizing side-lobe levels while maximizing the power efficiency of amplifiers through the use of isophoric array architectures.",
"These advantages have been exploited in cases where the desired beamshape is known a priori, including satellite communication and radio astronomy applications [2], [3].",
"In MIMO systems, adaptive beamforming is employed, thus there is no a priori knowledge on the desired beamshape nor the element excitations, since these are both channel-state dependent and dynamically adapted to improve the link quality and/or capacity.",
"Most of the available aperiodic array synthesis methods are therefore not suitable or readily applicable to such beamforming systems.",
"A recent study found that the array aperiodicity introduced by small random errors in the antenna element placement can be beneficial for the performance of massive multi-user (MU) MIMO communication systems [4].",
"However, to fully understand its advantages, it is necessary to examine and synthesize optimal aperiodic array layouts.",
"To the best of the author's knowledge, this it the first time that aperiodic array synthesis is proposed and studied in this context.",
"Figure: Illustration of an M×KM\\times K MU-MIMO system in downlink, where MM and KK are the number of Base Station (BS) antenna elements and User Equipments (UE), respectively.This paper introduces an innovative deterministic-statistical approach for the synthesis of optimal aperiodic array antennas as base stations for MU-MIMO applications.",
"We first derive the statistical distribution of the element excitations for a dense regular array for a preselected adaptive beamforming algorithm and propagation environment, which is then used as a density taper [5] to identify a reduced set of optimal array element locations.",
"This paper presents: (i) a mathematical formulation of the proposed approach, and; (ii) numerical results illustrating the effects of different propagation environments, as well as the number of base station antenna elements and users, on the the performance improvements of aperiodic arrays over regular ones.",
"For these comparisons we employ the Zero Forcing adaptive beamformer."
],
[
"MU-MIMO System Model",
"In the following subsections we introduce: 1) the MU-MIMO antenna system model for the downlink scenario; 2) the radio propagation channel model; 3) the key communication link performance metrics, employed to evaluate system performance such as the SINR and sum-rate capacity, and; 4) the figures of merit to evaluate power level variations of amplifiers due to adaptive beamforming.",
"Finally, in 5), the uplink duality is briefly outlined."
],
[
"Downlink",
"Consider the downlink of an $M\\times K$ single-cell narrowband MU-MIMO system, as shown in Fig.",
"REF .",
"The BS is equipped with $M$ antennas serving $K$ single-antenna UEs (User Equipments), with $M\\ge K$ .",
"Let $\\mathsf {x} \\in \\mathbb {C}^{M\\times 1}$ be the signal transmitted from the BS array with normalized power $||\\mathsf {x}||^2=1$ .",
"The received signal $\\mathsf {y} \\in \\mathbb {C}^{K\\times 1}$ at the $K$ UEs can be expressed as $\\mathsf {y}=\\sqrt{\\text{SNR}}\\mathsf {H}\\mathsf {x}+\\mathsf {n},$ where $\\text{SNR}$ is the average per-user signal-to-noise ratio (SNR), $\\mathsf {H}\\in \\mathbb {C}^{K\\times M}$ is the downlink MU-MIMO channel matrix and $\\mathsf {n}\\in \\mathbb {C}^{K\\times 1}$ is the additive white Gaussian noise at the users, here assumed having zero-mean and unit variance."
],
[
"Channel Model",
"The channel matrix $\\mathsf {H}$ captures the radio propagation conditions between the antenna ports of the BS and that of the multiple UEs.",
"In our model, the BS antenna is a linear array of $M$ Huygens sources serving $K$ UEs in a $120^\\circ $ cell sector.",
"Each UE is represented by a linearly polarized antenna with a random orientation and position, see Fig.",
"REF .",
"The MIMO system can be characterized in two extreme propagation environments: (i) the Rich Isotropic MultiPath (RIMP) environment [6], with a very large number of random uniformly distributed scattered waves, and; (ii) the Random Line of Sight (RLOS) environment [7], with a single direct wave with random properties due to arbitrary position and orientation of the UE.",
"To characterize these, as well as in-between environments, the problem was modeled by associating to each UE a set of waves (from 1 for RLOS up to 20 for RIMP) with a uniform random distribution in angle of arrival, amplitude, phase and polarization.",
"Note that each wave thus represents either the LOS component or a strong scatterer.",
"Simulations are repeated $10^6$ times to produce accurate statistics, such as the cumulative probability distribution (CDFs), the average and the variance."
],
[
"SINR and Sum Rate Capacity",
"We assume perfect channel state information at the transmitter, so that $\\mathsf {H}$ is known at the BS.",
"In linear precoding, the transmitted signal can then be expressed as $\\mathsf {x}=\\sqrt{\\beta }\\mathsf {W}\\mathsf {q}$ , where $\\mathsf {W} \\in \\mathbb {C}^{M\\times K}$ is the precoding matrix, $\\mathsf {q} \\in \\mathbb {C}^{K\\times 1}$ is the intended transmitted signal, $\\beta =1/\\operatorname{tr}(\\mathsf {W}\\mathsf {W}^\\dagger )$ is the power normalization constant, and $^\\dagger $ denotes the conjugate transpose operation.",
"Since we expect an interference-limited scenario, a Zero Forcing (ZF) beamformer is assumed to suppress the interference between UEs.",
"Accordingly, $\\mathsf {W}=\\mathsf {H}^\\dagger (\\mathsf {H}\\mathsf {H}^\\dagger )^{-1}$ .",
"The downlink signal-to-interference-plus-noise ratio (SINR) of the $k$ th user can be expressed as [8] $\\text{SINR}_k=\\frac{\\beta \\text{SNR}|\\mathsf {H}_{k,:}\\mathsf {W}_{:,k}|^2}{\\beta \\text{SNR}\\sum _{j\\ne k}^K{|\\mathsf {H}_{k,:}\\mathsf {W}_{:,j}|^2}+1}$ where $\\mathsf {X}_{i,:}$ and $\\mathsf {X}_{:,j}$ indicate the $i$ th row and $j$ th column of the matrix $\\mathsf {X}$ , respectively.",
"It is worthwhile to note that Eq (REF ) is valid for any pre-coding matrix.",
"The ergodic sum rate capacity of the total MU-MIMO system is then defined as [9] $\\text{SR}=\\sum _{k=1}^{K}{\\operatorname{\\mathrm {E}}[ \\log _2{(1+\\text{SINR}_k}) ]}.$ Figure: Illustration of the proposed dual-step aperiodic array design procedure, as described in Sec.",
"."
],
[
"Amplifier Output Power Levels",
"The pre-coding matrix $\\mathsf {W}$ is changed adaptively in time as it depends on $\\mathsf {H}$ , and thus different BS antenna excitation coefficients $\\mathsf {x}$ are obtained at every time instant for each UE.",
"Therefore, to study the statistics of the amplifier output power levels we introduce $\\mathsf {\\mu }&=\\operatorname{\\mathrm {E}}\\left[\\left|\\sum _{k=1}^{K}{\\mathsf {W}_{:,k}}\\right|^2\\right]; &\\mathsf {\\sigma }^2&=\\operatorname{\\mathrm {Var}}\\left[\\left|\\sum _{k=1}^{K}{\\mathsf {W}_{:,k}}\\right|^2\\right]$ where $\\mathsf {\\mu }$ and $\\mathsf {\\sigma }$ are the vectors describing the average power and its variance for each PA (power amplifier), when transmitting to all users at the same time.",
"When transmitting to one user only at a time instead, the above expressions reduce to $\\mathsf {\\mu }_k=\\operatorname{\\mathrm {E}}[|\\mathsf {W}_{:,k}|^2]$ and $\\mathsf {\\sigma }_k^2=\\operatorname{\\mathrm {Var}}[|\\mathsf {W}_{:,k}|^2]$ , however identical distribution are found in both scenarios.",
"We define the power spread as $\\text{PS}=\\max (\\mathsf {\\mu }+\\mathsf {\\sigma }^2)/\\min (\\mathsf {\\mu }-\\mathsf {\\sigma }^2)$ where PS=0dB represents the ideal constant and uniform power level for all PAs, while larger values indicate a stronger variation of the excitations across the array and/or channel realizations.",
"In practice, in the downlink scenario, PAs are deployed at each of the BS antenna ports to provide the desired radiated power.",
"Solid state PAs are designed for a fixed maximum output power and typically operate most efficiently at saturation [10].",
"It is thus desirable to have all power amplifiers operating at a uniform power level (thus lower PS values), but this is difficult to reach due to the high degree of adaptivity of MIMO systems."
],
[
"Uplink",
"Similar expressions to (REF ) apply to the uplink scenario, where $\\mathsf {y}\\in \\mathbb {C}^{M\\times 1}$ is the received signal at the BS and $\\mathsf {x} \\in \\mathbb {C}^{K\\times 1}$ is the signal transmitted from the UEs.",
"The decoded signal can instead be expressed as $\\mathsf {\\tilde{y}}=\\mathsf {D}\\mathsf {y}$ , where $\\mathsf {D}$ is the decoding matrix.",
"The $\\text{SINR}$ expression (REF ) is modified by the asymmetry of the link, however the SR expression (REF ) is the same as the one for the uplink.",
"We refer, e.g., to [11] for a more complete formulation."
],
[
"Design Methodology",
"In the following we propose a novel design approach to the synthesis of aperiodic arrays tailored to the MU-MIMO type scenario."
],
[
"Aperiodic array design",
"The newly proposed synthesis method is based on a combined statistical analysis and a density taper approach, where the knowledge of the statistical distribution of the excitations is used to determine the optimal array layout, see Fig.",
"REF .",
"Firstly, we densely sample the desired aperture with a regular array of Huygens sources to represent a generic aperture field distribution.",
"Then, we simulate the dense array in the desired propagation environment, as described in Section REF , and compute the resulting average powers $\\mathsf {\\mu }$ [see (REF )] of these array elements.",
"This will form our reference power distribution.",
"Subsequently, the aperiodic layout is synthesized through the density taper approach [5], i.e., elements are located with a density proportional to the reference power distribution.",
"Mathematically, the antenna positions are obtained as $x_m=i((m-1)\\Delta I)^{-1} \\quad m=1,\\ldots M$ where $i(x)=\\int _0^x {\\mu (\\tau ) \\text{d}\\tau }$ is the auxiliary cumulative distribution derived from $\\mu (x):[0,X_\\text{max}]$ , $\\Delta I=i(X_\\text{max})/(M-1)$ is its equipartition and $i^{-1}$ denotes the inverse operation.",
"As shown in Fig.",
"REF , starting from the reference distribution $\\mu (x)$ , the antenna positions are easily found as the intersection points between $i(x)$ and its equipartitions."
],
[
"Aperiodic array gains",
"For the assessment of the performance of the aperiodic array, all results are presented in comparison with the respective uniform array, having the same aperture, antenna type and total number of antenna elements.",
"Besides individual performance curves, relative gain curves are plotted to indicate the improvement of the aperiodic over the regular array.",
"For instance, in terms of the link quality, the SINR Gain (SINRG) is introduced, $\\text{SINRG}=[\\text{SINR}^\\text{aperiodic}/\\text{SINR}^\\text{regular}]^{\\text{SNR}=0\\text{dB}}_{\\text{CDF=5\\%}}$ as the SINR difference between the two arrays for an SNR of 0dB evaluated at the 5% level of the CDF (i.e.",
"for 95% of the users).",
"The same SINRG is obtained for each of the UE streams due to the random nature of the scenario.",
"Similarly, regarding the amplifier output powers, the Power Spread Compression (PSC) is defined as the power spread difference between the two arrays, i.e., $\\text{PSC}=[\\text{PS}^\\text{regular}/\\text{PS}^\\text{aperiodic}].$ Both the SINRG and the PSC are positive (in dB) when the aperiodic array outperforms the regular one."
],
[
"Results",
"In the following we discuss: 1) the effect of the propagation environment, 2) the uplink/downlink duality, 3) the impact of the aperiodic layout on the SINR and the SR, as well as 4) on the amplifier output powers.",
"Different systems are compared, whose sizes range from $8\\times 2$ up to $16\\times 8$ ($M\\times K$ ), and where the solid and dashed curves are for the regular and aperiodic array case, respectively.",
"In all cases the array aperture is $(M-1)\\lambda $ .",
"The SINR CDF, Eq.",
"(REF ), and the SR, Eq.",
"(REF ), are used to discuss the link reliability and the capacity, respectively.",
"Finally, in 5), large scale massive MIMO systems are examined in terms of the relative gains as defined in Sec.",
"REF ."
],
[
"LOS- to RIMP-dominated environments",
"We study scattering effects by changing the number of plane waves associated with each UE (see Sec.",
"REF ).",
"Fig.",
"REF compares the SINR CDFs for an $8\\times 2$ aperiodic and regular array, when moving from an RLOS- to RIMP-dominated environment.",
"Note that in all SINR CDF plots, rightmost and steepest curves are preferred since then higher SINR values are more probable.",
"For RLOS (1 random wave per UE), the aperiodic array offers the largest gain (SINRG=3dB), while it progressively reduces for increasing scattering until the two curves overlap for the RIMP environment (10-20 random waves per UE); Thus the aperiodic array always exhibits superior or identical SINRG performance.",
"Evidently, the RLOS environment is the most favorable propagation condition for aperiodic arrays and therefore considered in the remainder of the paper."
],
[
"Uplink and Downlink",
"Similar conclusions are valid for downlink and uplink when it comes to the SINR gain.",
"To show this, the CDFs of the aperiodic and regular arrays in the two scenarios are plotted in Fig.",
"REF .",
"Although the distributions are different due to the different SINR expressions, the aperiodic SINR gain is evidently present and of comparable value in both cases.",
"Regarding the antenna excitations, identical power profiles $\\mu (x)$ are found, which is due to the pre- and de-coding matrices, one being the Hermitian of the other.",
"However, the efficiencies of the low-noise amplifiers in the receive mode are not as important as those for the PAs in the transmit case.",
"Note that, in the uplink scenario: (i) the CDF shows directly the required transmitted power for a minimum SINR to a certain user percentile, and; (ii), the SINR CDF curves are linearly dependent on the SNR [12].",
"We thus choose to show the rest of the SINR results only for the uplink scenario and for one SNR only."
],
[
"SINR and Capacity",
"As shown above, aperiodic arrays can improve the SINR ratio, and thus the capacity too.",
"In Fig.",
"REF the SINR CDFs for the aperiodic and regular arrays of different system sizes are compared.",
"Starting from the $8\\times 2$ system (SINRG=3dB), if we double the number of BS antennas, but keep the number of UEs the same, the SINR CDF improves (i.e.",
"moves to the right) as expected.",
"However, the SINRG decreases to 1dB.",
"If, on the other hand, the number of UEs is also doubled, and thus the ratio of the number of BS-to-UE antennas is kept constant, not only the CDFs improve but also the SINRG increases to 3.5dB.",
"Finally, if the UEs are further doubled, the SINRG exceeds 10dB: note how the $16\\times 8$ aperiodic array provides approximately the same per-user SINR of the $8\\times 2$ array, where the regular array case would instead lose 10dB.",
"Fig.",
"REF shows the SR capacities corresponding to the same array configurations as shown in Fig.",
"REF .",
"The system capacity increases with both the number of BS antennas and the number of UE, with the $16\\times 8$ aperiodic array having a 12% rate increase over the regular.",
"That is, both the user's 5-percentile SINR improves (which condition the link budget), as well as the capacity in more crowded scenarios."
],
[
"Amplifier power",
"Fig.",
"REF shows the normalized average antenna output powers of the regular and aperiodic arrays that were considered above.",
"For the regular array, increasing the BS antennas and the UE-to-BS ratio exacerbates the power unbalance between the edge and central elements.",
"The aperiodic array, on the other hand, exhibits a more uniform average port power among the elements: the $16\\times 8$ aperiodic array has a 2.8dB tapering reduction.",
"A similar trend is observed for the variance in antenna port powers as shown in Fig.",
"REF .",
"Here too, the highest variance is associated with the edge elements, irrespective of the system size.",
"The aperiodic array reduces the variance, albeit to a lesser extent for the central elements; hence, aperiodicity is mostly beneficial in ensuring a uniform average power allocation."
],
[
"Massive MU-MIMO",
"The link quality (SINR) and the power spread of PAs are strongly dependent on the system size.",
"It is therefore relevant to study the impact of the massive MIMO architecture on the SINRG and the PSC figures-of-merit.",
"Fig.",
"REF shows the SINRG as a function of the number of BS antennas and cell crowdedness, i.e., the number of UEs as a percentage of the BS antennas ($K/M*100$ ).",
"In Fig.",
"REF the PSC is shown as well.",
"In general, the SINRG can be very substantial, particularly in crowded cells and for a large number of BS antennas.",
"The PSC, on the other hand, demonstrates more moderate gains.",
"As an example: a moderately large aperiodic system of 64 BS antennas would experience an SINR increase of 3dB at 10% UEs, and more than 15dB at 30%, while having a PSC between 1 to 3dB.",
"In this manuscript we have investigated the advantages of aperiodic arrays for MU-MIMO applications.",
"We have introduced a simple aperiodic design method based on a hybrid statistical-density tapering approach.",
"We have then considered the effects on: (i) the link performance, and; (ii) the amplifier power spread with respect to classical regular arrays.",
"Results show that aperiodic MU-MIMO arrays provide the largest gains in line-of-sight dominated environments.",
"This reduces for increasing degree of scattering, however they are never inferior to the regular arrays.",
"This can be concluded in both the up and downlink case.",
"Aperiodic arrays are shown to be beneficial to the link performance, both to the users' 5 percentile SINR as well as the sum rate capacity.",
"Moreover, the aperiodic layout improves the amplifier´s efficiency owing to a more uniform average power among the antenna's power amplifiers.",
"Even for a relatively small $16\\times 8$ MU-MIMO system, it is possible to achieve a 10dB power budget improvement, a 12% capacity increase and a 3dB amplifier tapering reduction.",
"Finally, it is shown that larger and more crowded MU-MIMO systems benefit most by the array aperiodicity.",
"Results show that aperiodic arrays can provide a substantial gain in the link quality and capacity, especially in scenarios with large interference."
]
] | 1808.08321 |
[
[
"Swift spectra of AT2018cow: A White Dwarf Tidal Disruption Event?"
],
[
"Abstract The bright transient AT2018cow has been unlike any other known type of transient.",
"Its high brightness, rapid rise and decay and initially nearly featureless spectrum are unprecedented and difficult to explain using models for similar burst sources.",
"We present evidence for faint gamma-ray emission continuing for at least 8 days, and featureless spectra in the ultraviolet bands -- both unusual for eruptive sources.",
"The X-ray variability of the source has a burst-like character.",
"The UV-optical spectrum does not show any CNO line but is well described by a blackbody.",
"We demonstrate that a model invoking the tidal disruption of a 0.1 - 0.4 Msun Helium White Dwarf (WD) by a 100,000 to one million solar mass Black Hole (BH) located in the outskirts of galaxy Z~137-068 could provide an explanation for most of the characteristics shown in the multi-wavelength observations.",
"A blackbody-like emission is emitted from an opaque photosphere, formed by the debris of the WD disruption.",
"Broad features showing up in the optical/infrared spectra in the early stage are probably velocity broadened lines produced in a transient high-velocity outward moving cocoon.",
"The asymmetric optical/infrared lines that appeared at a later stage are emission from an atmospheric layer when it detached from thermal equilibrium with the photosphere, which undergoes more rapid cooling.",
"The photosphere shrinks when its temperature drops, and the subsequent infall of the atmosphere produced asymmetric line profiles.",
"Additionally, a non-thermal jet might be present, emitting X-rays in the 10-150 keV band."
],
[
"Introduction",
"The transient AT2018cow/ATLAS18qqn/SN2018cow was discovered at an offset of 6(1.7 kpc) from galaxy Z 137-068 [88] by the ATLAS wide-field survey [94] on 2018-06-16 10:35:38 UT (MJD 58285.44141, referred to in this paper as the discovery date $T_{\\rm d}$ ) at an AB magnitude $o$ = ${14.74\\pm 0.10}$ mag (the $o$ -band covers $560-820\\;{\\rm nm}$ ,http://www.fallingstar.com/specifications).",
"A previous observation by [30] on MJD 58282.172 (3.3 d before the discovery date) with the Palomar 48-inch in the $i$ -band did not detect a source down to a limiting magnitude of $ i > 19.5$ mag, while on MJD 58286 ($T_{\\rm d}$ +0.75 d) $i = 14.32\\pm 0.01$ mag, nearly 5 magnitudes brighter - a rapid rise.",
"Maximum light occurred at MJD 58286.9 [76]).",
"Spectroscopic follow-up by [72], and [74] using the SPRAT on the Liverpool Telescope (402-800 nm, with 2 nm resolution) on MJD 58287.951 ($T_{\\rm d}$ +1.56 d) found a smooth spectrum.",
"[47] reported the Ca II H and K absorption lines close to the redshift of the co-located galaxy, proving that the transient was near that galaxy.",
"Spectra taken on the Xinglong 2.16-m Telescope using the BFOSC showed weak broad bumps or dips in the spectrum [98], [45] which may be interpreted as highly velocity-broadened lines though Perley et at (2018a) considered the features as an absorption trough.",
"The velocity derived from the broadening of the presumably He emission was $\\approx 1.6\\times 10^4\\,{\\rm km\\,s}^{-1}$ on [76].",
"Intrinsic optical polarization was measured on days $T_{\\rm d}$ +4.9 and 5.9 d by [90].",
"At high energies the transient was detected by the Neil Gehrels Swift Observatory [32] XRT [14] in the $0.3-10$ keV band [81], NICER [69], NuSTAR [64], and INTEGRAL IBIS/SGRI [28].",
"A search for impulsive emission by Fermi/GBM [22], Fermi/LAT [50], the INSIGHT HXMT/HE [44] and Astrosat CZTI [87] was unsuccessful.",
"In the radio a search of pre-outburst data by [25] found $3\\sigma $ upper limits of $370\\,\\mu $ Jy at 3 GHz and $410\\,\\mu $ Jy at 1.4 GHz.",
"The transient was detected at 90 and 150 GHz on $T_{\\rm d}$ +4.5 d with a flux density of $\\approx $ 6 mJy at 90 GHz [103], at 350 GHz on day 5.8 with flux density of $30.2\\pm 1.8$ mJy/beam [89] and with a $5\\sigma $ detection at 15.5 GHz of 0.5 mJy on $T_{\\rm d}$ +6.3 d [10].",
"Further detections were reported on days $T_{\\rm d}$ +10 and 11 d at 9 GHz and 34 GHz, and on $T_{\\rm d}$ +12 also at 5.5 GHZ [23], [24].",
"We will adopt a distance to the transient consistent with it being associated with the nearby galaxy Z 137-068, which has a red-shift $z = 0.01414\\pm 0.00013$NED refcode 2007SDSS6.C...0000 The reddening towards the galaxy is low, $E(B-V) = 0.077$ [86] and $N_{\\rm H}$$_{\\rm galactic} = 6.57\\times 10^{20}\\,{\\rm cm}^{-2}$ [97].",
"Adopting cosmological parameters $H_0 = 71.0\\;{\\rm km~s}^{-1}{\\rm Mpc}^{-1}$ , $\\Omega _{\\rm m}=0.27$ , $\\Omega _v=0.73$ , [46] the distance is $60\\pm 4$ Mpcusing NED/IPAC.",
"During our studies and preparation of this paper three other studies were published in preprint form [76], [81], [73], and we discuss and use their results in our discussion of the nature of the transient whilst extending their analysis.",
"As in this paper, [73] proposed that the transient could be a TDE and they discussed constraints on the TDE properties using recent models.",
"In two further papers a more general analysis was made in terms of a central engine [41], [63], leaving open the nature of the source.",
"We discuss our observations, and present a model derived from the observations in terms of the tidal disruption of a He white dwarf by a non-stellar mass black hole, i.e.",
"a TDE-WD event, where the debris forms a photosphere which produces blackbody-like emission in the UV-optical bands and with emission lines formed above the photosphere.",
"Moreover, a rapidly expanding cocoon has become detached from the photosphere and envelops the system initially.",
"It produces very broad emission features attributed to velocity broadened lines, i.e., the bumps seen by [76] and [73].",
"Finally, a jet is associated with the event, responsible for the high-energy $\\gamma $ -ray and X-ray emission."
],
[
"Observations",
"Swift started pointed observations of AT2018cow with all three instruments: the Burst Alert Telescope [3]BAT has earlier coverage of AT2018cow from its non-pointed survey data, the X-Ray Telescope [14] and the UltraViolet & Optical Telescope [82], on MJD 58288.44 which was 3.0 d after the first detection, and continued with an intensive observing schedule over the following 2 months.",
"Unless said otherwise, the Swift data were reduced using HEAsoft-6.22 (XRT), 6.24 (UVOT) and the latest Swift CALDB or, for the UVOT grism data, with the uvotpy calibration and software [54].",
"The XRT spectra were obtained using the online XRT product generator at the UK Swift Science Data Centre [27].",
"We used the Galactic absorption N$_H$ from [97].",
"Figure: BAT-XRT-UVOT light curve.The UVOT magnitudes are given in six filters uvw2,uvm2,uvw1,u,buvw2, uvm2, uvw1, u, b, and vv starting at T d T_{\\rm d} and have been corrected for the galaxy background and have been binned to increase S/N.The BAT survey data panel includes the NuSTAR data projected into the BAT band, as well as the BAT survey quality processed data for 8-day periods.During the first 8-day period significant detections occur, thereafter the BAT count rate is consistent with no detection.The flaring seen in the XRT possibly lines up with an increase in the BAT flux prior to day T d T_{\\rm d}+8."
],
[
"The UV-optical light curve, and SED",
"The UVOT images were inspected for anomalies, like drift during the exposure.",
"Photometry was obtained using the standard HEAsoft-6.24 tools followed by a check that the source did not fall on one of the patches of reduced sensitivity; several observations had to be discarded.",
"A 3 aperture was used throughout; the standard aperture correction has been used as described in [75]; and the filter effective areas and zeropoints were from [8].",
"For the fitting of the photometry in Xspec the UVOT filter response curves were used so that the fits take the interplay between filter transmission and spectrum into account.",
"The galaxy emission in the aperture of 3 radius was determined from a UVOT observation at day 120, in order to correct the UVOT photometry for the host contribution.",
"The galaxy background was measured in all 6 UV/optical filters, but the transient was still judged to dominate the UVOT emission in the UV.",
"The UV background from the galaxy was estimated by determining that from the GALEX FUV and NUV filters using the gPhoton database [70].",
"There is a blue source near the transient which falls within the 3 aperture.",
"The GALEX PSF is larger than UVOT, so the bright source and contribution from the bulge of the galaxy will lead to an overestimate of the flux in the 3 radius aperture used.",
"An SED was built for the galaxy emission component and folded through the UVOT effective area curves to derive the following galaxy flux and magnitude within the aperture: $uvw2$ =20.46, $uvm2$ =20.37, $uvw1$ =19.70 (AB).",
"The host galaxy values from UVOT data taken at day 120 were $u$ =19.14, $b$ =18.50, and $v$ =17.92 (AB) to an accuracy of 0.08 mag.",
"The galaxy emission becomes important first in the UVOT $v$ band around day 12, and later in the bluer bands.",
"The UVOT photometry corrected for the galaxy background as described above can be found in Table REF .",
"The UVOT light curves show a chromatic decline, where the $uvw2$ , and $uvm2$ fall off slower than the optical $u$ , $b$ and $v$ bands, see Fig.",
"REF .",
"The first report that the optical-IR spectra resembled a blackbody was by [21] who reported a temperature of $9200\\pm 600$ K on day $T_{\\rm d}$ +1.7 d. A more detailed fit was made by [76] who discuss the UV-optical/infrared data from the first 17 d after discovery.",
"They also showed that the data can be fit well with a blackbody spectrum, with luminosity changing over an order of magnitude, blackbody temperature changing from 28,000 K to 14,000 K and a nearly constant radius of the photosphere at $5\\times 10^{14}\\,{\\rm cm}$ .",
"[73] subsequently modeled ground based photometry and spectra, and reported a slow but steady decline of the photospheric radius, but they also needed a power law component to fit excess emission in the infrared.",
"The IR excess emission could be synchrotron emission from non-thermal energetic electrons present in an optically thin coronal atmosphere above a dense photosphere.",
"Perley at al.",
"(2018a) noted that such IR synchrotron emitting electrons could produce radio synchrotron emission at a level consistent with the observation.",
"Analyses by Ho et al.",
"(2018) showed further support to the scenario that the IR excess emission and the radio emission are of the same origin.",
"Using Xspec we fit a blackbody model to our corrected UVOT photometry.",
"The results are given in Table REF and Fig.",
"REF .",
"Our analysis is confined to the well-calibrated UVOT data, with bad data removed, taking account of the filter throughput with wavelength, and correcting for the measured galaxy background.",
"However, the $v$ magnitudes are generally too bright due to the extra red power law emission component, and lead to a poor reduced $\\chi ^2$ .",
"We see a varied evolution during the first 13 d, see Fig.",
"REF , followed by a steady decline in radius.",
"Table: Results of the black-body model fits to the UVOT photometry (1700 - 6800 Å)Figure: Black-body model fit to the UVOT data in a log-log plot.",
"Note that the luminosity decays approximately as a power law with different slopes before and after day T d T_{\\rm d}+6.5.",
"After day T d T_{\\rm d}+44 the galaxy background emission in vv and bb do no longer allow a good fit to be made."
],
[
"The UVOT Spectra",
"Daily exposures in the UVOT grisms were obtained from $T_{\\rm d}$ +5 d onward, first in the UV grism ($170-430$ nm) until day $T_{\\rm d}$ +23.4 d; on day $T_{\\rm d}$ +24, 25, 30 and 34 d we obtained exposures in the more sensitive V grism ($270-620$ nm).",
"The UVOT grism images were closely examined for contamination by background sources using the summed UVOT UV filter images as well as by comparison of the position of the spectrum to zeroth orders from sources in images from the Digital Sky Survey.",
"The affected parts of the spectrum were removed from consideration.",
"To improve S/N, spectra were extracted with a narrow slit measuring 1.3 times the FWHM of a fitted Gaussian across the dispersion direction, and the extracted spectra taken close in time were averaged together.",
"A standard correction to the flux from the narrow slit was made to scale it to the calibrated response and a correction was made to account for the coincidence loss in the detector.",
"The resulting spectra (see Fig.",
"REF ) show little evidence for emission lines like in, for example, novae, nor the characteristic UV absorption features due to blended lines of singly ionised metals as seen in SNe.",
"Figure: The dereddened UVOT grism spectra, using E(B-V)=0.077E(B-V) = 0.077 and the law with R V =3.1R_V = 3.1.The 1σ\\sigma errors are indicated with shading.The features below 2200 Å are due to noise.Black-body fits are included, as well as the galaxy brightness in the 3 radius aperture used for photometry.Excess flux on day T d T_{\\rm d}+13 and 18.5 longer than 3000 Å is due to order overlap.Our first summed spectra from around day 6 are well-exposed, yet relatively featureless, just like the optical spectra in [76].",
"The dereddening straightened the bump in the observed spectra around 2175 Å which means that there is no evidence of dust intrinsic to the environment of the transient.",
"The June 25, day $T_{\\rm d}$ +9, spectrum shows features near 1910Å, which are probably due to a noise problem, since this feature would likely have been seen as second order emission.",
"Therefore the features are probably unrelated to the broad 4850 Å absorption (or broad emission around 5000 Å) feature which is seen to emerge in the ground-based spectra from [73] on day $9-13$ .",
"Our UV-grism spectra from June 27.5 to July 1.6 were combined, to give a spectrum for day 13 (June $29.5\\pm 2$ d).",
"In this spectrum a weak emission feature near the He II 2511 Å line is seen, as well as a broad feature near 2710 Å which could be due to He I or He II.",
"At wavelengths longer than 3000 Å second order overlap contamination is present.",
"We should note that the second order lines of N III] and C III] would be seen if there were any, and their absence in second order confirms that there is no line in the first order.",
"The UV spectrum from day 18.5 is contaminated by second order emission overlap for wavelengths longer than about 3300 Å, while below 2400 Å noise starts to dominate [54].",
"The dip near 2200 Å is likely due to noise.",
"At day $T_{\\rm d}$ +34 we got V-grism exposures which were of low signal to noise.",
"Usually, the UVOT spectra are extracted from each individual exposure and then the wavelength reference is corrected to match the spectra before summing.",
"However, the latter is not possible if the spectrum in each exposure is too weak.",
"An alternative is possible for exposures taken with the same spacecraft roll angle.",
"To get a better S/N for the day $T_{\\rm d}$ +34 spectrum, we cross-correlated the images and then summed the grism images of the spectrum, followed by a standard extraction, again using the narrow extraction slit [53].",
"The spectrum of day 34, which covers the range of 2820-5600 Å, shows undulations which resemble those shown for the longer wavelengths in the spectra from [73].",
"Below 3800 Å, which was not covered in the ground-based spectra, we see no evidence for strong emission lines from other elements, i.e., the Mg II 2800 Å emission line which is often found in late type stellar spectra is not seen; nor is there any sign of the O III 3134 Å line which is pumped by He II 304 Å nor of He II 3204.",
"Using the BB-fits to our photometry, we plot those over the UVOT spectra in Fig.",
"REF .",
"Whereas the UVOT spectra become quite noisy after day $T_{\\rm d}$ +20 and do not show any lines above the noise, it is of interest to mention that the He I spectral lines that emerge after $\\sim $ day 22 in the ground-based spectra from [73] show large asymmetries.",
"The blue wing is seen to be largely missing in the stronger unblended lines.",
"After the lines become visible above the continuum, the line emission at first peaks at a 3000 km s$^{-1}$ redshift, while moving to lower redshifts until the peak is at the rest wavelength at day 34.",
"We think those asymmetric line profiles are important for understanding the transient.",
"Figure: A log-log plot of the XRT light curve indicates that the evolution of the X-ray emission follows a broken power-law with a break at day T d T_{\\rm d}+21.Three fits are shown: fitting all data with a single PL (black dashed line), or with a broken PL (black line) as well as a fit to the data excluding obvious flaring points (blue line).The flaring points which are shown here in red."
],
[
"XRT analysis",
"The X-ray light curve shows a regular pattern of brightenings which have already been remarked upon by [81].",
"These rebrightenings follow a trend, either as flares above a certain base-level, or as the main constituent.",
"A simple PL fits to all data has $\\alpha = -1.43\\pm 0.08$ for a $\\chi ^2$ /d.o.f.",
"= 1316.1/87.",
"Alternatively, fitting all the data including the flares with a broken PL has a slope $\\alpha _1$ = $0.85\\pm 0.11$ , with a break in its slope at day $t_{\\rm break}$ = $T_{\\rm d}$$+24.8 \\pm 1.6$ , followed by a steeper decay with a slope $\\alpha _2$ = $2.90\\pm 0.35$ , and $\\chi ^2$ /d.o.f.",
"= 803.7/85.",
"We also made a fit trying to find a trend underlying the flares.",
"A base level can be defined by masking the points in a flare, fitting a broken power-law (PL) trend, and iteratively removing points that are too far from the trend by eye (see Fig.",
"REF ).",
"The best fit found is a broken power law with a slope $\\alpha _1$ = $0.89^{+0.22}_{-0.23}$ , which has a break in its slope at day $t_{\\rm break}$ = $T_{\\rm d}$ +20.3$^{+2.7}_{-3.8}$ , followed by a steeper decay with a slope $\\alpha _2$ = $2.65^{+0.57}_{-0.43}$ , and $\\chi ^2$ /d.o.f.",
"= 89.3/50.",
"In Fig.REF one can clearly see that the single power law does not fit as well at the beginning and end.",
"The same trend is thus present regardless of the removal of the flares.",
"Although the flares seem to be on top of a smoothly varying component, this cannot be determined for sure since the whole overall emission is decreasing over time, and it could just as well be continuous flaring which shows evolution.",
"We will investigate the temporal behaviour of the flares further in section REF ."
],
[
"BAT analysis",
"BAT is a coded aperture imaging instrument (Barthelmy et al.",
"2005).",
"A sky image can be constructed by deconcolving the detector plane image with the BAT mask aperture map [65].",
"We performed a special analysis of the BAT data.",
"This analysis utilizes the BAT survey data from June 1st to July 18th, 2018 (i.e., the available HEASARC data at the time of the analysis).",
"Even when the BAT has not been triggered by a GRB, it collects continuous survey data with time bins of $\\sim 300$ s [65].",
"The signal to noise ratios reported here are calculated using the source count and background variation estimated from sky images with different exposure time .",
"Note that due to the nature of the deconvolution technique, the resulting noise (background variation) is Gaussian instead of Poissonian.",
"These sky images are mosaic images created by adding up all the snapshot observations within the desired durations (i.e., two 8-day intervals and one 17-day period).",
"The mosaic technique adopted here is the same one that is used to create the BAT survey catalogs [95], [4], [71].",
"This analysis pipeline carefully takes care of many instrumental effects, such as potential contamination from bright sources, systematic noise introduced by differences between each detector (so-called “pattern noise”), and corrections for sources with a different partial coding fraction when creating a mosaic image from individual snapshot observations.",
"The analysis produces results in the following eight energy bands: $14-20$ keV, $20-24$ keV, $24-35$ keV, $35-50$ keV, $50-75$ keV, $75-100$ keV, $100-150$ keV, and $150-195$ keV.",
"Figure REF shows the daily BAT mask-weighted light curve in $14-195$ keV.",
"Note that we exclude data collected from June 3rd to June 13th, 2018, during which the BAT underwent maintenance and recovery activities and the calibration of survey data is uncertain.",
"A spectrum was created for the 17-d period starting June 16 (day 0) to July 2, 2018, as well as for two 8-day periods of June 16 $-$ June 23, 2018, and June 23 - July 1, 2018.",
"For the 17-d period the S/N was 3.3, and for the two 8-d periods, 3.95 and 1.31 respectively.",
"For new sources the BAT detection limit is a higher level of S/N of 5, so these $3-4\\;\\!\\sigma $ detections are marginal detections that do not stand on their own."
],
[
"The high energy spectra",
"The Swift XRT data up to day 27 have been discussed in [81] who fit an absorbed power law to the data.",
"Their spectral fits did not show any evidence for spectral evolution in the $0.3-10$ keV band, and no evidence for spectral evolution during the flares.",
"The latter is also consistent with no changes being seen in the hardness ratio.",
"Their estimate for the peak X-ray luminosity is $10^{43}\\,{\\rm erg\\,s}^{-1}$ .",
"We addressed the UVOT data in detail in section REF but here we want to address the question of whether the origin of the X-ray emission is related to the UV emission.",
"The emission in the UV-optical is well fitted by a hot thermal blackbody (BB) which we interpreted as optically thick emission from an ionised He sphere surrounding the source.",
"The optical luminosity of $\\sim 2 \\times 10^{44}$ erg s$^{-1}$ [76] is larger than the X-ray luminosity [81], and Compton scattering on the electrons in the atmosphere may produce an X-ray spectrum, so we investigate if that would be large enough to explain the observed X-ray luminosity.",
"We used the XSPEC tool to model the UVOT photometry together with the XRT data.",
"We used the data on day $T_{\\rm d}$ +21 (using the data from UVOT: MJD 58306.8; XRT MJD $58306.2\\pm 1.0$ ) to determine if the optically thick Compton scattering of the BB spectrum was consistent with the observed X-rays.",
"Since a simple BB gives a reasonable fit, the optical depth of the scattering atmosphere was set to one.",
"Fitting the combined UV-optical and X-ray spectral data with an optically thick ($\\tau \\approx 1$ ) Compton spectrum (compbb*zphabs*redden*phabs) fails to get a reasonable fit for the X-ray data which are underestimated: its $\\chi ^2 = 145.69\\,(64\\,{\\rm d.o.f.",
"})$ .",
"This suggests that the X-rays are not due to the same source as the blackbody emission.",
"A much better fit is obtained using a model of optically thick Compton scattered BB plus a power law (PL), (compbb + powerlaw)*redden*phabs*zphabs with $\\chi ^2 = 54.0\\,(61\\,{\\rm d.o.f.",
"})$ and which gives results very close to a BB+PL model with similar BB temperatures as in [76] for the low-energy part of the spectrum.",
"The photon index $\\beta $ of the day $T_{\\rm d}$ +21 unabsorbed X-ray spectrum is $\\beta = 1.57 \\pm 0.07$We parametrise the photon index as $n_{\\nu } \\sim t^{-\\alpha }\\nu ^{-\\beta }$.",
"We fitted a broken power law spectrum to the combined BAT and XRT spectral data in order to determine the high-energy losses.",
"We used the summed BAT data for the first 8-day interval of $T_{\\rm d}$ +0 to 8 d, and for XRT day $T_{\\rm d}$ +3 to 8 d. Assuming that this represents a detection, we obtain for the broken power law fit a photon index $\\beta _1 = 1.792 \\pm 0.065$ , a break at $6.57\\pm 1.64$ keV, and thereafter $\\beta _2 = 0.65\\pm 0.13$ , with a fixed $N_{\\rm H,galactic} = 6.57\\times 10^{20}\\,{\\rm cm}^{-2}$ , and $N_{\\rm H,intrinsic} = (1.9 \\pm 1.5)\\times 10^{20} {\\rm cm}^{-2}$ .",
"The model predicts a luminosity ratio between the BAT and XRT bands of $L_{\\rm x}(10-200\\,{\\rm keV})/L_{\\rm x}(0.3-10\\,{\\rm keV}) = 29.8$ , while the goodness of fit $\\chi ^2 = 82.4\\,(90 {\\rm d.o.f.",
"})$ .",
"Figure: The combined XRT and NuSTAR spectra fit.",
"The first NuSTAR observation (blue) and the 2nd NuSTAR observation (black) are shown with their XRT counterparts (green and red respectively).In addition to Swift observations, NuSTAR [38] observed AT2018cow on four occassions during the period reported upon here.",
"NuSTAR observes in an energy range of 3-79 keV, overlapping the energy range of both XRT and BAT.",
"As NuSTAR has a greater sensitivity than BAT, it allows us to validate the quality of our XRT + BAT spectral fit.",
"We analysed the NuSTAR data utilizing the standard extraction methods, utilizing the HEAsoft nupipeline and nuproducts tools, and in cases where XRT and NuSTAR observed simultaneously, we include both data in the fit.",
"The first NuSTAR observation occurred on MJD 58292.7, $\\sim 7.3$ days after discovery.",
"By simultaneously fitting XRT and NuSTAR data we find that, similar to the previously reported BAT + XRT fit, it requires a broken power-law model to fit the data.",
"This fit gives $\\beta _1 = 1.67 \\pm 0.04$ , with a spectral break at $12.7 \\pm 0.9$ keV, followed by $\\beta _2 = 0.50 \\pm 0.10$ .",
"This model adequately ($\\chi ^2$ /d.o.f.",
"= 572.6/511) describes the data, see Fig.",
"REF .",
"We note that although the spectral indices are similar to the BAT + XRT fit, the energy at which the spectral break occurs is higher, and importantly, outside of the XRT 0.3 - 10 keV energy range.",
"It is not clear if this difference is instrumental or due to the different time periods over which the spectra were collected.",
"Utilizing this fit we derive a luminosity ratio between the BAT and XRT bands of $L_x$ (10-200 keV)/$L_x$ (0.3-10 keV) = 18.6.",
"Further NuSTAR spectra taken at $T_{\\rm d}$ +16.2, 27.8 and 36.15 show a marked change in the spectral shape, as the hard component disappears and the combined XRT and NuSTAR data for each observation can be well fit by a single power-law model from 0.5 to 79 keV.",
"Based on the spectral softening seen in NuSTAR, it is clear that AT2018cow would not have been detected by BAT after this hard component turned off.",
"However the relatively low statistics of the BAT light-curve means that it is not possible to estimate when this hard component turned off, although we note that after $T_{\\rm d}$ +8 days there are no statistically significant detections of AT2018cow by BAT either in 1-day or 8-day integrated data.",
"These results are consistent with the report of a “hard X-ray bump\" at day 7.7 which disappears by day 17 by [63]."
],
[
"Search for characteristic or periodic time scales",
"In the XRT photon-counting (PC) data there are some flares visible.",
"To investigate whether these were periodic, semi-periodic, or burst-like, we conducted a structure function analysis, following the prescription described in [85].",
"The analysis reveals a weak, not very significant, characteristic time scale around 4 hours for count rates binned on 100 s. A Lomb-Scargle period search was repeated in several ways, initially by using data binned per orbit and binned every 100s.",
"We determined the probability that the power of the periodogram was obtained by chance.",
"We simulated the light curve using the same observation times as the original data, but resampling the count rate with replacement in a Monte Carlo simulation.",
"After $10^5$ iterations the values at $3\\sigma $ and $5\\sigma $ were extracted from the resulting power distribution at each frequency.",
"Periods of about 3.7 d and 90 min are found in the periodograms, above the $5\\sigma $ level for the data binned per orbit and every 100s.",
"The 90 min period is due to the Swift orbit.",
"In order to determine if the long period is robust, we repeated the analysis using detrended data, i.e., by excluding the long term trend using a broken PL fit.",
"The analysis was repeated and found consistent results.",
"However, when we split the detrended XRT data into three equal time segments and repeated the Monte Carlo Lomb-Scargle analysis, we find that while a consistent period is found in the latter two thirds of the data, this period is not present in the first third of the data, suggesting the period is quasi-periodic/temporary and not an inherent property of the system.",
"A Monte-Carlo Lomb-Scargle period search in the $uvw2$ band shows no evidence for periodicity except at the orbital period of the Swift satellite.",
"Finally, we performed a wavelet analysis [29] of the XRT data.",
"We found no periodic signal, see Fig.REF ; instead we see that at certain times there is a burst of activity.",
"To investigate whether the variability is wave- or burst-like we also calculated Pearson's moment coefficient of skewness of the amplitude.",
"We disregarded the first ten data points, which cause a large skew, much larger than the rest of the time series presents because of the initial large drop in brightness.",
"The changes in count rate over the mean trend are significantly skewed with a coefficient of 0.67, which shows that the brightenings are burst-like, not periodic.",
"Figure: Wavelet analysis of the XRT PC data shows no period but bursty behaviour.",
"The Y-axis shows the time-scales (in days) searched for."
],
[
"AT2018cow compared to other transients",
"In Fig.",
"REF we compare Swift UV spectra which shows the early (day 5) spectrum of AT2018cow stands out when compared to a SN Ia (SN2011fe at day 4), or a SN IIp (SN2012aw) at day 5; SN IIp are brighter in the UV than SN Ia.",
"The SN spectra show broad absorptions which are mostly due to singly ionised metals.",
"The recent superluminous SN2017egm at day 24 displays a rather flat spectrum in contrast, but is not as UV-bright as AT2018cow.",
"A comparison to a CO-type and an ONeMg-type nova (V339 Del, V745 Sco, respectively) shows the strong UV emission lines from the expanding novae shell which are typically from enhanced abundances of C, N, O, Ne and Mg.",
"The distance and optical magnitude imply an intrinsic brightness of the transient at maximum (on $T_{\\rm d}$ +1.46 d) , $L_{\\rm bol} \\approx 1.7\\times 10^{44}\\,{\\rm erg\\,s}^{-1}$ , [76] which is large for a SN, and excludes a kilonova type event because it is too bright.",
"Fig.",
"REF compares the absolute magnitudes of AT2018cow in the $uvw1$ and $v$ filters to other UV-bright objects: GRB060218/SN2006aj [16], the shock breakout and subsequent SN 2016gkg [1], the superluminous supernova (SLSN) 2017egm [6], the SLSN or tidal disruption event (TDE) ASASSN-15lh [26], [12], [58], and the TDE ASASSN-14ae [43].",
"Photometry for all of these objects has been uniformly reduced using the Swift Optical Ultraviolet Supernova Archive [11]; with distances and estimated explosion dates taken from the cited papers.",
"Figure: UVOT grism spectra of various transients illustrate the difference with AT2018cow.The high luminosity of AT2018cow strains models for SNe like SN2006aj and SN 2016gkg.",
"Yet, the UV-optical emission is chromatic and presents a thermal spectrum, which is like a SN [62].",
"Fast evolving luminous transients (FELT) which show a rapid rise and fast decay are proposed to arise when a supernova runs into external material thus lighting up a large area all at once [79].",
"The FELT spectra show narrow emission lines from the re-ionised circum-stellar matter.",
"AT2018cow shows broad lines in its spectra [76] and thus is not a FELT.",
"The X-ray flux ($0.3-10$ keV) at $T_{\\rm d}$ +3 d is $10^{42}\\,{\\rm erg\\,s}^{-1} \\le L_{\\rm x} \\le 10^{43}\\,{\\rm erg\\,s}^{-1}$ [80].",
"Though the brightness is similar to a typical GRB, we do not know any GRBs with $\\gamma $ -ray emission in the BAT ($14-195$ keV) continuing for as long as 8 days.",
"However, the power law decay of the X-rays with index $\\alpha _1 = 0.9$ throughout, as well as the PL spectrum with $\\beta = 1.6$ are similar to those associated with an off-axis jet [102].",
"The X-ray emission is therefore possibly due to a magnetic dominated jet of the kind we saw in Swift J1644 [85], but it could also be that there is a more energetic GRB jet that we missed, while just seeing off-axis emission from that jet.",
"The non-thermal emission from a jet may also explain the early detection in the radio [103], [10] and point to a low CSM density.",
"Based on the X-Ray luminosity this could be a GRB; however, the luminosity ratio at day $T_{\\rm d}$ +3 d, $10 \\le L_{\\rm opt}/L_{\\rm x} \\le 100$ , is large; in GRBs 11 hours after the trigger the ratio is less than 10 [5].",
"If it is a GRB, it is not a common type of GRB.",
"In AGN X-ray flares are also commonly seen.",
"However, the bursty flaring seen in the X-rays is closer to those of TDEs, e.g., [52].",
"A comparison to a study of the peak X-ray flux in AGN and TDE also suggests the transient is a TDE [2].",
"As the comparison of the light curves in Fig.",
"REF shows, AT2018cow displays a faster evolution than the SLSN 2017egm, the SLSN or TDE ASASSN-15lh, or the TDE ASASSN-14ae, but its peak luminosity is in the same range.",
"Figure: The UV light curve of AT2018cow compared to those of the other UV-bright objects.Tidal disruption is a possible explanation of the observations.",
"The observation of an emission source of radius $\\approx 10^{15}$ cm as inferred from the blackbody fit (see Table REF ) which was formed within $\\approx 3$ d suggests a rapid, energetic event as would be the case for a tidal disruption by a black hole.",
"The star being disrupted can therefore not be too large.",
"Initially the observations showed only the spectral lines of He which suggest the small star could be a WD.",
"Many WD have a magnetic field, and the formation of an energetic jet can be mediated by the remnants of the magnetic field after the outer parts of the WD have been tidally removed and also block the energy generated by accretion from escaping for at least part of the sphere.",
"We explore in the following the jet and UV-optical debris resulting from the tidal disruption.",
"The power-law behaviour in the X- and $\\gamma $ -rays of the “afterglow” of AT2018cow brings to mind a jet.",
"No high energy detector in orbit (Integral SPI-ACS and IBIS/Veto, Fermi GBM and LAT, INSIGHT HXMT/HE, ASTROSAT CZTI, MAXI GSC, Swift BAT) saw a prompt gamma-ray flash, with upper limits of a few times 10$^{-6}$ erg cm$^{-2}$ s$^{-1}$ for a short 0.1 s bin and of approximately $2 \\times 10^{-7}$ erg cm$^{-2}$ s$^{-1}$ for a 10 s bin size at 10 keV to 100 MeV energies [84], , , , , [92], [59] , so it is not a common GRB jet.",
"The upper limit corresponds to a few times $10^{49}$ erg s$^{-1}$ in the 1 keV to 10 MeV range, much higher than the brightness found later in the optical and X-rays, so there might have been undetected prompt $\\gamma $ -ray emission.",
"We found in section REF that the gamma-ray emission was not very energetic and of long duration.",
"However, the slope of the light curve of $\\alpha \\approx 0.9$ prior to the break in the light curve at day 21, and the slope of the spectrum of $\\beta = 1.6$ are indicative of a synchrotron dominated jet.",
"We therefore investigated if the observed high-energy emission might be used to constrain an association with a GRB-like event.",
"We can assume that the energy in the jet is imparted in the early stages of the disruption, and is likely to be of similar magnitude, or somewhat less, then the energy that propels out the massive debris which reached a radius of $5\\times 10^{14}$ cm in about a day.",
"Using the photospheric density derived in section REF we get a constraint for the energy in the jet and the kinetic energy in the debris ejecta: $E_{\\rm jet} \\le 1.4\\times 10^{50}$ erg.",
"Based on the missing prompt $\\gamma $ -ray emission, if a GRB occurred, it must have been an “off-axis” event: if the opening angle of the ejecta that generate the unseen GRB jet emission is $\\theta _{\\rm j}$ , the observer is placed at an angle $\\theta _{\\rm obs} > \\theta _{\\rm j}$ .",
"This way, the gamma-ray emission of the ultra-relativistic outflow was beamed away from the observer.",
"The afterglow is instead visible because, in this phase, while the Lorentz factor $\\gamma $ of the ejecta is lower than in the prompt emission from the core jet, the observer is within the cone of the beamed emission.",
"Analytical modeling [35] and numerical modeling indicates that, an early decay slope $\\alpha \\simeq 0.9$ like the one observed can be obtained if $\\theta _{\\rm obs} \\simeq 1.25\\;\\!",
"\\theta _{\\rm j}$ .",
"An afterglow seen off-axis is also considerably weaker than that on-axis; the above mentioned models indicate that, compared to the core jet, the flux decreases by a factor $\\psi \\simeq 0.1$ .",
"The hard spectral $\\beta < 2$ index of the X-ray emission can be understood if $\\nu _{\\rm m} < \\nu _{\\rm x} < \\nu _{\\rm c}$ , where $\\nu _{\\rm m}$ and $\\nu _{\\rm c}$ are the synchrotron peak and cooling frequency, respectively.",
"In this configuration, $\\beta = 1 + (p-1)/2$ , where $p$ is the index of the power-law energy distribution of the electrons.",
"In GRB afterglows $p < 3$ is usually seen, hence $\\beta < 2$ .",
"In the following, we will assume $p=2.2$ , i.e.",
"the value of the decay slope given by deriving this parameter from $\\beta $ in the case just described.",
"Given all the conditions above, the flux in the X-ray band $F$ at $10^{18}$ Hz (4.1 keV) is $F & = & 5.1\\times 10^{-14} \\ \\left( \\frac{E_{\\rm kin}}{10^{50}\\;\\!",
"{\\rm erg}} \\right)^{1.3}\\left({\\frac{\\epsilon _{\\rm e}}{0.1} }\\right)^{1.2}\\nonumber \\\\& & \\times \\ \\left({\\frac{\\epsilon _{\\rm B}}{0.01} }\\right)^{0.8}\\left(\\frac{ n }{{\\rm cm}^{-3}}\\right)^{0.5} \\ {\\rm erg\\, cm}^{-2} {\\rm s}^{-1}\\ .\\nonumber $ [101], where $E_{\\rm kin}$ , $\\epsilon _{\\rm e}$ , $\\epsilon _{\\rm B}$ and $n$ are the kinetic energy of the ejecta (assuming isotropy), the fraction of energy in radiating electrons and the fraction of energy in magnetic field, and the number density of protons in the circum-expansion medium.",
"With the measured flux (0.3 - 10 keV) of $7.6 \\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ at $T_{\\rm d}$ +21, and adding the correction factor for off-axis emission $\\psi $ , the expression above gives $\\left(\\frac{\\psi \\, E_{\\rm kin}}{10^{50}\\;\\!",
"{\\rm erg}}\\right)^{1.3}\\left(\\frac{\\epsilon _{\\rm e}}{0.1}\\right)^{1.2}\\left(\\frac{\\epsilon _{\\rm B}}{0.01}\\right)^{0.8}\\left(\\frac{n}{{\\rm cm}^{-3}}\\right)^{0.5} \\simeq 12 \\ ,\\nonumber $ which can be easily satisfied for typical values of parameters found in GRB afterglows [83].",
"A consequence of collimated ejecta is that the observer should, at a given epoch, see a break simultaneously in the X-ray and in the optical band light curves.",
"As the Lorentz factor of the ejecta decreases, more and more emitting surface becomes visible to the observer; however, when $\\gamma \\simeq (\\theta _{\\rm obs} + 2 \\theta _{\\rm j})^{-1}$ , no further emitting surface can be seen by the observer [104].",
"As a result, a steepening - “jet break” - of the afterglow light-curve ensues.",
"Jet breaks should be a-chromatic, as in the case of AT2018cow.",
"They are detected in GRB afterglows [77], [96].",
"Post jet break decay slopes are $\\alpha \\simeq p$ , which equates to $\\alpha \\simeq 2.2$ in our case.",
"In our light-curves, we do not see such a fast decay up to $\\simeq 21$ d. Modifying eq.",
"19 of [100] to take into account the off-axis position of the observer, we derive a jet opening angle $\\theta _j & = & 0.39 \\left[\\left(\\frac{E_{\\rm kin}}{10^{50}\\;\\!",
"{\\rm erg}} \\right)\\left( \\frac{{\\rm cm}^{-3}}{n} \\right)\\right]^{-1/8}{\\rm rad} \\ , \\nonumber $ which depends only very weakly on the kinetic energy in the ejecta and number density of the circum-expansion media.",
"Correcting $E_{\\rm kin}$ for the jet opening angle we find that sufficient energy is available to the jet as it is less than the kinetic energy in the debris."
],
[
"Emission from a shock running into the circum-system medium",
"For several days after the event a high velocity outflow, at a few percent of the speed of light, is inferred from the smooth broad features in the spectrum [76], [73].",
"The high-energy emission could be due to that initial outflow shocking a pre-existing circum-system medium.",
"However, the source of such a circum-system medium (CSM) is not clear since mass loss from a WD is negligible and an origin from the black hole would require a previous interaction not too far in the past.",
"The observations from [76], [73] show in the optical evidence of the high velocity outflow up to about day 9; we also notice continued evidence for $\\gamma $ -ray emission for the first 8 days, perhaps longer; and the X-ray emission shows a steady decline during that time, though with flares.",
"The brightness of the high-energy emissions is such that a considerable CSM density is needed, however, we can place a limit to the CSM density using that the intrinsic column density from the fits to the X-ray spectra observations $N_{\\rm H} < 10^{20}$ cm$^{-2}$ , that the high velocity expansion is at $3\\times 10^4$ km s$^{-1}$ and that $\\gamma $ -rays take place over at least 8 days, so the CSM density must be lower than $5\\times 10^4$ cm$^{-3}$ [41].",
"Assuming optically thin radiative cooling [39] and a temperature of 7 keV for the shocked gas, the total radiative loss would be less than $2.4\\times 10^{33}$ erg s$^{-1}$ which falls short of the observed X-ray luminosity by several orders of magnitude."
],
[
"Condition for tidal disruption of a low-mass helium white dwarf",
"We propose that AT2018cow is probably caused by a TDE involving a mostly Helium white dwarf (hereafter He WD) or, alternatively, the remnant core of an evolved star, interacting with a black hole.",
"The WD would be of spectral type DA with a thin Hydrogen atmosphere or of type DB, and likely is a field white dwarf.",
"The radius of tidal disruption of a star of radius $R_*$ and mass $M_*$ by a black hole of mass $M_{\\rm bh}$ is roughly given by $R_{\\rm t} & \\approx & R_* \\left( \\frac{M_{\\rm bh}}{M_*} \\right)^{1/3} \\nonumber $ [40], [51], [60] The mass-radius relation of white dwarfs may be expressed as $\\left( \\frac{R_{\\rm wd}}{R_\\odot }\\right)^a \\left(\\frac{M_{\\rm wd}}{M_\\odot } \\right)^b & = & \\xi ~ f(M_{\\rm wd},R_{\\rm wd}) \\ .",
"\\nonumber $ For low-mass He WDs, where general relativistic effects are unimportant, $a = 3$ , $b = 1$ , $\\xi \\approx 2.08 \\times 10^{-6}$ and $f(M_{\\rm wd},R_{\\rm wd}) \\approx 1$ [18].",
"This implies that $R_{\\rm t} & \\approx & R_\\odot ~ \\xi ^{1/3} \\left(\\frac{M_{\\rm bh}}{M_\\odot } \\right)^{1/3} \\left(\\frac{M_\\odot }{M_{\\rm wd}} \\right)^{2/3} \\ .",
"\\nonumber $ The ratio of the radius of tidal disruption $R_{\\rm t}$ to the Schwarzschild radius of the black hole $R_{\\rm s}~ (= 2 GM_{\\rm bh}/c^2)$ is given by $\\frac{R_{\\rm t}}{R_{\\rm s}} & \\approx & \\frac{\\xi ^{1/3}}{2} \\left[ \\frac{R_\\odot c^2}{G M_\\odot } \\right] \\left( \\frac{M_\\odot }{M_{\\rm bh}}\\right)^{2/3} \\left( \\frac{M_\\odot }{M_{\\rm wd}}\\right)^{2/3} \\nonumber \\\\& = & 3.01 \\times 10^3 \\left( \\frac{M_\\odot }{M_{\\rm bh}}\\right)^{2/3} \\left( \\frac{M_\\odot }{M_{\\rm wd}}\\right)^{2/3} \\ .",
"\\nonumber $ For a TDE to occur requires $R_{\\rm t} > R_{\\rm s}$ .",
"Thus, setting a lower limit of $0.1\\;\\!M_\\odot $ to the mass of the He WD immediately constrains the mass of the black hole to be $<1.3 \\times 10^6\\!\\; M_\\odot $ ."
],
[
"Energetic considerations and constraints on the system parameters",
"While setting a lower limit to the WD mass gives the upper limit to the black-hole mass from the TDE criteria, the luminosity produced in the event provides a means to constrain the minimum black hole mass.",
"In order for the model to avoid self-contradiction, the lower limit to the black hole mass as inferred from the observed luminosity must not be larger than the upper limit to the black hole mass derived from the TDE criterion.",
"Thus, the two will serve as independent assessments of the validity of the He WD TDE scenario, Without losing much generality, we consider a spherical accretion of a neutral plasma (the debris of the disrupted star) into the black hole.",
"For a luminosity $L$ generated in the accretion process, a radiative force $F_{\\rm rad}$ will be generated and act on the charged particles in the inflowing plasma.",
"The radiative force acting on an electron is simply $F_{\\rm rad} & = & \\frac{ \\sigma _{\\rm Th}L}{4\\pi r^2 c} \\ , \\nonumber $ where $\\sigma _{\\rm Th}$ is the Thomson cross section and $\\bar{\\nu }$ is the characteristic frequency of the photons.",
"On the other hand, the gravitational force acting onto the plasma per electron is $F_{\\rm g} & = & \\frac{GM_{\\rm bh}(x\\;\\!m_{\\rm b}+m_{\\rm e})}{r^2} \\ , \\nonumber $ where $m_{\\rm b}$ is the mass of the baryons (which are the constituent protons and neutrons of the nuclei).",
"Assuming a pure helium plasma (note though, that there can be a substantial amount of Hydrogen in a DA WD), $x\\approx 2$ .",
"Equating $F_{\\rm rad} = F_{\\rm g}$ gives the critical (Eddington) luminosity $L_{\\rm Edd} & \\approx & 2.56 \\times 10^{38} \\left( \\frac{M_{\\rm bh}}{M_\\odot } \\right) {\\rm erg~s}^{-1} \\ .",
"\\nonumber $ The corresponding Eddington mass accretion rate is given by ${\\dot{m}}_{\\rm Edd} = L_{\\rm Edd}/\\lambda c^2$ , where the efficiency parameter $\\lambda $ is of the order of $\\sim 0.1$ for accreting black holes.",
"Similarly, we may obtain the effective rate of mass accretion, the inflow that powers the radiation, and that may be expressed as ${\\dot{m}}_{\\rm in} = L_{\\rm bol}/\\lambda c^2$ , where $L_{\\rm bol}$ is the bolometric luminosity of the radiation.",
"Critical Eddington accretion and super-Eddington accretion are generally accompanied by a strong radiatively driven mass outflow.",
"Thus, we have ${\\dot{m}}_{\\rm in} = {\\dot{m}}_{\\rm tot} - {\\dot{m}}_{\\rm out}= \\eta ~{\\dot{m}}_{\\rm tot}$ , where $\\eta $ is the fractional amount of inflow material which contributes to the production of radiation and ${\\dot{m}}_{\\rm tot}$ is the total mass that is available for the accretion process in the tidal disruption event It is useful to define a parameter $\\zeta = {{\\dot{m}}_{\\rm tot}}/{{\\dot{m}}_{\\rm Edd}}$ to indicate how much the Eddington accretion limit is violated.",
"With this parameter we may express the bolometric luminosity of the accretion-outflow process as $L_{\\rm bol} & = & L_{\\rm Edd} \\frac{{\\dot{m}}_{\\rm in}}{{\\dot{m}}_{\\rm Edd}} \\nonumber \\\\& = & 2.56 \\times 10^{38}~ \\eta \\;\\!",
"\\zeta \\left( \\frac{M_{\\rm bh}}{M_\\odot } \\right) {\\rm erg~s}^{-1} \\ .",
"\\nonumber $ It follows that $\\left(\\frac{M_{\\rm bh}}{M_\\odot } \\right) & \\approx & \\frac{7.8\\times 10^5}{\\eta \\;\\!",
"\\zeta }\\left(\\frac{ L_{\\rm bol} }{2.0\\times 10^{44}\\;\\!",
"{\\rm erg~s}^{-1}} \\right) \\ .",
"\\nonumber $ for given $\\eta $ and $L_{\\rm bol}$ , $M_{\\rm bh}(\\zeta ) > M_{\\rm bh}(\\zeta _{\\rm max})$ .",
"The maximum degree of violation of the Eddington limit in the subsequent accretion process after the WD disruption, $\\zeta _{\\rm max}$ , sets the lower mass limit of the black hole that is allowed for the TDE.",
"The remaining task now is to determine the parameter $\\zeta $ empirically using the observations."
],
[
"The scattering photosphere",
"The strong radiatively driven outflow in the critical- or super-Eddington regime will inevitably create a dense Thomson/Compton scattering photosphere.",
"For a BH mass $\\le 10^6$ M$_\\odot $ the photosphere likely resides in the outflow [91].",
"Imposing an opacity $\\tau _{\\rm sc} \\approx 1$ for a photosphere gives $n_{\\rm e} & \\approx & \\left( \\sigma _{\\rm Th} r_{\\rm ph}\\right)^{-1} \\nonumber \\\\& = & 3.0 \\times 10^9\\left(\\frac{r_{\\rm ph}}{5.0\\times 10^{14}\\;\\!",
"{\\rm cm}} \\right)^{-1} \\;\\!",
"{\\rm cm}^{-3} \\ , \\nonumber $ where $n_{\\rm e}$ is the mean electron density in the photosphere and $r_{\\rm ph}$ is the characteristic photospheric radius.",
"For a fully ionised helium plasma, the baryon (i.e.",
"proton and neutron) number density $n_{\\rm b} = 2\\;\\!",
"n_{\\rm e}$ .",
"The total mass enclosed in the ionised helium scattering photosphere is $m(< r_{\\rm ph}) & \\sim & \\frac{4\\pi }{3} {r_{\\rm ph}}^3 n_{\\rm b} m_{\\rm b} \\nonumber \\\\& \\approx & 5.2\\times 10^{30} \\left( \\frac{r_{\\rm ph}}{5.0\\times 10^{14}\\;\\!",
"{\\rm cm}} \\right)^2\\;\\!",
"{\\rm g} \\ , \\nonumber \\nonumber $ which is only a small fraction of the mass of the He WD in disruption.",
"The thermal energy contained in the scattering photosphere is roughly given by $E_{\\rm th} (< r_{\\rm ph}) & \\approx & \\frac{4\\pi }{3}\\;\\!",
"{r_{\\rm ph}}^3\\left[\\;\\!\\frac{3}{2} k_{\\rm B} \\left(n_{\\rm e}T_{\\rm e}+n_{\\rm b} T_{\\rm b} \\right)\\;\\!\\right] \\ .",
"\\nonumber $ Assuming thermal equilibrium between the baryons and the electrons, $T_{\\rm b} = T_{\\rm e} = T$ , and $E_{\\rm th} (< r_{\\rm ph}) & \\approx &6\\pi \\;\\!",
"{r_{\\rm ph}}^3 ( n_{\\rm e} k_{\\rm B} T ) \\nonumber \\\\& = & 1.1 \\times 10^{47} \\left( \\frac{r_{\\rm ph}}{5.0\\times 10^{14}\\;\\!",
"{\\rm cm}} \\right)^2\\left( \\frac{k_{\\rm B}T}{10~{\\rm keV}} \\right)\\;\\!",
"{\\rm erg} \\ , \\nonumber $ which is also a small fraction of the total energy produced in the TDE.",
"The photospheric radius is the boundary at which the outflow beyond it will become transparent.",
"At the photospheric radius the mass outflow rate is ${\\dot{m}}_{\\rm out} (r_{\\rm ph}) & \\sim & 4\\pi \\;\\!",
"{r_{\\rm ph}}^2 \\big [ \\;\\!",
"n_{\\rm b} m_{\\rm b}\\;\\!",
"v(r_{\\rm ph}) \\;\\!",
"\\big ] \\nonumber \\\\& \\approx & 6.3\\times 10^{24} \\left( \\frac{r_{\\rm ph}}{5.0\\times 10^{14}\\;\\!",
"{\\rm cm}} \\right) \\left( \\frac{ v(r_{\\rm ph})}{2000\\;\\!",
"{\\rm km~s}^{-1}} \\right)\\;\\!",
"{\\rm g~s}^{-1} \\ , \\nonumber $ where $v(r_{\\rm ph})$ is the outflow speed at the photospheric radius.",
"Since this is a supersonic outflow, this mass will be lost to the system.",
"Recall that $L_{\\rm bol} = \\lambda \\;\\!",
"{\\dot{m}}_{\\rm in} c^2$ and ${\\dot{m}}_{\\rm in} = \\eta \\;\\!",
"{\\dot{m}}_{\\rm tot}$ .",
"We then have $L_{\\rm bol} & \\approx & 5.7 \\times 10^{44} \\times \\left( \\frac{\\eta }{1-\\eta }\\right) \\left(\\frac{\\lambda }{0.1}\\right) \\nonumber \\\\& & \\left( \\frac{r_{\\rm ph}}{5.0\\times 10^{14}\\;\\!",
"{\\rm cm}}\\right)\\left( \\frac{ v(r_{\\rm ph})}{2000\\;\\!",
"{\\rm km~s}^{-1}} \\right)\\;\\!",
"{\\rm erg~s}^{-1} \\ , \\nonumber $ The constraints on the system parameters based on the above considerations can be seen in Fig.",
"REF .",
"For a TDE to occur, the value of $\\eta \\zeta $ must be larger than 0.5 as restricted by the upper mass limit of the black hole which is about $1.3 \\times 10^6M_\\odot $ .",
"Moreover, the black hole would be more massive than $1.3 \\times 10^5 M_\\odot $ for $\\eta \\zeta < 5$ , i.e.",
"the Eddington mass accretion limit is not too strongly violated.",
"More specifically, if we set $\\lambda = 0.1$ , $r_{\\rm ph} = 5\\times 10^{14}\\;\\!",
"{\\rm cm}$ and $v(r_{\\rm ph}) \\approx 2000\\;\\!",
"{\\rm km~s}^{-1}$ , then a bolometric luminosity of $2\\times 10^{44}\\;\\!",
"{\\rm erg~s}^{-1}$ will give $\\eta \\sim 0.26$ .",
"For a $10^6\\;\\!M_\\odot $ black hole, $\\zeta \\sim 3$ and ${\\dot{m}}_{\\rm in}/{\\dot{m}}_{\\rm Edd} \\sim 0.78$ ; whereas for a $5\\times 10^5M_\\odot $ accreting black hole, $\\zeta \\sim 6$ and ${\\dot{m}}_{\\rm in}/{\\dot{m}}_{\\rm Edd} \\sim 1.6$ .",
"These values are plausable for a black hole being practically forced-fed in a TDE.",
"Note that the BH could be rotating.",
"Some modification of the analysis in which we have adopted a Schwarzschild BH would be required in order to take account of a smaller event horizon for a Kerr black hole of the same mass.",
"For instance, the tidal disruption radius could be 50% smaller for a prograde entry of the disurpted star but could be larger for a retrograde entry .",
"Thus, a black hole mass higher than the limit set by the analysis of the WD TDE with a Schwarzschild BH would be allowed if the BH is rotating.",
"Moreover, relativistic effects would be non-negligible when the WD penetrates to a distance comparable with the BH gravitational radius regardless of whether the BH is rapidly spinning or not.",
"Nonetheless, the uncertainty that this would introduce would be of a $\\sim 10\\%$ level .",
"Figure: Constraints to the systems parameters for a tidal disruption eventin a log(R t /R s )\\log \\;\\!",
"(R_{\\rm t}/R_{\\rm s}) vs log(M bh /M ⊙ )\\log \\;\\!",
"(M_{\\rm bh}/M_\\odot ) plot,where M bh M_{\\rm bh} is the black hole mass and R s R_{\\rm s} is the Schwarzschild radius.The inclined coloured lines correspond to thetidal disruption radii for helium white dwarfsof masses 0.1, 0.2, 03 and 0.4M ⊙ 0.4\\;\\!",
"M_\\odot (from top to bottom respectively).Tidal disruption is allowed when R t >R s R_{\\rm t} > R_{\\rm s},and the critical condition where R t =R s R_{\\rm t} = R_{\\rm s}is indicated by the dotted-dashed horizontal line.The vertical lines indicate the lower bounds of the black hole massesfor the parameters ηζ=5\\eta \\zeta = 5, 1, 0.5 and 0.1(from left to right respectively)."
],
[
"Evolution of the photosphere and its atmosphere",
"The optical/infrared spectra of the source show a certain amount of excess emission above a continuum, which has been well-fit by a blackbody spectrum, and there is also clear evidence of distinctive emission lines in the later stages [73].",
"During the late stages (day $T_{\\rm d}$ +11 and later) there is obvious evidence of He emission as indicated in the sequence of spectra presented in [76] and also in [73].",
"The later stages also show lines from the H Balmer series.",
"Spectral lines due He-burning, i.e., C, N, and O, were however absent, as in our UV spectra.",
"For an evolved star, the presence of a substantial amount of He, together with the lack of CNO elements would require that the star involved in the TDE cannot be a very massive star.",
"Accepting the dominant presence of He in forming the spectrum, and interpreting the broad bumps seen in both the spectra shown in [76] and [73] during the 8 day-long initial stage as velocity broadened He I emission, a large outflow velocity reaching a substantial fraction of the speed of light would be required.",
"We argue that these lines were probably emitted from a fast expanding shock heated cocoon produced by the TDE, see Fig.",
"REF .",
"Such a high expansion velocity is in fact consistent with our proposed TDE induced scattering photosphere scenario, as the photosphere would need to be inflated to a radius of $\\sim 5\\times 10^{14}\\,{\\rm cm}$ within 3 d, which requires $V \\sim 0.1c$ .",
"Note that however the expanding cocoon would detach from the scattering photosphere quickly and it would become depleted in density during its outward propagation.",
"At the same time the thermalised photosphere would be maintained by fall-back; infall of the debris of the disrupted star providing the fuel by accretion into the black hole.",
"Figure: A schematic illustration (not in scale) of the model of the disrupted WD debris around the BH of AT2018cow.The debris of the tidal disruption forms a photosphere, its associated enveloping atmosphere, and a high velocity outward expanding transient cocoon.The line and continuum emission formation regions have been indicated as respectively, infall-atmosphere and photosphere.The atmosphere, which is optically thin, eventually falls back in while the photosphere cools and recedes inward and the atmosphere emits line profiles that loose their blue wings due to that infall.The γ\\gamma -ray and X-ray emitting jet is not included.Kinetic energy will dominate the expanding cocoon energy balance while its ionised plasma emits the very broad He lines whose intensity drops when the density of the cocoon decreases with expansion.",
"As shown in section REF the shock of the interaction of the cocoon with the CSM is not sufficient to power the observed high-energy emission.",
"Note that the spectral evolution in the optical/infrared band observed by [73] provides information about the thermal evolution of the scattering photosphere and its atmosphere.",
"While the scattering photosphere produces the (black body) continuum, the lines originate from a surrounding lower-density atmosphere.",
"The He I 5876 Å line is not expected to be heavily contaminated by other lines, and here we use it to illustrate the line formation process and the evolution of the line profile in terms of the photospheric and atmospheric emission processes.",
"We start with summarising the key features regarding the line strength development and the profile evolution from the [73] observation.",
"We ignore the broad bumps present in the early stage spectra and SEDs, as they were formed in the rapidly expanding cocoon instead of being associated with the more stationary optical/infrared photosphere.",
"The He I 5876 Å emission line was not obvious in the spectra before day 11.",
"It began to emerge, with a symmetric broad profile centred at a frequency red-ward of the rest-frame line frequency.",
"Though the line strength relative to the continuum increased, in reality both the continuum and the total line flux actually decrease quite substantially.",
"The line starts showing asymmetry with its peak migrating blue-ward, toward the rest-frame line frequency while the blue wing is decreasing in intensity.",
"By day 33 and afterwards its profile becomes extremely asymmetric, peaking sharply almost exactly at the rest-frame line frequency, but only emission from the red wing is present.",
"Almost no line flux remains in the frequencies blue-ward of the rest-frame line frequency.",
"There is however no evidence of a P-Cygni absorption feature, an indicator of cooler outflow surrounding or within the line formation region, .",
"These line properties can be explained nicely with a simple two-zone model in which a photosphere, which is optically thick to both the line and the continuum, is enveloped by a lower density atmosphere, which is relatively opaque to the line emission but much less so to the continuum emission.",
"The continuum is being emitted from the photosphere whose boundary is defined by an optical depth $\\tau _{\\rm con} = 1$ , and the line is being generated in the atmosphere, mostly at an optical depth $\\tau _{\\rm line} = 1$ when the atmosphere is opaque to the line emission.",
"For the special case when the photosphere and atmosphere are in local thermal equilibrium, the intensities of the line and its neighbouring continuum will be the same, characterised by a Planck function at their equilibrium temperatures, i.e.",
"$T_{\\rm ph} = T_{\\rm at}$ .",
"It follows that $B_\\nu (T_{\\rm ph})\\vert _{\\rm con} = B_\\nu (T_{\\rm at})\\vert _{\\rm line}$ , and hence no emission features but only a smooth thermal blackbody continuum will appear in the spectrum.",
"Emission lines emerge only when the temperature of the line emitting atmosphere becomes higher than the thermal temperature of the continuum emitting photosphere.",
"If the effective thermal temperature of the photosphere drops very substantially, because of a rapid cooling, while the temperature of the lower-density atmosphere decreases at a slower rate, due to a less efficient cooling, the atmosphere would become hotter than the photosphere below.",
"As a consequence, emission lines appear in the spectra together with the blackbody continuum from the photosphere.",
"With this emission line formation mechanism in mind, we now readily explain the extreme asymmetric profile of the He I 5876 Å line that had almost no emission blue-ward of the rest-frame line frequency and the overall reduction in the total line flux with time, by means of (1) the radial collapse of both of the photosphere and its atmosphere and (2) the cooling of the both the photosphere and its atmosphere and the difference in their relative rates.",
"Figure: Monte-Carlo simulation of the profile of theHe I 5876 Å line formed in a geometrically thin atmosphere(i.e.",
"its velocity structure is assumed uniform).The atmosphere is optically thin to the continuum but not to the He I emission line.Note that the simulated line profile shows most emission above the continuum to the red wavelengths relative to the line rest wavelength (indicated with a vertical dashed line), while there is no contribution to the blue line wing.The reason is that the line is formed in infall back to the centre.This profile simulates well the characteristicshape of the observed line profile for days 37 and 44 in Fig.",
"4 of The peak emission of the line at the rest-frame frequency will be contributed mostly by the limb region of the atmosphere where the line-of-sight projection of the infall velocity is essentially zero; the reddest emission in the line wing will be contributed by the atmosphere above the central region of the photospheric disk, where the line-of-sight projection of the infall velocity was the largest.",
"Although the atmosphere is optically thick to the line it should be sufficiently optically thin that scattered photons from the whole visible atmosphere can escape.",
"Otherwise, the line emission from the limb will be suppressed by absorption of line photons due to the longer path length through the atmosphere.",
"In Fig.",
"REF we show that such an asymmetric line profile can be produced in simulations of line emission from a thin collapsing atmosphere of a finite thickness which is transparent to the continuum at frequencies near the rest-frame line frequency.",
"What remains to be explained now is why the line at first appeared to be reasonably symmetric with a peak to the red and later became extremely asymmetric and yet the line peak was at a frequency that was always red-ward of the rest-frame line frequency.",
"We attribute the evolution of the line profile as being due to various radiative transfer effects, their convolution and combination.",
"Among the effects, one is the line-of-sight attenuation in the atmosphere not sufficiently transparent to the continuum which is partly caused by the small difference in scale height between the regions of formation of continuum and line.",
"The effects of the centre to limb variation in path length on the line formation also needs consideration.",
"To quantify the shift of the peak frequency and the development of the line profile with the change in the line and continuum opacities within the atmosphere and taking into consideration the thermal and dynamical structures of the photosphere and the atmosphere will require detailed radiative transfer calculations, which are beyond the scope of the work.",
"We therefore leave the line formation mechanism in a collapsing inhomogeneous plasma sphere to a separated future study.",
"Nevertheless, from a simple geometrical consideration, we may see that the optical depth on a curved atmospheric surface would vary according to $\\tau \\sim 1/\\cos \\Theta $ where $\\Theta $ is angle between the line-of-sight and the normal to the atmospheric surface.",
"Thus, the expectation is that the optical depth of the line would not show dramatic variations across the surface of a collapsing atmosphere as the one considered here.",
"The symmetric profile of the He I lines observed in the source shortly after they emerge is expected therefore to evolve into an asymmetric profile on a timescale comparable to the timescale on which the thermal coupling between the atmosphere and the photosphere becomes inefficient and the respective scale heights start to differ."
],
[
"AT2018cow compared to other TDE",
"Tidal disruption events can be quite energetic in their high energy emissions.",
"For example, the well studied Swift J1644+57 transient [15] with large variability has peak isotropic X-ray luminosities exceeding $10^{48}$ erg s$^{-1}$ , several orders of magnitude larger than the isotropic X-ray luminosity of AT2018cow, and also with a much harder spectrum.",
"Spectra of the TDE PS1-10jh [34] were well-fitted by a galaxy model, a $3\\times 10^4$ K BB, and prominent broad He II emission lines on day $T_{\\rm d}$ +22 indicative of velocities $9000\\pm 700\\,{\\rm km\\,s}^{-1}$ .",
"AT2018cow also has a BB and He lines, but the emission lines in PS1-10jh were well-defined.",
"The light curve of PS1-10jh showed a slow brightening to a maximum around day 80 while the estimated peak luminosity is of the order of $10^{44}\\,{\\rm erg\\,s}^{-1}$ [34].",
"Where the peak luminosity is similar, the light curve in PS1-10jh is quite different, indicative of an event with much larger intrinsic time scales.",
"Two UV studies of TDE for iPTF16fnl [13], and for ASASSN-14li [20] find their HST UV spectra are without the C III] line, but the N III] 1750 Å line as well as higher excitation permitted lines of C and N are seen in the spectra that they discuss.",
"This was seen as an indication that the interacting star was around a solar mass main sequence star.",
"The absence of N III] 1750 Å in our UV spectra is consistent with a He WD.",
"A He-star, the stripped core of a one-time heavier star might seem possible also, but usually the core is not He through and through but envelops a CNO nucleus."
],
[
"On the absence of lines of certain elements",
"One might wonder whether the absence of lines of C II, C III, N II, N III, O II in the UV are a result of an ionisation effect.",
"We can discuss this in terms of a steady-state atmosphere, like a stellar photosphere, or in terms of a photosphere in a turbulent medium, like in nova ejecta.",
"In the stellar context, ionisation of the CNO elements is either due to a high photospheric temperature, or due to non-radiative heating in a low density chromosphere above the photosphere.",
"Emission lines result due to the rising excitation temperature above the photosphere, and lines of multiple stages of ionisation can be found in strengths proportional to the emission measure of the chromosphere, [48].",
"The emission measure includes the elemental abundance and emission volume.",
"A turbulent atmosphere also can have an embedded photosphere and emission lines of multiple ionisations can be found.",
"An example would be ejecta in a nova.",
"It is quite common to find a range of ionisations in astrophysical objects.",
"In AT2018cow we interpret the spectrum as due to a photosphere at a temperature of 25,000 to 13,000 K, in which ions of singly and doubly ionised CNO would be expected.",
"It is surrounded by a lower density medium which shows at late times He and H emission lines that also indicate excitation temperatures in the 10,000-30,000 K range.",
"As we argued before, line formation depends on the temperature of the line-forming region compared to that of the continuum.",
"The evident He emission lines in the spectra prove that the excitation temperature in the lower density medium is sufficient to produce CNO lines provided that C, N, and O are abundant.",
"Models of the ionisation of the ejecta can be used to explore scenarios that might explain the absence of certain spectral lines at certain times from the TDE onset, like for example in [99].",
"In a hydrodynamical model of PS1-10jh [37] describe how it is possible to explain the absence of certain spectral lines due to zones of ionisation in the ejecta.",
"However, their model is for a case where the radiation pressure is negligible and that results in a completely different ionisation structure in the debris disk.",
"Our approach for understanding AT2018cow starts with the observed luminosity, temperature, line-broadening and line-shape to explain the data in terms of a spherical cloud due to radiation pressure which results from a fast and energetic TDE.",
"Assuming, like in [37] that the radial extent was just due to hydrodynamics and limited by the escape speed would require a super-massive BH for AT2018cow.",
"Therefore the ionisation structure of the AT2018cow debris cloud is likely very different."
],
[
"The host galaxy",
"The nearby galaxy, see Fig.",
"REF , likely hosts an active galactic nucleus which might explain that the historic photometry at, e.g., $0.64-0.67\\,\\mu {\\rm m}$ ranges by three orders of magnitude in brightness from $6.8\\times 10^{-5}-0.1$ Jy [31], and [55].",
"The $u$ band image taken at the CFHT Megacam instrument [7] shows extended faint emission around Z 137-068 which look like H II regions.",
"We retrieved the UKIDSS [56] WFCAM [19] images in $J$ and $K$ taken by the UKIRT from the archive at ROE.",
"We also inspected GALEX FUV and NUV images, which due to the lower resolution show emission overlap between the galaxy bulge and the UV bright source.",
"The UV bright source is clearly offset from the location of AT2018cow, however.",
"The mass of the galaxy disk and bulge have been estimated as $\\log (M_{\\rm disk}/M_\\odot ) = 9.575$ and $\\log (M_{\\rm bulge}/M_\\odot ) = 8.904$ .",
"Random uncertainties in the bulge mass are typically 0.15 dex, with additional systematic uncertainties of up to 60% [67].",
"The ratio of bulge mass to the mass of a central BH has been determined from known observations of central BH in galaxies.",
"The recent [66] correlation gives $M_{\\rm bh} =1.8\\times 10^6 M_\\odot $ with uncertainties of about 0.3 dex in the BH mass.",
"The uncertainties may be larger since the relations are typically derived for more massive galaxies.",
"This suggests that the central BH mass in the galaxy is slightly larger than the upper limit to the BH mass associated with a possible TDE in this transient, but it might be up to 10$^7 M_\\odot $ when errors are considered.",
"The transient is offset from the nearby Z 137-068 galaxy by 6.0which translates to $> 1.7$ kpc at the distance of 60.0 Mpc.",
"The extra emission near the location of the transient may be due to a a foreground object related to the excess nebulosity in the south-west area of the galaxy, since the narrow Ca II lines in the spectrum of [76] were found at a redshift of $z = 0.0139$ instead of $0.01414$ for Z 137-068.",
"The difference between z=0.0139 and 0.01414 is 0.00024.",
"The velocity difference is therefore 72 km s$^{-1}$ .",
"This is well within the rotation velocity of the galaxy, so it might be in Z 137-068, or it could be a satellite galaxy of it, or it could be of order 1 Mpc distant if $\\Delta z$ is due to the Hubble flow.",
"The positions of the late time spectral lines seem consistent with the transient association with Z 137-068, but could also be consistent with the smaller redshift of 0.0139 since the broad and asymmetric line profiles prevent an accurate redshift determination.",
"The negligible intrinsic $N_{\\rm H}$ column density suggest that the transient is located in a low IGM density environment.",
"One possibility is that the BH is in a globular cluster (GC) associated with the host galaxy.",
"However, black hole masses inside GCs are estimated from $10^3 - 4\\times 10^4 M_\\odot $ , i.e., intermediate mass BH (IMBH) [78].",
"A BH mass of $4\\times 10^4 M_\\odot $ is below range of the mass we expect based on our analysis in Section REF .",
"The old population of a GC could also have a higher WD proportion than the average galaxy.",
"In some TDE the presence of He was taken by some investigators as a sign of a WD rather than a MS or giant star impacting a black hole [52], but their observed time scales are much longer than AT2018cow.",
"Therefore the odds are that most of those are due to the interaction of a main sequence or giant star with a BH."
],
[
"Discussion",
"We show that this event is unlike other transients in its combination of brightness, its evolutionary timescales, its spectrum, its abundances, and proceed to use the observed characteristics to constrain its properties in terms of a model of a Helium white dwarf TDE.",
"Our observations point to a large debris cloud which became larger than 33 AU within 1-3 days, with embedded photosphere.",
"Around that a lower density atmosphere showing infall, a transient high-velocity cocoon being blown off and also, possibly, a jet.",
"The choice of a White Dwarf TDE rather than a main sequence star TDE is suggested by the size of the debris cloud being larger and its formation time shorter than would be expected in the disruption of a main sequence or giant branch star, whose larger radius and lower density mean that disruption takes place at larger scales, resulting in smaller debris scales.",
"Fitting disruption of a main sequence star to these observations would therefore require an extreme and unlikely accretion luminosity [73].",
"In the first days of the event we see fewer flares in the X-rays than later.",
"At the same time the blackbody luminosity light curve shows a flatter decay until day 6.",
"Judging from those two facts and considering that both the jet and the photosphere are powered from the accretion, it may be that the initial accretion was nearly continuous and stable but became intermittent when the accretion rate decreased, suggesting two accretion phases with the latter being more variable.",
"There is a possibility that in WD TDEs nuclear burning takes place [49].",
"The UVOT UV spectra of novae who have ejecta of similar temperatures generally show strong emission lines of N II, N III, and C III, but in this case there is no sign of any emission from these lines in the UV spectra taken in the period of $T_{\\rm d}$ +5$-$ 20 d. This suggests that the abundance of these elements was not enhanced which is consistent with the models in [49] likely for a 0.2 M$_\\odot $ or even less massive WD.",
"If the BH is rotating rapidly, and the WD approached the BH from outside the plane of its rotation, the WD debris is likely not limited to a thin disk.",
"It may initially be a 3-D spiral shape as in [93].",
"It is likely that the debris will be ionised and turbulent due to the large luminosity from accretion and will generate a strong magnetic field [36], which would lead to a further thickening of the debris due to turbulent pressure.",
"Magnetic fields in the TDE of a rotating BH would also play a role in jet formation similar to the model described in [57].",
"The luminosity at optical maximum and thereafter is caused by the large projected area of the debris atmosphere.",
"Although that may mean that we observe a disk-like cloud face-on, the odds are that we observe an extended, roughly spherical, debris cloud at a different angle, especially since the X-ray emission is consistent with an off-axis jet.",
"A possible reason that the debris is extended in all directions by the time of optical maximum, is that relativistic precession, turbulent pressure, magnetic field amplification, dissipation and instability are causing expansion of geometrically thin debris.",
"We are not aware of numerical models of TDEs that include also the physics of magnetic field generation, current systems, and their evolution in a relativistic environment which might be the missing element in explaining why AT2018cow developed so rapidly and why the debris cloud would be extended.",
"Our analysis of the BAT $\\gamma $ -ray and XRT X-ray emission could be explained by a jet associated with the TDE.",
"The observed bursty peaks in the X-ray luminosity are also seen in other TDE as would be expected from an unsteady accretion process driving a variable jet.",
"However, the total column density to the X-ray source is low, and the spectrum does not notably change during the bursts.",
"The low column density suggests that the line of sight to the source of the X-ray emission is not obscured by the tidal debris.",
"In such circumstances the variability in the X-rays might directly probe the accretion history onto the BH.",
"The initial optical polarisation seen day $T_{\\rm d}$ +4.6, 4.7, see [90], is possibly due to a slightly non-spherical shape of the line emission from the cocoon, and its disappearance is likely linked to the evolution of the cocoon.",
"Apart from WDs, we consider whether some other low mass star might be the cause of this TDE and have a short time-scale of interaction due to their low mass.",
"Very low mass ($M < 0.6\\;\\!",
"M_{\\odot }$ ) main sequence stars are not expected to have burned more than 70% of their hydrogen [68] by the end of their evolution.",
"The large amount of H in the tidal disruptions of a very low mass main sequence star would be at odds with the relative strength of the He lines in the spectrum and initial absence of similarly broadened H lines.",
"The He core from a stripped late type star might be a possible candidate.",
"He-cores can be formed by mass transfer in binaries, but observationally they would appear as He WDs.",
"Therefore the most likely candidate for the star in this tidal disruption is a WD.",
"The proposed nature of this transient as being possibly a magnetar [76], [41], a SN [81], or a TDE of a solar mass star [73] are either unlikely or do not explain as many observations as our model of a WD TDE.",
"By contrast, a low mass WD (e.g.",
"DB or even a DA WD) is considered a good candidate for AT2018cow, as they consist mostly of He.",
"WD are common; they comprise of about 12% of the Galactic stellar population.",
"The “mean” WD mass is about $0.6\\;\\!M_\\odot $ .",
"The Gaia DR2 data has showed about 5% of WD in the sample have mass less than $0.3\\;\\!M_\\odot $ (Gentille Fusillo, personal communication).",
"(For the Gaia DR2 WD catalogue, see [33].)",
"A recent study of WDs within 20 pc of the Sun [42] gave a less biased, but more uncertain, estimate that 2 out of 137 WDs have mass $< 0.3 M_\\odot $ .",
"About 0.5% of the galactic stellar population would therefore be in the low mass WDs.",
"Thus, assuming that a similar percentage holds in the host galaxy, the chance encounter of a low-mass WD by a massive black hole would not be an extremely rare event in comparison with other stellar types.",
"However, that does not take into account the effects of the change of loss cone radius with stellar density [61].",
"Such considerations are however difficult to make since the environment of the event is not known because it is not in the centre of a galaxy.",
"We propose a simple model that explains the observations of the emergent line profiles and show that infall of the lower density atmosphere surrounding the shrinking photosphere can explain the line profile shape and evolution.",
"The presence of infall as inferred from the line profiles, may be due to the infall being confined to closed magnetic field.",
"In the Sun chromospheric and transition region lines (formed at temperatures of 7000-50000 K) show on average a redshift [9] which is due to the higher density in filamentary structures.",
"A high velocity outflow may in the presence of a magnetic field co-exist with infall into the photosphere.",
"Detailed modeling of the hydrodynamics and thermodynamics, atomic physics and radiative transfer is needed to work out this simple model in more detail.",
"The Swift data of AT2018cow, supported by other studies and reports suggests that possibly this was the tidal disruption of a He WD on a relatively small non-stellar mass black hole, resulting in a large, hot, ionised debris cloud emitting a thermal spectrum with weak, broad line emission.",
"A jet may also be associated with the event and emit a non-thermal X-ray spectrum; it is not as luminous as those seen in long GRBs.",
"The X-ray component due to optically thick Compton scattering on the hot debris cloud is negligible, and X-ray emission from a shock caused when a high velocity cocoon encounters the circum-system medium is estimated to be non-detectable.",
"We explain the multi-wavelength temporal behaviours of the source with a WD-TDE model, and the observations give a constraint for the WD mass to be $\\approx 0.1 - 0.4\\,M_\\odot $ and the BH mass to be $\\sim 1.3 \\times 10^5 - 1.3 \\times 10^6\\,M_\\odot $ .",
"The model also predicts the total accreted rate of mass accretion is $\\approx 8.5\\times 10^{24}{\\rm g\\,s}^{-1}$ , see section REF , while we can use $\\zeta \\sim 3-6$ to also derive the ejected mass loss rate is $\\approx 6.3\\times 10^{24}\\,{\\rm g\\,s}^{-1}$ .",
"The model is consistent with the observed bursty peaks in the X-ray luminosity which are also seen in other TDE as would be expected from an unsteady accretion process."
],
[
"Acknowledgements",
"We acknowledge the efforts of the Swift planners.",
"Swift and NuSTAR Data were retrieved from the Swift and NuSTAR archive at HEASARC/GSFC, and from the UK Swift Science Data Centre.",
"We also used the CFHT archive hosted at the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency, and the WFCAM UKIRT data from the UKIDSSDR10PLUS data release from the WFCAM archive at the Royal Observatory Edinburgh.",
"This work has been supported by the UK Space Agency under grant ST/P002323/1 and the UK Science and Technology Facilities Council under grant ST/N00811/1.",
"QH is supported by a UCL MSSL Summer Research Studentship.",
"SRO gratefully acknowledges the support of the Leverhulme Trust Early Career Fellowship.",
"SC acknowledges the support of under ASI-INAF contract I/004/11/1.",
"We benefited from a very useful review by the referee, and thank him/her for their suggestions.",
"A"
]
] | 1808.08492 |
[
[
"Rain Streak Removal for Single Image via Kernel Guided CNN"
],
[
"Abstract Rain streak removal is an important issue and has recently been investigated extensively.",
"Existing methods, especially the newly emerged deep learning methods, could remove the rain streaks well in many cases.",
"However the essential factor in the generative procedure of the rain streaks, i.e., the motion blur, which leads to the line pattern appearances, were neglected by the deep learning rain streaks approaches and this resulted in over-derain or under-derain results.",
"In this paper, we propose a novel rain streak removal approach using a kernel guided convolutional neural network (KGCNN), achieving the state-of-the-art performance with simple network architectures.",
"We first model the rain streak interference with its motion blur mechanism.",
"Then, our framework starts with learning the motion blur kernel, which is determined by two factors including angle and length, by a plain neural network, denoted as parameter net, from a patch of the texture component.",
"Then, after a dimensionality stretching operation, the learned motion blur kernel is stretched into a degradation map with the same spatial size as the rainy patch.",
"The stretched degradation map together with the texture patch is subsequently input into a derain convolutional network, which is a typical ResNet architecture and trained to output the rain streaks with the guidance of the learned motion blur kernel.",
"Experiments conducted on extensive synthetic and real data demonstrate the effectiveness of the proposed method, which preserves the texture and the contrast while removing the rain streaks."
],
[
"Introduction",
"Outdoor vision systems are frequently affected by bad weather conditions, one of which is the rain.",
"Because of the high motion velocities and the light scattering, raindrops usually introduce bright streaks into the images or videos acquired by cameras.",
"This undesirable interference will degrade the performance of various computer vision algorithms [1], such as object detection [2], event detection [3], action recognition [4], and scene analysis [5].",
"Therefore, alleviating the effects from the rain is an essential task and has been investigated extensively.",
"Fig.",
"REF exhibits one example of the single image rain streak removal.",
"Figure: An example of rain streak removal for a real-world rainy image.",
"Top-left: the rainy image.",
"Top-right: the derain result by DID .",
"Button-left: the derain result by DDN .",
"Button-right: the derain result by the proposed KGCNN.Figure: The flowchart of the proposed KGCNN single image rain streak removal framework.",
"1) The rainy image is decomposed into the texture component and the structure component.",
"2) A rainy patch is feed to the parameter net to obtain the angle and the length of the motion blur kernel.",
"3) The motion blur kernel is stretched to the degradation map.",
"3) The rainy patch and the degradation map is then transmitted into the derain net, whose output is the rain streak patch.",
"4) Finally, derain image is obtained by the subtraction of the rainy image and the rain streak image.Most of the traditional methods always focus on the discriminative characteristics of the rain streaks and the clean background, for instance, the high-frequency property [8], [9], directionality [10], [11], [12], [13], [14] and repeatability [15] of the rain streaks and the piecewise smoothness [16], [17], [15], [13], [14] of the background (without loss of generality, we denote the rain-free content as “background” throughout this paper).",
"It is common for the model based methods to elaborately tailor an optimization model with the hand-crafted regularizer expressing the prior knowledge.",
"Although these model based methods go into much depth on the distinct characteristics of the rains streaks, they are often insufficient to cover the majority of the important factors in a real rainy scene since the degradation of rain streaks can be very complex.",
"The traditional learning based methods attempt to overcome this shortage by inferring the discriminative dictionaries[18], the GMMs [16], [17], the stochastic distributions[19] or the convolutional filters [20], [21], [22] from the data.",
"Benefit from the complex representation ability of the convolutional neural network, the deep learning techniques [23], [24], [6], [25], [26], [27], [20], [28] further leverage the data to the most extent, and obtain promising results.",
"As it can be seen in Fig.",
"REF , state-of-the-art de-raining algorithms [7], [6] tend to obtain the under-derain result (bottom-left) or the over-derain result (top-right).",
"We attribute these phenomena to the fact that it is challenging for deraining methods, even the deep learning based methods, to distinguish the rain streaks and the line pattern textures (e.g.",
"the grass in Fig.",
"REF ).",
"The rain streaks removal performance of the neural network can be heighten by adopting deeper architecture as [7] or elaborately designing the architecture, in which the contexture information are taken into account, as [27], [6].",
"However, it is still difficult to purposefully enhance the capacity of the neural network to face the fore-mentioned challenge.",
"Meanwhile, anther question is that can we address this challenge and achieve promising performance with the common and simple neural network structures?",
"In this paper, referring to [1], in which it was pointed out that the appearance of the rain streaks is mainly related to the motion blur mechanism, we propose a novel degradation model of the rain streak interference taking the motion blur procedure into account.",
"By modeling the degradation of rain streaks as the motion blur, we are able to utilize two important distinct characteristics of the rain streaks, i.e., the repeatability and the directionality.",
"The line pattern textures do not possess the same generation mechanism as the rain streaks'.",
"Therefore, this modeling strategy would contribute to distinguish the rain streaks and the line pattern textures.",
"After modeling the rain streaks with motion blur kernel, the questions come to 1) how to infer the motion kernel from the data, and 2) how to utilize the information provided by the motion blur kernel when deraining.",
"In our approach, we assume that the rain streaks in a small patch approximately share the same motion blur kernel.",
"At the beginning, a rainy patch is feed to the parameter net, a plain 6-layer network, to infer the angle and the length of the motion blur kernel.",
"To enable the learned motion blur kernels to participate in the subsequent deraining process, we adopt the dimensionality stretching strategy [29], which stretched the motion blur kernels to degradation maps with the same spatial resolution as the detail patches.",
"Then, the detail patch together with the degradation map is input into a common 26 layer ResNet, whose output is a patch of rain streaks.",
"Finally, we obtain the derain results by subtracting the rain streaks from rainy image.",
"The core idea of our framework is exploiting the generation mechanism of the rain streaks to guide rain streak removal, and the flowchart of our framework is illustrated in Fig.",
"REF .",
"Contributions: The contributions of this paper mainly include four aspects.",
"We build a novel rain streak generation model which takes the motion blur kernel into account.",
"This modeling strategy enables us to utilize the repeatability and the directionality of the rain streaks.",
"A sub-net, i.e., the parameter net, is built to learn the parameters (length and angle) of the motion blur kernel.",
"Unlike existing methods using sub-net to embed the contextual information, our sub-net is designed to exploit the generation information of the rain streaks.",
"We propose an effective kernel guided CNN (KGCNN) framework, in which the network structures are common and simple, for rain streak removal.",
"Within this framework, the automatically learned motion blur kernel thoroughly guides the process of rain streak removal.",
"Extensive experiments are conducted on publicly available real and synthetic data.",
"Qualitative and quantitative comparisons with existing state-of-the-art methods are presented.",
"The results show that the KGCNN removes rain streaks well while keeping the texture and the contrast of background.",
"The organization of this paper is as follows.",
"We provide an overview of the existing deraining methods in Section .",
"Section gives the detailed architecture of the proposed KGCNN.",
"In Section , experimental results on the synthetic data and the real-world data are reported.",
"Finally, we draw conclusions in Section ."
],
[
"Related Works",
"In the past decades, numerous methods have been proposed to improve the visibility of images/videos captured with rain streak interference.",
"[8], [9], [15], [18], [17], [16], [30], [31], [21], [10], [11], [12], [32], [33], [27], [34], [35], [6], [22], [23], [24], [36], [37], [38], [39], [40], [41], [13], [14], [19], [20], [42], [25], [26], [43], [28].",
"Traditionally, these methods can be divided into two categories: single image based methods and multiple-images/videos based methods.",
"Nevertheless, the explosive development of the deep learning brings in a novel branch, i.e., the deep learning methods."
],
[
"Single Image Based Methods",
"For the single image derain task, Kang et al.",
"[8] decomposed a rainy image into low-frequency (LF) and high-frequency (HF) components using a bilateral filter and then performed morphological component analysis (MCA)-based dictionary learning and sparse coding to separate the rain streaks in the HF component.",
"To alleviate the loss of the details when learning HF image bases, Sun et al.",
"[9] tactfully exploited the structural similarity of the derived HF image bases.",
"Kim et al.",
"[44] took advantage of the nonlocal similarity.",
"Chen et al.",
"[15] considered the similar and repeated patterns of the rain streaks and the smoothness of the rain-free content.",
"Sparse coding and dictionary learning were adopted by Luo et al.",
"[18] and Son et al.",
"[45].",
"In their results, the details of backgrounds were well preserved.",
"The recent work by Li et al.",
"[17] utilized the Gaussian mixture model (GMM) patch priors for rain streak removal, with the ability to account for rain streaks of different orientations and scales.",
"Meanwhile, the directional property of rain streaks received a lot of attention in [10], [30], [11], [12] and these methods achieved promising performances.",
"Ren et al.",
"[33] removed the rain streaks from the image recovery perspective.",
"Wang et al.",
"[32] took advantage of the image decomposition and dictionary learning."
],
[
"Video Based Methods",
"Garg et al.",
"[36] first raised a video rain streak removal method with a comprehensive analysis of the visual effects of rain on an imaging system.",
"Since then, many approaches have been proposed for the video rain streak removal task and obtained good performances with different rain circumstances.",
"Tripathi et al.",
"[37] took the spatiotemporal properties into consideration.",
"In [15], the similarity and repeatability of rain streaks were considered, and a generalized low-rank appearance model was proposed.",
"Chen et al.",
"[46] considered the highly dynamic scenes.",
"Whereafter, Kim et al.",
"[39] considered the temporal correlation of rain streaks and the low-rank nature of clean videos.",
"Santhaseelan et al.",
"[40] detected and removed the rain streaks based on phase congruency features.",
"Additionally, comprehensive early existing video based methods were reviewed in [38].",
"You et al.",
"[41] took the raindrops adhered to a windscreen or window glass into account.",
"In [13] and [14], a novel tensor-based video rain streak removal approach was proposed by considering the directional property.",
"The rain streaks and the clean background were stochastically modeled as a mixture of Gaussians by Wei et al.",
"[19].",
"The convolutional sparse coding (CSC), which has shown its ability in image cartoon-texture decomposition [22], was also adopted by Li et al.",
"[20] for the video rain streaks removal.",
"Ren et al.",
"[42] addressed the video desnow and derain task based on matrix decomposition."
],
[
"Deep Learning Based Methods",
"The deep learning based method was first applied to derain in [47], in which a 3-layer convolutional neural network (CNN) was designed to remove static raindrops and dirt spots from pictures taken through window glass.",
"Fu et al.",
"[34] was the first to successfully tailor a deep CNN for the rain streak removal task.",
"Moreover, in [7], Fu et al.",
"designed the deep detail network (DDN) to further improved the performance by adopting the well-known deep residual network (ResNet) [48] structure.",
"Pan et al.",
"[49] simultaneously operated on the texture component and the structure component.",
"Yang et al.",
"[27] added a binary map, which reflects the contextual information, in the rain streak observation model and constructed a deep network that jointly detected and removed rain streaks.",
"Meanwhile, the increasingly popular Generative Adversarial Networks (GAN) was first used in [35] for rain streak removal and recently applied to the task of dealing with adherent raindrop [23].",
"In [25], Chen et al.",
"proposed a CNN Framework for the video rain streak removal task, while the recurrent neural network was adopted by Liu et al [26].",
"For jointly rain-density estimation and derain, Zhang et al.",
"[6] raised a density aware multi-stream densely connected convolutional neural network (DID).",
"In [24], both the rain component and the background component are considered to remove rain streaks.",
"Fan et al.",
"[50] developed a residual-guide feature fusion network, which was detachable to meet different rainy conditions.",
"A lightweight pyramid of networks was proposed in [51], using the domain-specific knowledge to simplify the learning process.",
"In this section, we will give our rain streak observation model and subsequently clarify the detail architecture of the proposed derain framework.",
"As exhibited in Fig.",
"REF , there are mainly three parts, i.e., the parameter net, the dimensionality stretching, and the derain net.",
"The main stream is 1) decomposing the rainy image into texture component and structure component; 2) processing the patches in the texture component using the parameter net, the PCA operation, and the derain net; 3) subtracting the obtained rain streaks and obtaining the derain result."
],
[
"Observation Model",
"As mentioned previously, the rain streaks can be approximately viewed as sharing the same motion blur kernel.",
"Hence the basic unit of our observation model is the patch.",
"Similar to many existing methods, the rainy image is modeled as a linear superposition: $\\mathbf {O} = \\mathbf {B} + \\mathbf {R},$ where $\\mathbf {O}$ , $\\mathbf {B}$ , and $\\mathbf {R}\\in \\mathbb {R}^{m\\times n} $ are patches of the observed rainy image, the underlying background (i.e.",
"the background) and the rain streaks, respectively.",
"After taking the motion blur into consideration, the observation model in Eq.",
"(REF ) turns to be: $\\mathbf {O} = \\mathbf {B} + \\mathbf {K}(\\theta ,l) \\otimes \\mathbf {M},$ where $\\theta $ and $l$ are respectively the angle and length of the motion blur kernel $\\mathbf {K}\\in \\mathbb {R}^{p\\times p}$ , $\\mathbf {M}$ is the raindrops mask, and $\\otimes $ denotes the convolution operation.",
"Because of the high velocity of the raindrops, the appearance of the rain streaks are mostly linear.",
"Hence using the angle $\\theta $ and the length $l$ to characterize the motion blur kernel of the rain streaks is reasonable and its advantage illustrated in the next subsection.",
"Meanwhile, many existing methods conduct the rain streak removal procedures on the detail component [34], [7], [49] or the high-frequency (HF) component [8], [9].",
"Following this research line, we adopt the guided filter method in [52] as the low-pass filter because it is simple and fast to implementAs discussed in [34], the choice of low-pass filter is not limited to guided filtering..",
"The rainy patch is decomposed into two parts the texture component $ \\mathbf {O}_{\\text{T}}$ (denoted as “detail component” in [34], [7], [49]) and the structure component $\\mathbf {O}_{\\text{S}}$ , and they satisfy $\\mathbf {O} = \\mathbf {O}_{\\text{S}} + \\mathbf {O}_{\\text{T}}$ .",
"The advantages of processing on the texture component have been fully discussed in [34], [7].",
"In order to facilitate the readers, we briefly bring them herein.",
"It can be found in Fig.",
"REF that the texture component consists of all rain streaks, i.e., $\\mathbf {O}_{\\text{T}} = \\mathbf {B}_{\\text{T}}+\\mathbf {R}$ , so that training and testing on the texture component $\\mathbf {O}_{\\text{T}}$ is sufficient and compact.",
"Meanwhile, the texture component is sparser and the range of the values is significantly decreased compared to the pixels in the original image domain.",
"This also decreases the mapping range of the neural network, making the network focus on the important information.",
"After the decomposition in Eq.",
"(REF ), the observation model becomes $\\begin{aligned}\\mathbf {O}_{\\text{T}} = \\mathbf {B}_{\\text{T}} + \\mathbf {K}(\\theta ,l) \\otimes \\mathbf {M},\\end{aligned}$ where $\\mathbf {B}_{\\text{T}}\\in \\mathbb {R}^{m \\times n\\times c}$ is the rain free content of the texture component and the goal turns to estimate the clean texture part and separate the rain streaks from the rainy texture component.",
"In this work, considering the benefits of processing on the texture component, we attempt to design and train a CNN derainer $\\mathcal {F}_\\text{D}(\\cdot ; \\Theta _\\text{D})$ , which maps the texture $\\mathbf {O}_{\\text{T}}$ patch into the rain streaks patch $\\mathbf {R} = \\mathbf {K}(\\theta ,l) \\otimes \\mathbf {M}$ .",
"Modeling the rain streaks with the motion blur kernel $\\mathbf {K}$ maintains two advantages.",
"One is that two important factors, i.e., the length $l$ and the angle $\\theta $ , of the rain streak appearance are uniformly encoded by the motion blur kernel.",
"Another one is that the repeatability of the rain streaks allows us to easily infer the two parameters from an input texture patch.",
"In the next subsection, we would present the detail of how to estimate the parameters and embed the learned motion blur kernel to the deraining procedure."
],
[
"The Parameter Sub-Network",
"Since the CNN has shown its overwhelming superiority on feature extraction, we plan to use a CNN to learn the motion blur kernel.",
"Initially, given a CNN $\\mathcal {F}_\\text{K}(\\cdot ;\\Theta _\\text{K}): \\mathbb {R}^{m\\times n} \\rightarrow \\mathbb {R}^{p\\times p}$ , which maps the input texture patch to the motion blur kernel, with network parameter $\\Theta _\\text{K}$ , the loss function for training this CNN architecture is $L_\\text{K}(\\Theta _\\text{K})= \\frac{1}{n} \\sum \\limits _{i = 1}^{n} \\Vert \\mathcal {F}_\\text{K}(\\mathbf {O}_\\text{T}^i;\\Theta _\\text{K}) - \\mathbf {K}^i \\Vert _F^2,$ where $\\Vert \\cdot \\Vert _F$ denotes the Frobenius norm and $i$ index the patches and motion blur kernels.",
"However, the performance of $\\mathcal {F}_\\text{K}(\\cdot )$ is not satisfactory.",
"Without the fully connect layer, it dose not converge.",
"As we pointed out above, the motion blur kernel within the generation of rain streaks is conclusively decided by two parameters, i.e., the angle $\\theta $ and length $l$ .",
"This indicates that the intrinsic information lies in a parameter space with much lower dimension than the convolution filter space.",
"Working directly on the low dimension information can not only facilitate the task of motion blur kernel estimation, but also prevent possibly overfitting, which would be verified in the experimental part.",
"Therefore, we adopt the CNN $\\mathcal {F}_\\text{P}(\\cdot ;\\Theta _\\text{P}): \\mathbb {R}^{m\\times n} \\rightarrow \\mathbb {R}^{2}$ , which maps the input texture patch to the parameter vector, with network parameter $\\Theta _\\text{P}$ and the loss function thereof for training turns to: $L_\\text{P}(\\Theta _\\text{P}) = \\frac{1}{n} \\sum \\limits _{i = 1}^{n} \\Vert \\mathcal {F}_\\text{P}(\\mathbf {O}_\\text{T}^i;\\Theta _\\text{P})) - \\mathbf {p}^i \\Vert _F^2,$ where $\\mathbf {p} = [\\theta ,l]^\\top $ is the parameter vector.",
"The architecture of $\\mathcal {F}_\\text{P}(\\cdot )$ (denoted as “parameter net”) is exhibited in Fig.",
"REF .",
"Once the parameters $\\theta $ and $l$ are determined, the motion blur kernel $\\mathbf {K}$ is unique."
],
[
"Dimensionality Stretching",
"After maintaining the motion blur kernel $\\mathbf {K}$ , the question comes to how to utilize the motion blur kernel when deraining.",
"In general, the input of the derain net $\\mathcal {F}_\\text{D}(\\cdot ; \\Theta _\\text{D})$ , which would be detailed in the next subsection, is supposed to be the texture patch together with the motion blur kernel learned from this texture patch, since the motion blur kernel consists of the prior knowledge of the rain streaks.",
"If we simply splice the texture patch $\\mathbf {O}_\\text{T}\\in \\mathbb {R}^{m\\times n}$ and motion blur kernel $\\mathbf {K} \\in \\mathbb {R}^{p\\times p}$ , weight sharing in CNN makes that the texture patch could not get the entire information of the motion blur kernel.",
"Hence, a dimensionality stretching operation in [29] is necessary.",
"The dimensionality stretching strategy is schematically illustrated in Fig.",
"REF .",
"At the beginning, the motion blur kernel $\\mathbf {K}$ is vectorized into a vector $\\mathbf {k}\\in \\mathbb {R}^{p^2}$ .",
"After the vectorization, $\\mathbf {k}$ is projected onto a $t$ -dimensional linear space by the principal component analysis (PCA) technique.",
"Then the projected vector $\\mathbf {k}_t\\in \\mathbb {R}^{t}$ is stretched into degradation maps $\\mathcal {M}\\in \\mathbb {R}^{m \\times n \\times t}$ .",
"All values in the $j$ -th horizontal slice with size $m \\times n$ of the 3-dimensional $\\mathcal {M}$ are same as the $j$ -th element of $\\mathbf {k}_t$ .",
"By doing so, the degradation maps then can be concatenated with the texture patch, making CNN possible to handle the two inputs.",
"However, different from [29], in the case of plain motion blur kernel with only two parameters, we found that $t$ is still too large compared with number of channels of the texture image.",
"Meanwhile, since that working on texture component leads to the pixel values being close to zero, the information of the texture image may be drowned in the information of motion blur kernel with a relatively large $t$ .",
"To tackle this issue, degradation maps will be concatenated with the texture image after the first convolutional layer in the derain net, as shown in Fig.",
"REF ."
],
[
"Derain Net",
"As previously mentioned in Sec , instead of elaborately designing the architecture, we resort to the typical ResNet structure.",
"A cascade of $3\\times 3$ convolutional layers are applied to perform the deraining.",
"Each layer is composed of three types of operations, including convolution (denoted as “Conv”), rectified linear units [53] (denoted as “ReLU”), and batch normalization [54] (denoted as “BN”).",
"We still use Frobenius norm in the loss function, which is $L_\\text{D}(\\Theta _\\text{D}) = \\frac{1}{n} \\sum \\limits _{i = 1}^{n} \\Vert f_d(\\mathbf {O}_{texture}^i, \\mathbf {K}^i - \\mathbf {R}^i \\Vert _F^2,$ where $\\mathbf {R}$ is rain streaks.",
"After subtracting the rain streaks $\\mathbf {R}$ from the rainy image $\\mathbf {O}$ , we could get the background.",
"Disscusion: As we mentioned above, distinguishing the rain streaks and the line pattern textures is important but challenging.",
"In this work, we face this challenge by exploiting the generation mechanism of the rain streaks to guide the rain streak removal.",
"Within our framework, the generation mechanism of the rain streaks is taken into consideration, and the prior knowledge of the rain streaks, i.e., the angle and the length of the motion blur kernel, are automatically learned.",
"The embedding of the motion blur kernel into the derain net, which maintains a plain ResNet structure, greatly enhances the performance (see the comparisons in Sec.",
"REF ).",
"To some extent, the utilization of the motion blur kernel in our method can be viewed as the traditional optimization model utilizing the regularizer to express the prior knowledge.",
"To evaluate the performance of the proposed KGCNN framework, we test it on both synthetic and real-world rainy images.",
"The networks are trained on synthesized rainy images.",
"We compare our KGCNN with six state-of-the-art methods, including three traditional methods: the unidirectional global sparse model (UGSMhttp://www.escience.cn/people/dengliangjian/index.html) [11], the discriminative sparse coding method (DSChttp://www.math.nus.edu.sg/~matjh/research/ research.htm) [18], and the method using layer prior (LPhttp://yu-li.github.io/) [16], as well as three deep learning based methods: the density-aware multi-stream deraining dense network (DIDhttps://github.com/hezhangsprinter/DID-MDN) [6], a plain convolutional neural network deraining method (CNNhttps://xueyangfu.github.io/projects/tip2017.html) [34], and the deep detail network (DDNhttps://xueyangfu.github.io/projects/cvpr2017.html) [7]."
],
[
"Rainy Images Simulation",
"With the observation model in Eq.",
"(REF ), the synthetic rainy images are generated by the following steps.",
"(1) Transform the background from RGB color space to YUV color spacehttps://en.wikipedia.org/wiki/YUV.",
"(2) Generate the raindrops mask $\\mathbf {M}$ by adding salt and pepper noise with signal-noise ratio from 0.9 to 1.0 to a zero matrix with the same size as the Y channel of the background, and adding a Gaussian blur with standard variance from 0.2 to 0.5.",
"(3) Generate the motion blur kernel $\\mathbf {K}$ with angle $\\theta $ sampled from $[45^\\circ ,135^\\circ ]$ and length $l$ varing from 15 to 30.",
"(5) Directly added the generated rain streaks $\\mathbf {R} = \\mathbf {K}\\otimes \\mathbf {M}$ to the background on Y channel, and the intensity values greater than 1 are set as 1.",
"(6) Finally, transform the image back to RGB color space."
],
[
"Experiments Setting",
"For fair comparisons, we use the default parameters in the codes for traditional methods and the default trained models for the deep learning methods.",
"Since existing rainy datasets do not consist of the information of the motion blur kernel, we train our networks only on our synthetic data.",
"The patch size is set as $64\\times 64 \\times 3$ .",
"Guided filter with radius 15 and regularization 1 is selected to decompose the rain images.",
"By preserving $99\\%$ of the energy, the kernel is projected onto a space of dimension 162.",
"Because of the full connection, the input image should be split into several patches for experiments.",
"We use Adam [55] optimizers with learning rate 0.01.",
"Our model is trained and tested on Python 3.5.2 with TensorFlow 1.0.1 framework on a desktop of GPU NVIDIA GeForce GTX 1060 with 6GB.",
"For other compared methods based in Matlab, they are running on Matlab 2017A.",
"Figure: Rain streak removal results by different methods on one rainy image of the Rain12 dataset."
],
[
"Synthesized Data",
"In this subsection, we evaluate performance of different state-of-the-art methods on the synthetic rainy images.",
"Three datasets are selected: 1) the benchmark dataset provided by Dr. Yu Li using the rain streaks rendering technique in [56] (denoted as Rain12), 2) 3 synthetic rainy images by our simulating method, and 3) several synthetic rainy images in [11].",
"Due to the limit of space, we only show partial results in this section, and please see more results in the supplementary materials.",
"For quantitative comparisons, we adopt the peak signal to noise ratio (PSNR), structure similarity index (SSIM) [57], feature similarity index (FSIM) [58], universal image quality index (UIQI) [59], and gradient magnitude similarity deviation (GMSD, smaller is better) [60] as the quality metrics of the deraining results.",
"Particularly, since the compared methods are implemented with different programming languages (or platforms), e.g., UGSM with Matlab and CNN with Python, we save all output images of different methods as png format, then reload them in Matlab and compute the corresponding quantitative results on RGB color space.",
"To show that KGCNN could remove rain streaks while keeping the texture and the contrast of background, we show the rain streak images (residual images between rainy images and resulted images).",
"Normalization is performed to the rain streak images so that we could distinguish whether the proposed method changes texture and contrast significantly or removes rain streaks completely.",
"For instance, if the rain streak images are too bright, it indicates the method significantly changes intensity contrast.",
"For the first dataset, Fig.",
"REF shows the visual results, local close-up images and rain streak images on one synthetic rainy image of the Rain12 dataset.",
"We can see that the proposed KGCNN method could remove the rain streaks completely while other approaches fail to do so (see local close-up images for better comparisons in Fig.",
"REF ).",
"Especially, it is easy to see that the obtained rain streaks by the proposed approach do not contain the structures of background, which indicates KGCNN has a very good ability for rain streak removal.",
"From the perspective of quantitative results, KGCNN method performs best for the 12 synthesized images, compared with other six state-of-the-art methods (see Table REF for more details).",
"Table: Quantitative comparisons of rain streak removal results by DID , DSC , LP , UGSM , CNN , DDN , and KGCNN on the Rain12 (average value).For the second dataset, we generate another 3 synthesized rainy images (road, night, and street) for test.",
"Some of resulted derain images by different methods are selected to be shown in Fig.",
"REF and Fig.",
"REF .",
"The visual results also demonstrate that the KGCNN method not only removes rain streaks completely, but also preserves the background information well.",
"We report the quantitative performance of the derain results obtained by different approaches in Table REF , which shows the superiority of the KGCNN method.",
"Figure: Rain streak removal results by different methods on 3 synthetic rain images (road, night, and street) by our simulating method.",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.Figure: The rain streak images of the rain streak removal results by different methods on 3 synthetic rain images (road, night, and street) by our simulating method.",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.Table: Quantitative comparisons of rain streak removal results by DID , DSC , LP , UGSM , CNN , DDN , and KGCNN on 3 synthetic rainy images by our simulating method.UGSM performs quite competitively for Rain12.Therefore, it is necessary to take more test images from UGSM (tree, panda, and bamboo) to compare the rain removal ability of the proposed method and UGSM method.",
"In addition, based on the code provided by the authors in [11], we also change the rain streaks' angles of these images from UGSM to generate three new synthesize images (tree2, panda2, and bamboo2) for testing.",
"Fig.",
"REF and Fig.",
"REF respectively present the visual and rain streak results of these images from [11], which indicates the superiority of the proposed method.",
"Figure: Rain streak removal results by different methods on 3 synthetic rainy images (tree, panda, and bamboo) selected from .",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.Figure: The rain streak images of the rain streak removal results by different methods on 3 synthetic rainy images (tree, panda, and bamboo) selected from .",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.In Fig.",
"REF and Fig.",
"REF , rain streaks with very large angles are added to background to formulate rainy images.",
"Noting that the UGSM is based on directional priors, which is quite effective to the case of vertical rain streaks, but less effective for the case of oblique rain streaks.",
"Therefore, from Fig.",
"REF and Fig.",
"REF , we can know that the proposed KGCNN method performs significantly better than UGSM method, both visually and quantitatively.",
"Moreover, the KGCNN method also exhibits better ability of rain streak removal, compared with other state-of-the-art methods.",
"Table REF also demonstrate the effectiveness of our method from the perspective of quantitative results.",
"Figure: Rain streak removal results by different methods on 3 new synthetic rainy images (tree2, panda2, and bamboo2).",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.Figure: The rain streak images of the rain streak removal results by different methods on 3 new synthetic rainy images (tree2, panda2, and bamboo2).",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) DID , (d) DSC , (e) LP , (f) UGSM , (g) CNN , (h) DDN , and (i) KGCNN.Table: Quantitative comparisons of rain streak removal results by DID , DSC , LP , UGSM , CNN , DDN , and KGCNN on 3 synthetic rainy images selected in ."
],
[
"Real-world data",
"For real-world data, since the ground truth images are unknown, we do not give the quantitative comparisons and only evaluate the performance of different methods visually, including the derain images and the rain streak images.",
"From Fig.",
"REF , the methods of DID, DDN, and KGCNN exhibit similar visual results and remove rain streaks completely, while other approaches fail to remove all rain streaks.",
"In addition, from the rain streak images in Fig.",
"REF , the methods of DID and DDN fail to separate the rain streaks and background texture well and leave some background texture to rain streaks, which demonstrates that the networks of DID and DDN could not distinguish background texture and the rain streaks well.",
"Moreover, the DID method also changes image contrast significantly.",
"Figure: Rain streak removal results and rain streak images by different methods on real rainy images.",
"From left to right: (a) the rainy images, the results by (b) DID , (c) DSC , (d) LP , (e) UGSM , (f) CNN , (g) DDN , and (h) KGCNN.In Fig.",
"REF , the KGCNN method removes the rain streaks completely while other approaches still exist obvious rain streaks.",
"From the rain streak images, our method could separate rain streaks excellently, while other method leaves some background texture to the separated rain streaks.",
"Especially the visual result by DID method shows a little of blur effect due to the loss of the texture information.",
"Particularly, readers can find more results in Fig.",
"REF which also verifies the superiority of the proposed KGCNN method.",
"Figure: Rain streak removal results and rain streak images by different methods on real rainy images.",
"From left to right: (a) the rainy images, the results by (b) DID , (c) DSC , (d) LP , (e) UGSM , (f) CNN , (g) DDN , and (h) KGCNN.Figure: Rain streak removal results and rain streak images by different methods on real rainy images.",
"From left to right: (a) the rainy images, the results by (b) DID , (c) DSC , (d) LP , (e) UGSM , (f) CNN , (g) DDN , and (h) KGCNN."
],
[
"Influence of kernel in the KGCNN method",
"In this paper, we propose a kernel guided CNN method for the image rain streak removal application.",
"The kernel plays a very important role to the KGCNN method.",
"There are still two problems.",
"(1) Dose the derain net output the rain streaks only using the information of rainy image and ignoring the kernel information?",
"(2) Dose the derain net work better if we retain it without the input of kernel?",
"To show the influence of the kernel in our method, we discard the kernel guided assumption to see the results what will happen with Rain12.",
"We use KGCNN to represent the proposed method, KGCNN$^a$ to represent the proposed method with kernel information being zero, and KGCNN$^b$ to represent the detrain net trained individually with our training data.",
"Fig.",
"REF shows the visual results there methods.",
"It is easy to know that the kernel plays an import role in KGCNN.",
"The derain net dose use the kernel to output the rain streaks (see the result of KGCNN$a$ ) and even we train the derain net individually, the result is still good enough (see the result of KGCNN$^b$ ).",
"The quantitative results in Table REF also demonstrate the similar conclusion.",
"In summary, the kernel guided assumption is quite important to the framework of the KGCNN method.",
"Figure: Rain streak removal results by different methods on 3 selected images from Rain12 dataset.",
"From left to right: (a) the background, (b) the rainy images, the derain results by (c) KGCNN a ^a, (d) KGCNN b ^b, and (d) KGCNN.Table: Quantitative comparisons of rain streak removal results by KGCNN a ^a, KGCNN b ^b, and KGCNN on Rain12 (average value)."
],
[
"Discussions on the depth and breadth",
"Increasing the depth of network or increasing the filter number of network can improve a network's capacity.",
"We also investigate the optimal network design to achieve the best derain results.",
"In this section, we test the impact of network depth and width of KGCNN on Rain12.",
"Especially, we test for depth $\\in \\lbrace 18,26,34\\rbrace $ and filter number $\\in \\lbrace 24,36,48\\rbrace $ .",
"Table REF shows the average values of the quantitative results.",
"As is clear, adding more hidden layers achieves better results over increasing the number of filters per layer while increasing computational time.",
"But we could see that there is over-fitting when depth is 34 and filter numbers is 48.",
"To balance the performance between avoiding the over-fitting and reducing the computation, we choose 26 as depth and 36 as filter number for our experiments above.",
"Table: Quantitative comparisons of rain streak removal results by different depth and filter number on Rain12(average value)."
],
[
"Conclusion",
"We have presented a deep learning architecture called KGCNN for removing rain streaks for single images.",
"Using guided kernel on the texture component, our approach learns the mapping function between rain image on detail component and rain streaks.",
"We show that convolutional neural networks, a technology widely used for high-level vision task, with guided kernel can also be exploited to successfully deal with natural images under bad weather conditions.",
"We also show that KGCNN noticeably outperforms other state-of-the-art methods with respect to image quality.",
"In addition, by using guided kernel, we are able to show that we do not need a very complex network to perform rain streak removal."
],
[
"Acknowledgment",
"The research is supported by NSFC (61876203, 61772003, 61702083) and the Fundamental Research Funds for the Central Universities (ZYGX2016J132, ZYGX2016KYQD142, ZYGX2016J129).",
"Thank the authors of DID [6], DSC [18], LP [61], UGSM [11], CNN [34], DDN [7] for providing the code."
]
] | 1808.08545 |
[
[
"Intertwined Spin and Orbital Density Waves in MnP Uncovered by Resonant\n Soft X-ray Scattering"
],
[
"Abstract Unconventional superconductors are often characterized by numerous competing and even intertwined orders in their phase diagrams.",
"In particular, the electronic nematic phases, which spontaneously break rotational symmetry and often simultaneously involve spin, charge and/or orbital orders, appear conspicuously in both the cuprate and iron-based superconductors.",
"The fluctuations associated with these phases may provide the exotic pairing glue that underlies their high-temperature superconductivity.",
"Helimagnet MnP, the first Mn-based superconductor under pressure, lacks high rotational symmetry.",
"However our resonant soft X-ray scattering (RSXS) experiment discovers novel helical orbital density wave (ODW) orders in this three-dimensional, low-symmetry system, and reveals intertwined ordering phenomena in unprecedented detail.",
"In particular, a ODW forms with half the period of the spin order and fully develops slightly above the spin ordering temperature, their domains develop simultaneously, yet the spin order domains are larger than those of the ODW, and they cooperatively produce another ODW with 1/3 the period of the spin order.",
"These observations provide a comprehensive picture of the intricate interplay between spin and orbital orders in correlated materials, and they suggest that nematic-like physics ubiquitously exists beyond two-dimensional and high-symmetry systems, and the superconducting mechanism of MnP is likely analogous to those of cuprate and iron-based superconductors."
],
[
"Intertwined Spin and Orbital Density Waves in MnP Uncovered by Resonant Soft X-ray Scattering B. Y. Pan State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China School of Physics and Optoelectronic Engineering, Ludong University, Yantai, Shandong 264025, China H. Jang J.-S. Lee Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA R. Sutarto F. He Canadian Light Source, Saskatoon, Saskatchewan S7N 2V3, Canada J. F. Zeng Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Y. Liu Center for Correlated Matter, Zhejiang University, Hangzhou, 310058, China X. W. Zhang Y. Feng Y. Q. Hao State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China J. Zhao State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, China H. C. Xu State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China Z. H. Chen Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai Synchrotron Radiation Facility, Shanghai 201800, China J. P. Hu$^\\dag $ Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China Collaborative Innovation Center of Quantum Matter, Beijing 100084, China D. L. Feng$^\\ast $ State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Unconventional superconductors are often characterized by numerous competing and even intertwined orders in their phase diagrams.",
"In particular, the electronic nematic phases, which spontaneously break rotational symmetry and often simultaneously involve spin, charge and/or orbital orders, appear conspicuously in both the cuprate and iron-based superconductors.",
"The fluctuations associated with these phases may provide the exotic pairing glue that underlies their high-temperature superconductivity.",
"Helimagnet MnP, the first Mn-based superconductor under pressure [1], [2], [3], lacks high rotational symmetry.",
"However our resonant soft X-ray scattering (RSXS) experiment discovers novel helical orbital density wave (ODW) orders in this three-dimensional, low-symmetry system, and reveals intertwined ordering phenomena in unprecedented detail.",
"In particular, a ODW forms with half the period of the spin order and fully develops slightly above the spin ordering temperature; their domains develop simultaneously, yet the spin order domains are larger than those of the ODW; and they cooperatively produce another ODW with 1/3 the period of the spin order.",
"These observations provide a comprehensive picture of the intricate interplay between spin and orbital orders in correlated materials, and they suggest that nematic-like physics ubiquitously exists beyond two-dimensional and high-symmetry systems, and the superconducting mechanism of MnP is likely analogous to those of cuprate and iron-based superconductors.",
"In strongly correlated electron systems, the kinetic and interaction energies of electrons compete, which often couples the charge, spin, and orbital degrees of freedoms, resulting in a variety of complex quantum phases.",
"Nematic order is one of the most important forms of intertwined order, which exists in both the two known families of high temperature superconductors — cuprates and iron-based superconductors.",
"In iron pnictides, e.g.",
"LaFeAsO and BaFe$_2$ As$_2$[4], [5], [6], the nematic orbital order often emerges at a higher temperature than the collinear antiferromagnetic order, and with half the period, indicating that the nematicity may be strongly tied to magnetic fluctuations.",
"In the stripe phase of the cuprates, the charge order forms with half the period of the spin order[7], [8].",
"Moreover, unlike magnetic order, the nematic order is time-reversal invariant, allowing it to directly couple to orders arising from charge and orbital degrees of freedom.",
"Furthermore, it also appears that superconductivity is maximized in the region with the strongest nematic fluctuations.",
"In the iron pnictides, for example, the orbital and/or spin fluctuations associated with the nematic order are suggested to promote superconductivity [9], [10].",
"All these essential phenomena suggest that the electronic nematic phase stands as a bridge between high temperature superconductivity and magnetism.",
"Figure: Basic properties and X-ray diffraction peaks of MnP.",
"(a) Magnetic susceptibility and resistance data of the MnP single crystal sample.",
"(b) Schematic illustration of the helical spin order and the RSXS scattering geometry.",
"The spin rotates about 21 ∘ ^{\\circ } between Mn1 and Mn2, and between Mn3 and Mn4, giving an incommensurate double helical spin order, or a helical spin density wave (SDW), with a period of about 8.6cc.",
"Mn1 and Mn3 spins form one spin helix, and Mn2 and Mn4 spins forms the other.The incident X-ray is linearly polarized.",
"(c) X-ray absorption spectroscopy (XAS) of MnP (solid black line) and three reference compounds MnO (dashed red line), Mn 2 _2O 3 _3 (dashed olive line), and MnO 2 _2 (dashed blue line) around the Mn LL-edge.",
"The reference spectra were obtained on beam line 8.0.3 at the Advanced Light Source (ALS) and are plotted here offset by 1.55 eV in order to match our MnP spectrum obtained at the SSRL.",
"The Mn LL-edge XAS of MnP is characteristic of Mn 2+ ^{2+} ions.",
"(d) Scattering peaks around q 1 q_1=(0 0 0.116), q 2 q_2=(0 0 0.234), and q 3 q_3=(0 0 0.352) at resonant energies (black dots) and non-resonant energy (red dots, E=655 eV) at 20 K. q 1 q_1 and q 2 q_2 signals were measured by a Schottky barrier photodiode, and the q 3 q_3 signals were measured by a quantum-efficient channeltron.",
"The three resonant peaks represent the strongest resonant signals at each q position (see Fig.",
"3).",
"(e) Schematic diagram of the location of the diffraction peaks in MnP including lattice peaks (red stars), magnetic peaks (blue dots), and the newly discovered resonance peaks in this paper (green and magenta dots).Figure: Temperature dependences of the diffractions peaks in MnP.",
"(a) (0 0 2) lattice peak measured at 2180 eV from 20 to 60 K. (b-d), q 1 q_1 L 3 L_3 (E = 637 eV, π\\pi polarization), q 2 q_2 L 2 L_2 (E = 648.8 eV, π\\pi polarization), and q 3 q_3 L 3 L_3 (E = 636.6 eV, π\\pi polarization) resonance peaks at different temperatures, measured on cooling.",
"The solid lines in each panel are Gaussian (for q 1 q_1 and q 2 q_2 resonance peaks) or Lorentz (for q 3 q_3 resonance peaks) fittings with linear background.",
"The shadowed area in the upper panel of (c) is to indicate the weak peak intensity at the highest measured temperature.",
"(e) Temperature dependence of the peak area of q 1 q_1, q 2 q_2 and q 3 q_3 near the helical transition, revealing their differing transition temperatures.",
"The data were obtained in the cooling processes.",
"Solid lines are guides to the eye.",
"(f) Temperature dependences of the q 1 q_1, q 2 q_2/2, and q 3 q_3/3.",
"Here, we used a photodiode detector for q 1 q_1, a channeltron detector for q 2 q_2, and a Greateyes CCD for q 3 q_3.",
"(g) and (h) are the temperature dependences of the peak area and 1/FWHM for q 1 q_1 (black circles) q 2 q_2 (red squares) resonance peaks in the cooling measurements, respectively.",
"Log scale were used for the vertical axes and the solid lines are guides to the eye.Under pressure, the helical magnetic compound CrAs and MnP have been recently found to be the first Cr- and Mn- based superconductor, respectively [1], [11].",
"For example, under ambient pressure, MnP first enters a ferromagnetic state at $T_C$ = 290 K, then a metamagnetic transition at $T_S$ = 50 K switches it into a double helical magnetic state with moments lying in the $ab$ plane (Fig.",
"1(a, b)).",
"Such a complex magnetic behavior suggests that MnP is most likely an unconventional superconductor like the cuprate and iron pnictides.",
"However, not only has the material a helical magnetic state that differs from those of cuprates and iron-based superconductors (Fig.",
"1(b)), but its lattice structure is three dimensional and does not possess any high symmetry rotational axis.",
"Moreover, the spin, charge and orbital degrees of freedom in MnP could be highly interconnected.",
"Therefore, MnP provides a novel playground to probe the relation between unconventional superconductivity, magnetism, and possibly nematicity in the vicinity of a helical spin order on a low-symmetry lattice.",
"Resonant soft X-ray scattering (RSXS), which can be viewed as a combination of x-ray absorption (XAS) and x-ray emission (XES) spectroscopies with x-ray scattering, provides a direct and powerful probe of the ordering in 3$d$ transition metal compounds[12], [13], [14].",
"To elucidate the electronic state of MnP, we first present its XAS spectrum at Mn $L$ -edge in Fig.",
"1(c), together with that of MnO as a fingerprint of the Mn$^{2+}$ valence state[15], Mn$_2$ O$_3$ as a typical Mn$^{3+}$ spectrum, and MnO$_2$ representing Mn$^{4+}$ .",
"Evidently, MnP has the typical absorption spectrum of a Mn$^{2+}$ state with the half filling high spin configuration.",
"To our knowledge, this is the first XAS measurement on MnP.",
"The Mn$^{2+}$ state in MnP is contrary to the previously speculated Mn$^{3+}$ valence in this material[16].",
"The spin moment on each Mn site is only $\\sim $ 1.3$\\mu _B$ [17]; this value is significantly reduced from the localized spin-only moment $\\sim $ 5$\\mu _B$ for the 3d$^5$ configuration in Mn$^{2+}$ by Hund's rule.",
"This low spin moment may arise from quantum fluctuations and hybridization[18], [19], as is the case for the iron-based superconductors[20], [21].",
"It should be noted that for a half-filled electronic system, the orbital moment usually should be quenched, thus prohibiting spin-orbital coupling.",
"However, orbital anisotropy/ordering have been found in several half-filled systems due to hybridization anisotropy[22] and vacancy modulation[23].",
"Therefore, orbital angular momentum may be partially restored.",
"Figure 1(b) shows the scattering geometry, where $bc$ was used as the scattering plane.",
"The incident X-ray is either horizontally ($\\pi $ ) or vertically ($\\sigma $ ) polarized.",
"In this configuration, $\\sigma $ polarization has electric field along the $a$ axis and momentum transfer along the (0 0 $L$ ) direction.",
"When helical magnetic order emerges below $T_S$ = 50 K, its spin moments (blue arrows) lie in the $ab$ plane with magnetic propagation wavevector (0 0 0.117)[24].",
"In order to avoid specular reflection, we used a MnP single crystal with a (101) surface and tilted the sample by 48.4$^{\\circ }$ in our RSXS experiments.",
"We made an exhaustive search for scattering signals at the Mn $L$ -edge along (0 0 $L$ ) at 20 K. By varying $L$ , X-ray polarization, photon energy, and detection mode, we discovered resonance peaks at $q_1$ =(0 0 0.11634$\\pm $ 0.00002), $q_2$ =(0 0 0.23470$\\pm $ 0.00003), and $q_3$ =(0 0 0.35239$\\pm $ 0.00004).",
"$q_2$ and $q_3$ wavevectors approximately double and triple that of $q_1$ , respectively.",
"Figure 1(d) shows the scans for the strongest resonance peaks at $q_1$ (E = 637 eV, $\\pi $ polarization, black dots), $q_2$ (E = 648.2 eV, $\\sigma $ polarization, black dots), $q_3$ (E = 641.2 eV, $\\pi $ polarization, black dots), and the corresponding non-resonant scans at 655 eV (red dots).",
"$q_1$ is perfectly consistent with the magnetic diffraction peak q$_m$ = (0 0 $\\delta $ ) in MnP, thus should correspond to helical magnetism.",
"It should be noted that the magnetic diffraction peaks of MnP only appear at [$H$ $K$ $L\\pm \\delta $ ] positions in which [$H$ $K$ $L$ ] represent the Bragg peaks from the lattice, a law universal to helical magnets that has been verified by neutron diffraction in MnP [25], FeP[26], and CrAs[27].",
"It is thus clear that the new diffraction peaks at $q_2$ and $q_3$ discovered by RSXS are not from magnetism.",
"The diffraction positions of lattice, magnetic, and our newly discovered peaks in reciprocal space are illustrated in Fig. 1(e).",
"Next, we study the temperature dependences of the diffraction peaks.",
"First, we take the (0 0 2) lattice diffraction peak as a reference.",
"It does not show any observable temperature dependence across $T_S$ and down to 20 K (Fig.",
"2(a)), indicating no structure transition in this temperature range.",
"In contrast, $q_1$ , $q_2$ , and $q_3$ peaks exhibit drastic temperature dependences.",
"Figures 2(b-d) show the three diffraction peaks taken with their corresponding resonant scattering conditions, at $q_1$ (E = 637 eV, $\\pi $ polarization), $q_2$ (E = 648.8 eV, $\\pi $ polarization), and $q_3$ (E = 636.6 eV, $\\pi $ polarization) at different temperatures in a cooling sequence, respectively.",
"As can be seen, not only does the peak intensity rapidly grow across $T_S$ , but the propagation wave vector moves to higher $q$ with decreasing temperature.",
"The peak areas of the $q_1$ , $q_2$ and $q_3$ diffraction peaks are plotted as a function of temperature in Fig.",
"2(e), all showing a jump just around 50 K, consistent with a first order transition (the hysteresis behavior of the $q_1$ peak can be found in Supplementary Fig. S4(a)).",
"Interestingly, the $q_1$ peak is fully developed at a slightly lower temperature than the $q_2$ peak, while the full-development temperature of the $q_3$ peak lies in between or similar to the $q_2$ peak.",
"This observation can be justified by more temperature dependence measurements (Supplementary Fig. S4(b)).",
"In Fig.",
"2(f), we show the wavevector evolution with temperature, in which $q_2$ and $q_3$ are divided by 2 and 3, respectively, in order to scale with $q_1$ .",
"The propagation wavevectors of the three resonance peaks all show pronounced temperature dependence below $T_S$ , which is typical for an incommensurate electronic order.",
"Throughout the measured temperature range, $q_2$ and $q_3$ are approximately at the 2$q_1$ and 3$q_1$ positions within the experimental accuracy, respectively, indicating that these electronic orders are interconnected with each other.",
"According to the upper panels of Fig.",
"2(b-d), there are detectable scattering intensities above $T_S$ .",
"The $q_1$ and $q_2$ peaks persist up to the highest measured temperatures, i.e., 58.5K and 54.5 K, respectively (upper panels of Fig.",
"2(b) and Fig. 2(c)).",
"However, $q_3$ peak intensity is not detectable above 52 K (upper panel of Fig.",
"2(d)) due to its weak intensity and insufficient sensitivity of the detector.",
"To reveal the detailed evolution at high temperatures, Fig.",
"2(g) plots the peak areas as a function of temperature in the cooling process in log scale, and Fig.",
"2(h) shows the temperature dependences of the correlation length, $\\xi _i$ , which is defined as 1/FWHM, reflecting the average domain size along $c$ .",
"Above 50K, there is a similar slow-growing behavior for both the $q_1$ and $q_2$ peaks in Figs.",
"2(g) and 2(h).",
"The peak intensities are two orders of magnitude lower than their full values, thus they are due to fluctuating orders, and the long tails into high temperatures are likely related to local strain distributions.",
"A similar behavior has been observed for the nematic order in iron pnictides under uniaxial strain [5].",
"With decreasing temperature, $\\xi _1$ and $\\xi _2$ quickly increase almost identically before they saturate at $T_1^F$ and $T_2^F$ , respectively, indicating the peaks are from the same domain and have the same onset temperature, $T^E$ , which can be higher than 58.5K, the highest measured temperature.",
"Below $T_2^F$ , $\\xi _1$ continues to increase until $T_1^F$ , so it is larger than $\\xi _2$ at low temperatures.",
"Meanwhile, the peak intensities show sudden jumps to their fully-developed values just before $T_1^F$ and $T_2^F$ as well.",
"Therefore, $T_1^F$ , $T_2^F$ and $T_3^F$ are defined as the temperatures that the $q_1$ , $q_2$ and $q_3$ peaks are fully developed, respectively.",
"Note that, between $T_2^F$ and $T_1^F$ , the $q_1$ peak intensity is low, indicating that the spin order is still fluctuating while the order corresponding to $q_2$ is already static.",
"In addition, we found that $\\xi _3$ is almost identical to $\\xi _2$ .",
"These behaviors contain rich information on the evolutions of the orders corresponding the diffraction peak, which will be discussed later.",
"Figure: Resonance profiles at q 1 q_1, q 2 q_2, and q 3 q_3 with TT = 20 K. The resonance profiles at each resonance position are plotted as a function of X-ray energy around the Mn LL-edge and reciprocal lattice vector (0 0 LL), with σ\\sigma (upper panels) or π\\pi (lower panels) linearly polarized incident photons.",
"(a, b) q 1 q_1 resonance profile.",
"A photodiode detector was used to measure the scattering intensity I σ I_{\\sigma } and I π I_{\\pi } which are comparable in intensity.",
"(c-f) q 2 q_2 and q 3 q_3 resonance profiles measured by a channeltron detector, due to their relatively weak intensity.",
"For q 2 q_2, the I σ I_{\\sigma } maximum is ∼\\sim 3 times of the I π I_{\\pi } maximum.",
"For q 3 q_3, the I σ I_{\\sigma } maximum is about 80% of the I π I_{\\pi } maximum.",
"All data shown here are raw scattering intensity without background subtraction or absorption correction.",
"The color bars indicates scattering intensity in arbitrary unit.To comprehensively elucidate the nature of the three resonant peaks at $q_1$ , $q_2$ , and $q_3$ , we have plotted in Fig.",
"3 the resonant profiles around the three $q$ positions with $T$ = 20 K, i. e., the scattering intensities as a function of reciprocal lattice (0 0 $L$ ), X-ray energy, and incident photon polarization.",
"Since there was no polarization analyzer before the detector, the polarizations of the scattered photons were not distinguished.",
"That is, using self-explanatory subscript convention, the detected scattered intensities for two different incident photon polarizations $I_{\\pi }=I_{\\pi \\pi }+I_{\\pi \\sigma } \\hspace{28.45274pt}I_{\\sigma }=I_{\\sigma \\pi }+I_{\\sigma \\sigma }.$ Clearly, the resonant profiles of $q_1$ , $q_2$ , and $q_3$ are very different from each other in maximal intensity, resonant energy, and polarization dependence.",
"For example, the resonant energies for the maximum peaks at $q_1$ , $q_2$ , and $q_3$ are 637 eV ($\\pi $ polarization), 648.2 eV ($\\sigma $ polarization), and 641.2 eV ($\\pi $ polarization), respectively.",
"The maximal intensity at $q_3$ is $\\sim $ 23 times weaker than that at $q_2$ , and the maximal intensity for $q_2$ is $\\sim $ 30 times weaker than that of $q_1$ , based on the data of the resonance peaks taken with the same detector.",
"Figure: q-integrated intensity I q I_{q} of the three resonance peaks shown in Fig.",
"3.",
"(a) I q1 I_{q1} with σ\\sigma (black line) and π\\pi (red line) polarizations.",
"I q2 I_{q2} and I q3 I_{q3} are plotted in (b) and (c), respectively.Figure: An illustration of the intertwined orders in MnP.",
"(a) The helical spin density wave with a period of ∼\\sim 8.6c8.6c is shown with Mn ions with equal spacing in the c direction.",
"(b) The orbital density wave with 1/2 the period is represented by an exaggerated mixture of d xy d_{xy} and d x 2 -y 2 d_{x^2-y^2} orbitals that rotates together with the spin, but is symmetric under 180 ∘ ^\\circ rotation.",
"(c) The orbital density wave with 1/3 the period is represented by an exaggerated mixture of d xz d_{xz} and d yz d_{yz} orbitals that rotates three times as fast as the spin.",
"(d) Cartoon showing that short-ranged and fluctuating SDW (represented by the blue area) and ODW domains (represented by the texture) share the same region in MnP for T E ≥T^E \\ge T≥T 2 F \\ge T_2^F.",
"(e) When T 2 F ≥T_2^F \\ge T≥T 1 F \\ge T_1^F, the ODW is fully developed and stops to grow with decreased temperature, while SDW domains continue to grow.",
"There is regions with fluctuating spin order but without orbital order.",
"(f) When T≤T 1 F T \\le T_1^F , the spin order is fully developed.The $q$ -integrated resonant profiles are compared in Fig.",
"4, as a function of energy and polarization.",
"The resonant profile of the $q_1$ peak behaves similarly in both polarizations, consistent with its origin in resonant magnetic scattering.",
"As illustrated by a simple analysis, magnetic scattering will result in $\\pi \\pi $ , $\\pi \\sigma $ and $\\sigma \\pi $ scatterings with comparable intensity[28], [29].",
"The magnetic scattering matrix element is related to the dipole selection rules, which have the same origin as soft x-ray magnetic dichroism.",
"This explains it having the strongest intensity amongst all three peaks.",
"For the $q_2$ peak, $I_{\\sigma }$ is about triple $I_{\\pi }$ .",
"Moreover, the energy positions of the $q_2$ profile differ between the $\\sigma $ and $\\pi $ polarizations, which is unlikely from magnetic scattering.",
"The resonance profile of the $q_3$ peak shows moderate polarization dependence.",
"The saturation of the $q_1$ peak intensity below $T_1^F$ represents the full development of the incommensurate helical spin density wave.",
"Remarkably, the weak $q_2$ peak is fully developed slightly above $T_1^F$ , when the $q_1$ peak intensity is still two orders of magnitude smaller than its full value.",
"It implies that this is not a simple second-order harmonic of the spin density wave but an indication of a hidden order which is induced by magnetic fluctuations.",
"Applying Landau theory to the phase transition and using general symmetry analysis, we extract the nature of these orders by assuming that the Landau free-energy functional depends only on the fundamental Fourier components of the two different electronic orders.",
"The $q_1$ peak corresponds to the double helical spin density wave, $\\vec{S}_{q_1,a}\\propto (i,1,0)$ where $ a=1,2 $ denote the two helices.",
"As $q_2\\approx 2q_1$ , the hidden order, $\\Pi _{q_2}$ , must be coupled to a quadratic term of $\\vec{S}_{q_1} $ .",
"If we take $\\Pi _{q_2}$ to be a scalar, a natural choice of $\\Pi _{q_2}$ is the charge density wave, $\\rho _{q_2}$ .",
"The lowest order coupling in Landau theory can be written as $H_{c}=\\sum _{ab}\\lambda _{c,ab} \\rho _{q_2}(\\vec{S}^*_{q_1,a} \\cdot \\vec{S}^*_{q_1,b})+h.c.$ .",
"However, such an unequal charge distribution at different Mn sites could not be found in our LDA calculations shown in the supplementary materials.",
"Instead, an orbital distribution with half the periodicity of the magnetic order can be explicitly obtained in the calculation.",
"Here, we suggest that the hidden order must be a orbital density wave (ODW), which can be induced by the orbital redistribution in developing the double helical spin density wave.",
"In a simple helically ordered state, it is well known that an orbital redistribution can be induced by spin-orbital coupling in the presence of the crystal field, and such an orbital redistribution only depends on $|\\vec{S}_{q_1}|$[30], giving the X-ray magnetic linear dichroism effect (XMLD).",
"In this case, spin-parallel and -antiparallel Mn atoms would have identical orbital (wave function) distribution, which explains why $q_2$ corresponds to an order with a period half that of the magnetic peak.",
"This situation resembles the stripes in cuprates, but in MnP both orders are incommensurate with respect to the lattice.",
"We observe resonance at the Mn $2p$ to $3d$ transition, and orbital order of $3d$ electrons gives an electronic orbital order, ${\\Pi }_{\\alpha \\alpha }$ .",
"Thus a ODW is simultaneously developed in the helical spin state.",
"Now, in the case of double-helical spin order, we argue that a tiny orbital modulation can be induced by spin fluctuations even before the static helical spin order is developed.",
"In general, the coupling between the ODW and the double-helical spin order in Landau theory can be written as $H_{Q}=\\sum _{\\alpha }{\\Pi }_{\\alpha \\alpha ,q_2} [\\lambda ^\\alpha _{Q,1} ((S^{\\alpha *}_{q_1,1})^2 +(S^{\\alpha *}_{q_1,2})^2) + 2\\lambda ^\\alpha _{Q,2} S^{\\alpha *}_{q_1,1}S^{\\alpha *}_{q_1,2}]+h.c..$ With this coupling, there is a new phase in which $<S^{\\alpha *}_{q_1,a}>=0$ but $<S^{\\alpha *}_{q_1,1}S^{\\alpha *}_{q_1,2}>\\ne 0$ , which describes the locking of the magnetic fluctuations between the two helices.",
"In general, this phase could exist slightly above $T_s$ , thus explaining the intriguing full development of the $q_2$ peak at a slightly higher temperature than the $q_1$ peak.",
"The linear coupling between the ODW and $S^{\\alpha *}_{q_1,1}S^{\\alpha *}_{q_1,a}$ must result in $<{\\Pi }_{\\alpha \\alpha ,q_2}>\\ne 0$ if $<S^{\\alpha *}_{q_1,1}S^{\\alpha *}_{q_1,2}>\\ne 0$ .",
"This argument is generally known as “order by disorder\" and has been used to explain the nematicity in FeAs-based superconductors in which a similar phenomenon has been observed in the parent compounds, e.g.",
"BaFe$_2$ As$_2$[5], [6], where the nematic orbital order emerges at a slightly higher temperature than the collinear spin order due to spin fluctuations.",
"Since the spins lie in the $ab$ plane in the double helical phase, the orbital moment should also be in the $ab$ plane from the coupling $H_Q$ .",
"This explains the observed polarization dependence of the $q_2$ peak and also suggests that the charge redistribution mainly occurs in the $d_{x^2-y^2}$ and $d_{xy}$ orbitals (the orbital basis in the octahedral coordination with these two orbitals in the $ab$ -plane is used here for the ease of explanation, while we note Mn ions are in a tilted and distorted octahedrons made of P ions).",
"The observation of the weak $q_3$ peak also lends strong support to the presence of hidden ODW order since it represents the harmonics generated by the coupling between the ODW and the helical spin order.",
"The intensity of the $q_3$ peak is less than 1/600 of that of the SDW peak, yet it saturates at a slightly higher temperature.",
"This remarkable behavior suggests that the $q_3$ peak is also non-magnetic in nature.",
"Moreover, since it is slightly stronger for $\\pi $ -polarized than $\\sigma $ -polarized incident photons (Fig.",
"4(c)), it should involve $d_{xz}$ , $d_{yz}$ , and possibly $d_{3z^2-r^2}$ orbitals.",
"The simplest possibility for the $q_3$ peak would the rearrangement or modulation of these orbitals induced by the coupling between the ODW ($q_2$ ) and the helical SDW ($q_1$ ) as a third-order effect.",
"This represents another unique ODW that is observed here for the first time.",
"In Figs.",
"5(a-c), we summarize the observed orders.",
"The helical spin density wave is shown in Fig.",
"5(a), represented by spins rotated by equally-spaced in-plane angles.",
"In Fig.",
"5(b), the in-plane ODW follows the SDW with half the period, represented by a charge distribution whose axis follows the orientation of the spin.",
"In Fig.",
"5(c), the out-of-plane ODW rotates with one third of the SDW period.",
"These helical orbital density waves in MnP are discovered for the first time by RSXS, and they are intertwined with the helical spin density wave.",
"It is particularly noteworthy that the temperature dependencies in Fig.",
"2 further illustrate the intricate relation between the spin and orbital orders.",
"The short-ranged and fluctuating orbital and spin orders share the same domain as they both emerge and grow upon lowering the temperature (Fig.5(d)).",
"This starts from quite high temperatures, likely related to strain distributions in the system.",
"Upon further cooling, as shown in Fig.",
"5(e), the ODW order freezes and its domains are fixed after passing its fully-developed temperature $T_2^F$ .",
"Meanwhile, the spin order is still fluctuating and its domains keep expanding, since the intensity of the spin order peak is still a few percent of its full value at low temperature.",
"As a result, there are regions with fluctuating spin order but no orbital order in this temperature range.",
"Because the orbital order is fairly weak, as shown by the weak diffraction peak intensity, the orbital distribution can be influenced or pined by local strain or defects.",
"Therefore, the orbitals in these regions are disordered, and its configuration could not follow the helical spin rotation.",
"When temperature is further lowered below $T_1^F$ , the spin order is fully developed and its domains stop to grow, as illustrated in Fig. 5(f).",
"We note that because the out-of-plane ODW in Fig.",
"5(c) is the descendant of the SDW and the in-plane ODW, its domain is identical to that of the in-plane ODW.",
"A similar observation was made in Pr$_{0.6}$ Ca$_{0.4}$ MnO$_{3}$ , whose spin order correlation length is longer than its charge/orbital order correlation length, which was attributed to likely decoupling between spin and charge/orbital orders[31].",
"The diffraction peak intensity of its orbital order is also much weaker than the spin order in Pr$_{0.6}$ Ca$_{0.4}$ MnO$_{3}$ , although its spin ordering temperature is lower than that of the charge/orbital order.",
"So the picture revealed in Figs.",
"5(d-f) through our detailed temperature dependence are likely ubiquitous for systems with both spin and orbital ordering, which explains the difference in the correlation lengths in Pr$_{0.6}$ Ca$_{0.4}$ MnO$_{3}$ .",
"Our findings provide an unprecedented picture on the intricate interplay between spin and orbital orders in MnP, and show that intertwined ordering and nematic-like ordering is ubiquitous to the phase diagrams of unconventional superconductors, even in the case of low symmetry and incommensurate ordering.",
"The extraordinary helical ODWs found here may provide a foundation for understanding the complex behaviors of spin/orbital order in helimagnets and correlated systems in general, and for understanding the unconventional superconductivity in MnP and other related materials.",
"We acknowledge fruitful discussions with prof. J. L. Luo.",
"This work is supported in part by the National Key Research and Development Program of China (Grant No.",
"2016YFA0300200 and No.",
"2017YFA0303104), the National Natural Science Foundation of China (Grant No.",
"11804137), the Science and Technology Commission of Shanghai Municipality (Grant No.",
"15ZR1402900), and the Natural Science Foundation of Shandong Province (Grant No.",
"ZR2018BA026).",
"The experiments were conducted at the REIXS beamline of the Canadian Light Source (CLS) under Proposal No.",
"24-7906, and at beamline 13-3 of Stanford Synchrotron Radiation Lightsource (SSRL); SSRL is operated by the US DOE Office of Basic Energy Science.",
"$^\\dag $ jphu@iphy.ac.cn $^\\ast $ dlfeng@fudan.edu.cn Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION"
]
] | 1808.08562 |
[
[
"The effect of retardation in the random networks of excitable nodes\n embeddable in the Euclidean space"
],
[
"Abstract Some features of random networks with excitable nodes that are embeddable in the Euclidean space are not describable in terms of the conventional integrate and fire model (IFM) alone, and some further details should be involved.",
"In the present paper we consider the effect of the retardation, i.e.",
"the time that is needed for a signal to traverse between two agents.",
"This effect becomes important to discover the differences between e.g.",
"the neural networks with low and fast axon conduct times.",
"We show that the inclusion of the retardation effects makes some important changes in the statistical properties of the system.",
"It considerably suppresses/restricts the amplitude of the possible oscillations in the random network.",
"Additionally, it causes the critical exponents in the critical regime to considerably change."
],
[
"Introduction",
"Complex networks today have a wide applications in science, ranging from neuroscience [1] and intelligent signal processing [2], to the social networks and World Wide Web [3].",
"In the neural networks, the theories of adoptive optimizing control can be served as a basis for the learning process which, in the behavioral sense, is driven by changes in the expectations about the future salient events such as rewards and punishments [4].",
"There are many models to explain the experimental neuronal avalanches [5] which are based on the Hodgkin-Huxley model [6].",
"The criticality is a key factor in brain, since it improves the learning [7], optimizes the dynamic range [8], [9], [10], [11], makes information processing efficient [12], and leads to optimal transmission and storage of information [13].",
"In the current state of research on the complex networks with excitable agents, the communications between the interacting agents are supposed to be instantaneous [2].",
"Instantaneous here means that the conduction time is independent of the length of the connection, i.e.",
"two signals traversing two unequal connections in length have the same travel period.",
"Apparently the length should be meaningful here, and the network should have the capability of being embedded in the Euclidean space.",
"Many instantaneous artificial neural networks (which is a type of massively parallel computing architecture based on brain-like information encoding) with a vast range of applications have been invented, like signal processing and also pattern recognition [2].",
"In the systems which are embedded in the Euclidean space, when the speed of the signal (whatever it is) between two agents is very higher than the characteristic speeds in the system (resulting from the speed of the activities of the agents), this approach works well, as can be seen in the partial success of the instantaneous models in describing some experiments on brain [5], [14], like the self-organized criticality mode of brain activity [15], [16], the chaos for balanced excitatory and inhibitory activity [17], the neuronal coherence [18], and the synchronization of cortical activities [19].",
"Also theoretical explanation of the neuronal avalanches which are seen in the cerebral cortex (in which the spontaneous neural activities occur at the critical state) are based on such an instantaneous dynamics [20].",
"The instantaneous mechanisms which have also been proposed to explain the signal propagation in neocortical neurons based on the repetitions of spontaneous patterns of synaptic inputs should be added to this list [21].",
"There are however some situations that this speed is not that high, and one should take the retardation effects into account.",
"Here the various time scales play a vital role.",
"For example, there are many time scales for the neurons in primary auditory cortex of cats, ranging from hundreds of milliseconds to tens of seconds [22].",
"For a general argument on the time scales see [23].",
"In Ref.",
"[24] the hierarchy of time scales in the brain has been considered and analyzed.",
"These time scales are not however necessarily fixed, and in some situations they can be tuned.",
"An example such a tuning of time scale of neuronal activities has been reported in [25], for which millisecond time scales has been achieved.",
"A tangible example of the importance of the time scales and conduction times (and correspondingly the speed of signal) is the neural systems whose constituents (neurons) are lacking the myelin sheath in which the nerve conduction velocity in the avalanche pulse dynamics are not that high [26].",
"Axon conduction time is definitely a relevant quantity in these networks.",
"This nerve conduction velocity can also be precisely regulated with internal mechanisms, correct exertion of motor skills, sensory integration and cognitive functions [27].",
"In a neuronal population if the conduction velocity is low or equivalently the length of the axons is comparable with the speed of signal times the characteristic times of neuronal activities, the retardation effects become important.",
"The retardation in a nervous system is the effect in which the present activity of neuron in related to the activity of its neighboring neuron at $\\delta t=r/v$ times ago in which $r$ is the distance between two neurons and $v$ is the speed of the signal.",
"These retardation effects are expected to be an important in the neural systems with low-speed neurons.",
"For the neuronal cells with the Myelin sheath around their axons (a fatty insulating later that surrounds the nerve cells of jawed vertebrates, or gnathostomes) the speed of the signal is higher than that ones without myelin sheath.",
"This causes a lot of differences in these systems, which separates jawed vertebrates from the invertebrates.",
"In vertebrates, the rapid transmission of signals along nerve fibers is made possible by the myelination of axons and the resulting salutatory conduction in between nodes of Ranvier [27].",
"Among the vertebrates also, the speed of neuronal signals are more or less higher for more intelligent species.",
"Myelination not only maximizes conduction velocity, but also provides a means to systematically regulate conduction times in the nervous system, which to date, has not been understood well [27].",
"Node assembly, internode distance and the diameter of axon, which are controlled by myelination glia, determine the speed of signals along axons.",
"All of these show the importance of the retardation effects in real neural systems.",
"In this paper we consider these retardation effects for a random network with refractory period, i.e.",
"the agents are prevented to send a signal immediately after spiking.",
"Our numerical results show that the retardation effects, not only change the critical behaviors, but also decrease the oscillatory behaviors of the system.",
"By analyzing the branching ration and other statistical tests we show that the point (in terms of largest eigenvalue of the adjacency matrix $\\lambda $ ) at which the critical behaviors starts, the interval of critical behaviors and the point at which the bifurcation begins is just the same as the instantaneous random system.",
"The tuning of the signal propagation in such random networks in therefore promising for controlling some undesirable oscillatory responses.",
"The paper has been organized as follows: In the next section we explain the effects of the retardations and also the method to enter it in the calculations.",
"The SEC.",
"has been devoted to the numerical results and the explanation of the behaviors of the model.",
"We close the paper by a conclusion."
],
[
"Retardation effects",
"In this section we consider a random undirected graph with $N$ excitable nodes.",
"Each two nodes are connected with the probability $q$ , which results to the average node degree $\\left\\langle k\\right\\rangle =qN$ .",
"The connections are weighted with quenched random numbers $w_{i,j}$ (between nodes $i$ and $j$ which are connected), whose distribution is uniform in the interval $[0,2\\sigma ]$ , in which $\\sigma $ is an external parameter.",
"The state of a node ($i$ ) at time $t$ is described by $A_i(t)$ which is called the activity, assuming two values: active $A_i=1$ or quiescent $A_i=0$ .",
"The aggregate activity at time $t$ is defined as $x(t)\\equiv \\sum _{i=1}^N A_i(t)$ , which is commonly used for analyzing the statistical properties and also the stability of the system.",
"According to the integrate and fire model the $i$ th node at time $t$ becomes active depending on the aggregate input signal: $p(A_i(t)=1)=f\\left( \\sum _{j}w_{i,j}A_j(t-1)\\right)$ in which $f$ is a dynamical (monotonically increasing) map which yields the probability that a node becomes active based on the input signal to that node, and is commonly chosen to be: $f(y)={\\left\\lbrace \\begin{array}{ll}y, & 0\\le y \\le 1\\\\1 & y>1\\end{array}\\right.",
"}$ This dynamics is known to be dictated by the largest eigenvalue of the adjacency matrix $w_ij$ , namely $\\lambda $ [11], [28].",
"For the random graph that has been considered in this work, this eigenvalue is equal to $\\lambda =\\sigma \\left\\langle k\\right\\rangle =\\sigma qN$ [28], [29], [30].",
"For $\\lambda <1$ the system has an attractor $x=0$ , i.e.",
"some (stochastic) time after the external local drive, all nodes of the system become inactivate.",
"The completely inverse behavior is seen for $\\lambda >1$ in which the perturbation grows with time, reaching $x=N$ at some stochastic time.",
"The intersection between these two intervals, i.e.",
"$\\lambda =0$ is known to be critical for which some power-law behaviors occur for e.g.",
"the avalanches ($\\equiv $ an overall process between starting and ending an activity).",
"Let us define $S$ as the avalanche size ($\\equiv $ the total number of activities in an avalanche), $M$ as the avalanche mass ($\\equiv $ the total number of distinct nodes which have been activated (at least once) in an avalanche), $D$ as the avalanche duration ($\\equiv $ the total time interval of an avalanche), and $x$ the active nodes at a given time.",
"Then the fingerprint of the criticality can be found in the power-law behavior of the distribution functions, i.e.",
"$N(\\zeta )\\sim \\zeta ^{-\\tau _{\\zeta }}$ in which $\\zeta =S,M,D,x$ .",
"Also the critical point is detectable in terms of the branching ratio which is defined as the conditional expectation value $b(X)\\equiv E\\left[\\frac{x_{t+1}}{X} |x_t=X \\right] $ , i.e.",
"the expectation value of $x_{t+1}/X$ conditioned to have $x_t=X$ .",
"For the critical system $\\lim _{X\\rightarrow 0}b(X)=0$ , and also $\\lim _{X\\rightarrow 0}\\frac{\\text{d}b(X)}{\\text{d}X}<0$[30].",
"These two conditions state that when $X\\rightarrow 0$ , $b(X)$ approaches to zero from the negative values.",
"It also determines the possible fixed points of the model in hand by the condition $b(X^*)=1$ .",
"Now let us explain the network with refractory period (which has been done in Ref.",
"[30]) and also the retardation effects.",
"It is shown that the inclusion of the refractory period in the dynamics has some nontrivial effects, like the extension of critical interval, bifurcation, and non-trivial fixed points [30].",
"The retarded integrate and fire model is defined by the following non-linear dynamical equation: $p(A_i(t)=1)=\\delta _{A_i(t-1),0}f(\\text{sum}(A_i(t)))$ in which $\\text{sum}(A_i(t))$ is the integrated effect which has arrived to the $i$ th site at time $t$ , taking into account the retardation effects.",
"Also $\\delta _{A_i(t-1),0}$ is unity if $A_i(t-1)=0$ and is zero otherwise, i.e.",
"it is the effect of the refractory period in the node.",
"The sum-function sums the integrated retarded weighted signals, and is defined as: $\\text{sum}(A_i(t))\\equiv \\sum _{t^{\\prime }=0}^t\\sum _{j=1}^N w_{i,j}A_j(t^{\\prime })G_{i,j}(t,t^{\\prime })$ in which we have defined $G_{i,j}(t,t^{\\prime })\\equiv \\delta (t^{\\prime },t-\\frac{|i-j|}{v})$ as the retarded Green function, and $|i,j|$ is the distance of $i$ th and $j$ the nodes.",
"Therefore, this dynamic works for networks that are embedded in the Euclidean space, that is supposed to be two-dimensional in this study.",
"It is notable that it is not the only way to define $G_{i,j}(t,t^{\\prime })$ .",
"For example, one can take into account the dissipation of the signal as a function of the length or the time.",
"By inserting this into Eq.",
"REF , one finds that: $p\\left( A(t)=1\\right) =\\delta _{A_i(t-1),0}f\\left[ \\sum _{j=1}^Nw_{i,j}A_j\\left( t-\\frac{|i-j|}{v}\\right)\\right]$ This function carries instantaneously the effects of retardation and the refractory period, and $w_{i,j}$ is a periory known quenched stochastic variable that was introduced above."
],
[
"Numerical details",
"For building the host random network, we simply choose randomly two nodes and connect them.",
"We repeat this $q\\frac{N(N-1)}{2}$ times ($\\frac{N(N-1)}{2}$ being the total possible links in the system).",
"In Fig.",
"REF we have shown schematically a random graph that has been embedded to two dimensions.",
"The Fig.",
"REF shows the histogram of the lengths of the connections ($P(L)$ ) for $N=50^2, 100^2$ and $150^2$ Erdos-Renyi network (embedded in the Euclidean space) which have their peaks at $L=\\frac{1}{2}\\sqrt{N}$ as expected.",
"In Ref.",
"[30] it has been shown that the model with refractory period shows three relevant regimes: subcritical regime ($\\lambda <1$ ), extended critical regime $1\\le \\lambda \\le 2$ , and period-2 oscillatory regime $\\lambda >2$ .",
"We examine the effect of retardation in all of these regimes.",
"We also set the signal velocity to unity, i.e.",
"$v\\equiv 1$ , since it depends on the scale of the system.",
"In addition to the activity-dependent branching ratio $b(X)$ , two kinds of quantities are processed: the distribution function of $\\zeta $ and the scaling behaviors $\\eta \\sim \\eta ^{\\gamma _{\\eta \\zeta }}$ in which $\\eta ,\\zeta =S,M,D$ .",
"Figure: (Color online): (a) The schematic pattern of a highlighted agent in the random network embedded in a two-dimensional space.",
"(b) The distribution of the length between nodes in a simulated random network for N=100 2 N=100^2 (main panel), N=50 2 N=50^2 (lower inset) and N=150 2 N=150^2 (upper inset).We should be careful about the definitions.",
"For $\\lambda \\le 1$ the stable fixed point is $x^*=0$ , whereas for $1<\\lambda \\le 2$ $x^*\\ne 0$ will be attractor of the dynamics (these fixed points should be obtained by the condition $b(x^*)=1$ , and also $\\frac{\\text{d}b}{\\text{d}X}|_{X=x^*}<0$ ).",
"Therefore, for $\\lambda \\le 1$ we have some well-defined avalanches (avalanche $\\equiv $ the process in the time interval in which the activity starts from and ends on zero).",
"For $1<\\lambda \\le 2$ however we should define the avalanche in another way, since the process does not end, and $x$ fluctuates around $x^*\\ne 0$ .",
"In this case we define a threshold $X=x^*$ and define the avalanche as the process which starts from and ends on this threshold.",
"The time series for $\\lambda =0.9, 0.987, 1.5$ and $3.75$ have been shown as an instance in Fig.",
"REF for the dynamical system with retardation effects.",
"It is seen that for $\\lambda <1$ the stable fixed point is zero, and for $1<\\lambda = 1.5<2$ non-zero stable fixed point arises, and also for $\\lambda =3.75>2$ the system is in the oscillatory phase.",
"Figure: The time series for the activity (with retardation effects) for various rates of λ\\lambda .All of our analysis In the remaining of the paper will be compared with the results for the system without retardation effects."
],
[
"measures and results",
"We first start with the activity-dependent branching ration $b(X)$ .",
"In Fig.",
"REF we have shown this function for both retarded and instantaneous dynamical systems.",
"We have defined $M_c$ by the relation $b(M_c)=1$ .",
"The main panel shows $b(M)$ in terms of $M-M_c$ , from which we see that the spoles at the points in which the graphs cross $b(M)=1$ is negative.",
"This confirms that the found points are stable fixed points.",
"In the left inset the $\\lambda $ dependence of $M_c$ has been compared for $N=50^2, 100^2$ and $150^2$ networks.",
"It is notable that the mean field results for the instantaneous (retardation-free) dynamical systems reveals that $M_c^{\\text{spontaneous}}=N(1-\\frac{1}{\\lambda })$ .",
"The data in this inset is consistent with the mean field result for instantaneous dynamical system, but the (absolute value of) slope of the graphs are very higher, demonstrating that the dynamics (towards the fixed points) is more fast.",
"Figure: (Color online): (a) The branching ratio b(M)b(M) for retarded dynamics (main panel) and instantaneous dynamics (upper inset).",
"Left inset: M c M_c in terms of λ\\lambda for N=100 2 N=100^2.",
"Right inset: The same for N=50 2 N=50^2.",
"The distribution function P(x)P(x) for retarded dynamics (b) and instantaneous dynamics (c).",
"The inset of (b): f(λ)≡x ¯ upperbranch -x ¯ lowerbranch f(\\lambda )\\equiv \\bar{x}_{\\text{upper branch}}-\\bar{x}_{\\text{lower branch}} in terms of λ\\lambda for the retarded dynamics.One of the most serious difference of the retarded and instantaneous dynamical systems arise from their behaviors in the oscillatory regime $\\lambda >2$ .",
"The Fig.",
"REF and Fig.",
"REF show the distribution of the activity $x$ for instantaneous and retarded dynamical systems respectively.",
"In the Fig.",
"REF the two branches of the oscillations are evident which rapidly grow with $\\lambda $ .",
"To quantify this, we have plotted $f(\\lambda )\\equiv \\bar{x}_{\\text{upper branch}}-\\bar{x}_{\\text{lower branch}}$ , whose numerical value shows the amplitude of the oscillations.",
"$\\bar{x}$ is the average value of $x$ .",
"We see that it rapidly grows with increasing $\\lambda $ .",
"The Fig.",
"REF however shows a different behavior for the same $\\lambda $ s. The growth of this amplitude is meaningfully smaller than that for Fig.",
"REF .",
"For example, for $\\lambda =35$ the gap between two branches is less than 100, whereas for instantaneous dynamical system, it becomes of order 7000.",
"Here we see that the inclusion of the retardation effects, restrains the oscillatory behaviors of the random networks with excitable nodes.",
"For neural networks (when is modelled by random excitable nodes) this effect sounds promising for controlling undesirable activity oscillations.",
"Figure: (Color online): (a) The log-log plot of P(x)P(x) for both retarded and instantaneous dynamics for λ\\lambda in the onset of the critical region.",
"Inset: The same for the sub-critical regime.",
"(b) The plot of P(x)P(x) for both retarded and instantaneous dynamics for 1<λ≤21<\\lambda \\le 2.",
"Lower inset: the same for N=50 2 N=50^2.",
"Upper inset: the same for the instantaneous dynamical system.",
"Left inset: the standard deviation ζ\\zeta in terms of λ\\lambda .The same graphs have been shown for the $\\lambda $ s for the onset of criticality, i.e.",
"in the vicinity of $\\lambda =1$ (Fig.",
"REF ) and in the critical interval, i.e.",
"$1<\\lambda \\le 2$ (Fig.",
"REF ).",
"We see from Fig.",
"REF that for $\\lambda =1$ (or in its vicinity) the exponents of the retarded and instantaneous dynamical systems are nearly the same for the $x$ variable.",
"In the subcritical case however (the inset) $P(x)$ behaves logarithmically for both systems with different (non-universal) slopes.",
"The same has been sketched for $1<\\lambda \\le 2$ in Fig.",
"REF , whose left inset shows that the standard deviations $\\zeta $ grows monotonically with $\\lambda $ .",
"This increase is faster for larger $N$ s. Figure: (Color online) The log-log plot of the distribution functions of (a) the avalanche duration DD, (b) the avalanche mass MM, and (c) the avalanche size SS for N=100 2 N=100^2.",
"Lower insets: the same graph for N=50 2 N=50^2.",
"Upper insets: The same for instantaneous dynamical system.",
"(d) The log-log plot of S-TS-T diagram.",
"Upper inset: The log-log plot of S-MS-M diagram.",
"Lower inset: The log-log plot of M-TM-T diagram.The retardation is a relevant factor for the statistics of the avalanche duration $D$ .",
"More precisely $\\tau _D(\\lambda =1)$ is very different for retarded and instantaneous dynamics.",
"In the Fig.",
"REF we see that $\\tau _D(\\lambda =1)^{\\text{retarded}}\\approx \\tau _D(\\lambda =1)^{\\text{simulataneous}}+1=2.82\\pm 0.1$ .",
"The same exponents for $M$ show an agreement with the instantaneous dynamical system for $\\lambda $ in the onset of critical region, i.e.",
"$\\tau _M(\\lambda =1)^{\\text{retarded}}\\approx \\tau _M(\\lambda =1)^{\\text{simulataneous}}=1.44\\pm 0.2$ (Fig.",
"REF ).",
"For the avalanche size $S$ we have $\\tau _S(\\lambda =1)^{\\text{retarded}}=2.23\\pm 0.1$ and $\\tau _S(\\lambda =1)^{\\text{instantaneous }}=1.47\\pm 0.2$ (Fig.",
"REF ).",
"Note that the determination of the onset of the critical region for a given $N$ has itself an uncertainty and should be determined by analyzing the branching ratio.",
"For example, as the system size $N$ decreases, this value also decreases, e. g. For $N=50^2$ $\\lambda ^{\\text{onset}}= 0.95\\pm 0.02$ .",
"This itself generates a systematic error in the determination of the exponents on the onset of the criticality.",
"Now let us consider the scaling properties of the statistical variables.",
"This has been done in Fig.",
"REF for all possible scaling quantities.",
"As is explicit in this graph, the scaling between $S$ and $T$ ($T$ being duration of avalanche here) is displaced and the corresponding exponents changes from $1.87\\pm 0.05$ (for instantaneous dynamical system) to $1.74\\pm 0.05$ (for retarded dynamical system) which is out of its error bar, and the change is meaningful.",
"The same occurs for the $\\gamma _{M-T}$ (lower inset), i.e.",
"it changes from $1.77\\pm 0.05$ (for instantaneous dynamical system) to $1.64\\pm 0.05$ (for retarded dynamical system).",
"Interestingly the $\\gamma _{SM}$ does not change considerably and remains on $1.03\\pm 0.05$ .",
"The critical exponents have been gathered in Table REF and are compared to the instantaneous dynamical system.",
"Table: The critical exponents in the onset of criticality of two models: retarded dynamical system (RDS) and instantaneous dynamical system (IDS) for N=100 2 N=100^2."
],
[
"Conclusion",
"In this paper we have addressed the problem of the effect of retardation in random networks with excitable nodes.",
"The retardation effects have been brought into the calculations using the Eq.",
"REF , which is mixed by the refractory period.",
"We analyzed the branching ratio which yields the possible intervals of distinct behaviors, like the sub-critical, critical and oscillatory behaviors.",
"Our calculations demonstrated that the oscillations are remarkably suppressed by the retardation.",
"This can be a promising effect in the systems that such oscillations are undesirable.",
"Also the critical exponents of the retarded dynamical systems are meaningfully different from that for the instantaneous dynamical systems.",
"The numerical amounts of these exponents can be found in Table REF ."
]
] | 1808.08495 |
[
[
"A power-law upper bound on the correlations in the 2D random field Ising\n model"
],
[
"Abstract As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small.",
"This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary.",
"Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field.",
"Our analysis proceeds through a streamlined and quantified version of the Aizenman-Wehr proof of the Imry-Ma rounding effect."
],
[
"The Imry-Ma phenomenon for the $2D$ RFIM and its quantification",
"A first-order phase transition is one associated with phase coexistence, in which an extensive system admits at least two thermal equilibrium states which differ in their bulk densities of an extensive quantity.",
"The thermodynamic manifestation of such a transition is the discontinuity in the derivative of the extensive system's free energy with respect to one of the coupling constants which affect the system's energy.",
"At zero temperature, this would correspond to the existence of two infinite-volume ground states which differ in the bulk average of a local quantity.",
"In what is known as the Imry-Ma [17] phenomenon, in two-dimensional systems any first-order transition is rounded off upon the introduction of arbitrarily weak static, or quenched, disorder in the parameter which is conjugate to the corresponding extensive quantity.",
"Our goal here is to present quantitive estimates of this effect, strengthening the previously proven infinite-volume statement [5] by: i) upper bounds on the dependence of the local density on a finite-volume's boundary conditions, and ii) related bounds on the correlations among local quenched expectations, which are asymptotically independent functions of the quenched disorder.",
"The present discussion takes place in the context of the random-field Ising model.",
"In this case the original discontinuity is in the bulk magnetization, i.e.",
"volume average of the local spin $\\sigma _u$ , and it occurs at zero magnetic field ($h=0$ ).",
"Since $h$ is the conjugate parameter to the magnetization, the relevant disorder for the Imry-Ma phenomenon is given by site-independent random field $(\\varepsilon \\eta _u)$ .",
"More explicitly, the system consists of Ising spin variables $\\lbrace \\sigma _u\\rbrace _{u\\in \\mathbb {Z}^d}$ , associated with the vertices of the $d$ -dimensional lattice $\\mathbb {Z}^d$ , with the Hamiltonian $H(\\sigma ) := - \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subseteq \\mathbb {Z}^d\\end{array} }J_{u,v}\\, \\sigma _u \\sigma _v - \\sum _{v\\in \\mathbb {Z}^d} (h+ \\varepsilon \\,\\eta _v) \\sigma _v \\,,$ and ferromagnetic translation-invariant coupling constants $\\mathcal {J} = \\lbrace J_{u,v}\\rbrace $ ($J_{u,v} = J_{v,u} = J_{u-v,\\operatorname{\\mathbf {0}}}\\ge 0$ ).",
"For convenience we focus on the case that the $(\\eta _v)$ are independent standard Gaussians.",
"However it is expected, and for many of the key results proven true, that the model's essential features are similar among all independent, identically distributed $(\\eta _v)$ whose common distribution has a continuous component.",
"The main result presented here is the proof that in the two-dimensional case at any temperature $T\\ge 0$ , the effect on the local quenched magnetization of the boundary conditions at distance $L $ away decays by at least a power law ($1/L^\\gamma $ ).",
"This may be viewed as a quantitative extension of the uniqueness of the Gibbs state theorem [4], [5].",
"It also implies a similar bound on correlations within the infinite-volume Gibbs state.",
"A weaker upper bound, at the rate $1/\\sqrt{\\log \\log L}$ , was recently presented in [10], derived there by other means.",
"More explicitly: as the first question it is natural to ask whether the addition of random field terms in the Hamiltonian (REF ) changes the Ising model's phase diagram, whose salient feature is the phase transition which for $d>1$ occurs at $h=0$ and low enough temperatures, $T< T_c$ .",
"The initial prediction of Y. Imry and S-K Ma [17] was challenged by other arguments, however it was eventually proven to be true: For $d\\ge 3$ the RFIM continues to have a first-order phase transition at $h=0$ [15], [8], whereas in two dimensions at any $\\varepsilon \\ne 0$ the model's bulk mean magnetization has a unique value for each $h$ , and by implication it varies continuously in $h$ at any temperature, including $T=0$ [4], [5].",
"Through the FKG property [13] of the RFIM one may also deduce that in two dimensions, at any temperature $T\\ge 0$ and for almost every realization of the random field $\\eta = (\\eta _v)_{v\\in \\mathbb {Z}^2}$ , the system has a unique Gibbs state.",
"For $T=0$ this translates into uniqueness of the infinite-volume ground state configuration, i.e.",
"configuration(s) for which no flip of a finite number of spins results in lower energy.",
"Additional background and pedagogical review of the RFIM may be found in [7].",
"Seeking quantitative refinements of the above statement, we consider here the dependence of the finite-volume quenched magnetization $\\langle \\sigma _v\\rangle ^{\\Lambda , \\tau }$ on the boundary conditions $\\tau $ placed on the exterior of a domain $\\Lambda $ .",
"We denote by $\\langle -\\rangle ^{ \\Lambda ,\\tau }$ the finite volume “$\\tau $ state” quenched thermal average and by $\\mathbb {E}{}$ the further average over the random field (both defined explicitly in Section ).",
"Due to the model's FKG monotonicity property the finite volume Gibbs states at arbitrary boundary conditions are bracketed between the $+$ and the $-$ state.",
"Hence the relevant order parameter is $ \\begin{split}m(L) \\equiv m(L;T, \\mathcal {J}, h, \\epsilon ) \\, := \\, \\frac{1}{2} \\left[\\mathbb {E}[\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), +} ] \\ - \\ \\mathbb {E}[\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), -}]\\right] \\,\\end{split}$ where $\\Lambda _u(L):=\\lbrace v\\in \\mathbb {Z}^2\\,\\colon \\, d(u,v)\\le L\\rbrace \\,\\quad \\mbox{,}\\quad \\ \\Lambda (L) = \\Lambda _\\textbf {0}(L) \\,,$ with $d(u,v)$ the graph distance on $\\mathbb {Z}^2$ and $\\textbf {0}:=(0,0)$ .",
"Theorem 1.1 In the two-dimensional random-field Ising model with a finite-range interaction $\\mathcal {J}$ and independent standard Gaussian random field $(\\eta _v)$ , for any temperature $T\\ge 0$ , uniform field $h\\in \\mathbb {R}$ , and field intensity $\\varepsilon >0$ there exist $C=C(\\mathcal {J}, T, \\varepsilon ) >0$ and $\\gamma = \\gamma (\\mathcal {J}, T, \\varepsilon )>0$ such that for all large enough $L$ $ m(L; T , \\mathcal {J}, h, \\epsilon ) \\le \\frac{C}{L^{\\gamma }} \\, .$ For the nearest-neighbor interaction $J_{u,v} = J \\, \\delta _{d(u,v),1}$ the proof yields $\\gamma := 2^{-10}\\cdot \\chi \\left(\\frac{50 J }{\\varepsilon }\\right)$ in terms of the tail of the Gaussian distribution function: $\\chi (t):=2\\int _t^\\infty \\phi (s) \\, ds \\, ,\\qquad \\phi (s) = \\frac{1}{\\sqrt{2\\pi }}e^{-s^2/2}\\, .$ The phenomenon and the arguments discussed in the proof are somewhat simpler to present in the limit of zero temperature, where the quenched random field is the only source of disorder.",
"We therefore start by proving Theorem REF for this case, emphasizing the setting of nearest-neighbor interaction.",
"Then, in Section we present the changes by which the argument extends to $T>0$ .",
"With minor adjustments of the constants, discussed in Section , the natural extension of the statement to translation-invariant pair interactions of finite range is also valid."
],
[
"Direct implications",
"By the FKG inequality (see Section REF ), the difference whose mean is the order parameter is non-negative for any $\\eta $ (and all $T\\ge 0$ , $h\\in \\mathbb {R}$ ), $\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), +} - \\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), -} \\ \\ge \\ 0 \\,.$ Hence the bound on the mean (REF ) implies (through Markov's inequality) that this quantifier of sensitivity to boundary condition is similarly small with high probability.",
"The order parameter $m(L)$ controls also the covariances of: i) the spins under the infinite-volume quenched Gibbs states $\\langle -\\rangle \\equiv \\langle -\\rangle _{T,\\mathcal {J}, h, \\varepsilon \\eta }$ , and ii) of the infinite-volume quenched Gibbs state magnetization $\\langle \\sigma _u\\rangle $ under the random field fluctuations, over which the average is denoted by $\\mathbb {E}(-)$ .",
"To express these statements we denote $\\begin{aligned}\\langle \\sigma _u;\\sigma _v\\rangle &:= \\ \\langle \\sigma _u \\sigma _v\\rangle - \\langle \\sigma _u\\rangle \\, \\langle \\sigma _v\\rangle \\\\[1ex]\\mathbb {E}(\\langle \\sigma _u \\rangle ; \\langle \\sigma _v\\rangle )& := \\mathbb {E}\\Big (\\langle \\sigma _u \\rangle \\, \\langle \\sigma _v\\rangle \\Big ) - \\mathbb {E}(\\langle \\sigma _u \\rangle ) \\, \\, \\mathbb {E}(\\langle \\sigma _v \\rangle )\\\\&\\,\\,= \\mathbb {E}\\Big (\\big [\\langle \\sigma _u \\rangle - \\mathbb {E}(\\langle \\sigma _0\\rangle ) \\big ]\\,\\big [\\langle \\sigma _v \\rangle - \\mathbb {E}(\\langle \\sigma _0\\rangle ) \\big ] \\Big )\\,.\\end{aligned} $ Each of these truncated correlations is non-negative: in the former case due to the FKG property of the RFIM, and in the latter due to monotonicity of $\\langle \\sigma _u \\rangle $ in $\\eta $ and the Harris/FKG inequality for product measures.",
"As we prove below (Lemma REF ), for pairs $\\lbrace u,v\\rbrace \\in \\mathbb {Z}^2$ , if $d(u,v)>\\ell $ then $\\mathbb {E}{(\\langle \\sigma _u;\\sigma _v\\rangle )} \\ \\le \\ 2\\, m(\\ell ;T, \\mathcal {J}, h, \\epsilon )$ while if $ d(u,v) \\ge 2 \\ell + R(\\mathcal {J})$ , with $R(\\mathcal {J}) := \\max \\lbrace d(u,v) \\, : \\, J_{u,v} \\ne 0\\rbrace $ (the interaction's range) then $\\mathbb {E}(\\langle \\sigma _u \\rangle ; \\langle \\sigma _v\\rangle ) \\ \\le \\ 4\\, m(\\ell ;T, \\mathcal {J}, h, \\epsilon ) \\,.$ The comment made above in relation to (REF ), applies also here: The non-negativity of $\\langle \\sigma _u;\\sigma _v\\rangle $ , together with (REF ), implies that with high probability it does not exceed $m(\\ell ;T, \\mathcal {J}, h, \\epsilon ) $ by a large multiple.",
"The proof of (REF ) and (REF ) does not require the analysis which is developed in this paper.",
"It is therefore postponed to Section .",
"For (REF ) of particular interest is $h=0$ and $T=0$ .",
"In this case $\\langle \\sigma _u\\rangle $ coincides with the infinite-volume ground state configuration $\\widehat{\\sigma }_u(\\eta ) $ which, as is already known, is unique for almost all $\\eta $ .",
"By the spin-flip symmetry $\\mathbb {E}(\\widehat{\\sigma }_u ) =0$ , and the bound (REF ) translates into: $0\\le \\mathbb {E}(\\widehat{\\sigma }_u \\widehat{\\sigma }_v) \\ \\le \\ 4\\,m(\\ell ;0, \\mathcal {J}, 0, \\epsilon )\\,.$"
],
[
"A remaining question",
"As we shall discuss in greater detail in Appendix , at high enough disorder, i.e.",
"large enough $\\varepsilon $ , the order parameter $m(L)$ decays exponentially fast in $L$ .",
"Our results do not resolve the question of whether the two-dimensional model exhibits a disorder-driven phase transition, at which the decay rate changes from exponential to a power law, as the disorder is lowered (possibly even at $T=0$ ).",
"This remains among the interesting open problems concerning the Imry-Ma phenomenon in two dimensions, on which more is said in the open problem Section ."
],
[
"The Gibbs measure",
"Discussing the RFIM on $\\mathbb {Z}^2$ we shall use the following terminology.",
"Two vertices are deemed adjacent, $u\\sim v$ , if they differ by a unit vector.",
"The graph distance on $\\mathbb {Z}^2$ is denoted $d(u,v)$ and the graph ball of radius $L$ around $u$ is denoted $\\Lambda _u(L)$ , with $\\Lambda (L)$ standing for $\\Lambda _\\textbf {0}(L)$ , as before Theorem REF .",
"The edge boundary of a subset $\\Lambda \\subset \\mathbb {Z}^2$ (which is used in decoupling estimates) is denoted $\\partial _{\\text{e}}\\Lambda := \\lbrace (u,v)\\,\\colon \\, u\\in \\Lambda ,\\, v\\in \\mathbb {Z}^2\\setminus \\Lambda ,\\, J_{u,v} \\ne 0 \\rbrace $ and the external boundary (which is used when imposing boundary conditions) is $\\partial _{\\text{v}}\\Lambda :=\\lbrace v\\in \\mathbb {Z}^2\\setminus \\Lambda \\,\\colon \\,\\exists u\\in \\Lambda , J_{u,v} \\ne 0 \\rbrace \\,.$ Figure: A subset of ℤ 2 \\mathbb {Z}^2 of the form of Λ u (ℓ)\\Lambda _u(\\ell ) and its two boundary sets: the edge boundary ∂ e Λ u \\partial _{\\text{e}} \\Lambda _u and the vertex (external) boundary ∂ v Λ u \\partial _{\\text{v}}\\Lambda _u, both drawn for the case of the nearest-neighbor interaction.The RFIM Gibbs equilibrium state in the finite subset $\\Lambda \\subset \\mathbb {Z}^2$ , at specified values of the parameters $(T, \\mathcal {J}, h, \\varepsilon ) $ , the random field $\\eta $ , and a configuration of boundary spin values $\\tau :\\partial _{\\text{v}}\\Lambda \\rightarrow \\lbrace -1,1\\rbrace $ , is the probability measure over $\\Omega _\\Lambda = \\lbrace -1,1\\rbrace ^\\Lambda $ given by $\\mathbb {P}^{\\Lambda , \\tau }(\\sigma ) := \\frac{1}{Z^{\\Lambda , \\tau }}e^{-\\frac{1}{T} H^{\\Lambda ,\\tau }(\\sigma )},$ where $H^{\\Lambda ,\\tau }(\\sigma ) := -\\sum _{u,v\\in \\Lambda } J_{u,v} \\sigma _u \\sigma _v - \\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda } J_{u,v} \\sigma _u \\tau _v -\\sum _{v\\in \\Lambda }(h+ \\varepsilon \\eta _v) \\sigma _v\\,$ and $Z^{\\Lambda ,\\tau }$ is the corresponding normalizing factor (the “partition function”).",
"The associated expectation operator is denoted $\\langle -\\rangle ^{ \\Lambda ,\\tau }$ .",
"The notation $\\mathbb {P}^{\\Lambda , \\pm }$ or $\\langle -\\rangle ^{ \\Lambda ,\\pm }$ indicates that $\\tau $ is the corresponding uniform configuration $\\tau \\equiv +1$ or $\\tau \\equiv -1$ .",
"The notation $\\mathbb {P}{}$ and $\\mathbb {E}{}$ is used for the probability and expectation operators, respectively, of the further average over the random field.",
"At $T=0$ , the measure $\\mathbb {P}^{\\Lambda , \\tau }$ is supported on the almost-surely unique configuration which minimizes $H^{\\Lambda ,\\tau }$ .",
"These ground-state configurations, which depend on $\\varepsilon \\eta $ and $(\\mathcal {J},h)$ , are denoted here by $\\sigma ^{\\Lambda , \\tau } = (\\sigma ^{\\Lambda , \\tau }_v)_{v \\in \\Lambda }$ (The dependence on $\\eta $ is not displayed, but it is in the focus of the discussion.)"
],
[
"Monotonicity properties",
"In our discussion we shall take advantage of the known monotonicity property of the ferromagnetic Ising model, which is that its Gibbs equilibrium states as well as the ground-state configurations, at given $\\mathcal {J}$ , $h$ and $\\varepsilon $ , are increasing functions of the local field variables $\\eta $ and of the boundary spin configuration $\\tau $ .",
"The statement is a known consequence of the FKG inequality [13].",
"The $T=0$ version can also be seen through a more direct argument.",
"Thus, for any region $\\Lambda $ and pairs of boundary conditions $\\tau ^-, \\tau ^+ :\\partial _{\\text{v}}\\Lambda \\rightarrow \\lbrace -1,1\\rbrace $ : $\\text{$\\tau ^-\\le \\tau ^+ \\quad \\Longrightarrow \\quad \\mathbb {P}^{\\Lambda ,\\tau ^+}$ stochastically dominates $\\mathbb {P}^{\\Lambda ,\\tau ^-}$}$ where an inequality between configurations is to be interpreted as holding pointwise.",
"(Unlike $\\mathbb {R}$ , the configuration space is only partially ordered, but that suffices for our purpose.)",
"The following special case is noted for later reference $\\text{$\\tau ^-\\le \\tau ^+ \\quad \\Longrightarrow \\quad \\langle \\sigma _v\\rangle ^{\\Lambda ,\\tau ^-}\\le \\langle \\sigma _v\\rangle ^{\\Lambda ,\\tau ^+}$ for each $v\\in \\Lambda $}.$ By related reasoning, the Gibbs state at $+$ (or $-$ ) boundary conditions is stochastically decreasing (and correspondingly increasing) in its dependence on $\\Lambda $ .",
"In particular, for each $v \\in \\Lambda _1 \\subset \\Lambda _2 \\subset \\mathbb {Z}^2$ : $\\text{$\\langle \\sigma _v\\rangle ^{\\Lambda _1,+} \\ge \\langle \\sigma _v\\rangle ^{\\Lambda _2,+}\\quad $ and $\\quad \\langle \\sigma _v\\rangle ^{\\Lambda _1,-} \\le \\langle \\sigma _v\\rangle ^{\\Lambda _2,-}$}.$ The above inequalities hold also at $T=0$ , where $\\sigma ^{\\Lambda ,\\tau }_v$ substitutes for $\\langle \\sigma _v\\rangle ^{\\Lambda ,\\tau }$ .",
"It is convenient to note this explicitly for later reference: $&\\tau ^-\\le \\tau ^+ \\quad \\Longrightarrow \\quad \\sigma ^{\\Lambda ,\\tau ^-}\\le \\sigma ^{\\Lambda ,\\tau ^+},\\\\&\\sigma ^{\\Lambda _1,+}_v \\ge \\sigma ^{\\Lambda _2,+}_v \\quad \\mbox{and} \\quad \\sigma ^{\\Lambda _1,-}_v \\le \\sigma ^{\\Lambda _2,-}_v,\\\\&\\sigma ^{\\Lambda _1,+} - \\sigma ^{\\Lambda _1,-} \\, \\ge \\,\\sigma ^{\\Lambda _2, +} - \\sigma ^{\\Lambda _2, -} \\, \\ge \\, 0\\,,$ with the second and third assertions holding for $v \\in \\Lambda _1 \\subset \\Lambda _2 \\subset \\mathbb {Z}^2$ ."
],
[
"Proof of the main result for $T=0$",
"We start with the zero-temperature case of Theorem REF as it already contains the main features of the problem while being technically simpler.",
"For a further simplification, we consider first the nearest-neighbor interaction (REF ).",
"The extension to finite-range interactions will follow in Section ."
],
[
"Influence/disagreement percolation",
"Due to the monotonicity of the ground state in the boundary conditions, the order parameter $m(L)$ which is defined in (REF ) can be viewed as the probability that the difference of the boundary conditions at distance $L$ from a site $v$ “percolates” to $v$ : $m(L) = m(L; 0, \\mathcal {J}, h, \\epsilon ) \\ = \\ \\mathbb {P}\\left(\\sigma ^{\\Lambda (L),+}_{\\mathbf {0}}> \\sigma ^{\\Lambda (L),-}_{\\mathbf {0}}\\right) \\,.$ Remark: Disagreement percolation provides a concrete manifestation of the influence of the boundary condition.",
"The terms disagreement percolation and influence percolation are almost interchangeable: the former referring to specific manifestations of the latter.",
"The term percolation is called for since the influence/disagreement spreads only along connected sets.",
"To learn about $m(L)$ we find it useful to consider the following functions of the disorder: $D_{\\ell }(\\eta ) \\ := \\ \\sum _{v\\in \\Lambda (\\ell )} \\mathbb {1}[ \\sigma ^{\\Lambda (3\\ell ),+}_v\\ne \\sigma ^{\\Lambda (3\\ell ),-}_v]\\, ,$ the number of sites in $\\Lambda (\\ell )$ to which the difference of the boundary conditions imposed on the boundary of $\\Lambda (3\\ell )$ has “percolated”, and $B_{\\ell }(\\eta ) \\ := \\ \\sum _{(u,v)\\in \\partial _{\\text{e}} \\Lambda (2\\ell )} J_{u,v} \\, \\mathbb {1}[\\lbrace \\sigma ^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ),+}_u\\ne \\sigma ^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ),-}_u\\rbrace \\cap \\lbrace \\sigma ^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ),+}_v\\ne \\sigma ^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ),-}_v\\rbrace ]\\, .$ The latter is the combined strength of the edges crossing a separating surface at half the distance of $\\Lambda (\\ell )$ to the boundary of $\\Lambda (3\\ell )$ , which contribute to the surface tension."
],
[
"The surface tension",
"One may learn about the probability distribution of the disagreement set $D_\\ell $ through consideration of the surface tension, which for scale $\\ell $ (always a positive integer) is defined as $ \\mathcal {T}_{\\ell }(\\eta ) := \\\\-\\left[ \\mathcal {E}^{+,+}(\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )) + \\mathcal {E}^{-,-}(\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )) - \\mathcal {E}^{+,-}(\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )) - \\mathcal {E}^{-,+}(\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )) \\right] \\,.$ Here $\\mathcal {E}^{s,s^{\\prime }}(\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )) $ denotes the minimal value of the Hamiltonian $H^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ), \\tilde{\\tau }}$ (see (REF )) over spin configurations satisfying the boundary conditions $\\tilde{\\tau }_v \\, = \\, {\\left\\lbrace \\begin{array}{ll} s & v\\in \\partial _{\\text{v}}\\Lambda (3\\ell ) \\\\s^{\\prime } & v\\in \\partial _{\\text{v}}(\\mathbb {Z}^2\\setminus \\Lambda (\\ell ))\\end{array}\\right.}",
"\\, .$ Our analysis proceeds by contrasting a natural upper bound on the surface tension, with the analysis of the not-improbable fluctuations of $\\mathcal {T}_\\ell (\\eta )$ .",
"For the upper bound we have: Theorem 3.1 In the RFIM with nearest-neighbor interaction, for each configuration of the random field: $\\mathcal {T}_{\\ell }(\\eta )\\ \\le \\ 4 \\, B_{\\ell } (\\eta ) \\ \\le \\ 8 J \\, |\\partial _{\\text{v}}\\Lambda (2 \\ell )| \\,.$ Let $A$ be the set of vertices in $\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )$ on which there is equality between the ground-state configurations with $++$ and $--$ boundary conditions.",
"The monotonicity property (REF ) implies that all ground states on $\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )$ must coincide on $A$ .",
"Consider making two modifications to the Hamiltonian in the domain $\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )$ : First, rigidly restrict the spin values at all vertices in $A$ to their common value in these ground states.",
"This clearly has no effect on the energies of the ground-state configurations considered above.",
"Second, remove the energy terms corresponding to bonds in $\\partial _{\\text{e}}\\Lambda (2\\ell )$ whose endpoints do not intersect $A$ .",
"This change may affect the energy of each of the four ground states by at most $ B_{\\ell }(\\eta )$ .",
"Once both changes are made, the Hamiltonian decomposes into a sum of two terms, in whose minimization there is no interaction between the effects of the two components of the boundary.",
"Thus the surface tension based on the modified Hamiltonian vanishes.",
"It follows that $\\mathcal {T}_\\ell (\\eta ) \\ \\le 4\\, B_\\ell (\\eta ) $ as claimed in the first inequality in (REF ).",
"The second is its elementary consequence.",
"The upper bound which (REF ) yields on $\\mathcal {T}_\\ell $ will be contrasted with the implications of the following representation.",
"Theorem 3.2 For the RFIM with IID Gaussian random fields, the surface tension bears the following relation with disagreement percolation: $\\mathcal {T}_{\\ell }(\\eta ) \\ = \\ 2 \\varepsilon \\,\\int _\\mathbb {R}D_\\ell (\\eta ^{(t)}) \\, dt \\ = \\ \\ \\frac{2 \\varepsilon }{\\sqrt{|\\Lambda (\\ell )|}}\\, \\mathbb {E}_{\\widehat{\\eta }} \\left( \\frac{D_{\\ell }(\\eta ) }{ \\phi (\\widehat{\\eta }) } \\right) \\, ,$ where: $\\eta ^{(t)}$ is defined by adding a uniform field of intensity $t$ in $\\Lambda (\\ell )$ , $ \\eta ^{(t)}_v := {\\left\\lbrace \\begin{array}{ll}\\eta _v + t&v\\in \\Lambda (\\ell )\\\\\\eta _v&\\text{otherwise}\\end{array}\\right.}",
"\\, .$ the variable $\\widehat{\\eta }$ is defined as $\\widehat{\\eta }:= \\frac{1}{\\sqrt{| \\Lambda (\\ell )|}} \\sum _{v\\in \\Lambda (\\ell )} \\eta _v \\, .$ $\\mathbb {E}_{\\widehat{\\eta }} $ represents an average over $\\widehat{\\eta }$ at fixed values of the other, orthogonal, Gaussian degrees of freedom which determine $\\eta $ .",
"$\\phi $ is the Gaussian density function (REF ).",
"(An alternative presentation of $\\mathbb {E}_{\\widehat{\\eta }}$ : decomposing $\\eta $ as a sum of two independent Gaussian fields $\\eta _1, \\eta _2$ with $\\eta _1 \\equiv \\frac{1}{\\sqrt{| \\Lambda (\\ell )|}}\\widehat{\\eta }$ on $\\Lambda (\\ell )$ , and $\\eta _1 \\equiv 0$ outside $\\Lambda (\\ell )$ , the operation $\\mathbb {E}_{\\widehat{\\eta }}$ represents conditional expectation, given $\\eta _2$ .)",
"To derive (REF ) we approach $\\mathcal {T}_\\ell (\\eta ) $ through another function, $G_\\ell (\\eta )$ , which has already played a key role in the proof of the absence of symmetry breaking in the two-dimensional RFIM [4], [5].",
"Its zero-temperature version corresponds to the difference in the ground-state energies in $\\Lambda (3\\ell )$ between the $+$ and $-$ boundary conditions: $G_\\ell (\\eta ) \\, := \\,-\\left[\\mathcal {E}^{+}(\\Lambda (3\\ell )) - \\mathcal {E}^{-}(\\Lambda (3\\ell ))\\right]$ with $\\mathcal {E}^{\\pm }(\\Lambda (3\\ell )):=H^{\\Lambda (3\\ell ), \\pm }(\\sigma ^{\\Lambda (3\\ell ),\\pm })$ .",
"The two functions are linked by the relation $\\mathcal {T}_\\ell (\\eta ) \\ = \\ \\lim _{t\\rightarrow \\infty }G_\\ell (\\eta ^{(t)}) - G_\\ell (\\eta ^{(-t)})$ with $\\eta ^{(t)}$ defined by adding a uniform field of intensity $t$ in $\\Lambda (\\ell )$ , as described in (REF ).",
"Equality (REF ) is based on the observation that if $|h+\\varepsilon \\eta _v| > 4J$ then $\\sigma ^{\\Lambda (3\\ell ),\\pm }_v $ are both given by ${\\rm {sign}}(h+\\varepsilon \\eta _v)$ (4 appears here as the number of neighbors of $v$ in $\\mathbb {Z}^2$ ).",
"The function $G_\\ell (\\eta )$ is Lipschitz continuous and non-decreasing in each of the coordinates of $(\\eta _v)$ , $v\\in \\Lambda (3\\ell )$ , with $\\frac{\\partial }{\\partial \\eta _v} G_\\ell (\\eta ) = \\varepsilon \\left[ \\sigma ^{\\Lambda (3\\ell ),+}_v(\\eta ) - \\sigma ^{\\Lambda (3\\ell ),-}_v(\\eta ) \\right] \\, =\\, 2 \\varepsilon \\, \\mathbb {1}_{\\sigma ^{\\Lambda (3\\ell ),+}_v(\\eta ) \\ne \\sigma ^{\\Lambda (3\\ell ),-}_v(\\eta ) }$ for Lebesgue-almost-every $\\eta $ .",
"Combining this with (REF ) one gets $\\begin{split}\\mathcal {T}_\\ell (\\eta ) &= \\varepsilon \\int _{-\\infty }^\\infty \\sum _{v \\in \\Lambda (\\ell )} \\left[ \\sigma _v^{\\Lambda (3\\ell ),+}(\\eta ^{(t)}) -\\sigma _v^{\\Lambda (3\\ell ),-}(\\eta ^{(t)}) \\right] \\, dt\\, = \\, 2\\varepsilon \\int _{-\\infty }^\\infty D_{\\ell }(\\eta ^{(t)})\\, dt\\, .\\end{split}$ The shift by $t$ affects the random field's normalized sum over $\\Lambda (\\ell )$ , which we denote by $\\hat{\\eta }= \\sum _{v\\in \\Lambda (\\ell )} \\eta _v/ \\sqrt{|\\Lambda (\\ell )|} $ but it does not affect the independently distributed degrees of freedom which as Gaussian variables are orthogonal to it, $\\hat{\\eta }^{(\\perp )}$ .",
"Writing $\\eta = (\\hat{\\eta }, \\hat{\\eta }^{(\\perp )}) $ and $t\\sqrt{ | \\Lambda (\\ell )|} = s$ the change $\\eta \\mapsto \\eta ^{(t)}$ corresponds to the shift $(\\hat{\\eta }, \\hat{\\eta }^{(\\perp )}) \\mapsto (\\hat{\\eta }+s, \\hat{\\eta }^{(\\perp )}) $ .",
"Since the component $\\hat{\\eta }$ has the standard Gaussian distribution, of density $\\phi (\\hat{\\eta })$ , the above integral can be rewritten as: $\\begin{split}&\\int _{-\\infty }^\\infty D_{\\ell }(\\eta ^{(t)})\\, dt =\\frac{1}{\\sqrt{ | \\Lambda (\\ell )|} } \\int _{-\\infty }^\\infty D_{\\ell }((\\hat{\\eta }+s , \\hat{\\eta }^{(\\perp )}) \\, d s= \\frac{1}{\\sqrt{ | \\Lambda (\\ell )|} } \\int _{-\\infty }^\\infty D_{\\ell }((s , \\hat{\\eta }^{(\\perp )}) \\, d s \\\\& = \\frac{1}{\\sqrt{ | \\Lambda (\\ell )|} } \\int _{-\\infty }^\\infty D_{\\ell }((s , \\hat{\\eta }^{(\\perp )})\\phi ( s)^{-1}\\cdot \\phi ( s) \\, d s = \\frac{1}{\\sqrt{ | \\Lambda (\\ell )|}}\\, \\mathbb {E}_{\\widehat{\\eta }} \\left( D_{\\ell }(\\eta ) \\,\\phi ( \\hat{\\eta })^{-1}\\right)\\, .\\end{split}$"
],
[
"Proof outline for the RFIM ground states",
"Influence percolation quantities appear in both the surface tension formula (REF ) and the upper bound (REF ).",
"The combination of these two yields the following relation, which underlies our analysis: $ \\frac{2 \\, \\mathbb {E}(B_\\ell (\\eta ))}{\\varepsilon \\sqrt{|\\Lambda (\\ell )|}}& \\ge & \\mathbb {E}\\left( \\frac{D_{\\ell }(\\eta ) }{| \\Lambda (\\ell )|}\\,\\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\,.$ To motivate the direction which the discussion is about to take, let us note that (REF ) allows a streamlined proof of the following statement, which is among the significant results established in [5].",
"Corollary 3.3 In the two-dimensional RFIM with Gaussian random field, for any $\\varepsilon \\ne 0$ , the system has a unique ground-state configuration.",
"The monotonicity relations () imply that as the domains $\\Lambda _n$ increase to $\\mathbb {Z}^2$ , the ground state $\\sigma ^{\\Lambda _n, +}$ converges pointwise to a limiting ground state $\\sigma ^+$ , which is, moreover, independent of the choice of exhausting sequence $\\Lambda _n$ .",
"the ground state $\\sigma ^-$ is defined similarly with $-$ boundary conditions.",
"The monotonicity relation (REF ) then shows that uniqueness of the ground state is equivalent to the vanishing of the quantity $m(\\infty ) := \\lim _{\\ell \\rightarrow \\infty } m(\\ell ) \\ = \\ \\mathbb {P}\\left(\\sigma ^{+}_v\\ne \\sigma ^{-}_v\\right) \\, ,$ where $v$ is an arbitrary point in $\\mathbb {Z}^2$ .",
"The monotonicity relation () further allows to deduce from (REF ) that $\\frac{C \\,J}{\\varepsilon }\\ge \\ \\mathbb {E}\\left( \\left[ \\frac{ 1}{ | \\Lambda (\\ell )|} \\sum _{v\\in \\Lambda (\\ell )} \\mathbb {1}[ \\sigma ^{+}_v\\ne \\sigma ^{-}_v]\\right]\\, \\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\, ,$ where $C>0$ is an absolute constant.",
"The pair of ground states $(\\sigma ^+, \\sigma ^-)$ form an ergodic process under translations (as a factor of the IID process $\\eta $ ).",
"This allows to conclude that in the limit $\\ell \\rightarrow \\infty $ the quantity $ \\frac{ 1}{ | \\Lambda (\\ell )|} \\sum _{v\\in \\Lambda (\\ell )} \\mathbb {1}[ \\sigma ^{+}_v\\ne \\sigma ^{-}_v] $ converges almost surely to its mean, which is $m(\\infty )$ .",
"Hence, using Fatou's lemma (for the second inequality) $ \\begin{split}\\frac{\\rm {C} \\, J}{\\varepsilon \\, } &\\ \\ge \\ \\lim _{\\ell \\rightarrow \\infty } \\mathbb {E}\\left( \\left[ \\frac{ 1 }{ | \\Lambda (\\ell )|} \\sum _{v\\in \\Lambda (\\ell )} \\mathbb {1}[ \\sigma ^{+}_v\\ne \\sigma ^{-}_v]\\right]\\, \\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\ \\ge \\ \\, \\, \\mathbb {E}\\left( \\frac{m(\\infty ) }{ \\phi (\\widehat{\\eta }) } \\right) \\ = \\\\&\\ =\\ m(\\infty )\\, \\int _{-\\infty } ^ \\infty 1\\, dx \\ = \\ m(\\infty ) \\cdot \\infty \\, .\\end{split}$ This can hold true only if $m(\\infty ) =0$ .",
"The ergodicity argument is of not much help for the finite-volume bounds which are sought here.",
"It may however be substituted by more quantitative estimates, which are derived below under the assumption that $m(\\ell ) \\rightarrow 0$ at only a sub-power slow rate.",
"To produce a contradiction which replaces (REF ) we shall first show that (REF ) implies the following anti-concentration bound.",
"Proposition 3.4 For each integer $\\ell \\ge 1$ , $ \\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}(D_\\ell )} < \\frac{1}{2}\\right)\\,\\ge \\, \\chi \\left(\\frac{4J}{\\varepsilon }\\cdot \\frac{|\\partial _{\\text{v}}\\Lambda (2\\ell )|}{\\sqrt{|\\Lambda (\\ell )|}}\\cdot \\frac{m(\\ell - 1)}{m(4\\ell )} \\right)\\,,$ where $\\chi $ is the standard Gaussian distribution's two-sided tail (REF ).",
"This bound (REF ) will be contrasted with a conditional concentration-of-measure estimate, derived through the following two steps.",
"For the convenience of presentation we summarize here the key statements, and postpone their proofs to the sections which follow.",
"I) Slow decay of a monotone sequence implies the existence of long stretches of somewhat comparable values: Proposition 3.5 For any monotone non-increasing sequence $(p_j)$ satisfying $0\\le p_j\\le 1$ , and any $\\alpha >0$ : if for some $k\\ge 1$ it holds that $p_k\\ge k^{-\\alpha }$ then there exists an integer $n$ in the range $\\sqrt{k}\\le n\\le k$ such that for all $1\\le j \\le n$ , $p_{n}\\le p_j\\le p_{n}\\left(\\frac{n}{j}\\right)^{2\\alpha } \\,.$ The proposition will be employed with $(m(j))$ as the sequence $(p_j)$ .",
"II) A conditional variance bound: Proposition 3.6 For each $0<\\alpha \\le \\frac{1}{4}$ there exists $L_0>0$ such that the following holds for all integer $L\\ge L_0$ .",
"If $m(L)\\ge L^{-2\\alpha }$ and $m(L) \\le m(j)\\le m(L) \\left(\\frac{L}{j}\\right)^{2\\alpha },\\quad 1\\le j\\le L$ then $ \\operatorname{Var}\\big (D_{ \\lfloor L/4\\rfloor }\\big ) \\ \\le \\ 241 \\cdot \\alpha \\cdot \\big (\\mathbb {E}\\left( D_{ \\lfloor L/4\\rfloor }\\right) \\big )^2.$ Combining Proposition REF and Proposition REF with the assumption of sub-power decay of $(m(j))$ shows the existence of an infinite sequence of $L$ s for which (REF ) and (REF ) hold.",
"With $\\ell =\\lfloor L/4\\rfloor $ , Chebyshev's inequality and (REF ) imply that along this sequence $ \\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}\\big ( D_\\ell \\big )} < \\frac{1}{2}\\right) \\,\\le \\, 1000\\alpha .$ At the same time, for $\\alpha \\rightarrow 0$ , the ratio $m(\\ell -1)/m(4\\ell )$ tends to 1 by (REF ), and the argument of $\\chi $ in (REF ) is bounded by $\\rm {Const.}",
"J/\\varepsilon $ , uniformly in $\\ell $ and $\\alpha $ .",
"Hence, for small enough $\\alpha >0$ , (REF ) is in contradiction with the anti-concentration bound (REF ).",
"The above line of reasoning allows to conclude that the initial assumption of sub-power decay is false.",
"A quantitative version of the argument, proving the zero-temperature case of Theorem REF , is presented in Section REF after the derivation of the above three propositions."
],
[
"The anti-concentration estimate",
"In the proof of Proposition REF we shall make use of the following variational principle.",
"Lemma 3.7 Let $w: \\mathbb {R}\\mapsto [0,\\infty ) $ be a symmetric ($w(-x) = w(x)$ ), non-increasing in $|x|$ , probability density function on $\\mathbb {R}$ , i.e.",
"satisfying $\\int _\\mathbb {R}w(x) dx =1$ .",
"Then, for any $ p \\in (0, 1]$ , $\\min \\Big \\lbrace \\int _\\mathbb {R}f(x)\\, dx \\, \\quad {\\Big |} \\, \\, 0\\le f \\le 1\\,,\\,\\int _\\mathbb {R}f(x) \\, w(x) \\, dx \\ = \\ 1- p {\\Big \\rbrace } \\,= \\, 2 \\, q$ where the variation is over measurable functions satisfying the stated conditions, and $q$ is the unique value related to $p$ by $\\int _{|x|> q} w(x) \\, dx\\ = \\ p.$ For each test function satisfying the conditions in (REF ), $ \\int _\\mathbb {R}f(x) dx &=& 2 q \\ - \\ \\int _{|x| \\le q} [1-f(x) ]\\, dx \\ + \\ \\int _{|x| > q} f(x) \\, dx \\\\&\\ge & 2 q \\ - \\ \\frac{1}{w(q)} \\left[ \\int _{|x| \\le q} [1-f(x) ] w(x) \\, dx \\ - \\ \\int _{|x| > q} f(x) \\, w(x) \\, dx \\right] \\\\&= & 2 q \\ + \\ \\frac{1}{w(q)} \\left[ \\int _{|x| \\le q} w(x) \\, dx \\ - \\ \\int _\\mathbb {R}f(x) w(x) \\, dx \\right] \\ = \\ 2q.$ Equality in (REF ) is attained for the indicator function $f(x) = \\mathbb {1}_{[-q,q]}(x)$ .",
"Remarks: 1) For a structural grasp of Lemma REF one may note that by a rearrangement argument it suffices to restrict the variation there to $f$ which are also symmetric and non-increasing in $|x|$ .",
"A convexity argument allows to further restrict to extreme points in the convex set of admissible functions.",
"These are functions satisfying the constraints but taking (almost everywhere) only the values 0 and 1.",
"These two conditions single out the indicator function $\\mathbb {1}_{[-q,q]}(x)$ (or $\\mathbb {1}_{(-q,q)}(x)$ ), and thereby imply that it is a minimizer for (REF ).",
"2) The assumptions of symmetry and monotonicity of the probability density $w(x)$ are not essential, and upon the natural reformulation of (REF ) can be omitted.",
"They are however satisfied by the Gaussian density function $\\phi (x)=e^{-x^2/2}/\\sqrt{2\\pi }$ .",
"The above will next be used to prove the stated estimate.",
"Let $A$ be the event $\\big \\lbrace \\eta : D_\\ell \\ge \\mathbb {E}(D_\\ell )/2 \\, \\big \\rbrace $ and let us denote its probability as $1-p$ , i.e.",
"$\\mathbb {P}\\Big ( D_\\ell \\ge \\frac{1}{2} \\mathbb {E}(D_\\ell ) \\Big ) \\ = \\ \\mathbb {P}(A) \\ = \\ 1-p \\,.$ From (REF ) one may deduce: $ \\frac{2 \\,}{\\varepsilon } \\, \\frac{ \\mathbb {E}(B_\\ell )}{ \\sqrt{|\\Lambda (\\ell )|}}\\, \\frac{|\\Lambda (\\ell )|}{\\mathbb {E}(D_\\ell )}& \\ge & \\frac{1}{2}\\, \\mathbb {E}\\left( \\mathbb {1}[A] \\,\\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\, .$ Expressed in terms of the conditional probability of $A$ , conditioned on $\\widehat{\\eta }$ , the term on the right is, by Lemma REF , $\\mathbb {E}\\left( \\mathbb {1}[A] \\,\\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\ & = &\\int _{-\\infty } ^\\infty \\mathbb {P}\\big ( A \\big | \\, \\widehat{\\eta }= x \\big ) \\, dx \\\\& \\ge &\\min \\Big \\lbrace \\int _{-\\infty } ^\\infty f(x) \\, dx \\, \\Big | \\, 0 \\le f\\le 1\\, , \\,\\int _\\mathbb {R}f(x) \\, \\phi (x) dx = 1-p \\Big \\rbrace \\\\[2ex]& \\ge &\\, 2 \\, q$ with $q$ defined by: $\\chi (q) \\ \\equiv \\ \\int _{|x|>q} \\phi (x) \\, dx \\ = \\ p \\,.$ Combining (REF ) with (REF ) we learn that $\\frac{2 \\,}{\\varepsilon } \\, \\frac{ \\mathbb {E}(B_\\ell )}{ \\sqrt{|\\Lambda (\\ell )|}}\\, \\frac{|\\Lambda (\\ell )|}{\\mathbb {E}(D_\\ell )}\\ \\ge \\ q \\,.$ Hence $\\chi \\Big (\\frac{2 \\,}{\\varepsilon } \\, \\frac{ \\mathbb {E}(B_\\ell )}{ \\sqrt{|\\Lambda (\\ell )|}}\\, \\frac{|\\Lambda (\\ell )|}{\\mathbb {E}(D_\\ell )} \\Big ) \\ \\le \\ \\chi (q) \\ = \\ p \\ = \\ \\mathbb {P}\\Big ( D_\\ell < \\frac{1}{2} \\mathbb {E}(D_\\ell ) \\Big ).$ To obtain the conclusion (REF ) of the proposition, it remains to note that, by the definitions (REF ), (REF ), (REF ) of $m(j)$ , $D_\\ell $ and $B_\\ell $ , together with the monotonicity inequality (), $\\mathbb {E}(B_\\ell ) &\\le 2 J \\, |\\partial _{\\text{v}}\\Lambda (2\\ell )|\\,m(\\ell -1),\\\\\\mathbb {E}(D_\\ell ) &\\ge |\\Lambda (\\ell )|\\,m(4\\ell ).$"
],
[
"Implications of slow decay",
"We next show that slow decay of a monotone sequence implies the existence of long stretches of somewhat comparable values.",
"Assume that for some $k$ and $\\alpha >0$ $p_k\\ge k^{-\\alpha } \\,.$ As $(p_j)$ is non-increasing we need only prove the right-hand inequality in (REF ).",
"Define a sequence $k =: k_0 > k_1 > \\cdots > k_t$ inductively by letting $k_m$ be the maximal integer in $(0,k_{m-1})$ such that $p_{k_m}> p_{k_{m-1}}\\left(\\frac{k_{m-1}}{k_m}\\right)^{2\\alpha }$ provided such an integer exists, and denoting by $t$ the first value of $m$ beyond which the construction cannot proceed.",
"By construction, for all $0< j\\le k_t$ : $p_j\\le p_{k_t}\\left(\\frac{k_t}{j}\\right)^{2\\alpha }$ .",
"If $t=0$ the claim follows with $n := k$ .",
"Otherwise, using (REF ), $1\\ge p_{k_t}>p_{k_{t-1}}\\left(\\frac{k_{t-1}}{k_t}\\right)^{2\\alpha }>p_{k_{t-2}}\\left(\\frac{k_{t-2}}{k_t}\\right)^{2\\alpha }>\\cdots >p_k\\left(\\frac{k}{k_t}\\right)^{2\\alpha }\\ge \\left(\\frac{\\sqrt{k}}{k_t}\\right)^{2\\alpha }$ so that $k_t\\ge \\sqrt{k}$ and the claim (REF ) holds true with $n:=k_t$ .",
"Next we turn to the implications of slow decay on the variance of the size of the disagreement set, $ \\operatorname{Var}(D_\\ell )$ .",
"Assume that $m(L)\\ge L^{-2\\alpha }$ , and that (REF ) holds for all $1\\le j\\le L$ .",
"Throughout the proof we set $\\ell := \\lfloor L/4\\rfloor .$ For $v\\in \\Lambda (\\ell )$ let $E_v$ denote the event $\\lbrace \\eta \\, :\\, \\sigma ^{\\Lambda (3\\ell ),+}_{v}(\\eta ) \\ne \\sigma ^{\\Lambda (3\\ell ),-}_v(\\eta )\\rbrace $ .",
"In this notation: $\\operatorname{Var}\\left(D_\\ell \\right) = \\sum _{v,w\\in \\Lambda (\\ell )} \\big [\\mathbb {P}(E_v\\cap E_w) - \\mathbb {P}(E_v)\\mathbb {P}(E_w)\\big ]\\, .$ We proceed to bound the terms in this sum.",
"By the FKG monotonicity () and the definition (REF ) of $(m(j))$ , for any site $v \\in \\Lambda (\\ell )$ , $\\mathbb {P}(E_v) \\ \\ge \\ m(4\\ell )\\ge m(L)$ and for any pair $v, w \\in \\Lambda (\\ell )$ , $v\\ne w$ , $\\mathbb {P}(E_v\\cap E_w) \\ \\le \\ m(r(v,w))^2$ with $r(v,w):=\\lfloor (d(v,w)-1)/2\\rfloor $ and $d(v,w)$ the distance between the two sites.",
"The bound (REF ) holds since if both $v$ and $w$ are affected by boundary conditions placed outside of $\\Lambda (3\\ell )$ then each spin is necessarily affected also by boundary conditions placed at distance $r(v,w)$ from the site.",
"However, these two events are independent, since they depend only on the random fields in a pair of disjoint neighborhoods of $v$ and $w$ .",
"For pairs at distance $d(v,w) \\ \\le \\ 2$ we shall employ the simpler bound: $\\mathbb {P}(E_v \\cap E_w) - \\mathbb {P}(E_v)\\mathbb {P}(E_w)\\le \\mathbb {P}(E_v) \\le m(2\\ell ).$ Thus under the assumption (REF ) we get $\\begin{split}\\operatorname{Var}\\left( D_\\ell \\right)&\\le |\\Lambda (2)|\\cdot |\\Lambda (\\ell )| m(2\\ell ) + \\sum _{\\begin{array}{c}v,w\\in \\Lambda (\\ell )\\\\d(v,w)\\ge 3\\end{array}} \\left(m(r(v,w))^2 - m(L)^2\\right)\\\\&\\le |\\Lambda (2)|\\cdot |\\Lambda (\\ell )| m(2\\ell ) + m(L)^2\\sum _{\\begin{array}{c}v,w\\in \\Lambda (\\ell )\\\\d(v,w)\\ge 3\\end{array}} \\left(\\left(\\frac{L}{r(v,w)}\\right)^{4\\alpha } - 1\\right).\\end{split}$ The sum in the last bound can be estimated through the observation that most pairs $v,w\\in \\Lambda (\\ell )$ are at distance of order $\\ell $ , in which case $\\frac{L}{r(v,w)}$ is of order 1.",
"As $\\alpha $ is small, for such pairs $\\big (\\frac{L}{r(v,w)}\\big )^{4\\alpha } - 1$ is of order $\\alpha $ .",
"This leads to a bound of order $\\alpha \\, m(L)^2 L^4$ on the variance, which in light of (REF ) is of the order $\\alpha \\Big (\\mathbb {E}(D_\\ell )\\Big )^2$ .",
"We proceed to make this argument precise.",
"We first note that $\\begin{split}&\\sum _{\\begin{array}{c}v,w\\in \\Lambda (\\ell )\\\\d(v,w)\\ge 3\\end{array}} \\left(\\left(\\frac{L}{r(v,w)}\\right)^{4\\alpha } - 1\\right) = \\sum _{j=1}^{\\ell }|\\lbrace (v,w)\\subseteq \\Lambda (\\ell )\\,:\\,r(v,w)=j\\rbrace | \\left(\\left(\\frac{L}{j}\\right)^{4\\alpha } - 1\\right)\\\\&\\le |\\Lambda (\\ell )| \\sum _{j=1}^{\\ell }32j \\left(\\left(\\frac{L}{j}\\right)^{4\\alpha } - 1\\right).\\end{split}$ For large $\\ell $ and $0<\\alpha \\le \\frac{1}{4}$ $\\begin{split}\\sum _{j=1}^{\\ell }j \\left(\\left(\\frac{L}{j}\\right)^{4\\alpha } - 1\\right) & \\le \\ \\int _{1}^{\\ell +1} L^{4\\alpha }x^{1-4\\alpha }dx - \\int _0^{\\ell } x dx \\\\&\\le \\ \\frac{\\ell ^2}{2}\\left[\\frac{2}{2-4\\alpha }\\cdot \\left(\\frac{L}{\\ell +1}\\right)^{4\\alpha }\\cdot \\left(\\frac{\\ell +1}{\\ell }\\right)^2-1\\right]\\ \\le \\ 15\\, \\alpha \\, \\ell ^2.\\end{split}$ Substituting this into (REF ), along with $|\\Lambda (r)|\\ge 2r^2$ , we conclude that $\\begin{split}\\operatorname{Var}\\left(D_\\ell \\right)&\\ \\le \\ |\\Lambda (2)|\\cdot |\\Lambda (\\ell )|m(2\\ell ) + m(L)^2 \\cdot |\\Lambda (\\ell )|\\cdot 32\\cdot 15\\alpha \\ell ^2\\\\&\\ \\le \\ |\\Lambda (2)|\\cdot |\\Lambda (\\ell )|m(2\\ell ) + 240 \\alpha \\left(m(L) |\\Lambda (\\ell )|\\right)^2.\\end{split}$ It remains to observe that, by (REF ), $\\mathbb {E}\\big ( D_\\ell \\big ) \\ \\ge \\ m(L) |\\Lambda (\\ell )|.$ Moreover, by our assumptions that $m(L)\\ge L^{-2\\alpha }$ and that (REF ) holds, $|\\Lambda (2)|\\cdot |\\Lambda (\\ell )|m(2\\ell )\\ \\le \\ |\\Lambda (2)|\\cdot |\\Lambda (\\ell )|m(L) \\left(\\frac{L}{2\\ell }\\right)^{2\\alpha } \\ \\le \\ \\alpha \\Big ( m(L)|\\Lambda (\\ell )| \\Big )^2\\ \\le \\ \\alpha \\Big ( \\mathbb {E}\\big ( D_\\ell \\big ) \\Big )^2$ for $\\ell $ sufficiently large (as a function of $\\alpha $ ).",
"This allows to rewrite (REF ) in the simpler form stated in the proposition: $\\operatorname{Var}\\left( D_\\ell \\right)\\le 241\\cdot \\alpha \\cdot \\Big (\\mathbb {E}\\big ( D_\\ell \\big ) \\Big )^2.$"
],
[
"Putting it all together: $T=0$ for the nearest-neighbor case",
"We now have all the tools for proving the assertion made in Theorem REF for zero temperature.",
"Recall from (REF ) that $\\gamma = 2^{-10}\\chi \\left(\\frac{50 J}{\\varepsilon }\\right)\\, .$ In view of (REF ), if Theorem REF does not hold at zero temperature for $J$ and $\\varepsilon $ then $\\limsup _{L\\rightarrow \\infty } L^{\\gamma }\\cdot m(L)=\\infty \\, ,$ which implies that $\\text{$m(M)\\ge M^{-\\gamma }$ for infinitely many $M$}\\, .$ We assume, in order to obtain a contradiction, that (REF ) holds.",
"Let $M\\ge 64$ , later chosen sufficiently large, be such that $m(M)\\ge M^{-\\gamma }$ .",
"Applying Proposition REF we see that there is an $8\\le \\sqrt{M}\\le L\\le M$ such that $m(L)\\le m(j)\\le m(L)\\left(\\frac{L}{j}\\right)^{2\\gamma } \\,,\\quad 1\\le j\\le L\\, .$ We consider the three domains, $\\Lambda (k\\ell )$ with $\\ell = \\lfloor L/4\\rfloor $ and $k=1,2,3$ .",
"Applying Proposition REF we obtain the anti-concentration inequality $\\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}\\big ( D_\\ell \\big )} < \\frac{1}{2}\\right)\\,\\ge \\, \\chi \\left(\\frac{4J}{\\varepsilon }\\cdot \\frac{|\\partial _{\\text{v}}\\Lambda (2\\ell )|}{\\sqrt{|\\Lambda (\\ell )|}}\\cdot \\frac{m(\\ell - 1)}{m(4\\ell )} \\right).$ The right-hand side may be simplified, using (REF ) together with the fact that $\\gamma <2^{-10}$ , and noting that the assumption $L\\ge 8$ implies that $|\\partial _{\\text{v}}\\Lambda (2\\ell )|=4(2\\ell +1)\\le 3L$ and $|\\Lambda (\\ell )|=1+2\\ell (\\ell +1)\\ge \\frac{L^2}{16}$ .",
"This yields $\\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}\\big ( D_\\ell \\big )} < \\frac{1}{2}\\right) \\,\\ge \\, \\chi \\left(\\frac{4J}{\\varepsilon }\\cdot \\frac{3L}{L/4}\\cdot \\left(\\frac{L}{\\ell -1}\\right)^{2\\gamma }\\right)\\ge \\chi \\left(\\frac{50J}{\\varepsilon }\\right)\\, .$ We shall now reach a contradiction by applying Proposition REF with $\\ell = \\lfloor L/4\\rfloor $ , noting that the assumptions of that proposition are verified by (REF ) and the fact that $\\sqrt{M}\\le L\\le M$ and $m(M)\\ge M^{-\\alpha }$ .",
"The proposition implies that for $L$ sufficiently large (obtained by choosing $M$ sufficiently large), we have the concentration bound, $\\operatorname{Var}\\left( D_\\ell \\right)\\le 241\\cdot \\gamma \\cdot \\Big (\\mathbb {E}\\big ( D_\\ell \\big ) \\Big )^2\\, .$ Chebyshev's inequality then shows that $\\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}\\big ( D_\\ell \\big )} < \\frac{1}{2}\\right) \\,\\le \\, 1000\\gamma \\, .$ As this contradicts (REF ) for the choice (REF ) of $\\gamma $ , we conclude that our initial assumption (REF ) must be false, implying that Theorem REF holds at zero temperature."
],
[
"Extension of the power-law upper bound to $T>0$",
"In this section we adapt the zero-temperature proof of Theorem REF to the positive temperature case.",
"Again, for simplicity, we focus first on the case of nearest-neighbor interaction with the extension to finite-range interactions to follow in Section ."
],
[
"Adjustments in the terminology",
"At positive temperature the relevant function of the random field and of the boundary conditions is not the single ground-state configuration but the corresponding Gibbs probability measure.",
"We proceed to explain how the proof is modified to account for this difference.",
"Influence/disagreement percolation.",
"The order parameter, which at $T=0$ was the disagreement percolation of (REF ) $m(j; 0, \\mathcal {J}, h, \\epsilon ) \\ = \\ \\mathbb {P}\\left(\\sigma ^{\\Lambda (j),+}_{\\mathbf {0}}> \\sigma ^{\\Lambda (j),-}_{\\mathbf {0}}\\right) \\,$ is replaced by the difference in the expected magnetization $m(j; T, \\mathcal {J}, h, \\epsilon ) \\ = \\ \\frac{1}{2} \\left[\\mathbb {E}[\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (j), +} ] \\ - \\ \\mathbb {E}[\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (j), -}]\\right] \\,.$ Let us comment in passing that the available monotone coupling of the $+$ and $-$ probability measures allows to present also the last expression as the probability of disagreement percolation.",
"However, to keep the discussion simple, we shall not stress this point.",
"Correspondingly, as a measure of the disagreement in $\\Lambda (\\ell )$ due to the difference in boundary conditions placed on $\\Lambda (3\\ell )$ we take $D_\\ell (\\eta ) \\ := \\ \\frac{1}{2}\\sum _{v\\in \\Lambda (\\ell )} \\left[\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), +}-\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), -}\\right]\\, .$ Recall also that, at $T=0$ , $B_\\ell (\\eta )/J$ counted the number of edges in the separating surface $\\partial _{\\text{e}}\\Lambda (2\\ell )$ which contribute to the surface tension.",
"At $T>0$ , we find it more convenient to count vertices rather than edges, leading to the definition $\\tilde{B}_{\\ell }(\\eta ) \\ := \\frac{J}{2}\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} \\left[\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell )\\setminus \\Lambda (\\ell ),+} - \\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell )\\setminus \\Lambda (\\ell ),-}\\right].$ Surface tension.",
"For $T>0$ , the role which is played by energy in the zero-temperature analysis is taken by the free energy, which for different combinations of the boundary conditions is defined as: $\\mathcal {F}^{s,s^{\\prime }}_\\ell := -T\\cdot \\log (Z^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ); s,s^{\\prime }})$ where $s$ and $s^{\\prime }$ indicate the $(\\pm )$ boundary conditions placed on the external boundary of $\\Lambda (3\\ell )$ and the internal boundary of $\\Lambda (\\ell )$ , respectively, and the partition function is $Z^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ); s,s^{\\prime }}=Z^{s,s^{\\prime }}=\\sum _{\\sigma :\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )\\rightarrow \\lbrace -1,1\\rbrace } \\exp \\left(-\\frac{1}{T} H^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ); s,s^{\\prime }}(\\sigma )\\right)$ with $H^{\\Lambda (3\\ell )\\backslash \\Lambda (\\ell ); s,s^{\\prime }} = H^{s,s^{\\prime }}$ the Hamiltonian incorporating the boundary conditions.",
"Following this prescription, the extension of the surface tension, of (REF ), to positive temperatures is $\\mathcal {T}_{\\ell }(\\eta ) \\ = \\ T \\, \\log \\left(\\frac{Z^{+,+} \\cdot Z^{-,-} }{Z^{+,-} \\cdot Z^{-,+} }\\right)\\,.$ A similar replacement takes place in the definition of the function $G_{\\ell }(\\eta )$ in (REF ) and it is straightforward to check that the relation (REF ) still holds."
],
[
"Extension of the proof to $T>0$",
"The zero-temperature bound of Theorem REF is modified into the following statement, in which we replace the references to the ground-state spins by their quenched averages and where, for simplicity, we have upper bounded a sum over $(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )$ (analogous to the one in Theorem REF ) by a sum over $v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )$ .",
"Theorem 4.1 In the RFIM with nearest-neighbor interaction, for any realization of the field $\\eta $ , $ \\mathcal {T}_{\\ell }(\\eta )\\le 8 \\tilde{B}_\\ell (\\eta )\\,.$ As in the $T=0$ case, the set $\\partial _{\\text{v}}\\Lambda (2\\ell )$ enters the discussion as a separating barrier between the inner and the outer boundary of $\\Lambda (3\\ell ) \\backslash \\Lambda (\\ell )$ .",
"Denoting the restriction of the spin configuration to this set by $\\tau :\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace $ , let $\\rho _+$ and, correspondingly, $\\rho _-$ be the two probability measures induced on it by the $(+,+)$ and $(-,-)$ boundary conditions.",
"More explicitly, $\\rho _+(\\tau ) \\ = \\ \\frac{Z^{+,+}_\\tau }{Z^{+,+}}\\,, \\, \\qquad \\rho _-(\\tau ) \\ = \\ \\frac{Z^{-,-}_\\tau }{Z^{-,-}}\\, ,$ with $Z^{s,s^{\\prime }}_\\tau $ the restricted partition functions $Z^{s,s^{\\prime }}_\\tau :=\\sum _{\\begin{array}{c}\\sigma :\\Lambda (3\\ell )\\backslash \\Lambda (\\ell )\\rightarrow \\lbrace -1,1\\rbrace \\\\\\sigma |_{\\partial _{\\text{v}}\\Lambda (2\\ell )} = \\tau \\end{array}} \\exp \\left(-\\frac{1}{T} H^{s,s^{\\prime }}(\\sigma )\\right).$ Considering first the $(+)$ case, let us note that $\\sum _{\\tau :\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace } \\rho _+(\\tau ) \\, \\frac{Z^{+,-}_\\tau }{Z^{+,+}_\\tau } \\ = \\ \\frac{Z^{+,-}}{Z^{+,+}}$ Hence, by Jensen's inequality (and the convexity of $-\\log (X)$ ), for each specified $\\eta $ (which is omitted in the following expression) $\\log \\left(\\frac{Z^{+,+}}{Z^{+,-}}\\right) \\ \\le \\ \\sum _{\\tau :\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace }\\, \\rho _+(\\tau ) \\, \\log \\left(\\frac{Z^{+,+}_\\tau }{Z^{+,-}_\\tau }\\right) \\ $ Combining the above with the analogous statement for $\\rho _-(\\tau ) $ we get: $\\mathcal {T}_{\\ell }(\\eta ) & = & T \\log \\left(\\frac{Z^{+,+}}{Z^{+,-}}\\cdot \\frac{Z^{-,-}}{Z^{-,+}}\\right) \\ \\le \\ \\\\& \\le & T\\left[\\sum _{\\tau :\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace }\\, \\rho _+(\\tau )\\, \\log \\left(\\frac{Z^{+,+}_\\tau }{Z^{+,-}_\\tau }\\right) \\ + \\ \\sum _{\\tau :\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace }\\, \\rho _-(\\tau ) \\log \\left(\\frac{Z^{-,-}_\\tau }{Z^{-,+}_\\tau }\\right)\\right]\\,.$ We now use the fact that the measure $\\mathbb {P}^{+,+}$ stochastically dominates $\\mathbb {P}^{-,-}$ , as in (REF ).",
"In particular, there exists a probability measure $\\rho (\\tau ^+,\\tau ^-)$ on pairs $\\tau ^+,\\tau ^-:\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace $ such that $\\tau ^+\\ge \\tau ^-$ pointwise, with probability 1, and the marginal distribution of each $\\tau ^s$ is given by $\\rho _s$ .",
"This coupling of measures allows to express (REF ) in the form $\\mathcal {T}_\\ell (\\eta )\\le T\\left[\\sum _{\\tau ^+,\\tau ^-:\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace }\\rho (\\tau ^+,\\tau ^-)\\log \\left(\\frac{Z^{+,+}_{\\tau ^+}}{Z^{+,-}_{\\tau ^+}}\\cdot \\frac{Z^{-,-}_{\\tau ^-}}{Z^{-,+}_{\\tau ^-}}\\right)\\right].$ The coupling of the measures allows to bound the quantity on the right in terms of the positive temperature version of the disagreement percolation.",
"The estimate is motivated by the observation that for every configuration $\\tau $ : $Z^{+,+}_{\\tau }\\cdot Z^{-,-}_{\\tau }\\ = \\ Z^{+,-}_{\\tau } \\cdot Z^{-,+}_{\\tau }\\,.$ The proof is through the bijection associating to each pair $(\\sigma ^{+,+},\\sigma ^{-,-})$ contributing to the double sum on the left the following pair $(\\sigma ^{+,-},\\sigma ^{-,+})$ contributing to the double sum on the right: ${\\sigma }^{+,-}_v:={\\left\\lbrace \\begin{array}{ll}\\sigma ^{+,+}_v&v\\in \\Lambda (3\\ell )\\setminus \\Lambda (2\\ell )\\\\\\sigma ^{-,-}_v&v\\in \\Lambda (2\\ell )\\setminus \\Lambda (\\ell )\\end{array}\\right.",
"},\\quad {\\sigma }^{-,+}_v:={\\left\\lbrace \\begin{array}{ll}\\sigma ^{-,-}_v&v\\in \\Lambda (3\\ell )\\setminus \\Lambda (2\\ell )\\\\\\sigma ^{+,+}_v&v\\in \\Lambda (2\\ell )\\setminus \\Lambda (\\ell )\\end{array}\\right.",
"}.$ At the common value of the configuration $\\tau $ over the separating set $\\partial _{\\text{v}}\\Lambda (2\\ell )$ , the sums of the corresponding energy terms in (REF ) match.",
"Thus terms with $\\tau ^+=\\tau ^-$ make no contribution to the sum (REF ).",
"For the more general case we note that when the restriction of $\\sigma ^{+,+}$ ($\\sigma ^{-,-}$ ) to $\\partial _{\\text{v}}\\Lambda (2\\ell )$ is $\\tau ^+$ ($\\tau ^-$ ) and $\\sigma ^{+,-}, \\sigma ^{-,+}$ are given by (REF ) then, with $\\tau ^+\\ge \\tau ^-$ , $\\begin{split}&-\\frac{1}{T}\\left(H^{+,+}(\\sigma ^{+,+}) + H^{-,-}(\\sigma ^{-,-}) - H^{+,-}(\\sigma ^{+,-}) - H^{-,+}(\\sigma ^{-,+})\\right)\\\\&=\\frac{J}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} \\left(\\sigma ^{+,+}_u\\sigma ^{+,+}_v + \\sigma ^{-,-}_u\\sigma ^{-,-}_v - \\sigma ^{+,+}_u\\sigma ^{-,-}_v - \\sigma ^{-,-}_u\\sigma ^{+,+}_v\\right)\\\\&=\\frac{J}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} \\left(\\sigma ^{+,+}_u - \\sigma ^{-,-}_u\\right)\\cdot \\left(\\sigma ^{+,+}_v - \\sigma ^{-,-}_v\\right)\\\\&=\\frac{J}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} \\left(\\sigma ^{+,+}_u - \\sigma ^{-,-}_u\\right)\\cdot \\left(\\tau ^+_v - \\tau ^-_v\\right)\\le \\frac{4J}{T}\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} \\left(\\tau ^+_v - \\tau ^-_v\\right),\\end{split}$ where the third equality uses the fact that if $(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )$ then $v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )$ and the inequality uses the fact that each vertex $v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )$ is incident to at most two edges $(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )$ and the fact that $\\tau ^+\\ge \\tau ^-$ pointwise.",
"Thus $\\frac{Z^{+,+}_{\\tau ^+}}{Z^{+,-}_{\\tau ^+}}\\cdot \\frac{Z^{-,-}_{\\tau ^-}}{Z^{-,+}_{\\tau ^-}}\\le \\exp \\left(\\frac{4J}{T}\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} \\left(\\tau ^+_v - \\tau ^-_v\\right)\\right).$ Finally, inserting this estimate in (REF ) we get $\\mathcal {T}_\\ell (\\eta )\\le 4J\\sum _{\\tau ^+,\\tau ^-:\\partial _{\\text{v}}\\Lambda (2\\ell )\\rightarrow \\lbrace -1,1\\rbrace }\\rho (\\tau ^+,\\tau ^-)\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} \\left(\\tau ^+_v - \\tau ^-_v\\right).$ Through the definition of $\\rho (\\tau ^+,\\tau ^-)$ the above reduces to the bound asserted in (REF ).",
"The representation of the surface tension given by Theorem REF , which enables a lower bound on its expected value at zero temperature, continues to hold at positive temperature with the exact same statement.",
"The proof also remains the same, upon replacing (REF ) and (REF ) with the analogous $\\frac{\\partial }{\\partial \\eta _v} G_\\ell (\\eta ) = \\varepsilon \\left[ \\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), +}-\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), -} \\right]$ and $\\mathcal {T}_\\ell (\\eta ) = \\varepsilon \\int _{-\\infty }^\\infty \\sum _{v \\in \\Lambda (\\ell )} \\left[ \\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), +}(\\eta ^{(t)})-\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), -}(\\eta ^{(t)}) \\right] \\, dt\\, = \\, 2\\varepsilon \\int _{-\\infty }^\\infty D_{\\ell }(\\eta ^{(t)})\\, dt\\, .$ Combining Theorem REF and (REF ) we obtain $\\frac{4 \\, \\mathbb {E}(\\tilde{B}_\\ell (\\eta ))}{\\varepsilon \\sqrt{|\\Lambda (\\ell )|}}& \\ge & \\mathbb {E}\\left( \\frac{D_{\\ell }(\\eta ) }{| \\Lambda (\\ell )|}\\,\\frac{1 }{ \\phi (\\widehat{\\eta }) } \\right) \\,$ which replaces (REF ) when $T>0$ .",
"The bound implies that Proposition REF continues to hold at positive temperature, with the exact same statement and with $2\\tilde{B}_\\ell $ replacing $B_\\ell $ throughout the proof (noting, in particular, that $2\\mathbb {E}(\\tilde{B}_\\ell ) \\le 2\\, J\\, |\\partial _{\\text{v}}\\Lambda (2\\ell )|\\,m(\\ell -1)$ holds instead of (REF )).",
"The upper bound on the variance of $D_\\ell $ , given for $T=0$ by Proposition REF , continues to hold exactly as stated also when $T>0$ .",
"In the proof, the indicator random variable of the event $E_v$ is replaced with the random variable $X_v := \\frac{1}{2}\\left[\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), +}-\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell ), -}\\right]\\,.$ This yields, e.g., the analogous equation to (REF ), $\\operatorname{Var}\\left(D_\\ell \\right) = \\sum _{v,w\\in \\Lambda (\\ell )} \\big [\\mathbb {E}(X_v\\cdot X_w) - \\mathbb {E}(X_v)\\mathbb {E}(X_w)\\big ]\\,$ and the analogous equation to (REF ), $\\mathbb {E}(X_v\\cdot X_w) \\ \\le \\ m(r(v,w))^2$ with $r(v,w)$ defined in (REF ).",
"The last inequality holds as, via the monotonicity property (REF ), $\\mathbb {E}(X_v\\cdot X_w) \\le \\frac{1}{4}\\mathbb {E}\\left[\\left(\\langle \\sigma _v\\rangle ^{\\Lambda _v(r(v,w)), +}-\\langle \\sigma _v\\rangle ^{\\Lambda _v(r(v,w)), -}\\right)\\left(\\langle \\sigma _v\\rangle ^{\\Lambda _w(r(v,w)), +}-\\langle \\sigma _v\\rangle ^{\\Lambda _w(r(v,w)), -}\\right)\\right]\\,$ after which one may rely on independence.",
"The end of the proof of Theorem REF , detailed in Section REF for the zero-temperature case, applies without change to prove the theorem at positive temperature."
],
[
"Extension to finite-range interactions",
"At $T=0$ , the proof for general finite-range interactions $\\mathcal {J}$ remains the same with the following minor changes, in which $C_k(\\mathcal {J})$ denote positive constants depending only on $\\mathcal {J}$ and $R(\\mathcal {J}) = \\max \\lbrace d(u,v) \\, : \\, J_{u,v} \\ne 0\\rbrace $ (the interaction's range).",
"The statement of Theorem REF is changed by replacing the bound $B_{\\ell } (\\eta ) \\le 2 J \\, |\\partial _{\\text{v}}\\Lambda (2 \\ell )|$ by $B_{\\ell } (\\eta ) \\ \\le \\ \\sum _{(u,v)\\in \\partial _{\\text{e}} \\Lambda (2\\ell )} J_{u,v}\\,.$ The condition $|h+\\varepsilon \\eta _v| > 4J$ appearing in the proof of Theorem REF is replaced by $|h+\\varepsilon \\eta _v| > \\sum _v J_{\\operatorname{\\mathbf {0}}, v}$ .",
"The bound (REF ) is replaced by $\\mathbb {E}(B_\\ell ) \\le C_1(\\mathcal {J})|\\partial _{\\text{v}}\\Lambda (2\\ell )|\\,m(\\ell -R(\\mathcal {J})).$ Consequently in (REF ), $4 J |\\partial _{\\text{v}}\\Lambda (2\\ell )|\\,m(\\ell -1)$ becomes $C_2(\\mathcal {J}) |\\partial _{\\text{v}}\\Lambda (2\\ell )|\\,m(\\ell -R(\\mathcal {J}))$ .",
"In the proof of Proposition REF , the definition of $r(v,w)$ in (REF ) is replaced by $r(v,w):=\\lfloor (d(v,w)-R(\\mathcal {J}))/2\\rfloor .$ The simple bound (REF ) is then used for pairs $v,w$ at distance $d(v,w)\\le R(\\mathcal {J}) + 1$ , leading to the factor $|\\Lambda (2)|$ appearing in the proof being replaced by $|\\Lambda (R(\\mathcal {J}) + 1)|$ .",
"The statement of Proposition REF is changed to allow $L_0$ to depend on $\\mathcal {J}$ (besides $\\alpha $ ).",
"The proof of Theorem REF given in Section REF is modified by taking into account the change described in item REF above in the constants appearing in Proposition REF .",
"Correspondingly, inequality (REF ) is modified to $\\mathbb {P}\\left(\\frac{D_\\ell }{\\mathbb {E}\\big ( D_\\ell \\big )} < \\frac{1}{2}\\right) \\,\\ge \\, \\chi \\left(\\frac{C_3(\\mathcal {J})}{\\varepsilon }\\cdot \\left(\\frac{L}{\\ell - R(\\mathcal {J})}\\right)^{2\\gamma }\\right)\\ge \\chi \\left(\\frac{C_4(\\mathcal {J})}{\\varepsilon }\\right)$ holding for $L$ sufficiently large, and the power $\\gamma $ appearing in the theorem is modified from its value in (REF ) to $\\gamma = 2^{-10}\\chi \\left(\\frac{C_4(\\mathcal {J})}{\\varepsilon }\\right).$ At $T>0$ , the argument extends to general finite-range interactions by applying the following changes: The definition of $\\tilde{B}_{\\ell }$ in (REF ) is modified to $\\tilde{B}_{\\ell }(\\eta )\\ := \\frac{1}{4}\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} J_v \\left[\\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell )\\setminus \\Lambda (\\ell ),+} - \\langle \\sigma _v\\rangle ^{\\Lambda (3\\ell )\\setminus \\Lambda (\\ell ),-}\\right]$ with $J_v\\ :=\\ \\sum _{u\\colon (u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} J_{u,v}\\,.$ The proof of Theorem REF is modified by replacing the inequality (REF ) with $\\begin{split}&-\\frac{1}{T}\\left(H^{+,+}(\\sigma ^{+,+}) + H^{-,-}(\\sigma ^{-,-}) - H^{+,-}(\\sigma ^{+,-}) - H^{-,+}(\\sigma ^{-,+})\\right)\\\\&=\\frac{1}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} J_{u,v}\\left(\\sigma ^{+,+}_u\\sigma ^{+,+}_v + \\sigma ^{-,-}_u\\sigma ^{-,-}_v - \\sigma ^{+,+}_u\\sigma ^{-,-}_v - \\sigma ^{-,-}_u\\sigma ^{+,+}_v\\right)\\\\&=\\frac{1}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} J_{u,v}\\left(\\sigma ^{+,+}_u - \\sigma ^{-,-}_u\\right)\\cdot \\left(\\sigma ^{+,+}_v - \\sigma ^{-,-}_v\\right)\\\\&=\\frac{1}{T}\\sum _{(u,v)\\in \\partial _{\\text{e}}\\Lambda (2\\ell )} J_{u,v}\\left(\\sigma ^{+,+}_u - \\sigma ^{-,-}_u\\right)\\cdot \\left(\\tau ^+_v - \\tau ^-_v\\right)\\le \\frac{2}{T}\\sum _{v\\in \\partial _{\\text{v}}\\Lambda (2\\ell )} J_v\\left(\\tau ^+_v - \\tau ^-_v\\right),\\end{split}$ with this change propagating to the next two displayed equations in the proof.",
"The changes analogous to those described for the $T=0$ case."
],
[
"Magnetization decoupling bounds",
"For completeness sake we enclose here proofs that the influence percolation probability $m(\\ell ,...)$ provides bounds on both the covariance between the quenched local magnetizations at distant sites and the spin - spin covariance within the Gibbs states at typical configurations of the random field, as was asserted in (REF ) and (REF ).",
"The arguments apply in the generality of the random-field Ising model on a general infinite transitive graph, in any of its infinite-volume Gibbs states.",
"Lemma 6.1 In the random field Ising model on a transitive graph, with spin-spin coupling of a finite range $R(\\mathcal {J})$ and any pair of vertices $\\lbrace u,v\\rbrace $ at distance $d(u,v)$ .",
"If $ d(u,v) > \\ell $ then $\\mathbb {E}{(\\langle \\sigma _u;\\sigma _v\\rangle )} \\ \\le \\ 2\\, m(\\ell ;T, \\mathcal {J}, h, \\epsilon ),$ while if $ d(u,v) \\ge 2\\ell +R(\\mathcal {J})$ then $\\rm {Cov}\\Big ( \\langle \\sigma _u \\rangle ; \\langle \\sigma _v\\rangle ) \\Big ) \\ := \\ \\mathbb {E}(\\langle \\sigma _u \\rangle ; \\langle \\sigma _v\\rangle ) \\ \\le \\ 4 \\, m(\\ell ;T, \\mathcal {J}, h, \\epsilon ) \\,.$ Proof: i) By the FKG monotonicity of the RFIM Gibbs states, the Gibbs conditional expectation of $\\sigma _u$ , conditioned on the configuration's restriction to the complement of the set $\\Lambda _u(\\ell )$ , satisfies, for any configuration of the random field $\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\ \\le \\ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), \\sigma _{\\Lambda _u(\\ell )^c}} \\ \\le \\ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} \\,.$ Averaging over $\\sigma _{\\Lambda _u(\\ell )^c} $ , one learns that also the infinite-volume expectation value is bracketed by $\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), \\pm } $ : $\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\ \\le \\ \\langle \\sigma _u\\rangle \\ \\le \\ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} \\,.$ The two equations imply: $\\Big | \\langle \\sigma _u\\rangle \\ - \\ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), \\sigma _{\\Lambda _u(\\ell )^c}} \\Big | \\ \\le \\ \\Big [ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} -\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\Big ]$ The covariance of the spins within the infinite-volume Gibbs state can be written as $\\begin{aligned}\\langle \\sigma _u;\\sigma _v\\rangle & =\\langle \\big ( \\sigma _u - \\langle \\sigma _u\\rangle \\big ) \\, \\, \\sigma _v\\rangle \\\\& =\\langle \\big ( \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), \\sigma _{\\Lambda _u(\\ell )^c}} - \\langle \\sigma _u\\rangle \\big ) \\, \\, \\sigma _v\\rangle \\end{aligned} $ where the second equation is by the state's Dobrushin-Lanford-Ruelle property and the assumption that $ d(u,v) > \\ell $ .",
"Combining (REF ) with (REF ) we learn that for any realization of the random field $\\Big | \\langle \\sigma _u;\\sigma _v\\rangle \\Big | \\ \\le \\ \\Big [ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} -\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\Big ].$ Averaging this relation over the disorder one gets (REF ).",
"ii) For the second covariance bound let $\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av} \\ := \\ \\frac{1}{2}\\Big [ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} +\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\Big ]$ and observe that since the random fields on which $\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av}$ and $\\langle \\sigma _v\\rangle ^{\\Lambda _v(\\ell ), av}$ depend belong to disjoint sets, their covariance vanishes: $\\rm {Cov}\\Big (\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av},\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av}\\Big ) \\ = \\ 0$ Furthermore, by (REF ), $\\Big | \\langle \\sigma _u\\rangle \\ - \\ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av} \\Big | \\ \\le \\ \\frac{1}{2}\\Big [ \\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), +} -\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), -} \\Big ] \\,.",
"\\\\[2ex]$ The claimed (REF ) then follows by a simple application of the general covariance bound: $\\Big | \\rm {Cov}(A,B) - \\rm {Cov}(\\widetilde{A}, \\widetilde{B}) \\Big | = \\Big |\\mathbb {E}[(A - \\widetilde{A})B]+\\mathbb {E}[\\widetilde{A}(B-\\widetilde{B})]+\\mathbb {E}[(\\widetilde{A} - A)]\\mathbb {E}[B]+\\mathbb {E}[\\widetilde{A}]\\mathbb {E}[(\\widetilde{B} - B)]\\Big | \\\\\\le 2 \\,\\Vert A-\\widetilde{A}\\Vert _1 \\cdot \\Vert B \\Vert _{\\infty } \\ + \\ 2 \\, \\Vert B-\\widetilde{B}\\Vert _1 \\cdot \\Vert \\widetilde{A} \\Vert _{\\infty }$ applied to $\\begin{aligned}A &= \\langle \\sigma _u\\rangle \\, , \\qquad \\widetilde{A}&=\\langle \\sigma _u\\rangle ^{\\Lambda _u(\\ell ), av} \\\\B&= \\langle \\sigma _v\\rangle \\, , \\qquad \\widetilde{B} &=\\langle \\sigma _v\\rangle ^{\\Lambda _v(\\ell ), av}\\end{aligned} $ for which, by (REF ) and the definition of $m(\\ell ;T, \\mathcal {J}, h, \\epsilon ) $ , $\\Vert A-\\widetilde{A}\\Vert _1 = \\Vert B-\\widetilde{B}\\Vert _1 \\ \\le \\ m(\\ell ;T, \\mathcal {J}, h, \\epsilon ).$ $\\Box $"
],
[
"Discussion and open questions",
"In summary: our study quantifies the analysis of [4], [5] that for each value of the external field the model's Hamiltonian almost surely has a unique infinite-volume ground state, and similarly unique positive-temperature Gibbs states.",
"The upper bounds proven here establish that the probability that the ground-state configuration depends on the quenched disorder at distance $\\ell $ away decays by at least an $\\varepsilon $ -dependent power, and exponentially fast if the disorder parameter is sufficiently large.",
"However, our understanding of the model remains incomplete.",
"Following is a selection of open questions, some with relevance for physics models and some as a challenge to probabilists of related interests.",
"Exponential vs. power-law decay.",
"As mentioned above, an open question of enduring interest is whether as the disorder parameter ($\\varepsilon / J$ ) is tuned down the ground state's dependence on the quenched disorder makes a transition from exponential decay to a power law.",
"Tentative but admittedly weak arguments have appeared for each of these possibilities ([14], [8] and [12]).",
"Also of interest is the corresponding question for the $O(N)$ symmetric models in dimensions $d\\le 4$ , the latter being the critical dimension for the Imry-Ma phenomenon in the presence of continuous symmetry.",
"Cluster dynamics.",
"Consider the RFIM dynamics in which a large system with a quenched random field is subject to a slowly varying uniform magnetic field $h$ .",
"For $|h|/\\varepsilon $ large enough, the ground state configuration is close to being constant, coinciding with the sign of $h$ .",
"As the uniform field is increased, starting from the sufficiently negative value, the corresponding ground state configuration changes in a sequence of flips, in which a cluster of $-$ spins flips to $+$ spins.",
"Thus the graph is partitioned into connected clusters of sites for which at the given random field $\\eta $ the spins flip at a common value of $h$ .",
"It can be shown that in two dimensions almost surely each flip involves only a finite number of sites, and the mean value of the size of the cluster which flips along with a preselected site is finite throughout the regime in which the ground state spins decorrelate exponentially fast.",
"Does the mean stay finite for arbitrarily small $\\varepsilon >0$ ?",
"RFIM with other random field distributions.",
"Our analysis focused on IID Gaussian disorder.",
"In contrast, the theorem of [4], [5] applies to a wide class of random field distributions.",
"The Gaussian structure allowed a short-cut in the proof of Theorem REF .",
"While we expect the results to be valid also well beyond this case, that is not done here.",
"Among the other distributions of interest are: A dilute coercive field, with $(\\eta _v)$ given by independent random variables with $\\mathbb {P}(\\eta _v=-\\infty )=\\mathbb {P}(\\eta _v=\\infty )=\\varepsilon $ and $\\mathbb {P}(\\eta _v = 0) = 1-2\\varepsilon $ .",
"This distribution was considered in [12] where an observation was initially made suggesting the possibility of a transition from exponential to power-law decay of correlations at low $\\varepsilon $ .",
"(However, subsequent considerations have weakened the case for that, cf.",
"also the discussion in [8]).",
"Bounded variables, e.g.",
"with $(\\eta _v)$ independent and uniformly distributed in $\\lbrace -1,1\\rbrace $ or $[-1,1]$ .",
"The former is of particular relevance for the case of $Q$ -state Potts models with random couplings, for which $\\sigma _v$ takes values in $\\lbrace 1,..., Q\\rbrace $ and the Hamiltonian is: $H_\\eta (\\sigma ) \\ =\\ - \\sum _{\\lbrace x,y\\rbrace \\in E(\\mathbb {Z}^2)} \\left(J + \\varepsilon \\eta _{x,y}\\right)\\mathbb {1}[\\sigma _x = \\sigma _y ] \\, .$ The uniform bound on $|\\eta |$ allows to keep the discussion separate from that of frustration effects.",
"The more general Imry-Ma phenomenon.",
"While the RFIM is a bellwether for the more general Imry-Ma phenomenon, the general case is a bit more complicated on two accounts.",
"The first is the lack of a-priori obvious pair of opposing boundary conditions for the definition of the order parameter.",
"That can be addressed, as was done in [5], by inducing the $\\pm $ states not through boundary conditions but throughout a mild shift of the uniform field beyond the corresponding boundary of the region under study, $h\\rightarrow h\\pm \\delta h$ with $\\delta h$ in the range $|\\Lambda (\\ell )|^{-1}\\ll \\delta h \\ll 1 \\, \\quad \\mbox{ (as $ \\ell \\rightarrow \\infty $)} \\,.$ (An alternative is to define the order parameter though a maximization of the difference induced by different boundary spin configurations.)",
"A potentially more substantial difference with the RFIM, is that in the general case the natural order parameter does not control the difference in the configurations, or measures, just in their (generalized) magnetizations.",
"The resolution of this complication may require some new technical ideas."
],
[
"Exponential decay at high disorder",
"As a rule of thumb it is generally expected that at high enough disorder, be it thermal or due to noisy environment, correlations decay exponentially fast.",
"Results in this vein for systems related to the RFIM can be found in the works of A. Berretti [6], J. Imbrie and J. Fröhlich [16], and F. Camia, J. Jiang and C.M.",
"Newman [9].",
"Let us present here an especially simple proof of such behavior for the $T=0$ case, i.e.",
"exponential decay of the correlations of the RFIM's ground state, and also of the principle that fast enough power-law decay implies exponential decay.",
"Theorem 1.1 For the RFIM on $\\mathbb {Z}^d$ with the nearest-neighbor interaction (REF ) and random field given by IID random variables $(\\eta _u)$ , if $\\mathbb {P}\\big (|h+\\varepsilon \\eta _\\textbf {0}| \\le 2d J \\big ) \\ < \\ p_c(d) \\,$ with $p_c(d)$ the critical density for site percolation on $\\mathbb {Z}^d$ , then $m(L;0,\\mathcal {J}, h,\\varepsilon )$ decays exponentially fast in $L$ .",
"At sites where $|h+\\varepsilon \\eta _v| > 2d J$ the ground-state configuration is dictated by the sign of the local field.",
"Hence disagreement percolation can propagate only along the sites with $|h+\\varepsilon \\eta _v| \\le 2d J$ .",
"In the regime described by (REF ) the exceptional sites form a sub-percolating point process, for which the connectivity probability is known to decay exponentially in the distance [2], [19].",
"A boosted version of the above simple argument allows to conclude that if on some scale $\\ell $ the probability of influence propagation is small enough ($1/\\ell ^{d-1}$ power law with a small prefactor) then on larger scales the influence decays exponentially fast.",
"An analogous statement holds also for $T>0$ , but for simplicity of presentation we present the proof for $T=0$ .",
"Theorem 1.2 For the RFIM on $\\mathbb {Z}^d$ with the nearest-neighbor interaction (REF ), there is a finite constant $c_0$ (depending only on $d$ ) with which: if for some $\\ell < \\infty $ $m(\\ell ;0,\\mathcal {J}, h,\\varepsilon ) \\, \\le \\, c_0 / \\ell ^{d-1}$ then for all $L <\\infty $ $m(L;0,\\mathcal {J}, h,\\varepsilon ) \\, \\le \\, C_1 \\, e^{- b L/ \\ell }$ with $C_1, b \\in (0,\\infty )$ which do not depend on $J$ , $h$ , $\\epsilon $ and $\\ell $ .",
"In particular, we learn that $m(L;0,\\mathcal {J}, h,\\varepsilon )$ cannot decay by a power law faster than $1/L$ without decaying exponentially.",
"In the following we say that a site $v\\in \\mathbb {Z}^2$ is sensitive to boundary conditions at distance $\\ell $ if $\\sigma ^{\\Lambda _v(\\ell ),+}_v \\ne \\sigma ^{\\Lambda _v(\\ell ),-}_v \\,.",
"$ For each $L> \\ell $ the event whose probability defines $ m(L) $ , $\\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), +} \\ \\ne \\ \\langle \\sigma _\\textbf {0}\\rangle ^{\\Lambda (L), -} \\, ,$ requires the existence of a path from $\\textbf {0}$ to the set $\\partial _{\\text{v}}\\Lambda (L-\\ell )$ along sites $v\\in \\mathbb {Z}^2$ at which the condition (REF ) holds.",
"Let now $\\mathcal {P}_\\ell $ be a partition of the vertex set of $\\mathbb {Z}^d$ into a $\\mathbb {Z}^d$ -like array of disjoint cubic blocks of side length $2\\ell $ , and consider the random set of blocks in this partition which contain at least one site for which (REF ) holds.",
"These block events are 1-step independent, in the sense that they are jointly independent for any collection of blocks of which no two are touching.",
"The probability that an individual block contains a site at which the condition (REF ) holds is trivially dominated by $|\\partial _{\\text{v}}\\Lambda (\\ell )| \\times m(\\ell )$ .",
"Adjusting the constant $c_0$ in assumption (REF ) the above probability can be made as small as convenient.",
"The claim then follows through a standard exponentially-decaying bound on the connectivity probability in 1-step independent percolation of small enough density."
],
[
"The Mandelbrot percolation analogy",
"The results presented above do not answer the question whether in two dimensions the exponential decay of correlations persists into arbitrarily small values of the disorder parameter, or whether the exponential decay turns into a power-law decay at low enough (but still non zero) $\\varepsilon $ .",
"Related to this is the question of what would be a sensible algorithm for the computation of the ground state $\\widehat{\\sigma }$ for a given random field, and how would it perform at very low disorder.",
"An intriguing perspective is provided by the following hierarchal algorithm.",
"It has the virtue of simplicity but also the drawback of being potentially misleading through over simplification.",
"It is formulated for the specific case $h=0$ and nearest-neighbor interaction.",
"Let $(\\mathcal {P}_n)$ , $n\\ge 0$ , be a sequence of nested partitions of $\\mathbb {Z}^2$ into square blocks, with the blocks in $\\mathcal {P}_n$ having side-length $3^n$ and the square containing $x$ denoted by $D_{n,x}$ .",
"For each $n, x$ we define the following as a large-field event in $D_{n,x}$ : $\\mathcal {F}_{n,x} \\ := \\ \\lbrace \\eta \\, : \\ \\varepsilon \\, \\big |\\eta (D_{n,x}) \\big | \\ > \\ J \\, |\\partial _{\\text{e}} D_{n,x}| \\rbrace \\,.$ where $\\eta (D) := \\sum _{x\\in D} \\eta _x $ is the total block field.",
"A relevant feature of two dimensions is that the probabilities of the large-field events are scale invariant: $\\mathbb {P}(\\mathcal {F}_{n,x}) \\ = \\ \\chi (4J/\\varepsilon ) \\ := p \\ \\approx \\exp [-8J^2/\\varepsilon ^2] \\, .$ For a given $x\\in \\mathbb {Z}^2$ the events $\\mathcal {F}_{n,x} $ are not strictly independent, however the sequence (in $n$ ) of the corresponding indicator functions is easily seen to be asymptotic, in probability, to a stationary and mixing sequence of random variables.",
"Let $n(x; \\eta )$ be the first non-negative integer for which large field is exhibited in $D_{n,x}$ .",
"Due to the above properties of the events $\\mathcal {F}_{n,x} $ for any $J,\\varepsilon >0$ almost surely $n(x; \\eta ) < \\infty $ for all $x$ .",
"Under $\\mathcal {F}_{0,x}$ , i.e.",
"in case the large-field event occurs at $x$ already on the smallest scale, the value of the ground-state configuration at $x$ is predictably given by $\\rm {sign} (\\eta _x)$ , i.e.",
"the sign of the field.",
"In case the $\\eta _x$ is itself not large enough to meet this criterion, but the site is separated from the boundary of a set $\\Lambda $ by a loop of sites for which the large-field events occur at scale $n=0$ , one may still conclude that the finite-volume ground state at $x$ does not depend on the boundary spin configuration $\\sigma _{\\partial _{\\text{v}}\\Lambda }$ .",
"Scaling up these observations, though along the way departing from rigor, we arrive at the following somewhat over-simplified algorithm for the assignment of a spin configuration $\\tau (\\eta )$ which may mimic the infinite-volume ground state $\\widehat{\\sigma }(\\eta )$ .",
"For each $x\\in \\mathbb {Z}^2$ let $k(x; \\eta ) $ be defined as the smallest $0\\le k< n(x; \\eta )$ for which $x$ is separated from infinity by a loop of sites with $n(x; \\eta ) \\le k$ , if such a $k$ exists, and otherwise set $k(x; \\eta )=n(x; \\eta )$ .",
"In the first case, i.e.",
"$n(x; \\eta ) = k(x;\\eta )$ , we let $\\tau (\\eta )_x = \\rm {sign} (\\eta (D(x))$ .",
"If $ k(x;\\eta ) < n(x; \\eta ) $ , the value of $\\tau (\\eta )_x$ is determined by minimizing the RFIM energy over the interior of the corresponding $x$ -encapsulating loop, with the previously constructed values serving as boundary conditions for $\\tau (\\eta )_x$ .",
"For the finite-volume version of the construction, in $\\Lambda \\subset \\mathbb {Z}^2$ , the above construction is modified by limiting the considerations of large-field events to cubes contained in $\\Lambda $ .",
"In the last step, unless $\\tau ^{\\Lambda ,\\pm }(\\eta )_x$ is defined already through such events, its calculation will incorporate the boundary conditions imposed at $\\partial _{\\text{v}}\\Lambda $ .",
"Under the above algorithm the influence of the boundary conditions on $\\tau (\\eta )_x$ percolates over sites for which the events $\\mathcal {F}_{n,x}$ did not yet occur.",
"For an idea on the probability that the influence percolates deep inside $\\Lambda $ one may take the further approximation in which the correlations between the indicator functions of nested events $\\mathcal {F}_{n,x} $ are ignored.",
"Under the latter approximation, the collection of sites not covered by any of the large-field events, has the distribution of the random fractal set discussed in Mandelbrot's “canonical curdling” model [18].",
"In particular, the influence-percolation process coincides with the Mandelbrot-percolation process at density $p$ given by (REF ).",
"Curiously, as was proven by Chayes-Chayes-Durrett [11], the Mandelbrot-percolation process does undergo a phase transition.",
"Its manifestation in the lattice version of the model is that the connectivity function decays exponentially fast for $p$ large enough, but at $p$ small the decay changes to a power law.",
"(The model is most appealing in its continuum, or “ultraviolet”, limit while our discussion is focused on its infinite-volume, or “infrared”, limit.",
"However in the analysis there is a simple relation between the two).",
"It should however be noted that for the finite-volume version of the construction, the existence of a path connecting $x$ to $\\partial _{\\text{v}}\\Lambda $ in the complement of the set of sites covered by large-field events is only a necessary condition for the dependence of $\\tau ^{\\Lambda ,\\pm }(\\eta )_x$ on the boundary conditions.",
"As its value is determined through the energy minimization conditioned on both the $\\pm $ boundary conditions and the randomly determined values along the large-field sets, the $\\pm $ boundary conditions may lose their effect on $\\tau (\\eta )_x$ even before the geometric disconnection of $x$ from $\\partial _{\\text{v}}\\Lambda $ .",
"Thus the Mandelbrot-percolation's phase transition does not preclude exponential decay of the $\\tau $ -analog of our finite-volume order parameter at all $p>0$ ."
],
[
"Acknowledgements",
"The work of MA was supported in part by the NSF grant DMS-1613296 and the Weston Visiting Professorship at the Weizmann Institute.",
"The work of RP was supported in part by Israel Science Foundation grant 861/15 and the European Research Council starting grant 678520 (LocalOrder).",
"We thank the Faculty of Mathematics and Computer Science and the Faculty of Physics at WIS for the hospitality enjoyed there during work on this project."
]
] | 1808.08351 |
[
[
"Stochastic Collocation with Non-Gaussian Correlated Parameters via a New\n Quadrature Rule"
],
[
"Abstract This paper generalizes stochastic collocation methods to handle correlated non-Gaussian random parameters.",
"The key challenge is to perform a multivariate numerical integration in a correlated parameter space when computing the coefficient of each basis function via a projection step.",
"We propose an optimization model and a block coordinate descent solver to compute the required quadrature samples.",
"Our method is verified with a CMOS ring oscillator and an optical ring resonator, showing 3000x speedup over Monte Carlo."
],
[
"Introduction",
"Stochastic spectral methods are popular techniques to quantify the impact of process variations in nano-scale chip design.",
"Various techniques, such as stochastic Galerkin [1], stochastic testing [2] and stochastic collocation [3], have achieved great success in electronic circuits [4], [5], [6] and photonics [7], and have shown significant speedup over Monte Carlo.",
"These techniques approximate a stochastic solution as a linear combination of some basis functions, providing a close-form surrogate model for fast statistical analysis and design automation.",
"Almost all previous stochastic spectral methods assume that the random parameters are mutually independent.",
"This is rarely true in practice.",
"Device geometric or electrical parameters influenced by the same fabrication steps are highly correlated; circuit-level performance parameters used in system-level analysis usually depend on each other.",
"In this paper, we focus on the non-Gaussian correlated parameters in Fig.",
"REF (c).",
"Karhunen-Loev̀e theorem is error-prone and not scalable.",
"Preprocessing techniques such as principal component analysis can only handle Gaussian density functions.",
"Our contributions.",
"We generalize stochastic collocation to non-Gaussian correlated cases by two steps: [leftmargin=*] We propose a new set of basis functions to capture the impact caused by non-Gaussian correlated parameters that cannot be handled by generalized polynomial chaos [8].",
"Previous integration methods such as sparse grid [9] or Gauss quadrature [10] do not work for non-Gaussian correlated cases.",
"Motivated by [11], [12], we propose an optimization solver to calculate the quadrature nodes and weights.",
"We also present a block coordinate descent method to improve the scalability of our solver.",
"We validate our algorithm by both electronic and photonic ICs, showing $3000\\times $ speedup over Monte Carlo.",
"Figure: Joint density for (a): independent Gaussian, (b): correlated Gaussian, (c): correlated non-Gaussian (e.g., a Gaussian-mixture distribution) cases."
],
[
"Review: Stochastic Collocation",
"Let ${\\xi }=[{\\xi }_1, \\cdots , {\\xi }_{d}] \\in \\mathbb {R}^{d}$ be $d$ random parameters describing process variations.",
"We aim at estimating the uncertainty of a performance metric $y({\\xi })$ (e.g., chip frequency or power).",
"Stochastic spectral methods approximate the solution by $y({\\xi }) \\approx \\sum \\limits _{|\\alpha |=0}^{p} {c_{\\alpha } \\Psi _{\\alpha } ({\\xi })} , \\; {\\rm with}\\; \\mathbb {E}\\left[{\\Psi }_{\\alpha } \\left( {\\xi }\\right)\\Psi _{\\beta }\\left( {\\xi }\\right)\\right]=\\delta _{\\alpha , \\beta }.$ Here $\\mathbb {E}$ denotes expectation, $\\delta $ denotes a Delta function, the basis functions $\\lbrace {\\Psi }_{\\alpha } \\left({\\xi }\\right)\\rbrace $ are orthonormal polynomials, $\\alpha =[\\alpha _1,\\cdots , \\alpha _{d}] \\in \\mathbb {N}^{d}$ indicates the highest polynomial order of each parameter in the corresponding basis.",
"The total polynomial order $|\\alpha |=\\alpha _1+\\ldots +\\alpha _d$ is bounded by $p$ , and thus the total number of basis functions is $N=(p+d)!/(p!d!",
")$ .",
"Projection-based stochastic collocation methods compute the coefficient $c_{\\alpha }$ via a numerical integration.",
"If one has $M$ quadrature nodes $\\lbrace {\\xi }_{k}\\rbrace _{k=1}^M$ and weights $\\lbrace w_{k}\\rbrace _{k=1}^M$ , then $c_{\\alpha }=\\mathbb {E} \\left[ y({\\xi }) {\\Psi }_{\\alpha } ({\\xi })\\right]\\approx \\sum \\limits _{k=1}^M {y({\\xi }_{k}) {\\Psi }_{\\alpha } ({\\xi }_{k}) w_{k} }.$ If ${\\xi }$ are mutually independent, then ${\\Psi }_{\\alpha } ({\\xi }_{k})$ may be chosen as the generalized polynomial chaos [8], and the quadrature nodes and weights can be calculated via sparse grid [9] and Gauss quadrature [10].",
"However, how to choose the basis functions and quadrature rule is an open question for non-Gaussian correlated cases.",
"Soize suggested a modification of generalized polynomial chaos [13], but the resulting basis functions are non-smooth and unstable [14]."
],
[
"Our basis functions",
"We adopt the Gram-Schmidt approach to calculate the basis function recursively.",
"Gram-Schmidt was originally used for vector orthogonalization in the Euclidean space, and the key difference here is to replace the vector inner product with a functional expectation.",
"Specifically, we first reorder the monomials ${\\xi }^{\\alpha }=\\xi _1^{\\alpha _1}\\ldots \\xi _d^{\\alpha _d}$ in the graded lexicographic order, and denote them as $\\lbrace p_j({\\xi })\\rbrace _{j=1}^N$ .",
"Then we set $\\Psi _1({\\xi }) = 1$ and calculate a set of orthonormal polynomials $\\lbrace \\Psi _{j}({\\xi })\\rbrace _{j=1}^N$ in the correlated parameter space recursively: $ &\\hat{\\Psi }_j({\\xi }) = p_j({\\xi })-\\sum _{i=1}^{j-1} \\mathbb {E}[ p_j({\\xi })\\Psi _i({\\xi })] \\Psi _i({\\xi }),\\\\&\\Psi _j({\\xi }) = \\frac{\\hat{\\Psi }_j({\\xi })}{\\sqrt{\\mathbb {E}[\\hat{\\Psi }^2_j({\\xi })]}},\\ j=2,\\ldots ,N.$ The most time-consuming step is to compute the expectations.",
"We adopt the functional tensor train approach developed in [14] to speed up this computation."
],
[
"An Optimization-Based Quadrature Rule",
"Having chosen the basis functions, we still need to determine the number and values of the quadrature nodes and weights in order to calculate the coefficient $c_{\\alpha }$ by (REF ).",
"Our proposed method is summarized in Algorithm REF , and we explain the key ideas as follows."
],
[
"An Optimization-Based Quadrature Rule",
"Motivated by [11], [12], we set up an optimization model to decide a proper quadrature rule.",
"Our method differs from [11] because the latter optimizes quadrature weights only.",
"Our method differs from [12] in the following sense: (1) we focus on non-Gaussian correlated uncertainty analysis; (2) we handle the nonnegative constraint of $\\mathbf {w}$ and the nonlinear function of ${\\xi }$ separately via a novel block coordinate descent framework.",
"Suppose that $y({\\xi })$ can be well approximated by an order-$p$ polynomial function, then the product term $ y({\\xi }) {\\Psi }_{\\alpha } ({\\xi })$ can be well approximated by order-$2p$ polynomials.",
"As a result, $\\mathbb {E} \\left[ y({\\xi }) {\\Psi }_{\\alpha } ({\\xi })\\right]$ can be accurately computed if we have a quadrature rule that can accurately estimate the integration of every basis function bounded by order $2p$ : $\\mathbb {E}[\\Psi _{j}({\\xi })]=&\\delta _{1j} \\approx \\sum _{k=1}^{M}\\Psi _j({\\xi }_k) w_k,\\ \\forall \\, j=1,\\ldots ,N_{2p},$ with $N_{2p}=\\binom{d+2p}{d}$ , $\\delta _{1j}=1$ if $j=1$ and $\\delta _{1j}=0$ otherwise.",
"This formulation can be rewritten as a nonlinear least-square $\\min _{\\bar{{\\xi }},\\mathbf {w}}\\quad \\Vert \\mathbf {\\Phi }(\\bar{{\\xi }})\\mathbf {w}-\\mathbf {e_1}\\Vert ^2,$ where $(\\mathbf {\\Phi }(\\bar{{\\xi }}))_{jk}=\\Psi _j({\\xi }_k)$ , $\\bar{{\\xi }}=[{\\xi }_1;\\ldots ;{\\xi }_M]\\in \\mathbb {R}^{Md}$ , $\\mathbf {w}=[w_1,\\ldots ,w_M]^T\\in \\mathbb {R}^{M}$ and $\\mathbf {e}_1=[1,0,\\ldots ,0]^T\\in \\mathbb {R}^{N_{2p}}$ ."
],
[
"A Block Coordinate Solver for (",
"The number of unknowns in (REF ) is $M(d+1)$ , which becomes large as $d$ increases.",
"To improve the scalability, we solve (REF ) by a block coordinate descent method.",
"The idea is to update the variables block-by-block: at the $t$ -th iteration, given $\\bar{{\\xi }}_t$ and $\\mathbf {w}_t$ , we firstly fix $\\bar{{\\xi }}^t$ and solve the $\\mathbf {w}$ -subproblem to update $\\mathbf {w}^{t+1}$ , then fix $\\mathbf {w}^{t+1}$ and solve the ${\\xi }$ -subproblem to get $\\bar{{\\xi }}^{t+1}$ .",
"$\\mathbf {w}$ -subproblem.",
"Suppose $\\bar{{\\xi }}^t=[{\\xi }_1^t;\\ldots ;{\\xi }_M^t]$ is fixed, then (REF ) reduces to a convex linear least-square problem $\\mathbf {w}^{t+1}=\\arg \\min _{\\mathbf {w}\\ge 0}\\quad \\Vert \\mathbf {\\Phi }(\\bar{{\\xi }}^t) \\mathbf {w}-\\mathbf {e}_1\\Vert ^2.$ Here, we require the quadrature weights to be nonnegative.",
"${\\xi }$ -subproblem.",
"When $\\mathbf {w}^{t+1}$ is fixed, we apply the Gaussian Newton method to the ${\\xi }$ -subproblem ${\\xi }_k^{t+1}={\\xi }_k^t+\\mathbf {d}_k^t, \\text{ where } \\lbrace \\mathbf {d}_k^t\\rbrace =\\arg \\min _{\\lbrace \\mathbf {d}_k\\rbrace } \\ \\Vert \\sum _{k=1}^M\\mathbf {G}_k^t\\mathbf {d}_k + \\mathbf {r}^t\\Vert ^2.$ Here, $\\mathbf {r}^t = \\mathbf {\\Phi }(\\bar{{\\xi }}^t)\\mathbf {w}^{t+1}-\\mathbf {e}_1 \\in \\mathbb {R}^{N_{2p}}$ denotes the residual, $\\mathbf {G}_k^t\\in \\mathbb {R}^{N_{2p}\\times d}$ is the Jacobian matrix of $\\mathbf {r}^t$ with respect to ${\\xi }_k^t$ .",
"[t] Extensions of stochastic collocation method to non-Gaussian correlated variations InputInput OutputOutput Step 1 Initialize the quadrature nodes and weight according to Section REF .",
"Step 2 Increase phase.",
"Update the quadrature nodes & weights by solving (REF ).",
"If the optimization fails to converge, increase the node number and go back to Step 1.",
"Step 3 Decrease phase.",
"Decrease the node number, and update them by solving (REF ) again.",
"Repeat Step 3 until no points can be deleted.",
"Return the optimal nodes and weights.",
"Step 4 Call a simulator to compute $\\lbrace y({\\xi }_k)\\rbrace _{k=1}^M$ .",
"Then compute the coefficients $\\lbrace c_{\\alpha }\\rbrace $ for all ${|\\alpha |\\le p}$ via (REF ).",
"The coefficients $\\lbrace c_{\\alpha }\\rbrace $ in (REF )."
],
[
"Implementation Details",
"A good initial guess for the quadrature nodes is important to ensure the success of our nonlinear least-square solver.",
"Therefore, we first generate some candidate nodes via a Monte Carlo method, and then cluster them via a complete-linkage clustering method [15].",
"In general, we do not know the optimal number of quadrature nodes a priori.",
"Our algorithm consists of two phases: firstly we increase the number of quadrature nodes until the condition (REF ) holds with high accuracy.",
"Then, we decrease the number of nodes by deleting the node with the least weight and refine them by solving (REF ), until the number of nodes is too small to achieve a required integration accuracy."
],
[
"Three-Stage CMOS Ring Oscillator",
"We first use our method to simulate the 3-stage CMOS ring oscillator in Fig.",
"REF .",
"This oscillator has a Gaussian mixture model describing the correlated non-Gaussian threshold voltages of 6 transistors.",
"We aim to obtain a 2nd-order expansion for its frequency by calling a periodic steady-state simulator repeatedly.",
"The obtained results in Fig.",
"REF shows the obtained coefficients for all basis functions.",
"The obtained density function using only 34 quadrature samples is almost identical with that from $10^5$ Monte Carlo simulations.",
"Figure: Schematic of a 3-stage CMOS ring oscillator.Figure: Numerical results of the CMOS ring oscillator.",
"(a) obtained coefficients/weights of our basis functions; (b) probability density functions of the oscillator frequency obtained by our proposed method and Monte Carlo (MC)."
],
[
"Coupled Ring Resonator Optical Filter",
"We further consider the filter designed with bus-coupled micro-ring resonatorsThe details of this benchmark can be found at https://kb.lumerical.com/en/pic_circuits_coupled_ring_resonator_filters.html shown in Fig.",
"REF (a).",
"Coupled ring resonator are widely used for wavelength filtering and modulation in photonic integrated circuits.",
"Here we consider a filter with 3 stages of ring resonators, and we use a Gaussian mixture model to describe the correlated non-Gaussian uncertainties in waveguide lengths $L_{12}$ , $L_{21}$ , $L_{23}$ and $L_{32}$ .",
"Figure: (a) Schematic of a 3-stage parallel-coupled ring resonator optical filter.",
"(b) The black line shows the nominal transmission function, and the thin grey lines show the effect of fabrication uncertainties on the waveguide lengths.A 2nd-order expansion is built to approximate the frequency-dependent power transmission function: $y(f,{\\xi })=\\sum _{|\\alpha |=0}^p c_{\\alpha }(f) \\Psi _{\\alpha }({\\xi })$ .",
"The computed mean value and standard derivation are shown in Fig.",
"REF .",
"Our method only uses 16 quadrature samples for simulation, and it is able to achieve the similar level of accuracy compared with Monte Carlo method using $10^5$ simulation samples.",
"Figure: Simulation results of the optical filter.",
"(a) obtained mean value of the power transmission rate; (b) standard deviation of the transmission rate."
],
[
"Conclusion",
"This paper has proposed a stochastic collocation approach to solve the challenging non-Gaussian correlated uncertainty quantification problems.",
"We have proposed an optimization method to calculate the quadrature rule used in the projection step.",
"Our method has achieved 3000$\\times $ speedup than Monte Carlo on a CMOS ring oscillator and an optical resonator."
]
] | 1808.08381 |
[
[
"Churn Intent Detection in Multilingual Chatbot Conversations and Social\n Media"
],
[
"Abstract We propose a new method to detect when users express the intent to leave a service, also known as churn.",
"While previous work focuses solely on social media, we show that this intent can be detected in chatbot conversations.",
"As companies increasingly rely on chatbots they need an overview of potentially churny users.",
"To this end, we crowdsource and publish a dataset of churn intent expressions in chatbot interactions in German and English.",
"We show that classifiers trained on social media data can detect the same intent in the context of chatbots.",
"We introduce a classification architecture that outperforms existing work on churn intent detection in social media.",
"Moreover, we show that, using bilingual word embeddings, a system trained on combined English and German data outperforms monolingual approaches.",
"As the only existing dataset is in English, we crowdsource and publish a novel dataset of German tweets.",
"We thus underline the universal aspect of the problem, as examples of churn intent in English help us identify churn in German tweets and chatbot conversations."
],
[
"Introduction",
"Identifying customers who intend to terminate their relation with a company is commonly known as churn detection.",
"This is very important for companies if we consider that attracting new customers is a time and cost-intensive task.",
"Therefore, it is often preferable for companies to focus on the existing customers in order to prevent losing them instead of trying to acquire new ones.",
"Traditionally, churn detection is based on tracking the user behavior and correlating it with the decision to churn.",
"The analysis of the user behavior typically includes metadata such as the subscription information, network usage or customer transactions [18], [6].",
"The behavior-based techniques thus require a significant amount of data that are not easily available.",
"In addition, there is a cold start problem with novel systems which may not have access to the background required for this type of analysis.",
"Figure: Overview of Overall Pipeline.The current trend for detecting churn intent is to focus on textual user statements.",
"This intent is sufficient evidence for the likely following churn decision of a user.",
"Moreover, it is an actionable insight, as it allows companies to allocate resources to prevent the likely customer churn decision.",
"Textual churn detection is only based on the current interaction between the user and the service provider.",
"As a result, no a priori knowledge of the customer background is needed, thus bypassing the cold start problem.",
"A text-based analysis of the intent to churn is even more relevant today in the context of the chatbot explosion [10], [7], [21], [1].",
"Chatbots are becoming one of the main means of textual communication with the evolution of automation processes.",
"This chatbot explosion aims at converting the usual human-to-human interaction into a human-to-machine one, which however comes at a high cost.",
"Concretely, companies have no longer a full grasp on their users' level of discontent, since the customer contact is handled by chatbots.",
"Adding a churn detection functionality to bots allows companies to spot cases where the discontent reaches a high level, and the user expresses an intent to churn.",
"This, in turn, becomes an actionable insight, as the bot can decide if human intervention is needed and route the conversation to a human agent.",
"Churn detection is hard, as it requires discriminating between the intent to switch to and switch from a service.",
"For instance in the tweet \"@MARKE das klingt gut zu den genannten Konditionen würde ich dann doch gern wechseln :)\" which translates as \"@BRAND the conditions sound good to me.",
"I would like to switch :)\", the intention is not churny for the brand this tweet is addressed to.",
"However in \"@MARKE Internet langsamer als gedrosseltes.",
"bin deshalb zu eurer konkurrenz gewechselt\" which translates as \"@BRAND Internet slower than throttled.",
"So I switched to your competitor\" the intention is churny.",
"In this paper, we claim that we can transfer knowledge about churn intent detection from social media to chatbot conversations and churn intent detection can work in a multilingual way for both social media and chatbot conversations.",
"We visualize the approach we adopt in Fig.",
"REF .",
"We start by creating churn intent detectors, that are based on a neural architecture, that exploits convolutional, recurrent and attention layers.",
"We compare the performance of our model with the existing state-of-the-art for churn detection in English microblogs [15] and validate that our classifier achieves top-notch performance in this task.",
"We also contribute by providing datasets in English and German for churn detection to the research community.",
"First, we collect and annotate a dataset with German tweets that refer to any German telecommunication brand (e.g., Vodafone and O2).",
"This dataset complements the already existing microblog based dataset released from [8].",
"Secondly, we create our own chatbot platform which helps us in building and annotating the first datasets in German and English for chatbot conversations.",
"We later use these datasets as evaluation sets in order to prove our claim that we can successfully transfer knowledge from data extracted from social media to chatbot conversations.",
"In addition, we contribute by showing that expressions of churn intent are language-independent.",
"The intuition is that if we train a classifier to detect churny intents in a language, this knowledge can help identify churn intents in a second language.",
"To make the computation lighter, we do not use translation but rely on multilingual embeddings.",
"Multilingual embeddings extend monolingual ones with the objective of mapping similar words from different languages closely together in a unified space.",
"We perform experiments and show that models trained on data coming from both languages are more accurate than language-specific ones.",
"This is true for both the social media and the chatbot corpora.",
"As a result, we demonstrate not only that churn intent models generalize across media, but also across languages.",
"Our findings have a major implication.",
"Concretely, we prove that knowing how a customer, writing in English, expresses discontent with a telecommunications company in the US helps the system detect the churn intent in simulated chatbot conversations written in German about a German operator.",
"We summarize our contributions as follows: [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] we present a neural-based model that achieve state-of-the-art model results for churn detection (Section REF ).",
"we create a first multilingual approach for churn intent detection using multilingual embeddings (Section REF ).",
"we show that churn detection patterns can be learned from social media content and successfully applied to chatbot conversations (Section REF ).",
"we publish a novel dataset for churn detection in German tweets (Section REF ).",
"and finally, we create the first German and English datasets for churn intent detection in chatbot conversations (Section REF ).",
"The paper continues with an outline of related work in Section .",
"Section describes our text classifier and our approach for multilingual word embeddings.",
"The dataset construction is detailed in Section .",
"We describe our experiments in Section and finally conclude in Section .",
"Related Work This work is an intersection of churn detection in social media, multilingual churn detection and churn detection in chatbots Therefore we present the related work for each domain separately.",
"As there are no direct applications of multilingual embeddings and knowledge transfer from social media to churn detection in chatbot conversations, we include other applications that inspired our work.",
"Churn in Social Media The first approach of performing churn detection relies on user metadata.",
"Metadata are information about the customer activity for a particular service.",
"[18] propose a method based on customer transactions over time to detect churn whereas [6] focus on user's session duration.",
"Such techniques have proven to be efficient but rely on the fact that we possess a large amount of data regarding the user behavior, a fact which is rarely true.",
"The second approach focuses on textual interactions such as in social media.",
"Here, no a priori knowledge of the customer actions is required since churn detection is solely based on textual interactions between the user and the company.",
"[8] distributed a labeled English dataset of tweets (hereafter denoted as $\\textnormal {EN}_T$ ) about telecommunication brands and provided a baseline for churn detection in social media.",
"[9], [15] worked on $\\textnormal {EN}_T$ .",
"[9] focused on the extraction of additional features from tweets.",
"They gathered information about the context of the tweet (e.g.",
"number of replies).",
"This contextual information was passed through a pre-trained RNN to generate new features and improve classification performance.",
"Unfortunately, this technique depends on the availability of additional data which is not always present and therefore does not scale well.",
"On the contrary, [15] focused only on tweets and achieved the best-known performance on textual churn detection.",
"They did so by performing text classification using a Convolutional Neural Networks (CNN) [13] enriched with rule-based features.",
"Even though this approach has proven to improve the score significantly, it directly limits the model to English applications.",
"Transfer from Social Media to Chatbots Previous work on churn intent detection is centered on social media while chatbots are slowly replacing human-to-human interaction and becoming the main way of communication between customers and brands.",
"Due to the novel aspect of the topic, there are no publicly available datasets related to churn detection in chatbot conversations, and therefore no previous work on that field exists.",
"[14] propose multiple sentiment-based reply models for chatbot conversation.",
"They trained their models on a Twitter sentiment analysis corpus [16] which is composed of 15M data points with labeled sentiment.",
"However, to the best of our knowledge, there is no work that uses churn detection in the context of chatbot conversations.",
"Multilingual Aspect Multilingual word embeddings have been applied in the context of tasks like Cross-Lingual Document Classification (CLDC) as in [12].",
"The authors evaluate the quality of multilingual embeddings they induced using parallel data to classify unlabeled documents in a target language using only labeled documents in a source language.",
"However, a comparison between the performance using monolingual versus multilingual data is missing.",
"We try to address this problem in our research.",
"Other downstream tasks which benefited from multilingual embeddings include Cross Language Sentiment Classification (CLSC) as in [11].",
"They train the bilingual embeddings jointly using the task data and its translation and show that the multilingual approach outperforms the monolingual experiments.",
"This gain in performance encouraged us to try this approach to churn detection.",
"To the best of our knowledge, there is no prior work leveraging multilingual embeddings for this task.",
"Methodology Social media includes a wide range of platforms, however, we choose to use Twitter.",
"We do so for the following reasons.",
"First, we would like to take advantage of the free and widely used Twitter API.",
"Secondly, we would like to compile and annotate a German dataset for churn detection in order to complement the already existing dataset of [8].",
"Twitter helps to this end with its flexible policy for data distribution which allows us to release our novel dataset effortlessly.",
"Churn intent detection can be seen as a classification task where the input is a text, and the output is one of two classes (churn and non churn).",
"Here, we adapt a new architecture, tailored to the nature of tweets (e.g., short text length) and also low data availability.",
"In addition, the churn intent detection problem is not tied to a single language or domain of application.",
"We analyze the synergies between churn intents in multiple languages and how multilingual embeddings can help us solve the problem at hand.",
"For chatbot applications, the intuition is that a model trained on the social media domain might be helpful in finding churn expressions in the context of chatbots.",
"Text Classification Architecture Figure: Architecture of CNN-GRU with Attention.Our churn detection architecture is a text classifier based on cascaded collaborative layers where different feature extractors and aggregators complement each other.",
"More precisely, we employ a combination of a CNN and a bidirectional Gated Recurrent Unit (BiGRU) to make use of both spatial and temporal dependencies in the data [19], [3].",
"On top of that, an attention mechanism [2] is employed in order to recognize which BiGRU outputs have higher weights of importance.",
"While CNN acts as n-grams feature extractors, GRU cells are used to take word order into consideration.",
"This is crucially important as the word order can play an important role to understand the context and detect something as churny or non-churny.",
"We use GRU since it is a lightweight and more computationally efficient version of Long Short-Term Memory (LSTM) networks that preserves a comparable performance without using a memory unit [4].",
"BiGRU is used instead of unidirectional GRU to preserve information from the past and future.",
"The overall view of the architecture is depicted in Fig.",
"REF .",
"Each sentence can be represented as an $n \\times m$ input matrix, where $n$ is the maximum number of words over all sentences (padding is performed to the length of the longest tweet) and $m$ is the number of features (i.e., dimensionality of word embeddings).",
"We apply dropout directly to the embedding matrix to reduce overfitting.",
"For each sentence matrix, we apply $f$ convolution filters of kernel size $k$ which result in $f$ vectors of size $n-k+1$ .",
"We then feed the extracted features to a BiGRU which traverses the sentence in both the forward and backward directions.",
"In the end, we apply a softmax activation function to get the final prediction.",
"Multilingual Churn Intent Detection We introduce the task of cross-lingual churn detection by aiming at detecting churn in any language.",
"More specifically, we train and test one single robust model by concatenating data coming from English and German using multilingual embeddings.",
"We rely on the assumption that using multilingual embeddings | as a mechanism to represent words coming from different languages into the same low dimensional vector space | can capture the semantic and syntactic similarities between the languages which help with transfer learning between them.",
"In a sense, languages which are resource rich in churn detection can help those which lack the features needed to build a strong classifier by their own.",
"Our aim with this multilingual approach is to bridge the gap between English and German and improve the performance of German for which data is not as strongly labeled.",
"We build our multilingual embeddings which map words from different languages into one joint vector space by learning translation of embeddings in the source space into the target space.",
"We set German as the source space and English as the target space.",
"We then learn the transformation matrix that aligns German to English.",
"In other words, this approach fine-tunes German embeddings by applying a linear transformation that maps them into the English space.",
"Due to the presence of compound words and high availability of training data, the embedding space for English allows for a richer representation of the semantics of individual words.",
"The availability of multiple bilingual dictionaries, where English is one of the languages, motivates us to choose English as a target language.",
"For that purpose, we adopt an offline approach to guarantee a fair comparison between monolingual and multilingual churn detection.",
"We do so to show clearly the added value of the multilingual approach where both monolingual and multilingual embeddings are initially trained using the same monolingual constraints.",
"According to [20], this transformation matrix can be learned analytically using the product of the left and right singular vectors obtained from SVD of the product of the source and target dictionary vectors $X_{D}$ and $Y_{D}$ .",
"Concretely, $W_{DE\\rightarrow {EN}}=U \\cdot V$ such that $X_{D} \\cdot Y_{D}=U \\cdot \\Sigma \\cdot V$ which was proven to have the same quality as those obtained via iterative optimization.",
"The product of U and V is the closed form solution that optimizes the transformation from the source to the target spaces [20].",
"Transfer from Social Media to Chatbots We make the assumption that tweets and chatbot conversations are similar to a certain extent.",
"Even if the language is mostly different, we believe that the parts that are relevant to churn detection stay the same.",
"In other words, if a model trained on tweets gives promising results on chatbot conversations, then it confirms that there is an underlying churn intent pattern that can be generalized across mediums.",
"Still, differences exist between the way costumers express themselves through social media and chatbot conversations.",
"Social media, and especially Twitter, tend to carry specific structures that might prevent our model from detecting churn in chatbot conversations.",
"To this end, we work towards removing domain specific features of the text in order to be able to transfer knowledge from Twitter to chatbots successfully.",
"Therefore, we first remove patterns such as RT, # and @ that are Twitter-specific.",
"Moreover, users usually start their message with the mention of the brand such as \"@X I want to switch to @Y!\"",
"where X is the targeted brand and Y any potential competitor.",
"However, this is rarely true for chatbot conversations.",
"We can generalize these examples by removing the mention of the source brand to obtain \"I want to switch to @Y!\"",
"where the targeted brand is implicitly known and therefore is more likely to represent a typical chatbot entry.",
"Churn Intent Datasets In this work, we use pairs of datasets from two different languages (English and German) with the certainty that churn detection is a universal problem and therefore does not depend on the language.",
"Each pair is composed of a Twitter and a chatbot conversations dataset denoted as $\\textnormal {Lang}_T$ and $\\textnormal {Lang}_C$ respectively.",
"$\\textnormal {Lang}$ is a 2-letter abbreviation of the source language.",
"As a result, we discuss the creation of 4 different datasets, namely $\\textnormal {EN}_T$ , $\\textnormal {EN}_C$ , $\\textnormal {DE}_T$ and $\\textnormal {DE}_C$ The created datasets are publicly available at https://github.com/swisscom/churn-intent-DE.",
"English Twitter Dataset ($\\textnormal {EN}_T$ ) The dataset is introduced by [8] and is composed of English tweets that show mentions of Verizon, AT&T, and T-Mobile (telecommunication brands).",
"Each tweet is associated with a source brand (name of the company that is targeted by the tweet) and a label (1 or 0 whether the content is churny or not).",
"Table REF tabulates the exact distribution of the data as a function of the source brand where ${\\textnormal {\\bf churn}}$ is the number of churny tweets associated to the brand and ${\\textnormal {\\bf non churn}}$ the number of non-churny ones.",
"Overall, the dataset contains 4339We only keep those with annotation confidence above 0.7 as in [8].",
"labeled tweets and is highly imbalanced regarding the distribution of churny/non-churny tweets.",
"Table: Distribution of English tweets along the different brands.",
"German Twitter Dataset ($\\textnormal {DE}_T$ ) Since there is no existing dataset for churn detection except for English, we create a novel German dataset.",
"As a first step, we crawl all mentions on Twitter of multiple telecommunication brands that are active in German-speaking countries for a period of six months.",
"The result is a large Twitter dataset, $\\textnormal {DE}_{T_{FULL}}$ , containing more than 160000 tweets.",
"However, labeling such a large corpus is extremely time intensive and would result in a waste of resources since the density of churny tweets is extremely low.",
"A solution to reduce the size of $\\textnormal {DE}_{T_{FULL}}$ is to apply filters composed of predefined keywords to isolate potential churny tweets and generate a sub-dataset of candidates, $\\textnormal {DE}_{T_{FILTER}}$ , as depicted in Fig.",
"REF .",
"Those keywords are manually selected and are assumed to be linked with or carry churny content.",
"A non-exhaustive list of used keywords is displayed in Table REF .",
"Figure: Creation of DE T \\textnormal {DE}_T and transition to chatbots.Table: Non-exhaustive list of word filters used to detect potential churny tweets in German.The resulting subset, $\\textnormal {DE}_{T_{FILTER}}$ , is given to annotation through a platform specifically created for this purpose.",
"All tweets are annotated by at least two annotators.",
"We keep in our dataset only the entries where both annotators agree on the label.",
"We train the first version of our model with the newly labeled subset and then apply it to our initial dataset $\\textnormal {DE}_{T_{FULL}}$ .",
"By selecting only predictions with high confidence, we can generate an additional subset, $\\textnormal {DE}_{T_{BOOT}}$ , of potential churny tweets.",
"This new subset has the advantage of not being biased by the predefined filter keywords as opposed to $\\textnormal {DE}_{T_{FILTER}}$ .",
"Therefore, we can reduce the overall bias of our dataset by labeling $\\textnormal {DE}_{T_{BOOT}}$ and concatenating it to $\\textnormal {DE}_{T_{FILTER}}$ .",
"The final result is German Twitter dataset as $\\textnormal {DE}_{T} = \\textnormal {DE}_{T_{FILTER}} + \\textnormal {DE}_{T_{BOOT}}$ .",
"The complete distribution of the labels of $\\textnormal {DE}_T$ is displayed in Table REF for comparison purposes with $\\textnormal {EN}_T$ .",
"Here, three main companies emerged from our dataset, namely O2, Vodafone and Telekom (all other brands are grouped in the table as Others).",
"It is interesting to note that the size and distribution of the labels of the German dataset is comparable to the English one which allows fair performance comparison across languages.",
"Table: Distribution of German tweets along the different brands.",
"Chatbot Conversations ($\\textnormal {EN}_C$ + $\\textnormal {DE}_C$ ) Our ultimate goal is to detect churn intent in chatbot conversations.",
"However, no English nor German labeled chatbot conversations are available for this purpose.",
"To overcome this problem, we create our own chatbot platform to gather data and build our German ($\\textnormal {DE}_C$ ) and English ($\\textnormal {EN}_C$ ) chatbot conversations.",
"Our platform consists of a basic interface where the user can enter text that is processed by the chatbot as depicted in Fig.",
"REF .",
"Figure: Annotation process using our platform.We want the user to enter customer service related examples and their ground truth (churn or non-churn) to create our dataset.",
"However, creating and labeling data is a tedious task for the user and might lower the quality of our text-label pairs.",
"Therefore, we choose to present the chatbot interface as a game to make it more user-friendly.",
"Firstly, the user is asked to enter a sentence that is either churny or non churny.",
"Then, the chatbot predicts the output using a model trained on social media and informs the user about the prediction.",
"Finally, the user can approve or disapprove the prediction of the chatbot using buttons.",
"In both cases, we get the ground truth of the text and are able of expanding our database and even giving feedback to the user accordingly.",
"A second annotator is then responsible for double-checking the labeled data coming from the chatbot.",
"We keep only the data points where the two annotators agree.",
"Note that we append the results to the databases ($\\textnormal {EN}_C$ + $\\textnormal {DE}_C$ ) as a function of the detected language of the input text.",
"We end up with two novel datasets for churn detection in chatbot conversations.",
"Table REF presents the distribution of the labels in both languages.",
"The two columns indicate the number of churny and non-churny examples in each dataset respectively.",
"Table: Distribution of labels in chatbot conversations for both languages (EN/DE).",
"Evaluation For textual churn detection, we design and report on the performance of three experiments: [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Training on $\\textnormal {EN}_{T}$ and testing on $\\textnormal {EN}_{T}$ using English monolingual embeddings.",
"Training on $\\textnormal {DE}_{T}$ and testing on $\\textnormal {DE}_{T}$ using German monolingual embeddings.",
"Training on $\\textnormal {(EN+DE)}_{T}$ and testing on $\\textnormal {EN}_{T}$ and $\\textnormal {DE}_{T}$ using multilingual embeddings.",
"For all experiments, a consistent model with the same hyper-parameters is used to ensure a fair comparison.",
"We employ 256 filters with a kernel size of 2 for the convolutional layer.",
"In addition, we set the number of GRU units to 128 and apply a dropout with a rate of 0.3.",
"Finally, we use the Adam optimizer with its default parameters.",
"To allow a fair comparison, 10-fold cross validation is used as in [8].",
"This ensures that the results are less affected by the train/test split and all models are trained until convergence for each fold.",
"In the end, the mean and standard deviation of macro precision, recall and F1-score are computed over the maximum of each fold.",
"We execute all experiments 20 times, test them under statistical dependence and reject with a threshold of $\\alpha =5\\%$ .",
"For chatbot conversations, we directly evaluate the best model trained on datasets from social media on chatbot conversation data.",
"We report the performance for the following three experiments in Section REF : [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Best model trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ using English embeddings.",
"Best model trained on $\\textnormal {DE}_{T}$ and tested on $\\textnormal {DE}_{C}$ using German embeddings.",
"Best model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ and $\\textnormal {DE}_{C}$ using multilingual embeddings.",
"Table: Performance comparison of our model on English against the current state-of-the-art .",
"EN and DE are scores for language dependent models using monolingual embeddings, whereas EN+DE is for system trained on both languages at the same time using multilingual word embeddings.",
"The indices TT stands for Twitter.Table: Results on chatbot conversations.",
"EN and DE are scores for language dependent models using monolingual embeddings.",
"EN+DE are for system trained on both languages at the same time using multilingual word embeddings.",
"We distinguish Twitter from chatbot dataset using respectively the indices TT and CC.Embeddings and Data Augmentation We use pre-trained 300-dimensional word embeddings for English and German from fastText [17].",
"The same distributional vectors used in monolingual experiments are employed in building multilingual embeddings.",
"We learn the alignment based on a train part of a ground truth bilingual dictionary consisting of 5000 German-English pairs [5].",
"We then apply dimensionality reduction on top of SVD by deleting the last few rows corresponding to a value threshold of 1 in the diagonal vector.",
"The threshold value is chosen to maximize the performance on the test part of the bilingual dictionary pairs used for learning the alignment from $DE$ to $EN$ .",
"We also replace all brands with either \"target\" or \"competitor\" to improve vocabulary coverage.",
"\"Target\" refers to the brand concerned by the churny content and \"competitor\" to all other brands mentioned in the text.",
"Finally, it is important to notice that if a tweet is churny for a specific brand, it is not churny for the other cited brands.",
"For example, \"@X I want switch to @Y!\"",
"is churny for brand X but not for Y.",
"We can, therefore, generate more examples where Y is replaced as \"target\".",
"We use this procedure for each fold to augment the training set.",
"Social Media Results Table REF contains the results for churn detection in social media.",
"The first row shows the results for training and testing on $\\textnormal {EN}_{T}$ data which allows us to compare our score to state-of-the-art results.",
"We outperform the previous performance from [15] and reach 85.88% using multilingual word embeddings.",
"Note that the standard deviation over the 10-fold cross validation is not provided by [15].",
"However, an increase of 2.03% of the mean still represents an important improvement over the state-of-the-art.",
"As a result, we prove that our novel architecture provides an efficient way to detect churn in social media.",
"We notice a significant improvement in the performance of Twitter data when both English and German tweets are aggregated and used for training with multilingual embeddings.",
"The advantage of our multilingual model is promising especially for German with an increase of $7.8\\%$ in F1-score.",
"English also benefits with a slight increase of $1.65\\%$ .",
"The better quality of the English word embeddings makes it easier for our model to identify the churn patterns, compared to German.",
"This explains the gap between the gain in performance for German compared to English, although we used two corpora comparable in size for both languages.",
"To gain more insights into why the multilingual approach improves the test performance in German, it is worth reconsidering the example introduced earlier: \"@MARKE das klingt gut zu den genannten Konditionen würde ich dann doch gern wechseln :)\".",
"This example is predicted as churny using German monolingual model, while it is not churny according to the multilingual model.",
"This can be explained by the fact that the German model could only rely on the presence of switch keyword, while the multilingual approach can learn more complex patterns that are present in both languages.",
"There is a similar example in English: \"I want to switch to @BRAND already\" that portrays more or less the same pattern.",
"Chatbot Results Table REF shows that for chatbot conversations we obtain results comparable to Twitter.",
"This proves that our model is able of capturing the structure of the churny tweets in both languages and generalize it to other applications (e.g., chatbot conversations).",
"Moreover, we observe that the performance of churn detection in English chatbot conversations also benefits from the multilingual approach.",
"Concretely, the model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ outperforms its monolingual counterpart trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ with an increase $2.34\\%$ in F1-score.",
"On the other hand, performance for German exhibits a marginal drop compared to its monolingual counterpart.",
"This can be due to the small number of conversation examples and their lack of variability which makes them more similar in structure to the training tweets.",
"Therefore, even a monolingual model would work well in this case.",
"A final observation is that the recall is usually higher than the precision on chatbot conversations.",
"This is noteworthy in our application since it is more important to reduce the number of false negative in churn prediction.",
"Indeed, it is better for companies to falsely detect churn intent (in case of false positives) than missing actual customers (in case of false negatives).",
"Conclusion Preventing customers from leaving a service is an essential topic for companies, as acquiring new customers is a time and cost-intensive procedure.",
"While previous work solely focuses on user behavior over time or social media, here, we propose a novel approach for churn intent detection in chatbot conversations.",
"In this paper, we work towards multilingual churn intent detection in chatbot conversation with knowledge transfer from Twitter datasets.",
"First, we release a novel dataset of German tweets for churn intent detection to complement the existing English one.",
"Moreover, we create and distribute a dataset for churn intent detection in chatbot conversations for both English and German.",
"We present a model based on a neural architecture that outruns the state-of-the-art performance on churn intent detection in social media.",
"Our experiments show that our model can generalize churn intent patterns learned from social media and successfully apply them to chatbot conversations, proving that we can transfer churn detection knowledge from Twitter to chatbots.",
"In addition, we prove that our model, trained using multilingual word embeddings, surpasses monolingual approaches.",
"This result highlights the universal facet of the problem, as examples of churn intent in English help us in identifying churn about German telecommunication brands in German tweets and chatbot conversations."
],
[
"Related Work",
"This work is an intersection of churn detection in social media, multilingual churn detection and churn detection in chatbots Therefore we present the related work for each domain separately.",
"As there are no direct applications of multilingual embeddings and knowledge transfer from social media to churn detection in chatbot conversations, we include other applications that inspired our work.",
"Churn in Social Media The first approach of performing churn detection relies on user metadata.",
"Metadata are information about the customer activity for a particular service.",
"[18] propose a method based on customer transactions over time to detect churn whereas [6] focus on user's session duration.",
"Such techniques have proven to be efficient but rely on the fact that we possess a large amount of data regarding the user behavior, a fact which is rarely true.",
"The second approach focuses on textual interactions such as in social media.",
"Here, no a priori knowledge of the customer actions is required since churn detection is solely based on textual interactions between the user and the company.",
"[8] distributed a labeled English dataset of tweets (hereafter denoted as $\\textnormal {EN}_T$ ) about telecommunication brands and provided a baseline for churn detection in social media.",
"[9], [15] worked on $\\textnormal {EN}_T$ .",
"[9] focused on the extraction of additional features from tweets.",
"They gathered information about the context of the tweet (e.g.",
"number of replies).",
"This contextual information was passed through a pre-trained RNN to generate new features and improve classification performance.",
"Unfortunately, this technique depends on the availability of additional data which is not always present and therefore does not scale well.",
"On the contrary, [15] focused only on tweets and achieved the best-known performance on textual churn detection.",
"They did so by performing text classification using a Convolutional Neural Networks (CNN) [13] enriched with rule-based features.",
"Even though this approach has proven to improve the score significantly, it directly limits the model to English applications.",
"Transfer from Social Media to Chatbots Previous work on churn intent detection is centered on social media while chatbots are slowly replacing human-to-human interaction and becoming the main way of communication between customers and brands.",
"Due to the novel aspect of the topic, there are no publicly available datasets related to churn detection in chatbot conversations, and therefore no previous work on that field exists.",
"[14] propose multiple sentiment-based reply models for chatbot conversation.",
"They trained their models on a Twitter sentiment analysis corpus [16] which is composed of 15M data points with labeled sentiment.",
"However, to the best of our knowledge, there is no work that uses churn detection in the context of chatbot conversations.",
"Multilingual Aspect Multilingual word embeddings have been applied in the context of tasks like Cross-Lingual Document Classification (CLDC) as in [12].",
"The authors evaluate the quality of multilingual embeddings they induced using parallel data to classify unlabeled documents in a target language using only labeled documents in a source language.",
"However, a comparison between the performance using monolingual versus multilingual data is missing.",
"We try to address this problem in our research.",
"Other downstream tasks which benefited from multilingual embeddings include Cross Language Sentiment Classification (CLSC) as in [11].",
"They train the bilingual embeddings jointly using the task data and its translation and show that the multilingual approach outperforms the monolingual experiments.",
"This gain in performance encouraged us to try this approach to churn detection.",
"To the best of our knowledge, there is no prior work leveraging multilingual embeddings for this task.",
"Methodology Social media includes a wide range of platforms, however, we choose to use Twitter.",
"We do so for the following reasons.",
"First, we would like to take advantage of the free and widely used Twitter API.",
"Secondly, we would like to compile and annotate a German dataset for churn detection in order to complement the already existing dataset of [8].",
"Twitter helps to this end with its flexible policy for data distribution which allows us to release our novel dataset effortlessly.",
"Churn intent detection can be seen as a classification task where the input is a text, and the output is one of two classes (churn and non churn).",
"Here, we adapt a new architecture, tailored to the nature of tweets (e.g., short text length) and also low data availability.",
"In addition, the churn intent detection problem is not tied to a single language or domain of application.",
"We analyze the synergies between churn intents in multiple languages and how multilingual embeddings can help us solve the problem at hand.",
"For chatbot applications, the intuition is that a model trained on the social media domain might be helpful in finding churn expressions in the context of chatbots.",
"Text Classification Architecture Figure: Architecture of CNN-GRU with Attention.Our churn detection architecture is a text classifier based on cascaded collaborative layers where different feature extractors and aggregators complement each other.",
"More precisely, we employ a combination of a CNN and a bidirectional Gated Recurrent Unit (BiGRU) to make use of both spatial and temporal dependencies in the data [19], [3].",
"On top of that, an attention mechanism [2] is employed in order to recognize which BiGRU outputs have higher weights of importance.",
"While CNN acts as n-grams feature extractors, GRU cells are used to take word order into consideration.",
"This is crucially important as the word order can play an important role to understand the context and detect something as churny or non-churny.",
"We use GRU since it is a lightweight and more computationally efficient version of Long Short-Term Memory (LSTM) networks that preserves a comparable performance without using a memory unit [4].",
"BiGRU is used instead of unidirectional GRU to preserve information from the past and future.",
"The overall view of the architecture is depicted in Fig.",
"REF .",
"Each sentence can be represented as an $n \\times m$ input matrix, where $n$ is the maximum number of words over all sentences (padding is performed to the length of the longest tweet) and $m$ is the number of features (i.e., dimensionality of word embeddings).",
"We apply dropout directly to the embedding matrix to reduce overfitting.",
"For each sentence matrix, we apply $f$ convolution filters of kernel size $k$ which result in $f$ vectors of size $n-k+1$ .",
"We then feed the extracted features to a BiGRU which traverses the sentence in both the forward and backward directions.",
"In the end, we apply a softmax activation function to get the final prediction.",
"Multilingual Churn Intent Detection We introduce the task of cross-lingual churn detection by aiming at detecting churn in any language.",
"More specifically, we train and test one single robust model by concatenating data coming from English and German using multilingual embeddings.",
"We rely on the assumption that using multilingual embeddings | as a mechanism to represent words coming from different languages into the same low dimensional vector space | can capture the semantic and syntactic similarities between the languages which help with transfer learning between them.",
"In a sense, languages which are resource rich in churn detection can help those which lack the features needed to build a strong classifier by their own.",
"Our aim with this multilingual approach is to bridge the gap between English and German and improve the performance of German for which data is not as strongly labeled.",
"We build our multilingual embeddings which map words from different languages into one joint vector space by learning translation of embeddings in the source space into the target space.",
"We set German as the source space and English as the target space.",
"We then learn the transformation matrix that aligns German to English.",
"In other words, this approach fine-tunes German embeddings by applying a linear transformation that maps them into the English space.",
"Due to the presence of compound words and high availability of training data, the embedding space for English allows for a richer representation of the semantics of individual words.",
"The availability of multiple bilingual dictionaries, where English is one of the languages, motivates us to choose English as a target language.",
"For that purpose, we adopt an offline approach to guarantee a fair comparison between monolingual and multilingual churn detection.",
"We do so to show clearly the added value of the multilingual approach where both monolingual and multilingual embeddings are initially trained using the same monolingual constraints.",
"According to [20], this transformation matrix can be learned analytically using the product of the left and right singular vectors obtained from SVD of the product of the source and target dictionary vectors $X_{D}$ and $Y_{D}$ .",
"Concretely, $W_{DE\\rightarrow {EN}}=U \\cdot V$ such that $X_{D} \\cdot Y_{D}=U \\cdot \\Sigma \\cdot V$ which was proven to have the same quality as those obtained via iterative optimization.",
"The product of U and V is the closed form solution that optimizes the transformation from the source to the target spaces [20].",
"Transfer from Social Media to Chatbots We make the assumption that tweets and chatbot conversations are similar to a certain extent.",
"Even if the language is mostly different, we believe that the parts that are relevant to churn detection stay the same.",
"In other words, if a model trained on tweets gives promising results on chatbot conversations, then it confirms that there is an underlying churn intent pattern that can be generalized across mediums.",
"Still, differences exist between the way costumers express themselves through social media and chatbot conversations.",
"Social media, and especially Twitter, tend to carry specific structures that might prevent our model from detecting churn in chatbot conversations.",
"To this end, we work towards removing domain specific features of the text in order to be able to transfer knowledge from Twitter to chatbots successfully.",
"Therefore, we first remove patterns such as RT, # and @ that are Twitter-specific.",
"Moreover, users usually start their message with the mention of the brand such as \"@X I want to switch to @Y!\"",
"where X is the targeted brand and Y any potential competitor.",
"However, this is rarely true for chatbot conversations.",
"We can generalize these examples by removing the mention of the source brand to obtain \"I want to switch to @Y!\"",
"where the targeted brand is implicitly known and therefore is more likely to represent a typical chatbot entry.",
"Churn Intent Datasets In this work, we use pairs of datasets from two different languages (English and German) with the certainty that churn detection is a universal problem and therefore does not depend on the language.",
"Each pair is composed of a Twitter and a chatbot conversations dataset denoted as $\\textnormal {Lang}_T$ and $\\textnormal {Lang}_C$ respectively.",
"$\\textnormal {Lang}$ is a 2-letter abbreviation of the source language.",
"As a result, we discuss the creation of 4 different datasets, namely $\\textnormal {EN}_T$ , $\\textnormal {EN}_C$ , $\\textnormal {DE}_T$ and $\\textnormal {DE}_C$ The created datasets are publicly available at https://github.com/swisscom/churn-intent-DE.",
"English Twitter Dataset ($\\textnormal {EN}_T$ ) The dataset is introduced by [8] and is composed of English tweets that show mentions of Verizon, AT&T, and T-Mobile (telecommunication brands).",
"Each tweet is associated with a source brand (name of the company that is targeted by the tweet) and a label (1 or 0 whether the content is churny or not).",
"Table REF tabulates the exact distribution of the data as a function of the source brand where ${\\textnormal {\\bf churn}}$ is the number of churny tweets associated to the brand and ${\\textnormal {\\bf non churn}}$ the number of non-churny ones.",
"Overall, the dataset contains 4339We only keep those with annotation confidence above 0.7 as in [8].",
"labeled tweets and is highly imbalanced regarding the distribution of churny/non-churny tweets.",
"Table: Distribution of English tweets along the different brands.",
"German Twitter Dataset ($\\textnormal {DE}_T$ ) Since there is no existing dataset for churn detection except for English, we create a novel German dataset.",
"As a first step, we crawl all mentions on Twitter of multiple telecommunication brands that are active in German-speaking countries for a period of six months.",
"The result is a large Twitter dataset, $\\textnormal {DE}_{T_{FULL}}$ , containing more than 160000 tweets.",
"However, labeling such a large corpus is extremely time intensive and would result in a waste of resources since the density of churny tweets is extremely low.",
"A solution to reduce the size of $\\textnormal {DE}_{T_{FULL}}$ is to apply filters composed of predefined keywords to isolate potential churny tweets and generate a sub-dataset of candidates, $\\textnormal {DE}_{T_{FILTER}}$ , as depicted in Fig.",
"REF .",
"Those keywords are manually selected and are assumed to be linked with or carry churny content.",
"A non-exhaustive list of used keywords is displayed in Table REF .",
"Figure: Creation of DE T \\textnormal {DE}_T and transition to chatbots.Table: Non-exhaustive list of word filters used to detect potential churny tweets in German.The resulting subset, $\\textnormal {DE}_{T_{FILTER}}$ , is given to annotation through a platform specifically created for this purpose.",
"All tweets are annotated by at least two annotators.",
"We keep in our dataset only the entries where both annotators agree on the label.",
"We train the first version of our model with the newly labeled subset and then apply it to our initial dataset $\\textnormal {DE}_{T_{FULL}}$ .",
"By selecting only predictions with high confidence, we can generate an additional subset, $\\textnormal {DE}_{T_{BOOT}}$ , of potential churny tweets.",
"This new subset has the advantage of not being biased by the predefined filter keywords as opposed to $\\textnormal {DE}_{T_{FILTER}}$ .",
"Therefore, we can reduce the overall bias of our dataset by labeling $\\textnormal {DE}_{T_{BOOT}}$ and concatenating it to $\\textnormal {DE}_{T_{FILTER}}$ .",
"The final result is German Twitter dataset as $\\textnormal {DE}_{T} = \\textnormal {DE}_{T_{FILTER}} + \\textnormal {DE}_{T_{BOOT}}$ .",
"The complete distribution of the labels of $\\textnormal {DE}_T$ is displayed in Table REF for comparison purposes with $\\textnormal {EN}_T$ .",
"Here, three main companies emerged from our dataset, namely O2, Vodafone and Telekom (all other brands are grouped in the table as Others).",
"It is interesting to note that the size and distribution of the labels of the German dataset is comparable to the English one which allows fair performance comparison across languages.",
"Table: Distribution of German tweets along the different brands.",
"Chatbot Conversations ($\\textnormal {EN}_C$ + $\\textnormal {DE}_C$ ) Our ultimate goal is to detect churn intent in chatbot conversations.",
"However, no English nor German labeled chatbot conversations are available for this purpose.",
"To overcome this problem, we create our own chatbot platform to gather data and build our German ($\\textnormal {DE}_C$ ) and English ($\\textnormal {EN}_C$ ) chatbot conversations.",
"Our platform consists of a basic interface where the user can enter text that is processed by the chatbot as depicted in Fig.",
"REF .",
"Figure: Annotation process using our platform.We want the user to enter customer service related examples and their ground truth (churn or non-churn) to create our dataset.",
"However, creating and labeling data is a tedious task for the user and might lower the quality of our text-label pairs.",
"Therefore, we choose to present the chatbot interface as a game to make it more user-friendly.",
"Firstly, the user is asked to enter a sentence that is either churny or non churny.",
"Then, the chatbot predicts the output using a model trained on social media and informs the user about the prediction.",
"Finally, the user can approve or disapprove the prediction of the chatbot using buttons.",
"In both cases, we get the ground truth of the text and are able of expanding our database and even giving feedback to the user accordingly.",
"A second annotator is then responsible for double-checking the labeled data coming from the chatbot.",
"We keep only the data points where the two annotators agree.",
"Note that we append the results to the databases ($\\textnormal {EN}_C$ + $\\textnormal {DE}_C$ ) as a function of the detected language of the input text.",
"We end up with two novel datasets for churn detection in chatbot conversations.",
"Table REF presents the distribution of the labels in both languages.",
"The two columns indicate the number of churny and non-churny examples in each dataset respectively.",
"Table: Distribution of labels in chatbot conversations for both languages (EN/DE).",
"Evaluation For textual churn detection, we design and report on the performance of three experiments: [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Training on $\\textnormal {EN}_{T}$ and testing on $\\textnormal {EN}_{T}$ using English monolingual embeddings.",
"Training on $\\textnormal {DE}_{T}$ and testing on $\\textnormal {DE}_{T}$ using German monolingual embeddings.",
"Training on $\\textnormal {(EN+DE)}_{T}$ and testing on $\\textnormal {EN}_{T}$ and $\\textnormal {DE}_{T}$ using multilingual embeddings.",
"For all experiments, a consistent model with the same hyper-parameters is used to ensure a fair comparison.",
"We employ 256 filters with a kernel size of 2 for the convolutional layer.",
"In addition, we set the number of GRU units to 128 and apply a dropout with a rate of 0.3.",
"Finally, we use the Adam optimizer with its default parameters.",
"To allow a fair comparison, 10-fold cross validation is used as in [8].",
"This ensures that the results are less affected by the train/test split and all models are trained until convergence for each fold.",
"In the end, the mean and standard deviation of macro precision, recall and F1-score are computed over the maximum of each fold.",
"We execute all experiments 20 times, test them under statistical dependence and reject with a threshold of $\\alpha =5\\%$ .",
"For chatbot conversations, we directly evaluate the best model trained on datasets from social media on chatbot conversation data.",
"We report the performance for the following three experiments in Section REF : [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Best model trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ using English embeddings.",
"Best model trained on $\\textnormal {DE}_{T}$ and tested on $\\textnormal {DE}_{C}$ using German embeddings.",
"Best model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ and $\\textnormal {DE}_{C}$ using multilingual embeddings.",
"Table: Performance comparison of our model on English against the current state-of-the-art .",
"EN and DE are scores for language dependent models using monolingual embeddings, whereas EN+DE is for system trained on both languages at the same time using multilingual word embeddings.",
"The indices TT stands for Twitter.Table: Results on chatbot conversations.",
"EN and DE are scores for language dependent models using monolingual embeddings.",
"EN+DE are for system trained on both languages at the same time using multilingual word embeddings.",
"We distinguish Twitter from chatbot dataset using respectively the indices TT and CC.Embeddings and Data Augmentation We use pre-trained 300-dimensional word embeddings for English and German from fastText [17].",
"The same distributional vectors used in monolingual experiments are employed in building multilingual embeddings.",
"We learn the alignment based on a train part of a ground truth bilingual dictionary consisting of 5000 German-English pairs [5].",
"We then apply dimensionality reduction on top of SVD by deleting the last few rows corresponding to a value threshold of 1 in the diagonal vector.",
"The threshold value is chosen to maximize the performance on the test part of the bilingual dictionary pairs used for learning the alignment from $DE$ to $EN$ .",
"We also replace all brands with either \"target\" or \"competitor\" to improve vocabulary coverage.",
"\"Target\" refers to the brand concerned by the churny content and \"competitor\" to all other brands mentioned in the text.",
"Finally, it is important to notice that if a tweet is churny for a specific brand, it is not churny for the other cited brands.",
"For example, \"@X I want switch to @Y!\"",
"is churny for brand X but not for Y.",
"We can, therefore, generate more examples where Y is replaced as \"target\".",
"We use this procedure for each fold to augment the training set.",
"Social Media Results Table REF contains the results for churn detection in social media.",
"The first row shows the results for training and testing on $\\textnormal {EN}_{T}$ data which allows us to compare our score to state-of-the-art results.",
"We outperform the previous performance from [15] and reach 85.88% using multilingual word embeddings.",
"Note that the standard deviation over the 10-fold cross validation is not provided by [15].",
"However, an increase of 2.03% of the mean still represents an important improvement over the state-of-the-art.",
"As a result, we prove that our novel architecture provides an efficient way to detect churn in social media.",
"We notice a significant improvement in the performance of Twitter data when both English and German tweets are aggregated and used for training with multilingual embeddings.",
"The advantage of our multilingual model is promising especially for German with an increase of $7.8\\%$ in F1-score.",
"English also benefits with a slight increase of $1.65\\%$ .",
"The better quality of the English word embeddings makes it easier for our model to identify the churn patterns, compared to German.",
"This explains the gap between the gain in performance for German compared to English, although we used two corpora comparable in size for both languages.",
"To gain more insights into why the multilingual approach improves the test performance in German, it is worth reconsidering the example introduced earlier: \"@MARKE das klingt gut zu den genannten Konditionen würde ich dann doch gern wechseln :)\".",
"This example is predicted as churny using German monolingual model, while it is not churny according to the multilingual model.",
"This can be explained by the fact that the German model could only rely on the presence of switch keyword, while the multilingual approach can learn more complex patterns that are present in both languages.",
"There is a similar example in English: \"I want to switch to @BRAND already\" that portrays more or less the same pattern.",
"Chatbot Results Table REF shows that for chatbot conversations we obtain results comparable to Twitter.",
"This proves that our model is able of capturing the structure of the churny tweets in both languages and generalize it to other applications (e.g., chatbot conversations).",
"Moreover, we observe that the performance of churn detection in English chatbot conversations also benefits from the multilingual approach.",
"Concretely, the model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ outperforms its monolingual counterpart trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ with an increase $2.34\\%$ in F1-score.",
"On the other hand, performance for German exhibits a marginal drop compared to its monolingual counterpart.",
"This can be due to the small number of conversation examples and their lack of variability which makes them more similar in structure to the training tweets.",
"Therefore, even a monolingual model would work well in this case.",
"A final observation is that the recall is usually higher than the precision on chatbot conversations.",
"This is noteworthy in our application since it is more important to reduce the number of false negative in churn prediction.",
"Indeed, it is better for companies to falsely detect churn intent (in case of false positives) than missing actual customers (in case of false negatives).",
"Conclusion Preventing customers from leaving a service is an essential topic for companies, as acquiring new customers is a time and cost-intensive procedure.",
"While previous work solely focuses on user behavior over time or social media, here, we propose a novel approach for churn intent detection in chatbot conversations.",
"In this paper, we work towards multilingual churn intent detection in chatbot conversation with knowledge transfer from Twitter datasets.",
"First, we release a novel dataset of German tweets for churn intent detection to complement the existing English one.",
"Moreover, we create and distribute a dataset for churn intent detection in chatbot conversations for both English and German.",
"We present a model based on a neural architecture that outruns the state-of-the-art performance on churn intent detection in social media.",
"Our experiments show that our model can generalize churn intent patterns learned from social media and successfully apply them to chatbot conversations, proving that we can transfer churn detection knowledge from Twitter to chatbots.",
"In addition, we prove that our model, trained using multilingual word embeddings, surpasses monolingual approaches.",
"This result highlights the universal facet of the problem, as examples of churn intent in English help us in identifying churn about German telecommunication brands in German tweets and chatbot conversations."
],
[
"Methodology",
"Social media includes a wide range of platforms, however, we choose to use Twitter.",
"We do so for the following reasons.",
"First, we would like to take advantage of the free and widely used Twitter API.",
"Secondly, we would like to compile and annotate a German dataset for churn detection in order to complement the already existing dataset of [8].",
"Twitter helps to this end with its flexible policy for data distribution which allows us to release our novel dataset effortlessly.",
"Churn intent detection can be seen as a classification task where the input is a text, and the output is one of two classes (churn and non churn).",
"Here, we adapt a new architecture, tailored to the nature of tweets (e.g., short text length) and also low data availability.",
"In addition, the churn intent detection problem is not tied to a single language or domain of application.",
"We analyze the synergies between churn intents in multiple languages and how multilingual embeddings can help us solve the problem at hand.",
"For chatbot applications, the intuition is that a model trained on the social media domain might be helpful in finding churn expressions in the context of chatbots."
],
[
"Text Classification Architecture",
"Our churn detection architecture is a text classifier based on cascaded collaborative layers where different feature extractors and aggregators complement each other.",
"More precisely, we employ a combination of a CNN and a bidirectional Gated Recurrent Unit (BiGRU) to make use of both spatial and temporal dependencies in the data [19], [3].",
"On top of that, an attention mechanism [2] is employed in order to recognize which BiGRU outputs have higher weights of importance.",
"While CNN acts as n-grams feature extractors, GRU cells are used to take word order into consideration.",
"This is crucially important as the word order can play an important role to understand the context and detect something as churny or non-churny.",
"We use GRU since it is a lightweight and more computationally efficient version of Long Short-Term Memory (LSTM) networks that preserves a comparable performance without using a memory unit [4].",
"BiGRU is used instead of unidirectional GRU to preserve information from the past and future.",
"The overall view of the architecture is depicted in Fig.",
"REF .",
"Each sentence can be represented as an $n \\times m$ input matrix, where $n$ is the maximum number of words over all sentences (padding is performed to the length of the longest tweet) and $m$ is the number of features (i.e., dimensionality of word embeddings).",
"We apply dropout directly to the embedding matrix to reduce overfitting.",
"For each sentence matrix, we apply $f$ convolution filters of kernel size $k$ which result in $f$ vectors of size $n-k+1$ .",
"We then feed the extracted features to a BiGRU which traverses the sentence in both the forward and backward directions.",
"In the end, we apply a softmax activation function to get the final prediction."
],
[
"Multilingual Churn Intent Detection",
"We introduce the task of cross-lingual churn detection by aiming at detecting churn in any language.",
"More specifically, we train and test one single robust model by concatenating data coming from English and German using multilingual embeddings.",
"We rely on the assumption that using multilingual embeddings | as a mechanism to represent words coming from different languages into the same low dimensional vector space | can capture the semantic and syntactic similarities between the languages which help with transfer learning between them.",
"In a sense, languages which are resource rich in churn detection can help those which lack the features needed to build a strong classifier by their own.",
"Our aim with this multilingual approach is to bridge the gap between English and German and improve the performance of German for which data is not as strongly labeled.",
"We build our multilingual embeddings which map words from different languages into one joint vector space by learning translation of embeddings in the source space into the target space.",
"We set German as the source space and English as the target space.",
"We then learn the transformation matrix that aligns German to English.",
"In other words, this approach fine-tunes German embeddings by applying a linear transformation that maps them into the English space.",
"Due to the presence of compound words and high availability of training data, the embedding space for English allows for a richer representation of the semantics of individual words.",
"The availability of multiple bilingual dictionaries, where English is one of the languages, motivates us to choose English as a target language.",
"For that purpose, we adopt an offline approach to guarantee a fair comparison between monolingual and multilingual churn detection.",
"We do so to show clearly the added value of the multilingual approach where both monolingual and multilingual embeddings are initially trained using the same monolingual constraints.",
"According to [20], this transformation matrix can be learned analytically using the product of the left and right singular vectors obtained from SVD of the product of the source and target dictionary vectors $X_{D}$ and $Y_{D}$ .",
"Concretely, $W_{DE\\rightarrow {EN}}=U \\cdot V$ such that $X_{D} \\cdot Y_{D}=U \\cdot \\Sigma \\cdot V$ which was proven to have the same quality as those obtained via iterative optimization.",
"The product of U and V is the closed form solution that optimizes the transformation from the source to the target spaces [20]."
],
[
"Transfer from Social Media to Chatbots",
"We make the assumption that tweets and chatbot conversations are similar to a certain extent.",
"Even if the language is mostly different, we believe that the parts that are relevant to churn detection stay the same.",
"In other words, if a model trained on tweets gives promising results on chatbot conversations, then it confirms that there is an underlying churn intent pattern that can be generalized across mediums.",
"Still, differences exist between the way costumers express themselves through social media and chatbot conversations.",
"Social media, and especially Twitter, tend to carry specific structures that might prevent our model from detecting churn in chatbot conversations.",
"To this end, we work towards removing domain specific features of the text in order to be able to transfer knowledge from Twitter to chatbots successfully.",
"Therefore, we first remove patterns such as RT, # and @ that are Twitter-specific.",
"Moreover, users usually start their message with the mention of the brand such as \"@X I want to switch to @Y!\"",
"where X is the targeted brand and Y any potential competitor.",
"However, this is rarely true for chatbot conversations.",
"We can generalize these examples by removing the mention of the source brand to obtain \"I want to switch to @Y!\"",
"where the targeted brand is implicitly known and therefore is more likely to represent a typical chatbot entry."
],
[
"Churn Intent Datasets",
"In this work, we use pairs of datasets from two different languages (English and German) with the certainty that churn detection is a universal problem and therefore does not depend on the language.",
"Each pair is composed of a Twitter and a chatbot conversations dataset denoted as $\\textnormal {Lang}_T$ and $\\textnormal {Lang}_C$ respectively.",
"$\\textnormal {Lang}$ is a 2-letter abbreviation of the source language.",
"As a result, we discuss the creation of 4 different datasets, namely $\\textnormal {EN}_T$ , $\\textnormal {EN}_C$ , $\\textnormal {DE}_T$ and $\\textnormal {DE}_C$ The created datasets are publicly available at https://github.com/swisscom/churn-intent-DE."
],
[
"English Twitter Dataset ($\\textnormal {EN}_T$ )",
"The dataset is introduced by [8] and is composed of English tweets that show mentions of Verizon, AT&T, and T-Mobile (telecommunication brands).",
"Each tweet is associated with a source brand (name of the company that is targeted by the tweet) and a label (1 or 0 whether the content is churny or not).",
"Table REF tabulates the exact distribution of the data as a function of the source brand where ${\\textnormal {\\bf churn}}$ is the number of churny tweets associated to the brand and ${\\textnormal {\\bf non churn}}$ the number of non-churny ones.",
"Overall, the dataset contains 4339We only keep those with annotation confidence above 0.7 as in [8].",
"labeled tweets and is highly imbalanced regarding the distribution of churny/non-churny tweets.",
"Table: Distribution of English tweets along the different brands."
],
[
"German Twitter Dataset ($\\textnormal {DE}_T$ )",
"Since there is no existing dataset for churn detection except for English, we create a novel German dataset.",
"As a first step, we crawl all mentions on Twitter of multiple telecommunication brands that are active in German-speaking countries for a period of six months.",
"The result is a large Twitter dataset, $\\textnormal {DE}_{T_{FULL}}$ , containing more than 160000 tweets.",
"However, labeling such a large corpus is extremely time intensive and would result in a waste of resources since the density of churny tweets is extremely low.",
"A solution to reduce the size of $\\textnormal {DE}_{T_{FULL}}$ is to apply filters composed of predefined keywords to isolate potential churny tweets and generate a sub-dataset of candidates, $\\textnormal {DE}_{T_{FILTER}}$ , as depicted in Fig.",
"REF .",
"Those keywords are manually selected and are assumed to be linked with or carry churny content.",
"A non-exhaustive list of used keywords is displayed in Table REF .",
"Figure: Creation of DE T \\textnormal {DE}_T and transition to chatbots.Table: Non-exhaustive list of word filters used to detect potential churny tweets in German.The resulting subset, $\\textnormal {DE}_{T_{FILTER}}$ , is given to annotation through a platform specifically created for this purpose.",
"All tweets are annotated by at least two annotators.",
"We keep in our dataset only the entries where both annotators agree on the label.",
"We train the first version of our model with the newly labeled subset and then apply it to our initial dataset $\\textnormal {DE}_{T_{FULL}}$ .",
"By selecting only predictions with high confidence, we can generate an additional subset, $\\textnormal {DE}_{T_{BOOT}}$ , of potential churny tweets.",
"This new subset has the advantage of not being biased by the predefined filter keywords as opposed to $\\textnormal {DE}_{T_{FILTER}}$ .",
"Therefore, we can reduce the overall bias of our dataset by labeling $\\textnormal {DE}_{T_{BOOT}}$ and concatenating it to $\\textnormal {DE}_{T_{FILTER}}$ .",
"The final result is German Twitter dataset as $\\textnormal {DE}_{T} = \\textnormal {DE}_{T_{FILTER}} + \\textnormal {DE}_{T_{BOOT}}$ .",
"The complete distribution of the labels of $\\textnormal {DE}_T$ is displayed in Table REF for comparison purposes with $\\textnormal {EN}_T$ .",
"Here, three main companies emerged from our dataset, namely O2, Vodafone and Telekom (all other brands are grouped in the table as Others).",
"It is interesting to note that the size and distribution of the labels of the German dataset is comparable to the English one which allows fair performance comparison across languages.",
"Table: Distribution of German tweets along the different brands."
],
[
"Chatbot Conversations ($\\textnormal {EN}_C$ + {{formula:2e33ff11-a75a-46a8-b35d-062c5ee66d86}} )",
"Our ultimate goal is to detect churn intent in chatbot conversations.",
"However, no English nor German labeled chatbot conversations are available for this purpose.",
"To overcome this problem, we create our own chatbot platform to gather data and build our German ($\\textnormal {DE}_C$ ) and English ($\\textnormal {EN}_C$ ) chatbot conversations.",
"Our platform consists of a basic interface where the user can enter text that is processed by the chatbot as depicted in Fig.",
"REF .",
"Figure: Annotation process using our platform.We want the user to enter customer service related examples and their ground truth (churn or non-churn) to create our dataset.",
"However, creating and labeling data is a tedious task for the user and might lower the quality of our text-label pairs.",
"Therefore, we choose to present the chatbot interface as a game to make it more user-friendly.",
"Firstly, the user is asked to enter a sentence that is either churny or non churny.",
"Then, the chatbot predicts the output using a model trained on social media and informs the user about the prediction.",
"Finally, the user can approve or disapprove the prediction of the chatbot using buttons.",
"In both cases, we get the ground truth of the text and are able of expanding our database and even giving feedback to the user accordingly.",
"A second annotator is then responsible for double-checking the labeled data coming from the chatbot.",
"We keep only the data points where the two annotators agree.",
"Note that we append the results to the databases ($\\textnormal {EN}_C$ + $\\textnormal {DE}_C$ ) as a function of the detected language of the input text.",
"We end up with two novel datasets for churn detection in chatbot conversations.",
"Table REF presents the distribution of the labels in both languages.",
"The two columns indicate the number of churny and non-churny examples in each dataset respectively.",
"Table: Distribution of labels in chatbot conversations for both languages (EN/DE)."
],
[
"Evaluation",
"For textual churn detection, we design and report on the performance of three experiments: [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Training on $\\textnormal {EN}_{T}$ and testing on $\\textnormal {EN}_{T}$ using English monolingual embeddings.",
"Training on $\\textnormal {DE}_{T}$ and testing on $\\textnormal {DE}_{T}$ using German monolingual embeddings.",
"Training on $\\textnormal {(EN+DE)}_{T}$ and testing on $\\textnormal {EN}_{T}$ and $\\textnormal {DE}_{T}$ using multilingual embeddings.",
"For all experiments, a consistent model with the same hyper-parameters is used to ensure a fair comparison.",
"We employ 256 filters with a kernel size of 2 for the convolutional layer.",
"In addition, we set the number of GRU units to 128 and apply a dropout with a rate of 0.3.",
"Finally, we use the Adam optimizer with its default parameters.",
"To allow a fair comparison, 10-fold cross validation is used as in [8].",
"This ensures that the results are less affected by the train/test split and all models are trained until convergence for each fold.",
"In the end, the mean and standard deviation of macro precision, recall and F1-score are computed over the maximum of each fold.",
"We execute all experiments 20 times, test them under statistical dependence and reject with a threshold of $\\alpha =5\\%$ .",
"For chatbot conversations, we directly evaluate the best model trained on datasets from social media on chatbot conversation data.",
"We report the performance for the following three experiments in Section REF : [noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt] Best model trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ using English embeddings.",
"Best model trained on $\\textnormal {DE}_{T}$ and tested on $\\textnormal {DE}_{C}$ using German embeddings.",
"Best model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ and $\\textnormal {DE}_{C}$ using multilingual embeddings.",
"Table: Performance comparison of our model on English against the current state-of-the-art .",
"EN and DE are scores for language dependent models using monolingual embeddings, whereas EN+DE is for system trained on both languages at the same time using multilingual word embeddings.",
"The indices TT stands for Twitter.Table: Results on chatbot conversations.",
"EN and DE are scores for language dependent models using monolingual embeddings.",
"EN+DE are for system trained on both languages at the same time using multilingual word embeddings.",
"We distinguish Twitter from chatbot dataset using respectively the indices TT and CC."
],
[
"Embeddings and Data Augmentation",
"We use pre-trained 300-dimensional word embeddings for English and German from fastText [17].",
"The same distributional vectors used in monolingual experiments are employed in building multilingual embeddings.",
"We learn the alignment based on a train part of a ground truth bilingual dictionary consisting of 5000 German-English pairs [5].",
"We then apply dimensionality reduction on top of SVD by deleting the last few rows corresponding to a value threshold of 1 in the diagonal vector.",
"The threshold value is chosen to maximize the performance on the test part of the bilingual dictionary pairs used for learning the alignment from $DE$ to $EN$ .",
"We also replace all brands with either \"target\" or \"competitor\" to improve vocabulary coverage.",
"\"Target\" refers to the brand concerned by the churny content and \"competitor\" to all other brands mentioned in the text.",
"Finally, it is important to notice that if a tweet is churny for a specific brand, it is not churny for the other cited brands.",
"For example, \"@X I want switch to @Y!\"",
"is churny for brand X but not for Y.",
"We can, therefore, generate more examples where Y is replaced as \"target\".",
"We use this procedure for each fold to augment the training set."
],
[
"Social Media Results",
"Table REF contains the results for churn detection in social media.",
"The first row shows the results for training and testing on $\\textnormal {EN}_{T}$ data which allows us to compare our score to state-of-the-art results.",
"We outperform the previous performance from [15] and reach 85.88% using multilingual word embeddings.",
"Note that the standard deviation over the 10-fold cross validation is not provided by [15].",
"However, an increase of 2.03% of the mean still represents an important improvement over the state-of-the-art.",
"As a result, we prove that our novel architecture provides an efficient way to detect churn in social media.",
"We notice a significant improvement in the performance of Twitter data when both English and German tweets are aggregated and used for training with multilingual embeddings.",
"The advantage of our multilingual model is promising especially for German with an increase of $7.8\\%$ in F1-score.",
"English also benefits with a slight increase of $1.65\\%$ .",
"The better quality of the English word embeddings makes it easier for our model to identify the churn patterns, compared to German.",
"This explains the gap between the gain in performance for German compared to English, although we used two corpora comparable in size for both languages.",
"To gain more insights into why the multilingual approach improves the test performance in German, it is worth reconsidering the example introduced earlier: \"@MARKE das klingt gut zu den genannten Konditionen würde ich dann doch gern wechseln :)\".",
"This example is predicted as churny using German monolingual model, while it is not churny according to the multilingual model.",
"This can be explained by the fact that the German model could only rely on the presence of switch keyword, while the multilingual approach can learn more complex patterns that are present in both languages.",
"There is a similar example in English: \"I want to switch to @BRAND already\" that portrays more or less the same pattern."
],
[
"Chatbot Results",
"Table REF shows that for chatbot conversations we obtain results comparable to Twitter.",
"This proves that our model is able of capturing the structure of the churny tweets in both languages and generalize it to other applications (e.g., chatbot conversations).",
"Moreover, we observe that the performance of churn detection in English chatbot conversations also benefits from the multilingual approach.",
"Concretely, the model trained on $\\textnormal {(EN+DE)}_{T}$ and tested on $\\textnormal {EN}_{C}$ outperforms its monolingual counterpart trained on $\\textnormal {EN}_{T}$ and tested on $\\textnormal {EN}_{C}$ with an increase $2.34\\%$ in F1-score.",
"On the other hand, performance for German exhibits a marginal drop compared to its monolingual counterpart.",
"This can be due to the small number of conversation examples and their lack of variability which makes them more similar in structure to the training tweets.",
"Therefore, even a monolingual model would work well in this case.",
"A final observation is that the recall is usually higher than the precision on chatbot conversations.",
"This is noteworthy in our application since it is more important to reduce the number of false negative in churn prediction.",
"Indeed, it is better for companies to falsely detect churn intent (in case of false positives) than missing actual customers (in case of false negatives)."
],
[
"Conclusion",
"Preventing customers from leaving a service is an essential topic for companies, as acquiring new customers is a time and cost-intensive procedure.",
"While previous work solely focuses on user behavior over time or social media, here, we propose a novel approach for churn intent detection in chatbot conversations.",
"In this paper, we work towards multilingual churn intent detection in chatbot conversation with knowledge transfer from Twitter datasets.",
"First, we release a novel dataset of German tweets for churn intent detection to complement the existing English one.",
"Moreover, we create and distribute a dataset for churn intent detection in chatbot conversations for both English and German.",
"We present a model based on a neural architecture that outruns the state-of-the-art performance on churn intent detection in social media.",
"Our experiments show that our model can generalize churn intent patterns learned from social media and successfully apply them to chatbot conversations, proving that we can transfer churn detection knowledge from Twitter to chatbots.",
"In addition, we prove that our model, trained using multilingual word embeddings, surpasses monolingual approaches.",
"This result highlights the universal facet of the problem, as examples of churn intent in English help us in identifying churn about German telecommunication brands in German tweets and chatbot conversations."
]
] | 1808.08432 |
[
[
"Superconductivity near a nematic quantum critical point -- the interplay\n between hot and lukewarm regions"
],
[
"Abstract We present a strong coupling dynamical theory of the superconducting transition in a metal near a QCP towards $Q = 0$ nematic order.",
"We use a fermion-boson model, in which we treat the ratio of effective boson-fermion coupling and the Fermi energy as a small parameter $\\lambda$.",
"We solve, both analytically and numerically, the linearized Eliashberg equation.",
"Our solution takes into account both strong fluctuations at small momentum transfers $\\sim \\lambda k_F$ and weaker fluctuations at large momentum transfers.",
"The strong fluctuations determine $T_c$, which is of order $\\lambda^2 E_F$ for both s- and d- wave pairing.",
"The weaker fluctuations determine the angular structure of the superconducting order parameter $F(\\theta_k)$ along the Fermi surface, separating between hot and lukewarm regions.",
"In the hot regions $F(\\theta_k)$ is largest and approximately constant.",
"Beyond the hot region, whose width is $\\theta_h\\sim\\lambda^{1/3}$, $F(\\theta_k)$ drops by a factor $\\lambda^{4/3}$.",
"The s- and d- wave states are not degenerate but the relative difference $(T_c^s-T_c^d)/T_c^s\\sim\\lambda^2$ is small."
],
[
"Supplementary material",
"Our supplemenary material has two parts.",
"The first part gives a more detailed derivation of our results on angular variation of the gap function $F(\\theta _k)$ in both hot and lukewarm regions, and on the resulting splitting of critical temperatures $T^{s,d}_c$ between $s-$ wave $d-$ wave modes, Eq.",
"(REF ).",
"The second part discusses the numerical methods used to determine the critical temperature at the QCP, Eq.",
"(REF ), and to verify our analytic results.",
"In the main part of the paper, we noted that the critical temperature is, to first approximation, determined by the local, frequency dependent, gap equation (REF ).",
"In order to determine the angular behavior, we approximated the full gap equation (REF ) by an effective one dimensional integral equation where we replaced the frequency terms in the gap equation by their typical value $\\omega _n,\\omega _m \\sim T_c$ , and summed over the Matsubara frequencies.",
"The result is Eq.",
"(REF ) which we reproduce here for clarity, $F(\\theta _k) = \\frac{3 \\sqrt{3}\\lambda }{4} \\int \\frac{d \\theta _q}{\\pi } \\frac{F(\\theta _k + \\theta _q) |\\theta _q|}{|\\theta _q|^3 + \\lambda ^3}f^2\\left(\\theta _k+\\frac{\\theta _q}{2}\\right).$ Eq.",
"(REF ) neglects several angular terms, namely the angular dependency of the fermionic and bosonic self-energies, see Eqs.",
"(REF ), (REF ).",
"We have verified that neglecting these terms doesn't affect the final result.",
"Eq.",
"(REF ) has been the property that if we neglect the dependence of $F$ and $f^2$ on $\\theta _q$ , it is fulfilled trivially.",
"To determine the width of the hot region gap we assume that $F = F(\\theta _k/\\theta _{h})$ is a function of a single scaling parameter $\\theta _{h}$ , and analyze it for $1 \\gg \\theta _k \\gg \\theta _{h}$ .",
"The r.h.s.",
"of Eq.",
"(REF ) simplifies to, $0 &\\approx -F(x)\\theta _{h}^2x^2/2 + \\frac{3 \\sqrt{3}\\lambda }{4\\pi \\theta _{h}} \\int dy \\frac{F(y)}{|x-y|^2},$ where $x = \\theta _k/\\theta _{h} \\gg 1$ , but $\\theta _{h}^2x^2 \\ll 1$ .",
"The first term is the local contribution from $\\theta _q \\sim \\lambda $ , and the second term is the induced gap from the nearby hot region at $\\theta _q \\sim -\\theta _k$ .",
"It is easy to see that for $\\theta _{h}^3 = \\frac{3\\sqrt{3}\\lambda }{2\\pi }$ we obtain a dimensionless equation (for $x\\gg 1$ ), $F(x) = \\frac{1}{x^2}\\int dy \\frac{F(y)}{(x-y)^2}$ with a solution, $F(x) \\approx a F(0)/x^4,$ where $a$ is a constant of order one.",
"Our results are equivalent to Eqs.",
"(REF ),(REF ).",
"Eq.",
"(REF ) also demonstrates that near the lukewarm regions $\\theta _k \\sim 1$ , $F(\\theta _k) \\sim F(0)\\theta _{h}^4 \\propto F(0) \\lambda ^{4/3}.$ In order to obtain the transition temperatures for $s-$ wave and $d-$ wave gaps, we again reduce Eq.",
"(REF ) to an effective 1D equation.",
"We account for the expected temperature differences by introducing different eigenvalues for $s-$ wave and $d-$ wave solutions $\\eta _s(T), \\eta _d(T)$ , i.e., $\\eta _{s,d}F(\\theta _k)_{s,d} = \\frac{3 \\sqrt{3}\\lambda }{4} \\int \\frac{d \\theta _q}{\\pi } \\frac{F_{s,d}(\\theta _k + \\theta _q) |2\\sin \\theta _q/2|}{|2\\sin \\theta _q/2|^3 + \\lambda ^3}f^2\\left(\\theta _k+\\frac{\\theta _q}{2}\\right).$ We assume and then verify that $(T_c^s-T_c^d)\\ll T_c$ , and expand the $\\eta $ 's near $T_c^s,T_c^d$ , to obtain, $\\eta ^{s,d}_c(T_c) \\approx 1 + \\alpha _{s,d}\\frac{T^{s,d}_c-T_c}{T_c},$ where $T_c$ is the solution, Eq.",
"(REF ), of the local gap equation (REF ).",
"Then we have $1 - \\frac{T_c^d}{T_c^s} \\approx \\frac{\\eta _s-1}{\\alpha _d} - \\frac{\\eta _d - 1}{\\alpha _s}.$ In order to evaluate $\\eta _{s,d}$ we again account for the two contributions from the r.h.s.",
"of Eq.",
"(REF ), one coming from the local contribution $\\theta _q \\sim 0$ , and the other coming from far regions, $|\\theta _q| \\gg \\theta _{h}$ .",
"The local contribution is larger, but doesn't differentiate between $s-$ wave and $d-$ wave, which will be determined by the nonlocal contribution.",
"If we consider the behavior at a hot region, say $\\theta _k = 0$ , then the nonlocal contribution will come mostly from the hot regions at $\\theta _q = \\pm \\pi /2$ .",
"Therefore we have, $\\eta _{s,d} F(\\theta _k = 0) &\\approx F(0) \\pm 2\\int \\frac{3\\sqrt{3}\\lambda }{8\\pi }\\int d\\theta _q F(\\theta _q)f^2\\left(\\frac{\\pi }{4}+\\frac{\\theta _q}{2}\\right) \\nonumber \\\\&\\approx F(0) \\pm a\\lambda \\theta _{h}^3F(0).$ where in the integration we shifted $\\theta _q \\rightarrow \\theta _q\\pm \\pi /2$ .",
"In the second line, one $\\theta _{h}$ in the last term on the right comes from width of the hot region, and another $\\theta _{h}^2$ comes from expanding the form-factor, $f^2(\\pi /4+\\theta _q/2) \\approx \\theta _q^2/4$ .",
"$a$ is a constant of order one.",
"Eq.",
"(REF ) implies a splitting $\\eta _s - \\eta _d \\sim \\lambda ^2$ , which is second order in $\\lambda $ .",
"Such splitting is much smaller than what we would naively expect, namely a difference of order $\\lambda $ .",
"We therefore need to verify that there is no other contribution that is equivalent or larger.",
"To this end we re-iterate Eq.",
"(REF ), and obtain for $\\theta _k = 0$ , $\\lambda _{s,d}^2 F(0) &= \\left(\\frac{3 \\sqrt{3}\\lambda }{4}\\right)^2 \\int \\frac{d \\theta _q}{\\pi }\\frac{d \\theta _q^{\\prime }}{\\pi } \\frac{F(\\theta _q+\\theta _q^{\\prime }) |2\\sin \\theta _q^{\\prime }/2|}{|2\\sin \\theta _q^{\\prime }/2|^3 + \\lambda ^3}f^2\\left(\\frac{\\theta _q+\\theta _q^{\\prime }}{2}\\right)\\frac{|2\\sin \\theta _q/2|}{|2\\sin \\theta _q/2|^3 + \\lambda ^3} f^2\\left(\\frac{\\theta _q}{2}\\right) \\nonumber \\\\&\\approx F(0)(1 \\pm 2a \\lambda \\theta _{h}^3 + b_\\pm \\lambda ^2 \\theta _{h})$ Here $b_\\pm $ are constants of order one.",
"The final term comes from one of two contributions: (a) $\\theta _q \\sim 0$ but $0 \\ll |\\theta _q^{\\prime }| \\ll \\pi /2$ , or vice versa.",
"This is a contribution from the lukewarm region.",
"(b) $0 \\ll |\\theta _q|,|\\theta _q^{\\prime }| \\ll \\pi /2$ , but $|\\theta _q + \\theta _q^{\\prime }| \\sim 0,\\pi /2$ .",
"This is a contribution from the hot regions.",
"Regardless of origin, the final contribution is clearly smaller than the second term, and so, going back to Eq.",
"(REF ), we find that the split in $T_c^s, T_c^d$ scales with $\\lambda ^2$ .",
"Eq.",
"(REF ) is equivalent to Eq.",
"(REF ) in the main text.",
"We performed numerical analysis of the two gap equations we studied in the main text: both the full 2D Eliashberg equation, Eq.",
"(REF ), and the local gap equation, Eq.",
"(REF ).",
"All of our solutions were obtained in MATLAB 2017.",
"We solved the local gap equation by numerically finding the largest eigenvalue of the operator on the r.h.s.",
"of Eq.",
"(REF ).",
"We solved for using an increasing series of Matsubara frequencies, and then performed finite-size scaling.",
"The result is shown in Fig.",
"REF and was reported in Eq.",
"(REF ) of the main text.",
"Figure: Scaling of T c T_c in the local gap equation as a function of number of Matsubara frequencies included in the summation.",
"The solid red line is a fit to a+bexp(-cx)a + b \\exp (-c x).",
"The extrapolated result is reported inEq.",
"() of the main text.We solved the full 2D Eliashberg gap equation for a variety of of system sizes in both angle discretization and Matsubara frequencies, $N_\\theta = 2^7-2^9$ , $N_M = 2^3-2^6$ , and a variety of couplings, $\\lambda = 0.025-0.25$ .",
"All computations were performed using the resources of the Minnesota Supercomputing Institute (MSI).",
"We confirmed numerically the calculated scaling of the hot region width and decay, Eqs.",
"(REF ), (REF ).",
"We also confirmed that the eigenvalue splitting between $s-$ wave and $d-$ wave solutions of the full equation followed the same scaling as the one we found from the 1D equation, Eq.",
"(REF ).",
"We also confirmed the expected height of the gap in the lukewarm region, Eq.",
"(REF )."
]
] | 1808.08635 |
[
[
"The $K\\bar{K}^*$ interaction in the unitary coupled-channel\n approximation"
],
[
"Abstract The $K\\bar{K}^*$ interaction Lagrangian is constructed when the $SU(3)$ hidden gauge symmetry is taken into account, and then the $K\\bar{K}^*$ potential is obtained.",
"In the low energy region, the $K\\bar{K}^*$ potential mainly comes from the contribution of the $t-$channel interaction by exchanging $\\rho$,$\\omega$ and $\\varphi$ mesons, respectively.",
"The $K\\bar{K}^*$ amplitude is investigated by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation, where the loop function of the vector and pseudoscalar mesons are evaluated in the dimensional regularization scheme, and the contribution of the longitudinal part of the intermediate vector meson propagator is included in the calculation.",
"Finally, it is found that a resonance state of $K\\bar{K}^*$ is generated in the isospin $I=0$ sector, which might correspond to the $f_1(1420)$ particle in the review of the particle data group(PDG).",
"Moreover, in the isospin $I=1$ sector, a pole of the $K\\bar{K}^*$ amplitude is detected at $1425-i316$MeV on the complex plane of the total energy in the center of mass system, which is higher than the $K\\bar{K}^*$ threshold.",
"Thus this pole might be a resonance state of $K\\bar{K}^*$ although no counterpart has been found in the PDG review."
],
[
"Introduction",
"Quantum Chromodynamics(QCD) is considered to be the fundamental theory to describe strong interactions.",
"In the high energy region, the results obtained with QCD are in good agreement with the experimental data because of the asymptotic freedom.",
"However, the strong coupling constant increases rapidly when the energy becomes lower, and the method of the perturbation expansion will not be applied, so other methods are developed to deal with the non-perturbative QCD effect, such as Lattice QCD theory, QCD sum rules, the constituent quark model, and the chiral unitary method.",
"In the chiral unitary method, the Bethe-Salpeter equation is solved while the unitarity of the amplitude is reserved.",
"Thus the resonance state of the interacting hadrons could be generated dynamically.",
"In the low energy region, it has achieved great success in the study of the interaction of hadrons[1], [2], [3].",
"The vector meson component can be introduced in the interaction Lagrangian when the hidden gauge symmetry is taken into account[4], [5], [6], [7].",
"Along this clue, the pseudoscalar meson and vector meson interaction[8], the vector meson and vector meson interaction[9], [10],the vector meson and baryon octet interaction[11], [12], and the vector meson and baryon decuplet interaction in the SU(3) flavor space are studied in the unitary coupled-channel approximation[13], [14].",
"In the present work, the $K\\bar{K}^*$ interaction will be evaluated in the $SU(3)$ flavor space when the hidden gauge symmetry is considered, and then we will examine whether the resonance state with a $s\\bar{s}$ quark pairs can be generated dynamically by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation.",
"The article is organized as follows: In Section , the $K\\bar{K}^*$ potential is constructed in the SU(3) flavor space when the hidden gauge symmetry is taken into account.",
"In Section , the Bethe-Salpeter equation is discussed in the case of the $K\\bar{K}^*$ interaction.",
"Especially, the unitarity of the amplitude is emphasized when the longitudinal part of the intermediate vector meson propagator is included in the calculation.",
"The results are given in Section , and it is summarized in Section ."
],
[
"$K\\bar{K}^*$ potential",
"According to the hidden gauge symmetry, the interaction Lagrangian of the vector meson and the pseudoscalar meson can be written as ${\\cal L}_{VPP}=-i g \\langle V_\\mu [P, \\partial ^\\mu P ] \\rangle , $ where $g=\\frac{M_V}{2f_\\pi }$ , $f_\\pi =93$ MeV is the pion decay constant, $M_V$ is the mass of vector meson, and $P=\\left(\\begin{array}{ccc}\\frac{1}{\\sqrt{2}}\\pi ^0+\\frac{1}{\\sqrt{6}}\\eta & \\pi ^+ & K^+ \\\\\\pi ^- & -\\frac{1}{\\sqrt{2}}\\pi ^0+\\frac{1}{\\sqrt{6}}\\eta & K^0 \\\\K^- & \\bar{K}^0 & -\\frac{2}{\\sqrt{6}}\\eta \\end{array}\\right),$ and $V_\\mu =\\left(\\begin{array}{ccc}\\frac{\\omega }{\\sqrt{2}}+\\frac{\\rho ^0}{\\sqrt{2}} & \\rho ^+ & K^{*+} \\\\\\rho ^- & \\frac{\\omega }{\\sqrt{2}}-\\frac{\\rho ^0}{\\sqrt{2}} & K^{*0} \\\\K^{*-} & \\bar{K}^{*0} & \\varphi \\end{array}\\right)_\\mu $ are the pseudoscalar meson matrix and the vector meson matrix in the $SU(3)$ flavor space, respectively[8].",
"The vector meson Lagrangian takes the form of $\\mathcal {L}_V=-\\frac{1}{4}\\langle V_{\\mu \\nu }V^{\\mu \\nu } \\rangle ,$ where $V_{\\mu \\nu }=\\partial _{\\mu }V_{\\nu }-\\partial _{\\nu }V_{\\mu }-ig[V_{\\mu },V_{\\nu }].",
"$ According to Eqs.",
"(REF ) and (REF ), the interaction Lagrangian of three vector mesons can be written as $\\mathcal {L}_{VVV}=ig \\langle (\\partial _{\\mu }V_{\\nu }-\\partial _{\\nu }V_{\\mu })V^{\\mu }V^{\\nu } \\rangle .",
"$ Thus the $\\bar{K}^*\\bar{K}^*\\rho $ , $\\bar{K}^*\\bar{K}^*\\omega $ , and $\\bar{K}^*\\bar{K}^*\\varphi $ vertices can be obtained easily.",
"Figure: The interaction of K ¯ * K→K ¯ * K\\bar{K}^*K\\rightarrow \\bar{K}^*K, (a)Vector meson exchange, (b) π\\pi and η\\eta exchange.The $K\\bar{K}^*$ potential is composed of $t-$ channel and $u-$ channel interactions, as shown in Fig.",
"REF .",
"For the $t-$ channel interaction, there are three intermediate mesons exchanged, i.e., $\\omega $ , $\\rho $ and $\\varphi $ , respectively.",
"In this work, we study the scattering of $K\\bar{K}^*$ in the low energy region, so the momentum of the intermediate meson is ignored.",
"From the Feynman diagram in Fig.REF (a), we can see that the $t-$ channel interaction of $K\\bar{K}^*$ can be divided into two parts, and the one is the vertex of $\\bar{K}^*\\bar{K}^*$ and a vector meson, and the other is the vertex of $\\bar{K}K$ and a vector meson.",
"Therefore, we can calculate these two kinds of vertices and then combine them together to obtain the $t-$ channel potential of $K\\bar{K}^*$ , which can be written as $V_{ij}&=&C_{ij}\\frac{1}{f^2_{\\pi }}(p_1+p_2)\\cdot (k_1+k_2)\\varepsilon \\cdot \\varepsilon ^* \\nonumber \\\\&=&\\tilde{V}_{ij}g^{\\mu \\nu } \\varepsilon _\\mu \\varepsilon ^*_\\nu ,$ with $\\tilde{V}_{ij}=C_{ij}\\frac{1}{f^2_{\\pi }}(p_1+p_2)\\cdot (k_1+k_2).",
"$ In Eq.",
"(REF ), the label $i$ denotes the initial state and the label $j$ stands for the final state, $p_1(k_1)$ and $p_2(k_2)$ represent the initial and final momenta of the $\\bar{K}^*(K)$ meson, and $\\varepsilon $ and $\\varepsilon ^*$ are polarization vectors of the initial and final $\\bar{K}^*$ mesons, respectively.",
"Since the values of $\\rho $ , $\\omega $ and $\\varphi $ meson masses are similar to each other, we assume that $M_V\\approx m_{\\rho }\\approx m_{\\omega }\\approx m_{\\varphi }$ , then the coefficients $C_{ij}$ of the $K\\bar{K}^*$ potential take the simple values as listed in Table REF .",
"Table: The coefficients C ij C_{ij} in the KK ¯ * K\\bar{K}^* interaction,C ji =C ij C_{ji}=C_{ij}.If Mandelstam variables $s=(p_1+k_1)^2$ , $t=(k_2-k_1)^2$ and $u=(p_2-k_1)^2$ are adopted, the kernel in Eq.",
"(REF ) can be written as $\\tilde{V}_{ij}&=&C_{ij}\\frac{1}{f^2_{\\pi }} (s-u) \\nonumber \\\\&=&C_{ij}\\frac{1}{f^2_{\\pi }} \\left( 2s+t-2(M_K^2+M_{K^*}^2) \\right).$ Around the $K\\bar{K}^*$ threshold, Mandelstam variable $t=(k_2-k_1)^2$ is assumed to be zero, so Eq.",
"(REF ) can be simplified as $ \\tilde{V}_{ij}&=&C_{ij}\\frac{2}{f^2_{\\pi }} \\left( s-M_K^2-M_{K^*}^2 \\right).",
"$ According to the interaction Lagrangian of the vector meson and the pseudoscalar meson in Eq.",
"(REF ), the $u-$ channel potential of the $K\\bar{K}^*$ interaction via a pion or an $\\eta $ meson exchange in Fig.",
"REF b can be obtained as $V_{\\alpha ij}&=&D_{\\alpha ij}g^2(q-k_1)\\cdot \\varepsilon ^*\\frac{1}{q^2-m^2_{\\alpha }}(q-k_2)\\cdot \\varepsilon \\nonumber \\\\&=&4 D_{\\alpha ij}g^2 q\\cdot \\varepsilon ^*\\frac{1}{q^2-m^2_{\\alpha }}q\\cdot \\varepsilon ,$ where $\\alpha $ represents $\\pi $ or $\\eta $ , $D_{\\alpha ij}$ is the interaction coefficient, $q=p_2-k_1=p_1-k_2$ is the momentum of the intermediate meson, $p_1\\cdot \\varepsilon =0$ and $p_2\\cdot \\varepsilon ^*=0$ .",
"The zero component of the polarization vector of the $\\bar{K}^*$ meson is in inverse proportion to the $\\bar{K}^*$ meson mass, thus it can be neglected, so we have $ q \\cdot \\varepsilon ^* \\sim |\\vec{q}|, $ and $ q \\cdot \\varepsilon \\sim |\\vec{q}|.",
"$ If the three-momentum of the initial and final mesons is neglected, just as we have done in this work and other works of ours, the $u-$ channel potential of $K\\bar{K}^*$ is trivial.",
"Therefore, only the the $t-$ channel potential of the $K\\bar{K}^*$ interaction is taken into account when the Bethe-Salpeter equation is solved.",
"In addition to the $K\\bar{K}^*$ channel, there are other channels with strangeness zero, i.e., $\\pi \\rho $ , $\\pi \\omega $ , $\\pi \\varphi $ , $\\eta \\rho $ , $\\eta \\omega $ , $\\eta \\varphi $ .",
"However, the $\\pi \\rho $ threshold is far lower than the energy region where we are interested, and the elastic scattering amplitude of the other five channels are all zero.",
"Thus the influence of these six channels are neglected when the $K\\bar{K}^*$ interaction is studied in this work."
],
[
"The Bethe-Salpeter equation",
"The Bethe-Salpeter equation can be written as $T&=&V+VGV+VGVGV+... \\nonumber \\\\&=&V+VGT,$ where $T$ is the scattering amplitude, $V$ is the interaction kernel, and $G$ is the loop function, which is a diagonal matrix.",
"In the interaction of the pseudoscalar meson and the vector meson, the loop diagram in the Bethe-Salpeter equation can be written as $G_l(s)&=&i\\int \\frac{d^4 q}{(2\\pi )^4} \\frac{-g_{\\mu \\nu }+\\frac{q_\\mu q_\\nu }{M_a^2}}{q^2-M_a^2+i\\varepsilon }\\frac{1}{(P-q)^2-M_b^2+i\\varepsilon }\\nonumber \\\\&=&-g_{\\mu \\nu }G_{ab}(s)-\\frac{g^{\\mu \\nu }}{M_a^2}H^{00}_{ab}(s)-\\frac{P^{\\mu }P^{\\nu }}{M^2_a}H^{11}_{ab}(s),$ where $M_a$ and $M_b$ are masses of the vector meson and the pseudoscalar meson, respectively, and $P=p_1+k_1=p_2+k_2$ is the total momentum of the system.",
"The term relevant to the transverse part of the propagator of the intermediate vector meson is $G_{ab}(s)=i\\int \\frac{d^4q}{(2\\pi )^4}\\frac{1}{q^2-M_a^2+i\\varepsilon }\\frac{1}{(P-q)^2-M_b^2+i\\varepsilon },$ and the functions $H^{00}_{ab}(s)$ and $H^{11}_{ab}(s)$ correspond to the contribution of the longitudinal part of the propagator of the intermediate vector meson, $ g^{\\mu \\nu } H^{00}_{ab}(s)+P^\\mu P^\\nu H^{11}_{ab}(s)=\\frac{1}{i} \\int \\frac{d^4 q}{(2\\pi )^4} \\frac{q^\\mu q^\\nu }{(q^2-M_a^2+i\\varepsilon )[(P-q)^2-M_b^2+i\\varepsilon ]}.",
"$ In the dimensional regularization scheme, the function $G_{ab}(s)$ takes the form of $G_{ab}(s)&=&i \\int \\frac{d^4 q}{(2\\pi )^4} \\frac{1}{q^2-M_a^2+i\\epsilon } \\frac{1}{(P-q)^2-M_b^2+i\\epsilon } \\nonumber \\\\&=& \\frac{1}{16 \\pi ^2} \\left\\lbrace a_l(\\mu ) + \\ln \\frac{M_a^2}{\\mu ^2} +\\frac{M_b^2-M_a^2 + s}{2s} \\ln \\frac{M_b^2}{M_a^2} + \\right.\\nonumber \\\\ & & \\phantom{\\frac{2 M_a}{16 \\pi ^2}} +\\frac{\\bar{q}_l}{\\sqrt{s}} \\left[\\ln (s-(M_a^2-M_b^2)+2\\bar{q}_l\\sqrt{s})+\\ln (s+(M_a^2-M_b^2)+2\\bar{q}_l\\sqrt{s}) \\right.",
"\\\\& & \\left.",
"\\phantom{\\frac{2 M_a}{16 \\pi ^2} +\\frac{\\bar{q}_l}{\\sqrt{s}}} \\left.",
"\\hspace*{-8.5359pt}-\\ln (-s+(M_a^2-M_b^2)+2\\bar{q}_l\\sqrt{s})-\\ln (-s-(M_a^2-M_b^2)+2\\bar{q}_l\\sqrt{s}) \\right] \\right\\rbrace , \\nonumber $ with the square of the total energy of the system $s=P^2$ and the three-momentum of the intermediate particles in the center of mass frame $\\bar{q}_l=\\frac{\\sqrt{s-(M_a+M_b)^2}\\sqrt{s-(M_a-M_b)^2}}{2\\sqrt{s}}.$ The forms of $H^{00}_{ab}(s)$ and $H^{11}_{ab}(s)$ can be found in the appendix part of Ref.",
"[15], [16].",
"The loop function $G_l(s)$ can be written as $G_l(s)=g^{\\mu \\nu }\\tilde{G}_l(s),$ with $\\tilde{G}_l(s)=-\\left(G_{ab}(s)+\\frac{1}{M_a^2}H^{00}_{ab}(s)+\\frac{s}{4M_a^2}H^{11}_{ab}(s)\\right),$ approximately.",
"Replacing the potential in Eq.",
"(REF ) and the loop function in Eq.",
"(REF ) into the Bethe-Salpeter equation, we obtain $\\tilde{T}g^{\\mu \\nu }=\\tilde{V}g^{\\mu \\nu }+\\tilde{V}g^{\\mu \\alpha }~g_{\\alpha \\beta }\\tilde{G}~\\tilde{V}g^{\\beta \\nu }+..., $ and thus $\\tilde{T}&=&\\tilde{V}+\\tilde{V}\\tilde{G}\\tilde{V}+... \\nonumber \\\\&=&[1-\\tilde{V}\\tilde{G}]^{-1} \\tilde{V}.",
"$ The amplitude $\\tilde{T}$ is unitary when the Bethe-Salpeter equation is solved.",
"In addition, if the effect of the $\\bar{K}^*$ decay width is taken into account in the calculation, the loop function $\\tilde{G}_l(s)$ must be replaced by $\\tilde{\\tilde{G}}_l(s)$ obtained as $\\tilde{\\tilde{G}}_l(s)=\\frac{1}{N}\\int _{(M_a-2\\Gamma )^2}^{(M_a+2\\Gamma )^2}d \\tilde{m}^2~\\frac{1}{\\pi }~\\frac{M_a\\Gamma }{(\\tilde{m}^2-M_a^2)^2+M_a^2 \\Gamma ^2}\\tilde{G}_l(s,\\tilde{m}^2,M_b^2),$ with $N=\\int _{(M_a-2\\Gamma )^2}^{(M_a+2\\Gamma )^2}d \\tilde{m}^2~\\frac{1}{\\pi }~\\frac{M_a\\Gamma }{(\\tilde{m}^2-M_a^2)^2+M_a^2 \\Gamma ^2},$ where the $\\bar{K}^*$ decay width $\\Gamma =50$ MeV for the process of $\\bar{K}^* \\rightarrow \\bar{K}\\pi $ , and the equation $\\delta (\\tilde{m}^2-M_a^2)=\\lim _{\\Gamma \\rightarrow 0^+}~\\frac{1}{\\pi }~\\frac{M_a \\Gamma }{(\\tilde{m}^2-M_a^2)^2+M_a^2\\Gamma ^2}$ is used."
],
[
"Results",
"The $K\\bar{K}^*$ pair with isospin $I=0$ takes the form of $|K\\bar{K}^*\\rangle =\\frac{1}{\\sqrt{4}}(|\\bar{K}^0K^{*0}\\rangle +|K^-K^{*+}\\rangle -|K^0\\bar{K}^{*0}\\rangle -|K^+K^{*-}\\rangle ),$ with $K^{-}=-|1/2,-1/2\\rangle $ and $K^{*-}=-|1/2,-1/2\\rangle $ , where the C-parity of the $K\\bar{K}^*$ pair is assumed to be positive since $CKC^{-1}=K^{\\dag }$ and $CK^*C^{-1}=-K^{*\\dag }$ .",
"According to Eqs.",
"(REF ) and (REF ), the kernel of the $K\\bar{K}^*$ interaction in the isospin $I=0$ sector can be written as $\\tilde{V}^{I=0}_{K\\bar{K}^*\\rightarrow K\\bar{K}^*}&=&\\frac{3}{2}~\\frac{1}{f^2_{\\pi }}(s-M^2_K-M^2_{K^*}) .",
"$ Actually, since only non-strange vector mesons act as the intermediate particles in the $t-$ channel $K\\bar{K}^*$ interaction, the kernel in Eq.",
"(REF ) is independent on the C-parity of the $K\\bar{K}^*$ pair according to the coefficients listed in Table REF .",
"Therefore, the Bethe-Salpeter equation can be solved.",
"Assuming the subtraction constant $a=-2$ and the regularization scale $\\mu =600$ MeV, a pole of the squared amplitude $|T|^2$ appears at $1394-i83$ MeV on the complex energy plane of $\\sqrt{s}$ , which is higher than the $K\\bar{K}^*$ threshold, and lies in the second Riemann sheet.",
"This pole can be regarded as a $K\\bar{K}^*$ resonance state, and it is more possible to correspond to the $f_1(1420)$ particle in the review of the Particle Data Group[17], whose mass and decay width are $m=1426.4\\pm 0.9$ MeV and $\\Gamma =54.9\\pm 2.6$ MeV, respectively.",
"According to the formula to include the $\\bar{K}^*$ decay width in Ref.",
"[11], we obtain a pole at the position of $1394-i75$ MeV on the complex energy plane, which is similar to the result calculated with Eq.",
"(REF ).",
"In the isospin $I=1$ sector, the $K\\bar{K}^*$ wave function with the positive C-parity is constructed as $|K\\bar{K}^{*}\\rangle =\\sqrt{\\frac{1}{4}}(|\\bar{K}^0K^{*0}\\rangle -|K^-K^{*+}\\rangle -|K^0\\bar{K}^{*0}\\rangle +|K^+K^{*-}\\rangle ).$ Similarly, the $K\\bar{K}^*$ kernel with isospin $I=1$ can be written as $ \\tilde{V}^{I=1}_{K\\bar{K}^*\\rightarrow K\\bar{K}^*} &=&\\frac{1}{2}~\\frac{1}{f^2_{\\pi }}(s-M^2_K-M^2_{K^*}) .$ With the kernel in Eq.",
"(REF ), a pole of the squared amplitude is generated at the position of $1425-i316$ MeV on the complex energy plane of $\\sqrt{s}$ .",
"Since it is above the $K\\bar{K}^*$ threshold, it can be regarded as a $K\\bar{K}^*$ resonance state with isospin $I=1$ although its decay width is too large and no counterpart is found in the PDG review.",
"If the decay width of the $\\bar{K}^*$ is taken into account with the formula in Ref.",
"[11], the pole is detected at $1432-i330$ MeV on the complex energy plane of $\\sqrt{s}$ , and the decay width of this resonance state is still large."
],
[
"Summary",
"The $K\\bar{K}^*$ interaction is discussed when the hidden gauge symmetry is taken into account, and it is found that the $t-$ channel potential via a non-strange vector meson exchange plays an important role in the $K\\bar{K}^*$ interaction.",
"The Bethe-Salpeter equation is solved in the isospin space, and the contribution of the longitudinal part of the propagator of the intermediate vector meson is taken into account.",
"In the isospin $I=0$ sector, a resonance state is generated at $1394-i83$ MeV on the complex energy plane, which is above the $K\\bar{K}^*$ threshold, and might correspond to the $f_1(1420)$ particle in the PDG review.",
"In the isospin $I=1$ sector, a resonance state is generated at $1425-i316$ MeV on the complex energy plane.",
"The decay width of this resonance is too large, and it implies that this resonance is unstable.",
"No counterpart is found in the PDG data.",
"Moreover, the effect of the decay width of the intermediate $\\bar{K}^*$ meson is included in the calculation with two kinds of formulas, and it manifests that the results are similar to each other.",
"Bao-Xi Sun would like to thank Han-Qing Zheng for useful discussions."
]
] | 1808.08358 |
[
[
"The hyperbolic Ernst equation in a triangular domain"
],
[
"Abstract The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain.",
"We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann--Hilbert problem.",
"The formulation of the Riemann--Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform.",
"Our results are valid also for boundary data whose derivatives are unbounded at the triangle's corners---this level of generality is crucial for the application to colliding gravitational waves.",
"Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann--Hilbert formalism can be explicitly inverted at the boundary.",
"In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary."
],
[
"Introduction",
"Half a century after Einstein presented his theory of relativity, F. J. Ernst made the remarkable discovery that, in the presence of one space-like and one time-like Killing vector, the entire solution of the vacuum Einstein field equations reduces to solving a single equation for a complex-valued function $\\mathcal {E}$ of two variables [6].",
"This single equation, now known as the (elliptic) Ernst equation, has proved instrumental in the study and construction of stationary axisymmetric spacetimes, cf.",
"[15].",
"It later became clear that a similar reduction of Einstein's equations is possible also in the presence of two space-like Killing vectors, a situation relevant for the description of two colliding plane gravitational waves [1].",
"In this case the associated equation is known as the hyperbolic Ernst equation and can be written in the form $(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(\\mathcal {E}_{xy} - \\frac{\\mathcal {E}_x + \\mathcal {E}_y}{2(1-x-y)}\\right) = \\mathcal {E}_x \\mathcal {E}_y,$ where the Ernst potential $\\mathcal {E}(x,y)$ is a complex-valued function of the two real variables $(x,y)$ and subscripts denote partial derivatives.",
"The problem of finding the nonlinear interaction of two plane gravitational waves following their collision has a distinguished history going back to the work of Khan and Penrose [13], Szekeres [25], Nutku and Halil [23], and Chandrasekhar and coauthors [1], [2]; see the monograph [11] for further references and historical remarks.",
"In terms of the Ernst potential, this collision problem reduces to a Goursat problem for equation (REF ) in the triangular region $D$ defined by (see Figure REF ) $D = \\lbrace (x,y) \\in {R}^2\\, | \\, x \\ge 0, \\; y \\ge 0, \\; x+y < 1\\rbrace .$ More precisely, the problem can be formulated as follows (see [11] and the appendix): ${\\left\\lbrace \\begin{array}{ll}\\text{Given complex-valued functions $\\mathcal {E}_0(x)$, $x \\in [0, 1)$, and $\\mathcal {E}_1(y)$, $y \\in [0,1)$, }\\\\\\text{find a solution $\\mathcal {E}(x,y)$ of the hyperbolic Ernst equation (\\ref {ernst}) in $D$}\\\\\\text{such that $\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1)$ and $\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1)$.}\\end{array}\\right.",
"}$ In this paper, we use the integrable structure of equation (REF ) and Riemann–Hilbert (RH) techniques to analyze the Goursat problem (REF ).",
"We present four main results, denoted by Theorem REF -REF : Theorem REF is a solution representation result: Assuming that the given data satisfy the following conditions for some $n \\ge 2$ : ${\\left\\lbrace \\begin{array}{ll}\\mathcal {E}_0, \\mathcal {E}_1 \\in C([0,1)) \\cap C^n((0,1)),\\\\\\text{$x^\\alpha \\mathcal {E}_{0x}, y^\\alpha \\mathcal {E}_{1y} \\in C([0,1))$ for some $\\alpha \\in [0,1)$},\\\\\\mathcal {E}_0(0) = \\mathcal {E}_1(0) = 1,\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0$ for $x \\in [0,1)$},\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}_1(y) > 0$ for $y~\\in [0,1)$},\\end{array}\\right.",
"}$ and the Goursat problem (REF ) has a solution (in the precise sense specified in Definition REF ), we give a representation formula for this solution.",
"This formula is given in terms of the solution of a corresponding RH problem whose formulation only involves the given boundary data.",
"Theorem REF is a uniqueness result: Assuming that the given data satisfy the conditions (REF ) for some $n \\ge 2$ , we show that the solution of the Goursat problem (REF ) is unique, if it exists.",
"Theorem REF is an existence and regularity result: Assuming that the given data satisfy the conditions (REF ) for some $n \\ge 2$ , we show that there exists a unique solution $\\mathcal {E}$ of the problem (REF ) whenever the associated RH problem has a solution, and this $\\mathcal {E}$ has the same regularity as the given data.",
"In the case of collinearly polarized waves, this yields existence for general data; for noncollinearly polarized waves, a small-norm assumption is also needed.",
"Theorem REF provides exact formulas for the singular behavior of the solution $\\mathcal {E}$ near the boundary for data satisfying (REF ).",
"Figure: NO_CAPTIONWe emphasize that the assumptions (REF ) allow for functions $\\mathcal {E}_0(x)$ and $\\mathcal {E}_1(y)$ whose derivatives blow up as $x$ and $y$ approach the origin.",
"This level of generality is necessary for the application to gravitational waves.",
"Indeed, in order for the problem (REF ) to be relevant in the context of gravitational waves, it turns out that the solution should obey the conditions (see [11] and the appendix) $& \\lim _{x \\downarrow 0} x^\\alpha |\\mathcal {E}_x(x,y)| = \\frac{m_1\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} && \\text{for each $y \\in [0,1)$},\\\\& \\lim _{y \\downarrow 0} y^\\alpha |\\mathcal {E}_y(x,y)| = \\frac{m_2\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} && \\text{for each $x \\in [0,1)$},$ where $m_1$ and $m_2$ are real constants such that $m_1, m_2 \\in [1, \\sqrt{2})$ and $\\alpha = 1/2$ .",
"Remarkably, for data with a singular behavior at the origin of the form given in (REF ), the singular integral operator underlying the RH formalism can be explicitly inverted in the limit of small $x$ or $y$ .",
"This leads to the characterization of the boundary behavior given in Theorem REF .",
"In particular, it implies the following important conclusion for the collision of gravitational waves: A solution $\\mathcal {E}(x,y)$ of the Goursat problem for (REF ) fulfills () iff the boundary data are such that $\\lim _{x \\downarrow 0} x^\\alpha |\\mathcal {E}_{0x}(x)|$ and $\\lim _{y \\downarrow 0} y^\\alpha |\\mathcal {E}_{1y}(y)|$ lie in the interval $[1, \\sqrt{2})$ .",
"The assumptions $\\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0$ and $\\text{\\upshape Re\\,}\\mathcal {E}_1(y) > 0$ in (REF ) are natural because in the context of gravitational waves the real part of the Ernst potential is automatically strictly positive.",
"The assumption $\\mathcal {E}_0(0) = \\mathcal {E}_1(0)$ in (REF ) expresses the compatibility of the boundary values at the origin.",
"If $\\mathcal {E}$ is a solution of (REF ), then so is $a\\mathcal {E} + ib$ for any choice of the real constants $a$ and $b$ .",
"Thus, since $\\mathcal {E}(0,0) \\ne 0$ as a consequence of the assumption $\\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0$ , there is no loss of generality in assuming that $\\mathcal {E}(0,0) = 1$ .",
"The analysis of a boundary or initial-boundary value problem for an integrable equation is usually complicated by the fact that not all boundary values are known for a well-posed problem cf.",
"[8].",
"This issue does not arise for (REF ) which is a Goursat problem.",
"This means that the presented solution is as effective as the solution of the initial value problem via the inverse scattering transform for an equation such as the KdV or nonlinear Schrödinger equation.",
"Despite its great importance in the context of gravitational waves, there are few results in the literature on the Goursat problem (REF ).",
"In fact, rather than solving a given initial or boundary value problem, most of the literature on the Ernst equation has dealt with the generation of new exact solutions via solution-generating techniques, cf.",
"[11], [16], [15].",
"Solving an initial or boundary value problem is much more difficult than generating particular solutions.",
"In fact, even if a large class of particular solutions are known, the problem of determining which of these solutions satisfies the given initial and boundary conditions remains a highly nonlinear problem, often as difficult as the original problem.",
"As noted by Griffiths [11], “What would be much more significant would be to find a practical way to determine the solution in the interaction region for an arbitrary set of initial conditions.” Regarding the problem of determining the interaction of two colliding plane waves from arbitrary initial conditions, important first progress was made in a series of papers by Hauser and Ernst, see [12].",
"Their approach is based on the so-called Kinnersley $H$ -potential [14] rather than on equation (REF ).",
"In terms of the $2\\times 2$ -matrix valued Kinnersley potential $H(r,s)$ , the problem of determining the spacetime metric in the interaction region can be formulated as a Goursat problem in the triangular region $\\Delta = \\lbrace (r,s) \\in {R}^2 \\, | -1 \\le r < s \\le 1\\rbrace $ for the equation (see Eq.",
"(2.10) in [12]) $2(s-r) H_{rs} \\Omega - [H_r \\Omega , H_s\\Omega ] = 0, \\qquad \\Omega = \\begin{pmatrix} 0 & i \\\\ -i & 0 \\end{pmatrix}.$ Hauser and Ernst were able to relate the solution of this problem to the solution of a homogeneous Hilbert problem.",
"The analysis of [12] relies, at least implicitly, on the fact that equation (REF ) admits the Lax pair (see Eq.",
"(3.1) in [12]) $P_r = \\frac{H_r \\Omega }{2(\\tau - r)}P, \\qquad P_s= \\frac{H_s \\Omega }{2(\\tau - s)}P,$ where $P(r,s,\\tau )$ is a $2 \\times 2$ -matrix valued eigenfunction and $\\tau \\in is the spectral parameter.$ More recently, the authors of [9] have addressed the Goursat problem in the triangle $\\Delta $ for the equation $2(s-r)g_{rs} + g_r - g_s + (r-s)(g_rg^{-1}g_s + g_sg^{-1}g_r) = 0,$ where $g(r,s)$ is a $2~\\times 2$ -matrix valued function.",
"Equation (REF ) is related to the hyperbolic Ernst equation (REF ) as follows: Letting $g(r,s) = \\frac{s-r}{2\\text{\\upshape Re\\,}\\mathcal {E}} \\begin{pmatrix} |\\mathcal {E}|^2 & \\text{\\upshape Im\\,}\\mathcal {E} \\\\\\text{\\upshape Im\\,}\\mathcal {E} & 1 \\end{pmatrix},$ equation (REF ) reduces to the scalar equation $(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(\\mathcal {E}_{rs} - \\frac{\\mathcal {E}_r - \\mathcal {E}_s}{2(r-s)}\\right) = \\mathcal {E}_r \\mathcal {E}_s,$ which is related to equation (REF ) by the change of variables $y = (r+1)/2$ and $x = (1-s)/2$ .",
"Through a clever series of steps, the authors of [9] express the solution of (REF ) in terms of the solution of a RH problem.",
"Our approach here is inspired by the recent works [17] and [19] on the elliptic Ernst equation.",
"We have also drawn some inspiration from [9] and [12], although in contrast to these references, we analyze equation (REF ).",
"Two further differences between the present work and [9] are: It is assumed in [9] that the solution is $C^2$ on all of $\\Delta $ up to and including the non-diagonal part of the boundary.",
"However, as explained above (see equation ()), the Ernst potentials relevant for gravitational waves have boundary values $\\mathcal {E}(x,0)$ and $\\mathcal {E}(0,y)$ whose derivatives are not continuous (actually unbounded) at the origin.",
"Here we allow for such singularities in $\\mathcal {E}_x(x,0)$ and $\\mathcal {E}_y(0,y)$ .",
"These singularities transfer, in general, into singularities of the associated eigenfunction solutions of the Lax pair, and the rigorous treatment of all these singularities was one of the main challenges of the present work.",
"The normalization condition for the RH problem derived in [9] involves the solution itself; hence the solution representation is not effective.",
"We circumvent this problem by defining the eigenfunctions on a Riemann surface $\\mathcal {S}_{(x,y)}$ with branch points at $x$ and $1-y$ .",
"The Riemann surface $\\mathcal {S}_{(x,y)}$ is dynamic in the sense that it depends on the spatial point $(x,y)$ .",
"This dependence on $(x,y)$ creates some technical difficulties which we handle by introducing a map $F_{(x,y)}$ from $\\mathcal {S}_{(x,y)}$ to the standard Riemann sphere which takes the two moving branch points to the two fixed points $-1$ and 1.",
"After transferring the RH problem to the Riemann sphere in this way, we can analyze it using techniques from the theory of singular integral equations.",
"In the traditional implementation of the inverse scattering transform, the two equations in the Lax pair are treated separately—usually the spatial part of the Lax pair is first used to define the scattering data and the temporal part is then used to determine the time evolution.",
"The Goursat problem (REF ) does not fit this pattern, so a different approach is required; this is one reason why the solution of the problem (REF ) has proved elusive.",
"Actually, the approach in [9] was one of the first implementations of a general framework for the analysis of boundary value problems for integrable PDEs now known as the unified transform or Fokas method [7].",
"In this method the two equations in the Lax pair are analyzed simultaneously rather than separately.",
"The ideas of this method play an important role also in this paper.",
"It is an interesting open problem to investigate whether existence and uniqueness results for (REF ) can be obtained also via functional analytic techniques.",
"As was explained already in Chapter IV of Goursat's original treatise [10], existence and uniqueness results for Goursat problems for linear hyperbolic PDEs can be established by means of successive approximations and Riemann's method (see also [3]).",
"It is possible to extend these ideas to prove existence theorems also for certain nonlinear Goursat problems [24], [27].",
"However, even in the linear case, these theorems tend to assume that $\\lbrace \\mathcal {E}, \\mathcal {E}_x, \\mathcal {E}_y, \\mathcal {E}_{xy}\\rbrace $ are all continuous [24], [10], [3], or at least that the boundary values are Lipschitz [27].",
"These conditions fail for the assumptions (REF ) relevant for gravitational waves.",
"Let us finally point out that many exact solutions describing colliding plane gravitational waves are known (see e.g.",
"[23], [2], [26], [5]) and that there is a growing literature on colliding gravitational waves which are not necessarily plane (see e.g.",
"[20])."
],
[
"Organization of the paper",
"We begin by establishing some notation in Section .",
"Our main results (Theorems REF -REF ) are stated in Section .",
"In Section , as preparation for the general case, we analyze the special case in which the colliding waves have collinear polarization.",
"In this case, the problem reduces to a problem for the so-called Euler-Darboux equation.",
"We prove a theorem for this equation (Theorem REF ) which is analogous to Theorem REF -REF .",
"In Section , we discuss the Lax pair of equation (REF ) and analyze the spectral data as well as the uniqueness of the solution of the corresponding RH problem.",
"In Section , we present the proofs of Theorem REF -REF .",
"Section contains two short examples and the appendix contains some background on the origin of the Goursat problem (REF ) in the context of colliding gravitational waves."
],
[
"Organization of the paper",
"We begin by establishing some notation in Section .",
"Our main results (Theorems REF -REF ) are stated in Section .",
"In Section , as preparation for the general case, we analyze the special case in which the colliding waves have collinear polarization.",
"In this case, the problem reduces to a problem for the so-called Euler-Darboux equation.",
"We prove a theorem for this equation (Theorem REF ) which is analogous to Theorem REF -REF .",
"In Section , we discuss the Lax pair of equation (REF ) and analyze the spectral data as well as the uniqueness of the solution of the corresponding RH problem.",
"In Section , we present the proofs of Theorem REF -REF .",
"Section contains two short examples and the appendix contains some background on the origin of the Goursat problem (REF ) in the context of colliding gravitational waves."
],
[
"Notation",
"We introduce notation that will be used throughout the paper.",
"We let $D$ denote the triangular region defined in (REF ) and displayed in Figure REF .",
"Given $\\delta > 0$ , we let $D_\\delta $ denote the slightly smaller triangular region obtained by removing a narrow strip along the diagonal of $D$ as follows (see Figure ): $D_\\delta = \\lbrace (x,y) \\in D \\, | \\, x+y < 1-\\delta \\rbrace ,$ The interiors of $D$ and $D_\\delta $ will be denoted by $\\operatorname{int}D$ and $\\operatorname{int}D_\\delta $ , respectively.",
"The Riemann sphere will be denoted by $\\hat{ = \\lbrace \\infty \\rbrace .",
"}\\begin{figure}\\bigskip \\begin{center}\\begin{overpic}[width=.4]{Ddelta.pdf}\\put (25,28){ D_\\delta }\\put (102.5,9.7){\\small x}\\put (10,100){\\small y}\\put (85,4.5){\\small 1}\\put (65,4){\\small 1-\\delta }\\put (6,82){\\small 1}\\put (-4.5,72){\\small 1-\\delta }\\end{overpic}\\begin{figuretext}The triangle D_\\delta defined in (\\ref {Ddeltadef}).\\end{figuretext}\\end{center}\\end{figure}$"
],
[
"The Riemann surface $\\mathcal {S}_{(x,y)}$",
"For each $(x,y) \\in D$ , we let $\\mathcal {S}_{(x,y)}$ denote the Riemann surface consisting of all points $P := (\\lambda , k) \\in 2$ such that $\\lambda ^2 = \\frac{k - (1-y)}{k - x}$ together with two points $\\infty ^+=(1,\\infty )$ and $\\infty ^-=(-1,\\infty )$ at infinity and a branch point $x \\equiv (\\infty ,x)$ which make the surface compact.",
"The surface $\\mathcal {S}_{(x,y)}$ is two-sheeted in the sense that to each $k \\in \\hat{ \\setminus \\lbrace x, 1-y\\rbrace , there correspond exactly two values of \\lambda .",
"We introduce a branch cut in the complex k-plane from x to 1-y and, for k \\in \\hat{\\setminus [x,1-y], we let k^+ and k^- denote the corresponding points on the upper and lower sheet of \\mathcal {S}_{(x,y)}, respectively.",
"By definition, the upper (lower) sheet is characterized by \\lambda \\rightarrow 1 (\\lambda \\rightarrow -1) as k \\rightarrow \\infty .",
"Writing \\lambda (x,y,P) for the value of \\lambda corresponding to the point P \\in \\mathcal {S}_{(x,y)}, we have{\\begin{@align}{1}{-1}\\lambda (x,y,k^+) = \\sqrt{\\frac{k - (1-y)}{k - x}} = -\\lambda (x,y,k^-), \\qquad k \\in \\hat{ \\setminus [x, 1-y],}\\end{@align}where the sign of the square root in (\\ref {lambdasqrt}) is chosen so that \\lambda (x,y,k^+) has positive real part.",
"}\\begin{figure}\\bigskip \\bigskip \\begin{center}\\begin{overpic}[width=.9]{kzmap.pdf}\\put (18.9,17.8){\\small x}\\put (32.4,17.2){\\small 1-y}\\put (19,7){\\small x}\\put (32.4,6){\\small 1-y}\\put (71.5,9.6){\\small -1}\\put (87.6,9.6){\\small 1}\\put (100.8,11.7){\\small \\text{\\upshape Re\\,}z}\\put (79,25){\\small \\text{\\upshape Im\\,}z}\\put (80,-4){\\small \\hat{}\\put (24,-4){\\small \\mathcal {S}_{(x,y)}}\\put (53.5,18.5){\\small F_{(x,y)}}}\\bigskip \\bigskip \\medskip \\begin{figuretext}The map F_{(x,y)}:k \\mapsto z = \\frac{1+ \\lambda }{1 - \\lambda } is a biholomorphism from the two-sheeted Riemann surface \\mathcal {S}_{(x,y)} to the Riemann sphere \\hat{ = \\lbrace \\infty \\rbrace .",
"It maps the branch points x and 1-y to z = -1 and z = 1, respectively, and the upper (lower) sheet to the outside (inside) of the unit circle.",
"}\\end{figuretext}\\end{overpic}\\end{center}\\subsection {The map F_{(x,y)}}For each point (x,y) \\in D, \\mathcal {S}_{(x,y)} is a compact genus zero Riemann surface with branch points at k = x and k = 1-y.",
"In order to fix the locations of these branch points, we introduce a new variable z byz = \\frac{1+ \\lambda }{1 - \\lambda },and let F_{(x,y)}:\\mathcal {S}_{(x,y)} \\rightarrow \\hat{ be the map that sends P ~to z, i.e.,F_{(x,y)}(P) = \\frac{1+ \\lambda (x,y,P)}{1 - \\lambda (x,y,P)}, \\qquad P \\in \\mathcal {S}_{(x,y)}.For each (x,y) \\in D, F_{(x,y)} is a biholomorphism (i.e.",
"a bijective holomorphic function whose inverse is also holomorphic) from \\mathcal {S}_{(x,y)} to \\hat{ which maps the two branch points x and 1-y to z = -1 and z = 1, respectively, see Figure \\ref {kzmap.pdf}.",
"}}\\end{figure}\\subsection {The contours \\Sigma and \\Gamma }For each (x,y) \\in D, we let \\Sigma _0 \\equiv \\Sigma _0(x,y) denote the shortest path from 0^+ to 0^- in \\mathcal {S}_{(x,y)}, and we let \\Sigma _1 \\equiv \\Sigma _1(x,y) denote the shortest path from 1^- to 1^+ in \\mathcal {S}_{(x,y)}.",
"More precisely,{\\begin{@align}{1}{-1}\\Sigma _0 = [0,x]^+ \\cup [x,0]^-, \\qquad \\Sigma _1 = [1,1-y]^- \\cup [1-y,1]^+ ,\\end{@align}}where, for a subset S of the complex plane, we use the notation S^\\pm = \\lbrace k^\\pm \\in \\mathcal {S}_{(x,y)}\\,| \\, k \\in S\\rbrace to denote the sets in the upper and lower sheets of \\mathcal {S}_{(x,y)} which project onto S, see Figure \\ref {Sigma01.pdf}.We write \\Sigma := \\Sigma _0 \\cup \\Sigma _1 for the union of \\Sigma _0 and \\Sigma _1.",
"}}Given $ (x,y) D$, we let $ 0 0(x,y)$ and $ 1 1(x,y)$ denote two clockwise nonintersecting smooth contours in the complex $ z$-plane which encircle the real intervals\\begin{subequations}{\\begin{@align}{1}{-1}F_{(x,y)}(\\Sigma _0) = \\bigg [-\\frac{\\sqrt{1-y}+ \\sqrt{x}}{\\sqrt{1-y} - \\sqrt{x}}, -\\frac{\\sqrt{1-y}- \\sqrt{x}}{\\sqrt{1-y}+ \\sqrt{x}}\\bigg ]\\end{@align}}and{\\begin{@align}{1}{-1}F_{(x,y)}(\\Sigma _1) =\\bigg [\\frac{\\sqrt{1-x} - \\sqrt{y}}{\\sqrt{1-x} + \\sqrt{y}}, \\frac{\\sqrt{1-x} + \\sqrt{y}}{\\sqrt{1-x} - \\sqrt{y}}\\bigg ],\\end{@align}}\\end{subequations}respectively, but which do not encircle zero, see Figure \\ref {Gamma.pdf}.",
"We let $ (x,y)$ denote the union $ := 0 1$ of $ 0$ and $ 1$.$ Figure: NO_CAPTIONFigure: NO_CAPTION"
],
[
"Nontangential limits and function spaces",
"Let $\\Gamma \\subset be a piecewise smooth contour.For an analytic function $ m: , we denote the nontangential boundary values of $m$ from the left and right sides of $\\Gamma $ by $m_+$ and $m_-$ respectively.",
"Given a subset $S \\subset {R}^n$ , $n \\ge 1$ , we let $C(S)$ denote the space of complex-valued continuous functions on $S$ .",
"If $S$ is open, we define $C^n(S)$ as the space of complex-valued functions on $S$ which are $n$ times continuously differentiable, i.e., all partial derivatives of order $\\le n$ exist and are continuous.",
"By $\\mathcal {B}(X,Y)$ , we denote the space of bounded linear maps from a Banach space $X$ to another Banach space $Y$ equipped with the standard operator norm; if $X = Y$ , we write $\\mathcal {B}(X) \\equiv \\mathcal {B}(X,X)$ ."
],
[
"Main results",
"We adopt the following notion of a $C^n$ -solution of the Goursat problem (REF ).",
"Definition 3.1 Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ , be complex-valued functions.",
"A function $\\mathcal {E}:D \\rightarrow {R}$ is called a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ if ${\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D) \\cap C^n(\\operatorname{int}(D)),\\\\\\text{$\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (\\ref {ernst}) in $\\operatorname{int}(D)$,}\\\\\\text{$x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D)$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1)$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1)$,}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D$.}",
"\\end{array}\\right.",
"}$ We next state the four main results of the paper (Theorem REF -REF ), which all address different aspects of the Goursat problem (REF ).",
"In the formulation of Theorem REF -REF , it is assumed that $n \\ge 2$ is an integer and that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ , are two complex-valued functions satisfying the assumptions in (REF ) for a fixed $\\alpha \\in [0,1)$ .",
"The first theorem provides a representation formula for the solution in terms of the given boundary data via a RH problem.",
"Theorem 1 (Representation formula) If $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ , then this solution can be expressed in terms of the boundary values $\\mathcal {E}_0(x)$ and $\\mathcal {E}_1(y)$ by $\\mathcal {E}(x,y) = \\frac{1 + (m(x,y,0))_{11} - (m(x,y,0))_{21}}{1 + (m(x,y,0))_{11} + (m(x,y,0))_{21}},$ where $m(x,y,z)$ is the unique solution of the $2 \\times 2$ -matrix RH problem ${\\left\\lbrace \\begin{array}{ll}\\text{$m(x, y, \\cdot )$ is analytic in $\\Gamma $},\\\\\\text{$m_+(x, y, z) = m_-(x, y, z) v(x, y, z)$ for all $z \\in \\Gamma $},\\\\\\text{$m(x,y,z) = I + O(z^{-1})$ as $z \\rightarrow \\infty $},\\end{array}\\right.",
"}$ and the jump matrix $v(x,y,z)$ is defined as follows: Let $\\Phi _0$ and $\\Phi _1$ be the unique solutions of the linear Volterra integral equations $\\Phi _0(x,k^\\pm ) = I + \\int _0^x (\\mathsf {U}_0\\Phi _0)(x^{\\prime }, k^\\pm ) dx^{\\prime }, \\qquad x \\in [0, 1), \\ k \\in [0,1],\\\\ \\Phi _1(y,k^\\pm ) = I + \\int _0^y (\\mathsf {V}_1\\Phi _1)(y^{\\prime }, k^\\pm ) dy^{\\prime }, \\qquad y \\in [0, 1), \\ k \\in [0,1], $ where $\\mathsf {U}_0$ and $\\mathsf {V}_1$ are defined by $& \\mathsf {U}_0(x,k^\\pm ) = \\frac{1}{2 \\text{\\upshape Re\\,}\\mathcal {E}_0(x)} \\begin{pmatrix} \\overline{\\mathcal {E}_{0x}(x)} & \\lambda (x,0,k^\\pm ) \\overline{\\mathcal {E}_{0x}(x)} \\\\\\lambda (x,0,k^\\pm ) \\mathcal {E}_{0x}(x) & \\mathcal {E}_{0x}(x) \\end{pmatrix},\\\\& \\mathsf {V}_1(y,k^\\pm ) = \\frac{1}{2 \\text{\\upshape Re\\,}\\mathcal {E}_0(y)} \\begin{pmatrix} \\overline{\\mathcal {E}_{1y}(y)} & \\frac{1}{\\lambda (0,y,k^\\pm )} \\overline{\\mathcal {E}_{1y}(y)} \\\\\\frac{1}{\\lambda (0,y,k^\\pm )} \\mathcal {E}_{1y}(y) & \\mathcal {E}_{1y}(y)\\end{pmatrix}.$ Then $v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad (x,y) \\in D.$ Theorem REF establishes uniqueness of the $C^n$ -solution.",
"Theorem 2 (Uniqueness) The $C^n$ -solution $\\mathcal {E}(x,y)$ of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ is unique, if it exists.",
"In fact, the value of $\\mathcal {E}$ at a point $(x,y) \\in D$ is uniquely determined by the boundary values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0 \\le x^{\\prime } \\le x$ and $0 \\le y^{\\prime } \\le y$ .",
"Theorem REF establishes existence of a $C^n$ -solution—in the collinear case, for general data; otherwise under a small-norm assumption.",
"Theorem 3 (Existence and regularity) For each $\\delta > 0$ , the following three existence and regularity results hold: Suppose the $2 \\times 2$ -matrix RH problem (REF ) has a solution for all $(x,y)\\in D_\\delta $ .",
"Then there exists a $C^n$ -solution of the Goursat problem for (REF ) in $D_\\delta $ with data $\\lbrace \\mathcal {E}_0|_{[0,1-\\delta )},\\mathcal {E}_1|_{[0,1-\\delta )} \\rbrace $ .",
"Whenever the $L^1$ -norms of $\\mathcal {E}_{0x}/(\\text{\\upshape Re\\,}\\mathcal {E}_0)$ and $\\mathcal {E}_{0y}/(\\text{\\upshape Re\\,}\\mathcal {E}_1)$ on $[0,1-\\delta )$ are sufficiently small, there exists a $C^n$ -solution of the Goursat problem for (REF ) in $D_\\delta $ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ .",
"If $\\mathcal {E}_0, \\mathcal {E}_1 >0$ on $[0,1-\\delta )$ , i.e., if the incoming waves are collinearly polarized, then there exists a $C^n$ -solution of the Goursat problem for (REF ) in $D_\\delta $ with data $\\lbrace \\mathcal {E}_0|_{[0,1-\\delta )},\\mathcal {E}_1|_{[0,1-\\delta )} \\rbrace $ .",
"Remark 3.2 Part $(a)$ of Theorem REF shows that the solution $\\mathcal {E}(x,y)$ exists and has the same regularity as the given data as long as the associated RH problem has a solution.",
"By taking $\\delta > 0$ arbitrarily small, we see that the same statement holds also in all of $D$ .",
"Theorem REF establishes explicit formulas for the singular behavior of the solution near the boundary in terms of the given data.",
"Theorem 4 (Boundary behavior) Let $\\alpha \\in (0,1)$ and $n \\ge 2$ be an integer.",
"Let $\\mathcal {E}(x,y)$ be a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ .",
"Let $m_1, m_2 \\in denote the values of these functions at the origin, i.e.,{\\begin{@align}{1}{-1}m_1 = \\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_{0x}(x), \\qquad m_2 = \\lim _{y \\downarrow 0} y^\\alpha \\mathcal {E}_{1y}(y).\\end{@align}}Then the solution~ $ E(x,y)$ has the following behavior near the boundary:\\begin{subequations}{\\begin{@align}{1}{-1}& \\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y) = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} && \\text{for each $y \\in [0,1)$},\\\\ & \\lim _{y \\downarrow 0} y^\\alpha \\mathcal {E}_y(x,y) = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_0(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} && \\text{for each $x \\in [0,1)$}.\\end{@align}}\\end{subequations}In particular,{\\begin{@align*}{1}{-1}& \\lim _{x \\downarrow 0} x^\\alpha |\\mathcal {E}_x(x,y)| = |m_1| \\frac{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} && \\text{for each $y \\in [0,1)$},\\\\& \\lim _{y \\downarrow 0} y^\\alpha |\\mathcal {E}_y(x,y)| = |m_2| \\frac{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} && \\text{for each $x \\in [0,1)$}.\\end{@align*}}$ Remark 3.3 Theorem REF yields the following important result for the collision of plane gravitational waves: A solution $\\mathcal {E}(x,y)$ of the Goursat problem for (REF ) fulfills the gravitational wave boundary conditions () if and only if the boundary data $\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ and $\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ are such that $\\lim _{x \\downarrow 0} x^{\\alpha } |\\mathcal {E}_{0x}(x)|$ and $\\lim _{y \\downarrow 0} y^{\\alpha } |\\mathcal {E}_{1y}(y)|$ belong to the real interval $[1, \\sqrt{2})$ .",
"In particular, the behavior of $\\mathcal {E}_{x}(x,0)$ and $\\mathcal {E}_{y}(0,y)$ at the origin fully determines whether the functions $\\mathcal {E}_x(x,y)$ and $\\mathcal {E}_y(x,y)$ have the appropriate singular behavior near the edges $\\partial D \\cap \\lbrace x =0\\rbrace $ and $\\partial D \\cap \\lbrace y =0\\rbrace $ ."
],
[
"Collinearly polarized waves",
"Before turning to the general case, it is useful to first consider the special case in which the Ernst potential $\\mathcal {E}$ is strictly positive.",
"In the context of gravitational waves, this corresponds to the important situation when the two colliding waves have collinear polarization, see [11]."
],
[
"The Euler-Darboux equation",
"If the Ernst potential $\\mathcal {E}$ is strictly positive, we can write $\\mathcal {E}(x,y) = e^{-V(x,y)}$ , where $V(x,y)$ is a real-valued function.",
"A simple computation then shows that $\\mathcal {E}$ satisfies the Ernst equation (REF ) if and only if $V$ satisfies the linear hyperbolic equation $V_{xy} - \\frac{V_x + V_y}{2(1-x-y)} = 0,$ which is a version of the Euler-Darboux equation [22].",
"Since (REF ) is a linear equation, we can, without loss of generality, assume that $V$ is real-valued and that $V(0,0) =0$ .",
"Remark 4.1 (Linear limit) In addition to being a reformulation of (REF ) in the special case of collinearly polarized waves, equation (REF ) can also be viewed as the linearized version of (REF ).",
"Indeed, substituting $\\mathcal {E}(x,y) = 1 + \\epsilon V(x,y) + O(\\epsilon ^2)$ into (REF ) and considering the terms of $O(\\epsilon )$ , we see that (REF ) is the linear limit of (REF ).",
"The analysis of the Euler-Darboux equation (REF ) presented in this section serves two purposes.",
"First, it is used to prove the part of Theorem REF regarding existence in the collinearly polarized case.",
"Second, it turns out that the more difficult case of noncollinearly polarized solutions can be analyzed following steps which are conceptually very similar to—but technically more difficult than—those involved in the analysis of the collinear case.",
"In fact, the analysis of (REF ) presented in later sections strongly relies on the insight gained in this section.",
"We are interested in the following Goursat problem for (REF ) in the triangle $D$ : Given $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ , find a solution $V(x,y)$ of (REF ) in $D$ such that $V(x,0) = V_0(x)$ for $x \\in [0,1)$ and $V(0,y) = V_1(y)$ for $y \\in [0,1)$ .",
"We introduce a notion of $C^n$ -solution of this problem as follows.",
"Definition 4.2 Let $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ , be real-valued functions and $\\alpha \\in [0,1)$ .",
"We define a function $V:D \\rightarrow {R}$ to be a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace V_0, V_1\\rbrace $ if ${\\left\\lbrace \\begin{array}{ll}V \\in C(D) \\cap C^n(\\operatorname{int}(D)),\\\\\\text{$V(x,y)$ satisfies the Euler-Darboux equation (\\ref {linearernst}) in $\\operatorname{int}(D)$,}\\\\\\text{$x^\\alpha V_x, y^\\alpha V_y, x^\\alpha y^\\alpha V_{xy} \\in C(D)$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$V(x,0) = V_0(x)$ for $x \\in [0,1)$,}\\\\\\text{$V(0,y) = V_1(y)$ for $y \\in [0,1)$.}\\end{array}\\right.",
"}$ The following theorem establishes the unique existence of a solution of the Goursat problem for (REF ) in $D$ .",
"It also provides a representation for the solution in terms of the boundary data and characterizes the singular behavior near the boundary.",
"Theorem 5 (Solution of the Euler-Darboux equation in a triangle) Let $n \\ge 2$ be an integer.",
"Let $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ , be two real-valued functions such that ${\\left\\lbrace \\begin{array}{ll}V_0, V_1 \\in C([0,1)) \\cap C^n((0,1)),\\\\\\text{$x^\\alpha V_{0x}, y^\\alpha V_{1y} \\in C([0,1))$ for some $\\alpha \\in [0,1)$,}\\\\V_0(0) = V_1(0) = 0.\\end{array}\\right.",
"}$ Then there exists a unique $C^n$ -solution $V(x,y)$ of the Goursat problem for (REF ) in $D$ with data $\\lbrace V_0, V_1\\rbrace $ .",
"Moreover, this solution is given in terms of the boundary values $V_0(x)$ and $V_1(y)$ by $V(x,y) = -\\frac{1}{2}m(x,y,0), \\qquad (x,y) \\in D,$ where $m(x,y,z)$ is the unique solution of the following scalar RH problem: ${\\left\\lbrace \\begin{array}{ll}\\text{$m(x, y, \\cdot )$ is analytic in $\\Gamma $},\\\\\\text{$m_+(x, y, z) = m_-(x, y, z) + v(x, y, z)$ for all $z \\in \\Gamma $},\\\\\\text{$m(x,y,z) = O(z^{-1})$ as $z \\rightarrow \\infty $},\\end{array}\\right.",
"}$ and the jump $v(x,y,z)$ is defined by $v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in \\Gamma _1,\\end{array}\\right.",
"}$ with $\\Phi _0(x,k^\\pm ) = \\int _0^x \\lambda (x^{\\prime },0,k^\\pm ) V_{0x}(x^{\\prime }) dx^{\\prime }, \\qquad x \\in [0, 1), \\ k \\in [0,1],\\\\\\Phi _1(y,k^\\pm ) = \\int _0^y \\frac{1}{\\lambda (0,y^{\\prime },k^\\pm )} V_{1y}(y^{\\prime }) dy^{\\prime }, \\qquad y \\in [0, 1), \\ k \\in [0,1].$ Furthermore, if $\\alpha \\in (0,1)$ is such that the functions $x^\\alpha V_{0x}$ and $y^\\alpha V_{1y}$ are continuous on $[0, 1)$ and $m_1 := \\lim _{x \\downarrow 0} x^\\alpha V_{0x}(x), \\qquad m_2 := \\lim _{y \\downarrow 0} y^\\alpha V_{1y}(y),$ then the solution $V(x,y)$ has the following behavior near the boundary: $& \\lim _{x \\downarrow 0} x^\\alpha V_x(x,y) = \\frac{m_1}{\\sqrt{1-y}} && \\text{for each $y \\in [0,1)$},\\\\& \\lim _{y \\downarrow 0} y^\\alpha V_y(x,y) = \\frac{m_2}{\\sqrt{1-x}} && \\text{for each $x \\in [0,1)$}.$ Remark 4.3 The scalar RH problem (REF ) has the unique solution $m(x,y,z) = \\frac{1}{2\\pi i} \\int _{\\Gamma } \\frac{v(x,y,z^{\\prime })}{z^{\\prime }-z} dz^{\\prime }.$ Hence the solution $V(x,y)$ can be expressed in terms of $v$ by $V(x,y) = -\\frac{1}{4\\pi i} \\int _{\\Gamma } \\frac{v(x,y,z)}{z} dz.$ Collapsing the contour $\\Gamma $ in (REF ) onto the intervals in () and changing variables from $z$ to $k$ leads to the following representation for the solution in terms of Abel type integrals: $\\nonumber V(x,y) = &\\; \\frac{1}{\\pi } \\int _0^x \\frac{\\sqrt{1-k}}{\\sqrt{(1-y-k)(x-k)}} \\bigg (\\int _0^k \\frac{V_{0x}(x^{\\prime })}{\\sqrt{k - x^{\\prime }}}dx^{\\prime }\\bigg ) dk\\\\ & + \\frac{1}{\\pi } \\int _{1-y}^1 \\frac{\\sqrt{k}}{\\sqrt{(k - (1-y))(k-x)}} \\bigg (\\int _0^{1-k} \\frac{V_{1y}(y^{\\prime })}{\\sqrt{1 - y^{\\prime } - k}} dy^{\\prime }\\bigg ) dk$ for $(x,y) \\in D$ .",
"Formulas analogous to (REF ) for equation (REF ) have been derived in [12] and [9].",
"Remark 4.4 The representation (REF ) can be found more directly by formulating a RH problem for $\\Phi $ on $\\mathcal {S}_{(x,y)}$ with jump across $\\Sigma $ .",
"This is essentially the approach adopted in [9].",
"The representation (REF ) has the advantage that it is explicit in its dependence on $V_0$ and $V_1$ , but it has the disadvantage that the integrands are singular at some of the endpoints of the integration intervals.",
"These singularities complicate the verification that $V$ satisfies the appropriate regularity and boundary conditions, especially in the situation relevant for gravitational waves where $V_{0x}$ and $V_{1y}$ are singular at the origin.",
"For the nonlinear equation (REF ), this becomes a serious complication.",
"For this reason, we have formulated the RH problems in Theorem REF and Theorem REF in terms of the contour $\\Gamma $ (which avoids the problematic endpoints of the intervals in ()) rather than in terms of a contour running along the real axis.",
"However, the representation (REF ) allows for applying more classical techniques.",
"This approach is used in [21] to compute an asymptotic expansion of the solution near the diagonal of $D$ .",
"Remark 4.5 In [25] there was derived an alternative integral formula for the solution of the Goursat problem for the Euler–Darboux equation by applying Riemann's classical method [3], [10].",
"Whereas the representation (REF ) relies on Abel integrals, the expression of [25] is given in terms of the Legendre function $P_{-1/2}$ of order $-1/2$ .",
"Remark 4.6 In order to emphasize the analogy between (REF ) and its linearized version (REF ), we will use the same symbols in this section for the various linearized quantities as we use elsewhere for the corresponding quantities of the nonlinear problem.",
"Many quantities which are matrices in the noncollinear case reduce to scalar quantities in the collinear case.",
"For example, in other sections $\\Phi $ will denote a $2 \\times 2$ -matrix valued eigenfunction, but in this section $\\Phi $ is a scalar-valued eigenfunction."
],
[
"Proof of Theorem ",
"The proof of Theorem REF is divided into three parts.",
"In the first part, we prove uniqueness and establish the solution representation formula (REF ).",
"In the second part, we prove existence.",
"In the third part, we consider the boundary behavior."
],
[
"Proof of uniqueness and of (",
"Let $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ be real-valued functions satisfying (REF ) for some $n \\ge 2$ and $\\alpha \\in [0, 1)$ .",
"Suppose that $V(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace V_0, V_1\\rbrace $ .",
"We will show that $V(x,y)$ can be expressed in terms of $V_0$ and $V_1$ by (REF ).",
"Equation (REF ) admits the Lax pair ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y,k) = \\lambda V_x(x,y), \\\\\\Phi _y(x,y,k) = \\frac{1}{\\lambda } V_y(x,y),\\end{array}\\right.",
"}$ where $\\Phi (x,y,k)$ is an eigenfunction, $\\lambda = \\lambda (x,y,k)$ is defined by (REF ), and $k$ is a complex spectral parameter.",
"Indeed, using the relations $\\lambda _x = \\frac{\\lambda }{2(k-x)}= \\frac{(1- \\lambda ^2)\\lambda }{2(1-x-y)}, \\qquad \\lambda _y = \\frac{1}{2(k-x)\\lambda }= \\frac{(1- \\lambda ^2)}{2(1-x-y)\\lambda },$ it is straightforward to check that the compatibility condition $\\Phi _{xy} = \\Phi _{yx}$ of (REF ) is equivalent to (REF ).",
"The occurrence of $\\lambda $ in (REF ) implies that the spectral parameter is naturally considered as an element of the Riemann surface $\\mathcal {S}_{(x,y)}$ .",
"Thus, we will henceforth view $\\Phi (x,y,\\cdot )$ as a function defined on $\\mathcal {S}_{(x,y)}$ and write $\\Phi (x,y,P)$ for the value of $\\Phi $ at $P = (\\lambda , k) \\in \\mathcal {S}_{(x,y)}$ .",
"We emphasize, however, that the partial derivatives $\\Phi _x(x,y,P)$ and $\\lambda _x(x,y,P)$ (resp.",
"$\\Phi _y(x,y,P)$ and $\\lambda _y(x,y,P)$ ) are still computed with $(y,k)$ (resp.",
"$(x,k)$ ) held fixed (and $\\lambda $ allowed to change).",
"The basic idea in what follows is to write (REF ) in the differential form $d\\Phi = W$ , where $W$ denotes the one-form $W = \\lambda V_x dx + \\frac{1}{\\lambda } V_y dy$ , and then define a solution $\\Phi $ of (REF ) by $\\Phi (x,y, k^\\pm ) = \\int _{(0,0)}^{(x,y)} W(x^{\\prime },y^{\\prime },k^\\pm ), \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}Since the one-form W is closed, the integral on the right-hand side is independent of path.",
"However, since W in general is singular on the boundary of D, we need to be more careful when defining \\Phi .",
"We therefore choose to define \\Phi using the specific contour which consists of the horizontal segment from (0,0) to (x,0) followed by the vertical segment from (x,0) to (x,y) (see the left half of Figure \\ref {Dcontoursfig}), that is, we define{\\begin{@align}{1}{-1}\\nonumber &\\Phi (x,y, k^\\pm ) = \\int _0^x \\lambda (x^{\\prime },0,k^\\pm ) V_x(x^{\\prime },0) dx^{\\prime } + \\int _0^y \\lambda (x,y^{\\prime },k^\\pm )^{-1} V_y(x,y^{\\prime }) dy^{\\prime },\\\\ &\\hspace{227.62204pt} (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}Since x^\\alpha V_x, y^\\alpha V_y \\in C(D), the integrals on the right-hand side of (\\ref {linearPhidef}) are well-defined.",
"The next lemma establishes several properties of \\Phi .",
"}$ Figure: NO_CAPTIONLemma 4.7 (Solution of Lax pair equations) The function $\\Phi (x,y,P)$ defined in (REF ) has the following properties: $\\Phi $ can be alternatively expressed using the contour consisting of the vertical segment from $(0,0)$ to $(0,y)$ followed by the horizontal segment from $(0,y)$ to $(x,y)$ (see the right half of Figure REF ): $\\nonumber &\\Phi (x,y, k^\\pm ) = \\int _0^y \\lambda (0,y^{\\prime },k^\\pm )^{-1} V_y(0,y^{\\prime }) dy^{\\prime } + \\int _0^x \\lambda (x^{\\prime },y,k^\\pm ) V_x(x^{\\prime },y) dx^{\\prime },\\\\&\\hspace{227.62204pt} (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}$ For each $k \\in \\hat{ \\setminus [0,1], the function (x,y) \\mapsto \\Phi (x,y,k^+) is continuous on D and is C^n on~ \\operatorname{int}D.}\\item For each $ k [0,1]$, the functions$$(x,y) \\mapsto x^\\alpha \\Phi _x(x,y,k^+), \\quad (x,y) \\mapsto y^\\alpha \\Phi _y(x,y,k^+), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha \\Phi _{xy}(x,y,k^+),$$are continuous on $ D$.$ $\\Phi $ obeys the symmetries ${\\left\\lbrace \\begin{array}{ll} \\Phi (x,y,k^+) = -\\Phi (x, y, k^-),\\\\\\Phi (x,y,k^\\pm ) = \\overline{\\Phi (x, y,\\bar{k}^\\pm )},\\end{array}\\right.}",
"\\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}$ For each $(x,y) \\in D$ , $\\Phi (x,y,P)$ extends continuously to an analytic function of $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma $ , where $\\Sigma = \\Sigma _0 \\cup \\Sigma _1$ is the contour defined in ().",
"$\\Phi (x,y,\\infty ^+) = V(x,y)$ for $(x,y) \\in D$ .",
"Let $(x,y) \\in D$ .",
"In order to prove $(a)$ , we need to show that the expression $\\nonumber & \\int _0^y \\big [\\lambda (0,y^{\\prime },k^\\pm )^{-1} V_y(0,y^{\\prime }) - \\lambda (x,y^{\\prime },k^\\pm )^{-1} V_y(x,y^{\\prime })\\big ] dy^{\\prime }\\\\ & +\\int _0^x \\big [\\lambda (x^{\\prime },y,k^\\pm ) V_x(x^{\\prime },y) - \\lambda (x^{\\prime },0,k^\\pm ) V_x(x^{\\prime },0)\\big ] dx^{\\prime }$ vanishes.",
"Since $x^\\alpha V_x, y^\\alpha V_y, x^\\alpha y^\\alpha V_{xy} \\in C(D)$ , the function $\\lambda (\\cdot ,y^{\\prime },k^\\pm )^{-1} V_y(\\cdot ,y^{\\prime })$ is absolutely continuous on the compact interval $[0,x]$ for each $y^{\\prime } \\in (0, y]$ .",
"Similarly, the function $\\lambda (x^{\\prime },\\cdot ,k^\\pm ) V_x(x^{\\prime },\\cdot )$ is absolutely continuous on $[0,y]$ for each $x^{\\prime } \\in (0, x]$ .",
"Hence, we can write (REF ) as $\\nonumber & -\\int _0^y \\int _0^x \\frac{\\partial }{\\partial x^{\\prime }} \\big [\\lambda (x^{\\prime },y^{\\prime },k^\\pm )^{-1} V_y(x^{\\prime },y^{\\prime })\\big ] dx^{\\prime } dy^{\\prime }\\\\ & +\\int _0^x \\int _0^y \\frac{\\partial }{\\partial y^{\\prime }} \\big [\\lambda (x^{\\prime },y^{\\prime },k^\\pm ) V_x(x^{\\prime },y^{\\prime })\\big ] dy^{\\prime } dx^{\\prime }.$ Since $V$ is a solution of (REF ), the Lax pair compatibility condition $(\\lambda V_x)_y = (\\lambda ^{-1} V_y)_x$ is satisfied for $(x,y) \\in \\operatorname{int}D$ .",
"The assumption $x^\\alpha V_x, y^\\alpha V_y, x^\\alpha y^\\alpha V_{xy} \\in C(D)$ implies that $V_x, V_y, V_{xy} \\in L^1(D_\\delta )$ for each $\\delta > 0$ .",
"Hence Fubini's theorem implies that the expression in (REF ) vanishes.",
"This proves $(a)$ .",
"Moreover, if $k \\in \\hat{ \\setminus [0,1], then it follows from (\\ref {linearPhidef}) and (\\ref {linearPhidef2}) that \\Phi is a continuous function of (x,y) \\in D and a C^n-function of (x,y) \\in \\operatorname{int}D, which proves (b).",
"}Let $ k [0,1]$.",
"Then{\\begin{@align}{1}{-1}x^\\alpha \\Phi _x(x,y,k^+) = x^\\alpha \\lambda (x,y,k^+) V_x(x,y).\\end{@align}}The assumption $ xVx C(D)$ implies that the right-hand side of (\\ref {linearxalphaPhix}) is a continuous function of $ (x,y) D$.Similarly, we see that $ yy(x,y,k+)$ and$$x^\\alpha y^\\alpha \\Phi _{xy}(x,y,k^+) = \\frac{x^\\alpha y^\\alpha V_x(x,y)}{2(k-x) \\lambda (x,y,k^+)}+ x^\\alpha y^\\alpha \\lambda (x,y,k^+) V_{xy}(x,y)$$are continuous functions of $ (x,y) D$.",
"This proves $ (c)$.$ The symmetries in $(d)$ are a consequence of the symmetries $\\lambda (x,y,k^+) = -\\lambda (x,y,k^-), \\qquad \\lambda (x,y,k^\\pm ) = \\overline{\\lambda (x,y,\\bar{k}^\\pm )},$ and the definition (REF ) of $\\Phi $ .",
"To prove $(e)$ , we note that $\\lambda (x^{\\prime },0,k^+)$ is an analytic function of $k \\in \\hat{ \\setminus [x^{\\prime },1] and \\lambda (x,y^{\\prime },k^+)^{-1} is an analytic function of k \\in \\hat{ \\setminus [x, 1-y^{\\prime }].",
"It follows that \\Phi (x,y, k^+) and \\Phi (x,y, k^-) = -\\Phi (x,y, k^+) are analytic functions of k \\in \\hat{ \\setminus [0,1].Moreover, since\\lambda (x,y, (k+i0)^+) = \\lambda (x,y,(k-i0)^-), \\qquad k \\in (x, 1-y),we have{\\begin{@align*}{1}{-1}\\Phi (x,y, (k + i0)^+)= \\Phi (x,y,(k-i0)^-), \\qquad (x,y) \\in D, \\ k \\in (x, 1-y).\\end{@align*}}This shows that the values of \\Phi on the upper and lower sheets of \\mathcal {S}_{(x,y)} fit together across the branch cut; hence \\Phi extends to an analytic function of P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .",
"This proves (e).",
"}}To prove (f), we note that \\lambda (x,y,\\infty ^+) = 1 for all (x,y) \\in D, which gives{\\begin{@align}{1}{-1}\\Phi (x,y, k^\\pm ) = \\int _0^x V_{0x}(x^{\\prime }) dx^{\\prime } + \\int _0^y V_y(x,y^{\\prime }) dy^{\\prime }.\\end{@align}}Let \\delta > 0.",
"Since V_{0x} \\in L^1((1-\\delta )), V_0 belongs to the Sobolev space W^{1,1}((0,1-\\delta )).",
"Hence V_0 is absolutely continuous on (0, 1-\\delta ).",
"Using that V_0\\in C([0, 1)), we see that V_0 is absolutely continuous on the compact interval [0, 1-\\delta ].",
"Hence,{\\begin{@align}{1}{-1}\\int _0^x V_{0x}(x^{\\prime }) dx^{\\prime } = V(x,0)-V(0,0), \\qquad x \\in [0,1 - \\delta ).\\end{@align}}Moreover, since V_y \\in L^1(D_\\delta ), we have V_y(x, \\cdot ) \\in L^1((0,1-x-\\delta )) for a.e.",
"x \\in [0,1-\\delta ).",
"Hence V(x, \\cdot ) \\in W^{1,1}((0,1-x-\\delta )) for a.e.",
"x \\in [0,1-\\delta ).",
"Since V is also continuous on D, we conclude that V(x, \\cdot ) is absolutely continuous on the compact interval [0,1-x-\\delta ] for a.e.",
"x \\in [0,1-\\delta ).Hence,{\\begin{@align}{1}{-1}\\int _0^y V_y(x,y^{\\prime }) dy^{\\prime } = V(x,y)-V(x,0), \\qquad (x,y) \\in D_\\delta .\\end{@align}}}Hence, substituting (\\ref {linearPhiVx}) and (\\ref {linearPhiVy}) into (\\ref {linearPhiVxVy}) yields$$\\Phi (x,y, k^\\pm ) = V(x,0)-V(0,0) + V(x,y) - V(x,0).$$Since $ V(0,0) = 0$, part $ (f)$ follows.$ Lemma 4.8 For each $(x,y) \\in D$ , $P \\mapsto \\Phi (x,y,P) - \\Phi (x,0, P) \\quad \\text{and} \\quad P \\mapsto \\Phi (x,y,P) - \\Phi (0,y,P)$ extend continuously to analytic functions $\\mathcal {S}_{(x,y)} \\setminus \\Sigma _1 \\rightarrow and $ S(x,y) 0 , respectively.",
"Remark 4.9 The point $P$ in (REF ) belongs to $\\mathcal {S}_{(x,y)}$ whereas the maps $\\Phi (x,0,\\cdot )$ and $\\Phi (0,y,\\cdot )$ are defined on $\\mathcal {S}_{(x,0)}$ and $\\mathcal {S}_{(0,y)}$ , respectively.",
"The interpretation of equation (REF ) therefore deserves a comment of clarification: If $(x,y)$ and $(\\tilde{x}, \\tilde{y})$ are two points in $D$ and $F$ is a map from $\\mathcal {S}_{(x,y)}$ to some space $X$ , then $F$ naturally induces a map $\\tilde{F}$ from $\\mathcal {S}_{(\\tilde{x}, \\tilde{y})} \\setminus \\big ([0,1]^+ \\cup [0,1]^-\\big )$ to $X$ according to $\\tilde{F}(k^\\pm ) = F(k^\\pm )$ for $k \\in \\hat{ \\setminus [0,1].We sometimes, as in (\\ref {linearphiminusphi}) (and also in (\\ref {jumpdef})), identify these two maps and simply write F for \\tilde{F}.",
"}$ Fix $(x,y) \\in D$ .",
"Let $U$ be an open set in $\\mathcal {S}_{(x,y)} \\setminus \\Sigma _0$ .",
"Then $\\Phi (x,y, P) - \\Phi (0,y,P) = \\int _0^x \\lambda (x^{\\prime },y,P) V_x(x^{\\prime },y) dx^{\\prime }, \\qquad P \\in U,$ where the values of $\\Phi (0,y,P)$ and $\\lambda (x^{\\prime },y,P)$ in (REF ) are to be interpreted as in Remark REF .",
"Since $P \\mapsto \\lambda (x^{\\prime },y,P) = \\sqrt{\\frac{k - (1-y)}{k - x^{\\prime }}}$ defines an analytic map $U \\rightarrow for each $ x' [0,x]$, the map (\\ref {linearphiPmap}) is also analytic for $ P U$.",
"This establishes the desired statement for the second map in (\\ref {linearphiminusphi}); the proof for the first map is similar.$ Figure: NO_CAPTIONLet $\\Omega _0$ , $\\Omega _1$ , and $\\Omega _\\infty $ denote the three open components of $\\hat{ \\setminus \\Gamma chosen so that (see Figure \\ref {Omegas.pdf}){\\begin{@align}{1}{-1}-1 \\in \\Omega _0, \\qquad 1 \\in \\Omega _1, \\qquad \\infty \\in \\Omega _\\infty .\\end{@align}}}\\begin{lemma}The complex-valued function m(x,y,z) defined by{\\begin{@align}{1}{-1}m(x,y,z) =-V(x,y) + \\Phi \\big (x,y,F_{(x,y)}^{-1}(z)\\big ) - {\\left\\lbrace \\begin{array}{ll} \\Phi \\big (x,0,F_{(x,y)}^{-1}(z)\\big ), \\;& z \\in \\Omega _0, \\\\\\Phi \\big (0,y,F_{(x,y)}^{-1}(z)\\big ), & z \\in \\Omega _1, \\\\0, & z \\in \\Omega _\\infty ,\\end{array}\\right.}",
"\\; (x,y)\\in D,\\end{@align}}satisfies the RH problem (\\ref {linearRHm}) and the relation (\\ref {linearVrecover}) for each (x,y) \\in D.\\end{lemma}{\\begin{xmlelement*}{proof}Since F_{(x,y)} is a biholomorphism \\mathcal {S}_{(x,y)}~\\rightarrow \\hat{, we infer from Lemmas \\ref {linearclaim1} and \\ref {linearclaim2} that m(x,y, \\cdot ) is analytic in \\hat{ \\setminus \\Gamma and m(x,y,z) =O(z^{-1}) as z\\rightarrow \\infty for each (x,y) \\in D. The jump condition in (\\ref {linearRHm}) holds as a consequence of the definition (\\ref {linearjumpdef}) of v(x,y,z) and the fact that\\Phi _0(x,k) = \\Phi (x,0,k), \\qquad \\Phi _1(y,k) = \\Phi (0,y,k).Finally, since 0 \\in \\Omega _\\infty and F_{(x,y)}^{-1}(0) = \\infty ^-, (\\ref {linearmVPhi}) and Lemma \\ref {linearclaim1} yieldm(x,y,0) = - V(x,y) + \\Phi (x,y, \\infty ^-) = - 2V(x,y).This proves (\\ref {linearVrecover}).",
"}}\\end{xmlelement*}We have showed that if V(x,y) is a C^n-solution of the Goursat problem for (\\ref {linearernst}) in D with data \\lbrace V_0, V_1\\rbrace , then V(x,y) can be expressed in terms of V_0 and V_1 by (\\ref {linearVrecover}).",
"This also proves that the solution V is unique if it exists, and completes the first part of the proof.",
"}$ Proof of existence The second part of the proof is devoted to proving existence.",
"Let us therefore suppose that $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"We will construct a solution $V(x,y)$ of the associated Goursat problem as follows: Using the given data $V_0$ and $V_1$ , we define $\\Phi _0(x,P)$ and $\\Phi _1(x,P)$ by (REF ).",
"Then we define the jump matrix $v$ by (REF ) and let $m(x,y,z)$ denote the unique solution of the RH problem (REF ).",
"Finally, we show that the function $V(x,y)$ defined in terms of $m(x,y,0)$ via (REF ) constitutes a $C^n$ -solution of the Goursat problem in $D$ with data $\\lbrace V_0, V_1\\rbrace $ .",
"The proof proceeds through a series of lemmas.",
"Lemma 4.10 (Solution of the $x$ -part) The eigenfunction $\\Phi _0(x,P)$ defined in (REF ) has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function x \\mapsto \\Phi _0(x,k^+) is continuous on [0,1) and is C^n on~ (0,1).",
"}\\item $ 0$ obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _0(x,k^+) = -\\Phi _0(x, k^-),\\\\\\Phi _0(x,k^\\pm ) = \\overline{\\Phi _0(x, \\bar{k}^\\pm )},\\end{array}\\right.}",
"\\qquad x \\in [0,1), \\ k \\in \\hat{ \\setminus [0, 1].",
"}\\end{@align}}\\item For each~ $ x [0, 1)$, $ 0(x,P)$ extends continuously to an analytic function of $ P S(x,0) 0$.$ $\\Phi _0(x,\\infty ^+) = V_0(x)$ for $x \\in [0, 1)$ .",
"For each $x \\in (0,1)$ , $\\Phi _{0x}(x,P)$ is an analytic function of $P \\in \\mathcal {S}_{(x,0)}$ except for a simple pole (at most) at the branch point $k = x$ .",
"For each $x_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [0, x_0],{\\begin{@align}{1}{-1}x \\mapsto \\big (k \\mapsto \\Phi _0(x,k^+)\\big )\\end{@align}}is a continuous map [0, x_0) \\rightarrow L^\\infty (K) and a C^n-map (0, x_0) \\rightarrow L^\\infty (K).",
"Moreover,x \\mapsto x^\\alpha \\Phi _{0x}(x,k^+) and x \\mapsto \\Phi _{0k}(x,k^+) are continuous maps [0, x_0) \\rightarrow L^\\infty (K).",
"}$ If we note that $\\Phi _0(x,P)$ is analytic at the points $1^\\pm \\in \\mathcal {S}_{(x,0)}$ for each $x \\in [0, 1)$ , the properties $(a)$ -$(d)$ follow immediately by setting $y = 0$ in Lemma REF .",
"Moreover, since $\\Phi _{0x}(x,k^\\pm ) = \\lambda (x,0,k^\\pm ) V_{0x}(x)$ for $x \\in (0,1)$ , property $(e)$ follows from the definition of $\\lambda $ .",
"It remains to prove $(f)$ .",
"Fix $x_0 \\in (0,1)$ and let $K$ be a compact subset $\\hat{ \\setminus [0, x_0].The function \\lambda (x,0,\\cdot ) is bounded on \\mathcal {S}_{(x,0)} except for a simple pole at k = x.Hence, for x_1, x_2 \\in [0, x_0),{\\begin{@align*}{1}{-1}& \\sup _{k \\in K} \\big |\\Phi _0(x_2, k^+) - \\Phi _0(x_1, k^+)\\big |= \\sup _{k \\in K} \\bigg |\\int _{x_1}^{x_2} \\lambda (x,0,k^+) V_{0x}(x) dx\\bigg |\\\\& \\le \\bigg ( \\sup _{k \\in K} \\sup _{x \\in [0, x_0)} |\\lambda (x,0,k^+)|\\bigg ) \\int _{x_1}^{x_2} |V_{0x}(x)| dx\\le C \\int _{x_1}^{x_2} |V_{0x}(x)| dx,\\end{@align*}}where the right-hand side tends to zero as x_2 \\rightarrow x_1 because V_{0x} \\in L^1((0,x_0)).This shows that the map (\\ref {linearxkphimap}) is continuous [0, x_0) \\rightarrow L^\\infty (K).",
"}If $ x (0, x0)$, then{\\begin{@align*}{1}{-1}\\sup _{k \\in K} \\bigg | &\\frac{\\Phi _0(x+h, k^+) - \\Phi _0(x,k^+)}{h} - \\Phi _{0x}(x, k)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | \\frac{1}{h} \\int _x^{x+h} \\lambda (x^{\\prime }, 0, k) V_{0x}(x^{\\prime })~dx^{\\prime } - \\Phi _{0x}(x,k)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | \\lambda (\\xi , 0, k) V_{0x}(\\xi )~ - \\lambda (x, 0, k) V_{0x}(x)\\bigg |,\\end{@align*}}where $$ lies between $ x$ and $ x+h$.",
"As $ h 0$, the right-hand side goes to zero.",
"Hence (\\ref {linearxkphimap}) is differentiable as a map $ (0,x0) L(K)$ and the derivative satisfies $ 0x(x,k+) = (x,0,k+)V0x(x)$.",
"The same argument with $ k$ instead of $$ implies continuity of $ x 0k(x,k)$.$ The map $x \\mapsto \\big (k \\mapsto \\lambda (x,0,k^+)\\big )$ is $C^\\infty $ from $(0, x_0)$ to $L^\\infty (K)$ and $V_{0x}$ is $C^{n-1}$ on $(0,1)$ .",
"Hence the map $x \\mapsto \\big (k \\mapsto \\lambda (x,0,k^+)V_{0x}(x)\\big )$ is $C^{n-1}$ from $(0, x_0)$ to $L^\\infty (K)$ .",
"It follows that () is a $C^n$ -map $(0, x_0) \\rightarrow L^\\infty (K)$ .",
"Moreover, equation () evaluated at $y = 0$ implies $x \\mapsto x^\\alpha \\Phi _{0x}(x,k^+)$ is continuous $[0, x_0) \\rightarrow L^\\infty (K)$ .",
"This proves $(f)$ and completes the proof of the lemma.",
"In the same way that we constructed the eigenfunction $\\Phi _0(x,k)$ of the $x$ -part, we can construct an eigenfunction $\\Phi _1(y,k)$ of the $y$ -part.",
"Lemma 4.11 (Solution of the $y$ -part) The eigenfunction $\\Phi _1(y,P)$ defined in () has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function y \\mapsto \\Phi _1(y,k^+) is continuous on [0,1) and is C^n on~ (0,1).",
"}\\item $ 1$ obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _1(y,k^+) = -\\Phi _1(y, k^-),\\\\\\Phi _1(y,k^\\pm ) = \\overline{\\Phi _1(y, \\bar{k}^\\pm )},\\end{array}\\right.}",
"\\qquad y \\in [0,1), \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}\\item For each~ $ y [0, 1)$, $ 1(y,P)$ extends continuously to an analytic function of $ P S(0,y) 1$.$ $\\Phi _1(y,\\infty ^+) = V_1(y)$ for $y \\in [0, 1)$ .",
"For each $y \\in (0,1)$ , $\\Phi _{1y}(y,P)$ is an analytic function of $P \\in \\mathcal {S}_{(0,y)}$ except for a simple pole at the branch point $k = 1- y$ .",
"For each $y_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [1-y_0, 1],{\\begin{@align}{1}{-1}y \\mapsto \\big (k \\mapsto \\Phi _1(y,k^+)\\big )\\end{@align}}is a continuous map [0, y_0] \\rightarrow L^\\infty (K) and a C^n-map (0, y_0) \\rightarrow L^\\infty (K).",
"Moreover,y\\mapsto y^\\alpha \\Phi _{1y}(y,k^+) and xy\\mapsto \\Phi _{1k}(y,k^+) are continuous maps [0, x_0) \\rightarrow L^\\infty (K).",
"}$ The proof is analogous to that of Lemma REF .",
"Recall from the definition in Section that the contour $\\Gamma \\equiv \\Gamma (x,y)$ consists of two nonintersecting clockwise loops $\\Gamma _0$ and $\\Gamma _1$ which encircle the intervals $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ respectively, but which do not encircle the origin.",
"We are free to choose $\\Gamma _0$ and $\\Gamma _1$ as long as these requirements are met.",
"It turns out to be convenient to choose $\\Gamma _0$ and $\\Gamma _1$ independent of $(x,y)$ .",
"However, we see from () that the intervals $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ get arbitrarily close to the origin as $(x,y)$ approaches the diagonal edge $x+y=1$ of $D$ .",
"Hence we cannot take $\\Gamma $ independent of $(x,y)$ for all $(x,y) \\in D$ .",
"However, if we restrict ourselves to points $(x,y)$ which lie in the slightly smaller triangle $D_\\delta $ , $\\delta > 0$ , defined in (REF ), then we can choose $\\Gamma $ independent of $(x,y)$ .",
"Figure: NO_CAPTIONThus, fix $\\delta \\in (0,1)$ and choose $\\epsilon > 0$ so small that $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ are contained in the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, for all $(x,y) \\in D_\\delta $ .",
"Fix two smooth nonintersecting clockwise contours $\\Gamma _0$ and $\\Gamma _1$ in the complex $z$ -plane which encircle once the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, but which do not encircle zero, see Figure REF .",
"Suppose also that $\\Gamma _0$ and $\\Gamma _1$ are invariant under the involutions $z \\mapsto z^{-1}$ and $z \\mapsto \\bar{z}$ .",
"Let $\\Gamma = \\Gamma _0 \\cup \\Gamma _1$ and, using this particular choice of $\\Gamma $ , define $V(x,y)$ for $(x,y) \\in D_\\delta $ by (REF ), i.e., $V(x,y) = -\\frac{1}{4\\pi i} \\int _{\\Gamma } \\frac{v(x,y,z)}{z} dz,$ where $v(x,y,z)$ is given by (REF ).",
"We will show that ${\\left\\lbrace \\begin{array}{ll}V \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\\\text{$V(x,y)$ satisfies the Euler-Darboux equation (\\ref {linearernst}) in $\\operatorname{int}(D_\\delta )$,}\\\\\\text{$x^\\alpha V_x, y^\\alpha V_y, x^\\alpha y^\\alpha V_{xy} \\in C(D_\\delta )$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$V(x,0) = V_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$V(0,y) = V_1(y)$ for $y \\in [0,1-\\delta )$.}\\end{array}\\right.",
"}$ Since $\\delta > 0$ can be chosen arbitrarily small, this will complete the proof of the theorem.",
"Consider the family of scalar RH problems given in (REF ) parametrized by the two parameters $(x,y) \\in D_\\delta $ .",
"For each $(x,y) \\in D_\\delta $ , the unique solution of (REF ) is given by $m(x,y,z) = \\frac{1}{2\\pi i} \\int _{\\Gamma } \\frac{v(x,y,z^{\\prime })}{z^{\\prime }-z} dz^{\\prime }, \\qquad (x,y) \\in D_\\delta , \\ z \\in \\hat{ \\setminus \\Gamma .",
"}$ Lemma 4.12 The map $(x,y) \\mapsto v(x,y, \\cdot )$ is continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha v_x(x,y, \\cdot ), \\qquad (x,y) \\mapsto y^\\alpha v_x(x,y, \\cdot ), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha v_{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"The map $(x,y) \\mapsto v(x,y, \\cdot )$ is continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^\\infty (\\Gamma )$ as a consequence of part $(f)$ of Lemmas REF and REF .",
"Furthermore, $x^\\alpha v_x(x,y, z) = {\\left\\lbrace \\begin{array}{ll} x^\\alpha \\Phi _{0x}\\big (x, F_{(x,y)}^{-1}(z)\\big ) + x^\\alpha \\Phi _{0k}\\big (x, F_{(x,y)}^{-1}(z)\\big ) \\Big (\\frac{d}{dx}F_{(x,y)}^{-1}(z)\\Big ), \\quad & z \\in \\Gamma _0,\\\\x^\\alpha \\Phi _{1k}\\big (y, F_{(x,y)}^{-1}(z)\\big ) \\Big (\\frac{d}{dx}F_{(x,y)}^{-1}(z)\\Big ), \\quad & z \\in \\Gamma _1.\\end{array}\\right.",
"}$ Part $(f)$ of Lemma REF implies that the terms $x^\\alpha \\Phi _{0x}\\big (x, F_{(x,y)}^{-1}(\\cdot )\\big )$ and $\\Phi _{0k}\\big (x, F_{(x,y)}^{-1}(\\cdot )\\big )$ are continuous $D_\\delta \\rightarrow L^\\infty (\\Gamma _0))$ .",
"Similarly, part $(f)$ of Lemma REF implies that the term $\\Phi _{1k}\\big (y, F_{(x,y)}^{-1}(\\cdot )\\big )$ is continuous $D_\\delta \\rightarrow L^\\infty (\\Gamma _1))$ .",
"We conclude that $(x,y) \\mapsto x^\\alpha v_x(x,y, \\cdot )$ is continuous $D_\\delta \\rightarrow L^\\infty (\\Gamma )$ .",
"The other two maps in (REF ) are treated in a similar way.",
"Lemma 4.13 The solution $m(x,y,z)$ defined in (REF ) has the following properties: For each point $(x,y) \\in D_\\delta $ , $m(x,y,\\cdot )$ obeys the symmetries $m(x,y,z) = m(x,y,0) - m(x,y,z^{-1}) = \\overline{m(x,y,\\bar{z})}, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}$ For each $z \\in \\hat{\\setminus \\Gamma , the map (x,y) \\mapsto m(x,y,z) is continuous from D_\\delta to and is C^n from \\operatorname{int}D_\\delta to .",
"}\\item For each~ $ z $, the three maps$$(x,y) \\mapsto x^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto y^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha m_{xy}(x,y, z),$$are continuous from $ D$ to $ .",
"The symmetries in () and () show that $v$ satisfies ${\\left\\lbrace \\begin{array}{ll} v(x,y,z) = -v(x, y, z^{-1}),\\\\v(x,y,z) = \\overline{v(x, y, \\bar{z})},\\end{array}\\right.}",
"\\qquad z \\in \\Gamma , \\ (x,y) \\in D_\\delta .$ These symmetries imply that $m(x,y,0) - m(x,y,z^{-1})$ and $\\overline{m(x,y,\\bar{z})}$ satisfy the same RH problem as $m(x,y,z)$ .",
"Hence, by uniqueness, (REF ) holds.",
"This proves $(a)$ .",
"For each $z \\in \\hat{\\setminus \\Gamma , the mapf \\mapsto \\int _\\Gamma \\frac{f(z^{\\prime })}{z^{\\prime } - z} dz^{\\prime }is a bounded linear map L^\\infty (\\Gamma ) \\rightarrow .Hence properties (b) and (c) follow immediately from (\\ref {linearmsolution}) and Lemma \\ref {linearclaim3E}.",
"}$ Given a contour $\\gamma \\subset , we use the notation $ N()$ to denote an open tubular neighborhood of $$.",
"We extend the definition (\\ref {linearjumpdef}) of $ v$ to a tubular neighborhood $ N() = N(0) N(1)$ of $$ as follows, see Figure~ \\ref {Gammatubular.pdf}:{\\begin{@align}{1}{-1}v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in N(\\Gamma _0),\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in N(\\Gamma _1),\\end{array}\\right.",
"}\\qquad (x,y) \\in D_\\delta .\\end{@align}}We choose $ N()$ so narrow that it does not intersect the intervals $ [--1, -]$ and $ [, -1]$.",
"Then, for each $ (x,y) D$, $ v(x,y,)$ is an analytic function of $ z N()$.Using the notation $ z(x,y,P) := F(x,y)(P)$, we can write (\\ref {linearjumpdef2}) as{\\begin{@align}{1}{-1}v(x,y,z(x,y,P)) = {\\left\\lbrace \\begin{array}{ll} \\Phi _0(x, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _0)\\big ), \\\\\\Phi _1(y, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _1)\\big ),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .\\end{@align}}We define functions $ f0(x,y,z)$ and $ f1(x,y,z)$ for $ (x,y) D$ by{\\begin{@align*}{1}{-1}f_0(x,y,z) = m_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z), \\qquad z \\in \\hat{ \\setminus \\Gamma ,\\\\f_1(x,y,z) = m_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z), \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}}Moreover, we let n_0(x,y,z) and n_1(x,y,z) denote the functions given by\\begin{subequations}{\\begin{@align}{1}{-1}n_0(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_0(x,y,z) + \\Phi _{0x}\\big (x,F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Omega _0, \\\\f_0(x,y,z), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}and{\\begin{@align}{1}{-1}n_1(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_1(x,y,z) + \\Phi _{1y}\\big (y,F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Omega _1, \\\\f_1(x,y,z), & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}\\end{@align}}\\end{subequations}\\end{@align*}}$ Figure: NO_CAPTIONLemma 4.14 For each $(x,y) \\in \\operatorname{int}D_\\delta $ , it holds that $n_0(x,y,z)$ is an analytic function of $z \\in \\hat{ \\setminus \\lbrace -1\\rbrace and has at most a simple pole at z = -1.\\item n_1(x,y,z) is an analytic function of z \\in \\hat{ \\setminus \\lbrace 1\\rbrace and has at most a simple pole at z = 1.\\item n_0(x,y,\\infty ) = 0 and n_0(x,y,0) = -2V_x(x,y).\\item n_1(x,y,\\infty ) = 0 and n_1(x,y,0) = -2V_y(x,y).",
"}}{\\begin{xmlelement*}{proof}Let (x,y) \\in \\operatorname{int}D_\\delta .The function{\\begin{@align}{1}{-1}z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) = -\\frac{1-z}{1+z} \\frac{z}{1 - x - y}\\end{@align}}is analytic for z \\in \\hat{\\setminus \\lbrace -1, \\infty \\rbrace with simple poles at z = -1 and z = \\infty .Equation (\\ref {linearmsolution}) implies that m_z(x,y, z) = O(z^{-2}) and m_x(x,y, z) = O(z^{-1}) as z \\rightarrow \\infty .",
"Hence f_0(x,y,z) is analytic at z = \\infty .It follows that f_0(x,y,z) is analytic for all z \\in \\hat{\\setminus (\\Gamma \\cup \\lbrace -1\\rbrace ) with a simple pole at z = -1 at most.Now f_0 has continuous boundary values on \\Gamma and satisfies the following jump condition across \\Gamma :{\\begin{@align}{1}{-1}f_{0+}(x,y,z) = f_{0-}(x,y,z) + v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z), \\qquad z \\in \\Gamma .\\end{@align}}Differentiating (\\ref {linearvzPhi0}) with respect to x and y and evaluating the resulting equations at k = F_{(x,y)}^{-1}(z), we find, for (x,y) \\in \\operatorname{int}D_\\delta ,{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)),\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _0),\\end{@align}} and{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{1y}(x,F_{(x,y)}^{-1}(z)),\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _1).\\end{@align}} Using the first equations in (\\ref {linearvxzxvza}) and (\\ref {linearvxzxvzb}) in (\\ref {linearf0jump}), we conclude that f_0 is analytic across \\Gamma _1 and has the following jump across \\Gamma _0:{\\begin{@align}{1}{-1}f_{0+}(x,y,z) = f_{0-}(x,y,z) + \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)), \\qquad z \\in \\Gamma _0.\\end{@align}}Consequently, n_0 is analytic across \\Gamma .",
"Furthermore, by Lemma \\ref {linearclaim1E}, \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)) is analytic for z \\in \\hat{\\setminus \\lbrace -1\\rbrace with at most a simple pole at z = -1.",
"It follows that n_0 satisfies (a).",
"The proof of (b) is similar and relies on the second equations in (\\ref {linearvxzxvza}) and (\\ref {linearvxzxvzb}).",
"}Using (\\ref {linearzx}) in the definition (\\ref {linearndefa}) of n_0, we can write{\\begin{@align}{1}{-1}n_0(x,y,z) = f_0(x,y,z) = m_x(x,y,z) -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z(x,y,z), \\qquad z \\in \\Omega _\\infty .\\end{@align}}Since m_z(x,y, z) = O(z^{-2}) and m_x(x,y, z) = O(z^{-1}) as z \\rightarrow \\infty , this gives n_0(x,y,\\infty ) = 0.",
"On the other hand, evaluating (\\ref {linearn0f0}) at z = 0, we find n_0(x,y,0) = m_x(x,y,0) = -2V_x(x,y).",
"This proves (c); the proof of (d) is analogous.",
"}}\\end{xmlelement*}Equation (\\ref {linearmVPhi}) suggests that we define a function \\Phi (x,y,P) for (x,y) \\in D_\\delta and P \\in F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)} by{\\begin{@align}{1}{-1}\\Phi (x,y,P) = V(x,y) + m(x,y,F_{(x,y)}(P)).\\end{@align}}}$ Lemma 4.15 The function $\\Phi $ defined in () satisfies the Lax pair equations ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y,P) = \\lambda (x,y,P) V_x(x,y), \\\\\\Phi _y(x,y,P) = \\frac{1}{\\lambda (x,y,P)} V_y(x,y),\\end{array}\\right.",
"}$ for $(x,y) \\in \\operatorname{int}D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ .",
"The analyticity structure of $n_0$ established in Lemma REF implies that there exists a function $C(x,y)$ independent of $z$ such that $n_0(x,y,z) = \\frac{C(x,y)}{z+1}, \\qquad z \\in \\hat{.",
"}We determine C(x,y) by evaluating (\\ref {linearn0C}) at z = 0.",
"By Lemma \\ref {linearclaim5E} (d), this gives C(x,y) = -2V_x(x,y).It follows that{\\begin{@align}{1}{-1}n_0 = -\\frac{2V_x(x,y)}{z+1}, \\qquad (x,y) \\in D_\\delta , \\ z \\in \\hat{.",
"}\\end{@align}Note that we did not exclude that n_0 is free of singularities.",
"In this case we have C=-2V_x=0 by Lemma \\ref {linearclaim5E}.",
"}Differentiating (\\ref {linearphidef}) with respect to x and using (\\ref {linearn0f0}) and (\\ref {linearnoutside}), we find, for P \\in F_{(x,y)}^{-1}(\\Omega _\\infty ),{\\begin{@align*}{1}{-1}\\Phi _x(x,y,P)& = V_x(x,y) + f_0(x,y,z(x,y,P))= V_x(x,y) - \\frac{2V_x(x,y)}{z(x,y,P) +1}.\\end{@align*}}Since1 - \\frac{2}{z+1} = \\lambda ,this yields the first equation in (\\ref {linearphilax}).",
"A similar argument gives the second equation in (\\ref {linearphilax}).",
"This proves the lemma.$ Lemma 4.16 The real-valued function $V:D \\rightarrow {R}$ defined by (REF ) has the properties listed in (REF ).",
"The function $V(x,y) = -\\frac{1}{2}m(x,y,0)$ is real-valued by (REF ).",
"Moreover, by part $(b)$ of Lemma REF , the map $(x,y) \\mapsto m(x,y,0)$ is continuous from $D_\\delta $ to $ and is $ Cn$ from $ intD$ to $ .",
"Hence $V \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta )$ .",
"Similarly, part $(c)$ of Lemma REF implies that $x^\\alpha V_x, y^\\alpha V_y, x^\\alpha y^\\alpha V_{xy} \\in C(D_\\delta )$ .",
"Let $P = (\\lambda , k)$ be a point in $F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ .",
"For each fixed $k \\in \\hat{ with k^+ \\in F_{(x,y)}^{-1}(\\Omega _\\infty ), the map (x,y) \\rightarrow \\Phi (x,y,k^+) is C^n from \\operatorname{int}D_\\delta to .By Lemma \\ref {linearclaim6E}, it satisfies the Lax pair equations (\\ref {linearphilax}).",
"Since~ n \\ge 2, it follows that{\\begin{@align*}{1}{-1}0 &= \\Phi _{xy}(x,y,P) - \\Phi _{yx} (x,y,P)\\\\& = \\lambda _y V_x + \\lambda V_{xy}+ \\frac{\\lambda _x}{\\lambda ^2} V_y - \\frac{1}{\\lambda } V_{xy}\\\\& = \\frac{1}{2\\lambda (k-x)} (V_x + V_y) + \\bigg (\\lambda - \\frac{1}{\\lambda }\\bigg )V_{xy}\\\\& = \\frac{1}{2\\lambda (k-x)} \\big (V_x + V_y - 2(1-x-y)V_{xy}\\big ), \\qquad (x,y) \\in \\operatorname{int}D_\\delta .\\end{@align*}}It follows that V(x,y) satisfies Euler-Darboux equation (\\ref {linearernst}) for (x,y) \\in \\operatorname{int}D_\\delta .",
"}Finally, we show that $ V(x,0) = V0(x)$ for $ x [0, 1-)$; the proof that $ V(0,y) = V1(y)$ for $ y [0,1-)$ is similar.By definitions (\\ref {linearV2}) and (\\ref {linearjumpdef}) of $ V$ and $ v$, we have$$V(x,0) = -\\frac{1}{4\\pi i} \\int _{\\Gamma } \\frac{v(x,0,z)}{z} dz= -\\frac{1}{4\\pi i} \\int _{\\Gamma _0} \\frac{\\Phi _0(x,F_{(x,0)}^{-1}(z))}{z} dz, \\qquad x \\in [0, 1-\\delta ).$$But $ 0(x,F(x,0)-1(z))$ is analytic for $ z [--1, -]$ by Lemma \\ref {linearclaim1E}, so using Cauchy^{\\prime }s formula to compute the contributions from ~$ z = 0$ and $ z = $, we find{\\begin{@align*}{1}{-1}V(x,0) & = - \\frac{1}{2}\\Phi _0\\big (x,F_{(x,0)}^{-1}(0)\\big ) + \\frac{1}{2}\\Phi _0\\big (x,F_{(x,0)}^{-1}(\\infty )\\big )\\\\& = -\\frac{1}{2}\\Phi _0(x, \\infty ^-) + \\frac{1}{2}\\Phi _0(x, \\infty ^+)= \\Phi _0(x, \\infty ^+) = V_0(x), \\qquad x \\in [0, 1-\\delta ).\\end{@align*}}This completes the proof of the lemma.",
"Since $ > 0$ was arbitrary, it also completes the proof of existence.$ Proof of boundary behavior Let $V_0(x)$ , $x \\in [0, 1)$ , and $V_1(y)$ , $y \\in [0,1)$ be real-valued functions satisfying (REF ) for some $n \\ge 2$ and some $\\alpha \\in (0, 1)$ .",
"Suppose $V(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace V_0, V_1\\rbrace $ and define $m_1, m_2 \\in {R}$ by (REF ).",
"By (REF ), we have $V(x,y) = -\\frac{1}{4\\pi i} \\int _{\\Gamma _0} \\frac{\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big )}{z} dz-\\frac{1}{4\\pi i} \\int _{\\Gamma _1} \\frac{\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big )}{z} dz.$ Hence $\\nonumber V_x(x,y) = & -\\frac{1}{4\\pi i} \\int _{\\Gamma _0} \\frac{\\Phi _{0x}\\big (x, F_{(x,y)}^{-1}(z)\\big )}{z} dz-\\frac{1}{4\\pi i} \\int _{\\Gamma _0} \\frac{\\Phi _{0k}\\big (x, F_{(x,y)}^{-1}(z)\\big )}{z} \\Big (\\frac{d}{dx}F_{(x,y)}^{-1}(z)\\Big )dz\\\\ & -\\frac{1}{4\\pi i} \\int _{\\Gamma _1} \\frac{\\Phi _{1k}\\big (y, F_{(x,y)}^{-1}(z)\\big )}{z} \\Big (\\frac{d}{dx}F_{(x,y)}^{-1}(z)\\Big ) dz.$ Now $k = F_{(x,y)}^{-1}(z) = -\\frac{x (z-1)^2+(y-1) (z+1)^2}{4 z},$ so $\\frac{d}{dx}F_{(x,y)}^{-1}(z) = -\\frac{(z-1)^2}{4 z}, \\qquad \\frac{d}{dy}F_{(x,y)}^{-1}(z) = -\\frac{(z+1)^2}{4 z}.$ It follows from Lemma REF and Lemma REF that the last two integrals on the right-hand side of (REF ) remain bounded as $x \\downarrow 0$ .",
"Moreover, $\\lim _{x \\downarrow 0} x^\\alpha \\Phi _{0x}\\big (x, F_{(x,y)}^{-1}(z)\\big )& = \\lim _{x \\downarrow 0} x^\\alpha \\lambda (x,0, F_{(x,y)}^{-1}(z)) V_{0x}(x)= m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)).$ Using that $F_{(0,y)}^{-1}(z)= -\\frac{(y-1) (z+1)^2}{4 z}$ , we find $\\lambda (0,0, F_{(0,y)}^{-1}(z)) = \\sqrt{\\frac{1}{(z+1)^2}\\left(z-\\frac{1-\\sqrt{y}}{1+\\sqrt{y}}\\right)\\left(z-\\frac{1+\\sqrt{y}}{1-\\sqrt{y}}\\right)},$ where the square roots have positive (negative) real part for $|z| > 1$ ($|z| < 1$ ).",
"Thus $\\nonumber \\lim _{x \\downarrow 0} x^\\alpha \\Phi _{0x}\\big (x, F_{(x,y)}^{-1}(z)\\big )=\\frac{-m_1}{z+1} \\sqrt{\\left(z-\\frac{1-\\sqrt{y}}{1+\\sqrt{y}}\\right)\\left(z-\\frac{1+\\sqrt{y}}{1-\\sqrt{y}}\\right)},$ where the square root has a branch cut along the interval $[\\frac{1-\\sqrt{y}}{1+\\sqrt{y}}, \\frac{1+\\sqrt{y}}{1-\\sqrt{y}}]$ and the branch is fixed so that the root has positive real part for $z < 0$ .",
"Hence $\\nonumber \\lim _{x \\downarrow 0} x^\\alpha V_x(x,y) = &\\; \\frac{m_1}{4\\pi i} \\int _{\\Gamma _0} \\frac{1}{z+1} \\sqrt{\\left(z-\\frac{1-\\sqrt{y}}{1+\\sqrt{y}}\\right) \\left(z-\\frac{1+\\sqrt{y}}{1-\\sqrt{y}}\\right)}\\frac{dz}{z}\\\\\\nonumber = & -\\frac{m_1}{2} \\underset{z = -1}{\\text{\\upshape Res\\,}} \\frac{1}{z+1} \\sqrt{\\left(z-\\frac{1-\\sqrt{y}}{1+\\sqrt{y}}\\right) \\left(z-\\frac{1+\\sqrt{y}}{1-\\sqrt{y}}\\right)}\\frac{1}{z}\\\\\\nonumber = & -\\frac{m_1}{2} \\frac{-2}{\\sqrt{1-y}} = \\frac{m_1}{\\sqrt{1-y}}.$ This proves (REF ); the proof of () is similar.",
"Thus the proof of Theorem REF is complete.",
"Lax pair and eigenfunctions In this section we introduce a Lax pair for (REF ) and define appropriate eigenfunctions in preparation for the proofs of Theorems REF -REF .",
"Lax pair The hyperbolic Ernst equation (REF ) admits the Lax pair ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y, k) = \\mathsf {U}(x,y, k) \\Phi (x,y, k),\\\\\\Phi _y(x,y, k) = \\mathsf {V}(x,y,k) \\Phi (x,y,k),\\end{array}\\right.",
"}$ where $k$ is the spectral parameter, the function $\\Phi (x,y, k)$ is a $2 \\times 2$ -matrix valued eigenfunction, and the $2\\times 2$ -matrix valued functions $\\mathsf {U}(x,y,k)$ and $\\mathsf {V}(x,y,k)$ are defined as follows: $\\mathsf {U} = \\frac{1}{\\mathcal {E} + \\bar{\\mathcal {E}}} \\begin{pmatrix} \\bar{\\mathcal {E}}_x & \\lambda \\bar{\\mathcal {E}}_x \\\\\\lambda \\mathcal {E}_x & \\mathcal {E}_x \\end{pmatrix}, \\qquad \\mathsf {V} = \\frac{1}{\\mathcal {E} + \\bar{\\mathcal {E}}} \\begin{pmatrix} \\bar{\\mathcal {E}}_y & \\frac{1}{\\lambda } \\bar{\\mathcal {E}}_y \\\\\\frac{1}{\\lambda } \\mathcal {E}_y & \\mathcal {E}_y\\end{pmatrix},$ with $\\lambda $ given by (REF ).",
"We write the Lax pair (REF ) in terms of differential forms as $d\\Phi = W\\Phi ,$ where $W$ is the closed one-form $W = \\mathsf {U}dx + \\mathsf {V}dy.$ As in Section , we will view the map $\\Phi (x,y,\\cdot )$ as being defined on the Riemann surface $\\mathcal {S}_{(x,y)}$ and write $\\Phi (x,y,P)$ for the value of $\\Phi $ at $P = (\\lambda , k) \\in \\mathcal {S}_{(x,y)}$ .",
"Spectral analysis Suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Let $\\mathsf {U}_0$ and $\\mathsf {V}_1$ be given by (REF ), i.e., $\\mathsf {U}_0$ and $\\mathsf {V}_1$ denote the functions $\\mathsf {U}$ and $\\mathsf {V}$ evaluated at $y = 0$ and $x = 0$ , respectively.",
"Let $\\Phi _0(x,P)$ and $\\Phi _1(y,P)$ be the eigenfunctions defined in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ via the Volterra integral equations (REF ).",
"Lemma 5.1 (Solution of the $x$ -part) The eigenfunction $\\Phi _0(x,P)$ defined via the Volterra integral equation (REF ) has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function x \\mapsto \\Phi _0(x,k^+) is continuous on [0,1) and is C^n on (0,1).",
"Furthermore, for each x \\in [0,1), the function k \\mapsto \\Phi _0(x,k^+) is analytic on \\hat{ \\setminus [0,1].",
"}\\item \\Phi _0 obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _0(x,k^+) = \\sigma _3\\Phi _0(x, k^-)\\sigma _3,\\\\\\Phi _0(x,k^\\pm ) = \\sigma _1\\overline{\\Phi _0(x, \\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad x \\in [0,1), \\ k \\in \\hat{ \\setminus [0, 1].",
"}\\end{@align}}\\item For each~ x \\in [0, 1), \\Phi _0(x,P) extends continuously to an analytic function of P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.",
"}\\item The value of $ 0$ at $ P = +$ is given by{\\begin{@align}{1}{-1}\\Phi _0(x,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}_0(x)} & 1 \\\\ \\mathcal {E}_0(x) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad x \\in [0, 1).\\end{@align}}$ The determinant of $\\Phi _0$ is given by $\\det \\Phi _0(x,P) = \\text{\\upshape Re\\,}\\mathcal {E}_0(x), \\qquad x \\in [0,1), \\ P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.$ For each $x_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [0, x_0],{\\begin{@align}{1}{-1}x \\mapsto \\big (k \\mapsto \\Phi _0(x,k^+)\\big )\\end{@align}}is a continuous map [0, x_0) \\rightarrow L^\\infty (K) and a C^n-map (0, x_0) \\rightarrow L^\\infty (K).",
"Moreover, the mapx \\mapsto \\big ( k \\mapsto x^\\alpha \\Phi _{0x}(x,k^+) \\big ) is continuous [0, x_0) \\rightarrow L^\\infty (K).",
"}$ We first use successive approximations to show that the integral equation $\\Phi _0(x, k^+) = I + \\int _0^x \\mathsf {U}_0(x^{\\prime },k^+) \\Phi _0(x^{\\prime },k^+) dx^{\\prime }, \\qquad x \\in [0,1),$ has a unique solution for each $k \\in \\hat{ \\setminus [0,1].",
"Let K be a compact subset of \\hat{ \\setminus [0,1].Let \\Phi _0^{(0)} = I and define \\Phi _0^{(j)}(x,k^+) for j \\ge 1 inductively by{\\begin{@align*}{1}{-1}& \\Phi _0^{(j+1)}(x,k^+) = \\int _0^x \\mathsf {U}_0(x^{\\prime },k^+) \\Phi _0^{(j)}(x^{\\prime },k^+) dx^{\\prime }, \\qquad x \\in [0,1), \\ k \\in K.\\end{@align*}}Then{\\begin{@align}{1}{-1}\\Phi _0^{(j)}(x,k^+) = &\\; \\int _{0 \\le x_1 \\le \\cdots \\le x_j \\le x} \\mathsf {U}_0(x_j, k^+) \\mathsf {U}_0(x_{j-1}, k^+)\\cdots \\mathsf {U}_0(x_1, k^+) dx_1 \\cdots dx_j.\\end{@align}}The function \\lambda (x,0,k^+) is analytic for k \\in \\hat{ \\setminus [x,1]; in particular, it is a bounded function of k \\in K for each fixed x \\in [0,1).",
"In view of the assumptions (\\ref {E0E1assumptions}), this implies\\Vert \\mathsf {U}_0(x,k^+)\\Vert _{L^1([0,x])} < C(x), \\qquad x \\in [0,1),\\ k \\in K,where the function~ C(x) is bounded on each compact subset of [0,1).Thus{\\begin{@align}{1}{-1} |\\Phi _0^{(j)}(x,k^+)| \\le & \\frac{1}{j!}",
"\\Vert \\mathsf {U}_0(\\cdot , k^+)\\Vert _{L^1([0, x])}^j\\le \\frac{1}{j!}",
"C(x)^j, \\qquad x \\in [0,1), \\ k \\in K.\\end{@align}}Hence the series{\\begin{@align}{1}{-1} \\Phi _0(x,k^+) = \\sum _{j=0}^\\infty \\Phi _0^{(j)}(x,k^+)\\end{@align}}converges absolutely and uniformly for k \\in K and x in compact subsets of [0,1) to a continuous solution \\Phi _0(x,k^+) of (\\ref {phixk}).",
"The fact that x \\mapsto \\Phi _0(x,k^+) \\in C^n((0,1)) follows from differentiating x\\mapsto \\Phi _0^{(j)}(x,k^+) and applying estimates similar to (\\ref {phi0jestimate}) to the derivative.Differentiating (with respect to k) under the integral sign in (\\ref {xPhijiterated}), we see that k \\mapsto \\Phi _0^{(j)}(x, k^+) is analytic on \\operatorname{int}K for each j; the uniform convergence then proves that k \\mapsto \\Phi _0(x, k^+) is analytic on \\operatorname{int}K.A similar argument applies to the integral equation defining \\Phi _0(x,k^-).",
"We conclude that the functions \\Phi _0(x,k^+) and \\Phi _0(x,k^-) are well-defined for x \\in [0,1) and k \\in \\hat{ \\setminus [0,1] and are analytic functions of k \\in \\hat{ \\setminus [0,1] for each fixed x.",
"}We next show uniqueness.",
"Assume that \\tilde{\\Phi }_0 is another solution of the Volterra equation (\\ref {phixk}) such that x \\mapsto \\Phi _0(x,k^\\pm ) is continuous on [0,1), respectively, and let \\Psi =\\Phi _0 - \\tilde{\\Phi }_0.",
"Then \\Psi is a solution of the homogeneous equation\\Psi (x,k^\\pm ) = \\int _0^x \\mathsf {U}_0(x^{\\prime },k^\\pm ) \\Psi (x^{\\prime },k^\\pm ) dx^{\\prime }.Iterating this yields{\\begin{@align*}{1}{-1}\\Psi (x,k^\\pm ) &= \\int _0^x \\mathsf {U}_0(x_j,k^\\pm ) \\int _{0}^{x_j} \\mathsf {U}_0(x_{j-1},k^\\pm ) \\cdots \\int _{0}^{x_2} \\mathsf {U}_0(x_{1},k^\\pm )\\Psi (x_1,k^\\pm ) \\,dx_1 \\ldots dx_n\\\\&= \\int _{0 \\le x_1 \\le \\cdots \\le x_j \\le x} \\mathsf {U}_0(x_j, k^\\pm ) \\mathsf {U}_0(x_{j-1}, k^\\pm )\\cdots \\mathsf {U}_0(x_1, k^\\pm )\\Psi (x_1,k^\\pm ) dx_1 \\cdots dx_j.\\end{@align*}}Hence, as in the proof of existence, we get the estimate|\\Psi (x,k^\\pm ) | \\le \\sup _{x^{\\prime }\\in [0,x]} |\\Psi (x^{\\prime },k^\\pm ) | \\frac{\\Vert \\mathsf {U}_0(\\cdot ,k^\\pm ) \\Vert _{L^1([0,x])}^j}{j!}",
"\\rightarrow 0, \\qquad j \\rightarrow \\infty ,which yields \\Psi = 0.",
"This proves (a).",
"}The symmetries (\\ref {lambdasymm}) of \\lambda show that\\mathsf {U}_0(x,k^+) = \\sigma _3\\mathsf {U}_0(x, k^-)\\sigma _3, \\qquad \\mathsf {U}_0(x,k^+) = \\sigma _1\\overline{\\mathsf {U}_0(x,\\bar{k}^+)}\\sigma _1.Hence \\sigma _3\\Phi _0(x, k^-)\\sigma _3 and \\sigma _1\\overline{\\Phi _0(x, \\bar{k}^+)}\\sigma _1 satisfy the same Volterra equation as \\Phi _0(x,k^+).",
"By uniqueness, all three functions must be equal.",
"This proves (b).",
"}We next show that \\Phi _0(x, k^\\pm ) can be continuously extended across the branch cut to an analytic function on \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.Since \\mathsf {U}_0(x,k^\\pm ) has continuous boundary values on the interval (x, 1), the above argument (applied with a K that reaches up to the boundary) shows that \\Phi _0(x,k^\\pm ) also has continuous boundary values on (x, 1).Moreover, since\\lambda (x, 0, (k+i0)^+) = \\lambda (x,0,(k-i0)^-), \\qquad k \\in (x, 1),the boundary functions \\Phi (x,0, (k + i0)^+) and \\Phi (x,0, (k - i0)^-) satisfy the same integral equation, so by uniqueness they are equal:{\\begin{@align*}{1}{-1}\\Phi (x,y, (k + i0)^+) = \\Phi (x,y,(k-i0)^-), \\qquad (x,y) \\in D, \\ k \\in (x, 1).\\end{@align*}}Hence the values of \\Phi _0 on the upper and lower sheets of \\mathcal {S}_{(x,0)} fit together across the branch cut (x,1), showing that \\Phi _0 extends to an analytic function of P \\in \\mathcal {S}_{(x,0)} \\setminus \\big (\\Sigma _0 \\cup \\lbrace 1\\rbrace \\big ).",
"But \\lambda (x,0,P) is bounded in a neighborhood of the branch point 1, hence the possible singularity of \\Phi _0(x,P) at this point must be removable.",
"This shows that \\Phi _0 satisfies (c).",
"}}Since $ (x,y,+) = 1$, $ 0(x,+)$ satisfies the equation$$\\Phi _{0x}(x, \\infty ^+) = \\frac{1}{2\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} \\begin{pmatrix} \\overline{\\mathcal {E}_{0x}(x)} & \\overline{\\mathcal {E}_{0x}(x)} \\\\\\mathcal {E}_{0x}(x) & \\mathcal {E}_{0x}(x) \\end{pmatrix} \\Phi _0(x,\\infty ^+), \\qquad x \\in [0, 1).$$This equation has the two linearly independent solutions$$\\begin{pmatrix} \\overline{\\mathcal {E}_0(x)}\\\\ \\mathcal {E}_0(x) \\end{pmatrix} \\quad \\text{and}~\\quad \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}.$$Hence there exists a constant matrix $ A$ such that$$\\Phi _0(x, \\infty ^+) = \\begin{pmatrix} \\overline{\\mathcal {E}_0(x)} & 1 \\\\ \\mathcal {E}_0(x) & -1 \\end{pmatrix} A, \\qquad x \\in [0, 1).$$We determine $ A$ by evaluating this equation at $ x = 0$ and using that $ E0(0) = 1$ and $ 0(0, +) = I$.",
"This yields (\\ref {Phi0atinftyplus}) and proves $ (d)$.$ The proof of $(e)$ relies on the general identity $(\\ln \\det B)_x = \\text{\\upshape tr\\,}(B^{-1} B_x),$ where $B = B(x)$ is a differentiable matrix-valued function taking values in $GL(n, $ .",
"We find $(\\ln \\det \\Phi _0)_x = \\text{\\upshape tr\\,}(\\Phi _0^{-1} \\mathsf {U}_0 \\Phi _0) = \\text{\\upshape tr\\,}\\mathsf {U}_0 = \\frac{\\text{\\upshape Re\\,}\\mathcal {E}_{0x}}{\\text{\\upshape Re\\,}\\mathcal {E}_0} = (\\ln \\text{\\upshape Re\\,}\\mathcal {E}_0)_x.$ This relation is valid at least for small $x$ because $\\Phi _0(0, k^\\pm ) = I$ is invertible.",
"In fact, since $\\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0$ for $x \\in [0,1)$ by assumption (REF ), it extends to all of $[0,1)$ and we infer that, for $P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0$ , $\\det \\Phi _0(x, P) = C(P) \\text{\\upshape Re\\,}\\mathcal {E}_0(x), \\qquad x \\in [0,1),$ where $C(P) \\in is independent of $ x$.",
"Evaluation at $ x = 0$ gives $ CP = 1$.",
"This proves $ (e)$.$ It remains to prove $(f)$ .",
"Fix $x_0 \\in (0,1)$ and let $K$ be a compact subset of $\\hat{ \\setminus [0, x_0].The function \\lambda (x,0,\\cdot ) is bounded on \\mathcal {S}_{(x,0)} except for a simple pole at k = x.Hence,{\\begin{@align*}{1}{-1}& \\sup _{k \\in K} \\big |\\Phi _0(x_2, k^+) - \\Phi _0(x_1, k^+)\\big |= \\sup _{k \\in K} \\bigg |\\int _{x_1}^{x_2} (\\mathsf {U}_0\\Phi _0)(x, k^+) dx\\bigg |\\\\& \\le \\bigg ( \\sup _{k \\in K} \\sup _{x \\in [0, x_0)} |x^\\alpha \\mathsf {U}_0(x,k^+)|\\bigg ) \\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha }\\Phi _0(x, k^+)| dx \\right) \\\\&\\le C \\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha }\\Phi _0(x, k^+)| dx \\right), \\qquad x_1, x_2 \\in [0, x_0),\\end{@align*}}where the right-hand side tends to zero as x_2 \\rightarrow x_1, because{\\begin{@align*}{1}{-1}\\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha } \\Phi _0(x, k^+)| dx \\right)& \\le \\sum _{j=0}^{\\infty } \\frac{1}{j!}",
"\\sup _{k\\in K} \\Vert \\mathsf {U}_0(\\cdot ,k^+)\\Vert _{L^1([0,x_0])}^j \\int _{x_1}^{x_2} x^{-\\alpha } dx\\\\& \\le \\frac{e^{C(x_0)}(x_2^{1-\\alpha }-x_1^{1-\\alpha })}{1-\\alpha },\\end{@align*}}where C(x_0) is chosen as in the proof of (a).This shows that the map (\\ref {xkphimap}) is continuous [0, x_0) \\rightarrow L^\\infty (K).If x \\in (0, x_0), then{\\begin{@align*}{1}{-1}\\sup _{k \\in K} \\bigg | &\\frac{\\Phi _0(x+h, k^+) - \\Phi _0(x,k^+)}{h} - \\Phi _{0x}(x, k^+)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | \\frac{1}{h} \\int _x^{x+h} (\\mathsf {U}_0\\Phi _0)(x^{\\prime }, k^+)~dx^{\\prime } - \\Phi _{0x}(x,k^+)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | (\\mathsf {U}_0\\Phi _0)(\\xi , k^+)~ - (\\mathsf {U}_0\\Phi _0)(x, k^+)\\bigg |,\\end{@align*}}where \\xi lies between x and x+h.",
"As h \\rightarrow 0, the right-hand side goes to zero.",
"Hence (\\ref {xkphimap}) is differentiable as a map (0,x_0) \\rightarrow L^\\infty (K) and the derivative satisfies \\Phi _{0x}(x,k^+) = \\mathsf {U}_0(x,k^+)\\Phi _{0}(x).Furtermore, the mapx \\mapsto \\big (k \\mapsto \\lambda (x,0,k^+)\\big )is C^\\infty from (0, x_0) to L^\\infty (K) and \\mathcal {E}_{0} is C^{n} on (0,1).",
"Hence the mapx \\mapsto \\big (k \\mapsto \\mathsf {U}_0(x, k^+)\\big )is C^{n-1} from (0, x_0) to L^\\infty (K).It follows that (\\ref {xkphimap}) is a C^n-map (0, x_0) \\rightarrow L^\\infty (K).",
"}Finally, since$$x^\\alpha \\Phi _{0x}(x,k^+) = x^\\alpha \\mathsf {U}_0(x, k^+) \\Phi _0(x,k^+)$$we see that $ x x0x(x,k+)$ is continuous $ [0, x0) L(K)$.This proves $ (f)$ and completes the proof of the lemma.$ Lemma 5.2 (Solution of the $y$ -part) The eigenfunction $\\Phi _1(y,P)$ is well-defined for $y \\in [0,1)$ and $P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1$ and has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function y \\mapsto \\Phi _1(y,k^+) is continuous on [0,1) and is C^n on~ (0,1).",
"Furthermore, for each y \\in [0,1), the function k \\mapsto \\Phi _0(x,k^+) is analytic on \\hat{ \\setminus [0,1].",
"}\\item \\Phi _1 obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _1(y,k^+) = \\sigma _3\\Phi _1(y, k^-)\\sigma _3,\\\\\\Phi _1(y,k^\\pm ) = \\sigma _1\\overline{\\Phi _1(y, \\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad y \\in [0,1), \\ k \\in \\hat{ \\setminus [0, 1].",
"}\\end{@align}\\item For each~ y \\in [0, 1), \\Phi _1(y,P) is an analytic function of P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1.\\item The value of \\Phi _1 at P = \\infty ^+ is given by{\\begin{@align}{1}{-1}\\Phi _1(y,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}_1(y)} & 1 \\\\ \\mathcal {E}_1(y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad y \\in [0, 1).\\end{@align}}}}\\item The determinant of $ 1$ is given by$$\\det \\Phi _1(y,P) = \\text{\\upshape Re\\,}\\mathcal {E}_1(y), \\qquad y \\in [0,1), \\ P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1.$$$ For each $y_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [0, y_0],{\\begin{@align}{1}{-1}y \\mapsto \\big (k \\mapsto \\Phi _1(y,k^+)\\big )\\end{@align}}is a continuous map [0, y_0) \\rightarrow L^\\infty (K) and a C^n-map (0, y_0) \\rightarrow L^\\infty (K).",
"Moreover, the mapy \\mapsto \\big ( k \\mapsto y^\\alpha \\Phi _{1y}(y,k^+)\\big ) is continuous [0, y_0) \\rightarrow L^\\infty (K).",
"}$ The proof is similar to that of Lemma REF .",
"Uniqueness The following lemma ensures uniqueness of the solution of the RH problem (REF ).",
"The proof relies on the fact that the determinant of the jump matrix $v$ defined in (REF ) is constant on each of the subcontours $\\Gamma _0$ and $\\Gamma _1$ .",
"Lemma 5.3 Suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Then, for each $(x,y)\\in D$ , the solution $m(x,y,\\cdot )$ of the RH problem (REF ) is unique, if it exists.",
"Moreover, $\\det m(x,y,z) = 1, \\qquad (x,y)\\in D, \\ z \\in \\Omega _\\infty .$ Fix $(x,y) \\in D$ .",
"By (REF ) and the definition (REF ) of $v$ , we have $\\det v(x,y,z) = {\\left\\lbrace \\begin{array}{ll} \\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0, \\quad & z \\in \\Gamma _0, \\\\\\text{\\upshape Re\\,}\\mathcal {E}_1(y) > 0, \\quad & z \\in \\Gamma _1.\\end{array}\\right.",
"}$ Hence $\\sqrt{\\det v(x,y,z)} = {\\left\\lbrace \\begin{array}{ll} c_0(x), \\quad & z \\in \\Gamma _0, \\\\c_1(y), & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad (x,y) \\in D,$ where the two functions $c_0(x) > 0$ and $c_1(y) > 0$ are independent of $z$ .",
"The function $m(x,y,\\cdot )$ is a solution of the RH problem (REF ) if and only if the function $\\tilde{m}(x,y,\\cdot )$ defined by $\\tilde{m}(x,y,z) = {\\left\\lbrace \\begin{array}{ll} c_0(x) m(x,y,z), \\quad & z \\in \\Omega _0, \\\\c_1(y) m(x,y,z), & z \\in \\Omega _1, \\\\m(x,y,z), & z \\in \\Omega _\\infty , \\end{array}\\right.",
"}$ satisfies the RH problem ${\\left\\lbrace \\begin{array}{ll}\\text{$\\tilde{m}(x, y, \\cdot )$ is analytic in $\\Gamma $},\\\\\\text{$\\tilde{m}_+(x,y,z) = \\tilde{m}_-(x,y,z) \\tilde{v}(x,y,z)$ for all $z \\in \\Gamma $},\\\\\\text{$\\tilde{m}(x, y, z) = I + O(z^{-1})$ as $z\\rightarrow \\infty $},\\end{array}\\right.",
"}$ where $\\tilde{v}(x,y,z) = {\\left\\lbrace \\begin{array}{ll} \\frac{1}{c_0(x)}v(x,y,z), & z \\in \\Gamma _0, \\\\\\frac{1}{c_1(y)} v(x,y,z), \\quad & z \\in \\Gamma _1.\\end{array}\\right.",
"}$ But $\\det \\tilde{v}(x,y,z) = 1$ for all $z \\in \\Gamma $ ; hence the solution $\\tilde{m}(x,y,\\cdot )$ is unique and $\\det \\tilde{m} =1$ .",
"It follows that the solution $m$ is unique and that $\\det m(x,y,z) = \\det \\tilde{m}(x,y,z) = 1$ for $z \\in \\Omega _\\infty $ .",
"Proofs of main results In this section, we use the lemmas from the previous section to prove Theorem REF –REF .",
"Proofs of Theorem 1 & 2 Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ be complex-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ .",
"We will show that $\\mathcal {E}(x,y)$ can be uniquely expressed in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ by (REF ).",
"The idea in what follows is to introduce a solution $\\Phi $ of (REF ) as the solution of the integral equation $\\Phi (x,y,k^\\pm ) = I + \\int _{(0,0)}^{(x,y)} (W\\Phi )(x^{\\prime },y^{\\prime }, k^\\pm ).$ However, since $W$ in general is singular on the boundary of $D$ , we need to be more careful with the definition.",
"We therefore instead define $\\Phi $ as the solution of $\\Phi (x, y, k^+) = \\Phi _0(x, k^+) + \\int _0^y (\\mathsf {V} \\Phi )(x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}$ Lemma 6.1 (Solution of Lax pair equations) The function $\\Phi (x,y,P)$ defined in (REF ) has the following properties: $\\Phi (x,y,k^\\pm )$ is a well-defined $2\\times 2$ -matrix valued function of $(x,y) \\in D$ and $k \\in \\hat{ \\setminus [0,1] which also satisfies the alternative Volterra integral equation:{\\begin{@align}{1}{-1}\\Phi (x, y, k^+) = \\Phi _1(y, k^+) + \\int _0^x (\\mathsf {U} \\Phi )(x^{\\prime },y,k^+) dx^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}}\\item For each $ k [0,1]$, the function $ (x,y) (x,y,k+)$ is continuous on $ D$ and is $ Cn$ on~ $ intD$.$ For each $k \\in \\hat{ \\setminus [0,1], the functions(x,y) \\mapsto x^\\alpha \\Phi _x(x,y,k^+), \\quad (x,y) \\mapsto y^\\alpha \\Phi _y(x,y,k^+), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha \\Phi _{xy}(x,y,k^+),are continuous on D.}\\item $$ obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi (x,y,k^+) = \\sigma _3\\Phi (x, y, k^-)\\sigma _3,\\\\\\Phi (x,y,k^\\pm ) = \\sigma _1\\overline{\\Phi (x, y,\\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}\\item For each~ point $ (x,y) D$, $ (x,y,P)$ extends continuously to an analytic function of $ P S(x,y) $, where $ = 0 1$ is the contour defined in (\\ref {Sigma01def}).$ The value of $\\Phi $ at $P = \\infty ^+$ is given by $\\Phi (x,y,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad (x,y) \\in D.$ The determinant of $\\Phi $ is given by $\\det \\Phi (x,y,P) = \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0, \\qquad (x,y) \\in D, \\ P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .$ By Lemma REF the lemma holds for $y = 0$ , i.e., the function $\\Phi (x,0,P)$ is well-defined and the properties $(a)$ -$(d)$ are satisfied when $x = 0$ or $y=0$ .",
"In order to see that $\\Phi $ is well-defined also for $(x,y)$ in the interior of $D$ , we note that (REF ) implies $\\Phi (x, y, k^+) = \\Phi (x,0, k^+) + \\int _0^y \\mathsf {V}(x,y^{\\prime },k^+) \\Phi (x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}The same type of successive approximation argument already used in the proof of Lemma \\ref {xpartlemma} shows that the Volterra equation (\\ref {Phidefxy}) has a unique solution for each fixed x \\in (0,1) and each k \\in \\hat{ \\setminus [0,1], and that this solution \\Phi (x,y,P) extends continuously to an analytic function of P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .",
"This proves (b).",
"}In order to prove (a), it remains to deduce the alternative representation (\\ref {Phidef2}).",
"Note that \\Phi _y = V\\Phi by definition and{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)&=\\Phi _x(x,0,k^+) + \\int _{0}^y x\\Phi (x,y^{\\prime },k^+)+ _x(x,y^{\\prime },k^\\pm )dy^{\\prime }.\\end{@align*}}Since \\mathcal {E} is a solution of the Goursat problem, we havex=\\mathsf {U}_y +[\\mathsf {U},,and, moreover, \\Phi _x(x,0,k^+) =\\mathsf {U}\\Phi (x,0,k^+).",
"Now a straightforward calculation shows{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)= \\mathsf {U}\\Phi (x,y,k^+) + \\int _{0}^y _x(x,y^{\\prime },k^\\pm )-\\Phi (x,y^{\\prime },k^+) dy^{\\prime }.\\end{@align*}}Thus the function \\tilde{\\Phi }=\\Phi _x -\\mathsf {U}\\Phi is the unique solution of the Volterra integral equation\\tilde{\\Phi }(x,y^{\\prime },k^+) = \\int _0^y {\\Phi }(x,y^{\\prime },k^+)dy^{\\prime }giving \\tilde{\\Phi }=0.",
"This implies \\Phi _x = \\mathsf {U}\\Phi .",
"Consequently, \\Phi , defined by (\\ref {Phidef}), is an eigenfunction for the Lax pair equations (\\ref {lax}).",
"The difference between (\\ref {Phidef}) and (\\ref {Phidef2}) is given by{\\begin{@align*}{1}{-1}&\\Phi _0(x,k^+) - \\Phi _1(y,k^+) +\\int _0^y (x,y^{\\prime },k^+)dy^{\\prime } - \\int _0^x \\mathsf {U}\\Phi (x^{\\prime },y,k^+)dx^{\\prime }\\\\=& \\, \\int _0^y \\int _0^x ()_x(x^{\\prime },y^{\\prime },k^+)dx^{\\prime } dy^{\\prime } - \\int _0^x \\int _0^y (\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+)dy^{\\prime }dx^{\\prime }\\\\=& \\, \\int _0^x \\int _0^y ()_x(x^{\\prime },y^{\\prime },k^+)-(\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+) dy^{\\prime } dx^{\\prime }\\end{@align*}}and ()_x = (\\mathsf {U}\\Phi )_y is the compatibility condition for the Lax pair.",
"Hence the two representations (\\ref {Phidef}) and (\\ref {Phidef2}) are equal.This proves (a).$ The symmetries (REF ) of $\\lambda $ show that $W(x,y,k^+) = \\sigma _3W(x,y, k^-)\\sigma _3, \\qquad W(x,y,k^+) = \\sigma _1\\overline{W(x,y,\\bar{k}^+)}\\sigma _1.$ Since $\\lambda (x,y,\\infty ^+) = 1$ , $\\Phi (x,y,\\infty ^+)$ satisfies the equation $\\Phi _y(x, y, \\infty ^+) = \\frac{1}{2\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} \\begin{pmatrix} \\overline{\\mathcal {E}_y(x,y)} & \\overline{\\mathcal {E}_y(x,y)} \\\\\\mathcal {E}_y(x,y) & \\mathcal {E}_y(x,y) \\end{pmatrix} \\Phi (x,y,\\infty ^+), \\qquad (x,y) \\in D.$ Using the above equations and arguing as in the proof of Lemma REF , the statements $(c)$ , $(d)$ , $(e)$ , $(f)$ , and $(g)$ follow from equation (REF ) and the corresponding statements in Lemma REF .",
"Part $(g)$ of Lemma REF implies that the inverse matrix $\\Phi (x,y,P)^{-1}$ is well-defined for $(x,y) \\in D$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma $ .",
"Lemma 6.2 For each $(x,y) \\in D$ , $P \\mapsto \\Phi (x,y,P)\\Phi (x,0, P)^{-1} \\quad \\text{and} \\quad P \\mapsto \\Phi (x,y,P)\\Phi (0,y,P)^{-1}$ are analytic functions of $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _0$ , respectively.",
"Let $U$ be an open set in $\\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ .",
"Multiplying (REF ) by $\\Phi (x,0,P)^{-1}$ from the right, we find $\\nonumber & \\Phi (x, y, P)\\Phi (x,0,P)^{-1} = I + \\int _0^y \\mathsf {V}(x,y^{\\prime },P) \\Phi (x,y^{\\prime },P)\\Phi (x,0,P)^{-1} dy^{\\prime },\\\\ & \\hspace{227.62204pt} (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}where the values of \\Phi (x,0,P) and \\lambda (x,y^{\\prime },P) in (\\ref {phiPmap}) are to be interpreted as in Remark \\ref {tilderemark}.",
"SinceP \\mapsto \\lambda (x,y^{\\prime },P)^{-1} = \\sqrt{\\frac{k - x}{k - (1-y^{\\prime })}}is an analytic map U \\rightarrow for each y^{\\prime }, so is \\mathsf {V}(x,y^{\\prime },\\cdot ).",
"It follows that the solution \\Phi (x, y, P)\\Phi (x,0,P)^{-1} of (\\ref {phiPmap}) also is analytic for P \\in U.",
"This establishes the desired statement for the first map in (\\ref {phiminusphi}); the proof for the second map is similar.$ Let $\\Omega _0$ , $\\Omega _1$ , and $\\Omega _\\infty $ denote the three components of $\\hat{ \\setminus \\Gamma defined in (\\ref {Omegadef}) and displayed in Figure \\ref {Omegas.pdf}.",
"}$ Lemma 6.3 The $2\\times 2$ -matrix valued function $m(x,y,z)$ defined for $(x,y)\\in D$ by $m(x,y,z) =\\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi \\big (x,y,F_{(x,y)}^{-1}(z)\\big ) \\times {\\left\\lbrace \\begin{array}{ll} \\Phi \\big (x,0,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _0, \\\\\\Phi \\big (0,y,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _1, \\\\I, & z \\in \\Omega _\\infty ,\\end{array}\\right.",
"}$ satisfies the RH problem (REF ) and the relation (REF ) for each $(x,y) \\in D$ .",
"Since $F_{(x,y)}$ is a biholomorphism $\\mathcal {S}_{(x,y)}~\\rightarrow \\hat{, we infer from Lemma \\ref {claim1} together with Lemma \\ref {claim2} that m(x,y, \\cdot ) is analytic in \\Gamma and that m(x,y,z) \\rightarrow I as z \\rightarrow \\infty for each (x,y) \\in D. The jump condition in (\\ref {RHm}) holds as a consequence of the definition (\\ref {jumpdef}) of v(x,y,z) and the fact that\\Phi _0(x,k) = \\Phi (x,0,k), \\qquad \\Phi _1(y,k) = \\Phi (0,y,k).Finally, since 0 \\in \\Omega _\\infty and F_{(x,y)}^{-1}(0) = \\infty ^-, the first symmetry in (\\ref {phisymmetries}) yields{\\begin{@align}{1}{-1}m(x,y,0) = \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi (x,y, \\infty ^-)= \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\sigma _3 \\Phi \\big (x,y, \\infty ^+\\big ) \\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phiatinftyplus}) for \\Phi \\big (x,y, \\infty ^+\\big ), the (11) and (21) entries of (\\ref {mxy0}) give{\\begin{@align*}{1}{-1}(m(x,y,0))_{11} = \\frac{1 + \\mathcal {E}(x,y) \\overline{\\mathcal {E}(x,y)}}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}, \\qquad (m(x,y,0))_{21} = \\frac{(1 - \\mathcal {E}(x,y))(1 + \\overline{\\mathcal {E}(x,y)})}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}.\\end{@align*}}Solving these two equations for \\mathcal {E} and \\bar{\\mathcal {E}}, we find (\\ref {Erecover}).",
"}$ We have showed that if $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ , then $\\mathcal {E}(x,y)$ can be expressed in terms of the function $m$ defined in (REF ) via equation (REF ).",
"By Lemma REF , this function $m(x,y,z)$ is the unique solution of the RH-problem (REF ) whose formulation involves only the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ .",
"As a consequence, the value of the solution $\\mathcal {E}$ at $(x,y)$ is uniquely determined by the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ , if it exists.",
"This completes the proofs of Theorem REF and REF .",
"Proof of Theorem REF This subsection is devoted to proving Theorem REF regarding existence.",
"Let us therefore suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Define $\\Phi _0(x,P)$ and $\\Phi _1(y,P)$ in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ via the Volterra equations (REF ).",
"Then $\\Phi _0$ and $\\Phi _1$ have the properties listed in Lemma REF and Lemma REF .",
"Let $\\delta \\in (0,1)$ and let $D_\\delta $ be the triangle defined in (REF ).",
"As in the proof of Theorem REF , choose $\\epsilon > 0$ so small that $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ are contained in the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, for all $(x,y) \\in D_\\delta $ .",
"Fix two smooth nonintersecting clockwise contours $\\Gamma _0$ and $\\Gamma _1$ in the complex $z$ -plane which encircle the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, but which do not encircle zero, see Figure REF .",
"Suppose $\\Gamma _0$ and $\\Gamma _1$ are invariant under the involutions $z \\mapsto z^{-1}$ and $z \\mapsto \\bar{z}$ .",
"Let $\\Gamma = \\Gamma _0 \\cup \\Gamma _1$ and consider the family of RH problems given in (REF ) parametrized by the two parameters $(x,y) \\in D_\\delta $ .",
"We will show that if (REF ) has a (unique) solution $m(x,y,z)$ for each $(x,y) \\in D_\\delta $ , then the function $\\mathcal {E}(x,y)$ defined in terms of $m$ via equation (REF ) satisfies ${\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\\\text{$\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (\\ref {ernst}) in $\\operatorname{int}(D_\\delta )$,}\\\\\\text{$x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta )$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}$ We next list some facts about Cauchy integrals that we will use throughout the proof.",
"If $h \\in L^2(\\Gamma )$ , then the Cauchy transform $\\mathcal {C}h$ is defined by $(\\mathcal {C}h)(z) = \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{h(z^{\\prime })}{z^{\\prime } - z} dz^{\\prime }, \\qquad z \\in \\Gamma ,$ We denote the nontangential boundary values of $\\mathcal {C}f$ from the left and right sides of $\\Gamma $ by $\\mathcal {C}_+ f$ and $\\mathcal {C}_-f$ respectively.",
"Then $\\mathcal {C}_+$ and $\\mathcal {C}_-$ are bounded operators on $L^2(\\Gamma )$ and $\\mathcal {C}_+ - \\mathcal {C}_- = I$ .",
"Let $w(x,y,z) = v(x,y,z) - I$ .",
"We define the operator $\\mathcal {C}_w: L^2(\\Gamma ) + L^\\infty (\\Gamma ) \\rightarrow L^2(\\Gamma )$ by $\\mathcal {C}_{w}(f) = \\mathcal {C}_-(f w).$ Then $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C \\Vert w\\Vert _{L^\\infty (\\Gamma )},$ where $C = \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}$ .",
"We henceforth assume that the RH problem (REF ) has a solution for all $(x,y) \\in D_\\delta $ or, equivalently, that $I - \\mathcal {C}_{w} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"For each $(x,y) \\in D_\\delta $ , we have $v \\in C(\\Gamma )$ and $v, v^{-1} \\in I + L^2(\\Gamma ) \\cap L^\\infty (\\Gamma )$ .",
"The theory of singular integral equations then implies that the solution of the RH problem (REF ) is given by (see e.g.",
"[4] or [18]) $m = I + \\mathcal {C}(\\mu w),$ where the $2\\times 2$ -matrix valued function $\\mu (x,y,\\cdot )$ is defined by $\\mu = I + (I - \\mathcal {C}_w)^{-1}\\mathcal {C}_w I \\in I + L^2(\\Gamma ).$ Equation (REF ) can be written more explicitly as $m(x,y,z) = I + \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{(\\mu w)(x,y,s) ds}{s-z}, \\qquad (x,y) \\in D_\\delta , \\ z \\in \\hat{ \\setminus \\Gamma .",
"}$ Lemma 6.4 The map $ (x,y) \\mapsto w(x,y, \\cdot )$ is continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto y^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"For $N \\ge 0$ , let $C^N(K)$ denote the Banach space of functions on $K$ with continuous partial derivatives of order $\\le N$ equipped with the usual norm $\\Vert f\\Vert _{C^N(K)} = \\sup _{|\\alpha | \\le N} \\Vert D^\\alpha f\\Vert _{L^\\infty (K)}.$ By part $(f)$ of Lemma REF the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ): D_\\delta \\rightarrow C(K)$ is continuous for any compact set $K$ not intersecting $\\Sigma $ .",
"Moreover, assuming $F_{(x,y)}^{-1}(\\Gamma ) \\subset K $ , the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):D_\\delta \\rightarrow \\mathcal {B}(C(K), C(\\Gamma ))$ is continuous, because $\\sup _{\\Vert f\\Vert _{C(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f \\in C(K)$ on the compact set $K$ .",
"It follows that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ also is continuous.",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )): D_\\delta \\rightarrow C(\\Gamma _1)$ is continuous.",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is continuous from $D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If a sequence of holomorphic functions $f_n$ converges uniformly on an open set $\\Omega $ then the sequence of derivatives $f_n^{\\prime }$ converges uniformly on compact subsets of $\\Omega $ .",
"Fix $N \\ge n$ and let $K$ be a compact subset of $\\mathcal {S}_{(x,0)} \\setminus \\Sigma _0$ .",
"Then part $(f)$ of Lemma REF implies that the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ):\\operatorname{int}D_\\delta \\rightarrow C^N(K)$ is $C^n$ .",
"On the other hand, the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):\\operatorname{int}D_\\delta \\rightarrow \\mathcal {B}(C^N(K), C(\\Gamma ))$ is $C^n$ .",
"Indeed, the map is continuous because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f$ on the compact set $K$ .",
"Moreover, the map has a continuous partial derivative with respect to $x$ because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\bigg |\\frac{f(F_{(x+h,y)}^{-1}(z)) - f(F_{(x,y)}^{-1}(z))}{h} - \\frac{d}{dx}f(F_{(x,y)}^{-1}(z)) \\bigg | \\rightarrow 0$ as $h \\rightarrow 0$ by the mean-value theorem and the uniform continuity of the first partial derivatives of $f$ .",
"Similar arguments show that all partial derivatives of order $\\le n$ exist and are continuous.",
"We conclude that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ built from (REF ) and (REF ) is $C^n$ .",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _1)$ is $C^n$ .",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is $C^n$ as a map from $\\operatorname{int}D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If $z \\in \\Gamma _0$ , we have $w_x(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)) + \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{d x}F_{(x,y)}^{-1}(z),$ where $\\frac{d}{d x}F_{(x,y)}^{-1}(z)$ denotes the derivative of the $k$ -projection of $F_{(x,y)}^{-1}(z)$ , which is given by $\\frac{d}{d x}F_{(x,y)}^{-1}(z) = -\\frac{(z-1)^2}{4z}.$ Thus part $(f)$ of Lemma REF and of Lemma REF imply that $(x,y) \\mapsto x^\\alpha w_x(x,y,\\cdot )$ is a continuous map $D_\\delta \\rightarrow L^\\infty (\\Gamma )$ .",
"The maps $(x,y) \\mapsto y^\\alpha w_y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y,\\cdot )$ can be treated similarly.",
"Lemma 6.5 The map $(x,y) \\mapsto \\mu (x,y, \\cdot ) - I$ is continuous from $D_\\delta $ to $L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto y^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^2(\\Gamma )$ .",
"In view of the definition of $\\mu $ , the map (REF ) is given by $(x,y) \\mapsto (I - \\mathcal {C}_{w(x,y,\\cdot )})^{-1}\\mathcal {C}_-(w(x,y,\\cdot )).$ We note that the map $f \\mapsto I - \\mathcal {C}_f: L^\\infty (\\Gamma ) \\rightarrow \\mathcal {B}(L^2(\\Gamma ))$ is smooth by the estimate $\\Vert \\mathcal {C}_f\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C\\Vert f\\Vert _{L^\\infty (\\Gamma )},$ and that the linear map $f \\mapsto \\mathcal {C}_-f: L^2(\\Gamma ) \\rightarrow L^2(\\Gamma )$ is bounded.",
"Since (REF ) can be viewed as a composition of maps of the form (REF ), (REF ), and (REF ) together with the smooth inversion map $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , it follows that (REF ) is continuous $D_\\delta \\rightarrow L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Similarly, $(x,y) \\mapsto x^\\alpha \\mu _x(x,y,\\cdot )$ can be viewed as composition of the continuous maps (REF ), (REF ), $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , (REF ), and (REF ), and is hence continuous.",
"The maps $(x,y) \\mapsto y^\\alpha \\mu _y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y,\\cdot )$ can be treated analogously.",
"Lemma 6.6 The solution $m(x,y,z)$ of the RH problem (REF ) defined in (REF ) has the following properties: For each point $(x,y) \\in D_\\delta $ , $m(x,y,\\cdot )$ obeys the symmetries $m(x,y,z) = m(x,y,0)\\sigma _3m(x,y,z^{-1})\\sigma _3 = \\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}$ For each $z \\in \\hat{\\setminus \\Gamma , the map (x,y) \\mapsto m(x,y,z) is continuous from D_\\delta to {2 \\times 2} and is C^n from \\operatorname{int}D_\\delta to {2 \\times 2}.",
"}\\item For each~ $ z $, the three maps$$(x,y) \\mapsto x^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto y^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha m_{xy}(x,y, z),$$are continuous from $ D$ to $ 22$.$ The symmetries in () and () show that $v$ satisfies ${\\left\\lbrace \\begin{array}{ll} v(x,y,z) = \\sigma _3v(x, y, z^{-1})\\sigma _3,\\\\v(x,y,z) = \\sigma _1\\overline{v(x, y, \\bar{z})}\\sigma _1,\\end{array}\\right.}",
"\\qquad z \\in \\Gamma , \\ (x,y) \\in D_\\delta .$ These symmetries imply that $\\sigma _3 m(x,y,0)^{-1} m(x,y,z^{-1})\\sigma _3$ and $\\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1$ satisfy the same RH problem as $m(x,y,z)$ .",
"The symmetries in (REF ) follow by uniqueness.",
"Properties $(b)$ and $(c)$ follow from (REF ) together with the Lemmas REF and REF .",
"As in the proof of Theorem REF , we extend the definition (REF ) of $v$ to an open tubular neighborhood $N(\\Gamma ) = N(\\Gamma _0) \\cup N(\\Gamma _1)$ of $\\Gamma $ as follows, see Figure REF : $v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in N(\\Gamma _0),\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in N(\\Gamma _1),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We choose $N(\\Gamma )$ so narrow that it does not intersect the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ .",
"Then, for each $(x,y) \\in D_\\delta $ , $v(x,y,\\cdot )$ is an analytic function of $z \\in N(\\Gamma )$ .",
"Using the notation $z(x,y,P) := F_{(x,y)}(P)$ , we can write (REF ) as $v(x,y,z(x,y,P)) = {\\left\\lbrace \\begin{array}{ll} \\Phi _0(x, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _0)\\big ), \\\\\\Phi _1(y, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _1)\\big ),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We define functions $f_0(x,y,z)$ and $f_1(x,y,z)$ for $(x,y) \\in D_\\delta $ by $f_0(x,y,z) = \\big [m_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma ,\\\\f_1(x,y,z) = \\big [m_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}Moreover, we let n_0(x,y,z) and n_1(x,y,z) denote the functions given by\\begin{subequations}{\\begin{@align}{1}{-1}n_0(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_0(x,y,z) + m(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _0, \\\\f_0(x,y,z), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}and{\\begin{@align}{1}{-1}n_1(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_1(x,y,z) + m(x,y,z)\\mathsf {V}_1\\big (y,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _1, \\\\f_1(x,y,z), & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}\\end{@align}}\\end{subequations}}$ Lemma 6.7 For each $(x,y) \\in D_\\delta $ , it holds that $n_0(x,y,z)$ is an analytic function of $z \\in \\hat{ \\setminus \\lbrace -1\\rbrace and has at most a simple pole at z = -1.\\item n_1(x,y,z) is an analytic function of z \\in \\hat{ \\setminus \\lbrace 1\\rbrace and has at most a simple pole at z = 1.\\item n_0(x,y,\\infty ) = 0 and n_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.\\item n_1(x,y,\\infty ) = 0 and n_1(x,y,0) = m_y(x,y,0)m(x,y,0)^{-1}.",
"}}{\\begin{xmlelement*}{proof}By (\\ref {linearzx}) the function z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1, \\infty \\rbrace with simple poles at z = -1 and z = \\infty .Equation (\\ref {msolution}) implies that m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty .",
"Hence f_0(x,y,z) is analytic at z = \\infty .It follows that f_0(x,y,z) is analytic for all z \\in \\hat{\\setminus (\\Gamma \\cup \\lbrace -1\\rbrace ) with a simple pole at z = -1 at most.Now f_0 satisfies the following jump condition across \\Gamma :{\\begin{@align}{1}{-1}\\nonumber f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\big [v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z)\\big ]\\\\ & \\times v(x,y,z)^{-1} m_-(x,y,z)^{-1}, \\qquad z \\in \\Gamma .\\end{@align}}Differentiating (\\ref {vzPhi0}) with respect to x and y and evaluating the resulting equations at k = F_{(x,y)}^{-1}(z), we find{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)),\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _0),\\end{@align}} and{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{1y}(x,F_{(x,y)}^{-1}(z)),\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _1).\\end{@align}} Using the first equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}) in (\\ref {f0jump}), we conclude that f_0 is analytic across \\Gamma _1 and has the following jump across \\Gamma _0:{\\begin{@align}{1}{-1}f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) m_-(x,y,z)^{-1}, \\quad \\ z \\in \\Gamma _0.\\end{@align}}Thus n_0 is analytic across \\Gamma .",
"Furthermore, since \\lambda (x,y,k) is analytic on \\mathcal {S}_{(x,y)} except for a simple pole at the branch point k = x, the function \\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1\\rbrace with a simple pole at z = -1.",
"It follows that n_0 satisfies (a).",
"The proof of (b) is similar and relies on the second equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}).",
"}Using (\\ref {linearzx}) in the definition (\\ref {ndefa}) of n_0, we can write, for z \\in \\Omega _\\infty ,{\\begin{@align}{1}{-1}n_0(x,y,z) = f_0(x,y,z) = \\Big [m_x(x,y,z) -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z(x,y,z)\\Big ]m(x,y,z)^{-1}.\\end{@align}}Since m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty , it follows that n_0(x,y,\\infty ) = 0.",
"On the other hand, evaluating (\\ref {n0f0}) at z = 0, we findn_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.This proves (c); the proof of (d) is analogous.",
"}}\\end{xmlelement*}Let \\hat{m}(x,y) denote the function m(x,y,z)~ evaluated at z = 0, that is,\\hat{m}(x,y) = m(x,y,0).Evaluating the first symmetry in (\\ref {msymm}) at z = \\infty , we find{\\begin{@align}{1}{-1}I = \\hat{m}(x,y)\\sigma _3\\hat{m}(x,y)\\sigma _3.\\end{@align}}The unit determinant condition (\\ref {detmone}) implies that \\det \\hat{m} = 1.",
"Hence equation (\\ref {m0sigma3}) reduces to\\text{adj}(\\hat{m})=\\sigma _3 \\hat{m} \\sigma _3,where \\text{adj} denotes the adjugate matrix, which shows that \\hat{m}_{11} = \\hat{m}_{22}.A straightforward algebraic computation then yields{\\begin{@align}{1}{-1}\\hat{m}(x,y) = \\tilde{\\Phi }(x,y) \\sigma _3\\tilde{\\Phi }(x,y)\\sigma _3, \\qquad (x,y) \\in D_\\delta ,\\end{@align}}where the 2\\times 2-matrix valued function \\tilde{\\Phi }(x,y) is defined by{\\begin{@align}{1}{-1}\\tilde{\\Phi }(x,y) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}\\end{@align}}and the functions \\mathcal {E}(x,y) and \\overline{\\mathcal {E}(x,y)} are defined by{\\begin{@align}{1}{-1}\\mathcal {E} = \\frac{1 + \\hat{m}_{11} - \\hat{m}_{21}}{1 + \\hat{m}_{11} + \\hat{m}_{21}}, \\qquad \\bar{\\mathcal {E}} = -\\frac{1 - \\hat{m}_{11} + \\hat{m}_{21}}{1 - \\hat{m}_{11} - \\hat{m}_{21}}.\\end{@align}}The second symmetry in (\\ref {msymm}) evaluated at z = 0 implies{\\begin{@align}{1}{-1}\\hat{m}_{11} = \\overline{\\hat{m}_{22}}, \\qquad \\hat{m}_{12} = \\overline{\\hat{m}_{21}}.\\end{@align}}Recalling the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, it follows that \\bar{\\mathcal {E}} is the complex conjugate of \\mathcal {E}.",
"The next lemma shows, among other things, that \\mathcal {E} is free of singularities.",
"}\\begin{lemma}The function \\mathcal {E}(x,y) defined in (\\ref {Edef}) has the following properties:{\\begin{@align*}{1}{-1}{\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ),\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}\\end{@align*}}\\end{lemma}{\\begin{xmlelement*}{proof}By Lemma \\ref {claim4E}, the map (x,y) \\mapsto \\hat{m}(x,y) is continuous from D_\\delta to and is C^n from \\operatorname{int}D_\\delta to .The first equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) also has these regularity properties except possibly on the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = -1\\rbrace \\end{@align}}where the denominator vanishes.In the same way, the second equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) is regular away from the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = 1\\rbrace .\\end{@align}}Since the sets (\\ref {singular1}) and (\\ref {singular2}) are disjoint and closed in D_\\delta , we conclude that \\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ).",
"That x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ) follows by differentiating (\\ref {Edef}) and applying Lemma \\ref {claim4E}.\\end{xmlelement*}We next show that \\text{\\upshape Re\\,}\\mathcal {E} > 0 on D_\\delta .Equation (\\ref {Edef}) yields\\mathcal {E} + \\bar{\\mathcal {E}} = \\frac{4\\hat{m}_{21}}{(\\hat{m}_{11} +\\hat{m}_{21} )^2 -1}.In light of the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, this gives{\\begin{@align}{1}{-1}\\text{\\upshape Re\\,}\\mathcal {E} = \\frac{2(1 + \\hat{m}_{11})}{|1 + \\hat{m}_{11} + \\hat{m}_{12}|^2}.\\end{@align}}On the other hand, the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1 together with (\\ref {m0symm}) yield \\hat{m}_{11} \\in {R} and \\hat{m}_{11}^2 - |\\hat{m}_{12}|^2 = 1.",
"We infer that \\hat{m}_{11} \\in (-\\infty ,-1] \\cup [1, \\infty ).For (x,y) = (0,0) we have m(0,0,z) = I for all z, because the jump matrix v is the identity matrix.",
"In particular, \\hat{m}_{11}(0,0) = 1.",
"By continuity, this gives (\\hat{m}(x,y))_{11} \\ge 1 for all ~(x,y) \\in D_\\delta .",
"In view of (\\ref {reEm0}), it follows that \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0 on D_\\delta .",
"}Finally, we show that $ E(x,0) = E0(x)$ for $ x [0, 1-)$; the proof that $ E(0,y) = E1(y)$ for $ y [0,1-)$ is similar.For $ y = 0$, the definition (\\ref {jumpdef}) of $ v$ yields{\\begin{@align}{1}{-1}v(x,0,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\I, & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad x \\in [0, 1-\\delta ).\\end{@align}}It follows from part $ (c)$ of Lemma \\ref {xpartlemma} that the $ 22$-matrix valued function $ m0(y,z)$ defined for $ x [0,1-)$ by{\\begin{@align}{1}{-1}m_0(x,z) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _0, \\\\\\Phi _0\\big (x,F_{(x,0)}^{-1}(z)\\big ), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}satisfies the RH problem (\\ref {RHm}) associated with $ (x,y) = (x,0)$ for each $ x [0,1-)$.Furthermore, since $ 0 $ and $ F(x,y)-1(0) = -$, the first symmetry in (\\ref {phi0symmetries}) yields{\\begin{@align}{1}{-1}m_0(x,0) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\Phi _0\\big (x,\\infty ^-\\big )= \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\sigma _3\\Phi _0\\big (x,\\infty ^+\\big )\\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phi0atinftyplus}) for $ 0(x,+)$, the $ (11)$ and $ (21)$ entries of (\\ref {mx00}) give{\\begin{@align*}{1}{-1}(m_0(x,0))_{11} = \\frac{1 + \\mathcal {E}_0(x) \\overline{\\mathcal {E}_0(x)}}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}, \\qquad (m_0(x,0))_{21} = \\frac{(1 - \\mathcal {E}_0(x))(1 + \\overline{\\mathcal {E}_0(x)})}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}.\\end{@align*}}Solving these two equations for $ E0$ and $ E0$, we find{\\begin{@align}{1}{-1}\\mathcal {E}_0(x) = \\frac{1 + (m_0(x,0))_{11} - (m_0(x,0))_{21}}{1 + (m_0(x,0))_{11} + (m_0(x,0))_{21}}.\\end{@align}}But by uniqueness of the solution of the RH problem (\\ref {RHm}), we have $ m0(x,z) = m(x,0,z)$; hence, comparing (\\ref {E0recover}) with (\\ref {Erecover}), we deduce that $ E(x,0) = E0(x)$ for $ x [0, 1-)$.$ It only remains to show that $\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"The proof of this relies on the construction of an eigenfunction $\\Phi $ of the Lax pair.",
"Equations (REF ) and (REF ) suggest that we define $\\Phi (x,y,P)$ for $(x,y) \\in D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ by $\\Phi (x,y,P) = \\tilde{\\Phi }(x,y)m(x,y,F_{(x,y)}(P)),$ where $\\tilde{\\Phi }(x,y)$ is the function defined in ().",
"Lemma 6.8 The function $\\Phi $ defined in (REF ) satisfies the Lax pair equations ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y, P) = \\mathsf {U}(x,y, P) \\Phi (x,y, P),\\\\\\Phi _y(x,y, P) = \\mathsf {V}(x,y,P) \\Phi (x,y,P),\\end{array}\\right.",
"}$ for $(x,y) \\in \\operatorname{int}D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ .",
"The analyticity structure of $n_0$ established in Lemma REF implies that there exists a $2\\times 2$ -matrix valued function $C(x,y)$ independent of $z$ such that $n_0(x,y,z) = \\frac{C(x,y)}{z+1}, \\qquad z \\in \\hat{.",
"}We determine C(x,y) by evaluating (\\ref {n0C}) at z = 0.",
"By Lemma \\ref {claim5E}, this gives C(x,y) = \\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}.It follows that{\\begin{@align}{1}{-1}n_0 = \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1} = \\bigg (m_x -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z\\bigg )m^{-1}\\end{@align}}for (x,y) \\in D_\\delta and z \\in \\Omega _\\infty .$ Differentiating (REF ) with respect to $x$ and using (), we find, for $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ , $\\Phi _x(x,y,P)& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y)(m_x + z_x m_z)\\\\& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1}m(x,y,z)\\\\& = \\bigg (\\tilde{\\Phi }_x(x,y)\\tilde{\\Phi }(x,y)^{-1}+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z(x,y,P)+1}\\tilde{\\Phi }(x,y)^{-1}\\bigg )\\Phi (x,y,P)$ Substituting in the expressions () and () for $\\tilde{\\Phi }$ and $\\hat{m}$ in terms of $\\mathcal {E}$ , $\\bar{\\mathcal {E}}$ , and recalling that $1 - \\frac{2}{z+1} = \\lambda ,$ this yields the first equation in (REF ).",
"A similar argument gives the second equation in (REF ).",
"Lemma 6.9 The complex-valued function $\\mathcal {E}:D \\rightarrow {R}$ defined by (REF ) satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"Fix a point $P = (\\lambda , k)$ in $F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ .",
"By Lemma , the map $(x,y) \\mapsto \\Phi (x,y, P)$ is $C^n$ from $\\operatorname{int}D_\\delta $ to $ and satisfies the Lax pair equations (\\ref {philax}).",
"Since~ $ n 2$, it follows that $$ satisfies{\\begin{@align*}{1}{-1}\\Phi _{xy}(x,y,P) - \\Phi _{yx} (x,y,P) = 0, \\qquad (x,y) \\in \\operatorname{int}D_\\delta .\\end{@align*}}The $ (21)$-entry of this equation reads$$\\frac{(1-x-y)\\lambda }{2(\\text{\\upshape Re\\,}\\mathcal {E}(x,y))^2 (1-k-y)}\\bigg \\lbrace (\\text{\\upshape Re\\,}\\mathcal {E})\\bigg (\\mathcal {E}_{xy} - \\frac{\\mathcal {E}_x + \\mathcal {E}_y}{2(1-x-y)}\\bigg ) - \\mathcal {E}_x \\mathcal {E}_y\\bigg \\rbrace = 0.$$It follows that $ E(x,y)$ satisfies (\\ref {ernst}) for $ (x,y) intD$.This completes the proof of the lemma.$ Lemma REF completes the proof of part $(a)$ of Theorem REF .",
"The following lemma proves part (b).",
"Lemma 6.10 There exists a constant $c_\\delta > 0$ such that if $\\Vert \\mathcal {E}_0 / \\text{\\upshape Re\\,}\\mathcal {E}_0 \\Vert _{L^1([0,1-\\delta ))} , \\, \\Vert \\mathcal {E}_1/ \\text{\\upshape Re\\,}\\mathcal {E}_1 \\Vert _{L^1([0,1-\\delta ))} < c_\\delta ,$ then the linear operator $I - \\mathcal {C}_{w(x,y,\\cdot )} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"It follows from () and () that, by choosing $c_\\delta $ sufficiently small, equation (REF ) gives $|\\Phi _0(x,k^\\pm )-I| < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ and an analogous estimate holds for $|\\Phi _1(y,k^\\pm )-I|$ .",
"This yields $\\Vert w(x,y,\\cdot )\\Vert _{L^\\infty (\\Gamma )} < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ for all $(x,y) \\in D_\\delta $ whenever (REF ) holds.",
"Indeed, equation (REF ) implies $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\Vert w\\Vert _{L^\\infty (\\Gamma )} < 1$ for all $(x,y) \\in D_\\delta $ .",
"Hence $I - \\mathcal {C}_{w(x,y,\\cdot )}$ is invertible in $\\mathcal {B}(L^2(\\Gamma ))$ for each $(x,y) \\in D_\\delta $ .",
"For part (c) assume $\\mathcal {E}_0, \\mathcal {E}_1>0$ and write $V_0= -\\log \\mathcal {E}_0$ , $V_1 = -\\log \\mathcal {E}_1$ .",
"Then there exists a $C^n$ -solution $V(x,y)$ of the Goursat problem for the Euler-Darboux equation (REF ) with data $\\lbrace V_0,V_1 \\rbrace $ by Theorem REF .",
"Hence $\\mathcal {E}=e^{- V}$ is a $C^n$ -solution of the Goursat problem for (REF ) with data $\\lbrace \\mathcal {E}_0,\\mathcal {E}_1 \\rbrace $ .",
"This completes the proof of part (c) and hence of Theorem REF .",
"Proof of Theorem REF Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ , be complex-valued functions satisfying (REF ) for some $n \\ge 2$ and some $\\alpha \\in (0,1)$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ and define $m_1, m_2 \\in by (\\ref {m1m2def}).We will prove (\\ref {boundarylimita}); the proof of (\\ref {boundarylimitb}) is similar.$ By (REF ), we have $x^\\alpha \\mathcal {E}_x(x,y) = 2x^\\alpha \\frac{\\hat{m}_{21}(x,y) \\hat{m}_{11x}(x,y) - (1 + \\hat{m}_{11}(x,y))\\hat{m}_{21x}(x,y)}{(1 + \\hat{m}_{11}(x,y) + \\hat{m}_{21}(x,y))^2},$ where, as before, $\\hat{m}(x,y) = m(x,y,0)$ .",
"Thus, in order to compute $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y)$ , it is enough to compute $\\hat{m}(0,y)$ and $\\lim _{x \\downarrow 0} x^\\alpha \\hat{m}_x(x,y)$ .",
"Since $m = I + \\mathcal {C}(\\mu w)$ and $m_x = \\mathcal {C}(\\mu _x w) + \\mathcal {C}(\\mu w_x),$ this means that we are interested in the values of $w(0,y,z), \\quad \\mu (0,y,z),\\quad \\lim _{x \\downarrow 0} x^\\alpha w_x(x,y,z), \\quad \\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z).$ Lemma 6.11 We have $& w(0,y, z) = {\\left\\lbrace \\begin{array}{ll}0, & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y, F_{(0,y)}^{-1}(z)\\big ) - I, \\quad & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad y \\in [0,1),\\\\~& \\mu (0,y,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y,\\infty ^+\\big )^{-1} , & z \\in \\Gamma _1,\\end{array}\\right.",
"}\\quad y \\in [0,1),$ and $\\hat{m}(0,y) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1}\\sigma _3 \\Phi _1\\big (y,\\infty ^+\\big ) \\sigma _3, \\qquad y \\in [0,1).$ Equation (REF ) is immediate from (REF ).",
"Moreover, by (REF ), $ m(0,y,z) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _1, \\\\\\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}$ Equation () follows from (REF ) and the fact that $\\mu (x,y,z)=m_-(x,y,z)$ for $(x,y) \\in D$ and $z \\in \\Gamma $ .",
"Since $0 \\in \\Omega _0$ and $F_{(0,y)}^{-1}(0) = \\infty ^-$ , equation (REF ) follows by setting $z = 0$ in (REF ) and using the first symmetry in ().",
"Lemma 6.12 For $y \\in [0,1)$ , we have $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\begin{pmatrix} \\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) & m_1 \\end{pmatrix}, \\quad & z \\in \\Gamma _0,\\\\0, & z \\in \\Gamma _1,\\end{array}\\right.",
"}$ and $\\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z)= \\Pi (y,z), \\qquad z \\in \\Gamma _1,$ where the function $\\Pi (y,z)$ is defined by $\\Pi (y,z) = - \\frac{1}{\\sqrt{1-y}}\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}}{z+1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}.$ It follows from (REF ) and (REF ) that $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) =0$ for $z \\in \\Gamma _1$ and that, for $z\\in \\Gamma _0$ , $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y,z)& = \\lim _{x \\rightarrow 0} x^\\alpha \\bigg \\lbrace \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))+ \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{dx} F_{(x,y)}^{-1}(z)\\bigg \\rbrace \\\\~& = \\lim _{x \\rightarrow 0} x^\\alpha \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))= \\lim _{x \\rightarrow 0} x^\\alpha \\mathsf {U}_0(x,F_{(x,y)}^{-1}(z)).$ Recalling the definition (REF ) of $\\mathsf {U}_0$ , (REF ) follows.",
"To prove (REF ), we note that differentiation of the relation $\\mu = I + \\mathcal {C}_w \\mu $ gives $ \\mu _x = (I-\\mathcal {C}_w)^{-1}\\mathcal {C}_-(\\mu w_x).$ We first compute $\\lim _{x \\downarrow 0} \\mathcal {C}_-(\\mu x^\\alpha w_x)$ .",
"Equations () and (REF ) imply, for $z \\in \\Gamma _1$ , $\\nonumber &\\Big \\lbrace \\mathcal {C}_- \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ]\\Big \\rbrace (z)= \\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} }{2\\pi i}\\\\\\nonumber & \\times \\int _{\\Gamma _0} \\frac{\\Phi _1\\big (y,F_{(0,y)}^{-1}(z^{\\prime })\\big )\\frac{1}{2} \\Big ({\\begin{matrix}\\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) & m_1 \\end{matrix}}\\Big ) dz^{\\prime }}{z^{\\prime } -z}\\\\ & =- \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1(y,F_{(0,y)}^{-1}(z^{\\prime })) \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime }))\\left({\\begin{matrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{matrix}}\\right)}{2(z^{\\prime } -z)} =: \\tilde{\\Pi }(y,z).$ Recalling the expression (REF ) for $\\lambda (0,0, F_{(0,y)}^{-1}(z))$ and using that $F_{(0,y)}^{-1}(-1) = 0$ , we find $\\tilde{\\Pi }(y,z) = -\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0) \\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix} }{(z+1)\\sqrt{1-y}}.$ In view of (REF ), it only remains to show that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"We have, for $z \\in \\Gamma _1$ , $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)= \\frac{1}{2\\pi i} \\int _{\\Gamma _1}\\frac{\\Pi (y, z^{\\prime })\\big (\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I\\big )}{z^{\\prime } - z_-} dz^{\\prime }\\\\& = -\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{2\\pi i\\sqrt{1-y}}\\int _{\\Gamma _1}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z_-} \\frac{dz^{\\prime }}{z^{\\prime }+1}.$ Deforming the contour to infinity and using that $\\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z} \\frac{1}{z^{\\prime }+1} = -\\frac{\\Phi _1(y, 0) - I}{z+1}, $ a residue computation gives $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)=\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{\\sqrt{1-y}} \\frac{\\Phi _1(y, 0) - I}{z+1}.$ Simple algebra now shows that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"Lemma 6.13 For $y \\in [0,1)$ , we have $\\Phi _1(y, 0) = \\begin{pmatrix} e^{\\int _0^y \\frac{\\overline{\\mathcal {E}_{1y}(y^{\\prime })}}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} & 0 \\\\ 0 & e^{\\int _0^y \\frac{\\mathcal {E}_{1y}(y^{\\prime })}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\end{pmatrix}.$ Since 0 is a real branch point of the Riemann surface $\\Sigma _{(0,y)}$ , the symmetries () of $\\Phi _1$ imply that $\\Phi _1(y, 0^+) = \\Phi _1(y, 0^-) = \\sigma _3 \\Phi _1(y, 0^+)\\sigma _3 \\quad \\text{and} \\quad \\Phi _1(y, 0) = \\sigma _1\\overline{\\Phi _1(y, 0)}\\sigma _1.$ Hence $\\Phi _1(y, 0)$ has the form $\\Phi _1(y, 0) = \\begin{pmatrix} f(y) & 0 \\\\ 0 & \\overline{f(y)} \\end{pmatrix},$ where $f(y)$ is a function of $y$ .",
"Since $\\lambda (0, y, 0) = \\infty $ , we can determine $f(y)$ by solving the equation $\\Phi _{1y}(y, 0) = \\frac{1}{2 \\text{\\upshape Re\\,}\\mathcal {E}_{1}(y)} \\begin{pmatrix} \\overline{\\mathcal {E}_{1y}(y)} & 0 \\\\~0 & \\mathcal {E}_{1y}(y) \\end{pmatrix} \\Phi _1(y,0),$ which is a consequence of (REF ).",
"This gives the desired statement.",
"The following lemma completes the proof of Theorem REF .",
"Lemma 6.14 For $y \\in [0,1)$ , we have $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y) = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}}.$ We first compute $ \\lim _{x \\downarrow 0} x^\\alpha m_x(x,y,0)$ .",
"Proceeding as in the proof of (REF ), we find $\\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ](0)=-\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0)}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix},$ and $\\nonumber \\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu _x(x,y,\\cdot ) w(x,y,\\cdot )\\big ](0)= &\\; \\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}\\\\ & \\times \\left( \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3 \\right),$ where the derivation of (REF ) employs Lemma REF and Lemma REF as well as the residue calculation $-\\frac{1}{2\\pi i} \\int _{\\Gamma _1} \\frac{\\Phi _1(y, F_{(0,y)}^{-1}(z)) - I}{z} \\frac{dz}{z+1} = \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Adding (REF ) and (REF ) and recalling (REF ), we obtain $\\nonumber \\lim _{x \\rightarrow 0} x^\\alpha m_x(x,y,0)= & - \\frac{1}{\\sqrt{1-y}}\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\\\&\\times \\Phi _1\\big (y, 0\\big )^{-1}\\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Substituting (), (REF ), (REF ), and (REF ) into (REF ), long but straightforward computations yield ().",
"Examples We consider two examples of exact solutions—one with collinear polarization and one with noncollinear polarization.",
"For each example, we verify explicitly that the formulas () of Theorem REF on the behavior near the boundary are satisfied.",
"The Khan-Penrose solution The Khan-Penrose [13] solution is given by the potential $\\mathcal {E}(x,y)= \\frac{1+\\sqrt{x}\\sqrt{1-y}+\\sqrt{y}\\sqrt{1-x}}{1-\\sqrt{x}\\sqrt{1-y}-\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y) \\in D.$ Straightforward computations show that $m_1=1=m_2$ and $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{\\sqrt{1-y}}{(1-\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= \\frac{\\sqrt{1-x}}{(1-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ The Nutku-Halil solution One version of the Nutku-Halil [23] solution is given by $\\mathcal {E}(x,y)= \\frac{1-i\\sqrt{x}\\sqrt{1-y}+i\\sqrt{y}\\sqrt{1-x}}{1+i\\sqrt{x}\\sqrt{1-y}-i\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y)\\in D.$ In this case, $m_1=-i=-m_2$ and we compute $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{i\\sqrt{1-y}}{(i+\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= -\\frac{i\\sqrt{1-x}}{(i-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ appendix Gravitational waves and the hyperbolic Ernst equation It is shown in Eq.",
"(11.7) in [11] that the Ernst potential $\\mathcal {E}$ satisfies $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{uv} - U_u \\mathcal {E}_v - U_v \\mathcal {E}_u\\right) = 4\\mathcal {E}_u \\mathcal {E}_v.$ where $e^{-U(u,v)} = f(u) + g(v)$ and $f(u)$ and $g(v)$ are monotonically decreasing for positive argument and $f(0) = g(0) = 1/2$ .",
"(Note that Griffiths writes $Z$ for the Ernst potential.)",
"As suggested by Szekeres [25], it is possible to use $(f, g)$ as coordinates.",
"This leads to the equation $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{fg} + \\frac{\\mathcal {E}_f + \\mathcal {E}_g}{f+g}\\right) = 4\\mathcal {E}_f \\mathcal {E}_g,$ where $(f,g)$ belongs to the triangular region $\\bigg \\lbrace (f,g) \\in {R}^2 \\, \\bigg | \\, f \\le \\frac{1}{2}, \\; g \\le \\frac{1}{2}, \\; f + g > 0\\bigg \\rbrace .$ The change of variables $x = \\frac{1}{2} - g$ , $y = \\frac{1}{2} - f$ transforms (REF ) into (REF ).",
"In order for the solution to describe gravitational waves, the following boundary condition must be satisfied (Eq.",
"(7.15) in [11]; see also (11.23) in [11] but in (11.23) equation $(f,g)$ approaches the corner whereas in (7.15) the two edges are approached; also in (7.15) there is a factor $(f+g)$ missing; this factor comes from (7.9)) $& \\lim _{g \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -g\\Big ) (f+g)\\frac{|\\mathcal {E}_g|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_2}{2},\\\\& \\lim _{f \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -f\\Big ) (f+g) \\frac{|\\mathcal {E}_f|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"In terms of $(x,y)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\frac{x(1-x-y)|\\mathcal {E}_x|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}= \\frac{k_2}{2},\\\\& \\lim _{y \\rightarrow 0} \\frac{y(1-x-y)|\\mathcal {E}_y|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2} = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"That is, since $\\text{\\upshape Re\\,}\\mathcal {E} > 0$ , $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1),$ for some constants $m_1 , m_2 \\in [1, \\sqrt{2})$ .",
"If we assume that $\\mathcal {E} \\in C(D)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1).$ These are the conditions given in () with $\\alpha = 1/2$ .",
"In particular, $& \\mathcal {E}_{0x}(x) = \\frac{m_1 + o(1)}{\\sqrt{x}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{0}(x) \\sim 2m_1\\sqrt{x}, \\qquad x \\downarrow 0,\\\\& \\mathcal {E}_{1y}(y) = \\frac{m_2 + o(1)}{\\sqrt{y}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{1}(y) \\sim 2m_2\\sqrt{y}, \\qquad y \\downarrow 0,$ where $m_1, m_2 \\in [1, \\sqrt{2})$ .",
"Acknowledgement The authors acknowledge support from the European Research Council, Grant Agreement No.",
"682537, the Swedish Research Council, Grant No.",
"2015-05430, and the Göran Gustafsson Foundation, Sweden."
],
[
"Lax pair and eigenfunctions",
"In this section we introduce a Lax pair for (REF ) and define appropriate eigenfunctions in preparation for the proofs of Theorems REF -REF ."
],
[
"Lax pair",
"The hyperbolic Ernst equation (REF ) admits the Lax pair ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y, k) = \\mathsf {U}(x,y, k) \\Phi (x,y, k),\\\\\\Phi _y(x,y, k) = \\mathsf {V}(x,y,k) \\Phi (x,y,k),\\end{array}\\right.",
"}$ where $k$ is the spectral parameter, the function $\\Phi (x,y, k)$ is a $2 \\times 2$ -matrix valued eigenfunction, and the $2\\times 2$ -matrix valued functions $\\mathsf {U}(x,y,k)$ and $\\mathsf {V}(x,y,k)$ are defined as follows: $\\mathsf {U} = \\frac{1}{\\mathcal {E} + \\bar{\\mathcal {E}}} \\begin{pmatrix} \\bar{\\mathcal {E}}_x & \\lambda \\bar{\\mathcal {E}}_x \\\\\\lambda \\mathcal {E}_x & \\mathcal {E}_x \\end{pmatrix}, \\qquad \\mathsf {V} = \\frac{1}{\\mathcal {E} + \\bar{\\mathcal {E}}} \\begin{pmatrix} \\bar{\\mathcal {E}}_y & \\frac{1}{\\lambda } \\bar{\\mathcal {E}}_y \\\\\\frac{1}{\\lambda } \\mathcal {E}_y & \\mathcal {E}_y\\end{pmatrix},$ with $\\lambda $ given by (REF ).",
"We write the Lax pair (REF ) in terms of differential forms as $d\\Phi = W\\Phi ,$ where $W$ is the closed one-form $W = \\mathsf {U}dx + \\mathsf {V}dy.$ As in Section , we will view the map $\\Phi (x,y,\\cdot )$ as being defined on the Riemann surface $\\mathcal {S}_{(x,y)}$ and write $\\Phi (x,y,P)$ for the value of $\\Phi $ at $P = (\\lambda , k) \\in \\mathcal {S}_{(x,y)}$ ."
],
[
"Spectral analysis",
"Suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Let $\\mathsf {U}_0$ and $\\mathsf {V}_1$ be given by (REF ), i.e., $\\mathsf {U}_0$ and $\\mathsf {V}_1$ denote the functions $\\mathsf {U}$ and $\\mathsf {V}$ evaluated at $y = 0$ and $x = 0$ , respectively.",
"Let $\\Phi _0(x,P)$ and $\\Phi _1(y,P)$ be the eigenfunctions defined in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ via the Volterra integral equations (REF ).",
"Lemma 5.1 (Solution of the $x$ -part) The eigenfunction $\\Phi _0(x,P)$ defined via the Volterra integral equation (REF ) has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function x \\mapsto \\Phi _0(x,k^+) is continuous on [0,1) and is C^n on (0,1).",
"Furthermore, for each x \\in [0,1), the function k \\mapsto \\Phi _0(x,k^+) is analytic on \\hat{ \\setminus [0,1].",
"}\\item \\Phi _0 obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _0(x,k^+) = \\sigma _3\\Phi _0(x, k^-)\\sigma _3,\\\\\\Phi _0(x,k^\\pm ) = \\sigma _1\\overline{\\Phi _0(x, \\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad x \\in [0,1), \\ k \\in \\hat{ \\setminus [0, 1].",
"}\\end{@align}}\\item For each~ x \\in [0, 1), \\Phi _0(x,P) extends continuously to an analytic function of P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.",
"}\\item The value of $ 0$ at $ P = +$ is given by{\\begin{@align}{1}{-1}\\Phi _0(x,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}_0(x)} & 1 \\\\ \\mathcal {E}_0(x) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad x \\in [0, 1).\\end{@align}}$ The determinant of $\\Phi _0$ is given by $\\det \\Phi _0(x,P) = \\text{\\upshape Re\\,}\\mathcal {E}_0(x), \\qquad x \\in [0,1), \\ P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.$ For each $x_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [0, x_0],{\\begin{@align}{1}{-1}x \\mapsto \\big (k \\mapsto \\Phi _0(x,k^+)\\big )\\end{@align}}is a continuous map [0, x_0) \\rightarrow L^\\infty (K) and a C^n-map (0, x_0) \\rightarrow L^\\infty (K).",
"Moreover, the mapx \\mapsto \\big ( k \\mapsto x^\\alpha \\Phi _{0x}(x,k^+) \\big ) is continuous [0, x_0) \\rightarrow L^\\infty (K).",
"}$ We first use successive approximations to show that the integral equation $\\Phi _0(x, k^+) = I + \\int _0^x \\mathsf {U}_0(x^{\\prime },k^+) \\Phi _0(x^{\\prime },k^+) dx^{\\prime }, \\qquad x \\in [0,1),$ has a unique solution for each $k \\in \\hat{ \\setminus [0,1].",
"Let K be a compact subset of \\hat{ \\setminus [0,1].Let \\Phi _0^{(0)} = I and define \\Phi _0^{(j)}(x,k^+) for j \\ge 1 inductively by{\\begin{@align*}{1}{-1}& \\Phi _0^{(j+1)}(x,k^+) = \\int _0^x \\mathsf {U}_0(x^{\\prime },k^+) \\Phi _0^{(j)}(x^{\\prime },k^+) dx^{\\prime }, \\qquad x \\in [0,1), \\ k \\in K.\\end{@align*}}Then{\\begin{@align}{1}{-1}\\Phi _0^{(j)}(x,k^+) = &\\; \\int _{0 \\le x_1 \\le \\cdots \\le x_j \\le x} \\mathsf {U}_0(x_j, k^+) \\mathsf {U}_0(x_{j-1}, k^+)\\cdots \\mathsf {U}_0(x_1, k^+) dx_1 \\cdots dx_j.\\end{@align}}The function \\lambda (x,0,k^+) is analytic for k \\in \\hat{ \\setminus [x,1]; in particular, it is a bounded function of k \\in K for each fixed x \\in [0,1).",
"In view of the assumptions (\\ref {E0E1assumptions}), this implies\\Vert \\mathsf {U}_0(x,k^+)\\Vert _{L^1([0,x])} < C(x), \\qquad x \\in [0,1),\\ k \\in K,where the function~ C(x) is bounded on each compact subset of [0,1).Thus{\\begin{@align}{1}{-1} |\\Phi _0^{(j)}(x,k^+)| \\le & \\frac{1}{j!}",
"\\Vert \\mathsf {U}_0(\\cdot , k^+)\\Vert _{L^1([0, x])}^j\\le \\frac{1}{j!}",
"C(x)^j, \\qquad x \\in [0,1), \\ k \\in K.\\end{@align}}Hence the series{\\begin{@align}{1}{-1} \\Phi _0(x,k^+) = \\sum _{j=0}^\\infty \\Phi _0^{(j)}(x,k^+)\\end{@align}}converges absolutely and uniformly for k \\in K and x in compact subsets of [0,1) to a continuous solution \\Phi _0(x,k^+) of (\\ref {phixk}).",
"The fact that x \\mapsto \\Phi _0(x,k^+) \\in C^n((0,1)) follows from differentiating x\\mapsto \\Phi _0^{(j)}(x,k^+) and applying estimates similar to (\\ref {phi0jestimate}) to the derivative.Differentiating (with respect to k) under the integral sign in (\\ref {xPhijiterated}), we see that k \\mapsto \\Phi _0^{(j)}(x, k^+) is analytic on \\operatorname{int}K for each j; the uniform convergence then proves that k \\mapsto \\Phi _0(x, k^+) is analytic on \\operatorname{int}K.A similar argument applies to the integral equation defining \\Phi _0(x,k^-).",
"We conclude that the functions \\Phi _0(x,k^+) and \\Phi _0(x,k^-) are well-defined for x \\in [0,1) and k \\in \\hat{ \\setminus [0,1] and are analytic functions of k \\in \\hat{ \\setminus [0,1] for each fixed x.",
"}We next show uniqueness.",
"Assume that \\tilde{\\Phi }_0 is another solution of the Volterra equation (\\ref {phixk}) such that x \\mapsto \\Phi _0(x,k^\\pm ) is continuous on [0,1), respectively, and let \\Psi =\\Phi _0 - \\tilde{\\Phi }_0.",
"Then \\Psi is a solution of the homogeneous equation\\Psi (x,k^\\pm ) = \\int _0^x \\mathsf {U}_0(x^{\\prime },k^\\pm ) \\Psi (x^{\\prime },k^\\pm ) dx^{\\prime }.Iterating this yields{\\begin{@align*}{1}{-1}\\Psi (x,k^\\pm ) &= \\int _0^x \\mathsf {U}_0(x_j,k^\\pm ) \\int _{0}^{x_j} \\mathsf {U}_0(x_{j-1},k^\\pm ) \\cdots \\int _{0}^{x_2} \\mathsf {U}_0(x_{1},k^\\pm )\\Psi (x_1,k^\\pm ) \\,dx_1 \\ldots dx_n\\\\&= \\int _{0 \\le x_1 \\le \\cdots \\le x_j \\le x} \\mathsf {U}_0(x_j, k^\\pm ) \\mathsf {U}_0(x_{j-1}, k^\\pm )\\cdots \\mathsf {U}_0(x_1, k^\\pm )\\Psi (x_1,k^\\pm ) dx_1 \\cdots dx_j.\\end{@align*}}Hence, as in the proof of existence, we get the estimate|\\Psi (x,k^\\pm ) | \\le \\sup _{x^{\\prime }\\in [0,x]} |\\Psi (x^{\\prime },k^\\pm ) | \\frac{\\Vert \\mathsf {U}_0(\\cdot ,k^\\pm ) \\Vert _{L^1([0,x])}^j}{j!}",
"\\rightarrow 0, \\qquad j \\rightarrow \\infty ,which yields \\Psi = 0.",
"This proves (a).",
"}The symmetries (\\ref {lambdasymm}) of \\lambda show that\\mathsf {U}_0(x,k^+) = \\sigma _3\\mathsf {U}_0(x, k^-)\\sigma _3, \\qquad \\mathsf {U}_0(x,k^+) = \\sigma _1\\overline{\\mathsf {U}_0(x,\\bar{k}^+)}\\sigma _1.Hence \\sigma _3\\Phi _0(x, k^-)\\sigma _3 and \\sigma _1\\overline{\\Phi _0(x, \\bar{k}^+)}\\sigma _1 satisfy the same Volterra equation as \\Phi _0(x,k^+).",
"By uniqueness, all three functions must be equal.",
"This proves (b).",
"}We next show that \\Phi _0(x, k^\\pm ) can be continuously extended across the branch cut to an analytic function on \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0.Since \\mathsf {U}_0(x,k^\\pm ) has continuous boundary values on the interval (x, 1), the above argument (applied with a K that reaches up to the boundary) shows that \\Phi _0(x,k^\\pm ) also has continuous boundary values on (x, 1).Moreover, since\\lambda (x, 0, (k+i0)^+) = \\lambda (x,0,(k-i0)^-), \\qquad k \\in (x, 1),the boundary functions \\Phi (x,0, (k + i0)^+) and \\Phi (x,0, (k - i0)^-) satisfy the same integral equation, so by uniqueness they are equal:{\\begin{@align*}{1}{-1}\\Phi (x,y, (k + i0)^+) = \\Phi (x,y,(k-i0)^-), \\qquad (x,y) \\in D, \\ k \\in (x, 1).\\end{@align*}}Hence the values of \\Phi _0 on the upper and lower sheets of \\mathcal {S}_{(x,0)} fit together across the branch cut (x,1), showing that \\Phi _0 extends to an analytic function of P \\in \\mathcal {S}_{(x,0)} \\setminus \\big (\\Sigma _0 \\cup \\lbrace 1\\rbrace \\big ).",
"But \\lambda (x,0,P) is bounded in a neighborhood of the branch point 1, hence the possible singularity of \\Phi _0(x,P) at this point must be removable.",
"This shows that \\Phi _0 satisfies (c).",
"}}Since $ (x,y,+) = 1$, $ 0(x,+)$ satisfies the equation$$\\Phi _{0x}(x, \\infty ^+) = \\frac{1}{2\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} \\begin{pmatrix} \\overline{\\mathcal {E}_{0x}(x)} & \\overline{\\mathcal {E}_{0x}(x)} \\\\\\mathcal {E}_{0x}(x) & \\mathcal {E}_{0x}(x) \\end{pmatrix} \\Phi _0(x,\\infty ^+), \\qquad x \\in [0, 1).$$This equation has the two linearly independent solutions$$\\begin{pmatrix} \\overline{\\mathcal {E}_0(x)}\\\\ \\mathcal {E}_0(x) \\end{pmatrix} \\quad \\text{and}~\\quad \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}.$$Hence there exists a constant matrix $ A$ such that$$\\Phi _0(x, \\infty ^+) = \\begin{pmatrix} \\overline{\\mathcal {E}_0(x)} & 1 \\\\ \\mathcal {E}_0(x) & -1 \\end{pmatrix} A, \\qquad x \\in [0, 1).$$We determine $ A$ by evaluating this equation at $ x = 0$ and using that $ E0(0) = 1$ and $ 0(0, +) = I$.",
"This yields (\\ref {Phi0atinftyplus}) and proves $ (d)$.$ The proof of $(e)$ relies on the general identity $(\\ln \\det B)_x = \\text{\\upshape tr\\,}(B^{-1} B_x),$ where $B = B(x)$ is a differentiable matrix-valued function taking values in $GL(n, $ .",
"We find $(\\ln \\det \\Phi _0)_x = \\text{\\upshape tr\\,}(\\Phi _0^{-1} \\mathsf {U}_0 \\Phi _0) = \\text{\\upshape tr\\,}\\mathsf {U}_0 = \\frac{\\text{\\upshape Re\\,}\\mathcal {E}_{0x}}{\\text{\\upshape Re\\,}\\mathcal {E}_0} = (\\ln \\text{\\upshape Re\\,}\\mathcal {E}_0)_x.$ This relation is valid at least for small $x$ because $\\Phi _0(0, k^\\pm ) = I$ is invertible.",
"In fact, since $\\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0$ for $x \\in [0,1)$ by assumption (REF ), it extends to all of $[0,1)$ and we infer that, for $P \\in \\mathcal {S}_{(x,0)} \\setminus \\Sigma _0$ , $\\det \\Phi _0(x, P) = C(P) \\text{\\upshape Re\\,}\\mathcal {E}_0(x), \\qquad x \\in [0,1),$ where $C(P) \\in is independent of $ x$.",
"Evaluation at $ x = 0$ gives $ CP = 1$.",
"This proves $ (e)$.$ It remains to prove $(f)$ .",
"Fix $x_0 \\in (0,1)$ and let $K$ be a compact subset of $\\hat{ \\setminus [0, x_0].The function \\lambda (x,0,\\cdot ) is bounded on \\mathcal {S}_{(x,0)} except for a simple pole at k = x.Hence,{\\begin{@align*}{1}{-1}& \\sup _{k \\in K} \\big |\\Phi _0(x_2, k^+) - \\Phi _0(x_1, k^+)\\big |= \\sup _{k \\in K} \\bigg |\\int _{x_1}^{x_2} (\\mathsf {U}_0\\Phi _0)(x, k^+) dx\\bigg |\\\\& \\le \\bigg ( \\sup _{k \\in K} \\sup _{x \\in [0, x_0)} |x^\\alpha \\mathsf {U}_0(x,k^+)|\\bigg ) \\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha }\\Phi _0(x, k^+)| dx \\right) \\\\&\\le C \\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha }\\Phi _0(x, k^+)| dx \\right), \\qquad x_1, x_2 \\in [0, x_0),\\end{@align*}}where the right-hand side tends to zero as x_2 \\rightarrow x_1, because{\\begin{@align*}{1}{-1}\\sup _{k \\in K} \\left(\\int _{x_1}^{x_2} |x^{-\\alpha } \\Phi _0(x, k^+)| dx \\right)& \\le \\sum _{j=0}^{\\infty } \\frac{1}{j!}",
"\\sup _{k\\in K} \\Vert \\mathsf {U}_0(\\cdot ,k^+)\\Vert _{L^1([0,x_0])}^j \\int _{x_1}^{x_2} x^{-\\alpha } dx\\\\& \\le \\frac{e^{C(x_0)}(x_2^{1-\\alpha }-x_1^{1-\\alpha })}{1-\\alpha },\\end{@align*}}where C(x_0) is chosen as in the proof of (a).This shows that the map (\\ref {xkphimap}) is continuous [0, x_0) \\rightarrow L^\\infty (K).If x \\in (0, x_0), then{\\begin{@align*}{1}{-1}\\sup _{k \\in K} \\bigg | &\\frac{\\Phi _0(x+h, k^+) - \\Phi _0(x,k^+)}{h} - \\Phi _{0x}(x, k^+)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | \\frac{1}{h} \\int _x^{x+h} (\\mathsf {U}_0\\Phi _0)(x^{\\prime }, k^+)~dx^{\\prime } - \\Phi _{0x}(x,k^+)\\bigg |\\\\& \\le \\sup _{k \\in K} \\bigg | (\\mathsf {U}_0\\Phi _0)(\\xi , k^+)~ - (\\mathsf {U}_0\\Phi _0)(x, k^+)\\bigg |,\\end{@align*}}where \\xi lies between x and x+h.",
"As h \\rightarrow 0, the right-hand side goes to zero.",
"Hence (\\ref {xkphimap}) is differentiable as a map (0,x_0) \\rightarrow L^\\infty (K) and the derivative satisfies \\Phi _{0x}(x,k^+) = \\mathsf {U}_0(x,k^+)\\Phi _{0}(x).Furtermore, the mapx \\mapsto \\big (k \\mapsto \\lambda (x,0,k^+)\\big )is C^\\infty from (0, x_0) to L^\\infty (K) and \\mathcal {E}_{0} is C^{n} on (0,1).",
"Hence the mapx \\mapsto \\big (k \\mapsto \\mathsf {U}_0(x, k^+)\\big )is C^{n-1} from (0, x_0) to L^\\infty (K).It follows that (\\ref {xkphimap}) is a C^n-map (0, x_0) \\rightarrow L^\\infty (K).",
"}Finally, since$$x^\\alpha \\Phi _{0x}(x,k^+) = x^\\alpha \\mathsf {U}_0(x, k^+) \\Phi _0(x,k^+)$$we see that $ x x0x(x,k+)$ is continuous $ [0, x0) L(K)$.This proves $ (f)$ and completes the proof of the lemma.$ Lemma 5.2 (Solution of the $y$ -part) The eigenfunction $\\Phi _1(y,P)$ is well-defined for $y \\in [0,1)$ and $P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1$ and has the following properties: For each $k \\in \\hat{ \\setminus [0,1], the function y \\mapsto \\Phi _1(y,k^+) is continuous on [0,1) and is C^n on~ (0,1).",
"Furthermore, for each y \\in [0,1), the function k \\mapsto \\Phi _0(x,k^+) is analytic on \\hat{ \\setminus [0,1].",
"}\\item \\Phi _1 obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi _1(y,k^+) = \\sigma _3\\Phi _1(y, k^-)\\sigma _3,\\\\\\Phi _1(y,k^\\pm ) = \\sigma _1\\overline{\\Phi _1(y, \\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad y \\in [0,1), \\ k \\in \\hat{ \\setminus [0, 1].",
"}\\end{@align}\\item For each~ y \\in [0, 1), \\Phi _1(y,P) is an analytic function of P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1.\\item The value of \\Phi _1 at P = \\infty ^+ is given by{\\begin{@align}{1}{-1}\\Phi _1(y,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}_1(y)} & 1 \\\\ \\mathcal {E}_1(y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad y \\in [0, 1).\\end{@align}}}}\\item The determinant of $ 1$ is given by$$\\det \\Phi _1(y,P) = \\text{\\upshape Re\\,}\\mathcal {E}_1(y), \\qquad y \\in [0,1), \\ P \\in \\mathcal {S}_{(0,y)} \\setminus \\Sigma _1.$$$ For each $y_0 \\in (0,1)$ and each compact subset $K \\subset \\hat{ \\setminus [0, y_0],{\\begin{@align}{1}{-1}y \\mapsto \\big (k \\mapsto \\Phi _1(y,k^+)\\big )\\end{@align}}is a continuous map [0, y_0) \\rightarrow L^\\infty (K) and a C^n-map (0, y_0) \\rightarrow L^\\infty (K).",
"Moreover, the mapy \\mapsto \\big ( k \\mapsto y^\\alpha \\Phi _{1y}(y,k^+)\\big ) is continuous [0, y_0) \\rightarrow L^\\infty (K).",
"}$ The proof is similar to that of Lemma REF .",
"Uniqueness The following lemma ensures uniqueness of the solution of the RH problem (REF ).",
"The proof relies on the fact that the determinant of the jump matrix $v$ defined in (REF ) is constant on each of the subcontours $\\Gamma _0$ and $\\Gamma _1$ .",
"Lemma 5.3 Suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Then, for each $(x,y)\\in D$ , the solution $m(x,y,\\cdot )$ of the RH problem (REF ) is unique, if it exists.",
"Moreover, $\\det m(x,y,z) = 1, \\qquad (x,y)\\in D, \\ z \\in \\Omega _\\infty .$ Fix $(x,y) \\in D$ .",
"By (REF ) and the definition (REF ) of $v$ , we have $\\det v(x,y,z) = {\\left\\lbrace \\begin{array}{ll} \\text{\\upshape Re\\,}\\mathcal {E}_0(x) > 0, \\quad & z \\in \\Gamma _0, \\\\\\text{\\upshape Re\\,}\\mathcal {E}_1(y) > 0, \\quad & z \\in \\Gamma _1.\\end{array}\\right.",
"}$ Hence $\\sqrt{\\det v(x,y,z)} = {\\left\\lbrace \\begin{array}{ll} c_0(x), \\quad & z \\in \\Gamma _0, \\\\c_1(y), & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad (x,y) \\in D,$ where the two functions $c_0(x) > 0$ and $c_1(y) > 0$ are independent of $z$ .",
"The function $m(x,y,\\cdot )$ is a solution of the RH problem (REF ) if and only if the function $\\tilde{m}(x,y,\\cdot )$ defined by $\\tilde{m}(x,y,z) = {\\left\\lbrace \\begin{array}{ll} c_0(x) m(x,y,z), \\quad & z \\in \\Omega _0, \\\\c_1(y) m(x,y,z), & z \\in \\Omega _1, \\\\m(x,y,z), & z \\in \\Omega _\\infty , \\end{array}\\right.",
"}$ satisfies the RH problem ${\\left\\lbrace \\begin{array}{ll}\\text{$\\tilde{m}(x, y, \\cdot )$ is analytic in $\\Gamma $},\\\\\\text{$\\tilde{m}_+(x,y,z) = \\tilde{m}_-(x,y,z) \\tilde{v}(x,y,z)$ for all $z \\in \\Gamma $},\\\\\\text{$\\tilde{m}(x, y, z) = I + O(z^{-1})$ as $z\\rightarrow \\infty $},\\end{array}\\right.",
"}$ where $\\tilde{v}(x,y,z) = {\\left\\lbrace \\begin{array}{ll} \\frac{1}{c_0(x)}v(x,y,z), & z \\in \\Gamma _0, \\\\\\frac{1}{c_1(y)} v(x,y,z), \\quad & z \\in \\Gamma _1.\\end{array}\\right.",
"}$ But $\\det \\tilde{v}(x,y,z) = 1$ for all $z \\in \\Gamma $ ; hence the solution $\\tilde{m}(x,y,\\cdot )$ is unique and $\\det \\tilde{m} =1$ .",
"It follows that the solution $m$ is unique and that $\\det m(x,y,z) = \\det \\tilde{m}(x,y,z) = 1$ for $z \\in \\Omega _\\infty $ .",
"Proofs of main results In this section, we use the lemmas from the previous section to prove Theorem REF –REF .",
"Proofs of Theorem 1 & 2 Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ be complex-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ .",
"We will show that $\\mathcal {E}(x,y)$ can be uniquely expressed in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ by (REF ).",
"The idea in what follows is to introduce a solution $\\Phi $ of (REF ) as the solution of the integral equation $\\Phi (x,y,k^\\pm ) = I + \\int _{(0,0)}^{(x,y)} (W\\Phi )(x^{\\prime },y^{\\prime }, k^\\pm ).$ However, since $W$ in general is singular on the boundary of $D$ , we need to be more careful with the definition.",
"We therefore instead define $\\Phi $ as the solution of $\\Phi (x, y, k^+) = \\Phi _0(x, k^+) + \\int _0^y (\\mathsf {V} \\Phi )(x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}$ Lemma 6.1 (Solution of Lax pair equations) The function $\\Phi (x,y,P)$ defined in (REF ) has the following properties: $\\Phi (x,y,k^\\pm )$ is a well-defined $2\\times 2$ -matrix valued function of $(x,y) \\in D$ and $k \\in \\hat{ \\setminus [0,1] which also satisfies the alternative Volterra integral equation:{\\begin{@align}{1}{-1}\\Phi (x, y, k^+) = \\Phi _1(y, k^+) + \\int _0^x (\\mathsf {U} \\Phi )(x^{\\prime },y,k^+) dx^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}}\\item For each $ k [0,1]$, the function $ (x,y) (x,y,k+)$ is continuous on $ D$ and is $ Cn$ on~ $ intD$.$ For each $k \\in \\hat{ \\setminus [0,1], the functions(x,y) \\mapsto x^\\alpha \\Phi _x(x,y,k^+), \\quad (x,y) \\mapsto y^\\alpha \\Phi _y(x,y,k^+), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha \\Phi _{xy}(x,y,k^+),are continuous on D.}\\item $$ obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi (x,y,k^+) = \\sigma _3\\Phi (x, y, k^-)\\sigma _3,\\\\\\Phi (x,y,k^\\pm ) = \\sigma _1\\overline{\\Phi (x, y,\\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}\\item For each~ point $ (x,y) D$, $ (x,y,P)$ extends continuously to an analytic function of $ P S(x,y) $, where $ = 0 1$ is the contour defined in (\\ref {Sigma01def}).$ The value of $\\Phi $ at $P = \\infty ^+$ is given by $\\Phi (x,y,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad (x,y) \\in D.$ The determinant of $\\Phi $ is given by $\\det \\Phi (x,y,P) = \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0, \\qquad (x,y) \\in D, \\ P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .$ By Lemma REF the lemma holds for $y = 0$ , i.e., the function $\\Phi (x,0,P)$ is well-defined and the properties $(a)$ -$(d)$ are satisfied when $x = 0$ or $y=0$ .",
"In order to see that $\\Phi $ is well-defined also for $(x,y)$ in the interior of $D$ , we note that (REF ) implies $\\Phi (x, y, k^+) = \\Phi (x,0, k^+) + \\int _0^y \\mathsf {V}(x,y^{\\prime },k^+) \\Phi (x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}The same type of successive approximation argument already used in the proof of Lemma \\ref {xpartlemma} shows that the Volterra equation (\\ref {Phidefxy}) has a unique solution for each fixed x \\in (0,1) and each k \\in \\hat{ \\setminus [0,1], and that this solution \\Phi (x,y,P) extends continuously to an analytic function of P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .",
"This proves (b).",
"}In order to prove (a), it remains to deduce the alternative representation (\\ref {Phidef2}).",
"Note that \\Phi _y = V\\Phi by definition and{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)&=\\Phi _x(x,0,k^+) + \\int _{0}^y x\\Phi (x,y^{\\prime },k^+)+ _x(x,y^{\\prime },k^\\pm )dy^{\\prime }.\\end{@align*}}Since \\mathcal {E} is a solution of the Goursat problem, we havex=\\mathsf {U}_y +[\\mathsf {U},,and, moreover, \\Phi _x(x,0,k^+) =\\mathsf {U}\\Phi (x,0,k^+).",
"Now a straightforward calculation shows{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)= \\mathsf {U}\\Phi (x,y,k^+) + \\int _{0}^y _x(x,y^{\\prime },k^\\pm )-\\Phi (x,y^{\\prime },k^+) dy^{\\prime }.\\end{@align*}}Thus the function \\tilde{\\Phi }=\\Phi _x -\\mathsf {U}\\Phi is the unique solution of the Volterra integral equation\\tilde{\\Phi }(x,y^{\\prime },k^+) = \\int _0^y {\\Phi }(x,y^{\\prime },k^+)dy^{\\prime }giving \\tilde{\\Phi }=0.",
"This implies \\Phi _x = \\mathsf {U}\\Phi .",
"Consequently, \\Phi , defined by (\\ref {Phidef}), is an eigenfunction for the Lax pair equations (\\ref {lax}).",
"The difference between (\\ref {Phidef}) and (\\ref {Phidef2}) is given by{\\begin{@align*}{1}{-1}&\\Phi _0(x,k^+) - \\Phi _1(y,k^+) +\\int _0^y (x,y^{\\prime },k^+)dy^{\\prime } - \\int _0^x \\mathsf {U}\\Phi (x^{\\prime },y,k^+)dx^{\\prime }\\\\=& \\, \\int _0^y \\int _0^x ()_x(x^{\\prime },y^{\\prime },k^+)dx^{\\prime } dy^{\\prime } - \\int _0^x \\int _0^y (\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+)dy^{\\prime }dx^{\\prime }\\\\=& \\, \\int _0^x \\int _0^y ()_x(x^{\\prime },y^{\\prime },k^+)-(\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+) dy^{\\prime } dx^{\\prime }\\end{@align*}}and ()_x = (\\mathsf {U}\\Phi )_y is the compatibility condition for the Lax pair.",
"Hence the two representations (\\ref {Phidef}) and (\\ref {Phidef2}) are equal.This proves (a).$ The symmetries (REF ) of $\\lambda $ show that $W(x,y,k^+) = \\sigma _3W(x,y, k^-)\\sigma _3, \\qquad W(x,y,k^+) = \\sigma _1\\overline{W(x,y,\\bar{k}^+)}\\sigma _1.$ Since $\\lambda (x,y,\\infty ^+) = 1$ , $\\Phi (x,y,\\infty ^+)$ satisfies the equation $\\Phi _y(x, y, \\infty ^+) = \\frac{1}{2\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} \\begin{pmatrix} \\overline{\\mathcal {E}_y(x,y)} & \\overline{\\mathcal {E}_y(x,y)} \\\\\\mathcal {E}_y(x,y) & \\mathcal {E}_y(x,y) \\end{pmatrix} \\Phi (x,y,\\infty ^+), \\qquad (x,y) \\in D.$ Using the above equations and arguing as in the proof of Lemma REF , the statements $(c)$ , $(d)$ , $(e)$ , $(f)$ , and $(g)$ follow from equation (REF ) and the corresponding statements in Lemma REF .",
"Part $(g)$ of Lemma REF implies that the inverse matrix $\\Phi (x,y,P)^{-1}$ is well-defined for $(x,y) \\in D$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma $ .",
"Lemma 6.2 For each $(x,y) \\in D$ , $P \\mapsto \\Phi (x,y,P)\\Phi (x,0, P)^{-1} \\quad \\text{and} \\quad P \\mapsto \\Phi (x,y,P)\\Phi (0,y,P)^{-1}$ are analytic functions of $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _0$ , respectively.",
"Let $U$ be an open set in $\\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ .",
"Multiplying (REF ) by $\\Phi (x,0,P)^{-1}$ from the right, we find $\\nonumber & \\Phi (x, y, P)\\Phi (x,0,P)^{-1} = I + \\int _0^y \\mathsf {V}(x,y^{\\prime },P) \\Phi (x,y^{\\prime },P)\\Phi (x,0,P)^{-1} dy^{\\prime },\\\\ & \\hspace{227.62204pt} (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}where the values of \\Phi (x,0,P) and \\lambda (x,y^{\\prime },P) in (\\ref {phiPmap}) are to be interpreted as in Remark \\ref {tilderemark}.",
"SinceP \\mapsto \\lambda (x,y^{\\prime },P)^{-1} = \\sqrt{\\frac{k - x}{k - (1-y^{\\prime })}}is an analytic map U \\rightarrow for each y^{\\prime }, so is \\mathsf {V}(x,y^{\\prime },\\cdot ).",
"It follows that the solution \\Phi (x, y, P)\\Phi (x,0,P)^{-1} of (\\ref {phiPmap}) also is analytic for P \\in U.",
"This establishes the desired statement for the first map in (\\ref {phiminusphi}); the proof for the second map is similar.$ Let $\\Omega _0$ , $\\Omega _1$ , and $\\Omega _\\infty $ denote the three components of $\\hat{ \\setminus \\Gamma defined in (\\ref {Omegadef}) and displayed in Figure \\ref {Omegas.pdf}.",
"}$ Lemma 6.3 The $2\\times 2$ -matrix valued function $m(x,y,z)$ defined for $(x,y)\\in D$ by $m(x,y,z) =\\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi \\big (x,y,F_{(x,y)}^{-1}(z)\\big ) \\times {\\left\\lbrace \\begin{array}{ll} \\Phi \\big (x,0,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _0, \\\\\\Phi \\big (0,y,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _1, \\\\I, & z \\in \\Omega _\\infty ,\\end{array}\\right.",
"}$ satisfies the RH problem (REF ) and the relation (REF ) for each $(x,y) \\in D$ .",
"Since $F_{(x,y)}$ is a biholomorphism $\\mathcal {S}_{(x,y)}~\\rightarrow \\hat{, we infer from Lemma \\ref {claim1} together with Lemma \\ref {claim2} that m(x,y, \\cdot ) is analytic in \\Gamma and that m(x,y,z) \\rightarrow I as z \\rightarrow \\infty for each (x,y) \\in D. The jump condition in (\\ref {RHm}) holds as a consequence of the definition (\\ref {jumpdef}) of v(x,y,z) and the fact that\\Phi _0(x,k) = \\Phi (x,0,k), \\qquad \\Phi _1(y,k) = \\Phi (0,y,k).Finally, since 0 \\in \\Omega _\\infty and F_{(x,y)}^{-1}(0) = \\infty ^-, the first symmetry in (\\ref {phisymmetries}) yields{\\begin{@align}{1}{-1}m(x,y,0) = \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi (x,y, \\infty ^-)= \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\sigma _3 \\Phi \\big (x,y, \\infty ^+\\big ) \\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phiatinftyplus}) for \\Phi \\big (x,y, \\infty ^+\\big ), the (11) and (21) entries of (\\ref {mxy0}) give{\\begin{@align*}{1}{-1}(m(x,y,0))_{11} = \\frac{1 + \\mathcal {E}(x,y) \\overline{\\mathcal {E}(x,y)}}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}, \\qquad (m(x,y,0))_{21} = \\frac{(1 - \\mathcal {E}(x,y))(1 + \\overline{\\mathcal {E}(x,y)})}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}.\\end{@align*}}Solving these two equations for \\mathcal {E} and \\bar{\\mathcal {E}}, we find (\\ref {Erecover}).",
"}$ We have showed that if $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ , then $\\mathcal {E}(x,y)$ can be expressed in terms of the function $m$ defined in (REF ) via equation (REF ).",
"By Lemma REF , this function $m(x,y,z)$ is the unique solution of the RH-problem (REF ) whose formulation involves only the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ .",
"As a consequence, the value of the solution $\\mathcal {E}$ at $(x,y)$ is uniquely determined by the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ , if it exists.",
"This completes the proofs of Theorem REF and REF .",
"Proof of Theorem REF This subsection is devoted to proving Theorem REF regarding existence.",
"Let us therefore suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Define $\\Phi _0(x,P)$ and $\\Phi _1(y,P)$ in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ via the Volterra equations (REF ).",
"Then $\\Phi _0$ and $\\Phi _1$ have the properties listed in Lemma REF and Lemma REF .",
"Let $\\delta \\in (0,1)$ and let $D_\\delta $ be the triangle defined in (REF ).",
"As in the proof of Theorem REF , choose $\\epsilon > 0$ so small that $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ are contained in the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, for all $(x,y) \\in D_\\delta $ .",
"Fix two smooth nonintersecting clockwise contours $\\Gamma _0$ and $\\Gamma _1$ in the complex $z$ -plane which encircle the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, but which do not encircle zero, see Figure REF .",
"Suppose $\\Gamma _0$ and $\\Gamma _1$ are invariant under the involutions $z \\mapsto z^{-1}$ and $z \\mapsto \\bar{z}$ .",
"Let $\\Gamma = \\Gamma _0 \\cup \\Gamma _1$ and consider the family of RH problems given in (REF ) parametrized by the two parameters $(x,y) \\in D_\\delta $ .",
"We will show that if (REF ) has a (unique) solution $m(x,y,z)$ for each $(x,y) \\in D_\\delta $ , then the function $\\mathcal {E}(x,y)$ defined in terms of $m$ via equation (REF ) satisfies ${\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\\\text{$\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (\\ref {ernst}) in $\\operatorname{int}(D_\\delta )$,}\\\\\\text{$x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta )$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}$ We next list some facts about Cauchy integrals that we will use throughout the proof.",
"If $h \\in L^2(\\Gamma )$ , then the Cauchy transform $\\mathcal {C}h$ is defined by $(\\mathcal {C}h)(z) = \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{h(z^{\\prime })}{z^{\\prime } - z} dz^{\\prime }, \\qquad z \\in \\Gamma ,$ We denote the nontangential boundary values of $\\mathcal {C}f$ from the left and right sides of $\\Gamma $ by $\\mathcal {C}_+ f$ and $\\mathcal {C}_-f$ respectively.",
"Then $\\mathcal {C}_+$ and $\\mathcal {C}_-$ are bounded operators on $L^2(\\Gamma )$ and $\\mathcal {C}_+ - \\mathcal {C}_- = I$ .",
"Let $w(x,y,z) = v(x,y,z) - I$ .",
"We define the operator $\\mathcal {C}_w: L^2(\\Gamma ) + L^\\infty (\\Gamma ) \\rightarrow L^2(\\Gamma )$ by $\\mathcal {C}_{w}(f) = \\mathcal {C}_-(f w).$ Then $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C \\Vert w\\Vert _{L^\\infty (\\Gamma )},$ where $C = \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}$ .",
"We henceforth assume that the RH problem (REF ) has a solution for all $(x,y) \\in D_\\delta $ or, equivalently, that $I - \\mathcal {C}_{w} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"For each $(x,y) \\in D_\\delta $ , we have $v \\in C(\\Gamma )$ and $v, v^{-1} \\in I + L^2(\\Gamma ) \\cap L^\\infty (\\Gamma )$ .",
"The theory of singular integral equations then implies that the solution of the RH problem (REF ) is given by (see e.g.",
"[4] or [18]) $m = I + \\mathcal {C}(\\mu w),$ where the $2\\times 2$ -matrix valued function $\\mu (x,y,\\cdot )$ is defined by $\\mu = I + (I - \\mathcal {C}_w)^{-1}\\mathcal {C}_w I \\in I + L^2(\\Gamma ).$ Equation (REF ) can be written more explicitly as $m(x,y,z) = I + \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{(\\mu w)(x,y,s) ds}{s-z}, \\qquad (x,y) \\in D_\\delta , \\ z \\in \\hat{ \\setminus \\Gamma .",
"}$ Lemma 6.4 The map $ (x,y) \\mapsto w(x,y, \\cdot )$ is continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto y^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"For $N \\ge 0$ , let $C^N(K)$ denote the Banach space of functions on $K$ with continuous partial derivatives of order $\\le N$ equipped with the usual norm $\\Vert f\\Vert _{C^N(K)} = \\sup _{|\\alpha | \\le N} \\Vert D^\\alpha f\\Vert _{L^\\infty (K)}.$ By part $(f)$ of Lemma REF the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ): D_\\delta \\rightarrow C(K)$ is continuous for any compact set $K$ not intersecting $\\Sigma $ .",
"Moreover, assuming $F_{(x,y)}^{-1}(\\Gamma ) \\subset K $ , the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):D_\\delta \\rightarrow \\mathcal {B}(C(K), C(\\Gamma ))$ is continuous, because $\\sup _{\\Vert f\\Vert _{C(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f \\in C(K)$ on the compact set $K$ .",
"It follows that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ also is continuous.",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )): D_\\delta \\rightarrow C(\\Gamma _1)$ is continuous.",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is continuous from $D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If a sequence of holomorphic functions $f_n$ converges uniformly on an open set $\\Omega $ then the sequence of derivatives $f_n^{\\prime }$ converges uniformly on compact subsets of $\\Omega $ .",
"Fix $N \\ge n$ and let $K$ be a compact subset of $\\mathcal {S}_{(x,0)} \\setminus \\Sigma _0$ .",
"Then part $(f)$ of Lemma REF implies that the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ):\\operatorname{int}D_\\delta \\rightarrow C^N(K)$ is $C^n$ .",
"On the other hand, the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):\\operatorname{int}D_\\delta \\rightarrow \\mathcal {B}(C^N(K), C(\\Gamma ))$ is $C^n$ .",
"Indeed, the map is continuous because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f$ on the compact set $K$ .",
"Moreover, the map has a continuous partial derivative with respect to $x$ because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\bigg |\\frac{f(F_{(x+h,y)}^{-1}(z)) - f(F_{(x,y)}^{-1}(z))}{h} - \\frac{d}{dx}f(F_{(x,y)}^{-1}(z)) \\bigg | \\rightarrow 0$ as $h \\rightarrow 0$ by the mean-value theorem and the uniform continuity of the first partial derivatives of $f$ .",
"Similar arguments show that all partial derivatives of order $\\le n$ exist and are continuous.",
"We conclude that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ built from (REF ) and (REF ) is $C^n$ .",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _1)$ is $C^n$ .",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is $C^n$ as a map from $\\operatorname{int}D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If $z \\in \\Gamma _0$ , we have $w_x(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)) + \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{d x}F_{(x,y)}^{-1}(z),$ where $\\frac{d}{d x}F_{(x,y)}^{-1}(z)$ denotes the derivative of the $k$ -projection of $F_{(x,y)}^{-1}(z)$ , which is given by $\\frac{d}{d x}F_{(x,y)}^{-1}(z) = -\\frac{(z-1)^2}{4z}.$ Thus part $(f)$ of Lemma REF and of Lemma REF imply that $(x,y) \\mapsto x^\\alpha w_x(x,y,\\cdot )$ is a continuous map $D_\\delta \\rightarrow L^\\infty (\\Gamma )$ .",
"The maps $(x,y) \\mapsto y^\\alpha w_y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y,\\cdot )$ can be treated similarly.",
"Lemma 6.5 The map $(x,y) \\mapsto \\mu (x,y, \\cdot ) - I$ is continuous from $D_\\delta $ to $L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto y^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^2(\\Gamma )$ .",
"In view of the definition of $\\mu $ , the map (REF ) is given by $(x,y) \\mapsto (I - \\mathcal {C}_{w(x,y,\\cdot )})^{-1}\\mathcal {C}_-(w(x,y,\\cdot )).$ We note that the map $f \\mapsto I - \\mathcal {C}_f: L^\\infty (\\Gamma ) \\rightarrow \\mathcal {B}(L^2(\\Gamma ))$ is smooth by the estimate $\\Vert \\mathcal {C}_f\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C\\Vert f\\Vert _{L^\\infty (\\Gamma )},$ and that the linear map $f \\mapsto \\mathcal {C}_-f: L^2(\\Gamma ) \\rightarrow L^2(\\Gamma )$ is bounded.",
"Since (REF ) can be viewed as a composition of maps of the form (REF ), (REF ), and (REF ) together with the smooth inversion map $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , it follows that (REF ) is continuous $D_\\delta \\rightarrow L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Similarly, $(x,y) \\mapsto x^\\alpha \\mu _x(x,y,\\cdot )$ can be viewed as composition of the continuous maps (REF ), (REF ), $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , (REF ), and (REF ), and is hence continuous.",
"The maps $(x,y) \\mapsto y^\\alpha \\mu _y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y,\\cdot )$ can be treated analogously.",
"Lemma 6.6 The solution $m(x,y,z)$ of the RH problem (REF ) defined in (REF ) has the following properties: For each point $(x,y) \\in D_\\delta $ , $m(x,y,\\cdot )$ obeys the symmetries $m(x,y,z) = m(x,y,0)\\sigma _3m(x,y,z^{-1})\\sigma _3 = \\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}$ For each $z \\in \\hat{\\setminus \\Gamma , the map (x,y) \\mapsto m(x,y,z) is continuous from D_\\delta to {2 \\times 2} and is C^n from \\operatorname{int}D_\\delta to {2 \\times 2}.",
"}\\item For each~ $ z $, the three maps$$(x,y) \\mapsto x^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto y^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha m_{xy}(x,y, z),$$are continuous from $ D$ to $ 22$.$ The symmetries in () and () show that $v$ satisfies ${\\left\\lbrace \\begin{array}{ll} v(x,y,z) = \\sigma _3v(x, y, z^{-1})\\sigma _3,\\\\v(x,y,z) = \\sigma _1\\overline{v(x, y, \\bar{z})}\\sigma _1,\\end{array}\\right.}",
"\\qquad z \\in \\Gamma , \\ (x,y) \\in D_\\delta .$ These symmetries imply that $\\sigma _3 m(x,y,0)^{-1} m(x,y,z^{-1})\\sigma _3$ and $\\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1$ satisfy the same RH problem as $m(x,y,z)$ .",
"The symmetries in (REF ) follow by uniqueness.",
"Properties $(b)$ and $(c)$ follow from (REF ) together with the Lemmas REF and REF .",
"As in the proof of Theorem REF , we extend the definition (REF ) of $v$ to an open tubular neighborhood $N(\\Gamma ) = N(\\Gamma _0) \\cup N(\\Gamma _1)$ of $\\Gamma $ as follows, see Figure REF : $v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in N(\\Gamma _0),\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in N(\\Gamma _1),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We choose $N(\\Gamma )$ so narrow that it does not intersect the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ .",
"Then, for each $(x,y) \\in D_\\delta $ , $v(x,y,\\cdot )$ is an analytic function of $z \\in N(\\Gamma )$ .",
"Using the notation $z(x,y,P) := F_{(x,y)}(P)$ , we can write (REF ) as $v(x,y,z(x,y,P)) = {\\left\\lbrace \\begin{array}{ll} \\Phi _0(x, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _0)\\big ), \\\\\\Phi _1(y, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _1)\\big ),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We define functions $f_0(x,y,z)$ and $f_1(x,y,z)$ for $(x,y) \\in D_\\delta $ by $f_0(x,y,z) = \\big [m_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma ,\\\\f_1(x,y,z) = \\big [m_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}Moreover, we let n_0(x,y,z) and n_1(x,y,z) denote the functions given by\\begin{subequations}{\\begin{@align}{1}{-1}n_0(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_0(x,y,z) + m(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _0, \\\\f_0(x,y,z), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}and{\\begin{@align}{1}{-1}n_1(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_1(x,y,z) + m(x,y,z)\\mathsf {V}_1\\big (y,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _1, \\\\f_1(x,y,z), & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}\\end{@align}}\\end{subequations}}$ Lemma 6.7 For each $(x,y) \\in D_\\delta $ , it holds that $n_0(x,y,z)$ is an analytic function of $z \\in \\hat{ \\setminus \\lbrace -1\\rbrace and has at most a simple pole at z = -1.\\item n_1(x,y,z) is an analytic function of z \\in \\hat{ \\setminus \\lbrace 1\\rbrace and has at most a simple pole at z = 1.\\item n_0(x,y,\\infty ) = 0 and n_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.\\item n_1(x,y,\\infty ) = 0 and n_1(x,y,0) = m_y(x,y,0)m(x,y,0)^{-1}.",
"}}{\\begin{xmlelement*}{proof}By (\\ref {linearzx}) the function z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1, \\infty \\rbrace with simple poles at z = -1 and z = \\infty .Equation (\\ref {msolution}) implies that m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty .",
"Hence f_0(x,y,z) is analytic at z = \\infty .It follows that f_0(x,y,z) is analytic for all z \\in \\hat{\\setminus (\\Gamma \\cup \\lbrace -1\\rbrace ) with a simple pole at z = -1 at most.Now f_0 satisfies the following jump condition across \\Gamma :{\\begin{@align}{1}{-1}\\nonumber f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\big [v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z)\\big ]\\\\ & \\times v(x,y,z)^{-1} m_-(x,y,z)^{-1}, \\qquad z \\in \\Gamma .\\end{@align}}Differentiating (\\ref {vzPhi0}) with respect to x and y and evaluating the resulting equations at k = F_{(x,y)}^{-1}(z), we find{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)),\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _0),\\end{@align}} and{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{1y}(x,F_{(x,y)}^{-1}(z)),\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _1).\\end{@align}} Using the first equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}) in (\\ref {f0jump}), we conclude that f_0 is analytic across \\Gamma _1 and has the following jump across \\Gamma _0:{\\begin{@align}{1}{-1}f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) m_-(x,y,z)^{-1}, \\quad \\ z \\in \\Gamma _0.\\end{@align}}Thus n_0 is analytic across \\Gamma .",
"Furthermore, since \\lambda (x,y,k) is analytic on \\mathcal {S}_{(x,y)} except for a simple pole at the branch point k = x, the function \\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1\\rbrace with a simple pole at z = -1.",
"It follows that n_0 satisfies (a).",
"The proof of (b) is similar and relies on the second equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}).",
"}Using (\\ref {linearzx}) in the definition (\\ref {ndefa}) of n_0, we can write, for z \\in \\Omega _\\infty ,{\\begin{@align}{1}{-1}n_0(x,y,z) = f_0(x,y,z) = \\Big [m_x(x,y,z) -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z(x,y,z)\\Big ]m(x,y,z)^{-1}.\\end{@align}}Since m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty , it follows that n_0(x,y,\\infty ) = 0.",
"On the other hand, evaluating (\\ref {n0f0}) at z = 0, we findn_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.This proves (c); the proof of (d) is analogous.",
"}}\\end{xmlelement*}Let \\hat{m}(x,y) denote the function m(x,y,z)~ evaluated at z = 0, that is,\\hat{m}(x,y) = m(x,y,0).Evaluating the first symmetry in (\\ref {msymm}) at z = \\infty , we find{\\begin{@align}{1}{-1}I = \\hat{m}(x,y)\\sigma _3\\hat{m}(x,y)\\sigma _3.\\end{@align}}The unit determinant condition (\\ref {detmone}) implies that \\det \\hat{m} = 1.",
"Hence equation (\\ref {m0sigma3}) reduces to\\text{adj}(\\hat{m})=\\sigma _3 \\hat{m} \\sigma _3,where \\text{adj} denotes the adjugate matrix, which shows that \\hat{m}_{11} = \\hat{m}_{22}.A straightforward algebraic computation then yields{\\begin{@align}{1}{-1}\\hat{m}(x,y) = \\tilde{\\Phi }(x,y) \\sigma _3\\tilde{\\Phi }(x,y)\\sigma _3, \\qquad (x,y) \\in D_\\delta ,\\end{@align}}where the 2\\times 2-matrix valued function \\tilde{\\Phi }(x,y) is defined by{\\begin{@align}{1}{-1}\\tilde{\\Phi }(x,y) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}\\end{@align}}and the functions \\mathcal {E}(x,y) and \\overline{\\mathcal {E}(x,y)} are defined by{\\begin{@align}{1}{-1}\\mathcal {E} = \\frac{1 + \\hat{m}_{11} - \\hat{m}_{21}}{1 + \\hat{m}_{11} + \\hat{m}_{21}}, \\qquad \\bar{\\mathcal {E}} = -\\frac{1 - \\hat{m}_{11} + \\hat{m}_{21}}{1 - \\hat{m}_{11} - \\hat{m}_{21}}.\\end{@align}}The second symmetry in (\\ref {msymm}) evaluated at z = 0 implies{\\begin{@align}{1}{-1}\\hat{m}_{11} = \\overline{\\hat{m}_{22}}, \\qquad \\hat{m}_{12} = \\overline{\\hat{m}_{21}}.\\end{@align}}Recalling the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, it follows that \\bar{\\mathcal {E}} is the complex conjugate of \\mathcal {E}.",
"The next lemma shows, among other things, that \\mathcal {E} is free of singularities.",
"}\\begin{lemma}The function \\mathcal {E}(x,y) defined in (\\ref {Edef}) has the following properties:{\\begin{@align*}{1}{-1}{\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ),\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}\\end{@align*}}\\end{lemma}{\\begin{xmlelement*}{proof}By Lemma \\ref {claim4E}, the map (x,y) \\mapsto \\hat{m}(x,y) is continuous from D_\\delta to and is C^n from \\operatorname{int}D_\\delta to .The first equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) also has these regularity properties except possibly on the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = -1\\rbrace \\end{@align}}where the denominator vanishes.In the same way, the second equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) is regular away from the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = 1\\rbrace .\\end{@align}}Since the sets (\\ref {singular1}) and (\\ref {singular2}) are disjoint and closed in D_\\delta , we conclude that \\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ).",
"That x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ) follows by differentiating (\\ref {Edef}) and applying Lemma \\ref {claim4E}.\\end{xmlelement*}We next show that \\text{\\upshape Re\\,}\\mathcal {E} > 0 on D_\\delta .Equation (\\ref {Edef}) yields\\mathcal {E} + \\bar{\\mathcal {E}} = \\frac{4\\hat{m}_{21}}{(\\hat{m}_{11} +\\hat{m}_{21} )^2 -1}.In light of the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, this gives{\\begin{@align}{1}{-1}\\text{\\upshape Re\\,}\\mathcal {E} = \\frac{2(1 + \\hat{m}_{11})}{|1 + \\hat{m}_{11} + \\hat{m}_{12}|^2}.\\end{@align}}On the other hand, the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1 together with (\\ref {m0symm}) yield \\hat{m}_{11} \\in {R} and \\hat{m}_{11}^2 - |\\hat{m}_{12}|^2 = 1.",
"We infer that \\hat{m}_{11} \\in (-\\infty ,-1] \\cup [1, \\infty ).For (x,y) = (0,0) we have m(0,0,z) = I for all z, because the jump matrix v is the identity matrix.",
"In particular, \\hat{m}_{11}(0,0) = 1.",
"By continuity, this gives (\\hat{m}(x,y))_{11} \\ge 1 for all ~(x,y) \\in D_\\delta .",
"In view of (\\ref {reEm0}), it follows that \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0 on D_\\delta .",
"}Finally, we show that $ E(x,0) = E0(x)$ for $ x [0, 1-)$; the proof that $ E(0,y) = E1(y)$ for $ y [0,1-)$ is similar.For $ y = 0$, the definition (\\ref {jumpdef}) of $ v$ yields{\\begin{@align}{1}{-1}v(x,0,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\I, & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad x \\in [0, 1-\\delta ).\\end{@align}}It follows from part $ (c)$ of Lemma \\ref {xpartlemma} that the $ 22$-matrix valued function $ m0(y,z)$ defined for $ x [0,1-)$ by{\\begin{@align}{1}{-1}m_0(x,z) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _0, \\\\\\Phi _0\\big (x,F_{(x,0)}^{-1}(z)\\big ), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}satisfies the RH problem (\\ref {RHm}) associated with $ (x,y) = (x,0)$ for each $ x [0,1-)$.Furthermore, since $ 0 $ and $ F(x,y)-1(0) = -$, the first symmetry in (\\ref {phi0symmetries}) yields{\\begin{@align}{1}{-1}m_0(x,0) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\Phi _0\\big (x,\\infty ^-\\big )= \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\sigma _3\\Phi _0\\big (x,\\infty ^+\\big )\\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phi0atinftyplus}) for $ 0(x,+)$, the $ (11)$ and $ (21)$ entries of (\\ref {mx00}) give{\\begin{@align*}{1}{-1}(m_0(x,0))_{11} = \\frac{1 + \\mathcal {E}_0(x) \\overline{\\mathcal {E}_0(x)}}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}, \\qquad (m_0(x,0))_{21} = \\frac{(1 - \\mathcal {E}_0(x))(1 + \\overline{\\mathcal {E}_0(x)})}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}.\\end{@align*}}Solving these two equations for $ E0$ and $ E0$, we find{\\begin{@align}{1}{-1}\\mathcal {E}_0(x) = \\frac{1 + (m_0(x,0))_{11} - (m_0(x,0))_{21}}{1 + (m_0(x,0))_{11} + (m_0(x,0))_{21}}.\\end{@align}}But by uniqueness of the solution of the RH problem (\\ref {RHm}), we have $ m0(x,z) = m(x,0,z)$; hence, comparing (\\ref {E0recover}) with (\\ref {Erecover}), we deduce that $ E(x,0) = E0(x)$ for $ x [0, 1-)$.$ It only remains to show that $\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"The proof of this relies on the construction of an eigenfunction $\\Phi $ of the Lax pair.",
"Equations (REF ) and (REF ) suggest that we define $\\Phi (x,y,P)$ for $(x,y) \\in D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ by $\\Phi (x,y,P) = \\tilde{\\Phi }(x,y)m(x,y,F_{(x,y)}(P)),$ where $\\tilde{\\Phi }(x,y)$ is the function defined in ().",
"Lemma 6.8 The function $\\Phi $ defined in (REF ) satisfies the Lax pair equations ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y, P) = \\mathsf {U}(x,y, P) \\Phi (x,y, P),\\\\\\Phi _y(x,y, P) = \\mathsf {V}(x,y,P) \\Phi (x,y,P),\\end{array}\\right.",
"}$ for $(x,y) \\in \\operatorname{int}D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ .",
"The analyticity structure of $n_0$ established in Lemma REF implies that there exists a $2\\times 2$ -matrix valued function $C(x,y)$ independent of $z$ such that $n_0(x,y,z) = \\frac{C(x,y)}{z+1}, \\qquad z \\in \\hat{.",
"}We determine C(x,y) by evaluating (\\ref {n0C}) at z = 0.",
"By Lemma \\ref {claim5E}, this gives C(x,y) = \\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}.It follows that{\\begin{@align}{1}{-1}n_0 = \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1} = \\bigg (m_x -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z\\bigg )m^{-1}\\end{@align}}for (x,y) \\in D_\\delta and z \\in \\Omega _\\infty .$ Differentiating (REF ) with respect to $x$ and using (), we find, for $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ , $\\Phi _x(x,y,P)& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y)(m_x + z_x m_z)\\\\& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1}m(x,y,z)\\\\& = \\bigg (\\tilde{\\Phi }_x(x,y)\\tilde{\\Phi }(x,y)^{-1}+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z(x,y,P)+1}\\tilde{\\Phi }(x,y)^{-1}\\bigg )\\Phi (x,y,P)$ Substituting in the expressions () and () for $\\tilde{\\Phi }$ and $\\hat{m}$ in terms of $\\mathcal {E}$ , $\\bar{\\mathcal {E}}$ , and recalling that $1 - \\frac{2}{z+1} = \\lambda ,$ this yields the first equation in (REF ).",
"A similar argument gives the second equation in (REF ).",
"Lemma 6.9 The complex-valued function $\\mathcal {E}:D \\rightarrow {R}$ defined by (REF ) satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"Fix a point $P = (\\lambda , k)$ in $F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ .",
"By Lemma , the map $(x,y) \\mapsto \\Phi (x,y, P)$ is $C^n$ from $\\operatorname{int}D_\\delta $ to $ and satisfies the Lax pair equations (\\ref {philax}).",
"Since~ $ n 2$, it follows that $$ satisfies{\\begin{@align*}{1}{-1}\\Phi _{xy}(x,y,P) - \\Phi _{yx} (x,y,P) = 0, \\qquad (x,y) \\in \\operatorname{int}D_\\delta .\\end{@align*}}The $ (21)$-entry of this equation reads$$\\frac{(1-x-y)\\lambda }{2(\\text{\\upshape Re\\,}\\mathcal {E}(x,y))^2 (1-k-y)}\\bigg \\lbrace (\\text{\\upshape Re\\,}\\mathcal {E})\\bigg (\\mathcal {E}_{xy} - \\frac{\\mathcal {E}_x + \\mathcal {E}_y}{2(1-x-y)}\\bigg ) - \\mathcal {E}_x \\mathcal {E}_y\\bigg \\rbrace = 0.$$It follows that $ E(x,y)$ satisfies (\\ref {ernst}) for $ (x,y) intD$.This completes the proof of the lemma.$ Lemma REF completes the proof of part $(a)$ of Theorem REF .",
"The following lemma proves part (b).",
"Lemma 6.10 There exists a constant $c_\\delta > 0$ such that if $\\Vert \\mathcal {E}_0 / \\text{\\upshape Re\\,}\\mathcal {E}_0 \\Vert _{L^1([0,1-\\delta ))} , \\, \\Vert \\mathcal {E}_1/ \\text{\\upshape Re\\,}\\mathcal {E}_1 \\Vert _{L^1([0,1-\\delta ))} < c_\\delta ,$ then the linear operator $I - \\mathcal {C}_{w(x,y,\\cdot )} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"It follows from () and () that, by choosing $c_\\delta $ sufficiently small, equation (REF ) gives $|\\Phi _0(x,k^\\pm )-I| < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ and an analogous estimate holds for $|\\Phi _1(y,k^\\pm )-I|$ .",
"This yields $\\Vert w(x,y,\\cdot )\\Vert _{L^\\infty (\\Gamma )} < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ for all $(x,y) \\in D_\\delta $ whenever (REF ) holds.",
"Indeed, equation (REF ) implies $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\Vert w\\Vert _{L^\\infty (\\Gamma )} < 1$ for all $(x,y) \\in D_\\delta $ .",
"Hence $I - \\mathcal {C}_{w(x,y,\\cdot )}$ is invertible in $\\mathcal {B}(L^2(\\Gamma ))$ for each $(x,y) \\in D_\\delta $ .",
"For part (c) assume $\\mathcal {E}_0, \\mathcal {E}_1>0$ and write $V_0= -\\log \\mathcal {E}_0$ , $V_1 = -\\log \\mathcal {E}_1$ .",
"Then there exists a $C^n$ -solution $V(x,y)$ of the Goursat problem for the Euler-Darboux equation (REF ) with data $\\lbrace V_0,V_1 \\rbrace $ by Theorem REF .",
"Hence $\\mathcal {E}=e^{- V}$ is a $C^n$ -solution of the Goursat problem for (REF ) with data $\\lbrace \\mathcal {E}_0,\\mathcal {E}_1 \\rbrace $ .",
"This completes the proof of part (c) and hence of Theorem REF .",
"Proof of Theorem REF Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ , be complex-valued functions satisfying (REF ) for some $n \\ge 2$ and some $\\alpha \\in (0,1)$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ and define $m_1, m_2 \\in by (\\ref {m1m2def}).We will prove (\\ref {boundarylimita}); the proof of (\\ref {boundarylimitb}) is similar.$ By (REF ), we have $x^\\alpha \\mathcal {E}_x(x,y) = 2x^\\alpha \\frac{\\hat{m}_{21}(x,y) \\hat{m}_{11x}(x,y) - (1 + \\hat{m}_{11}(x,y))\\hat{m}_{21x}(x,y)}{(1 + \\hat{m}_{11}(x,y) + \\hat{m}_{21}(x,y))^2},$ where, as before, $\\hat{m}(x,y) = m(x,y,0)$ .",
"Thus, in order to compute $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y)$ , it is enough to compute $\\hat{m}(0,y)$ and $\\lim _{x \\downarrow 0} x^\\alpha \\hat{m}_x(x,y)$ .",
"Since $m = I + \\mathcal {C}(\\mu w)$ and $m_x = \\mathcal {C}(\\mu _x w) + \\mathcal {C}(\\mu w_x),$ this means that we are interested in the values of $w(0,y,z), \\quad \\mu (0,y,z),\\quad \\lim _{x \\downarrow 0} x^\\alpha w_x(x,y,z), \\quad \\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z).$ Lemma 6.11 We have $& w(0,y, z) = {\\left\\lbrace \\begin{array}{ll}0, & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y, F_{(0,y)}^{-1}(z)\\big ) - I, \\quad & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad y \\in [0,1),\\\\~& \\mu (0,y,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y,\\infty ^+\\big )^{-1} , & z \\in \\Gamma _1,\\end{array}\\right.",
"}\\quad y \\in [0,1),$ and $\\hat{m}(0,y) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1}\\sigma _3 \\Phi _1\\big (y,\\infty ^+\\big ) \\sigma _3, \\qquad y \\in [0,1).$ Equation (REF ) is immediate from (REF ).",
"Moreover, by (REF ), $ m(0,y,z) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _1, \\\\\\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}$ Equation () follows from (REF ) and the fact that $\\mu (x,y,z)=m_-(x,y,z)$ for $(x,y) \\in D$ and $z \\in \\Gamma $ .",
"Since $0 \\in \\Omega _0$ and $F_{(0,y)}^{-1}(0) = \\infty ^-$ , equation (REF ) follows by setting $z = 0$ in (REF ) and using the first symmetry in ().",
"Lemma 6.12 For $y \\in [0,1)$ , we have $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\begin{pmatrix} \\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) & m_1 \\end{pmatrix}, \\quad & z \\in \\Gamma _0,\\\\0, & z \\in \\Gamma _1,\\end{array}\\right.",
"}$ and $\\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z)= \\Pi (y,z), \\qquad z \\in \\Gamma _1,$ where the function $\\Pi (y,z)$ is defined by $\\Pi (y,z) = - \\frac{1}{\\sqrt{1-y}}\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}}{z+1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}.$ It follows from (REF ) and (REF ) that $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) =0$ for $z \\in \\Gamma _1$ and that, for $z\\in \\Gamma _0$ , $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y,z)& = \\lim _{x \\rightarrow 0} x^\\alpha \\bigg \\lbrace \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))+ \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{dx} F_{(x,y)}^{-1}(z)\\bigg \\rbrace \\\\~& = \\lim _{x \\rightarrow 0} x^\\alpha \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))= \\lim _{x \\rightarrow 0} x^\\alpha \\mathsf {U}_0(x,F_{(x,y)}^{-1}(z)).$ Recalling the definition (REF ) of $\\mathsf {U}_0$ , (REF ) follows.",
"To prove (REF ), we note that differentiation of the relation $\\mu = I + \\mathcal {C}_w \\mu $ gives $ \\mu _x = (I-\\mathcal {C}_w)^{-1}\\mathcal {C}_-(\\mu w_x).$ We first compute $\\lim _{x \\downarrow 0} \\mathcal {C}_-(\\mu x^\\alpha w_x)$ .",
"Equations () and (REF ) imply, for $z \\in \\Gamma _1$ , $\\nonumber &\\Big \\lbrace \\mathcal {C}_- \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ]\\Big \\rbrace (z)= \\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} }{2\\pi i}\\\\\\nonumber & \\times \\int _{\\Gamma _0} \\frac{\\Phi _1\\big (y,F_{(0,y)}^{-1}(z^{\\prime })\\big )\\frac{1}{2} \\Big ({\\begin{matrix}\\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) & m_1 \\end{matrix}}\\Big ) dz^{\\prime }}{z^{\\prime } -z}\\\\ & =- \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1(y,F_{(0,y)}^{-1}(z^{\\prime })) \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime }))\\left({\\begin{matrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{matrix}}\\right)}{2(z^{\\prime } -z)} =: \\tilde{\\Pi }(y,z).$ Recalling the expression (REF ) for $\\lambda (0,0, F_{(0,y)}^{-1}(z))$ and using that $F_{(0,y)}^{-1}(-1) = 0$ , we find $\\tilde{\\Pi }(y,z) = -\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0) \\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix} }{(z+1)\\sqrt{1-y}}.$ In view of (REF ), it only remains to show that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"We have, for $z \\in \\Gamma _1$ , $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)= \\frac{1}{2\\pi i} \\int _{\\Gamma _1}\\frac{\\Pi (y, z^{\\prime })\\big (\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I\\big )}{z^{\\prime } - z_-} dz^{\\prime }\\\\& = -\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{2\\pi i\\sqrt{1-y}}\\int _{\\Gamma _1}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z_-} \\frac{dz^{\\prime }}{z^{\\prime }+1}.$ Deforming the contour to infinity and using that $\\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z} \\frac{1}{z^{\\prime }+1} = -\\frac{\\Phi _1(y, 0) - I}{z+1}, $ a residue computation gives $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)=\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{\\sqrt{1-y}} \\frac{\\Phi _1(y, 0) - I}{z+1}.$ Simple algebra now shows that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"Lemma 6.13 For $y \\in [0,1)$ , we have $\\Phi _1(y, 0) = \\begin{pmatrix} e^{\\int _0^y \\frac{\\overline{\\mathcal {E}_{1y}(y^{\\prime })}}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} & 0 \\\\ 0 & e^{\\int _0^y \\frac{\\mathcal {E}_{1y}(y^{\\prime })}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\end{pmatrix}.$ Since 0 is a real branch point of the Riemann surface $\\Sigma _{(0,y)}$ , the symmetries () of $\\Phi _1$ imply that $\\Phi _1(y, 0^+) = \\Phi _1(y, 0^-) = \\sigma _3 \\Phi _1(y, 0^+)\\sigma _3 \\quad \\text{and} \\quad \\Phi _1(y, 0) = \\sigma _1\\overline{\\Phi _1(y, 0)}\\sigma _1.$ Hence $\\Phi _1(y, 0)$ has the form $\\Phi _1(y, 0) = \\begin{pmatrix} f(y) & 0 \\\\ 0 & \\overline{f(y)} \\end{pmatrix},$ where $f(y)$ is a function of $y$ .",
"Since $\\lambda (0, y, 0) = \\infty $ , we can determine $f(y)$ by solving the equation $\\Phi _{1y}(y, 0) = \\frac{1}{2 \\text{\\upshape Re\\,}\\mathcal {E}_{1}(y)} \\begin{pmatrix} \\overline{\\mathcal {E}_{1y}(y)} & 0 \\\\~0 & \\mathcal {E}_{1y}(y) \\end{pmatrix} \\Phi _1(y,0),$ which is a consequence of (REF ).",
"This gives the desired statement.",
"The following lemma completes the proof of Theorem REF .",
"Lemma 6.14 For $y \\in [0,1)$ , we have $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y) = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}}.$ We first compute $ \\lim _{x \\downarrow 0} x^\\alpha m_x(x,y,0)$ .",
"Proceeding as in the proof of (REF ), we find $\\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ](0)=-\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0)}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix},$ and $\\nonumber \\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu _x(x,y,\\cdot ) w(x,y,\\cdot )\\big ](0)= &\\; \\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}\\\\ & \\times \\left( \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3 \\right),$ where the derivation of (REF ) employs Lemma REF and Lemma REF as well as the residue calculation $-\\frac{1}{2\\pi i} \\int _{\\Gamma _1} \\frac{\\Phi _1(y, F_{(0,y)}^{-1}(z)) - I}{z} \\frac{dz}{z+1} = \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Adding (REF ) and (REF ) and recalling (REF ), we obtain $\\nonumber \\lim _{x \\rightarrow 0} x^\\alpha m_x(x,y,0)= & - \\frac{1}{\\sqrt{1-y}}\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\\\&\\times \\Phi _1\\big (y, 0\\big )^{-1}\\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Substituting (), (REF ), (REF ), and (REF ) into (REF ), long but straightforward computations yield ().",
"Examples We consider two examples of exact solutions—one with collinear polarization and one with noncollinear polarization.",
"For each example, we verify explicitly that the formulas () of Theorem REF on the behavior near the boundary are satisfied.",
"The Khan-Penrose solution The Khan-Penrose [13] solution is given by the potential $\\mathcal {E}(x,y)= \\frac{1+\\sqrt{x}\\sqrt{1-y}+\\sqrt{y}\\sqrt{1-x}}{1-\\sqrt{x}\\sqrt{1-y}-\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y) \\in D.$ Straightforward computations show that $m_1=1=m_2$ and $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{\\sqrt{1-y}}{(1-\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= \\frac{\\sqrt{1-x}}{(1-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ The Nutku-Halil solution One version of the Nutku-Halil [23] solution is given by $\\mathcal {E}(x,y)= \\frac{1-i\\sqrt{x}\\sqrt{1-y}+i\\sqrt{y}\\sqrt{1-x}}{1+i\\sqrt{x}\\sqrt{1-y}-i\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y)\\in D.$ In this case, $m_1=-i=-m_2$ and we compute $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{i\\sqrt{1-y}}{(i+\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= -\\frac{i\\sqrt{1-x}}{(i-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ appendix Gravitational waves and the hyperbolic Ernst equation It is shown in Eq.",
"(11.7) in [11] that the Ernst potential $\\mathcal {E}$ satisfies $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{uv} - U_u \\mathcal {E}_v - U_v \\mathcal {E}_u\\right) = 4\\mathcal {E}_u \\mathcal {E}_v.$ where $e^{-U(u,v)} = f(u) + g(v)$ and $f(u)$ and $g(v)$ are monotonically decreasing for positive argument and $f(0) = g(0) = 1/2$ .",
"(Note that Griffiths writes $Z$ for the Ernst potential.)",
"As suggested by Szekeres [25], it is possible to use $(f, g)$ as coordinates.",
"This leads to the equation $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{fg} + \\frac{\\mathcal {E}_f + \\mathcal {E}_g}{f+g}\\right) = 4\\mathcal {E}_f \\mathcal {E}_g,$ where $(f,g)$ belongs to the triangular region $\\bigg \\lbrace (f,g) \\in {R}^2 \\, \\bigg | \\, f \\le \\frac{1}{2}, \\; g \\le \\frac{1}{2}, \\; f + g > 0\\bigg \\rbrace .$ The change of variables $x = \\frac{1}{2} - g$ , $y = \\frac{1}{2} - f$ transforms (REF ) into (REF ).",
"In order for the solution to describe gravitational waves, the following boundary condition must be satisfied (Eq.",
"(7.15) in [11]; see also (11.23) in [11] but in (11.23) equation $(f,g)$ approaches the corner whereas in (7.15) the two edges are approached; also in (7.15) there is a factor $(f+g)$ missing; this factor comes from (7.9)) $& \\lim _{g \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -g\\Big ) (f+g)\\frac{|\\mathcal {E}_g|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_2}{2},\\\\& \\lim _{f \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -f\\Big ) (f+g) \\frac{|\\mathcal {E}_f|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"In terms of $(x,y)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\frac{x(1-x-y)|\\mathcal {E}_x|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}= \\frac{k_2}{2},\\\\& \\lim _{y \\rightarrow 0} \\frac{y(1-x-y)|\\mathcal {E}_y|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2} = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"That is, since $\\text{\\upshape Re\\,}\\mathcal {E} > 0$ , $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1),$ for some constants $m_1 , m_2 \\in [1, \\sqrt{2})$ .",
"If we assume that $\\mathcal {E} \\in C(D)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1).$ These are the conditions given in () with $\\alpha = 1/2$ .",
"In particular, $& \\mathcal {E}_{0x}(x) = \\frac{m_1 + o(1)}{\\sqrt{x}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{0}(x) \\sim 2m_1\\sqrt{x}, \\qquad x \\downarrow 0,\\\\& \\mathcal {E}_{1y}(y) = \\frac{m_2 + o(1)}{\\sqrt{y}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{1}(y) \\sim 2m_2\\sqrt{y}, \\qquad y \\downarrow 0,$ where $m_1, m_2 \\in [1, \\sqrt{2})$ .",
"Acknowledgement The authors acknowledge support from the European Research Council, Grant Agreement No.",
"682537, the Swedish Research Council, Grant No.",
"2015-05430, and the Göran Gustafsson Foundation, Sweden."
],
[
"Proofs of main results",
"In this section, we use the lemmas from the previous section to prove Theorem REF –REF ."
],
[
"Proofs of Theorem 1 & 2",
"Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ be complex-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ .",
"We will show that $\\mathcal {E}(x,y)$ can be uniquely expressed in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ by (REF ).",
"The idea in what follows is to introduce a solution $\\Phi $ of (REF ) as the solution of the integral equation $\\Phi (x,y,k^\\pm ) = I + \\int _{(0,0)}^{(x,y)} (W\\Phi )(x^{\\prime },y^{\\prime }, k^\\pm ).$ However, since $W$ in general is singular on the boundary of $D$ , we need to be more careful with the definition.",
"We therefore instead define $\\Phi $ as the solution of $\\Phi (x, y, k^+) = \\Phi _0(x, k^+) + \\int _0^y (\\mathsf {V} \\Phi )(x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}$ Lemma 6.1 (Solution of Lax pair equations) The function $\\Phi (x,y,P)$ defined in (REF ) has the following properties: $\\Phi (x,y,k^\\pm )$ is a well-defined $2\\times 2$ -matrix valued function of $(x,y) \\in D$ and $k \\in \\hat{ \\setminus [0,1] which also satisfies the alternative Volterra integral equation:{\\begin{@align}{1}{-1}\\Phi (x, y, k^+) = \\Phi _1(y, k^+) + \\int _0^x (\\mathsf {U} \\Phi )(x^{\\prime },y,k^+) dx^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}}\\item For each $ k [0,1]$, the function $ (x,y) (x,y,k+)$ is continuous on $ D$ and is $ Cn$ on~ $ intD$.$ For each $k \\in \\hat{ \\setminus [0,1], the functions(x,y) \\mapsto x^\\alpha \\Phi _x(x,y,k^+), \\quad (x,y) \\mapsto y^\\alpha \\Phi _y(x,y,k^+), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha \\Phi _{xy}(x,y,k^+),are continuous on D.}\\item $$ obeys the symmetries{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll} \\Phi (x,y,k^+) = \\sigma _3\\Phi (x, y, k^-)\\sigma _3,\\\\\\Phi (x,y,k^\\pm ) = \\sigma _1\\overline{\\Phi (x, y,\\bar{k}^\\pm )}\\sigma _1,\\end{array}\\right.}",
"\\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}\\end{@align}}\\item For each~ point $ (x,y) D$, $ (x,y,P)$ extends continuously to an analytic function of $ P S(x,y) $, where $ = 0 1$ is the contour defined in (\\ref {Sigma01def}).$ The value of $\\Phi $ at $P = \\infty ^+$ is given by $\\Phi (x,y,\\infty ^+) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}, \\qquad (x,y) \\in D.$ The determinant of $\\Phi $ is given by $\\det \\Phi (x,y,P) = \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0, \\qquad (x,y) \\in D, \\ P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .$ By Lemma REF the lemma holds for $y = 0$ , i.e., the function $\\Phi (x,0,P)$ is well-defined and the properties $(a)$ -$(d)$ are satisfied when $x = 0$ or $y=0$ .",
"In order to see that $\\Phi $ is well-defined also for $(x,y)$ in the interior of $D$ , we note that (REF ) implies $\\Phi (x, y, k^+) = \\Phi (x,0, k^+) + \\int _0^y \\mathsf {V}(x,y^{\\prime },k^+) \\Phi (x,y^{\\prime },k^+) dy^{\\prime }, \\qquad (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}The same type of successive approximation argument already used in the proof of Lemma \\ref {xpartlemma} shows that the Volterra equation (\\ref {Phidefxy}) has a unique solution for each fixed x \\in (0,1) and each k \\in \\hat{ \\setminus [0,1], and that this solution \\Phi (x,y,P) extends continuously to an analytic function of P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma .",
"This proves (b).",
"}In order to prove (a), it remains to deduce the alternative representation (\\ref {Phidef2}).",
"Note that \\Phi _y = V\\Phi by definition and{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)&=\\Phi _x(x,0,k^+) + \\int _{0}^y x\\Phi (x,y^{\\prime },k^+)+ _x(x,y^{\\prime },k^\\pm )dy^{\\prime }.\\end{@align*}}Since \\mathcal {E} is a solution of the Goursat problem, we havex=\\mathsf {U}_y +[\\mathsf {U},,and, moreover, \\Phi _x(x,0,k^+) =\\mathsf {U}\\Phi (x,0,k^+).",
"Now a straightforward calculation shows{\\begin{@align*}{1}{-1}\\Phi _x(x,y,k^+)= \\mathsf {U}\\Phi (x,y,k^+) + \\int _{0}^y _x(x,y^{\\prime },k^\\pm )-\\Phi (x,y^{\\prime },k^+) dy^{\\prime }.\\end{@align*}}Thus the function \\tilde{\\Phi }=\\Phi _x -\\mathsf {U}\\Phi is the unique solution of the Volterra integral equation\\tilde{\\Phi }(x,y^{\\prime },k^+) = \\int _0^y {\\Phi }(x,y^{\\prime },k^+)dy^{\\prime }giving \\tilde{\\Phi }=0.",
"This implies \\Phi _x = \\mathsf {U}\\Phi .",
"Consequently, \\Phi , defined by (\\ref {Phidef}), is an eigenfunction for the Lax pair equations (\\ref {lax}).",
"The difference between (\\ref {Phidef}) and (\\ref {Phidef2}) is given by{\\begin{@align*}{1}{-1}&\\Phi _0(x,k^+) - \\Phi _1(y,k^+) +\\int _0^y (x,y^{\\prime },k^+)dy^{\\prime } - \\int _0^x \\mathsf {U}\\Phi (x^{\\prime },y,k^+)dx^{\\prime }\\\\=& \\, \\int _0^y \\int _0^x ()_x(x^{\\prime },y^{\\prime },k^+)dx^{\\prime } dy^{\\prime } - \\int _0^x \\int _0^y (\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+)dy^{\\prime }dx^{\\prime }\\\\=& \\, \\int _0^x \\int _0^y ()_x(x^{\\prime },y^{\\prime },k^+)-(\\mathsf {U}\\Phi )_y(x^{\\prime },y^{\\prime },k^+) dy^{\\prime } dx^{\\prime }\\end{@align*}}and ()_x = (\\mathsf {U}\\Phi )_y is the compatibility condition for the Lax pair.",
"Hence the two representations (\\ref {Phidef}) and (\\ref {Phidef2}) are equal.This proves (a).$ The symmetries (REF ) of $\\lambda $ show that $W(x,y,k^+) = \\sigma _3W(x,y, k^-)\\sigma _3, \\qquad W(x,y,k^+) = \\sigma _1\\overline{W(x,y,\\bar{k}^+)}\\sigma _1.$ Since $\\lambda (x,y,\\infty ^+) = 1$ , $\\Phi (x,y,\\infty ^+)$ satisfies the equation $\\Phi _y(x, y, \\infty ^+) = \\frac{1}{2\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} \\begin{pmatrix} \\overline{\\mathcal {E}_y(x,y)} & \\overline{\\mathcal {E}_y(x,y)} \\\\\\mathcal {E}_y(x,y) & \\mathcal {E}_y(x,y) \\end{pmatrix} \\Phi (x,y,\\infty ^+), \\qquad (x,y) \\in D.$ Using the above equations and arguing as in the proof of Lemma REF , the statements $(c)$ , $(d)$ , $(e)$ , $(f)$ , and $(g)$ follow from equation (REF ) and the corresponding statements in Lemma REF .",
"Part $(g)$ of Lemma REF implies that the inverse matrix $\\Phi (x,y,P)^{-1}$ is well-defined for $(x,y) \\in D$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma $ .",
"Lemma 6.2 For each $(x,y) \\in D$ , $P \\mapsto \\Phi (x,y,P)\\Phi (x,0, P)^{-1} \\quad \\text{and} \\quad P \\mapsto \\Phi (x,y,P)\\Phi (0,y,P)^{-1}$ are analytic functions of $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ and $P \\in \\mathcal {S}_{(x,y)} \\setminus \\Sigma _0$ , respectively.",
"Let $U$ be an open set in $\\mathcal {S}_{(x,y)} \\setminus \\Sigma _1$ .",
"Multiplying (REF ) by $\\Phi (x,0,P)^{-1}$ from the right, we find $\\nonumber & \\Phi (x, y, P)\\Phi (x,0,P)^{-1} = I + \\int _0^y \\mathsf {V}(x,y^{\\prime },P) \\Phi (x,y^{\\prime },P)\\Phi (x,0,P)^{-1} dy^{\\prime },\\\\ & \\hspace{227.62204pt} (x,y) \\in D, \\ k \\in \\hat{ \\setminus [0,1].",
"}where the values of \\Phi (x,0,P) and \\lambda (x,y^{\\prime },P) in (\\ref {phiPmap}) are to be interpreted as in Remark \\ref {tilderemark}.",
"SinceP \\mapsto \\lambda (x,y^{\\prime },P)^{-1} = \\sqrt{\\frac{k - x}{k - (1-y^{\\prime })}}is an analytic map U \\rightarrow for each y^{\\prime }, so is \\mathsf {V}(x,y^{\\prime },\\cdot ).",
"It follows that the solution \\Phi (x, y, P)\\Phi (x,0,P)^{-1} of (\\ref {phiPmap}) also is analytic for P \\in U.",
"This establishes the desired statement for the first map in (\\ref {phiminusphi}); the proof for the second map is similar.$ Let $\\Omega _0$ , $\\Omega _1$ , and $\\Omega _\\infty $ denote the three components of $\\hat{ \\setminus \\Gamma defined in (\\ref {Omegadef}) and displayed in Figure \\ref {Omegas.pdf}.",
"}$ Lemma 6.3 The $2\\times 2$ -matrix valued function $m(x,y,z)$ defined for $(x,y)\\in D$ by $m(x,y,z) =\\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi \\big (x,y,F_{(x,y)}^{-1}(z)\\big ) \\times {\\left\\lbrace \\begin{array}{ll} \\Phi \\big (x,0,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _0, \\\\\\Phi \\big (0,y,F_{(x,y)}^{-1}(z)\\big )^{-1}, & z \\in \\Omega _1, \\\\I, & z \\in \\Omega _\\infty ,\\end{array}\\right.",
"}$ satisfies the RH problem (REF ) and the relation (REF ) for each $(x,y) \\in D$ .",
"Since $F_{(x,y)}$ is a biholomorphism $\\mathcal {S}_{(x,y)}~\\rightarrow \\hat{, we infer from Lemma \\ref {claim1} together with Lemma \\ref {claim2} that m(x,y, \\cdot ) is analytic in \\Gamma and that m(x,y,z) \\rightarrow I as z \\rightarrow \\infty for each (x,y) \\in D. The jump condition in (\\ref {RHm}) holds as a consequence of the definition (\\ref {jumpdef}) of v(x,y,z) and the fact that\\Phi _0(x,k) = \\Phi (x,0,k), \\qquad \\Phi _1(y,k) = \\Phi (0,y,k).Finally, since 0 \\in \\Omega _\\infty and F_{(x,y)}^{-1}(0) = \\infty ^-, the first symmetry in (\\ref {phisymmetries}) yields{\\begin{@align}{1}{-1}m(x,y,0) = \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\Phi (x,y, \\infty ^-)= \\Phi \\big (x,y, \\infty ^+\\big )^{-1} \\sigma _3 \\Phi \\big (x,y, \\infty ^+\\big ) \\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phiatinftyplus}) for \\Phi \\big (x,y, \\infty ^+\\big ), the (11) and (21) entries of (\\ref {mxy0}) give{\\begin{@align*}{1}{-1}(m(x,y,0))_{11} = \\frac{1 + \\mathcal {E}(x,y) \\overline{\\mathcal {E}(x,y)}}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}, \\qquad (m(x,y,0))_{21} = \\frac{(1 - \\mathcal {E}(x,y))(1 + \\overline{\\mathcal {E}(x,y)})}{\\mathcal {E}(x,y) + \\overline{\\mathcal {E}(x,y)}}.\\end{@align*}}Solving these two equations for \\mathcal {E} and \\bar{\\mathcal {E}}, we find (\\ref {Erecover}).",
"}$ We have showed that if $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ , then $\\mathcal {E}(x,y)$ can be expressed in terms of the function $m$ defined in (REF ) via equation (REF ).",
"By Lemma REF , this function $m(x,y,z)$ is the unique solution of the RH-problem (REF ) whose formulation involves only the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ .",
"As a consequence, the value of the solution $\\mathcal {E}$ at $(x,y)$ is uniquely determined by the values $\\mathcal {E}_0(x^{\\prime })$ and $\\mathcal {E}_1(y^{\\prime })$ for $0\\le x^{\\prime } \\le x$ and $0\\le y^{\\prime }\\le y$ , if it exists.",
"This completes the proofs of Theorem REF and REF .",
"Proof of Theorem REF This subsection is devoted to proving Theorem REF regarding existence.",
"Let us therefore suppose that $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ are real-valued functions satisfying (REF ) for some $n \\ge 2$ .",
"Define $\\Phi _0(x,P)$ and $\\Phi _1(y,P)$ in terms of $\\mathcal {E}_0$ and $\\mathcal {E}_1$ via the Volterra equations (REF ).",
"Then $\\Phi _0$ and $\\Phi _1$ have the properties listed in Lemma REF and Lemma REF .",
"Let $\\delta \\in (0,1)$ and let $D_\\delta $ be the triangle defined in (REF ).",
"As in the proof of Theorem REF , choose $\\epsilon > 0$ so small that $F_{(x,y)}(\\Sigma _0)$ and $F_{(x,y)}(\\Sigma _1)$ are contained in the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, for all $(x,y) \\in D_\\delta $ .",
"Fix two smooth nonintersecting clockwise contours $\\Gamma _0$ and $\\Gamma _1$ in the complex $z$ -plane which encircle the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ , respectively, but which do not encircle zero, see Figure REF .",
"Suppose $\\Gamma _0$ and $\\Gamma _1$ are invariant under the involutions $z \\mapsto z^{-1}$ and $z \\mapsto \\bar{z}$ .",
"Let $\\Gamma = \\Gamma _0 \\cup \\Gamma _1$ and consider the family of RH problems given in (REF ) parametrized by the two parameters $(x,y) \\in D_\\delta $ .",
"We will show that if (REF ) has a (unique) solution $m(x,y,z)$ for each $(x,y) \\in D_\\delta $ , then the function $\\mathcal {E}(x,y)$ defined in terms of $m$ via equation (REF ) satisfies ${\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\\\text{$\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (\\ref {ernst}) in $\\operatorname{int}(D_\\delta )$,}\\\\\\text{$x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta )$ for some $\\alpha \\in [0,1)$,}\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}$ We next list some facts about Cauchy integrals that we will use throughout the proof.",
"If $h \\in L^2(\\Gamma )$ , then the Cauchy transform $\\mathcal {C}h$ is defined by $(\\mathcal {C}h)(z) = \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{h(z^{\\prime })}{z^{\\prime } - z} dz^{\\prime }, \\qquad z \\in \\Gamma ,$ We denote the nontangential boundary values of $\\mathcal {C}f$ from the left and right sides of $\\Gamma $ by $\\mathcal {C}_+ f$ and $\\mathcal {C}_-f$ respectively.",
"Then $\\mathcal {C}_+$ and $\\mathcal {C}_-$ are bounded operators on $L^2(\\Gamma )$ and $\\mathcal {C}_+ - \\mathcal {C}_- = I$ .",
"Let $w(x,y,z) = v(x,y,z) - I$ .",
"We define the operator $\\mathcal {C}_w: L^2(\\Gamma ) + L^\\infty (\\Gamma ) \\rightarrow L^2(\\Gamma )$ by $\\mathcal {C}_{w}(f) = \\mathcal {C}_-(f w).$ Then $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C \\Vert w\\Vert _{L^\\infty (\\Gamma )},$ where $C = \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}$ .",
"We henceforth assume that the RH problem (REF ) has a solution for all $(x,y) \\in D_\\delta $ or, equivalently, that $I - \\mathcal {C}_{w} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"For each $(x,y) \\in D_\\delta $ , we have $v \\in C(\\Gamma )$ and $v, v^{-1} \\in I + L^2(\\Gamma ) \\cap L^\\infty (\\Gamma )$ .",
"The theory of singular integral equations then implies that the solution of the RH problem (REF ) is given by (see e.g.",
"[4] or [18]) $m = I + \\mathcal {C}(\\mu w),$ where the $2\\times 2$ -matrix valued function $\\mu (x,y,\\cdot )$ is defined by $\\mu = I + (I - \\mathcal {C}_w)^{-1}\\mathcal {C}_w I \\in I + L^2(\\Gamma ).$ Equation (REF ) can be written more explicitly as $m(x,y,z) = I + \\frac{1}{2\\pi i} \\int _\\Gamma \\frac{(\\mu w)(x,y,s) ds}{s-z}, \\qquad (x,y) \\in D_\\delta , \\ z \\in \\hat{ \\setminus \\Gamma .",
"}$ Lemma 6.4 The map $ (x,y) \\mapsto w(x,y, \\cdot )$ is continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto y^\\alpha w_x(x,y, \\cdot ), \\quad (x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^\\infty (\\Gamma )$ .",
"For $N \\ge 0$ , let $C^N(K)$ denote the Banach space of functions on $K$ with continuous partial derivatives of order $\\le N$ equipped with the usual norm $\\Vert f\\Vert _{C^N(K)} = \\sup _{|\\alpha | \\le N} \\Vert D^\\alpha f\\Vert _{L^\\infty (K)}.$ By part $(f)$ of Lemma REF the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ): D_\\delta \\rightarrow C(K)$ is continuous for any compact set $K$ not intersecting $\\Sigma $ .",
"Moreover, assuming $F_{(x,y)}^{-1}(\\Gamma ) \\subset K $ , the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):D_\\delta \\rightarrow \\mathcal {B}(C(K), C(\\Gamma ))$ is continuous, because $\\sup _{\\Vert f\\Vert _{C(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f \\in C(K)$ on the compact set $K$ .",
"It follows that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ also is continuous.",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )): D_\\delta \\rightarrow C(\\Gamma _1)$ is continuous.",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is continuous from $D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If a sequence of holomorphic functions $f_n$ converges uniformly on an open set $\\Omega $ then the sequence of derivatives $f_n^{\\prime }$ converges uniformly on compact subsets of $\\Omega $ .",
"Fix $N \\ge n$ and let $K$ be a compact subset of $\\mathcal {S}_{(x,0)} \\setminus \\Sigma _0$ .",
"Then part $(f)$ of Lemma REF implies that the map $(x,y) \\mapsto \\Phi _0(x,\\cdot ):\\operatorname{int}D_\\delta \\rightarrow C^N(K)$ is $C^n$ .",
"On the other hand, the map $(x,y) \\mapsto (f \\mapsto f(F_{(x,y)}^{-1}(\\cdot ))):\\operatorname{int}D_\\delta \\rightarrow \\mathcal {B}(C^N(K), C(\\Gamma ))$ is $C^n$ .",
"Indeed, the map is continuous because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\big |f(F_{(x,y)}^{-1}(z)) - f(F_{(x^{\\prime },y^{\\prime })}^{-1}(z))\\big | \\rightarrow 0$ as $(x^{\\prime },y^{\\prime }) \\rightarrow (x,y)$ by uniform continuity of $f$ on the compact set $K$ .",
"Moreover, the map has a continuous partial derivative with respect to $x$ because $\\sup _{\\Vert f\\Vert _{C^N(K)}=1} \\sup _{z \\in \\Gamma } \\bigg |\\frac{f(F_{(x+h,y)}^{-1}(z)) - f(F_{(x,y)}^{-1}(z))}{h} - \\frac{d}{dx}f(F_{(x,y)}^{-1}(z)) \\bigg | \\rightarrow 0$ as $h \\rightarrow 0$ by the mean-value theorem and the uniform continuity of the first partial derivatives of $f$ .",
"Similar arguments show that all partial derivatives of order $\\le n$ exist and are continuous.",
"We conclude that the composed map $(x,y) \\mapsto \\Phi _0(x,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _0)$ built from (REF ) and (REF ) is $C^n$ .",
"A similar argument shows that $(x,y) \\mapsto \\Phi _1(y,F_{(x,y)}^{-1}(\\cdot )):\\operatorname{int}D_\\delta \\rightarrow C(\\Gamma _1)$ is $C^n$ .",
"Recalling the definition (REF ) of $v$ , this shows that the map (REF ) is $C^n$ as a map from $\\operatorname{int}D_\\delta $ to $ L^\\infty (\\Gamma )$ .",
"If $z \\in \\Gamma _0$ , we have $w_x(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)) + \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{d x}F_{(x,y)}^{-1}(z),$ where $\\frac{d}{d x}F_{(x,y)}^{-1}(z)$ denotes the derivative of the $k$ -projection of $F_{(x,y)}^{-1}(z)$ , which is given by $\\frac{d}{d x}F_{(x,y)}^{-1}(z) = -\\frac{(z-1)^2}{4z}.$ Thus part $(f)$ of Lemma REF and of Lemma REF imply that $(x,y) \\mapsto x^\\alpha w_x(x,y,\\cdot )$ is a continuous map $D_\\delta \\rightarrow L^\\infty (\\Gamma )$ .",
"The maps $(x,y) \\mapsto y^\\alpha w_y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha w_{xy}(x,y,\\cdot )$ can be treated similarly.",
"Lemma 6.5 The map $(x,y) \\mapsto \\mu (x,y, \\cdot ) - I$ is continuous from $D_\\delta $ to $L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Moreover, the three maps $(x,y) \\mapsto x^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto y^\\alpha \\mu _x(x,y, \\cdot ), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y, \\cdot ),$ are continuous from $D_\\delta $ to $L^2(\\Gamma )$ .",
"In view of the definition of $\\mu $ , the map (REF ) is given by $(x,y) \\mapsto (I - \\mathcal {C}_{w(x,y,\\cdot )})^{-1}\\mathcal {C}_-(w(x,y,\\cdot )).$ We note that the map $f \\mapsto I - \\mathcal {C}_f: L^\\infty (\\Gamma ) \\rightarrow \\mathcal {B}(L^2(\\Gamma ))$ is smooth by the estimate $\\Vert \\mathcal {C}_f\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le C\\Vert f\\Vert _{L^\\infty (\\Gamma )},$ and that the linear map $f \\mapsto \\mathcal {C}_-f: L^2(\\Gamma ) \\rightarrow L^2(\\Gamma )$ is bounded.",
"Since (REF ) can be viewed as a composition of maps of the form (REF ), (REF ), and (REF ) together with the smooth inversion map $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , it follows that (REF ) is continuous $D_\\delta \\rightarrow L^2(\\Gamma )$ and $C^n$ from $\\operatorname{int}D_\\delta $ to $L^2(\\Gamma )$ .",
"Similarly, $(x,y) \\mapsto x^\\alpha \\mu _x(x,y,\\cdot )$ can be viewed as composition of the continuous maps (REF ), (REF ), $I - \\mathcal {C}_w \\mapsto (I - \\mathcal {C}_w)^{-1}$ , (REF ), and (REF ), and is hence continuous.",
"The maps $(x,y) \\mapsto y^\\alpha \\mu _y(x,y,\\cdot )$ and $(x,y) \\mapsto x^\\alpha y^\\alpha \\mu _{xy}(x,y,\\cdot )$ can be treated analogously.",
"Lemma 6.6 The solution $m(x,y,z)$ of the RH problem (REF ) defined in (REF ) has the following properties: For each point $(x,y) \\in D_\\delta $ , $m(x,y,\\cdot )$ obeys the symmetries $m(x,y,z) = m(x,y,0)\\sigma _3m(x,y,z^{-1})\\sigma _3 = \\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}$ For each $z \\in \\hat{\\setminus \\Gamma , the map (x,y) \\mapsto m(x,y,z) is continuous from D_\\delta to {2 \\times 2} and is C^n from \\operatorname{int}D_\\delta to {2 \\times 2}.",
"}\\item For each~ $ z $, the three maps$$(x,y) \\mapsto x^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto y^\\alpha m_x(x,y, z), \\qquad (x,y) \\mapsto x^\\alpha y^\\alpha m_{xy}(x,y, z),$$are continuous from $ D$ to $ 22$.$ The symmetries in () and () show that $v$ satisfies ${\\left\\lbrace \\begin{array}{ll} v(x,y,z) = \\sigma _3v(x, y, z^{-1})\\sigma _3,\\\\v(x,y,z) = \\sigma _1\\overline{v(x, y, \\bar{z})}\\sigma _1,\\end{array}\\right.}",
"\\qquad z \\in \\Gamma , \\ (x,y) \\in D_\\delta .$ These symmetries imply that $\\sigma _3 m(x,y,0)^{-1} m(x,y,z^{-1})\\sigma _3$ and $\\sigma _1\\overline{m(x,y,\\bar{z})}\\sigma _1$ satisfy the same RH problem as $m(x,y,z)$ .",
"The symmetries in (REF ) follow by uniqueness.",
"Properties $(b)$ and $(c)$ follow from (REF ) together with the Lemmas REF and REF .",
"As in the proof of Theorem REF , we extend the definition (REF ) of $v$ to an open tubular neighborhood $N(\\Gamma ) = N(\\Gamma _0) \\cup N(\\Gamma _1)$ of $\\Gamma $ as follows, see Figure REF : $v(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in N(\\Gamma _0),\\\\\\Phi _1\\big (y, F_{(x,y)}^{-1}(z)\\big ), & z \\in N(\\Gamma _1),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We choose $N(\\Gamma )$ so narrow that it does not intersect the intervals $[-\\epsilon ^{-1}, -\\epsilon ]$ and $[\\epsilon , \\epsilon ^{-1}]$ .",
"Then, for each $(x,y) \\in D_\\delta $ , $v(x,y,\\cdot )$ is an analytic function of $z \\in N(\\Gamma )$ .",
"Using the notation $z(x,y,P) := F_{(x,y)}(P)$ , we can write (REF ) as $v(x,y,z(x,y,P)) = {\\left\\lbrace \\begin{array}{ll} \\Phi _0(x, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _0)\\big ), \\\\\\Phi _1(y, P), \\quad & P \\in F_{(x,y)}^{-1}\\big (N(\\Gamma _1)\\big ),\\end{array}\\right.",
"}\\quad (x,y) \\in D_\\delta .$ We define functions $f_0(x,y,z)$ and $f_1(x,y,z)$ for $(x,y) \\in D_\\delta $ by $f_0(x,y,z) = \\big [m_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma ,\\\\f_1(x,y,z) = \\big [m_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) m_z(x,y,z)\\big ]m(x,y,z)^{-1}, \\qquad z \\in \\hat{ \\setminus \\Gamma .",
"}Moreover, we let n_0(x,y,z) and n_1(x,y,z) denote the functions given by\\begin{subequations}{\\begin{@align}{1}{-1}n_0(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_0(x,y,z) + m(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _0, \\\\f_0(x,y,z), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}and{\\begin{@align}{1}{-1}n_1(x,y,z) = {\\left\\lbrace \\begin{array}{ll}f_1(x,y,z) + m(x,y,z)\\mathsf {V}_1\\big (y,F_{(x,y)}^{-1}(z)\\big )m(x,y,z)^{-1}, \\quad & z \\in \\Omega _1, \\\\f_1(x,y,z), & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}\\end{@align}}\\end{subequations}}$ Lemma 6.7 For each $(x,y) \\in D_\\delta $ , it holds that $n_0(x,y,z)$ is an analytic function of $z \\in \\hat{ \\setminus \\lbrace -1\\rbrace and has at most a simple pole at z = -1.\\item n_1(x,y,z) is an analytic function of z \\in \\hat{ \\setminus \\lbrace 1\\rbrace and has at most a simple pole at z = 1.\\item n_0(x,y,\\infty ) = 0 and n_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.\\item n_1(x,y,\\infty ) = 0 and n_1(x,y,0) = m_y(x,y,0)m(x,y,0)^{-1}.",
"}}{\\begin{xmlelement*}{proof}By (\\ref {linearzx}) the function z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1, \\infty \\rbrace with simple poles at z = -1 and z = \\infty .Equation (\\ref {msolution}) implies that m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty .",
"Hence f_0(x,y,z) is analytic at z = \\infty .It follows that f_0(x,y,z) is analytic for all z \\in \\hat{\\setminus (\\Gamma \\cup \\lbrace -1\\rbrace ) with a simple pole at z = -1 at most.Now f_0 satisfies the following jump condition across \\Gamma :{\\begin{@align}{1}{-1}\\nonumber f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\big [v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z)\\big ]\\\\ & \\times v(x,y,z)^{-1} m_-(x,y,z)^{-1}, \\qquad z \\in \\Gamma .\\end{@align}}Differentiating (\\ref {vzPhi0}) with respect to x and y and evaluating the resulting equations at k = F_{(x,y)}^{-1}(z), we find{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{0x}(x,F_{(x,y)}^{-1}(z)),\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _0),\\end{@align}} and{\\begin{@align}{1}{-1}{\\left\\lbrace \\begin{array}{ll}v_x(x,y,z) + z_x\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = 0,\\\\v_y(x,y,z) + z_y\\big (x,y, F_{(x,y)}^{-1}(z)\\big ) v_z(x,y,z) = \\Phi _{1y}(x,F_{(x,y)}^{-1}(z)),\\end{array}\\right.}",
"\\quad z \\in N(\\Gamma _1).\\end{@align}} Using the first equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}) in (\\ref {f0jump}), we conclude that f_0 is analytic across \\Gamma _1 and has the following jump across \\Gamma _0:{\\begin{@align}{1}{-1}f_{0+}(x,y,z) = &\\; f_{0-}(x,y,z) + m_-(x,y,z)\\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) m_-(x,y,z)^{-1}, \\quad \\ z \\in \\Gamma _0.\\end{@align}}Thus n_0 is analytic across \\Gamma .",
"Furthermore, since \\lambda (x,y,k) is analytic on \\mathcal {S}_{(x,y)} except for a simple pole at the branch point k = x, the function \\mathsf {U}_0\\big (x,F_{(x,y)}^{-1}(z)\\big ) is analytic for z \\in \\hat{\\setminus \\lbrace -1\\rbrace with a simple pole at z = -1.",
"It follows that n_0 satisfies (a).",
"The proof of (b) is similar and relies on the second equations in (\\ref {vxzxvza}) and (\\ref {vxzxvzb}).",
"}Using (\\ref {linearzx}) in the definition (\\ref {ndefa}) of n_0, we can write, for z \\in \\Omega _\\infty ,{\\begin{@align}{1}{-1}n_0(x,y,z) = f_0(x,y,z) = \\Big [m_x(x,y,z) -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z(x,y,z)\\Big ]m(x,y,z)^{-1}.\\end{@align}}Since m_x(x,y, z) = O(z^{-1}) and m_z(x,y, z) = O(z^{-2}) as z \\rightarrow \\infty , it follows that n_0(x,y,\\infty ) = 0.",
"On the other hand, evaluating (\\ref {n0f0}) at z = 0, we findn_0(x,y,0) = m_x(x,y,0)m(x,y,0)^{-1}.This proves (c); the proof of (d) is analogous.",
"}}\\end{xmlelement*}Let \\hat{m}(x,y) denote the function m(x,y,z)~ evaluated at z = 0, that is,\\hat{m}(x,y) = m(x,y,0).Evaluating the first symmetry in (\\ref {msymm}) at z = \\infty , we find{\\begin{@align}{1}{-1}I = \\hat{m}(x,y)\\sigma _3\\hat{m}(x,y)\\sigma _3.\\end{@align}}The unit determinant condition (\\ref {detmone}) implies that \\det \\hat{m} = 1.",
"Hence equation (\\ref {m0sigma3}) reduces to\\text{adj}(\\hat{m})=\\sigma _3 \\hat{m} \\sigma _3,where \\text{adj} denotes the adjugate matrix, which shows that \\hat{m}_{11} = \\hat{m}_{22}.A straightforward algebraic computation then yields{\\begin{@align}{1}{-1}\\hat{m}(x,y) = \\tilde{\\Phi }(x,y) \\sigma _3\\tilde{\\Phi }(x,y)\\sigma _3, \\qquad (x,y) \\in D_\\delta ,\\end{@align}}where the 2\\times 2-matrix valued function \\tilde{\\Phi }(x,y) is defined by{\\begin{@align}{1}{-1}\\tilde{\\Phi }(x,y) = \\frac{1}{2} \\begin{pmatrix} \\overline{\\mathcal {E}(x,y)} & 1 \\\\ \\mathcal {E}(x,y) & -1 \\end{pmatrix}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}\\end{@align}}and the functions \\mathcal {E}(x,y) and \\overline{\\mathcal {E}(x,y)} are defined by{\\begin{@align}{1}{-1}\\mathcal {E} = \\frac{1 + \\hat{m}_{11} - \\hat{m}_{21}}{1 + \\hat{m}_{11} + \\hat{m}_{21}}, \\qquad \\bar{\\mathcal {E}} = -\\frac{1 - \\hat{m}_{11} + \\hat{m}_{21}}{1 - \\hat{m}_{11} - \\hat{m}_{21}}.\\end{@align}}The second symmetry in (\\ref {msymm}) evaluated at z = 0 implies{\\begin{@align}{1}{-1}\\hat{m}_{11} = \\overline{\\hat{m}_{22}}, \\qquad \\hat{m}_{12} = \\overline{\\hat{m}_{21}}.\\end{@align}}Recalling the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, it follows that \\bar{\\mathcal {E}} is the complex conjugate of \\mathcal {E}.",
"The next lemma shows, among other things, that \\mathcal {E} is free of singularities.",
"}\\begin{lemma}The function \\mathcal {E}(x,y) defined in (\\ref {Edef}) has the following properties:{\\begin{@align*}{1}{-1}{\\left\\lbrace \\begin{array}{ll}\\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ),\\\\x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ),\\\\\\text{$\\mathcal {E}(x,0) = \\mathcal {E}_0(x)$ for $x \\in [0,1-\\delta )$,}\\\\\\text{$\\mathcal {E}(0,y) = \\mathcal {E}_1(y)$ for $y \\in [0,1-\\delta )$.",
"}\\\\\\text{$\\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0$ for $(x,y) \\in D_\\delta $.}",
"\\end{array}\\right.",
"}\\end{@align*}}\\end{lemma}{\\begin{xmlelement*}{proof}By Lemma \\ref {claim4E}, the map (x,y) \\mapsto \\hat{m}(x,y) is continuous from D_\\delta to and is C^n from \\operatorname{int}D_\\delta to .The first equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) also has these regularity properties except possibly on the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = -1\\rbrace \\end{@align}}where the denominator vanishes.In the same way, the second equation in (\\ref {Edef}) shows that \\mathcal {E}(x,y) is regular away from the set{\\begin{@align}{1}{-1}\\lbrace (x,y) \\in D_\\delta \\, | \\, (\\hat{m}(x,y))_{11} + (\\hat{m}(x,y))_{21} = 1\\rbrace .\\end{@align}}Since the sets (\\ref {singular1}) and (\\ref {singular2}) are disjoint and closed in D_\\delta , we conclude that \\mathcal {E} \\in C(D_\\delta ) \\cap C^n(\\operatorname{int}D_\\delta ).",
"That x^\\alpha \\mathcal {E}_x, y^\\alpha \\mathcal {E}_y, x^\\alpha y^\\alpha \\mathcal {E}_{xy} \\in C(D_\\delta ) follows by differentiating (\\ref {Edef}) and applying Lemma \\ref {claim4E}.\\end{xmlelement*}We next show that \\text{\\upshape Re\\,}\\mathcal {E} > 0 on D_\\delta .Equation (\\ref {Edef}) yields\\mathcal {E} + \\bar{\\mathcal {E}} = \\frac{4\\hat{m}_{21}}{(\\hat{m}_{11} +\\hat{m}_{21} )^2 -1}.In light of the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1, this gives{\\begin{@align}{1}{-1}\\text{\\upshape Re\\,}\\mathcal {E} = \\frac{2(1 + \\hat{m}_{11})}{|1 + \\hat{m}_{11} + \\hat{m}_{12}|^2}.\\end{@align}}On the other hand, the relations \\hat{m}_{11} = \\hat{m}_{22} and \\det \\hat{m} = 1 together with (\\ref {m0symm}) yield \\hat{m}_{11} \\in {R} and \\hat{m}_{11}^2 - |\\hat{m}_{12}|^2 = 1.",
"We infer that \\hat{m}_{11} \\in (-\\infty ,-1] \\cup [1, \\infty ).For (x,y) = (0,0) we have m(0,0,z) = I for all z, because the jump matrix v is the identity matrix.",
"In particular, \\hat{m}_{11}(0,0) = 1.",
"By continuity, this gives (\\hat{m}(x,y))_{11} \\ge 1 for all ~(x,y) \\in D_\\delta .",
"In view of (\\ref {reEm0}), it follows that \\text{\\upshape Re\\,}\\mathcal {E}(x,y) > 0 on D_\\delta .",
"}Finally, we show that $ E(x,0) = E0(x)$ for $ x [0, 1-)$; the proof that $ E(0,y) = E1(y)$ for $ y [0,1-)$ is similar.For $ y = 0$, the definition (\\ref {jumpdef}) of $ v$ yields{\\begin{@align}{1}{-1}v(x,0,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _0\\big (x, F_{(x,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\I, & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad x \\in [0, 1-\\delta ).\\end{@align}}It follows from part $ (c)$ of Lemma \\ref {xpartlemma} that the $ 22$-matrix valued function $ m0(y,z)$ defined for $ x [0,1-)$ by{\\begin{@align}{1}{-1}m_0(x,z) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _0, \\\\\\Phi _0\\big (x,F_{(x,0)}^{-1}(z)\\big ), & z \\in \\Omega _1 \\cup \\Omega _\\infty ,\\end{array}\\right.",
"}\\end{@align}}satisfies the RH problem (\\ref {RHm}) associated with $ (x,y) = (x,0)$ for each $ x [0,1-)$.Furthermore, since $ 0 $ and $ F(x,y)-1(0) = -$, the first symmetry in (\\ref {phi0symmetries}) yields{\\begin{@align}{1}{-1}m_0(x,0) = \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\Phi _0\\big (x,\\infty ^-\\big )= \\Phi _0\\big (x,\\infty ^+\\big )^{-1}\\sigma _3\\Phi _0\\big (x,\\infty ^+\\big )\\sigma _3.\\end{@align}}Substituting in the expression (\\ref {Phi0atinftyplus}) for $ 0(x,+)$, the $ (11)$ and $ (21)$ entries of (\\ref {mx00}) give{\\begin{@align*}{1}{-1}(m_0(x,0))_{11} = \\frac{1 + \\mathcal {E}_0(x) \\overline{\\mathcal {E}_0(x)}}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}, \\qquad (m_0(x,0))_{21} = \\frac{(1 - \\mathcal {E}_0(x))(1 + \\overline{\\mathcal {E}_0(x)})}{\\mathcal {E}_0(x) + \\overline{\\mathcal {E}_0(x)}}.\\end{@align*}}Solving these two equations for $ E0$ and $ E0$, we find{\\begin{@align}{1}{-1}\\mathcal {E}_0(x) = \\frac{1 + (m_0(x,0))_{11} - (m_0(x,0))_{21}}{1 + (m_0(x,0))_{11} + (m_0(x,0))_{21}}.\\end{@align}}But by uniqueness of the solution of the RH problem (\\ref {RHm}), we have $ m0(x,z) = m(x,0,z)$; hence, comparing (\\ref {E0recover}) with (\\ref {Erecover}), we deduce that $ E(x,0) = E0(x)$ for $ x [0, 1-)$.$ It only remains to show that $\\mathcal {E}(x,y)$ satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"The proof of this relies on the construction of an eigenfunction $\\Phi $ of the Lax pair.",
"Equations (REF ) and (REF ) suggest that we define $\\Phi (x,y,P)$ for $(x,y) \\in D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ by $\\Phi (x,y,P) = \\tilde{\\Phi }(x,y)m(x,y,F_{(x,y)}(P)),$ where $\\tilde{\\Phi }(x,y)$ is the function defined in ().",
"Lemma 6.8 The function $\\Phi $ defined in (REF ) satisfies the Lax pair equations ${\\left\\lbrace \\begin{array}{ll}\\Phi _x(x,y, P) = \\mathsf {U}(x,y, P) \\Phi (x,y, P),\\\\\\Phi _y(x,y, P) = \\mathsf {V}(x,y,P) \\Phi (x,y,P),\\end{array}\\right.",
"}$ for $(x,y) \\in \\operatorname{int}D_\\delta $ and $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ .",
"The analyticity structure of $n_0$ established in Lemma REF implies that there exists a $2\\times 2$ -matrix valued function $C(x,y)$ independent of $z$ such that $n_0(x,y,z) = \\frac{C(x,y)}{z+1}, \\qquad z \\in \\hat{.",
"}We determine C(x,y) by evaluating (\\ref {n0C}) at z = 0.",
"By Lemma \\ref {claim5E}, this gives C(x,y) = \\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}.It follows that{\\begin{@align}{1}{-1}n_0 = \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1} = \\bigg (m_x -\\frac{1-z}{1+z} \\frac{z}{1 - x - y} m_z\\bigg )m^{-1}\\end{@align}}for (x,y) \\in D_\\delta and z \\in \\Omega _\\infty .$ Differentiating (REF ) with respect to $x$ and using (), we find, for $P \\in F_{(x,y)}^{-1}(\\Omega _\\infty )$ , $\\Phi _x(x,y,P)& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y)(m_x + z_x m_z)\\\\& = \\tilde{\\Phi }_x(x,y)m(x,y,F_{(x,y)}(z))+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z+1}m(x,y,z)\\\\& = \\bigg (\\tilde{\\Phi }_x(x,y)\\tilde{\\Phi }(x,y)^{-1}+ \\tilde{\\Phi }(x,y) \\frac{\\hat{m}_x(x,y)\\hat{m}(x,y)^{-1}}{z(x,y,P)+1}\\tilde{\\Phi }(x,y)^{-1}\\bigg )\\Phi (x,y,P)$ Substituting in the expressions () and () for $\\tilde{\\Phi }$ and $\\hat{m}$ in terms of $\\mathcal {E}$ , $\\bar{\\mathcal {E}}$ , and recalling that $1 - \\frac{2}{z+1} = \\lambda ,$ this yields the first equation in (REF ).",
"A similar argument gives the second equation in (REF ).",
"Lemma 6.9 The complex-valued function $\\mathcal {E}:D \\rightarrow {R}$ defined by (REF ) satisfies the hyperbolic Ernst equation (REF ) in $\\operatorname{int}(D_\\delta )$ .",
"Fix a point $P = (\\lambda , k)$ in $F_{(x,y)}^{-1}(\\Omega _\\infty ) \\subset \\mathcal {S}_{(x,y)}$ .",
"By Lemma , the map $(x,y) \\mapsto \\Phi (x,y, P)$ is $C^n$ from $\\operatorname{int}D_\\delta $ to $ and satisfies the Lax pair equations (\\ref {philax}).",
"Since~ $ n 2$, it follows that $$ satisfies{\\begin{@align*}{1}{-1}\\Phi _{xy}(x,y,P) - \\Phi _{yx} (x,y,P) = 0, \\qquad (x,y) \\in \\operatorname{int}D_\\delta .\\end{@align*}}The $ (21)$-entry of this equation reads$$\\frac{(1-x-y)\\lambda }{2(\\text{\\upshape Re\\,}\\mathcal {E}(x,y))^2 (1-k-y)}\\bigg \\lbrace (\\text{\\upshape Re\\,}\\mathcal {E})\\bigg (\\mathcal {E}_{xy} - \\frac{\\mathcal {E}_x + \\mathcal {E}_y}{2(1-x-y)}\\bigg ) - \\mathcal {E}_x \\mathcal {E}_y\\bigg \\rbrace = 0.$$It follows that $ E(x,y)$ satisfies (\\ref {ernst}) for $ (x,y) intD$.This completes the proof of the lemma.$ Lemma REF completes the proof of part $(a)$ of Theorem REF .",
"The following lemma proves part (b).",
"Lemma 6.10 There exists a constant $c_\\delta > 0$ such that if $\\Vert \\mathcal {E}_0 / \\text{\\upshape Re\\,}\\mathcal {E}_0 \\Vert _{L^1([0,1-\\delta ))} , \\, \\Vert \\mathcal {E}_1/ \\text{\\upshape Re\\,}\\mathcal {E}_1 \\Vert _{L^1([0,1-\\delta ))} < c_\\delta ,$ then the linear operator $I - \\mathcal {C}_{w(x,y,\\cdot )} \\in \\mathcal {B}(L^2(\\Gamma ))$ is bijective for each $(x,y) \\in D_\\delta $ .",
"It follows from () and () that, by choosing $c_\\delta $ sufficiently small, equation (REF ) gives $|\\Phi _0(x,k^\\pm )-I| < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ and an analogous estimate holds for $|\\Phi _1(y,k^\\pm )-I|$ .",
"This yields $\\Vert w(x,y,\\cdot )\\Vert _{L^\\infty (\\Gamma )} < \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))}^{-1}$ for all $(x,y) \\in D_\\delta $ whenever (REF ) holds.",
"Indeed, equation (REF ) implies $\\Vert \\mathcal {C}_w\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\le \\Vert \\mathcal {C}_-\\Vert _{\\mathcal {B}(L^2(\\Gamma ))} \\Vert w\\Vert _{L^\\infty (\\Gamma )} < 1$ for all $(x,y) \\in D_\\delta $ .",
"Hence $I - \\mathcal {C}_{w(x,y,\\cdot )}$ is invertible in $\\mathcal {B}(L^2(\\Gamma ))$ for each $(x,y) \\in D_\\delta $ .",
"For part (c) assume $\\mathcal {E}_0, \\mathcal {E}_1>0$ and write $V_0= -\\log \\mathcal {E}_0$ , $V_1 = -\\log \\mathcal {E}_1$ .",
"Then there exists a $C^n$ -solution $V(x,y)$ of the Goursat problem for the Euler-Darboux equation (REF ) with data $\\lbrace V_0,V_1 \\rbrace $ by Theorem REF .",
"Hence $\\mathcal {E}=e^{- V}$ is a $C^n$ -solution of the Goursat problem for (REF ) with data $\\lbrace \\mathcal {E}_0,\\mathcal {E}_1 \\rbrace $ .",
"This completes the proof of part (c) and hence of Theorem REF .",
"Proof of Theorem REF Let $\\mathcal {E}_0(x)$ , $x \\in [0, 1)$ , and $\\mathcal {E}_1(y)$ , $y \\in [0,1)$ , be complex-valued functions satisfying (REF ) for some $n \\ge 2$ and some $\\alpha \\in (0,1)$ .",
"Suppose $\\mathcal {E}(x,y)$ is a $C^n$ -solution of the Goursat problem for (REF ) in $D$ with data $\\lbrace \\mathcal {E}_0, \\mathcal {E}_1\\rbrace $ and define $m_1, m_2 \\in by (\\ref {m1m2def}).We will prove (\\ref {boundarylimita}); the proof of (\\ref {boundarylimitb}) is similar.$ By (REF ), we have $x^\\alpha \\mathcal {E}_x(x,y) = 2x^\\alpha \\frac{\\hat{m}_{21}(x,y) \\hat{m}_{11x}(x,y) - (1 + \\hat{m}_{11}(x,y))\\hat{m}_{21x}(x,y)}{(1 + \\hat{m}_{11}(x,y) + \\hat{m}_{21}(x,y))^2},$ where, as before, $\\hat{m}(x,y) = m(x,y,0)$ .",
"Thus, in order to compute $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y)$ , it is enough to compute $\\hat{m}(0,y)$ and $\\lim _{x \\downarrow 0} x^\\alpha \\hat{m}_x(x,y)$ .",
"Since $m = I + \\mathcal {C}(\\mu w)$ and $m_x = \\mathcal {C}(\\mu _x w) + \\mathcal {C}(\\mu w_x),$ this means that we are interested in the values of $w(0,y,z), \\quad \\mu (0,y,z),\\quad \\lim _{x \\downarrow 0} x^\\alpha w_x(x,y,z), \\quad \\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z).$ Lemma 6.11 We have $& w(0,y, z) = {\\left\\lbrace \\begin{array}{ll}0, & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y, F_{(0,y)}^{-1}(z)\\big ) - I, \\quad & z \\in \\Gamma _1,\\end{array}\\right.}",
"\\quad y \\in [0,1),\\\\~& \\mu (0,y,z) = {\\left\\lbrace \\begin{array}{ll}\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Gamma _0,\\\\\\Phi _1\\big (y,\\infty ^+\\big )^{-1} , & z \\in \\Gamma _1,\\end{array}\\right.",
"}\\quad y \\in [0,1),$ and $\\hat{m}(0,y) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1}\\sigma _3 \\Phi _1\\big (y,\\infty ^+\\big ) \\sigma _3, \\qquad y \\in [0,1).$ Equation (REF ) is immediate from (REF ).",
"Moreover, by (REF ), $ m(0,y,z) = \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\times {\\left\\lbrace \\begin{array}{ll} I, & z \\in \\Omega _1, \\\\\\Phi _1\\big (y,F_{(0,y)}^{-1}(z)\\big ), \\quad & z \\in \\Omega _0 \\cup \\Omega _\\infty .\\end{array}\\right.",
"}$ Equation () follows from (REF ) and the fact that $\\mu (x,y,z)=m_-(x,y,z)$ for $(x,y) \\in D$ and $z \\in \\Gamma $ .",
"Since $0 \\in \\Omega _0$ and $F_{(0,y)}^{-1}(0) = \\infty ^-$ , equation (REF ) follows by setting $z = 0$ in (REF ) and using the first symmetry in ().",
"Lemma 6.12 For $y \\in [0,1)$ , we have $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\begin{pmatrix} \\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z)) & m_1 \\end{pmatrix}, \\quad & z \\in \\Gamma _0,\\\\0, & z \\in \\Gamma _1,\\end{array}\\right.",
"}$ and $\\lim _{x \\downarrow 0} x^\\alpha \\mu _x(x,y,z)= \\Pi (y,z), \\qquad z \\in \\Gamma _1,$ where the function $\\Pi (y,z)$ is defined by $\\Pi (y,z) = - \\frac{1}{\\sqrt{1-y}}\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}}{z+1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}.$ It follows from (REF ) and (REF ) that $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y, z) =0$ for $z \\in \\Gamma _1$ and that, for $z\\in \\Gamma _0$ , $\\lim _{x \\rightarrow 0} x^\\alpha w_x(x,y,z)& = \\lim _{x \\rightarrow 0} x^\\alpha \\bigg \\lbrace \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))+ \\Phi _{0k}(x,F_{(x,y)}^{-1}(z))\\frac{d}{dx} F_{(x,y)}^{-1}(z)\\bigg \\rbrace \\\\~& = \\lim _{x \\rightarrow 0} x^\\alpha \\Phi _{0x}(x,F_{(x,y)}^{-1}(z))= \\lim _{x \\rightarrow 0} x^\\alpha \\mathsf {U}_0(x,F_{(x,y)}^{-1}(z)).$ Recalling the definition (REF ) of $\\mathsf {U}_0$ , (REF ) follows.",
"To prove (REF ), we note that differentiation of the relation $\\mu = I + \\mathcal {C}_w \\mu $ gives $ \\mu _x = (I-\\mathcal {C}_w)^{-1}\\mathcal {C}_-(\\mu w_x).$ We first compute $\\lim _{x \\downarrow 0} \\mathcal {C}_-(\\mu x^\\alpha w_x)$ .",
"Equations () and (REF ) imply, for $z \\in \\Gamma _1$ , $\\nonumber &\\Big \\lbrace \\mathcal {C}_- \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ]\\Big \\rbrace (z)= \\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} }{2\\pi i}\\\\\\nonumber & \\times \\int _{\\Gamma _0} \\frac{\\Phi _1\\big (y,F_{(0,y)}^{-1}(z^{\\prime })\\big )\\frac{1}{2} \\Big ({\\begin{matrix}\\bar{m}_1 & \\bar{m}_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) \\\\m_1 \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime })) & m_1 \\end{matrix}}\\Big ) dz^{\\prime }}{z^{\\prime } -z}\\\\ & =- \\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1(y,F_{(0,y)}^{-1}(z^{\\prime })) \\lambda (0,0, F_{(0,y)}^{-1}(z^{\\prime }))\\left({\\begin{matrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{matrix}}\\right)}{2(z^{\\prime } -z)} =: \\tilde{\\Pi }(y,z).$ Recalling the expression (REF ) for $\\lambda (0,0, F_{(0,y)}^{-1}(z))$ and using that $F_{(0,y)}^{-1}(-1) = 0$ , we find $\\tilde{\\Pi }(y,z) = -\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0) \\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix} }{(z+1)\\sqrt{1-y}}.$ In view of (REF ), it only remains to show that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"We have, for $z \\in \\Gamma _1$ , $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)= \\frac{1}{2\\pi i} \\int _{\\Gamma _1}\\frac{\\Pi (y, z^{\\prime })\\big (\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I\\big )}{z^{\\prime } - z_-} dz^{\\prime }\\\\& = -\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{2\\pi i\\sqrt{1-y}}\\int _{\\Gamma _1}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z_-} \\frac{dz^{\\prime }}{z^{\\prime }+1}.$ Deforming the contour to infinity and using that $\\underset{z^{\\prime } = -1}{\\text{\\upshape Res\\,}}\\frac{\\Phi _1\\big (y, F_{(0,y)}^{-1}(z^{\\prime })\\big ) - I}{z^{\\prime } - z} \\frac{1}{z^{\\prime }+1} = -\\frac{\\Phi _1(y, 0) - I}{z+1}, $ a residue computation gives $& (\\mathcal {C}_{w(0,y,\\cdot )} \\Pi )(z)=\\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}}{\\sqrt{1-y}} \\frac{\\Phi _1(y, 0) - I}{z+1}.$ Simple algebra now shows that $(I-\\mathcal {C}_w)\\Pi = \\tilde{\\Pi }$ .",
"Lemma 6.13 For $y \\in [0,1)$ , we have $\\Phi _1(y, 0) = \\begin{pmatrix} e^{\\int _0^y \\frac{\\overline{\\mathcal {E}_{1y}(y^{\\prime })}}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} & 0 \\\\ 0 & e^{\\int _0^y \\frac{\\mathcal {E}_{1y}(y^{\\prime })}{2\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\end{pmatrix}.$ Since 0 is a real branch point of the Riemann surface $\\Sigma _{(0,y)}$ , the symmetries () of $\\Phi _1$ imply that $\\Phi _1(y, 0^+) = \\Phi _1(y, 0^-) = \\sigma _3 \\Phi _1(y, 0^+)\\sigma _3 \\quad \\text{and} \\quad \\Phi _1(y, 0) = \\sigma _1\\overline{\\Phi _1(y, 0)}\\sigma _1.$ Hence $\\Phi _1(y, 0)$ has the form $\\Phi _1(y, 0) = \\begin{pmatrix} f(y) & 0 \\\\ 0 & \\overline{f(y)} \\end{pmatrix},$ where $f(y)$ is a function of $y$ .",
"Since $\\lambda (0, y, 0) = \\infty $ , we can determine $f(y)$ by solving the equation $\\Phi _{1y}(y, 0) = \\frac{1}{2 \\text{\\upshape Re\\,}\\mathcal {E}_{1}(y)} \\begin{pmatrix} \\overline{\\mathcal {E}_{1y}(y)} & 0 \\\\~0 & \\mathcal {E}_{1y}(y) \\end{pmatrix} \\Phi _1(y,0),$ which is a consequence of (REF ).",
"This gives the desired statement.",
"The following lemma completes the proof of Theorem REF .",
"Lemma 6.14 For $y \\in [0,1)$ , we have $\\lim _{x \\downarrow 0} x^\\alpha \\mathcal {E}_x(x,y) = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}}.$ We first compute $ \\lim _{x \\downarrow 0} x^\\alpha m_x(x,y,0)$ .",
"Proceeding as in the proof of (REF ), we find $\\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu (x,y,\\cdot ) w_x(x,y,\\cdot )\\big ](0)=-\\frac{\\Phi _1\\big (y,\\infty ^+\\big )^{-1} \\Phi _1(y,0)}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1 & 0 \\end{pmatrix},$ and $\\nonumber \\mathcal {C} \\big [\\lim _{x \\rightarrow 0} x^\\alpha \\mu _x(x,y,\\cdot ) w(x,y,\\cdot )\\big ](0)= &\\; \\frac{\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )}{\\sqrt{1-y}}\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\Phi _1\\big (y, 0\\big )^{-1}\\\\ & \\times \\left( \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3 \\right),$ where the derivation of (REF ) employs Lemma REF and Lemma REF as well as the residue calculation $-\\frac{1}{2\\pi i} \\int _{\\Gamma _1} \\frac{\\Phi _1(y, F_{(0,y)}^{-1}(z)) - I}{z} \\frac{dz}{z+1} = \\Phi _1(y,0) - \\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Adding (REF ) and (REF ) and recalling (REF ), we obtain $\\nonumber \\lim _{x \\rightarrow 0} x^\\alpha m_x(x,y,0)= & - \\frac{1}{\\sqrt{1-y}}\\Phi _1\\big (y, \\infty ^+\\big )^{-1}\\Phi _1\\big (y, 0\\big )\\begin{pmatrix} 0 & \\bar{m}_1 \\\\m_1& 0 \\end{pmatrix}\\\\&\\times \\Phi _1\\big (y, 0\\big )^{-1}\\sigma _3 \\Phi _1(y,\\infty ^+) \\sigma _3.$ Substituting (), (REF ), (REF ), and (REF ) into (REF ), long but straightforward computations yield ().",
"Examples We consider two examples of exact solutions—one with collinear polarization and one with noncollinear polarization.",
"For each example, we verify explicitly that the formulas () of Theorem REF on the behavior near the boundary are satisfied.",
"The Khan-Penrose solution The Khan-Penrose [13] solution is given by the potential $\\mathcal {E}(x,y)= \\frac{1+\\sqrt{x}\\sqrt{1-y}+\\sqrt{y}\\sqrt{1-x}}{1-\\sqrt{x}\\sqrt{1-y}-\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y) \\in D.$ Straightforward computations show that $m_1=1=m_2$ and $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{\\sqrt{1-y}}{(1-\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= \\frac{\\sqrt{1-x}}{(1-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ The Nutku-Halil solution One version of the Nutku-Halil [23] solution is given by $\\mathcal {E}(x,y)= \\frac{1-i\\sqrt{x}\\sqrt{1-y}+i\\sqrt{y}\\sqrt{1-x}}{1+i\\sqrt{x}\\sqrt{1-y}-i\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y)\\in D.$ In this case, $m_1=-i=-m_2$ and we compute $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{i\\sqrt{1-y}}{(i+\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= -\\frac{i\\sqrt{1-x}}{(i-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$ appendix Gravitational waves and the hyperbolic Ernst equation It is shown in Eq.",
"(11.7) in [11] that the Ernst potential $\\mathcal {E}$ satisfies $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{uv} - U_u \\mathcal {E}_v - U_v \\mathcal {E}_u\\right) = 4\\mathcal {E}_u \\mathcal {E}_v.$ where $e^{-U(u,v)} = f(u) + g(v)$ and $f(u)$ and $g(v)$ are monotonically decreasing for positive argument and $f(0) = g(0) = 1/2$ .",
"(Note that Griffiths writes $Z$ for the Ernst potential.)",
"As suggested by Szekeres [25], it is possible to use $(f, g)$ as coordinates.",
"This leads to the equation $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{fg} + \\frac{\\mathcal {E}_f + \\mathcal {E}_g}{f+g}\\right) = 4\\mathcal {E}_f \\mathcal {E}_g,$ where $(f,g)$ belongs to the triangular region $\\bigg \\lbrace (f,g) \\in {R}^2 \\, \\bigg | \\, f \\le \\frac{1}{2}, \\; g \\le \\frac{1}{2}, \\; f + g > 0\\bigg \\rbrace .$ The change of variables $x = \\frac{1}{2} - g$ , $y = \\frac{1}{2} - f$ transforms (REF ) into (REF ).",
"In order for the solution to describe gravitational waves, the following boundary condition must be satisfied (Eq.",
"(7.15) in [11]; see also (11.23) in [11] but in (11.23) equation $(f,g)$ approaches the corner whereas in (7.15) the two edges are approached; also in (7.15) there is a factor $(f+g)$ missing; this factor comes from (7.9)) $& \\lim _{g \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -g\\Big ) (f+g)\\frac{|\\mathcal {E}_g|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_2}{2},\\\\& \\lim _{f \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -f\\Big ) (f+g) \\frac{|\\mathcal {E}_f|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"In terms of $(x,y)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\frac{x(1-x-y)|\\mathcal {E}_x|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}= \\frac{k_2}{2},\\\\& \\lim _{y \\rightarrow 0} \\frac{y(1-x-y)|\\mathcal {E}_y|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2} = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"That is, since $\\text{\\upshape Re\\,}\\mathcal {E} > 0$ , $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1),$ for some constants $m_1 , m_2 \\in [1, \\sqrt{2})$ .",
"If we assume that $\\mathcal {E} \\in C(D)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1).$ These are the conditions given in () with $\\alpha = 1/2$ .",
"In particular, $& \\mathcal {E}_{0x}(x) = \\frac{m_1 + o(1)}{\\sqrt{x}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{0}(x) \\sim 2m_1\\sqrt{x}, \\qquad x \\downarrow 0,\\\\& \\mathcal {E}_{1y}(y) = \\frac{m_2 + o(1)}{\\sqrt{y}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{1}(y) \\sim 2m_2\\sqrt{y}, \\qquad y \\downarrow 0,$ where $m_1, m_2 \\in [1, \\sqrt{2})$ .",
"Acknowledgement The authors acknowledge support from the European Research Council, Grant Agreement No.",
"682537, the Swedish Research Council, Grant No.",
"2015-05430, and the Göran Gustafsson Foundation, Sweden."
],
[
"Examples",
"We consider two examples of exact solutions—one with collinear polarization and one with noncollinear polarization.",
"For each example, we verify explicitly that the formulas () of Theorem REF on the behavior near the boundary are satisfied."
],
[
"The Khan-Penrose solution",
"The Khan-Penrose [13] solution is given by the potential $\\mathcal {E}(x,y)= \\frac{1+\\sqrt{x}\\sqrt{1-y}+\\sqrt{y}\\sqrt{1-x}}{1-\\sqrt{x}\\sqrt{1-y}-\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y) \\in D.$ Straightforward computations show that $m_1=1=m_2$ and $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{\\sqrt{1-y}}{(1-\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= \\frac{\\sqrt{1-x}}{(1-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}} =\\frac{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$"
],
[
"The Nutku-Halil solution",
"One version of the Nutku-Halil [23] solution is given by $\\mathcal {E}(x,y)= \\frac{1-i\\sqrt{x}\\sqrt{1-y}+i\\sqrt{y}\\sqrt{1-x}}{1+i\\sqrt{x}\\sqrt{1-y}-i\\sqrt{y}\\sqrt{1-x}}, \\qquad (x,y)\\in D.$ In this case, $m_1=-i=-m_2$ and we compute $\\lim _{x \\downarrow 0} \\sqrt{x} \\mathcal {E}_x(x,y) &= \\frac{i\\sqrt{1-y}}{(i+\\sqrt{y})^2} = m_1 \\frac{e^{i\\int _0^y \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{1y}(y^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y^{\\prime })} dy^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_1(y)}{\\sqrt{1-y}},\\\\\\lim _{y \\downarrow 0} \\sqrt{y} \\mathcal {E}_y(x,y) &= -\\frac{i\\sqrt{1-x}}{(i-\\sqrt{x})^2} = m_2 \\frac{e^{i\\int _0^x \\frac{\\text{\\upshape Im\\,}\\mathcal {E}_{0x}(x^{\\prime })}{\\text{\\upshape Re\\,}\\mathcal {E}_1(x^{\\prime })} dx^{\\prime }} \\text{\\upshape Re\\,}\\mathcal {E}_0(x)}{\\sqrt{1-x}}.$"
],
[
"Gravitational waves and the hyperbolic Ernst equation",
"It is shown in Eq.",
"(11.7) in [11] that the Ernst potential $\\mathcal {E}$ satisfies $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{uv} - U_u \\mathcal {E}_v - U_v \\mathcal {E}_u\\right) = 4\\mathcal {E}_u \\mathcal {E}_v.$ where $e^{-U(u,v)} = f(u) + g(v)$ and $f(u)$ and $g(v)$ are monotonically decreasing for positive argument and $f(0) = g(0) = 1/2$ .",
"(Note that Griffiths writes $Z$ for the Ernst potential.)",
"As suggested by Szekeres [25], it is possible to use $(f, g)$ as coordinates.",
"This leads to the equation $2(\\text{\\upshape Re\\,}\\mathcal {E}) \\left(2\\mathcal {E}_{fg} + \\frac{\\mathcal {E}_f + \\mathcal {E}_g}{f+g}\\right) = 4\\mathcal {E}_f \\mathcal {E}_g,$ where $(f,g)$ belongs to the triangular region $\\bigg \\lbrace (f,g) \\in {R}^2 \\, \\bigg | \\, f \\le \\frac{1}{2}, \\; g \\le \\frac{1}{2}, \\; f + g > 0\\bigg \\rbrace .$ The change of variables $x = \\frac{1}{2} - g$ , $y = \\frac{1}{2} - f$ transforms (REF ) into (REF ).",
"In order for the solution to describe gravitational waves, the following boundary condition must be satisfied (Eq.",
"(7.15) in [11]; see also (11.23) in [11] but in (11.23) equation $(f,g)$ approaches the corner whereas in (7.15) the two edges are approached; also in (7.15) there is a factor $(f+g)$ missing; this factor comes from (7.9)) $& \\lim _{g \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -g\\Big ) (f+g)\\frac{|\\mathcal {E}_g|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_2}{2},\\\\& \\lim _{f \\rightarrow \\frac{1}{2}} \\bigg [\\Big (\\frac{1}{2} -f\\Big ) (f+g) \\frac{|\\mathcal {E}_f|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}\\bigg ] = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"In terms of $(x,y)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\frac{x(1-x-y)|\\mathcal {E}_x|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2}= \\frac{k_2}{2},\\\\& \\lim _{y \\rightarrow 0} \\frac{y(1-x-y)|\\mathcal {E}_y|^2}{(\\mathcal {E} + \\bar{\\mathcal {E}})^2} = \\frac{k_1}{2},$ for some constants $k_1, k_2 \\in [\\frac{1}{2}, 1)$ .",
"That is, since $\\text{\\upshape Re\\,}\\mathcal {E} > 0$ , $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x-y}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}(x,y)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1),$ for some constants $m_1 , m_2 \\in [1, \\sqrt{2})$ .",
"If we assume that $\\mathcal {E} \\in C(D)$ , these conditions become $& \\lim _{x \\rightarrow 0} \\sqrt{x} \\sqrt{1-y}\\frac{|\\mathcal {E}_x(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_1(y)} = \\sqrt{2k_2} = m_1, \\qquad y \\in [0,1),\\\\& \\lim _{y \\rightarrow 0} \\sqrt{y} \\sqrt{1-x}\\frac{|\\mathcal {E}_y(x,y)|}{\\text{\\upshape Re\\,}\\mathcal {E}_0(x)} = \\sqrt{2k_1} = m_2,\\qquad x \\in [0,1).$ These are the conditions given in () with $\\alpha = 1/2$ .",
"In particular, $& \\mathcal {E}_{0x}(x) = \\frac{m_1 + o(1)}{\\sqrt{x}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{0}(x) \\sim 2m_1\\sqrt{x}, \\qquad x \\downarrow 0,\\\\& \\mathcal {E}_{1y}(y) = \\frac{m_2 + o(1)}{\\sqrt{y}} \\quad \\text{i.e.}",
"\\quad \\mathcal {E}_{1}(y) \\sim 2m_2\\sqrt{y}, \\qquad y \\downarrow 0,$ where $m_1, m_2 \\in [1, \\sqrt{2})$ .",
"Acknowledgement The authors acknowledge support from the European Research Council, Grant Agreement No.",
"682537, the Swedish Research Council, Grant No.",
"2015-05430, and the Göran Gustafsson Foundation, Sweden."
]
] | 1808.08365 |
[
[
"Additive Volume of Sets Contained in Few Arithmetic Progressions"
],
[
"Abstract A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$.",
"The formula is known to hold when $T \\le 3k-4$, for some small range over $3k-4$ and for families of structured sets called chains.",
"In this paper we extend the formula to sets of every dimension and prove it for sets composed of three segments, giving structural results for the extremal case.",
"A weaker extension to sets composed of a bounded number of segments is also discussed."
],
[
"Introduction",
"Let $A\\subset {\\mathbb {Z}}$ be a finite set of integers.",
"The Minkowski sum of $A$ is $A+A=\\lbrace a+a^{\\prime }: a,a^{\\prime }\\in A\\rbrace $ .",
"The doubling of $A$ is the cardinality of $2A=A+A$ .",
"The Freiman–Ruzsa theorem giving the structure of sets of integers with small doubling is one of the central results in Additive Number Theory.",
"It states that a set $A$ with doubling $|2A|\\le c|A|$ is a dense set of a multidimensional arithmetic progression $P$ , where the density $|A|/|P|$ and dimension of $P$ depend only on $c$ , see Freiman [3], Bilu [1] and Ruzsa [13].",
"The estimation of the best lower bounds for the density of $A$ in $P$ was the object of a long series of papers and it was eventually brought to its essentially best values by Schoen [17].",
"At a conference in Toronto in 2008, Freiman proposed a precise formula for the largest possible volume of a set of integers $A \\subset {\\mathbb {Z}}$ with given doubling $T=|2A|$ in terms of a specific parametrization of the value of $T$ , see [5].",
"We start by recalling some definitions in order to state this conjecture.",
"Given abelian groups $G$ and $G^{\\prime }$ , two sets $A\\subset G$ and $B\\subset G^{\\prime }$ are Freiman isomorphic of order 2 ($F_2$ –isomorphic for short) if there is a bijection $\\phi :A\\rightarrow B$ such that, for every $x,y,z,t \\in A$ , we have $x+y=z+t \\quad \\Leftrightarrow \\quad \\phi (x)+\\phi (y)=\\phi (z)+\\phi (t).$ The additive dimension $\\dim (A)$ of a set $A \\subset {\\mathbb {Z}}$ is the largest $d \\in {\\mathbb {N}}$ such that there exists a set $B \\subset {\\mathbb {Z}}^d$ not contained in a hyperplane of ${\\mathbb {Z}}^d$ which is $F_2$ –isomorphic to $A$ .",
"Note that any $d$ –dimensional set $A$ of cardinality $k$ satisfies $d \\le k-1$ and that furthermore by results of Freiman [2] as well as Konyagin and Lev [11] we have $ (d+1)k-{d+1 \\atopwithdelims ()2} \\le |2A| \\le {k \\atopwithdelims ()2}+d+1.$ The volume ${\\rm {vol}}(A)$ of a $d$ –dimensional set $A$ is defined to be the minimum cardinality of the convex hull among all sets in ${\\mathbb {Z}}^d$ that are $F_2$ –isomorphic to $A$ .",
"We say that a set of integers $A \\subset {\\mathbb {Z}}$ is in normal form if $\\min (A)=0$ and $\\gcd (A)=1$ .",
"We call $\\tilde{A} = \\big ( A-\\min (A) \\big ) / \\gcd \\big ( A-\\min (A) \\big )$ the normalization of $A$ since it is a set in normal form and it is $F_2$ –isomorphic to $A$ .",
"Note that for any 1–dimensional set $A$ we have ${\\rm {vol}}(A) = \\max (\\tilde{A})+1.$ If $\\min (A)=0$ , then the reflection of $A$ is defined as $A^-=-A+\\max (A)$ .",
"The reflection of $A$ is certainly isomorphic to $A$ .",
"We are interested in obtaining upper bounds for the volume of a set $A$ of integers in terms of its cardinality $|A|$ , the cardinality of its doubling $|2A|$ and its dimension $\\dim (A)$ .",
"We denote the maximum volume of all sets $A$ of integers with cardinality $k$ , doubling $T$ and dimension $d$ by ${\\rm {vol}}(k,T,d) = \\max \\lbrace {\\rm {vol}}(A): A\\subset {\\mathbb {N}}\\;, |A|=k, |2A|=T \\text{ and } \\dim (A)=d\\rbrace .$ A set $A$ is extremal if ${\\rm {vol}}(A) = {\\rm {vol}}(|A|,|2A|,\\dim (A))$ .",
"The following is a more general and slightly reformulated version of the previously mentioned conjecture of Freiman, which can be traced back to [2].",
"Its notable addition is that it takes the dimension of a set into consideration.",
"Conjecture 1 Given any $k,T,d \\in {\\mathbb {N}}$ such that $T = (d+c) k - {d+c+1 \\atopwithdelims ()2} + d + b+1,$ where $1 \\le c \\le k-d-1 \\, \\text{ and } \\, 0 \\le b \\le k-d-c-1,$ we have ${\\rm {vol}}(k,T,d) = 2^{c-1} \\, (k-c+b)+1.$ Note that given any $k \\in {\\mathbb {N}}$ , $1 \\le d \\le k-1$ and $T \\in \\left\\lbrace (d+1)k-{d+1 \\atopwithdelims ()2},\\dots ,{k \\atopwithdelims ()2}+d+1 \\right\\rbrace $ , there are uniquely defined $c = c(k,T,d) \\quad \\text{and} \\quad b = b(k,T,d)$ subject to the boundary conditions stated in the conjecture, such that $T$ can be expressed as the right–hand side of (REF ).",
"It follows that Conjecture REF states a tight upper bound on the volume of any possible set of integers.",
"If $A$ has cardinality $|A| = k$ , dimension $\\dim (A) = d$ and doubling $|2A|=T$ then we call this uniquely determined $c=c(k,T,d)$ the doubling constant of $A$ .",
"There are examples showing that the right hand side in (REF ) is at least a lower bound for ${\\rm {vol}}(k,T,d)$ , see e.g.",
"[2], [7].",
"Thus an extremal set $A$ has volume $vol(A)\\ge vol(k,T,d).$ Furthermore, equality has been established in a few cases: by Freiman [2] for one–dimensional sets satisfying $T \\le 3k-4$ , that is either $c = 1$ and any admissible $b$ or $c = 2$ and $b = 0$ , with an additional structural description of extremal sets given in [4], by Freiman [2] as well as by Hamidoune and Plagne [8] for one–dimensional sets if $T = 3k-3$ , that is $c = 2$ and $b = 1$ , with a structural description of the extremal case due to Jin [9], by Freiman [2] for two–dimensional sets satisfying $k \\ge 10$ and $T \\le 10/3 \\, k - 6$ , that is $c = 1$ and $0 \\le b \\le k/3-2$ , with a structural description of any such set, by Jin [10] using tools from non-standard analysis in the case of large one–dimensional sets satisfying $T \\le (3+\\epsilon ) k$ , that is $c = 2$ and $0 \\le b \\le \\epsilon k$ , for some $\\epsilon > 0$ , by Stanchescu [15] for any $d$ –dimensional set satisfying $c = 1$ and $b = 0$ , by Freiman and Serra [7] for a class of one–dimensional sets called chains, which can be seen as extremal sets build by a greedy algorithm, and any admissible values of the doubling constant $c$ .",
"In order to give further evidence towards the validity of this conjecture, we consider sets composed of a given number of segments.",
"Throughout the paper we say that $A \\subset {\\mathbb {Z}}$ is the union of $s$ segments if $A = P_1 \\cup \\cdots \\cup P_s$ where each $P_i$ is a segment of length $k_i$ with $\\max P_i +1< \\min P_{i+1}$ for $1 \\le i < s$ and moreover $k_i > 1$ for some $1 \\le i \\le s$ .",
"Regarding the doubling of such a set, we have the upper bound $|2A|=\\left| \\cup _{1\\le i\\le j\\le s} (P_i+P_j)\\right|\\le \\sum _{1\\le i\\le j\\le s} (|P_i|+|P_j|-1)=(s+1)|A|-{s+1\\atopwithdelims ()2}.$ Equality holds if and only if the sums $P_i+P_j$ are pairwise distinct, in which case the set $A$ has dimension $s$ .",
"As previously mentioned, Conjecture REF has been proved for sets $A$ with doubling $|2A|\\le 3|A|-3$ , and the structure of extremal sets for this range of doubling is well understood.",
"In spite of many efforts, not much is known about the exact maximum volume of sets with doubling at least $3|A|-2$ .",
"This motivates us to consider sets composed of three segments and doubling larger than $3|A|-3$ .",
"Our main result is to show that the statement of Conjecture REF holds for sets $A$ composed of three segments, also giving a structural description of the extremal cases.",
"Theorem 1.1 Let $A$ be an extremal set with cardinality $k>7$ and doubling $|2A|>3k-4$ consisting of three segments.",
"Then $A$ is isomorphic to one of the following sets: $([0,k+b-2]\\setminus [1,b])\\cup \\lbrace 2(k+b-2)\\rbrace $ with $\\dim (A)=1$ , ${\\rm {vol}}(A)=2k+2b-3$ and $|2A|=3k-4+b$ , $([0, k+b+i-3]\\setminus [k-i-1,k+b-3]) \\cup \\lbrace 2(k+b-2)\\rbrace $ for $1\\le i\\le k/3$ and $2 \\le b \\le k-2i-1$ , with $\\dim (A)=1$ , ${\\rm {vol}}(A)=2k+2b-3$ and $|2A|=3k-4+b$ , $([0,k+b-2]\\setminus [1,b])\\times \\lbrace 0\\rbrace \\cup \\lbrace (0,1)\\rbrace \\subset {\\mathbb {Z}}^2$ with $\\dim (A)=2$ , ${\\rm {vol}}(A)=k+b$ and $|2A|=3k-3+b$ , or $([0,k_1-1]\\times \\lbrace (0,0)\\rbrace )\\cup ([0,k_2-1]\\times \\lbrace (0,1)\\rbrace )\\cup ([0,k_3-1]\\times \\lbrace (1,0)\\rbrace )\\subset {\\mathbb {Z}}^3$ with $k_1+k_2+k_3=k$ , $k_1, k_2, k_3\\ge 1$ , $\\dim (A)=3$ , ${\\rm {vol}}(A)=k$ and $|2A|=4k-6$ .",
"The extremal sets composed of three segments described in Theorem REF are illustrated for $k=11$ and $|2A|=3k-1$ in Figure REF .",
"Figure: Extremal sets, up to isomorphism, for k=11k=11 and |2A|=3k-1|2A|=3k-1.We also believe that the following statement, regarding an upper bound on the volume of 1–dimensional sets composed of few disjoint segments, should hold as well.",
"It is independent of the doubling of such sets and in fact one may derive it from Conjecture REF as the case with maximum doubling without too much effort.",
"Conjecture 2 Let $A$ be a 1–dimensional set and let $s$ be the minimum number of disjoint segments into which it can be decomposed.",
"If $s \\le |A|-1$ , then $ {\\rm {vol}}(A) \\le 2^{s-1} \\big ( |A| - s \\big )+1.$ The set $A_s=\\lbrace 0,1,\\ldots ,k-s\\rbrace \\cup \\lbrace 2^i(k-s), i=1,\\ldots ,s-1\\rbrace $ shows that this conjectured upper bound would be tight.",
"Note that, if we only know that a set $A$ is 1–dimensional, Freiman [2] gave ${\\rm {vol}}(A) \\le 2^{k-2}+1$ as an upper bound on the volume of $A$ .",
"In this paper, we show the validity of Conjecture REF for some small values of $s$ .",
"Proposition 1.2 The statement of Conjecture REF holds for $s \\le 4$ .",
"One may prove Conjecture REF for further moderate values of $s$ along the lines presented in the proof of Propositon REF , at the cost of a more involved case analysis.",
"A general proof of this conjecture avoiding the increase of cases would be of interest.",
"Outline.",
"The paper is organized as follows: in Section we discuss the dimension of sets composed of few segments.",
"We then we prove Proposition REF in Section and Theorem REF in Section ."
],
[
"The dimension of sets contained in few segments",
"Konyagin and Lev [11] established a formula for the dimension of a given set $A \\subset {\\mathbb {Z}}^m$ of cardinality $k$ .",
"Let us write $A = \\lbrace a_1, \\dots , a_k\\rbrace $ and introduce some necessary notation.",
"For $1 \\le i \\le k$ , let $\\mathbf {e}_i$ denote the vector in $\\mathbb {R}^k$ that has a one at coordinate $i$ and zero everywhere else.",
"$M_A$ denotes the integer valued matrix with $k$ columns obtained by listing as its rows all vectors $\\mathbf {e}_{i_1} + \\mathbf {e}_{i_2} - \\mathbf {e}_{i_3} - \\mathbf {e}_{i_4}$ for which $a_{i_1} + a_{i_2} = a_{i_3} + a_{i_4}$ holds and for which we do not have $i_1 = i_2 = i_3 = i_4$ .",
"The dimension of $A$ can be derived from the rank of $M_A$ using the following result.",
"Theorem 2.1 (Konyagin and Lev [11]) For any set $A \\subseteq \\mathbb {Z}^m$ we have $\\dim (A) = |A| - 1 - \\rm {rank}(M_A).$ Now let $A \\subset {\\mathbb {Z}}$ be a set which is the union of $s$ disjoint segments as in (REF ).",
"Given such a set $A$ , we denote by $S_A$ the integer valued matrix with $s$ columns obtained by listing in its rows all vectors $\\mathbf {e}_{j_1} + \\mathbf {e}_{j_2} - \\mathbf {e}_{j_3} - \\mathbf {e}_{j_4}$ for which $(A_{j_1} + A_{j_2}) \\cap (A_{j_3} + A_{j_4} )\\ne \\emptyset $ and for which we do not have $j_1 = j_2 = j_3 = j_4$ .",
"We derive the following Corollary from Theorem REF .",
"Corollary 2.2 Any $A \\subset {\\mathbb {Z}}$ that is the union of $1 \\le s \\le |A| - 1$ disjoint segments satisfies $\\dim (A) = s - \\rm {rank}(S_A).$ Every row in $M_A$ is associated with up to four (not all equal) elements $a_{i_1},a_{i_2},a_{i_3},a_{i_4}$ such that $a_{i_1} + a_{i_2} = a_{i_3} + a_{i_4}$ .",
"Let $a_{i_1} \\in P_{j_1}$ , $a_{i_2} \\in P_{j_2}$ , $a_{i_3} \\in P_{j_3}$ and $a_{i_4} \\in P_{j_4}$ where we may assume $j_1 \\le j_2$ and $j_3 \\le j_4$ as well as $\\min \\lbrace j_1,j_2\\rbrace \\le \\min \\lbrace j_3,j_4\\rbrace $ .",
"Furthermore, let $0 \\le y = \\# \\lbrace j : |P_j| = 1 \\rbrace < s$ denote the number of segments that are singletons.",
"We distinguish the following cases.",
"Case 1.",
"$\\# \\lbrace j_1,j_2,j_3,j_4\\rbrace = 1$ , that is $j_1 = j_2 = j_3 = j_4 = j$ for some $1 \\le j \\le s$ .",
"Since $P_j$ is one dimensional, by Theorem REF there are a total of $\\max \\lbrace |P_j| - 2,0\\rbrace $ linearly independent equations of this type for each $P_j$ .",
"As these equations only involve elements in $P_j$ and the segments are disjoint, it is clear that each equation is linearly independent from those of other segments, so we get a total of $|A| - 2s + y$ linearly independent equations of this type in $M_A$ .",
"On the other hand this case does not contribute to the rank of $S_A$ Case 2.",
"$\\# \\lbrace j_1,j_2,j_3,j_4\\rbrace = 2$ .",
"We distinguish two further cases.",
"Case 2.1.",
"$j_1 = j_3 < j_4 = j_2$ .",
"Segments of length one can only give trivial equations of this type not contributing to the rank of $M_A$ .",
"If $P_{i_0}<P_{i_1}<\\cdots <P_{i_{s-y}}$ are the $s-y$ segments which are not singletons, then equations of this type give us a total of $s - y - 1$ new linear independent ones on top of the ones given by Case 1, one for each pair $P_{i_0}, P_{i_j}$ and $j=2,\\ldots s-y$ , the remaining ones being linearly dependent with these.",
"Moreover this case does not contribute to the rank of $S_A$ .",
"Case 2.2.",
"$j_1<j_2=j_3=j_4$ or $j_1=j_2=j_3<j_4$ .",
"Each pair $P_{j}, P_{j^{\\prime }}$ with $j\\ne j^{\\prime }$ for which an equation of this type exists implies that $P_j\\cap P_{j^{\\prime }}$ intersects either $2P_j$ or $2P_{j^{\\prime }}$ and contributes one additional linear independent equation in $M_A$ on top of the above ones, and it contributes to one additional linear equation in $S_A$ as well.",
"Case 3.",
"$\\# \\lbrace j_1,j_2,j_3,j_4\\rbrace \\ge 3$ , that is $j_1 < j_3 \\le j_4 < j_2$ .",
"This implies that $P_{j_1} + P_{j_2}$ intersects $P_{j_3} + P_{j_4}$ .",
"Each such intersection contributes with one additional linear independent equation in $M_A$ and also on $S_A$ on top of the above ones.",
"Taken together it follows that ${\\rm {rank}}(M_A) = (|A| - 2s + y) + (s - y - 1) + {\\rm {rank}}(S_A)$ and therefore, by Theorem REF , $\\dim (A) = s - {\\rm {rank}}(S_A)$ .",
"We note that for $s \\ge 6$ there are sets for which $k_i = 1$ for all $i$ that are not covered by Corollary REF or Conjecture REF ."
],
[
"Proof of Proposition ",
"Let $A \\subset {\\mathbb {Z}}$ again be a set which is the union of $s$ segments.",
"We denote the interval separating the two consecutive segments $P_i$ and $P_{i+1}$ by $L_i = [\\max (P_i)+1,\\min (P_{i-1})-1]$ and write $\\ell _i = |L_i|$ for its cardinality.",
"It follows that $\\min P_i = \\sum _{j<i} (k_j+l_j) \\quad \\text{and} \\quad \\max (P_i) = \\min (P_i)+k_i-1.$ In order to prove Proposition REF , we will need the following lemma that gives us an inductive approach to Conjecture REF .",
"Lemma 3.1 Let $A$ be a 1–dimensional set of cardinality $k$ that is composed of $s \\le k-1$ disjoint segments $P_1,\\dots ,P_s$ such that $\\max P_i < \\min P_{i+1}$ for $1 \\le i < s$ .",
"If Conjecture REF holds for $s-1$ and ${\\rm {vol}}(A)>2^{s-1}(|A|-s)+1,$ then we must have $P_i+P_j<P_i+P_{j+1}, \\qquad 1\\le i\\le s,\\; 1\\le j<s.$ We observe that, for each $1 \\le j < s$ , the set $A \\cup L_j$ is 1–dimensional, consists of $s-1$ disjoint segments and has the same volume as $A$ .",
"By the assumption that Conjecture REF holds for $s-1$ , we must have ${\\rm {vol}}(A\\cup L_j)\\le 2^{s-2}(|A|+\\ell _j-s+1)+1.$ Hence, our assumption on ${\\rm {vol}}(A)$ implies that $\\ell _j\\ge |A|-s\\ge k_i+1 \\quad \\text{ for }1\\le i\\le s,\\; 1\\le j<s.$ In particular, we have $\\min (P_i+P_{j+1})-\\max (P_i+P_{j})=&(\\min P_{j+1}-\\max P_j)-(\\max P_i-\\min P_i)\\\\=&\\ell _j-k_i+1> 0,$ which implies the desired statement (REF ).",
"It follows that, under the hypothesis of Lemma REF , the only possible intersections between sums of two segments are of the form $(P_i+P_j)\\cap (P_{i^{\\prime }}+P_{j^{\\prime }}) \\quad \\text{where} \\quad i < i^{\\prime } \\le j^{\\prime } < j.$ By using (REF ) we next prove Proposition REF .",
"For $s=1$ the conjecture trivially holds.",
"For $s = 2$ , suppose that $A$ is composed of two segments and ${\\rm {vol}}(A)>2k-2$ .",
"By Lemma REF we have $2P_1<P_1+P_2<2P_2$ and therefore ${\\rm rank} (S_A) = 0$ , so that $A$ must be 2–dimensional by Corollary REF .",
"Suppose now that $s=3$ and that ${\\rm {vol}}(A)>4k-10$ .",
"By (REF ) the only possible intesections of sums of segments are $P_1+P_3$ and $P_2$ .",
"Again, $A$ must be 2–dimensional by Corollary REF .",
"Finally, suppose that $s=4$ and ${\\rm {vol}}(A) > 7k-30$ .",
"By (REF ) the only possible intersections of sets of the type $P_i + P_j$ are $P_1+P_3$ can intersect with $2P_2$ , $P_1+P_4$ can only intersect with at most one of $2P_2$ , $P_2+P_3$ or $2P_3$ , $P_2+P_4$ can only intersect with $2P_3$ .",
"By Corollary REF the only case of interest is if three intersections occur, so let us distinguish the following two cases.",
"Case 1.",
"$P_1+P_4$ intersects $P_2+P_3$ .",
"We note that the vector $(1,-1,-1,1)$ can be written as the sum of $(1,-2,1,0)$ and $(0,1,-2,1)$ and therefore Corollary REF again implies that $A$ is 2–dimensional.",
"Case 2.",
"$P_1+P_4$ intersects $2P_2$ or $2P_3$ .",
"It is clear that these cases are identical by symmetry, so let us assume the former.",
"We must have $P_1+P_4 \\cap 2P_2 \\ne \\emptyset & \\quad \\Leftrightarrow \\quad k_1 + k_2 + \\ell _1 \\ge k_3 + \\ell _2 + \\ell _3 + 2, \\\\P_1+P_3 \\cap 2P_2 \\ne \\emptyset & \\quad \\Leftrightarrow \\quad k_2 + k_3 + \\ell _2 \\ge \\ell _1 + 2, \\\\P_2+P_4 \\cap 2P_3 \\ne \\emptyset & \\quad \\Leftrightarrow \\quad k_3 + k_4 + \\ell _3 \\ge \\ell _2 + 2.$ Combining the first two inequalities gives $\\ell _3 \\le k_1 + 2k_2 - 4$ which combined with the third inequality gives $\\ell _2 \\le |A| + k_2 - 6$ which inserted into the second inequality gives $\\ell _1 \\le |A| + 2k_2 + k_3 - 8$ .",
"Taken together this would imply $\\ell = \\ell _1 + \\ell _2 + \\ell _3 \\le 2|A| + k_1 + 5k_2 + k_3 - 18 \\le 7|A| - 31$ in contradiction to the assumption that $\\ell =\\ell _1+\\ell _2+\\ell _3 \\ge 7|A| - 30$ ."
],
[
"Proof of Theorem ",
"We quote explicitly the so–called $(3k-4)$ –Theorem of Freiman [2] mentioned in the Introduction which will be used throughout the proof.",
"Theorem 4.1 (Freiman) Let $A\\subset {\\mathbb {Z}}$ in normal form with $a=\\max (A)$ and $k=|A|$ .",
"We have $|2A|\\ge \\min \\lbrace k+a, 3k-3\\rbrace .$ There are several versions of the above Theorem for the sum of distinct sets due to Freiman [2], Lev and Smeliansky [12] and Stanchescu [14].",
"We will use the following slightly weaker form of the one by Lev and Smeliansky [12].",
"Theorem 4.2 (Lev and Smelianski) Any two finite sets $A, B \\subset {\\mathbb {Z}}$ in normal form with $\\max (A) > \\max (B)$ satisfy $|A+B|\\ge \\min \\lbrace |A|+2|B|-2, \\max (A)+|B|\\rbrace .$ We make also use of the following result by Freiman [2].",
"Theorem 4.3 Let $A$ be a two dimensional set of cardinality $k>6$ with $|2A| = 3|A| - 3 + b$ .",
"If $A$ can not be covered by a set consisting of two lines with volume at most $k+b$ then $b \\ge |A|/3 - 2$ .",
"We will also use the following Lemma which handles the case of two segments.",
"Lemma 4.4 Let $A$ be an extremal set with cardinality $k$ composed by two segments.",
"Then $A$ is isomorphic to one of the following sets: $[0,k+b-1]\\setminus [1,b]$ , with $\\dim (A)=1$ , ${\\rm {vol}}(A)=k+b$ and $|2A|=2k-1+b$ for some $1\\le b\\le k-3$ , or $([0,k_1-1]\\times \\lbrace 0\\rbrace )\\cup ([0,k_2-1]\\times \\lbrace 1\\rbrace )$ with $k_1+k_2=k$ , $k_1,k_2\\ge 1$ , $\\dim (A)=2$ , ${\\rm {vol}}(A)=k$ and and $|2A|=3k-3$ .",
"If $\\dim (A)=2$ , then there must be no relation in the matrix $S_A$ in Corollary REF and we are led to Case (ii).",
"Suppose that $\\dim (A)=1$ and $A=P_1\\cup P_2$ , say $P_1=[0,k_1-1]$ and $P_2=[k_1+\\ell _1,k_1+\\ell _1+k_2-1]$ for some $k_1, k_2\\ge 1$ and $k=k_1+k_2$ and $\\ell _1\\ge 1$ .",
"Since $dim (A)=1$ , we may also assume that $ (P_1+P_2)\\cap 2P_2\\ne \\emptyset $ , so that $2A$ consists of the interval $[0,2(k+\\ell _1-1)]$ with a hole of some length $h\\ge 0$ .",
"Since $A$ is extremal and has volume $vol(A)=k+\\ell _1$ we have $|2A|=2k-1+\\ell _1$ , so that $h=\\ell _1$ .",
"Therefore $\\ell _1=\\min (P_1+P_2)-\\max (2P_1)-1=\\ell _1+1-k_1,$ which implies $k_1=1$ and gives Case (i).",
"Let $A$ consist of three segments $P_1$ , $P_2$ , and $P_3$ , separated by intervals of holes $L_1$ and $L_2$ .",
"We consider three cases according to the dimension of $A$ .",
"Case 1.",
"$\\dim (A)=3$ .",
"By Corollary REF we must have ${\\rm rank} (S_A) = 0$ , that is all $P_i + P_j$ are disjoint for $1 \\le i,j \\le 3$ , and we are led to Case (iv) of the Theorem.",
"Case 2.",
"$\\dim (A)=2$ .",
"It follows from Corollary REF that the matrix $S_A$ has ${\\rm rank} (S_A) = 1$ .",
"Up to isomorphisms we have two possibilities for the only independent relation in $S_A$ .",
"Case 2.1: $(P_1+P_3)\\cap 2P_2\\ne \\emptyset $ .",
"In this case we may assume that $P_1=\\lbrace (0,0),\\dots ,(0,k_1-1)\\rbrace ,\\; P_2=\\lbrace (1,0),\\dots ,(1,k_2-1)\\rbrace \\; \\text{and}\\; P_3=\\lbrace (2,\\ell ),\\dots ,(2,\\ell +k_3-1)\\rbrace $ for some $\\ell \\in {\\mathbb {N}}_0$ .",
"We know that $|2A| = 4k-6 - |(P_1+P_3) \\cap 2P_2|$ so that $b = k-3- |(P_1+P_3) \\cap 2P_2|$ .",
"We also have ${\\rm {vol}}(A) = k + \\max (\\lfloor (k_1+\\ell +k_3)/2 \\rfloor -k_2,0)$ .",
"Since $\\dim (A) = 2$ we must have $|(P_1+P_3) \\cap 2P_2| > 0$ and therefore $\\ell \\le 2k_2 - 2$ .",
"Now if $0 \\le \\ell \\le 2k_2 - k_1 - k_3$ then $b = k-3 - (k_1+k_3-1) = k_2-2$ and ${\\rm {vol}}(A) = k$ .",
"If $A$ is extremal, we therefore have $k_2 = 2$ so that $k_1+k_3 \\le 4$ and hence $k \\le 6$ .",
"If $\\max (2k_2 - k_1 - k_3,0) < \\ell \\le 2k_2 - 2$ then $b = k-3 - (2k_2 - 2 - \\ell + 1) = k-2 + 2k_2 - \\ell > \\max (\\lfloor (k_1+\\ell +k_3)/2 \\rfloor -k_2,0)$ so the set cannot be extremal.",
"Case 2.2: $2P_1\\cap (P_1+P_2) \\ne \\emptyset $ .",
"The case $(P_1+P_2) \\cap 2P_2 \\ne \\emptyset $ works likewise.",
"We may assume that $P_1= \\lbrace (0,0), (0,1),\\cdots , (0,k_1-1)\\rbrace ,\\; P_2= \\lbrace (0,k_1+\\ell _1), (0,k_1+\\ell _1+1),\\ldots ,(0,k_1+\\ell _1+k_2-1)\\rbrace ,$ with $k_1\\ge \\ell _1+2$ , and $P_3=\\lbrace (1,0),\\ldots ,(1,k_3-1)\\rbrace .$ Let $A_0=P_0 \\cup P_1$ and $A_1=P_2$ , so that $2A = 2A_0 \\cup (A_0+A_1) \\cup 2A_1,$ the union being disjoint.",
"We have $|2A_1| = 2k_3-1$ and, by Theorem REF , we also have $|2A_0| \\ge 2(k_1+k_2)-1+\\ell _1$ .",
"Moreover, it can be readily checked that $|A_0+A_1|={\\left\\lbrace \\begin{array}{ll} k_1+\\ell _1 + k_2 + (k_3-1) = k + \\ell _1 - 1, & \\text{if} \\; k_3>\\ell _1+1,\\\\ (k_1+k_3-1) + (k_2+k_3-1) = k + k_3 - 2, & \\text{otherwise}.\\end{array}\\right.",
"}$ It follows that $|2A| \\ge 3k-3 + \\ell _1 + \\min \\lbrace k_3-1,\\ell _1\\rbrace .$ As ${\\rm {vol}}(A) = k + \\ell _1$ , the set can only be extremal if $\\min (k_3-1,\\ell _1) = 0$ , which implies $k_3=1$ (as $\\ell _1\\ge 1$ ) and there is equality in (REF ), namely, if $|2A_0| = 2(k_1+k_2) - 1 + \\ell _1$ .",
"Applying Lemma REF to $A_0$ leads to Case (iii) of the Theorem.",
"Case 3.",
"$\\dim (A)=1$ .",
"We recall the notation $|P_1|=k_1,\\quad |P_2|=k_2,\\quad |P_3|=k_3,\\quad |L_1|=\\ell _1,\\quad |L_2|=\\ell _2,$ and $a = \\max (A)=k+\\ell -1.$ where $\\ell = \\ell _1+\\ell _2$ .",
"The six segments in $2A$ are detailed below for further reference: $2P_1 &= [0,2k_1-2],\\\\P_1+P_2 &= (k_1+\\ell _1)+[0,k_1+k_2-2],\\\\2P_2 &= 2(k_1+\\ell _1)+[0,2k_2-2],\\\\P_1+P_3 &= k_1+\\ell _1+k_2+\\ell _2+[0,k_1+k_3-2],\\\\P_2+P_3 & = 2(k_1+\\ell _1)+(k_2+\\ell _2)+[0,k_2+k_3-2],\\\\2P_3 &= 2(k_1+\\ell _1+k_2+\\ell _2)+[0,2k_3-2].$ Since $A$ is extremal, we have $a \\ge 2(k+b-2),$ so that $\\ell \\ge k+2b-3.$ We will use the following facts.",
"Claim 1 If $\\max (k_1,k_2) < \\ell _1 + 2$ then $2P_1$ , $P_1+P_2$ and $2P_2$ are pairwise disjoint.",
"If $\\max (k_1,k_2) \\ge \\ell _1+2$ then $P_1 \\cup P_2$ is 1–dimensional and $2(P_1 \\cup P_2)$ is a segment with a hole of length $h = \\max \\big \\lbrace \\ell _1 - \\min (k_1,k_2)+1,0 \\big \\rbrace .$ We note that $2P_1$ does not intersect $P_1+P_2$ if and only if $\\max (2P_1) < \\min (P_1+P_2)$ , which is equivalent to $k_1 < \\ell _1+2$ .",
"Likewise, $P_1+P_2$ does not intersect $2P_2$ if and only if $k_2 < \\ell _1+2$ , establishing the first part of the claim.",
"Assume without loss of generality that $k_1 \\le k_2$ and $k_2 \\ge \\ell _1+2$ .",
"Then $(P_1+P_2) \\cup 2P_2$ is a segment since the two parts intersect.",
"In particular, $P_1 \\cup P_2$ is 1–dimensional.",
"Moreover, either $2P_1\\cup (P_1+P_2)\\cup 2P_2$ is a segment or a segment with a hole of length $h=\\min (P_1+P_2)-\\max (2P_1)-1= \\ell _1 - k_1+1$ , establishing the second part of the claim.",
"By the above Claim, if both $\\max (k_1,k_2) < \\ell _1 + 2$ and $\\max (k_2,k_3) < \\ell _2 + 2$ , then the five segments in $2P_1\\cup (P_1+P_2)\\cup 2P_2 \\cup (P_2+P_3) \\cup 2P_3$ are pairwise disjoint.",
"Using Corollary REF it follows that ${\\rm rank} (S_A) \\le 1$ and hence $\\dim (A) \\ge 2$ , contradicting the assumption of this case.",
"We will therefore without loss of generality assume that $\\max (k_1,k_2) \\ge \\ell _1+2$ .",
"In this case we have Claim 2 $\\max \\lbrace k_2,k_3\\rbrace <\\ell _2+2$ .",
"Suppose on the contrary that $\\max \\lbrace k_2,k_3\\rbrace \\ge \\ell _2+2$ .",
"Then, using (REF ), $k+2b-3\\le \\ell \\le \\max \\lbrace k_1,k_2\\rbrace +\\max \\lbrace k_2,k_3\\rbrace -4$ which implies $\\max \\lbrace k_1,k_2\\rbrace =\\max \\lbrace k_2,k_3\\rbrace =k_2$ .",
"It follows that $\\max (2P_2)=2(k_1+l_1+k_2-1)\\ge 2(k_1+\\ell _1)+k_2+\\ell _2=\\min (P_2+P_3)$ .",
"Hence, the sets $2(P_1\\cup P_2)$ and $2(P_2\\cup P_3)$ overlap and, by Claim REF , $2A$ consists of the interval $[0,2a]$ with two holes of total length at most $\\max \\lbrace \\ell _1-k_1+1,0\\rbrace +\\max \\lbrace \\ell _2-k_3+1,0\\rbrace \\le \\ell .$ Therefore, by using $a=k+\\ell -1$ and (REF ), we obtain $|2A|\\ge 2a-\\ell +1\\ge 3k+2b-4$ and therefore $A$ is not extremal.",
"It follows from Claim REF and Claim REF that the three segments $2P_2, P_2+P_3, 2P_3$ are pairwise disjoint.",
"Since $A$ is one–dimensional, $2(P_1\\cup P_2)$ must intersect $P_1+P_3$ .",
"In particular, $\\max (2P_2)\\ge \\min (P_1+P_3)$ which yields $k_1+\\ell _1+k_2\\ge \\ell _2+2.$ Claim 3 $k_3=1$ .",
"Suppose on the contrary that $k_3>1$ .",
"We then have $\\ell _1>1$ , since otherwise (REF ) and (REF ) give $k_1+k_2\\ge \\ell _2+1\\ge k+2b-3$ and we get $k_3\\le 1$ .",
"Let $B=2(P_1\\cup P_2) \\cup (P_1+P_3)\\cup (P_2+P_3)$ .",
"We can write $2A$ as the disjoint union $2A=B \\cup 2P_3.$ Consider now the set $A^{\\prime }$ obtained from $A$ by replacing $\\min (P_3)$ with $\\max (P_1)+1$ if $k_1\\ge k_2$ and with $\\min (P_2)-1$ otherwise.",
"The resulting set is still composed of three disjoint segments, $A^{\\prime }=P^{\\prime }_1\\cup P^{\\prime }_2\\cup P^{\\prime }_3$ with $\\ell ^{\\prime }_1=\\ell _1-1$ , $\\ell _2^{\\prime } = \\ell _2 + 1$ and $\\min \\lbrace k_1^{\\prime },k^{\\prime }_2\\rbrace =\\min \\lbrace k_1,k_2\\rbrace $ .",
"We can write $2A^{\\prime }$ as the disjoint union $2A^{\\prime }=B^{\\prime } \\cup 2P_3^{\\prime },$ where $B^{\\prime }=2(P_1^{\\prime }\\cup P_2^{\\prime }) \\cup (P_1^{\\prime }+P_3^{\\prime })\\cup (P_2^{\\prime }+P_3^{\\prime })$ .",
"We have $|2P^{\\prime }_3|=|2P_3|-2$ .",
"Let us show that $|B^{\\prime }|\\le |B|+1$ .",
"By Claim REF , $|2(P^{\\prime }_1\\cup P^{\\prime }_2)|\\le |2(P_1\\cup P_2)|+1$ .",
"If $k_1\\ge k_2$ then $P_2^{\\prime }=P_2$ and $|P^{\\prime }_2+P^{\\prime }_3|=|P_2+P_3|-1$ , while $P^{\\prime }_1+P^{\\prime }_3=(P_1+P_3)+1$ .",
"If $P_1+P_3$ and $P_2+P_3$ are disjoint then the two last modifications compensate each other, while if they intersect then there is no change in the cardinality of their union.",
"Similarly, if $k_1<k_2$ then $P_1^{\\prime }=P_1$ and we loose one unit in $P^{\\prime }_1+P^{\\prime }_3$ while $P^{\\prime }_2+P^{\\prime }_3$ is translated one unit to the right from $P_2+P_3$ and again there is no change in the cardinality of the union of these two segments.",
"In either case, we get $|2A^{\\prime }|<|2A|$ so that, if $A^{\\prime }$ is one–dimensional it would have the same volume as $A$ contradicting that $A$ is extremal.",
"It follows that $A^{\\prime }$ must be 2–dimensional.",
"This implies $\\max (2P^{\\prime }_2)<\\min (P^{\\prime }_1+P_3^{\\prime })$ .",
"Since $\\max (2P_2)\\ge \\min (P_1+P_3)$ , we have equality in the last inequality.",
"Therefore, $|2A| = |2(P_1\\cup P_2)|+|P_3+A|-1.$ By Theorem REF we have $|P_3+A| \\ge |A|+2|P_3|-2=k+2k_3-2.$ Therefore, $|2A| &= |2(P_1\\cup P_2)|+|P_3+A|\\\\&\\ge (\\max (2P_2)+1-\\ell _1)+(k+2k_3-2)\\\\&= 2(k-k_3+\\ell _1-1)-\\ell _1+k+2k_3-2\\\\&=3k+\\ell _1-4,$ so that $\\ell _1\\le b$ .",
"But then, by (REF ), we have $\\ell =\\ell _1+\\ell _2\\le b+(k-k_3+b-2)=k-k_3+2b-2,$ contradicting (REF ).",
"Therefore $A$ could not have been extremal.",
"We can therefore assume $P_3=\\lbrace a\\rbrace $ .",
"It follows that $2A=2(P_1\\cup P_2)\\cup (a+A).$ Moreover, $\\min (P_2+P_3)-\\max (P_1+P_3)=\\ell _1-k_3+2=\\ell _1+1> 1.$ We next consider two cases.",
"Case 3.1: $k_1\\le k_2$ .",
"The sumset $2A$ can be written as the disjoint union $2A=B\\cup (P_2+P_3)\\cup 2P_3,$ where $B=2(P_1\\cup P_2)\\cup (P_1+P_3)$ is an interval with a hole of length $h=\\max \\lbrace \\ell _1-k_1+1,0\\rbrace $ .",
"Such a one–dimensional set with $k_1>1$ cannot be extremal since, by exchanging $\\max (P_1)$ by $\\min (P_2)-1$ we get a one–dimensional set with the same volume and smaller doubling.",
"It follows that $k_1=1$ .",
"By using (REF ), we get $\\max (2P_2)-\\max (P_1+P_3)=\\ell _1+k_2-\\ell _2-1\\ge 0$ .",
"In this case $2(k+\\ell _1-2)=\\max (2P_2)\\ge a = \\max (P_1+P_3)$ and, again by extremality, equality holds.",
"We thus have $|2A|=(a-\\ell _1+1)+(k-2)+1=3k+\\ell _1-4$ , leading to Case (i) of the Theorem.",
"Case 3.2: $k_1>k_2$ .",
"From $3k-4+b=|2A|&=|2(P_1\\cup P_2)|+|a+A|-|2(P_1\\cup P_2)\\cap (a+A)|\\\\&= 2(k-1+\\ell _1)-1- \\max \\lbrace \\ell _1 -k_2+1,0 \\rbrace +k \\\\& \\quad -|2(P_1\\cup P_2)\\cap (a+A)|,$ we obtain $|2(P_1\\cup P_2)\\cap (a+A)| = 2\\ell _1+1 - \\max \\lbrace \\ell _1 -k_2+1,0 \\rbrace - b.$ For this equality to hold, a necessary condition is $\\max (2P_2)-a+1\\ge 2\\ell _1+1 - \\max \\lbrace \\ell _1 -k_2+1,0 \\rbrace - b.$ By using $\\max (2P_2)=2(k-1)+2\\ell _1-2$ and $a=k+\\ell -1$ in (REF ), we obtain $\\ell \\le k+b+\\max \\lbrace \\ell _1-k_2+1,0\\rbrace -3.$ By (REF ) we have $\\ell _1\\ge k_2+b-1$ .",
"On the other hand, since $|2(P_1\\cup P_2)\\cap (a+A)| \\le |2P_2| = 2k_2-1$ , it follows from (REF ) that $\\ell _1\\le b+k_2-1$ .",
"Hence $\\ell _1=b+k_2-1$ , there is equality in (REF ) and $2P_2$ must be included in $P_1+P_3$ , so that $2k_2-1 = |2P_2| \\le |P_1+P_3| = k_1$ .",
"This gives Case (ii) of the Theorem."
]
] | 1808.08455 |
[
[
"Code Generation for Higher Inductive Types"
],
[
"Abstract Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity.",
"While work proceeds on type theories with a computational interpretation of univalence and higher inductive types, it is convenient to encode these structures in more traditional type theories with mature implementations.",
"However, these encodings involve a great deal of error-prone additional syntax.",
"We present a library that uses Agda's metaprogramming facilities to automate this process, allowing higher inductive types to be specified with minimal additional syntax."
],
[
"Introduction",
"Type theory unites programming and mathematics in a delightful synthesis, in which we can write programs and proofs in the same language.",
"Work on higher-dimensional type theory has revealed a beautiful higher-dimensional structure, lurking just beyond reach.",
"In particular, higher inductive types provide a natural encoding of many otherwise-difficult mathematical concepts, and univalence lets us work in our type theory the way we do on paper: up to isomorphism.",
"Homotopy type theory, however, is not yet done.",
"We do not yet have a mature theory or a mature implementation.",
"While work proceeds on prototype implementations of higher-dimensional type theories [26][25], much work remains before they will be as convenient for experimentation with new ideas as Coq, Agda, or Idris is today.",
"In the meantime, it is useful to be able to experiment with ideas from higher-dimensional type theory in our existing systems.",
"If one is willing to put up with some boilerplate code, it is possible to encode higher inductive types and univalence using postulated identities.",
"Boilerplate postulates, however, are not just inconvenient, they are also an opportunity to make mistakes.",
"Luckily, this boilerplate code can be mechanically generated using Agda's recent support for elaborator reflection [6], a paradigm for metaprogramming in an implementation of type theory.",
"An elaborator is the part of the implementation that translates a convenient language designed for humans into a much simpler, more explicit, verbose language designed to be easy for a machine to process.",
"Elaborator reflection directly exposes the primitive components of the elaborator to metaprograms written in the language being elaborated, allowing them to put these components to new uses.",
"Homotopy type theory has thus far primarily been applied to the encoding of mathematics, rather than to programming.",
"Nevertheless, there are a few applications of homotopy type theory to programming.",
"Applications such as homotopical patch theory [2] discuss a model of the core of the of Darcs [7] version control system using patch theory [17] encoded as a HIT.",
"Containers in homotopy type theory [12], [16] implement data structures such as multisets and cycles.",
"Automating the HIT boilerplate code allows more programmers to begin experimenting with programming with HITs.",
"Using Agda's elaborator reflection, we automatically generate the support code for many useful higher inductive types, specifically those that include additional paths between constructors, but not paths between paths, which is sufficient for treating various interesting examples on the programming side [2][21][32].",
"We automate the production of the recursion principles, induction principles, and their computational behavior.",
"Angiuli et al.",
"'s encoding of patch theory as a higher inductive type [2] requires approximately 1500 lines of code when represented using rewrite mechanism.",
"Using our library, the encoding can be expressed in just 70 lines.",
"This paper makes the following contributions: We describe the design and implementation of a metaprogram that automates an encoding of higher inductive types with one path dimension using Agda's new metaprogramming system.",
"We demonstrate applications of this metaprogram to examples from the literature, including both standard textbook examples of higher inductive types as well as larger systems, including both patch theory and specifying cryptographic schemes.",
"This metaprogram serves as an example of the additional power available in Agda's elaborator reflection relative to earlier metaprogramming APIs.",
"In Agda, we don't have built-in primitives to support the definition of higher inductive types.",
"In this paper, we use Agda's rewrite rules mechanism to define higher inductive types [29][30].",
"Unlike [30], we use basic modules, without parameters, to encode higher inductive types.",
"This is because Agda's reflection library does not have primitives to support introducing parameterized modules.",
"Homotopy type theory [1] is a research program that aims to develop univalent, higher-dimensional type theories.",
"A universe is univalent when equivalences between types are considered equivalent to identifications between types.",
"A type theory is univalent when every universe in the type theory is univalent; it is higher-dimensional when we allow non-trivial identifications that every structure in the theory must nevertheless respect.",
"Identifications between elements of a type are considered to be at the lowest dimension, while identifications between identifications at dimension $n$ are at dimension $n+1$ .",
"Voevodsky added univalence to type theories as an axiom, asserting new identifications without providing a means to compute with them.",
"While more recent work arranges the computational mechanisms of the type theory such that univalence can be derived, as is done in cubical type theories [26][25], we are concerned with modeling concepts from homotopy type theory in existing, mature implementations of type theory, so we follow Univalent Foundations Program [1] in modeling paths using Martin-Löf's identity type.",
"Higher-dimensional structure can arise from univalence, but it can also be introduced by defining new type formers that introduce not only introduction and elimination principles, but also new non-trivial identifications.",
"In homotopy type theories, one tends to think of types not as collections of distinct elements, but rather through the metaphor of topological spaces.",
"The individual elements of the type correspond with points in the topological space, and identifications correspond to paths in this space.",
"While work proceeds on the general schematic characterization of higher inductive types[10][32][8], it is convenient to syntactically represent the higher inductive types that we know are acceptable using a syntax similar to a traditional inductive type by providing its constructors (i.e.",
"its points); we additionally specify the higher-dimensional structure by providing additional constructors for paths.",
"For example, Figure REF describes |Circle|, which is a higher inductive type with one point constructor |base| and one non-trivial path constructor |loop|.",
"Figure: A specification of a higher inductive typeFigure: A HIT encoded using rewrite rulesFigure REF represents the implementation of Circle in Agda.",
"Inside module Circle, the type S and the constructors base and loop and the recursion and induction principles are declared as postulates.",
"recS ignores the path argument and simply computes to the appropriate answer for the point constructor.",
"The computation rule for point base is declared as a rewrite rule using |- REWRITE , ...-| pragma.",
"The computation rule for the path constructor loop is postulated using reduction rule βloop.",
"The operator ap is frequently referred to as cong, because it expresses that propositional equality is a congruence.",
"However, when viewed through a homotopy type theory lens, it is often called ap, as it describes the action of a function on paths.",
"In a higher inductive type, ap should compute new paths from old ones.",
"ap : A B : Set x y : A (f : A → B) (p : x ≡ y) → f x ≡ f y In addition to describing the constructors of the points and paths of S, Figure REF additionally demonstrates the dependent eliminator (that is, the induction rule) indS and its computational meaning.",
"The dependent eliminator relies on another operation on identifications, called transport, that coerces an inhabitant of a family of types at a particular index into an inhabitant at another index.",
"Outside of homotopy type theory, transport is typically called subst or replace, because it also expresses that substituting equal elements for equal elements is acceptable.",
"transport : A : Set x y : A → (P : A → Set) → (p : x ≡ y) → P x → P y In the postulated computation rule for indS, the function apd is the dependent version of ap: it expresses the action of dependent functions on paths.",
"apd : A : Set B : A → Set x y : A → (f : (a : A) → B a) → (p : x ≡ y) → transport B p (f x) ≡ f y"
],
[
"Agda Reflection",
"Agda [27] is a functional programming language with full dependent types and dependent pattern matching.",
"Agda's type theory has gained a number of new features over the years, among them the ability to restrict pattern matching to that subset that does not imply Streicher's Axiom K [28], which is inconsistent with univalence.",
"The convenience of programming in Agda, combined with the ability to avoid axiom K, makes it a good laboratory for experimenting with the idioms and techniques of univalent programming while more practical implementations of univalent type theories are under development.",
"Agda's reflection library enables compile-time metaprogramming.",
"This reflection library directly exposes parts of the implementation of Agda's type checker and elaborator for use by metaprograms, in a manner that is similar to Idris's elaborator reflection [22], [6] and Lean's tactic metaprogramming [23].",
"The type checker's implementation is exposed as effects in a monad called TC.",
"Agda exposes a representation of its syntax to metaprograms, including datatypes for expressions (called Term) and definitions (called Definition).",
"The primitives exposed in TC include declaring new metavariables, unifying two Terms, declaring new definitions, adding new postulates, computing the normal form or weak head normal form of a Term, inspecting the current context, and constructing fresh names.",
"This section describes the primitives that are used in our code generation library; more information on the reflection library can be found in the Agda documentation [5].",
"TC computations can be invoked in three ways: by macros, which work in expression positions, using the unquoteDecl operator in a declaration position, which can bring new names into scope, and using the unquoteDef operator in a declaration position, which can automate constructions using names that are already in scope.",
"This preserves the principle in Agda's design that the system never invents a name.",
"An Agda macro is a function of type $t_1$ → $t_2$ → $\\ldots $ → Term → TC ⊤ that is defined inside a macro block.",
"Macros are special: their last argument is automatically supplied by the type checker and consists of a Term that represents the metavariable to be solved by the macro.",
"If the remaining arguments are quoted names or Terms, then the type checker will automatically quote the arguments at the macro's use site.",
"At some point, the macro is expected to unify the provided metavariable with some other term, thus solving it.",
"Figure: A macro that unquotes its argumentFigure REF demonstrates a macro that quotes its argument.",
"The first step is to quote the quoted expression argument again, using quoteTC, yielding a quotation of a quotation.",
"This double-quoted expression is passed, using Agda's new support for Haskell-style do-notation, into a function that unifies it with the hole.",
"Because unification removes one layer of quotation, unify inserts the original quoted term into the hole.",
"The value of sampleTerm is lam visible (abs \"n\" (var 0 [])) The constructor lam represents a lambda, and its body is formed by the abstraction constructor abs that represents a scope in which a new name \"n\" is bound.",
"The body of the abstraction is a reference back to the abstracted name using de Bruijn index 0.",
"The unquoteTC primitive removes one level of quotation.",
"Figure REF demonstrates the use of unquoteTC.",
"The macro mc2 expects a quotation of a quotation and substitutes its unquotation for the current metavariable.",
"The unquoteDecl and unquoteDef primitives, which run TC computations in a declaration context, will typically introduce new declarations by side effect.",
"A function of a given type is declared using declareDef, and it can be given a definition using defineFun.",
"Similarly, a postulate of a given type is defined using declarePostulate.",
"Figure REF shows an Agda implementation of addition on natural numbers, while Figure REF demonstrates an equivalent metaprogram that adds the same definition to the context.",
"Figure: NO_CAPTIONFigure: Addition, defined by metaprogrammingIn Figure REF , declareDef declares the type of plus.",
"The constructor pi represents dependent function types, but a pattern synonym is used to make it shorter.",
"Similarly, def constructs references to defined names, and the pattern synonym |`Nat| abbreviates references to the defined name Nat, and vArg represents the desired visibility and relevance settings of the arguments.",
"Once declared, plus is defined using defineFun, which takes a name and a list of clauses, defining the function by side effect.",
"Each clause consists of a pattern and a right-hand side.",
"Patterns have their own datatype, while right-hand sides are Terms.",
"The name con is overloaded: in patterns, it denotes a pattern that matches a particular constructor, while in Terms, it denotes a reference to a constructor.",
"The next section introduces the necessary automation features by describing the automatic generation of eliminators for a variant on Dybjer's inductive families.",
"Section 4 then generalizes this feature to automate the production of eliminators for higher inductive types using the rewrite mechanism.",
"Section 5 revisits Angiuli et al.",
"'s encoding of Darcs's patch theory [2] and demonstrates that the higher inductive types employed in that paper can be generated succinctly using our libraryPlease see https://github.com/pavenvivek/WFLP-18."
],
[
"Code Generation for Inductive Types",
"An inductive type $D$ is a type that is freely generated by a finite collection of constructors.",
"The constructors of $D$ accept zero or more arguments and have $D$ as the co-domain.",
"The constructors can also take an element of type $D$ itself as an argument, but only strictly positively: any occurrences of the type constructor $D$ in the type of an argument to a constructor of $D$ must not be to the left of any arrows.",
"Type constructors can have a number of parameters, which may not vary between the constructors, as well as indices, which may vary.",
"In Agda, constructors are given a function type.",
"In Agda's reflection library, the constructor data-type of the datatype Definition stores the constructors of an inductive type as a list of Names.",
"The type of a constructor can be retrieved by giving its Name as an input to the getType primitive.",
"In this section, we discuss how to use the list of constructors and their types to generate code for the elimination rules of an inductive type."
],
[
"Non-dependent Eliminators",
"In Agda, we define an inductive type using data keyword.",
"A definition of an inductive datatype declares its type and specifies its constructors.",
"While Agda supports a variety of ways to define new data types, we will restrict our attention to the subset that corresponds closely to Dyber's inductive families.",
"In general, the definition of an inductive datatype $D$ with constructors $c_1 \\ldots c_n$ has the following form: $&\\textsf {\\textbf {data}}\\ D\\ (a_1 : A_1) \\ldots (a_n:A_n) : (i_1 : I_1) \\rightarrow \\ldots \\rightarrow (i_m : I_m) \\rightarrow \\textsf {Set}{}\\ \\textsf {\\textbf {where}} \\\\&\\hspace{8.5359pt} c_1 : \\Delta _1 \\rightarrow D \\, a_1 \\ldots a_n \\, e_{11} \\ldots e_{1m} \\\\&\\hspace{99.58464pt} \\vdots \\\\&\\hspace{8.5359pt} c_r : \\Delta _n \\rightarrow D \\, a_1 \\ldots a_n \\, e_{r1} \\ldots e_{rm}$ where the index instantiations $e_{k1} \\ldots e_{km}$ are expressions in the scope induced by the telescope $\\Delta _k$ .",
"Every expression in the definition must also be well-typed according to the provided declarations.",
"A telescope $\\Delta \\, = \\, (x_1 : B_1) \\ldots (x_n : B_n)$ is a sequence of types where later types may depend on elements of previous types.",
"Figure: Length-indexed listsAs an example, the datatype Vec (Figure REF ) represents lists of a known length.",
"There is one parameter, namely (A : Set), and one index, namely Nat.",
"The second constructor, _::_, has a recursive instance of Vec as an argument.",
"While inductive datatypes are essentially characterized by their constructors, it must also be possible to eliminate their inhabitants, exposing the information in the constructors.",
"This section describes an Agda metaprogram that generates a non-dependent recursion principle for an inductive type; section REF generalizes this technique to fully dependent induction principles.",
"For Vec, the recursion principle says that, in order to eliminate a Vec A n, one must provide a result for the empty Vec and a means for transforming the head and tail of a non-empty Vec combined with the result of recursion onto a tail into the desired answer for the entire Vec.",
"Concretely, the type of the recursor recVec is given as follows.",
"recVec : (A : Set) → n : Nat → Vec A n → (C : Set) → (base : C) → (step : n : Nat → (x : A) → (xs : Vec A n) → C → C) → C The recursor recVec maps the constructor [], which takes zero arguments, to base.",
"It maps (x :: xs) to (step x xs (recVec xs C base step)).",
"Because step is applied to a recursive call to the recursor, it takes one more argument than the constructor _::_.",
"Based on the schematic presentation of inductive types $D$ earlier in this section, we can define a schematic representation for their non-dependent eliminators $D_{\\mathit {rec}}$ .",
"$D_{\\mathit {rec}} :\\ & (a_1 : A_1) \\rightarrow \\ldots \\rightarrow (a_n : A_n) \\rightarrow \\\\& (i_1 : I_1) \\rightarrow \\ldots \\rightarrow (i_m : I_m) \\rightarrow \\\\& (\\mathit {tgt} : D\\ a_1 \\ldots a_n\\ i_1\\ \\ldots \\ i_n) \\rightarrow \\\\& (\\mathit {C} : \\textsf {Set}{}) \\rightarrow \\\\& (\\mathit {f_1} : \\Delta _1^\\prime \\rightarrow \\mathit {C}) \\rightarrow \\ldots \\rightarrow (\\mathit {f_r} : \\Delta _r^\\prime \\rightarrow \\mathit {C}) \\rightarrow \\\\& \\mathit {C}$ The type of $f_i$ , which is the method for fulfilling the desired type $C$ when eliminating the constructor $c_i$ , is determined by the type of $c_i$ .",
"The telescope $\\Delta _i^\\prime $ is the same as $\\Delta _i$ for non-recursive constructor arguments.",
"However, $\\Delta _i^\\prime $ binds additional variables when there are recursive occurrences of $D$ in the arguments.",
"For instance, if $\\Delta _i$ has an argument $(y : B)$ , where $B$ is not an application of $D$ or a function returning such an application, $\\Delta _i^\\prime $ binds $(y : B)$ directly.",
"If $B$ is an application of $D$ , then an additional binding $(y^\\prime : C)$ is inserted following $y$ .",
"Finally, if $B$ is a function type $\\Psi \\rightarrow D$ , the additional binding is $(y^\\prime : \\Psi \\rightarrow C)$ .",
"To construct the type of recVec, we need to build the types of base and step.",
"These are derived from the corresponding types of [] and _::_, which can be discovered using reflection primitives.",
"Since [] requires no arguments, its corresponding method is (base : C).",
"The constructor pi of type Term encodes the abstract syntax tree (AST) representation of _::_.",
"We can retrieve and traverse the AST of _::_, and add new type information into it to build a new type representing step.",
"Once the AST for step's type has been found, it is possible to build the type of recVec.",
"To quantify over the return type (C : Set), we use the Term constructor agda-sort to refer to Set.",
"In general, when automating the production of $D_{\\mathit {rec}}$ , all the information that is needed to produce the type signature is available in the TC monad by looking up $D$ 's definition.",
"The constructor data-type contains the number of parameters occurring in a defined type.",
"It also encodes the constructors of the type as a list of Names.",
"Metaprograms can retrieve the index count by using the type and the number of parameters.",
"The constructors of $D$ refer to the parameter and the index using de Bruijn indices.",
"The general schema for the computation rules corresponding to $D_{\\mathit {rec}}$ and constructors $c_1, \\ldots , c_n$ is as follows: $& D_\\mathit {rec}\\ a_1\\ \\ldots \\ a_n\\ i_1\\ \\ldots \\ i_m\\ (c_1\\ {\\Delta _1}) \\, C \\, f_1 \\ldots f_r = \\mathsf {RHS}\\left(f_{1}, \\Delta _{1}^\\prime \\right)\\\\& \\vdots \\\\& D_\\mathit {rec}\\ a_1\\ \\ldots \\ a_n\\ i_1\\ \\ldots \\ i_m\\ (c_r\\ {\\Delta _r}) \\, C \\, f_1 \\ldots f_r = \\mathsf {RHS}\\left(f_{r}, \\Delta _{r}^\\prime \\right)$ Here, ${\\Delta _j}^\\prime $ is the sequence of variables bound in $\\Delta _j$ .",
"$\\mathsf {RHS}$ constructs the application of the method $f_j$ to the arguments of $c_j$ , such that $C$ is satisfied.",
"It is defined by recursion on $\\Delta _j$ .",
"$\\mathsf {RHS}\\left(f_j, \\cdot \\right)$ is $f_j$ , because all arguments have been accounted for.",
"$\\mathsf {RHS}\\left(f_j, (y : B)\\Delta _k\\right)$ is $\\mathsf {RHS}\\left(f_j\\ y, \\Delta _k\\right)$ when $B$ does not mention $D$ .",
"$\\mathsf {RHS}\\left(f_j, (y : D) (y^\\prime : C)\\Delta _k\\right)$ is $\\mathsf {RHS}\\left(f_j\\ y\\ \\left(D_{\\mathit {rec}}\\ \\ldots y \\ldots \\right), \\Delta _k\\right)$ , where the recursive use of $D_{\\mathit {rec}}$ is applied to the recursive constructor argument as well as the appropriate indices, and the parameters, result type, and methods remain constant.",
"Higher-order recursive arguments are a generalization of first-order arguments.",
"Finally, $\\mathsf {RHS}\\left(f_j, (y : \\Psi \\rightarrow D) (y^\\prime : \\Psi \\rightarrow C)\\Delta _k\\right)$ is $\\mathsf {RHS}\\left(f_j\\ y\\ \\left(\\lambda \\overline{\\Psi } .",
"D_{\\mathit {rec}}\\ \\ldots \\left(y\\ \\overline{\\Psi } \\right) \\ldots \\right), \\Delta _k\\right)$ where the recursive use of $D_{\\mathit {rec}}$ is as before.",
"After declaring recVec's type using declareDef, it is time to define its computational meaning using the schematic rules defined above.",
"The computation rule representing the action of function recVec on [] and _::_ is defined using clause.",
"The first argument to clause encodes variables corresponding to the above type, and it also includes the abstract representation of the constructors [] and _::_ on which the pattern matching should occur.",
"The second argument to clause, which is of type Term, refers to the variables in the first argument using de Bruijn indices, and it encodes the output of recVec when the pattern matches.",
"The computation rules for recVec are given as follows.",
"recVec [] C base step = base recVec (x :: xs) C base step = step x xs (f xs C base step) generateRec (Figure REF ) build the computation and elimination rules respectively.",
"The recursion rule generated by generateRec is brought into scope using unquoteDecl.",
"The first argument to generateRec is the quoted Name of the recursor encoded inside Arg, and the second argument is the quoted Name of the inductive type.",
"Figure: NO_CAPTION"
],
[
"Dependent Eliminators",
"The dependent eliminator for a datatype, also known as the induction principle, is used to eliminate elements of a datatype when the type resulting from the elimination mentions the very element being eliminated.",
"The type of the induction principle for $D$ is: $D_{\\mathit {ind}} :\\ & (a_1 : A_1) \\rightarrow \\ldots \\rightarrow (a_n : A_n) \\rightarrow \\\\& (i_1 : I_1) \\rightarrow \\ldots \\rightarrow (i_m : I_m) \\rightarrow \\\\& (\\mathit {tgt} : D\\ a_1 \\ldots a_n\\ i_1\\ \\ldots \\ i_m) \\rightarrow \\\\& \\begin{array}{@{}l@{}l}(\\mathit {C} :\\ & (i_1 : I_1) \\rightarrow \\ldots \\rightarrow (i_m : I_m) \\rightarrow \\\\ & D\\ a_1 \\ldots a_n\\ i_1\\ \\ldots \\ i_n \\rightarrow \\textsf {Set}{})\\rightarrow \\end{array} \\\\& (\\mathit {f_1} : \\Delta _1^\\prime \\rightarrow \\mathit {C}\\ e_{11} \\ldots e_{1p}\\ (c_1\\ \\overline{\\Delta _1})) \\rightarrow \\\\& (\\mathit {f_r} : \\Delta _r^\\prime \\rightarrow \\mathit {C}\\ e_{r1} \\ldots e_{rp}\\ (c_r\\ \\overline{\\Delta _r})) \\rightarrow \\\\& \\mathit {C}\\ i_1\\ \\ldots \\ i_n\\ \\mathit {tgt}$ Unlike the non-dependent recursion principle $D_{\\mathit {rec}}$ , the result type is now computed from the target and its indices.",
"Because it expresses the reason that the target must be eliminated, the function $C$ is often referred to as the motive.",
"Similarly to $D_{\\mathit {rec}}$ , the type of each method $f_i$ is derived from the type of the constructor $c_i$ —the method argument telescope $\\Delta _k^\\prime $ is similar, except the arguments that represents the result of recursion now apply the motive $C$ to appropriate arguments.",
"If $\\Delta _i$ has an argument $(y : B)$ , where $B$ is not an application of $D$ or a function returning such an application, $\\Delta _i^\\prime $ still binds $(y : B)$ directly.",
"If $B$ is an application of $D$ to parameters $a\\ldots $ and indices $e\\ldots $ , then an additional binding $(y^\\prime : C\\ e\\ldots \\ y)$ is inserted following $y$ .",
"Finally, if $B$ is a function type $\\Psi \\rightarrow D\\ a\\ldots \\ e\\ldots $ , the additional binding is $(y^\\prime : \\Psi \\rightarrow C\\ e\\ldots (y\\ \\overline{\\Psi }))$ .",
"Following these rules, the induction principle for Vec can be defined as follows.",
"indVec : (A : Set) → n : Nat → (xs : Vec A n) → (C : n : Nat → Vec A n → Set) → (base : C []) → (step : n : Nat → (x : A) → (xs : Vec A n) → C xs → C (x :: xs)) → C xs Automating the production of the dependent eliminator is an extension of the procedure for automating the production of the non-dependent eliminator.",
"The computation rules for the induction principle are automated using the same approach as for the recursion principle.",
"The generation of induction principles is carried out using generateInd (Figure REF )."
],
[
"Code Generation for Higher Inductive Types",
"In Agda, there are no built-in primitives to support the definition of higher inductive types.",
"However, we can still define a higher inductive type using rewrite rules, as described in section REF .",
"In this section, we discuss the automation of code generation for the elimination and the computation rules of higher inductive types.",
"While the general formulation of higher inductive types is a subject of active research [8][9][11], we stick to a schema that follows the pattern of Basold et al.",
"'s [32] general rules for higher inductive types."
],
[
"Non-dependent Eliminators for HITs",
"The recursion principle of a higher inductive type $G$ maps the points and paths of $G$ to points and paths in an output type $C$ .",
"We extend the general schema of the recursion principle given in section REF by adding methods for path constructors (Figure REF ).",
"Figure: NO_CAPTIONThe schematic definition of $G_{\\mathit {rec}}$ supports only one-dimensional paths.",
"The type of the method $f_i$ for a point constructor $g_i$ in $G_{rec}$ is built the same way as for the normal inductive type $D$ , as described in section REF .",
"The code generator builds the type of $k_i$ , method for path constructor $p_i$ in $G_{rec}$ , by traversing the AST of $p_i$ .",
"The arguments of $k_i$ are handled the same way as for the point constructor's method $f_i$ .",
"During the traversal, the code generator uses the base type recursor $D_{rec}$ to map the point constructors $g_i$ of $G$ in the codomain of $p_i$ to $f_i$ .",
"Determining the computation rules corresponding to points $g_i$ is similar to the computation rules corresponding to constructors $c_i$ of the inductive type $D$ , except that there are additional methods to handle paths.",
"Paths compute new paths; the computation rules that govern the interaction of recursors and paths $p_i$ are named and postulated.",
"They identify the action of the recursor on the path with the corresponding method.",
"The computation rules corresponding to paths $p_i$ are postulated as given in Figure REF .",
"As an example, if the code for the circle HIT from section REF has been generated, and the type is called S, then the recursor needs a method for base and one for loop.",
"The method for base should be an inhabitant of C. If it is called cbase, then the method for loop should be a path cbase ≡ cbase.",
"The types of the path methods depend on the values of the point methods.",
"The code generator builds the type of loop's method by traversing the AST of loop's type, replacing references to point constructors with the result of applying the base type's recursor to the point methods.",
"The recursion rule recS follows this pattern.",
"recS : S → (C : Set) → (cbase : C) → (cloop : cbase ≡ cbase) → C The code generator builds the computation rule for the point constructor base using the same approach as described in section REF as if it were for the base type.",
"Additionally, it includes variables in the clause definition for the path constructor loop.",
"The code generator postulates the following computation rule βloop for the path constructor loop: βloop : (C : Set) → (cbase : C) → (cloop : cbase ≡ cbase) → ap (λ x → recS x C cbase cloop) loop ≡ cloop The application of function recS to the path loop substitutes the point base for the argument x and it evaluates to the path cloop in the output type C. In the tool, generateRecHit is used to build the elimination rule and the computation rules for points, and generateβRecHitPath is used to build the computation rules for paths (Figure REF ).",
"The third argument to generateRecHit is the base type's recursor built using generateβRec that constructs the computation rules for points using rewrite rules.",
"The parameter count is passed as the fourth argument."
],
[
"Dependent Eliminators for HITs",
"The dependent eliminator for a higher inductive type $G$ is a dependent function that maps an element $g$ of $G$ to an output type $C \\, g$ .",
"The general schema for the induction principle of $G$ is given in Figure REF .",
"Figure: NO_CAPTIONSimilar to $G_{rec}$ , the type of $f_i$ is built the same way as for the normal inductive type $D$ .",
"The code generator builds the type of the method for path constructor $p_i$ , called $k_i$ , in $G_{ind}$ , by traversing the AST of $p_i$ .",
"During the traversal, the code generator uses the base eliminator $D_{ind}$ to map the point constructors $g_i$ of $G$ in the codomain of $p_i$ to $f_i$ .",
"In the first argument to the identity type in the codomain of $k_i$ , the code generator adds an application of transport to the motive C and the path $p_i$ .",
"The arguments of $k_i$ are handled the same way as for $f_i$ .",
"The computation rules corresponding to paths $p_i$ are postulated as given in Figure REF .",
"For the type S with point constructor base and path constructor loop, to define a mapping indS : (x : S) → C x, we need cbase : C base and cloop : transport C loop cbase ≡ cbase, where cloop is a heterogeneous path transported over loop.",
"The code generator builds the type of cloop by adding relevant type information to the type of loop.",
"The type of the method for the path constructor cloop is derived by inserting a call to transport with arguments C, loop, and cbase.",
"The code generator applies the base eliminator to map the point base to cbase during the construction of the codomain of cloop.",
"The following declaration gives the type of indS.",
"indS : (circle : S) → (C : S → Set) → (cbase : C base) → (cloop : transport C loop cbase ≡ cbase) → C circle The computation rule for base, which defines the action of indS on base, is built using the same approach as for the non-dependent eliminator recS.",
"The postulated computation rule iβloop for the path loop uses apd which gives the action of dependent function indS on the path loop.",
"iβloop : (C : S → Set) → (cbase : C base) → (cloop : transport C loop cbase ≡ cbase) → apd (λ x → indS x C cbase cloop) loop ≡ cloop generateIndHit is used to build the elimination rule and the computation rules for points, and generateβIndHitPath is used to build the computation rules for paths (Figure REF )."
],
[
"Patch Theory Revisited",
"We reimplemented Angiuli et al.",
"'s patch theory [2] using our code generator in Agda.",
"We implemented basic patches such as the insertion of a string as line $l_1$ in a file and deletion of a line $l_2$ from a file.",
"The functions implementing insertion and deletion in the universe are not bijective.",
"So, we used Angiuli et al.",
"'s patch history approach to encode non-bijective functions.",
"According to this approach, we developed a separate higher inductive type History which serves as the types of patches.",
"We also implemented patches involving encryption or decryption with cryptosystems like RSA and Paillier.",
"In addition to easing the implementation difficulties of higher inductive types, the code generator greatly reduced the code size.",
"The type definitions shrank from around 1500 to around 70 lines, resulting in a 60% decrease in the overall number of lines of code in the development."
],
[
"Cryptographic Protocols",
"Vivekanandan [21] models certain cryptographic protocols using homotopy type theory, introducing a new approach to formally specifying cryptographic schemes using types.",
"The work discusses modeling cryptDB [3] using a framework similar to Angiuli et al.",
"'s patch theory.",
"CryptDB employs layered encryption techniques and homomorphic encryption.",
"We can implement cryptDB by modeling the database queries as paths in a higher inductive type and mapping the paths to the universe using singleton types [2].",
"The code generator can be applied to generate code for the higher inductive type representing cryptDB and its corresponding elimination and computation rules.",
"By using the code generator, we can decrease the length and increase the readability of the definitions, hopefully making it more accessible to the broad cryptographic community."
],
[
"Related Work",
"Kokke and Swierstra [4] implemented a library for proof search using Agda's old reflection primitives, from before it had elaborator reflection.",
"They describe a Prolog interpreter in the style of Stutterheim et al.",
"[18].",
"It employs a hint database and a customizable depth-first traversal, with lemmas to assist in the proof search.",
"Van der Walt and Swierstra [19] and van der Walt [20] discuss automating specific categories of proofs using proof by reflection.",
"A key component of this proof technique is a means for converting an expression into a quoted representation.",
"They automate this process, giving a user-defined datatype.",
"Van der Walt [20] also give an overview of Agda's old metaprogramming tools.",
"Datatype-generic programming [13][14][15] via universes allows defining a single function over an entire class of datatypes at once, saving developers the effort of implementing the operation for datatypes specific to their programs.",
"As Agda's reflection library evolves and the internal representation of datatypes changes, the tool described in this paper requires maintenance work.",
"A future direction would be to work with a universe extended with support for higher inductive types.",
"In such case, the only metaprograms necessary are those that convert to and from the universe.",
"The metaprograms automating the elimination rules do not need to change as long as the universe is kept the same.",
"Ongoing work on cubical type theories [26][25][24] provides a computational interpretation of univalence and HITs.",
"We strenuously hope that these systems quickly reach maturity, rendering our code generator obsolete.",
"In the meantime, however, these systems are not yet as mature as Agda."
],
[
"Conclusion and Future Work",
"We presented a code generator that generates the encodings of higher inductive types, developed using Agda's new support for Idris-style elaborator reflection.",
"In particular, the tool generates the dependent and non-dependent elimination rules and the computational rules for 1-dimensional higher inductive types.",
"This syntax is greatly simplified with respect to writing the encoding by hand.",
"We demonstrated an extensive reduction in code size by employing our tool.",
"Next, we intend to extend the tool to support higher-dimensional paths in the definition of HITs, bringing its benefits to a wider class of problems."
],
[
"Acknowledgements",
"The author is greatly indebted to David Christiansen for his contributions and advice, and the anonymous reviewers for their valuable review comments."
]
] | 1808.08330 |
[
[
"SYK/AdS duality with Yang-Baxter deformations"
],
[
"Abstract In this paper, based on the notion of SYK/AdS duality we explore the effects of Yang-Baxter (YB) deformations on the SYK spectrum at strong coupling.",
"In the first part of our analysis, we explore the consequences of YB deformations through the Kaluza-Klein (KK) reduction on $ (AdS_2)_{\\eta}\\times (S^1)/Z_2 $.",
"It turns out that the YB effects (on the SYK spectrum) starts showing off at \\textit{quadratic} order in $ 1/J $ expansion.",
"For the rest of the analysis, we provide an interpretation for the YB deformations in terms of bi-local/collective field excitations of the SYK model.",
"Using large $ N $ techniques, we evaluate the effective action upto quadratic order in the fluctuations and estimate $ 1/J^2 $ corrections to the correlation function at strong coupling."
],
[
"Overview and Motivation",
"Very recently, the SYK model [1]-[22] has been proposed as one of those handful examples (within the realm of AdS/CFT correspondence) where one might hope to solve the spectrum associated with the quantum mechanical system at strong couplingSee [23] for a nice comprehensive review.. Other than its solvability, this $ (1+0) $ D system of ($ N\\gg 1 $ ) fermions possesses two other remarkable features namely,-(I) the Lyapunov exponent associated with the out of time ordered four point correlators saturates the bound [24]-[26] and (II) the emerging scale invariance at low energies.",
"Understanding the dual gravitational counterpart [27]-[33] corresponding to the SYK model had always been challenging until very recently [34]-[36].",
"In [34], the authors provide a very solid evidence in favour of the dual 3D gravitational counterpart corresponding to the zero temperature version of the model where they explore the Kaluza-Klein (KK) tower associated with the scalar field excitations on $ AdS_2 \\times (S^1)/Z_2 $ .",
"In the strict IR ($ |J t |\\gg 1 $ ) limit, the metric along the compact third direction becomes constant, whereas on the other hand, away from the fixed point it acquires non trivial dependence on the $ AdS_2 $ coordinates through the dilaton profile.",
"In their analysis [34], the authors explore the $ 1/J $ corrections to the zero modes of the theory and compute the strong coupling corrections to the propagator that precisely matches to that with the earlier results in [8].",
"The purpose of the present paper is to explore and understand this duality conjecture in the presence of so called Yang-Baxter (YB) deformations [37]-[39] and also to understand its possible consequences on the corresponding bi-local excitations [13]-[14] associated with the SYK model at strong coupling.",
"The motivation behind our analysis strictly follows from holography where we start with the deformed $ AdS_2 $ version of the theory in the bulk [37] and lift it to three dimensions in order to compute the KK spectrum associated with the scalar field excitations in the dual gravitational counterpart.",
"Our analysis reveals that the holographic correspondence between the SYK model and its dual gravitational counterpart brings into a non trivial $ 1/J^2 $ corrections to the spectrum of the SYK model which has its origin in the YB deformations associated with the $ (AdS_2)_{\\eta }\\times (S^1)/Z_2 $ theory in ($ 2+1 $ ) D. For the rest of our analysis, we search for an interpretation of such deformations in terms of collective field excitations [40] associated with the SYK model.",
"Looking at the holographic side of the duality, we propose a possible YB scaling of the collective excitations within the SYK model and compute the effective action at quadratic order in the fluctuations (around IR critical point) and thereby the corresponding propagator at strong coupling ($ |Jt|\\gg 1 $ ).",
"Our analysis reveals that the effective action receives non trivial $ 1/J^2 $ corrections that has remarkable structural similarity to that with the corresponding quadratic action associated with scalar field excitations computed on the dual gravitational counterpart of the theory.",
"The organisation of the paper is as follows: In section 2, we briefly review the YB deformations and its implications on the Almheiri-Polchinski (AP) model [37]-[39].",
"In section 3, we compute the zero modes associated with the spectrum and estimate the $ 1/J^2 $ corrections to it.",
"In section 4, we provide a possible interpretation of the YB effects on the collective excitations within the SYK model.",
"Finally, we conclude in section 5."
],
[
"Basics",
"The primary motivation behind introducing the Yang-Baxter (YB) deformations (associated with non-linear $\\sigma $ -models, e.g; the Principal Chiral Model (PCM)) was the observation that the later is equivalent to a two-dimensional field theory (defined on some compact manifold $\\mathcal {M}$ ) embodied with a rank 2 symmetric tensor (metric) field ($ \\gamma ^{\\alpha \\beta } $ ) as well as an antisymmetric two-form.",
"These YB $\\sigma $ -models are characterized by some $\\mathbb {R}$ -linear operators [41], [42], [43] and seem to posses a left-symmetry together with the Poisson-Lie symmetry with respect to the right action of the group on itself.",
"These symmetries of the model could be associated with certain types of dualities embedded in its structureFor more details, the interested reader is encouraged to go through the references [41].. For any generic Lie group $\\mathcal {G}$ with Lie algebra $\\textsl {g}=Lie(\\mathcal {G})$ , the action corresponding to YB $\\sigma $ -models could be formally expressed asThe $ \\mathfrak {R} $ operator satisfies the so called Yang-Baxter (YB) equation (REF ) which we elaborate in the next secion in the context of deformations associated with the $ AdS_2 $ supercosets.",
"[41], [42], [43] , $S=-\\frac{1}{2}\\left(\\gamma ^{\\alpha \\beta }-\\epsilon ^{\\alpha \\beta }\\right) \\int _{-\\infty }^{\\infty } d\\tau \\int _{0}^{2\\pi }d\\sigma \\; \\text{Tr}\\left(J_{\\alpha }\\frac{1}{1-\\eta ~ \\mathfrak {R}_{\\mathfrak {g}}}J_{\\beta }\\right)$ where $(\\tau ,\\sigma )$ are the two-dimensional world-sheet coordinates together with the skew-symmetric tensor $\\epsilon ^{\\alpha \\beta }$ normalized as, $\\epsilon ^{\\tau \\sigma }=-\\epsilon ^{\\sigma \\tau }=1$ .",
"Here, $J_{\\alpha }=\\mathfrak {g}^{-1}\\partial _{\\alpha }\\mathfrak {g}$ is the left-invariant one-form expressed in terms of $\\mathfrak {g}(\\tau ,\\sigma )\\in \\mathcal {G}$ and the trace is defined over the fundamental representation of the algebra $\\textsl {g}$ .",
"It is indeed trivial to notice that in the limit of the vanishing ($\\eta \\rightarrow 0$ ) YB deformation one recovers the sigma model corresponding to that of the PCM.",
"As an additional fact, the YB deformed version of an integrable sigma model seems the preserve the integrable structure as wellThe $ AdS_2 $ supercoset model (that is realized as a solution in the framework of 2D dilaton gravity [27]-[33]) does not preserve integrability as the dual SYK turns out to be maximally chaotic.",
"As a consequence of this, the YB deformed version of it is not expected to be integrable as well..",
"The motivation behind introducing the YB deformations in the context of string sigma models stems from the fact that they play crucial role towards a profound understanding of the underlying dynamics in AdS/CFT correspondence.",
"It eventually includes a broader classes of stringy geometries [44] within the unified framework of gauge/string duality.",
"Referring back to the original Maldacena duality between type IIB super-strings propagating in $ AdS_5 \\times S^5 $ and that of $ \\mathcal {N}=4 $ SYM in 4D, the YB deformations applied to the corresponding string sigma model provide a non trivial generalization of the duality.",
"On the gauge theory side, these deformations could be realized as a $ q $ deformation of the (say for example $ SO(6) $ ) spin chain [45] which thereby preserves integrability like in the usual $ \\mathcal {N}=4 $ SYM [46].",
"On the other hand, on the gravity side one could generate a wider class of dual geometries depending on different types of solutions associated with the classical $ \\mathfrak {R} $ matrix [47]-[48].",
"Keeping the spirit of the above discussion, the purpose of the present paper is to generalize the notion of $ AdS_2/SYK $ duality [34] in the presence of YB deformations.",
"Very recently, the YB deformation of the Almheiri-Polchinski model has been constructed in [37] and the dual (deformed) SYK version of this gravity model is yet to be constructed.",
"The purpose of the present analysis is to fill up this gap and provide a systematic realization of the dual gauge theory at strong coupling."
],
[
"The deformed AP model",
"The purpose of this Section is to provide a brief introduction to the Almheiri-Polchinski (AP) model [29] and its Yang-Baxter (YB) form of deformations introduced very recently in [37].",
"The Yang-Baxter (YB) deformation of the $ AdS_2 $ metric is based on the usual notion of coset space formulation of the 2D non-linear sigma model that acts both on the metric as well as the anti-symmetric two form field.",
"For the metric part, the YB deformation was introduced as [37], $ds_{\\eta }^2 = 2 Tr\\left[ J \\frac{1}{1-2 \\eta \\mathfrak {R}_{\\mathfrak {g}}\\circ P}P(J)\\right] $ where, $ \\eta $ is the deformation parameter.",
"Here, the left invariant one form $ J(\\in \\mathfrak {sl}(2)) $ is defined as usual, $J= \\mathfrak {g}^{-1}d\\mathfrak {g}$ where $ \\mathfrak {g} $ is an element of $ SL(2)/U(1) $ .",
"The projection could be defined as follows[37], $P(X)=\\frac{Tr(X T_0)}{Tr(T_0 T_0)}T_0 +\\frac{Tr(X T_1)}{Tr(T_1 T_1)}T_1,~~X\\in \\mathfrak {sl}(2)$ together with the chain operation of the following form, $\\mathfrak {R}_{\\mathfrak {g}}(X)=\\mathfrak {g}^{-1}\\circ \\mathfrak {R}(\\mathfrak {g}X \\mathfrak {g}^{-1})\\circ \\mathfrak {g}$ where the linear operator $ \\mathfrak {R} $ satisfies the modified classical YB (mCYBE) equation of the following form, $[\\mathfrak {R}(X),\\mathfrak {R}(Y)]-\\mathfrak {R}([\\mathfrak {R}(X),Y]+[X,\\mathfrak {R}(Y)])=c.",
"[X,Y]$ where, $ c\\ne 0 $ for mCYBE and $ c=0 $ for homogeneous CYBE [37].",
"Taking the generators, $ T_I \\in \\mathfrak {so}(1,2) $ in the fundamental representation, $T_I \\sim \\sigma _I$ (where, $ \\sigma _I $ s are the Pauli matrices) a straightforward calculation of (REF ) reveals [37], $ds_{\\eta }^2 = \\mathcal {F}_{\\eta }(X.P)\\frac{-dt^2 +dz^2}{z^2}$ where, the function $ \\mathcal {F}_{\\eta }(X.P) $ could be formally expressed as, $\\mathcal {F}_{\\eta }(X.P)&=&\\frac{1}{1-\\eta ^2 (X.P)^2}\\nonumber \\\\X.P &=& \\frac{1}{z}(\\alpha +\\beta t +\\gamma (-t^2+z^2)).$ Here, $ \\alpha $ , $\\beta $ and $ \\gamma $ are three constant parameters of the theory that satisfy the following constraint, $\\beta ^2 +4 \\alpha \\gamma + 4c=0.$ Clearly in the limit, $ \\eta \\rightarrow 0 $ one recovers the usual $ AdS_2 $ metric as a part of the full background solutions corresponding to the AP model [29].",
"The major accomplishment of [37] was to embed the above YB deformed $ AdS_2 $ metric (REF ) as a solution within $ 1+1 $ D dilaton gravity systemIn their analysis, the authors set the matter part of the Lagrangian equal to zero [37]., $S^{(\\eta )}_{g}=\\frac{1}{16 \\pi G}\\int d^2x \\sqrt{-g}(\\Phi _{\\eta }^{2}R-U(\\Phi ^{2}_{\\eta }))$ with a particular type of $ \\eta $ modification introduced to the dilaton potential ($ U(\\Phi ) $ ) that drives the potential from its standard quadratic form to a hyperbolic function.",
"The resulting dilaton profile could be formally expressed as [37], $\\Phi _{\\eta }^{2}=\\frac{1}{2 \\eta }\\log \\left[\\frac{1+ \\eta (X.P)}{1-\\eta (X.P)} \\right]+1.$"
],
[
"3D holography",
"In this section we intend to build up a notion for 3D holography in the presence of YB deformations described above.",
"The first step towards this direction would be to uplift the deformed AP model (REF ) in one higher dimension by introducing an additional compact direction ($ \\Theta $ ) and compute the corresponding scalar spectrum associated with Kaluza-Klein (KK) modes [34]."
],
[
"A 3D uplift",
"We propose the 3D metric of the following form, $d\\mathsf {S}_{\\eta }^2 = \\mathsf {G}_{MN}dX^{M}dX^{N}=\\mathcal {F}_{\\eta }(X.P)\\frac{-dt^2 +dz^2}{z^2}+\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }d\\Theta ^{2}.$ A straightforward computation reveals, $\\frac{1}{16 \\pi G^{(3)}}\\int d^3 x\\sqrt{-\\mathsf {G}} \\mathsf {R}^{(3)}&=&\\frac{\\Sigma _{I}}{16 \\pi G^{(3)}}\\int d^2x\\sqrt{-g}\\sqrt{\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }}\\left( R +\\frac{z^2 \\mathcal {I}^{(\\eta )}}{4 \\mathcal {F}_{\\eta }(X.P)\\mathsf {G}^{(\\eta )2}_{\\Theta \\Theta }}\\right)\\nonumber \\\\&=& \\frac{\\Sigma _{I}}{16 \\pi G^{(3)}}\\int d^2x\\sqrt{-g}\\sqrt{\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }}\\left( R +\\varphi _{\\eta }\\right)$ whereHere, $ \\square ^{(2)}=-\\partial ^{2}_{t}+\\partial ^{2}_{z} $ is the 2D Laplacian., $\\mathcal {I}^{(\\eta )}&=& (\\partial _t \\mathsf {G}^{(\\eta )}_{\\Theta \\Theta })^{2}-(\\partial _z \\mathsf {G}^{(\\eta )}_{\\Theta \\Theta })^{2}+2\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }(-\\partial ^{2}_{t}+\\partial _z^{2})\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }\\nonumber \\\\&=& (\\partial _t \\mathsf {G}^{(\\eta )}_{\\Theta \\Theta })^{2}-(\\partial _z \\mathsf {G}^{(\\eta )}_{\\Theta \\Theta })^{2}+2\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }\\square ^{(2)}\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }$ and $ \\Sigma _{I} $ is the corresponding volume of the compact manifold.",
"As a trivial check of our analysis, we first notice that in the limit, $ \\eta \\rightarrow 0 $ and with the following choice of the metric, $\\mathsf {G}^{(\\eta =0)}_{\\Theta \\Theta }=\\left(\\frac{\\alpha +\\gamma \\left(z^2-t^2\\right)+\\beta t}{z}+1\\right)^2$ one correctly reproduces the desired form of the dilaton potential [37], $\\sqrt{\\mathsf {G}^{(\\eta =0)}_{\\Theta \\Theta }}\\varphi _{\\eta =0}=-U(\\Phi ^{2}_{\\eta =0} )=2(\\Phi ^{2}_{\\eta =0}-1).$ The above scenario generalizes quite non-trivially for non zero deformations, $\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }=\\left(\\frac{\\log \\left(\\frac{1+\\eta (X.P)}{1-\\eta (X.P)}\\right)}{2 \\eta }+1\\right)^2$ that reproduces the desired form of the dilaton potential [37], $\\sqrt{\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }}\\varphi _{\\eta }=-U(\\Phi ^{2}_{\\eta } ).$ Therefore like in the undeformed scenario, the YB deformed ($ 1+1 $ ) D dilaton gravity could be uplifted to $ (2+1) $ D with dilaton being the third direction.",
"The above background (REF ) would serve as the starting point of our subsequent analysis."
],
[
"Kaluza-Klein modes",
"The purpose of this Section is to obtain the KK spectrum of a single scalar field ($ \\phi $ ) over the $ \\eta $ deformed background (REF ) obtained previously.",
"We start with the scalar action of the following form, $S_{\\phi }=\\frac{1}{2}\\int d^3 x\\sqrt{-\\mathsf {G}}\\left[ -\\mathsf {G}^{MN}\\partial _M \\phi \\partial _N \\phi - m^{2}\\phi ^2 - V(\\Theta )\\phi ^2\\right]$ where, $ V(\\Theta )=V\\delta (\\Theta ) $ is the delta function potential as usual [34].",
"In the subsequent analysis we focus on the case with homogeneous CYBE which amounts of setting [37], $X.P = \\frac{\\alpha }{z},~~X^2=-1,~~P^2 =0$ and simplifies the background metric (REF ) as, $\\begin{split}d\\mathsf {S}_{\\eta }^2 &= \\mathsf {G}_{MN}dX^{M}dX^{N}=\\mathcal {F}_{\\eta }(z)\\frac{-dt^2 +dz^2}{z^2}+\\mathsf {G}^{(\\eta )}_{\\Theta \\Theta }(z)d\\Theta ^{2} \\\\\\mathcal {F}_{\\eta }(z) &=\\frac{1}{1-\\frac{\\eta ^2 \\alpha ^2}{z^2}}.\\end{split}$ Before we proceed further, a few important remarks regarding the $ \\eta $ deformed background (REF ) are in order.",
"One should notice that unlike the case for the usual $ AdS_2 $ background, the $ \\eta $ deformed background (REF ) exhibits a metric singularity (associated with the function $ \\mathcal {F}_{\\eta }(z) $ ) at a finite radial distance, $|z_B| =\\eta ~\\alpha \\equiv \\frac{\\eta }{J}$ which thereby naturally constraints our bulk calculations within this (radial) cutoff.",
"This turns out to be a very special feature for spacetimes associated with YB deformations where one imagines putting the so called singularity surface namely the holographic screen [49] at a finite radial distance $ z=z_B $ that eventually acts a boundary for the bulk spacetime.",
"The boundary theory is therefore considered to be living on this holographic screen.",
"Following the prescription of Gauge/Gravity duality, one could thereby imagine the above entity (REF ) as being the energy scale associated with the holographic RG flow.",
"Depending on the (radial) location of the holographic screen one is supposed to probe the physics associated with the dual field theory under the RG flow.",
"The UV fixed point of this RG flow is given by the condition, $| \\frac{\\eta }{J}| \\sim 0$ for which one recovers the metric corresponding to $ AdS_2 \\times S^1 /Z_2 $ .",
"We now move on to the other end point of the RG flow where we set, $ |\\frac{\\eta }{J}|=\\Lambda _{IR}\\gg 1 $ where $ \\Lambda _{IR} $ could be thought of as being that of the deep IR cutoff.",
"In this limit, a careful analysis reveals the 2D bulk metric of the following formFor the moment, we ignore the (compact) third direction as it becomes trivial near both the fixed points., $d\\mathsf {S}^{2}_{\\eta }\\Big |_{2D}\\approx \\frac{1}{|\\varepsilon |^{2\\varsigma }}(-d T^2 + d\\varepsilon ^{2})$ where, we have introduced the following change of variables, $\\varepsilon =1-\\frac{|\\Lambda _{IR}|}{z},~~ T=\\frac{t}{|\\Lambda _{IR}|},~~|\\varepsilon | \\ll 1$ together with the fact that the value corresponding to the dynamical critical exponent, $ \\varsigma =\\frac{1}{2} $ .",
"Therefore, in summary, the theory flows from a $ \\varsigma =1 $ UV conformal fixed point to a $ \\varsigma =\\frac{1}{2} $ Lifshitz fixed point in the deep IR.",
"In the following we (Fourier) decompose the scalar field as, $\\phi = \\int \\frac{dw}{2 \\pi }e^{-i w t}\\xi _{w}(z,\\Theta )$ which finally yields the scalar action of the following form, $S_{\\phi }=\\frac{1}{2}\\int dz d \\Theta \\int \\frac{dw}{2\\pi }\\xi _{-w}\\left( \\mathfrak {D}_{0}+\\mathfrak {D}^{(\\eta )} \\right)\\xi _{w}$ where, the individual operators could be formally expressed asKeeping the spirit of the earlier analysis [34], we have ignored all the higher order contributions beyond $ \\mathcal {O}(\\alpha ^2 \\eta ^2) $ ., $\\mathfrak {D}_{0}&=\\partial _{z}^{2}+\\omega ^{2}-\\frac{m^{2}}{z^{2}}-\\frac{1}{z^{2}}\\left(-\\partial _{\\Theta }^{2}+V(\\Theta ) \\right) \\\\\\mathfrak {D}^{(\\eta )}&=\\frac{\\alpha }{z}\\left[ \\partial _{z}^{2}+\\omega ^{2}-\\frac{1}{z}\\partial _{z} - \\frac{m^{2}}{z^{2}}\\left( 1+\\frac{\\alpha \\eta ^{2}}{z} \\right)\\right]+\\sum _{n=1,2} \\left(-1\\right)^{n}\\frac{\\alpha ^{n}}{z^{n+2}} \\mathsf {F}_{n}.$ Notice that like in the undeformed scenario [34], the zeroth order operator $ \\mathfrak {D}_{0} $ (and hence the associated Green's function) does not receive any $ \\eta $ corrections.",
"However, the operator at next to leading order modifies substantially due to YB deformations, $\\mathsf {F}_{n}= \\left(1+ \\eta ^{2(n-1)} \\right)\\partial _{\\Theta }^{2}+(-1)^{n-1}\\eta ^{2(n-1)}V(\\Theta ).",
"$ For $ n=1 $ one has the usual contribution as in the undeformed case.",
"Quite interestingly, the $ \\eta $ modification appears with, $ n=2 $ which is an effect associated to $ \\mathcal {O}(\\alpha ^2) $ .",
"Therefore, strictly at linear order in $ \\alpha (\\sim 1/J )$ [34] one should not expect to find any of the imprints of YB deformations on the SYK spectrum."
],
[
"Eigenfunctions of $ \\mathfrak {D}_{0} $",
"The eigenfunctions corresponding to the operator $ \\mathfrak {D}_{0} $ are clearly separable, $\\xi _{w}(z,\\Theta )=\\mathcal {Z}_{w}(z)\\mathfrak {f}_{\\mathfrak {K}}(\\Theta )$ where the function $ \\mathfrak {f}_{\\mathfrak {K}}(\\Theta )$ satisfies Schrodinger equation of the following form, $(-\\partial ^{2}_{\\Theta }+V \\delta (\\Theta ))\\mathfrak {f}_{\\mathfrak {K}}(\\Theta )=\\mathsf {E}\\mathfrak {f}_{\\mathfrak {K}}(\\Theta ).$ In our calculations, we stick to the parity even sector [34] of the wave function, $\\mathfrak {f}_{\\mathfrak {K}}(\\Theta )\\sim {\\left\\lbrace \\begin{array}{ll}\\mathsf {B} \\sin (\\mathfrak {K}(\\Theta -L)), &~(\\ 0<\\Theta <L) \\\\-\\mathsf {B} \\sin (\\mathfrak {K}(\\Theta +L)), & ~ (-L<\\Theta <0)\\end{array}\\right.",
"}$ with, $ \\mathsf {E}=\\mathfrak {K}^2 $ .",
"Clearly, with the above choice of the wave function (REF ) one ends up with the following boundary conditionsFor the moment we ignore the overall normalization constant., $\\begin{split}\\mathfrak {f}_{\\mathfrak {K}}(L)&=0=\\mathfrak {f}_{\\mathfrak {K}}(-L) \\\\\\mathfrak {f}_{\\mathfrak {K}}(+0)&=-\\sin (\\mathfrak {K}L)=\\mathfrak {f}_{\\mathfrak {K}}(-0).\\end{split}$ Next, integrating (REF ) within a small interval around $ \\Theta \\sim 0 $ we notice, $-\\partial _{\\Theta }\\mathfrak {f}_{\\mathfrak {K}}|_{-0}^{+0}+V \\mathfrak {f}_{\\mathfrak {K}}(0)=\\mathsf {E}(\\mathfrak {f}_{\\mathfrak {K}}(+0)-\\mathfrak {f}_{\\mathfrak {K}}(-0))$ which upon substitution of (REF ) yields the following transcendental equation, $-\\frac{2}{V}\\mathfrak {K}=\\tan (\\mathfrak {K}L)$ whose solutions could be formally denoted as $ \\mathsf {p}_a $ with $ a (=0,1,2,..)$ being an integer and $ 2a+1<\\mathsf {p}_a <2a+2 $ [34]."
],
[
"Eigenfunctions of $ \\mathfrak {D}^{(\\eta )} $",
"Likewise, we separate the eigenfunctions of the operator $\\mathfrak {D}^{(\\eta )}$ as, $\\chi _{\\omega }(z,\\Theta )=\\chi _{\\omega }(z)\\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{n}}(\\Theta )$ where, $\\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{n}}$ is the eigenfunction corresponding to the operator $ \\mathsf {F}_{n} $ defined in (REF ), $-\\mathsf {F}_1 \\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{1}}(\\Theta ) = \\tilde{\\mathsf {E}}_{1}\\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{1}}(\\Theta )\\nonumber \\\\\\mathsf {F}_2 \\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{2}}(\\Theta ) = \\tilde{\\mathsf {E}}_{2}\\mathfrak {f}_{\\tilde{\\mathfrak {K}}_{2}}(\\Theta ).$ Restricting ourselves to the parity even sector of the wave function (REF ) it is trivial to arrive at the following set of transcendental equations, $\\frac{2}{V}\\tilde{\\mathfrak {K}}_{1}&=&\\tan (\\tilde{\\mathfrak {K}}_{1}L)\\\\-\\frac{2}{V}\\tilde{\\mathfrak {K}}_{2}&=&\\frac{\\eta ^{2}}{1+\\eta ^{2}}\\tan (\\tilde{\\mathfrak {K}}_{2}L).$ Figure: Energy eigenvalue spectrum with V=3 V=3 and L=π 2 L=\\frac{\\pi }{2} .",
"The blue curve corresponds to eigenvalues (𝗊 1 (2) ) 2 ( \\mathsf {q}^{(2)}_{1}) ^2 and the red curve corresponds to eigenvalues (𝗊 0 (2) ) 2 ( \\mathsf {q}^{(2)}_{0}) ^2 .Notice that the second Eq.",
"() makes sense only in the presence of YB deformations ($ \\eta \\ne 0 $ ).",
"As we shall see, the eigenfunctions corresponding to $ \\mathsf {F}_{2} $ introduce a non trivial shift in the SYK spectrum at strong couplings.",
"We denote the solutions corresponding to the above set of equations (REF )-() as $ 2a+1<\\mathsf {q}^{(n)}_{a}<2a+2 $ where, $ n =1,2 $ refers to $ \\mathsf {F}_{1} $ and $ \\mathsf {F}_{2} $ respectively.",
"We focus our attention on the first two roots ($ 1<\\mathsf {q}^{(2)}_{0}<2 $ and $ 3<\\mathsf {q}^{(2)}_{1}<4 $ ) corresponding to $ \\mathsf {F}_{2} $ and explore their functional dependence with the deformation parameter ($ \\eta $ ).",
"To see this explicitly we plot the energy eigenvalues $ \\tilde{\\mathsf {E}}^{(a)}_{2}(\\sim (\\mathsf {q}^{(2)}_{a})^2 $ ) against the YB parameter ($ \\eta $ ) (see, Fig.",
"(REF )) which yields the following functional dependencies, $\\tilde{\\mathsf {E}}^{(0)}_{2} \\sim ( \\mathsf {q}^{(2)}_{0}) ^2\\approx \\mathcal {N}+\\eta ^{4/5},~~\\mathcal {N}=0.778626$ $\\tilde{\\mathsf {E}}^{(1)}_{2} \\sim ( \\mathsf {q}^{(2)}_{1}) ^2\\approx \\sum _{m=0}^{m=4}(-1)^m\\mathcal {C}_{m}\\eta ^{m},~~\\mathcal {C}_{m}\\in \\mathbb {R}.$ Finally, substituting these eigenvalues into (REF ) we obtain, $\\mathfrak {D}_{0}& = \\partial ^{2}_{z}+ w^2 -\\frac{(m^{2}+\\mathsf {p}^2_{a})}{z^2} , \\\\\\mathfrak {D}^{(\\eta )}& =\\frac{\\alpha }{z} \\left[\\partial ^2_z + w^{2} -\\frac{1}{z} \\partial _z-\\sum _{n=1,2}\\frac{\\alpha ^{n-1}}{z^{n+1}}\\left( m^{2}\\eta ^{2(n-1)}-\\mathsf {q}^{2(n)}_{a}\\right) \\right].",
"$"
],
[
"Green's functions",
"Given the desired choices for the parameters ($ V,L $ ) of the theory, we now compute the Green's function corresponding to the operator, $ \\mathfrak {D}\\equiv \\mathfrak {D}_{0}+\\mathfrak {D}^{(\\eta )} $ .",
"It is naturally expected that the effects of YB deformations on the SYK spectrum would appear through the Green's function corresponding to the operator $ \\mathfrak {D}^{(\\eta )} $ ."
],
[
"Zeroth order solution",
"The solution corresponding to the zeroth order propagator, $\\mathfrak {D}_{0}\\mathcal {G}^{(0)}_{w,w^{\\prime }}(z,z^{\\prime };\\Theta , \\Theta ^{\\prime })=-\\delta (z-z^{\\prime })\\delta (\\Theta - \\Theta ^{\\prime })\\delta (w+w^{\\prime })$ remains the same as in the undeformed scenarioSee Appendix A for details.[34].",
"Setting, $ \\Theta =0=\\Theta ^{\\prime } $ into (REF ) one finally obtains, $\\mathcal {G}^{(0)}(t,z ;t^{\\prime },z^{\\prime })=-|zz^{\\prime }|^{1/2}\\sum _{a=0}^{\\infty }|\\mathfrak {f}_{\\mathsf {p}_{a}}(0)|^2\\int \\frac{dw}{2 \\pi }e^{-iw(t-t^{\\prime })}\\mathcal {I}(z,z^{\\prime })$ where we note, $|\\mathfrak {f}_{\\mathsf {p}_{a}}(0)|^2 &=& \\mathsf {B}_{a}^{2}\\sin ^2(\\mathfrak {K}L)=\\frac{ \\mathsf {B}_{a}^{2}\\mathsf {p}^{2}_{a}}{(3/2)^2 +\\mathsf {p}^2_{a}}=\\frac{2 \\mathsf {p}_a}{3}\\mathsf {R}(\\mathsf {p}_{a})\\nonumber \\\\\\mathsf {R}(\\mathsf {p}_{a})&=&\\frac{3 \\mathsf {p}^{2}_{a}}{((3/2)^2 +\\mathsf {p}^2_{a})(\\pi \\mathsf {p}_{a}-\\sin (\\pi \\mathsf {p}_{a}))}.$ Setting, $ m^2 =-1/4 $ the integral $ \\mathcal {I}(z,z^{\\prime })$ could be expressed as a sum of two terms[34], $\\mathcal {I}_{1}=\\sum _{n=0}^{\\infty }\\frac{2 \\nu }{\\nu ^2 -\\mathsf {p}^{2}_{a}}J_{\\nu }(|wz|)J_{\\nu }(|wz^{\\prime }|)\\big |_{\\nu =3/2+2n}$ where, the index $ \\nu $ is real.",
"The above entity (REF ) has discrete poles at $ \\nu =\\mathsf {p}_{a} $ (on real $ \\nu $ axis) and is clearly valid for $ \\nu = 3/2+2n ~(n=0,1,2...) $ for which $ \\xi _{\\nu }=0 $ [7].",
"This corresponds to bound energy (eigen)states with eigenvalues precisely as the roots of the transcendental equation (REF ).",
"The second contribution to the integral $ \\mathcal {I}(z,z^{\\prime })$ comes from the scattering states [7] with $ \\nu =ir~ (0<r<\\infty ) $ that amount of settingThe resulting wave function is a linear superposition of these two states [7]., $\\mathcal {I}_{2}=-\\frac{i}{2}\\int _{0}^{i \\infty }\\frac{d \\nu }{\\sin (\\pi \\nu )}\\frac{\\nu }{\\nu ^2 - \\mathsf {p}^{2}_{a}}(\\mathcal {Q}_{\\nu }(z,z^{\\prime })+\\tilde{\\mathcal {Q}}_{\\nu }(z,z^{\\prime }))$ where, the individual entities in the integrand (REF ) could be formally expressed as, $\\mathcal {Q}_{\\nu }(z,z^{\\prime })=(J_{-\\nu }(|wz|)+\\xi _{-\\nu }J_{\\nu }(|wz|))J_{\\nu }(|wz^{\\prime }|)\\nonumber \\\\\\tilde{\\mathcal {Q}}_{\\nu }(z,z^{\\prime })=(J_{\\nu }(|wz|)+\\xi _{\\nu }J_{-\\nu }(|wz|))J_{-\\nu }(|wz^{\\prime }|).$ A trivial change in the variable, $ \\nu \\rightarrow -\\nu $ yields, $\\frac{i}{2}\\int _{0}^{-i \\infty }\\frac{d \\nu }{\\sin (\\pi \\nu )}\\frac{\\nu }{\\nu ^2 - \\mathsf {p}^{2}_{a}}\\tilde{\\mathcal {Q}}_{-\\nu }(z,z^{\\prime })=-\\frac{i}{2}\\int _{-i \\infty }^{0}\\frac{d \\nu }{\\sin (\\pi \\nu )}\\frac{\\nu }{\\nu ^2 - \\mathsf {p}^{2}_{a}}\\mathcal {Q}_{\\nu }(z,z^{\\prime })$ which thereby simplifies (REF ) as, $\\mathcal {I}_{2}=-\\frac{i}{2}\\int _{-i \\infty }^{i \\infty }\\frac{d \\nu }{\\sin (\\pi \\nu )}\\frac{\\nu }{\\nu ^2 - \\mathsf {p}^{2}_{a}}\\mathcal {Q}_{\\nu }(z,z^{\\prime };z>z^{\\prime }).$ We close the contour on the $ Re(\\nu )>0 $ plane which essentially extends along the positive real axis of the complex $ \\nu $ plane and extends to infinity along the imaginary axis.",
"Clearly there are two types of poles within this contour- (I) simple pole at $ \\nu =\\mathsf {p}_{a} $ and (II) the poles (distributed along the real axis) associated with $ \\xi _{-\\nu } $ at $ \\nu =3/2+2n $ , $\\mathcal {I}_{2}=-\\frac{\\pi }{2\\sin (\\pi \\mathsf {p}_{a})}\\mathcal {Q}_{\\nu =\\mathsf {p}_{a}}(z,z^{\\prime };z>z^{\\prime })-\\mathcal {I}_{1}.$ Combining (REF ) and (REF ) finally yields, $\\mathcal {G}^{(0)}(t,z ;t^{\\prime },z^{\\prime })=\\frac{|zz^{\\prime }|^{1/2}}{4}\\sum _{a=0}^{\\infty }\\frac{|\\mathfrak {f}_{\\mathsf {p}_{a}}(0)|^2}{\\sin (\\pi \\mathsf {p}_{a})}\\int dw e^{-iw(t-t^{\\prime })}\\mathcal {Q}_{\\nu =\\mathsf {p}_{a}}(z,z^{\\prime };z>z^{\\prime }).$"
],
[
"YB shift in the spectrum",
"The purpose of this Section is to compute the first order shift in the energy spectrum corresponding to $ \\mathfrak {D}^{(\\eta )} $ and explore the effects of YB deformations at next to leading order in the SYK coupling ($ \\alpha \\sim 1/J$ ).",
"A straightforward calculation reveals, $\\mathfrak {D}^{(\\eta )}\\sqrt{z}\\mathcal {Z}_{\\nu }(|wz|)=\\frac{\\alpha }{\\sqrt{z}}\\left( -\\frac{\\partial _{z}}{z}-\\sum _{n=1,2}\\frac{\\alpha ^{n-1}}{z^{n+1}}\\Delta ^{(n)}\\right) \\mathcal {Z}_{\\nu }(|wz|).$ where, we have introduced, $\\Delta ^{(n)}=\\left(\\frac{3}{4}-\\nu ^2 \\right) \\delta _{n,1}+m^2 \\eta ^{2(n-1)}-\\mathsf {q}^{2(n)}_{a}.$ In order to simplify (REF ), we further notice that, $\\partial _{z}\\mathcal {Z}_{\\nu }(|wz|)=\\partial _{z}J_{\\nu }(|w z|)+\\xi _{\\nu }\\partial _{z}J_{-\\nu }(|w z|)$ which by means of the following two identities, $\\partial _z J_{\\nu }(z)&=&\\frac{\\nu }{z}J_{\\nu }(z)-J_{\\nu +1}(z)\\nonumber \\\\\\partial _z J_{\\nu }(z)&=&-\\frac{\\nu }{z}J_{\\nu }(z)+J_{\\nu -1}(z)$ could be further simplified asOne has to replace, $ \\nu \\rightarrow -\\nu $ in the second of the identities in (REF )., $\\partial _{z}\\mathcal {Z}_{\\nu }(|wz|)=\\frac{\\nu }{|z|}\\mathcal {Z}_{\\nu }(|w z|)-|w| (J_{\\nu +1}(|wz|)-\\xi _{\\nu }J_{-\\nu -1}(|wz|)).$ Substituting (REF ) into (REF ) we find, $\\mathfrak {D}^{(\\eta )}\\sqrt{z}\\mathcal {Z}_{\\nu }(|wz|)=-\\frac{\\alpha }{\\sqrt{z}}\\left( \\frac{\\nu }{z^2}+\\sum _{n=1,2}\\frac{\\alpha ^{n-1}}{z^{n+1}}\\Delta ^{(n)}\\right) \\mathcal {Z}_{\\nu }(|wz|)\\nonumber \\\\+\\frac{\\alpha }{z^{3/2}}|w|(J_{\\nu +1}(|wz|)-\\xi _{\\nu }J_{-\\nu -1}(|wz|))$ which finally yields the matrix element of the following form, $\\int _{0}^{\\infty }dz\\sqrt{z}\\mathcal {Z}_{\\nu ^{\\prime }}^{\\ast }(|wz|)\\mathfrak {D}^{(\\eta )}\\sqrt{z}\\mathcal {Z}_{\\nu }(|wz|)=<\\mathsf {D}^{(\\eta )}_{1}>+<\\mathsf {D}^{(\\eta )}_{2}>$ where the individual matrix elements could be formally expressed as, $<\\mathsf {D}^{(\\eta )}_{1}>=-\\alpha (\\nu +\\Delta ^{(1)})\\int _{0}^{\\infty }\\frac{dz}{z^2}\\mathcal {Z}_{\\nu ^{\\prime }}^{\\ast }(|wz|)\\mathcal {Z}_{\\nu }(|wz|)\\nonumber \\\\-\\alpha ^{2}\\Delta ^{(2)}\\int _{0}^{\\infty }\\frac{dz}{z^3}\\mathcal {Z}_{\\nu ^{\\prime }}^{\\ast }(|wz|)\\mathcal {Z}_{\\nu }(|wz|)$ $<\\mathsf {D}^{(\\eta )}_{2}>=\\alpha |w|\\int _{0}^{\\infty }\\frac{dz}{z}\\mathcal {Z}_{\\nu ^{\\prime }}^{\\ast }(|wz|)(J_{\\nu +1}(|wz|)-\\xi _{\\nu }J_{-\\nu -1}(|wz|)).$ In order to evaluate the above integrals (REF ) we focus on the bound states with discrete energy eigenvalues corresponding to $ \\nu =3/2+2n $ [7].",
"This yields the followingWe have set, $ m^2 =-1/4 $ which corresponds to setting the mass of the KK scalar at its BF bound.",
"Also we have set, $ \\nu ^{\\prime } =\\nu =3/2 $ .",
"This corresponds to the zero modes of the spectrum., $<\\mathsf {D}^{(\\eta )}>=\\int _{0}^{\\infty }dz\\sqrt{z}\\mathcal {Z}_{\\nu ^{\\prime }}^{\\ast }(|wz|)\\mathfrak {D}^{(\\eta )}\\sqrt{z}\\mathcal {Z}_{\\nu }(|wz|)=\\frac{\\alpha |w|}{2 \\pi } (5/4+(\\mathsf {q}^{(1)}_{0})^{2})+\\Delta k_{YB}.\\nonumber \\\\$ The second term on the R.H.S.",
"of (REF ) could be repackaged as, $\\Delta k_{YB} = \\frac{\\alpha ^2 w^{2}}{30}(\\eta ^2 +4(\\mathcal {N}+\\eta ^{4/5})^{2})\\sim 1/J^2$ is precisely the YB contribution to the SYK spectrum that appears as a next to leading order effect in the SYK coupling ($ \\sim 1/J $ ).",
"Notice that, here $ \\Delta k_{YB} (\\ge 0)$ is manifestly positive definite (as a leading order effect in YB deformations) that shifts the pole of the propagator corresponding to the zero mode by an amount, $\\nu ^2 =\\left( \\frac{3}{2}\\right)^{2}+ <\\mathsf {D}^{(\\eta )}>$ which could be further expanded as a perturbation in the SYK coupling, $\\nu = \\frac{3}{2}+ \\frac{|w|}{6\\pi J}\\Delta \\nu _{(1)}+\\frac{w^2}{90 J^2}\\Delta \\nu _{(2)}+\\mathcal {O}(1/J^3).$ Here, we have introduced new variables as, $\\Delta \\nu _{(1)}&=&5/4+(\\mathsf {q}^{(1)}_{0})^{2}\\nonumber \\\\\\Delta \\nu _{(2)}&=&\\frac{30 J^2}{w^2}\\Delta k_{YB}-\\frac{5}{6 \\pi ^2}\\Delta \\nu _{(1)}^{2}.$ Notice that the shift $ \\Delta \\nu _{(2)} $ is purely a next to leading order ($ \\sim 1/J^2 $ ) contribution to the spectrum that comes into play in the presence of YB deformations which forces us to consider effects beyond linear order in the perturbation expansion.",
"This finally yields the zero mode contribution to the propagator (REF ) as, $\\Delta \\mathcal {G}^{(0)}(t,z ;t^{\\prime },z^{\\prime })&=&-\\frac{9 \\pi J \\mathsf {B}^{2}_{0}}{4 }\\frac{|zz^{\\prime }|^{1/2}}{\\Delta \\nu _{(1)}}\\int _{-\\infty }^{\\infty } \\frac{dw}{|w|}\\Omega ^{-1}(J,w, \\eta ) e^{-i|w|(t-t^{\\prime })}J_{3/2}(|wz|)J_{3/2}(|w z^{\\prime }|)\\nonumber \\\\\\Omega (J,w, \\eta )&=&1+\\frac{\\pi w}{15 J}\\frac{\\Delta \\nu _{(2)}}{•\\Delta \\nu _{(1)}}.$ Clearly, if one switches off the YB deformation and restricts upto leading order ($ \\sim 1/J $ ) in the perturbation series then, $\\Omega =1 $ which reproduces the known results of [8]."
],
[
"Bi-local holography and YB deformations",
"From the analysis in the previous section, it is indeed quite evident that there are precise holographic confirmations of the perturbative energy shift in the SYK model due to the presence of YB deformations in the bulk counterpart.",
"The purpose of this section is therefore to understand this deformation and in particular its consequences on the ($ 1+0 $ )D SYK model in terms of bi-local excitations [13] of the theory."
],
[
"Brief review of SYK",
"The SYK model consists of $N(\\gg 1)$ Majorana fermions with all-to-all interactions.",
"This is a quantum mechanical model where the interactions between the fermions are completely random, described by a random coupling constant $J_{ijkl}$ which exhibits a Gaussian distribution with zero mean ($ \\langle J_{ijkl} \\rangle =0$ ) and a non zero variance $\\langle J_{ijkl}J_{ijkl} \\rangle = \\frac{6 J^{2}}{N^{3}}$ [7].",
"After performing the disorder averaging, there is only one coupling constant left in the theory [7], $J^{2}=\\frac{N^{3}}{3!",
"}\\overline{J^{2}_{ijkl}}$ that appears in the effective action.",
"The Hamiltonian of the system could be formally expressed asThe generalization of this model for $ q $ point vertex is quite straightforward [8].",
"[7], $\\mathcal {H}_{\\text{SYK}}=\\frac{1}{4!",
"}\\sum _{i,j,k,l=1}^{N}J_{ijkl}\\chi _{i}\\chi _{j}\\chi _{k}\\chi _{l}$ which leads to the Lagrangian of the following form, $\\mathcal {L}_{SYK}= -\\frac{1}{2!",
"}\\sum _{i=1}^{N}\\chi _{i}\\partial _{\\tau }\\chi _{i} -\\frac{1}{4!",
"}\\sum _{i,j,k,l=1}^{N}J_{ijkl}\\chi _{i}\\chi _{j}\\chi _{k}\\chi _{l}.$ Here, $\\chi _{i}$ s ($i=1,2,\\cdots ,N$ ) are the so called Majorana fermions which satisfy the following anti-commutation relation, $\\left\\lbrace \\chi _{i}, \\, \\chi _{j} \\right\\rbrace = \\delta _{ij}.$ together with the fact that they are equal to their own antiparticles namely, $\\chi _{i}^{\\dagger }=\\chi _{i}$ .",
"One can usually calculate the free energy of the system by using the so called replica trickNotice that the replica trick is usually used in systems with quenched disorder in order to overcome the averaging over logarithms.",
"Moreover, there is an amount of ambiguity in taking the limit $n\\rightarrow 0$ since we start from an integer $n$ .",
"Nevertheless, the results with replica method matches well with other methods., $\\langle \\ln \\mathcal {Z} \\rangle _{J} = \\lim _{n\\rightarrow 0}\\frac{\\langle \\mathcal {Z}^{n}\\rangle _{J}-1}{n};\\qquad n\\in \\mathbb {Z}$ where, the partition function could be formally expressed as, $\\langle \\mathcal {Z}^{n}\\rangle _{J}=\\int \\mathcal {D}\\chi _i ~ T( e^{\\mathcal {S}^{(R)}}).$ The action corresponding to the replica SYK is given byThe effective replica action in (REF ) could be obtained after integrating over the random coupling, $ J_{ijkl} $ in the path integral [13].",
"The integral that one evaluates is Gaussian and is of the form, $ \\sim \\int \\mathcal {D}\\chi _i (..)\\int \\prod _{i,j,k,l=1}^{N} dJ_{ijkl}~e^{-N^3 J^2_{ijkl}/12 J^2} e^{\\frac{1}{4!",
"}J_{ijkl}\\chi _i \\chi _j \\chi _k \\chi _l}$ .",
"[13], $\\mathcal {S}^{(R)}=-\\frac{1}{2}\\sum _{a=1}^{n}\\sum _{i=1}^{N}\\int dt \\chi ^{a}_{i}(t)\\partial _{t}\\chi ^{a}_{i}(t)-\\frac{J^{2}}{8N^3}\\int dt_1 dt_2 \\sum _{a,b=1}^{n}\\left(\\sum _{i=1}^{N}\\chi ^{a}_{i}(t_1)\\chi ^{b}_{i}(t_2) \\right)^{4}$ where, the index $ a $ stands for the so called replica index.",
"In the large $ N $ limit, one introduces the bi-local (collective) fields as [13], $\\Psi (t_1 , t_2)=\\frac{1}{N}\\sum _{i=1}^{N}\\chi _i (t_1)\\chi _i (t_2)$ where, we have suppressed the replica indices for later convenience.",
"This yields the following path integral, $\\mathcal {Z}_{SYK}\\sim \\int \\prod _{t_1 ,t_2}\\mathcal {D}\\Psi (t_1 ,t_2) \\mu (\\Psi )e^{-\\mathcal {S}^{(R)}_{col}[\\Psi ]}$ with the SYK replica action expressed in terms of collective field excitations (REF ), $\\mathcal {S}^{(R)}_{col}[\\Psi ]=\\frac{N}{2}\\int dt \\partial _{t}\\Psi (t,t^{\\prime })\\Big |_{t^{\\prime }=t}+\\frac{N}{2}Tr\\log \\Psi -\\frac{J^2 N}{8}\\int dt_1 dt_2 \\Psi ^{4}(t_1 , t_2).$ Notice that since the collective field $ \\Psi \\sim \\mathcal {O}(N^0) $ , therefore the collective action in (REF ) is of $ \\sim \\mathcal {O}(N) $ .",
"The trace term in the action (REF ) takes care of the Jacobian that appears due to the change in the path integral measure from $ \\chi _i (t) \\rightarrow \\Psi (t_1 , t_2) $ .",
"In the so called IR ($ |J t| \\gg 1 $ ) limit one could ignore the kinematics of collective field excitations and therefore the collective action (REF ) is approximately reduced to, $\\mathcal {S}^{(IR)}_{col}[\\Psi ] \\approx \\frac{N}{2}Tr\\log \\Psi -\\frac{J^2 N}{8}\\int dt_1 dt_2 \\Psi ^{4}(t_1 , t_2)$ which clearly has the re-parametrization invariance of the following form, $t \\rightarrow \\tilde{t} = f(t);~~\\Psi (t_1 ,t_2) \\rightarrow (f^{\\prime }(t_1)f^{\\prime }(t_2))^{1/4}\\tilde{\\Psi }(\\tilde{t}_{1},\\tilde{t}_{2})$ characterizing the IR critical point at strong coupling."
],
[
"The effective action",
"The classical saddle point equation has a solution [13], $\\Psi _{0}(t_1 ,t_2)=-\\left(\\frac{1}{4 \\pi J^2} \\right)^{1/4} \\frac{sgn(t_{12})}{\\sqrt{|t_{12}|}};~~t_{12}=t_1 -t_2.$ Next, we turn on fluctuations around this IR fixed point and propose the following YB modification to the bi-local excitations, $\\Psi (t_1 ,t_2)=\\Psi _{0}(t_1 ,t_2) +\\sqrt{\\frac{2}{N}}\\zeta _{YB}$ where the YB fluctuations associated to collective excitationsSince the YB deformation is expected to be appearing as $ 1/J^2 $ corrections to the SYK spectrum therefore it should not modify the solution ($ \\Psi _{0}(t_1 ,t_2) $ ) corresponding to the IR critical point., $\\zeta _{YB} (t_1 ,t_2)=\\frac{\\zeta (t_1 ,t_2)}{\\mathcal {F}_{\\eta }(t_1 ,t_2)};~~ \\mathcal {F}_{\\eta }(t_1 ,t_2)=\\frac{1}{1-\\frac{4 \\eta ^2}{J^2 (t_1 -t_2)^{2}}}$ would precisely be identified with the corresponding scalar field d.o.f.",
"living in the YB deformed $ AdS_2 $ spacetime.",
"The collective action could be expanded around the IR critical point as, $\\mathcal {S}^{(R)}_{col}[\\Psi ]=\\mathcal {S}^{(R)}_{col}[\\Psi _{0}]+\\frac{\\delta ^{2}\\mathcal {S}^{(R)}_{col}}{\\delta \\zeta _{YB}(t_1 ,t_2)\\delta \\zeta _{YB}(t_3 ,t_4) }\\delta \\Psi (t_1 ,t_2) \\delta \\Psi (t_3 ,t_4)+\\mathcal {O}(1/\\sqrt{N})\\nonumber \\\\$ Clearly, the first term on the R.H.S.",
"of (REF ) is of $ \\mathcal {O}(N) $ .",
"Whereas, on the other hand, the second term is of $ \\mathcal {O}(N^0) $ which could be formally expressed as, $\\mathcal {S}^{(2)}=\\frac{1}{2}\\int dt_1 dt_2 dt_3 dt_4~ \\zeta _{YB}(t_1 ,t_2)\\mathcal {K}(t_1,t_2;t_3,t_4)\\zeta _{YB}(t_3 ,t_4)$ where, the kernel could be formally expressed as [13], $\\begin{split}\\mathcal {K}(X_{1},X_{2};X_{3},X_{4})&=-\\frac{1}{2}\\Big [\\Psi ^{-1}_{0}(X_{4},X_{1})\\Psi ^{-1}_{0}(X_{2},X_{3})-\\Psi ^{-1}_{0}(X_{4},X_{2})\\Psi ^{-1}_{0}(X_{1},X_{3})\\Big ] \\\\&-3J^{2}\\Psi _{0}(X_{1},X_{2})\\Psi _{0}(X_{3},X_{4})\\frac{1}{2}\\Big (\\delta _{X_{1},X_{3}}\\delta _{X_{2},X_{4}}+\\delta _{X_{1},X_{4}}\\delta _{X_{2},X_{3}} \\Big ).\\end{split}$ In the following we evaluate the first term inHere we have used the short hand notation, $ A \\star B = \\int dt A(t_a ,t)B(t ,t_b ) $ [13].",
"(REF ), $\\begin{split}-\\int dt_{1}dt_{2}&dt_{3}dt_{4} \\\\&\\frac{1}{2}\\zeta _{YB} (t_{1},t_{2})\\Big [\\Psi ^{-1}_{0}(t_{4},t_{1})\\Psi ^{-1}_{0}(t_{2},t_{3})-\\Psi ^{-1}_{0}(t_{4},t_{2})\\Psi ^{-1}_{0}(t_{1},t_{3})\\Big ]\\zeta _{YB} (t_{3},t_{4}) \\\\&=-\\frac{1}{2}\\int dt_{1}dt_{2}dt_{3}dt_{4}\\Big [ \\Psi ^{-1}_{0}(t_{4},t_{1})\\zeta _{YB} (t_{1},t_{2})\\Psi ^{-1}_{0}(t_{2},t_{3})\\zeta _{YB} (t_{3},t_{4}) \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +\\Psi ^{-1}_{0}(t_{4},t_{2})\\zeta _{YB} (t_{2},t_{1})\\Psi ^{-1}_{0}(t_{1},t_{3})\\zeta _{YB} (t_{3},t_{4})\\Big ] \\\\&=-\\int dt_{1}dt_{2}dt_{3}dt_{4}\\;\\Psi ^{-1}_{0}(t_{4},t_{1})\\zeta _{YB} (t_{1},t_{2})\\Psi ^{-1}_{0}(t_{2},t_{3})\\zeta _{YB} (t_{3},t_{4})\\\\&\\equiv -\\text{Tr}\\left( \\Psi _{0}^{-1}\\star \\zeta _{YB} \\star \\Psi _{0}^{-1}\\star \\zeta _{YB} \\right)\\end{split}$ where, in the second line of (REF ) we have used the antisymmetry properties associated with the collective fields namely, $\\Psi _{0}(X,Y)=-\\Psi _{0}(Y,X), \\qquad \\qquad \\zeta _{YB}(X,Y)=-\\zeta _{YB}(Y,X).$ This finally yields the effective quadratic action as, $\\mathcal {S}^{(2)}=-\\frac{1}{2}\\text{Tr}\\left( \\Psi _{0}^{-1}\\star \\zeta _{YB} \\star \\Psi _{0}^{-1}\\star \\zeta _{YB} \\right) -\\frac{3J^{2}}{2}\\int dt_{1}dt_{2}\\left(\\Psi _{0}(t_{1},t_{2})\\right)^{2}\\left(\\zeta _{YB}(t_{1},t_{2})\\right)^{2}.\\nonumber \\\\$ In order to match our results to that with the $ (AdS_2)_{\\eta } $ calculations in the preceeding section, we define the following coordinate transformations [13], $t=\\frac{1}{2}(t_1 +t_2);~~z=\\frac{1}{2}(t_1 -t_2)$ and expand the fluctuations a complete basis, $\\psi (t,z)\\equiv \\zeta _{YB}(t_1 , t_2)=\\sum _{\\nu w}\\tilde{\\psi }_{\\nu w}(t_1 ,t_2)b_{\\nu w}(t_{1},t_{2});~~\\tilde{\\psi }_{\\nu w}(t_1 ,t_2)=\\frac{\\psi _{\\nu w}}{\\mathcal {F}_{\\eta }(t_1 ,t_2)}$ that is defined as follows [13], $b_{\\nu w}(t_{1},t_{2})=e^{iw\\frac{t_{1}+t_{2}}{2}}\\text{sgn}(t_{1}-t_{2})\\mathcal {Z}_{\\nu }\\left(\\Big |w\\frac{t_{1}-t_{2}}{2}\\Big |\\right)$ where, $ \\mathcal {Z}_{\\nu }(|x|) $ is a linear combination of Bessel functions defined in (REF ).",
"Following the original prescription of [13], $b_{\\nu w}(t_1 ,t_2)=-\\frac{3J}{16 \\sqrt{\\pi }}\\frac{1}{g(\\nu )}\\int \\frac{dt_a dt_b}{|t_a -t_b|}\\Psi _0 (t_1 ,t_a)\\Psi _0 (t_2 ,t_b)b_{\\nu w}(t_a ,t_b)$ one could further re-express (REF ) asHere, $ \\tilde{g}(\\nu )=\\frac{1}{g(\\nu )}=-\\frac{2 \\nu }{3}\\cot (\\pi \\nu /2) $[13]., $\\psi (t,z) = -\\sum _{\\nu w}\\frac{3J \\tilde{g}(\\nu )}{16 \\sqrt{\\pi }}\\tilde{\\psi }_{\\nu w}\\int \\frac{dt_a dt_b}{|t_a -t_b|}\\Psi _0 (t_1 ,t_a) \\Psi _0 (t_2 ,t_b)b_{\\nu w}(t_a ,t_b).$ Using (REF ), one could evaluate the quadratic action (REF ) asSee Appendix B for details of the analysis., $\\mathcal {S}^{(2)}=\\frac{3J}{32 \\sqrt{\\pi }}\\sum _{\\nu }\\int dw~\\vartheta _{\\nu -w}N_{\\nu } (\\tilde{g}(\\nu )-1)\\vartheta _{\\nu w}+\\mathcal {O}(\\frac{\\eta ^4}{J^4})$ where, we have introduced, $\\vartheta _{\\nu w}\\approx \\psi _{\\nu w}\\left( 1-\\frac{\\eta ^2 w^2 }{8 \\pi J^2 (\\nu ^2 -1)}\\right).$ The above Eq.",
"(REF ) is one of the key findings of our paper.",
"It turns out that the YB scaling associated with the bi-local fields in the SYK model produces a non trivial shift in the effective action (associated with quadratic fluctuations) at next to leading order ($ \\sim 1/J^2 $ ) in the coupling which is thereby highly suppressed compared to that with the leading order ($ \\sim 1/J $ ) effects.",
"This observation is clearly compatible with our earlier findings in the previous section.",
"This finally leads to the correlation function, $\\mathcal {G}(t_1,t_2;t_{1}^{\\prime },t_2^{\\prime })=\\frac{16 \\sqrt{\\pi }}{3J}\\sum _{\\nu =3/2 +2n}\\int dw \\frac{b_{\\nu -w}(t_1,t_2)b_{\\nu w}(t_1^{\\prime },t_2^{\\prime })}{\\tilde{N}_{\\nu }(\\tilde{g}(\\nu )-1)}$ where, we have defined, $\\tilde{N}_{\\nu }=N_{\\nu }\\mathcal {F}_{\\eta }(t_1 ,t_1^{\\prime })\\mathcal {F}_{\\eta }(t_2 ,t_2^{\\prime })\\left( 1-\\frac{\\eta ^2 w^2 }{4 \\pi J^2 (\\nu ^2 -1)}\\right).$"
],
[
"The $ (AdS_{2})_{\\eta } $ spectrum",
"Consider an effective action for scalar fields on $ AdS_2 $ [13], $S_{\\varphi }&=&\\frac{1}{2}\\int d^2 x \\sqrt{|g|}\\left(-g^{ab}\\partial _{a}\\varphi _{m}\\partial _{b}\\varphi _{m}-\\left(p^{2}_{m}-\\frac{1}{4} \\right)\\varphi ^{2}_{m} \\right)\\nonumber \\\\&=&-\\frac{1}{2}\\int d^2 x \\sqrt{|g|} \\mathcal {L}_{\\varphi }$ where, the metric could be formally expressed as, $g_{ab}&=&diag (-\\mathcal {F}_{\\eta }(z)/z^2 , \\mathcal {F}_{\\eta }(z)/z^2 )\\nonumber \\\\\\mathcal {F}_{\\eta }(z)&=&\\frac{1}{1-\\frac{\\eta ^2 \\alpha ^2}{z^2}}.$ A straightforward calculation reveals, $\\sqrt{|g|} \\mathcal {L}_{\\phi }=-\\varphi _{m}\\left( \\square ^{(2)} -\\frac{\\mathcal {F}_{\\eta }(z)}{z^2}\\left( p^{2}_{m}-\\frac{1}{4}\\right)\\right)\\varphi _{m}.$ On the other hand, it is trivial to notice thatHere, $ D_B = z^2 \\partial ^{2}_{z}+z\\partial _{z}-z^2\\partial ^{2}_{t} $ is the Bessel differential operator.",
"[34], $\\nabla ^{a}\\nabla _{a}\\varphi _{m} = \\frac{z^2}{\\mathcal {F}_{\\eta }(z)}\\square ^{(2)}\\varphi _{m} =\\sqrt{z}\\mathrm {D}_{B}\\left(\\frac{\\tilde{\\varphi }_{m}}{\\sqrt{z}} \\right)-\\frac{\\tilde{\\varphi }_{m}}{4} \\tilde{\\mathcal {F}}_{\\eta }(z)$ where, we have rescaled the scalar field asThis precisely confirms that the bi-local fields in the SYK should also get appropriately rescaled in the presence of YB deformations., $\\tilde{\\varphi }_{m} =\\frac{\\varphi _{m}}{\\mathcal {F}_{\\eta }(z)}$ .",
"Notice that here we have introduced a new function, $\\tilde{\\mathcal {F}}_{\\eta }(z)=\\mathcal {F}_{\\eta }(z)\\left( 1-\\frac{17 \\alpha ^2 \\eta ^2}{z^2}\\right)$ that precisely goes to unity in the limit of the vanishing YB deoformations and thereby one recovers the original results of [34].",
"Based on the above analysis, we propose the following non local field redefinition, $\\varphi _{m} (t,z)= \\left( \\frac{3J}{8 \\sqrt{\\pi }}\\right)^{1/2} z^{1/2}\\sqrt{f(\\sqrt{D_B})}~\\tilde{\\varphi }_{m}(t,z).$ Substituting (REF ) into (REF ) we obtain, $S_{\\varphi }=\\frac{1}{2}\\frac{3J}{8 \\sqrt{\\pi }}\\int dt \\int _{0}^{\\infty }\\frac{dz}{z}\\tilde{\\varphi }_{m}(D_B -\\tilde{p}^{2}_{m})f\\tilde{\\varphi }_{m}$ where, the pole has been rescaled due to YB deformations as, $\\tilde{p}^{2}_{m}=\\mathcal {F}_{\\eta }(z)\\left(p^2_{m}-\\frac{\\eta ^2 \\alpha ^2}{4z^2} \\right).$ Implementing the definition [13], $\\tilde{g}(\\nu )-1=(\\nu ^2 -\\tilde{p}^{2}_{m})f(\\nu )$ one could further express (REF ) as, $S_{\\varphi }&=&\\frac{1}{2}\\frac{3J}{8 \\sqrt{\\pi }}\\int dt \\int _{0}^{\\infty }\\frac{dz}{z}\\tilde{\\varphi }_{m}\\left(\\tilde{g}(\\sqrt{D_B})-1 \\right) \\tilde{\\varphi }_{m}\\nonumber \\\\&=&\\frac{1}{2}\\frac{3J}{8 \\sqrt{\\pi }}\\int dt \\int _{0}^{\\infty }\\frac{dz}{z}\\tilde{\\Phi }_{m}\\left(\\tilde{g}(\\sqrt{D_B})-1 \\right) \\tilde{\\Phi }_{m}+\\mathcal {O}(\\eta ^4 \\alpha ^4)$ where, we have introduced, $\\tilde{\\Phi }_{m} \\approx \\varphi _{m}\\left( 1-\\frac{\\eta ^2 \\alpha ^2}{z^2}\\right).$ Notice that the above equation (REF ) clearly resemblance our previous finding in (REF ).",
"It is also worthwhile to mention that in the limit of the vanishing YB deformations, the quadratic action (REF ) precisely reproduces the previous findings of [13].",
"It is indeed interesting to notice that, as observed in the previous section, the YB deformations shifts the pole (REF ) by an amount that goes with the quadratic order ($ \\sim 1/J^2 $ ) in the inverse of the SYK coupling."
],
[
"Concluding remarks",
"In this paper, based on the notion of SYK/AdS correspondence, we explore the effects of Yang-Baxter (YB) deformations on the collective field excitations within the SYK model.",
"The motivation behind our analysis solely comes from the underlying holographic principle which strongly suggests a possible modification of the SYK spectrum at quadratic ($ 1/J^2 $ ) order in the SYK coupling.",
"Based on the notion of holography (namely, looking at the scalar fluctuations and their YB scaling in $ (AdS_2)_{\\eta } $ ) we propose a possible YB scaling of the bi-local fields in the SYK model and compute the effective action upto quadratic order in the fluctuations.",
"It would be really nice to understand this YB scaling in terms of $ 1/N $ diagrammatics and thereby the associated Feynman rules in terms of these newly defined collective excitations.",
"We hope to address some of these issues in the near future.",
"It is indeed a great pleasure to thank Kenta Suzuki for valuable correspondences on several technical aspects of the manuscript.",
"The authors would also like to convey their sincere thanks to Sumit R. Das, Antal Jevicki, Marika Taylor and Kenta Suzuki for their valuable comments on the draft.",
"AL would like to acknowledge the financial support from PUCV, Chile.",
"The work of DR was supported through the Newton-Bhahba Fund.",
"DR would like to acknowledge the Royal Society UK and the Science and Engineering Research Board India (SERB) for financial assistance."
],
[
"Evaluation of the Green's function $\\mathcal {G}^{(0)}_{w,w^{\\prime }}$",
"The zero-th order Green's function $\\mathcal {G}^{(0)}_{w,w^{\\prime }}$ is defined through the following equation, $\\mathfrak {D}_{0}\\mathcal {G}^{(0)}_{w,w^{\\prime }}(z,z^{\\prime };\\Theta , \\Theta ^{\\prime })=-\\delta (z-z^{\\prime })\\delta (\\Theta - \\Theta ^{\\prime })\\delta (w+w^{\\prime }).",
"$ Expressing the Green's function in a basis of orthonormal wave functions, $\\mathcal {G}^{(0)}_{w,w^{\\prime }}=\\sqrt{z}\\sum _{\\mathfrak {K},\\mathfrak {K}^{\\prime }}\\mathfrak {f}_{\\mathfrak {K}}(\\Theta )\\mathfrak {f}_{\\mathfrak {K}^{\\prime }}(\\Theta ^{\\prime })\\tilde{G}^{(0)}_{\\omega ,\\mathfrak {K};\\omega ^{\\prime },\\mathfrak {K}^{\\prime }}(z;z^{\\prime })$ and substituting back into (REF ) we find, $\\left( \\hat{\\mathcal {L}}_{B}-\\nu _{0}^{2} \\right)\\tilde{G}^{(0)}_{w,\\mathfrak {K};-w,\\mathfrak {K}}(z;z^{\\prime })=-z^{3/2}\\delta (z-z^{\\prime })$ where we have introduced, $\\hat{\\mathcal {L}}_{B}& =& z^2 \\partial ^2_{z}+z\\partial _z + w^2 z^2 \\\\\\nu _{0}^{2}&=&\\mathsf {p}_{a}^{2}+m^{2}+\\frac{1}{4}$ and used the orthonormality conditions for the wave functions, $\\sum _{\\mathfrak {K},\\mathfrak {K}^{\\prime }}\\mathfrak {f}_{\\mathfrak {K}}(\\Theta )\\mathfrak {f}_{\\mathfrak {K}^{\\prime }}(\\Theta ^{\\prime })=\\delta _{\\mathfrak {K},\\mathfrak {K}^{\\prime }}\\delta \\left(\\Theta -\\Theta ^{\\prime }\\right).$ We express the Green's function (REF ) in a basis of Bessel function, $\\tilde{G}^{(0)}_{w,\\mathfrak {K};-w,\\mathfrak {K}}(z;z^{\\prime }) =\\int d \\nu \\tilde{\\mathfrak {g}}^{(0)}_{\\nu }(z^{\\prime })\\mathcal {Z}_{\\nu }(| w z|)$ that satisfies the Bessel equation, $\\hat{\\mathcal {L}}_{B}\\mathcal {Z}_{\\nu }(| w z|) = \\nu ^2 \\mathcal {Z}_{\\nu }(| w z|).$ The most general solution to (REF ) could be formally expressed as [7], $\\mathcal {Z}_{\\nu }(|x|)=J_{\\nu }(|x|)+\\xi _{\\nu }J_{-\\nu }(|x|).$ Notice that while both functions converge at large $ |x|\\rightarrow \\infty $ , one of the solutions $ J_{-\\nu }(x) $ diverges for $ x\\sim 0 $ which thereby amounts of setting the coefficient, $\\xi _{\\nu }=\\frac{\\tan (\\pi \\nu /2)+1}{\\tan (\\pi \\nu /2)-1}=0\\Rightarrow ~~\\nu =3/2+2n.$ Substituting, (REF ) into (REF ) and using the completeness condition, $\\int \\frac{d \\nu }{N_{\\nu }}\\mathcal {Z}_{\\nu }^{\\ast }(|x|) \\mathcal {Z}_{\\nu }(|x^{\\prime }|) =x \\delta (x-x^{\\prime })$ it is in fact quite straightforward to show, $\\tilde{\\mathfrak {g}}^{(0)}_{\\nu }(z^{\\prime })=-\\frac{\\sqrt{z^{\\prime }}}{N_{\\nu }}\\mathcal {Z}_{\\nu }^{\\ast }(|wz^{\\prime }|)$ which finally yields the real space zeroth order Green's function, $\\mathcal {G}^{(0)}(t,z,\\Theta ;t^{\\prime },z^{\\prime },\\Theta ^{\\prime })&=&-|zz^{\\prime }|^{1/2}\\sum _{a=0}^{\\infty }\\mathfrak {f}_{\\mathsf {p}_{a}}(\\Theta )\\mathfrak {f}_{\\mathsf {p}_{a}}(\\Theta ^{\\prime })\\int \\frac{dw}{2 \\pi }e^{-iw(t-t^{\\prime })}\\mathcal {I}(z,z^{\\prime })\\nonumber \\\\\\mathcal {I}(z,z^{\\prime })&=&\\int \\frac{d \\nu }{N_{\\nu }}\\frac{\\mathcal {Z}_{\\nu }^{\\ast }(|wz|) \\mathcal {Z}_{\\nu }(|w z^{\\prime }|)}{\\nu ^2 -\\nu _{0}^{2}}.$"
],
[
"Evaluation of the quadratic action $\\mathcal {S}^{(2)}$",
"We divide the quadratic action (REF ) into following two parts, $\\mathcal {S}^{(2)}_{I}=-\\frac{1}{2}\\int dt_{1}dt_{2}dt_{3}dt_{4}\\;\\Psi ^{-1}_{0}(t_{4},t_{1})\\zeta _{YB} (t_{1},t_{2})\\Psi ^{-1}_{0}(t_{2},t_{3})\\zeta _{YB} (t_{3},t_{4})\\nonumber \\\\=-\\frac{1}{2}\\sum _{\\nu ,\\nu ^{\\prime }}\\int dw dw^{\\prime } dt_1 dt_2 dt_3 dt_4 \\Psi _0^{-1}(t_4 ,t_1) \\Psi _0^{-1}(t_2 ,t_3)~~~~~~~~~~~~~\\nonumber \\\\\\tilde{\\psi }_{\\nu w}(t_1 ,t_2)\\tilde{\\psi }_{\\nu ^{\\prime } w^{\\prime }}(t_3 ,t_4)b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_3, t_4)\\nonumber \\\\=\\frac{3J}{32 \\sqrt{\\pi }}\\sum _{\\nu ,\\nu ^{\\prime }}\\tilde{g}(\\nu ^{\\prime })\\int dw dw^{\\prime } dt_1 dt_2 dt_3 dt_4\\frac{dt_a dt_b}{|t_a -t_b|}\\Psi ^{-1}_{0}(t_{1},t_{4})\\Psi _{0}(t_{4},t_{b})\\nonumber \\\\\\Psi ^{-1}_{0}(t_{2},t_{3})\\Psi _{0}(t_{3},t_{a})\\tilde{\\psi }_{\\nu w}(t_1 ,t_2)\\tilde{\\psi }_{\\nu ^{\\prime } w^{\\prime }}(t_3 ,t_4)b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_b, t_a)\\nonumber \\\\=\\frac{3J}{32 \\sqrt{\\pi }}\\sum _{\\nu ,\\nu ^{\\prime }}\\tilde{g}(\\nu ^{\\prime })\\int dw dw^{\\prime } \\frac{dt_1 dt_2}{|t_1 -t_2|}\\psi _{\\nu w}\\psi _{\\nu ^{\\prime } w^{\\prime }}b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_1, t_2)\\nonumber \\\\-\\frac{3 \\eta ^2}{4\\sqrt{\\pi }J}\\sum _{\\nu ,\\nu ^{\\prime }}\\tilde{g}(\\nu ^{\\prime })\\int dw dw^{\\prime } \\frac{dt_1 dt_2}{|t_1 -t_2|^{3}}\\psi _{\\nu w}\\psi _{\\nu ^{\\prime } w^{\\prime }}b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_1, t_2)+\\mathcal {O}(\\frac{\\eta ^4}{J^3})$ where, we have used the orthogonalization condition [7], $\\int dt \\Psi ^{-1}_{0}(t_a ,t)\\Psi _{0}(t, t_b)=\\delta (t_a -t_b).$ Using (REF ) this could be further re-expressed as, $\\mathcal {S}^{(2)}_{I}=\\frac{3J}{32 \\sqrt{\\pi }}\\sum _{\\nu }\\int dw \\psi _{\\nu -w}N_{\\nu }\\left( 1-\\frac{\\eta ^2 w^2 }{4 \\pi J^2 (\\nu ^2 -1)}\\right) \\tilde{g}(\\nu )\\psi _{\\nu w}+\\mathcal {O}(\\frac{\\eta ^4}{J^4})$ where, we have performed the integral only for bound states with integer $ \\nu = \\frac{3}{2}+2n~(n=0,1,2..) $ .",
"$\\mathcal {S}^{(2)}_{II}=-\\frac{3J^{2}}{2}\\frac{1}{8}\\int dt_{1}dt_{2}\\left(\\Psi _{0}(t_{1},t_{2})\\right)^{2}\\left(\\zeta _{YB}(t_{1},t_{2})\\right)^{2}~~~~~~~~~~~~~~~\\nonumber \\\\=-\\frac{3J^{2}}{2}\\frac{1}{8}\\sum _{\\nu ,\\nu ^{\\prime }}\\int dw dw^{\\prime } dt_1 dt_2 \\left(\\Psi _{0}(t_{1},t_{2})\\right)^{2} \\tilde{\\psi }_{\\nu w}(t_1 , t_2)~~~~\\nonumber \\\\\\tilde{\\psi }_{\\nu ^{\\prime } w^{\\prime }}(t_1 ,t_2)b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_1 ,t_2)\\nonumber \\\\=-\\frac{3J^{2}}{2}\\frac{1}{8}\\sum _{\\nu ,\\nu ^{\\prime }}\\int dw dw^{\\prime } dt_1 dt_2 \\left(\\Psi _{0}(t_{1},t_{2})\\right)^{2} \\psi _{\\nu w}\\psi _{\\nu ^{\\prime } w^{\\prime }}~~~~~\\nonumber \\\\b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_1 ,t_2)\\nonumber \\\\+12\\eta ^{2}\\frac{1}{8}\\sum _{\\nu ,\\nu ^{\\prime }}\\int dw dw^{\\prime } \\frac{dt_1 dt_2}{|t_1 -t_2|^2} \\left(\\Psi _{0}(t_{1},t_{2})\\right)^{2} \\psi _{\\nu w}\\psi _{\\nu ^{\\prime } w^{\\prime }}\\nonumber \\\\b_{\\nu w}(t_1 ,t_2)b_{\\nu ^{\\prime } w^{\\prime }}(t_1 ,t_2)+\\mathcal {O}(\\frac{\\eta ^4}{J^3})\\nonumber \\\\=-\\frac{3J}{32 \\sqrt{\\pi }}\\sum _{\\nu }\\int dw\\psi _{\\nu -w}N_{\\nu }\\left( 1-\\frac{\\eta ^2 w^2 }{4 \\pi J^2 (\\nu ^2 -1)}\\right) \\psi _{\\nu w}+\\mathcal {O}(\\frac{\\eta ^4}{J^4})$ where, the factor $ 1/8 $ has been introduced in order to avoid the overcounting in the expansion of $ \\sum _{i,j=1}^{4} \\Psi _{0}(X_i ,X_j)^{2}\\eta (X_i , X_j)^{2}$ ."
]
] | 1808.08380 |
[
[
"Nonlinear $n$-term approximation of harmonic functions from shifts of\n the Newtonian Kernel"
],
[
"Abstract A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\\mathbb R}^d$ (Newtonian kernel).",
"The main goal of this article is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian kernel with poles outside $\\overline{B^d}$ in the harmonic Hardy spaces.",
"Optimal rates of approximation are obtained in terms of harmonic Besov spaces.",
"The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel."
],
[
"Introduction",
"The fundamental solution of the Laplace equation $\\frac{1}{|x|^{d-2}}$ in dimension $d>2$ or $\\ln \\frac{1}{|x|}$ if $d=2$ with $|x|$ being the Euclidean norm of $x\\in {\\mathbb {R}}^d$ is a basic building block in Potential theory.",
"As is customary, we shall term the harmonic function $\\frac{1}{|x|^{d-2}}$ or $\\ln \\frac{1}{|x|}$ “Newtonian kernel”.",
"The main purpose of this article is to study the nonlinear $n$ -term approximation of harmonic functions on the unit ball $B^d$ in ${\\mathbb {R}}^d$ from linear combinations of shifts of the Newtonian kernel.",
"More explicitly, the problem is for a given harmonic function $U$ on $B^d$ and $n\\ge 1$ to find $n$ locations $\\lbrace y_j\\rbrace $ in ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and coefficients $\\lbrace a_j\\rbrace $ in ${\\mathbb {C}}$ so that $a_0+\\sum _{j=1}^n \\frac{a_j}{|x-y_j|^{d-2}}\\quad \\hbox{if} \\;\\; d>2\\quad \\hbox{or}\\quad a_0+\\sum _{j=1}^n a_j\\ln \\frac{1}{|x-y_j|}\\quad \\hbox{if} \\;\\; d=2$ approximates $U$ with an optimal rate (near best) in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ , $0<p<\\infty $ .",
"Denote by $\\mathcal {N}_n$ the set of all harmonic functions on $B^d$ that can be represented in the form (REF ).",
"Here the points $\\lbrace y_j\\rbrace $ are allowed to vary with the function and hence $\\mathcal {N}_n$ is nonlinear.",
"Given $U\\in \\mathcal {H}^p(B^d)$ we denote $E_n(U)_{\\mathcal {H}^p}:=\\inf _{G\\in \\mathcal {N}_n}\\Vert U-G\\Vert _{\\mathcal {H}^p(B^d)}.$ We shall term $E_n(U)_{\\mathcal {H}^p}$ the best $n$ -term approximation of $U$ in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ from shifts of the Newtonian kernel as in (REF ).",
"Our goal is to study the rate of convergence of $\\lbrace E_n(U)_{\\mathcal {H}^p}\\rbrace $ and the smoothness spaces that govern this approximation process.",
"The same approximation problem is also important in the case when the function $U$ to be approximated is harmonic on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and the poles $\\lbrace y_j\\rbrace $ are in $B^d$ or $U$ is harmonic on ${\\mathbb {R}}^d_+$ and the poles $\\lbrace y_j\\rbrace $ are in ${\\mathbb {R}}^d_-$ .",
"The results of A. Pekarski [27], [28] on rational approximation of holomorphic functions on the unit disc in ${\\mathbb {C}}$ and also the results in [21] served as an inspiration and motivation for the development in this article.",
"An important motivation to us also comes from some applications of Potential theory.",
"In Geodesy people consider approximation of the gravitational (disturbing) potential using the potential of $n$ point masses.",
"A given potential $U$ is approximated by the potential of $n$ point charges in Electrostatics or by the potential of $n$ magnetic poles in Magnetism.",
"There is also a great deal of work done on the Method of Fundamental Solutions for the Dirichlet problem of the Laplace equation in Numerical Analysis.",
"This research is directly related to the problems we consider here.",
"The multipole method of V. Rokhlin and his collaborators (e.g.",
"[5], [13]) is also relevant to our undertaking.",
"We refer the reader to [2], [14], [19] for the basics of Potential theory.",
"The focus of this article is on the establishment of a direct (Jackson type) estimate for nonlinear $n$ -term approximation of functions in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ , $0<p<\\infty $ , from shifts of the Newtonian kernel.",
"As one can expect the harmonic Besov spaces on the ball $B^{s\\tau }_\\tau (\\mathcal {H}) \\quad \\hbox{with}\\quad 1/\\tau =s/(d-1)+1/p,~s>0,$ will be naturally involved in the approximation process.",
"The poor localization of the Newtonian kernel is the first obstacle to overcome in approximating from linear combinations of its shifts.",
"An important step forward in solving this approximation problem is the construction in [16] of highly localized summability kernels on the unit sphere ${{\\mathbb {S}}^{d-1}}$ in ${\\mathbb {R}}^d$ that are restrictions to the sphere of linear combinations of a fixed number of shifts of the Newtonian kernel just as in (REF ).",
"Note that the harmonic functions by their nature cannot be well localized in an open subset of ${\\mathbb {R}}^d$ , but they can be well localized on the boundary of such a set; typical examples are ${{\\mathbb {S}}^{d-1}}$ and ${\\mathbb {R}}^{d-1}$ .",
"To obtain our approximation result we proceed as follows: We first use the result from [16] to construct a pair of dual frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ for all spaces of interest on ${{\\mathbb {S}}^{d-1}}$ whose elements $\\lbrace \\theta _\\xi \\rbrace $ are linear combinations of a fixed number of shifts of the Newtonian kernel and are well localized.",
"Armed with these frames we apply an intermediate nonlinear $n$ -term approximation from $\\lbrace \\theta _\\xi \\rbrace $ to the boundary value function/distribution $f_U$ of the harmonic function $U$ to be approximated.",
"This leads us to the desired estimate by harmonic extension to $B^d$ of the approxiamant and using the fact that each $\\theta _\\xi $ is a finite linear combination of shifts of the Newtonian kernel.",
"Thus a major step in our development is to construct such a pair of dual frames.",
"More precisely, one of our main goals is to construct (see Theorem REF ) a pair of frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ for the Besov and Triebel-Lizorkin spaces $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ with parameters $(s, p, q)$ in the range ${\\mathcal {Q}}={\\mathcal {Q}}(A):=\\big \\lbrace (s, p, q): |s| \\le A,\\; A^{-1}\\le p \\le A, \\;\\hbox{and}\\; A^{-1}\\le q<\\infty \\big \\rbrace ,$ where $A >1$ is a fixed constant.",
"This construction employs the small perturbation method for construction of frames developed in [9] and relies on the kernels from [16].",
"While the basic ideas behind the construction of the frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ is relatively simple, some of the details become technical when applied to the specific case of this article.",
"For example, the requirement that $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ are frames for the class of Besov and Triebel-Lizorkin spaces $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ with parameters $(s, p, q)\\in {\\mathcal {Q}}(A)$ compels us to carefully trace the constants appearing in all relevant estimates.",
"The next several remarks will perhaps clarify some of the issues arising in our construction of the pair of frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ described above: (1) In applying the small perturbation method from [9] we use as a backbone a frame $\\lbrace \\psi _\\xi \\rbrace $ on ${{\\mathbb {S}}^{d-1}}$ from [25], which can characterize the Besov and Triebel-Lizorkin spaces $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ with complete range of parameters $(s, p, q)$ , i.e.",
"$s\\in {\\mathbb {R}}$ , $0<p,q<\\infty $ .",
"With the restriction that each frame element $\\lbrace \\theta _\\xi \\rbrace $ is a linear combination of a fixed number of shifts of the Newtonian kernel comes the natural limitation that the new frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ can characterize the Besov and Triebel-Lizorkin spaces $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ with parameters from ${\\mathcal {Q}}(A)$ (see(REF )).",
"Here $A>1$ can be arbitrarily large but is fixed and the number of shifts depends on $A$ .",
"(2) If the old frame $\\lbrace \\psi _\\xi \\rbrace $ is a basis, then the new frame $\\lbrace \\theta _\\xi \\rbrace $ is also a basis.",
"This is the case in dimension $d=2$ , where we use Meyer's periodic wavelet basis on ${\\mathbb {S}}^1$ .",
"As for now there are no convenient bases on ${{\\mathbb {S}}^{d-1}}$ when $d>2$ .",
"For this reason we work with frames, which are completely satisfactory for our purposes.",
"(3) The rotation group on ${{\\mathbb {S}}^{d-1}}$ is not commutative in dimensions $d\\ge 3$ , which is a major difference from the translation group in ${\\mathbb {R}}^{d-1}$ .",
"This is an essential obstacle in constructing highly localized linear combinations of a fixed number of shifts of the Newtonian kernel with vanishing moments on ${{\\mathbb {S}}^{d-1}}$ .",
"In order to overcome this difficulty we replace the vanishing moment conditions on the $\\varphi $ -transform of Frazier and Jawerth with small moment conditions, see e.g.",
"Propositions REF –REF and (REF ) in Theorem REF .",
"In general, the vanishing moment conditions are not valid for $\\theta _\\xi $ .",
"(4) We restrict the parameters to $p,q<\\infty $ for several reasons.",
"First, whenever $p,q<\\infty $ the Besov and Triebel-Lizorkin spaces $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ are separable and the finite sequences are dense in the respective Besov and Triebel-Lizorkin sequence spaces.",
"Also, the respective frame representations converge unconditionally.",
"These facts are important in the construction and utilization of the frames $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ .",
"Furthermore, as is well known, in general, nonlinear $n$ -term approximation from frames or bases in $L^\\infty $ or as in our case $\\mathcal {H}^\\infty $ is not quite natural.",
"Just as in Harmonic analysis one should work in BMO instead.",
"The intimate relation between the harmonic Hardy and Besov spaces on $B^d$ on the one hand and the Triebel-Lizorkin and Besov spaces of functions/distributions on ${{\\mathbb {S}}^{d-1}}$ on the other will play a critical role in our development.",
"Harmonic Besov and Triebel-Lizorkin spaces on $B^d$ and Besov and Triebel-Lizorkin spaces of distributions on ${{\\mathbb {S}}^{d-1}}$ with full range of parameters are treated in [15].",
"In particular, the equivalence of these spaces on $B^d$ and on its boundary ${{\\mathbb {S}}^{d-1}}$ is established in [15].",
"These equivalences enable us to mediate between spaces and frames on $B^d$ and on ${{\\mathbb {S}}^{d-1}}$ .",
"For example, it allows to transfer the constructed frame $\\lbrace \\theta _\\xi \\rbrace $ on ${{\\mathbb {S}}^{d-1}}$ to a frame on $B^d$ and approximation results from ${{\\mathbb {S}}^{d-1}}$ to $B^d$ .",
"Our main result in Theorem REF asserts that if $U\\in B^{s\\tau }_\\tau (\\mathcal {H})$ with $s>0$ and $1/\\tau =s/(d-1)+1/p$ , then $U\\in \\mathcal {H}^p(B^d)$ and $E_n(U)_{\\mathcal {H}^p} \\le c n^{-s/(d-1)}\\Vert U\\Vert _{B^{s\\tau }_\\tau (\\mathcal {H})}, \\quad n\\ge 1.$ We derive this estimate from a respective estimate for nonlinear $n$ -term approximation of functions/distributions from $\\lbrace \\theta _\\xi \\rbrace $ on the unit sphere ${{\\mathbb {S}}^{d-1}}$ .",
"Denote by $\\sigma _n(f)_{\\mathcal {F}_p^{02}}$ the best $n$ -term approximation of $f$ in the Triebel-Lizorkin space $\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})$ from the frame $\\lbrace \\theta _\\xi \\rbrace $ mentioned above.",
"We show that whenever $f~\\in ~B^{s\\tau }_\\tau ({{\\mathbb {S}}^{d-1}})$ with $1/\\tau =s/(d-1)+1/p$ , $0<p<\\infty $ , then $f\\in \\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})$ and $\\sigma _n(f)_{\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})} \\le c n^{-s/(d-1)}\\Vert f\\Vert _{B^{s\\tau }_\\tau ({{\\mathbb {S}}^{d-1}})}, \\quad n\\ge 1.$ As is well known the harmonic Hardy space $\\mathcal {H}^p(B^d)$ , $0<p<\\infty $ , can be identified with the Triebel-Lizorkin space $\\mathcal {F}_p^{02}=\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})$ of functions/distributions on ${{\\mathbb {S}}^{d-1}}$ , and hence (REF ) implies that for any harmonic function $U\\in B^{s\\tau }_\\tau (\\mathcal {H})$ $\\sigma _n(U)_{\\mathcal {H}^p} \\le c n^{-s/(d-1)}\\Vert U\\Vert _{B^{s\\tau }_\\tau (\\mathcal {H})}, \\quad n\\ge 1,$ where $\\sigma _n(U)_{\\mathcal {H}^p}$ stands for the best $n$ -term approximation of $U$ in $\\mathcal {H}^p(B^d)$ from the harmonic extension to $B^d$ of $\\lbrace \\theta _\\xi \\rbrace $ .",
"Finally, estimate (REF ) yields (REF ) taking into account that each frame element $\\theta _\\xi $ is a linear combination of a fixed number of shifts of the Newtonian kernel.",
"It is insightful to study the nonlinear approximation from functions as in (REF ) in the norms of the closely related to $\\mathcal {H}^p(B^d)$ harmonic Triebel-Lizorkin and Besov spaces $F_p^{0q}(B^d)$ and $B_p^{0q}(B^d)$ .",
"As shown in Theorem REF the nonlinear $n$ -term approximations in $F_p^{0q}(B^d)$ have the optimal rate $O(n^{-s/(d-1)})$ for any $0<q<\\infty $ , while the nonlinear $n$ -term approximation in $B_p^{sq}(B^d)$ achieves this optimal order only for $p\\le q<\\infty $ , see Theorems REF .",
"Bernstein inequality: Conjecture.",
"We conjecture that the following Bernstein type inequality is valid: Let $1<p<\\infty $ , $s>0$ , and $1/\\tau =s/(d-1)+1/p$ .",
"Then $\\Vert G\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})} \\le cn^{s/(d-1)}\\Vert G\\Vert _{\\mathcal {H}^p(B^d)}, \\quad \\forall G\\in \\mathcal {N}_n.$ If valid this estimate along with the Jackson estimate (REF ) would lead to a complete characterization of the rates of approximation (approximation spaces) of nonlinear $n$ -term approximation in $\\mathcal {H}^p(B^d)$ , $1<p<\\infty $ , from shifts of the Newtonian kernel.",
"It is natural to pose the question whether the approximation results of this paper hold when $p=\\infty $ .",
"We think that just as in the case of rational approximation analogues of these results are valid if $\\mathcal {H}^\\infty (B^d)$ is replaced by the harmonic BMO space on $B^d$ .",
"We shall not pursue this line of research in the present article.",
"Organization.",
"The outline of the paper is as follows.",
"In Section we introduce some basic notation and assemble background material about the maximal operator, spherical harmonics, and maximal $\\delta $ -nets; we also give some technical estimates on inner products of localized functions on the sphere.",
"Section presents some basic facts about harmonic Besov and Triebel-Lizorkin spaces on $B^d$ and ${{\\mathbb {S}}^{d-1}}$ developed in [15]; it also recalls the construction of frames for Besov and Triebel-Lizorkin spaces on ${{\\mathbb {S}}^{d-1}}$ and their frame decomposition developed in [25].",
"Section presents and somewhat refines the small perturbation method for construction of frames developed in [9].",
"Section deals with localization properties of the frame elements from § and highly localized kernels induced by shifts of the Newtonian kernel, developed in [16].",
"Section contains the construction of a pair of frames for Besov and Triebel-Lizorkin spaces whose elements are finite linear combinations of shifts of the Newtonian kernel.",
"Section is devoted to nonlinear $n$ -term approximation of functions in the harmonic Hardy spaces from shifts of the Newtonian kernel.",
"Section deals with nonlinear $n$ -term approximation in the exterior of the unit ball in ${\\mathbb {R}}^d$ and in the upper half space in ${\\mathbb {R}}^d$ from shifts of the Newtonian kernel.",
"Proofs of key estimates supporting our main results are given in Section .",
"Notation.",
"Throughout this article the constants $d$ , $M$ , and $K$ will appear frequently.",
"Here $d\\in {\\mathbb {N}}$ is the dimension of the space ${\\mathbb {R}}^d$ , $M>0$ determines decay rates, and $K\\in {\\mathbb {N}}$ is a parameter determining the upper bound of the order of derivatives required from some functions.",
"Positive constants will be denoted by $c$ and they may vary at every occurrence.",
"Most of these constants will depend only on $d, K, M$ .",
"By $C$ 's we denote numbers (constants) that also depend on parameters different from $d, K, M$ .",
"When we would like to trace the dependence of a constant $c$ on these parameters we use indexing, e.g.",
"$c_1,c_2$ , etc.",
"or indicate the dependence on parameters in parenthesis.",
"These indexed constants preserve their values throughout the article.",
"The relation $a\\sim b$ means that there exists a constant $c\\ge 1$ such that $c^{-1}a\\le b \\le ca$ ."
],
[
"Basic notation and simple inequalities",
"In this article we use standard notation.",
"Thus ${\\mathbb {R}}^d$ stands for the $d$ -dimensional Euclidean space.",
"The inner product of $x,y\\in {\\mathbb {R}}^d$ is denoted by $x\\cdot y =\\sum _{k=1}^d x_ky_k$ and the Euclidean norm of $x$ by $|x|=\\sqrt{x\\cdot x}$ .",
"We write ${\\mathbb {B}}(x_0,r):=\\lbrace x : |x-x_0|< r\\rbrace $ and set $B^d:={\\mathbb {B}}(0, 1)$ , the open unit ball in ${\\mathbb {R}}^d$ .",
"As usual ${\\mathbb {N}}_0$ stands for the set of non-negative integers.",
"For $\\beta =(\\beta _1,\\dots ,\\beta _d)\\in {\\mathbb {N}}_0^d$ the monomial $x^\\beta $ is defined by $x^\\beta :=x_1^{\\beta _1}\\dots x_d^{\\beta _d}$ and its degree is $|\\beta |:=\\beta _1+\\dots +\\beta _d$ .",
"The set of all polynomials in $x\\in {\\mathbb {R}}^d$ of total degree $n$ is denoted by $\\mathcal {P}_n^d$ .",
"We denote $\\partial _k:=\\partial /\\partial x_k$ and then $\\partial ^\\beta :=\\partial _1^{\\beta _1}\\dots \\partial _d^{\\beta _d}$ is a differential operator of order $|\\beta |$ , the gradient operator is $\\nabla :=(\\partial _1,\\dots , \\partial _d)$ , and $\\Delta :=\\partial _1^2+\\dots +\\partial _d^2$ stands for the Laplacian.",
"When necessary we indicate the variable of differentiation by a subscript, e.g.",
"$\\partial ^\\beta _x$ .",
"The unit sphere in ${\\mathbb {R}}^d$ is denoted by ${{\\mathbb {S}}^{d-1}}:=\\lbrace x : |x|=1\\rbrace $ .",
"We denote by $\\rho (x, y)$ the geodesic distance between $x, y\\in {{\\mathbb {S}}^{d-1}}$ , that is, $\\rho (x,y):=\\arccos (x\\cdot y)$ .",
"The open spherical cap (ball on the sphere) centred at $\\eta \\in {{\\mathbb {S}}^{d-1}}$ of radius $r$ is denoted by $B(\\eta ,r)=\\lbrace x\\in {{\\mathbb {S}}^{d-1}}: \\rho (\\eta ,x)<r\\rbrace $ .",
"We denote by $\\Delta _0$ the Laplace-Beltrami operator on ${{\\mathbb {S}}^{d-1}}$ .",
"As is well known (e.g.",
"[8]) $\\Delta _0$ has the decomposition $\\Delta _0 =\\sum _{1\\le i<\\ell \\le d}D_{i,\\ell }^2,\\quad D_{i,\\ell }=x_i\\partial _\\ell -x_\\ell \\partial _i,\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ For any function $g$ on ${{\\mathbb {S}}^{d-1}}$ we denote by $\\breve{g}$ its standard extension, defined by $\\breve{g}(x):=g(x/|x|) \\quad \\hbox{for}\\quad x\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace .$ As is well known (e.g.",
"[8] or [29]) for any $g\\in C^2({{\\mathbb {S}}^{d-1}})$ $\\Delta \\breve{g}(x)=\\Delta _0 g(x),\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ By definition $g\\in W_\\infty ^K({{\\mathbb {S}}^{d-1}})$ , $K\\in {\\mathbb {N}}_0$ , if $\\Vert \\partial ^\\beta \\breve{g}\\Vert _{L^\\infty ({\\mathbb {B}}(0,2)\\setminus {\\mathbb {B}}(0,1/2))} \\le c$ , $\\forall |\\beta |\\le K$ .",
"The Lebesgue measure on ${{\\mathbb {S}}^{d-1}}$ is denoted by $\\sigma $ and we set $|E|:=\\sigma (E)$ for a measurable set $E\\subset {{\\mathbb {S}}^{d-1}}$ .",
"Thus, $\\omega _d:=|{{\\mathbb {S}}^{d-1}}|=2\\pi ^{d/2}/\\Gamma (d/2)$ .",
"The inner product of $f,g\\in L^2({{\\mathbb {S}}^{d-1}})$ is given by $\\left\\langle f,g\\right\\rangle := \\int _{{{\\mathbb {S}}^{d-1}}} f(y)\\overline{g(y)}\\,d\\sigma (y).$ The nonstandard convolution of functions $F\\in L^\\infty [-1,1]$ and $g\\in L({{\\mathbb {S}}^{d-1}})$ is defined by $F*g(x):=\\left\\langle F(x\\cdot \\bullet ),\\overline{g}\\right\\rangle =\\int _{{{\\mathbb {S}}^{d-1}}}F(x\\cdot y)g(y) d\\sigma (y).$ We say that a function $f$ defined on ${{\\mathbb {S}}^{d-1}}$ is localized around $\\eta \\in {{\\mathbb {S}}^{d-1}}$ with dilation factor $N$ and decay rate $M>0$ if the estimate $|f(x)|\\le \\kappa N^{d-1}(1+N\\rho (\\eta ,x))^{-M},\\quad x\\in {{\\mathbb {S}}^{d-1}},$ holds for some constant $\\kappa >0$ independent of $N, x, \\eta $ .",
"The multiplier $N^{d-1}$ is used as part of the decay function in (REF ) in order to have $\\Vert f\\Vert _{L^1({{\\mathbb {S}}^{d-1}})}\\le c$ .",
"Namely, for $M> d-1$ we have $\\int _{{{\\mathbb {S}}^{d-1}}} \\frac{N^{d-1}}{(1+N\\rho (\\eta ,y))^M}d\\sigma (y)\\le c_0,\\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}},~\\forall N\\ge 1,$ where $c_0=c(d)/(M-d+1)$ depends only on $d$ and $M$ .",
"The weight function in the right-hand side of (REF ) also has the property: For any $\\eta _1,\\eta _2\\in {{\\mathbb {S}}^{d-1}}$ with $\\rho (\\eta _1,\\eta _2)\\le N^{-1}$ $(1+N\\rho (\\eta _2, x))^{-1}\\le 2(1+N\\rho (\\eta _1, x))^{-1},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}}.$ Indeed, $1+N\\rho (\\eta _1, x)\\le 1+N(\\rho (\\eta _1,\\eta _2)+\\rho (\\eta _2, x))\\le 2+N\\rho (\\eta _2, x)$ , which implies (REF ).",
"Another simple inequality that will be useful is: $|\\Delta _0^{K/2}x^\\beta |\\le c_6,\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},~0\\le |\\beta |\\le K,$ where $c_6$ depends only on $d$ and $K$ ."
],
[
"Spherical harmonics",
"The spherical harmonics will be our main vehicle in dealing with harmonic functions on the unit ball $B^d$ in ${\\mathbb {R}}^d$ .",
"Denote by $\\mathcal {H}_k$ the space of all spherical harmonics of degree $k$ on ${{\\mathbb {S}}^{d-1}}$ .",
"As is well known the dimension of $\\mathcal {H}_k$ is $N(k, d)= \\frac{2k+d-2}{k}\\binom{k+d-3}{k-1} \\sim k^{d-1}$ .",
"Furthermore, the spaces $\\mathcal {H}_k$ , $k=0, 1, \\dots $ , are orthogonal and $L^2({{\\mathbb {S}}^{d-1}}) =\\bigoplus _{k\\ge 0} \\mathcal {H}_k$ .",
"Let $\\lbrace Y_{k\\nu }: \\nu =1, \\dots , N(k, d)\\rbrace $ be a real-valued orthonormal basis for $\\mathcal {H}_k$ .",
"Then the kernel of the orthogonal projector onto $\\mathcal {H}_k$ is given by $Z_k(x\\cdot y) = \\sum _{\\nu =1}^{N(k, d)} Y_{k\\nu }(x)Y_{k\\nu }(y),\\quad x, y\\in {{\\mathbb {S}}^{d-1}}.$ As is well known (see e.g.",
"[8]) $Z_k(x\\cdot y)= \\frac{k+\\mu }{\\mu \\omega _d}\\,C^{\\mu }_k(x\\cdot y),\\quad x, y\\in {{\\mathbb {S}}^{d-1}},\\quad \\mu :=\\frac{d-2}{2},~d>2.$ Here $C_k^{\\mu }$ is the Gegenbauer (ultraspherical) polynomial of degree $k$ normalized by $C_k^{\\mu }(1)= \\binom{k+ 2\\mu -1}{k}$ .",
"The Gegenbauer polynomials are usually defined by the following generating function $(1-2uz+z^2)^{-\\mu }=\\sum _{k=0}^\\infty C^{\\mu }_k(u)z^k,\\quad |z|<1,~|u|<1.$ The polynomials $C_k^{\\mu }$ , $k=0, 1, \\dots $ , are orthogonal in the space $L^2([-1, 1], w)$ with weight $w(u):= (1-u^2)^{\\mu -1/2}$ , see [33] or [26].",
"In the case $d=2$ the kernel of the orthogonal projector onto $\\mathcal {H}_k$ takes the form $Z_0(x\\cdot y)=\\frac{1}{2\\pi },\\quad Z_k(x\\cdot y)=\\frac{1}{\\pi }T_k(x\\cdot y),\\quad k\\ge 1,$ where $T_k(u):=\\cos (n\\arccos u)$ is the $k$ -th degree Chebyshev polynomial of the first kind.",
"We refer the reader to [23], [31] for the basics of spherical harmonics.",
"As is well known (see e.g.",
"[8]) the spherical harmonics are eigenfunctions of the Laplace-Beltrami operator $\\Delta _0$ on ${{\\mathbb {S}}^{d-1}}$ , namely, $-\\Delta _0Y(x)=k(k+d-2)Y(x),\\quad ~x\\in {{\\mathbb {S}}^{d-1}},~\\forall Y\\in \\mathcal {H}_k.$ The set of all band-limited functions (i.e.",
"spherical polynomials) on ${{\\mathbb {S}}^{d-1}}$ of degree $\\le N$ will be denoted by $\\Pi _N$ , i.e.",
"$\\Pi _N :=\\bigoplus _{k=0}^N \\mathcal {H}_k$ .",
"The Poisson kernel on the unit ball $B^d$ is given by $P(y, x):=\\sum _{k= 0}^\\infty |x|^kZ_k\\Big (\\frac{x}{|x|}\\cdot y\\Big )=\\frac{1}{\\omega _d}\\frac{1-|x|^2}{|x-y|^d},\\quad |x| <1,\\; y\\in {{\\mathbb {S}}^{d-1}}.$ Kernels on the sphere ${{\\mathbb {S}}^{d-1}}$ of the form $\\Lambda _N(x\\cdot y) := \\sum _{k=0}^\\infty \\lambda (k/N)Z_k(x\\cdot y),\\quad x, y\\in {{\\mathbb {S}}^{d-1}},\\quad N\\ge 1,$ where $\\lambda \\in C^\\infty [0,\\infty )$ is compactly supported, will play a key role in this article.",
"Observe that in this case $\\Lambda _N(u) := \\sum _{k=0}^\\infty \\lambda (k/N)Z_k(u), \\quad u\\in [-1, 1],$ is simply a polynomial kernel.",
"The localization of this kernel is given in the following Theorem 2.1 Let $\\nu \\ge 0$ and $M\\in {\\mathbb {N}}$ .",
"Assume $\\lambda \\in C^\\infty [0, \\infty )$ , $\\Vert \\lambda ^{(m)}\\Vert _\\infty \\le \\kappa $ for $0\\le m\\le M$ and either $\\operatorname{supp}\\lambda \\subset [1/4, 4]$ or $\\operatorname{supp}\\lambda \\subset [0, 2]$ and $\\lambda (t)=1$ for $t\\in [0, 1]$ .",
"Then there exists a constant $c>0$ depending only on $M$ , $\\nu $ , and $d$ such that for any $N\\ge 1$ the kernel $\\Lambda _N$ from $(\\ref {def-Lam})-(\\ref {def-Lam-2})$ obeys $|\\Lambda _N^{(\\nu )}(\\cos \\theta )| \\le c \\kappa \\frac{N^{d-1+2\\nu }}{(1+N|\\theta |)^{M}},\\quad |\\theta |\\le \\pi ,$ and hence $|\\Lambda _N^{(\\nu )}(x\\cdot y)| \\le c \\kappa \\frac{N^{d-1+2\\nu }}{(1+N\\rho (x, y))^{M}},\\quad x, y\\in {{\\mathbb {S}}^{d-1}}.$ Furthermore, for $x, y, z\\in {{\\mathbb {S}}^{d-1}}$ $|\\Lambda _N(x\\cdot z)-\\Lambda _N(y\\cdot z)| \\le c\\kappa \\frac{\\rho (x, y) N^d}{(1+N\\rho (x, z))^{M}},\\quad \\hbox{if}\\quad \\rho (x, y) \\le N^{-1}.$ For a proof, see [24] and [25], also [17]."
],
[
"Maximal $\\delta $ -nets and cubature formulas on the sphere",
"For discretization of integrals and construction of frames on ${{\\mathbb {S}}^{d-1}}$ we shall need cubature formulas, which are naturally constructed using maximal $\\delta $ -nets on ${{\\mathbb {S}}^{d-1}}$ .",
"Definition.",
"Given $\\delta >0$ we say that a finite set $\\mathcal {Z}\\subset {{\\mathbb {S}}^{d-1}}$ is a maximal $\\delta $ -net on ${{\\mathbb {S}}^{d-1}}$ if (i) $\\rho (\\zeta _1,\\zeta _2)\\ge \\delta $ for all $\\zeta _1,\\zeta _2\\in \\mathcal {Z}$ , $\\zeta _1\\ne \\zeta _2$ , and (ii) $\\cup _{\\zeta \\in \\mathcal {Z}}B(\\zeta ,\\delta )={{\\mathbb {S}}^{d-1}}$ .",
"Clearly, a maximal $\\delta $ -net on ${{\\mathbb {S}}^{d-1}}$ exists for any $\\delta >0$ .",
"For every maximal $\\delta $ -net $\\mathcal {Z}$ it is easy to construct (see [6]) a disjoint partition $\\lbrace A_\\zeta \\rbrace _{\\zeta \\in \\mathcal {Z}}$ of ${{\\mathbb {S}}^{d-1}}$ consisting of measurable sets such that $B(\\zeta ,\\delta /2)\\subset A_\\zeta \\subset B(\\zeta ,\\delta ),\\quad \\zeta \\in \\mathcal {Z}.$ Two kinds of cubature formulas on ${{\\mathbb {S}}^{d-1}}$ will be utilized.",
"Simple cubature formulas on ${{\\mathbb {S}}^{d-1}}$ .",
"Let $0<\\gamma \\le 1$ be a parameter to be selected.",
"Let $\\mathcal {Z}_j\\subset {{\\mathbb {S}}^{d-1}}$ ($j\\in {\\mathbb {N}}$ ) be a maximal $\\delta _j$ -net on ${{\\mathbb {S}}^{d-1}}$ with $\\delta _j:=\\gamma 2^{-j+1}$ .",
"We shall use the cubature formula $\\int _{{{\\mathbb {S}}^{d-1}}} f(x) d\\sigma (x)\\approx \\sum _{\\zeta \\in \\mathcal {Z}_j} w_\\zeta f(\\zeta ), \\quad w_\\zeta :=|A_\\zeta |,$ where $A_\\zeta $ is from (REF ) with $\\mathcal {Z}=\\mathcal {Z}_j$ , $\\delta =\\delta _j$ .",
"The cubature (REF ) is apparently exact for all constants.",
"Evidently, $w_\\zeta =|A_\\zeta |\\sim (\\gamma 2^{-j+1})^{d-1}$ with constants of equivalence depending only on $d$ .",
"Note that (REF ) implies that the number of elements in $\\mathcal {Z}_j$ is $\\le c(d)(\\gamma ^{-1}2^j)^{d-1}$ .",
"Further, given $j\\in {\\mathbb {N}}$ we define a map $\\zeta $ from ${{\\mathbb {S}}^{d-1}}$ to $\\mathcal {Z}_j$ as follows: For every $y\\in {{\\mathbb {S}}^{d-1}}$ we set $\\zeta (y):=\\eta \\in \\mathcal {Z}_j$ if $y\\in A_\\eta $ .",
"We shall use this map in Lemmas REF and REF below.",
"Nontrivial cubature formulas on ${{\\mathbb {S}}^{d-1}}$ .",
"Let $\\mathcal {X}_j\\subset {{\\mathbb {S}}^{d-1}}$ be a maximal $\\delta _j$ -net on ${{\\mathbb {S}}^{d-1}}$ with $\\delta _j:=\\gamma 2^{-j+1}$ , $0<\\gamma <1$ , $j\\ge 1$ .",
"In [24] it is shown that there exist $\\gamma $ ($0<\\gamma <1$ ), depending only on $d$ , and weights $\\lbrace \\widetilde{w}_\\xi \\rbrace _{\\xi \\in \\mathcal {X}_j}$ , satisfying $c_7^{-1} 2^{-j(d-1)}\\le \\widetilde{w}_\\xi \\le c_7 2^{-j(d-1)}, \\quad \\xi \\in \\mathcal {X}_j,$ with constant $c_7$ depending only on $d$ , such that the cubature formula $\\int _{{\\mathbb {S}}^{d-1}}f(x)d\\sigma (x)\\approx \\sum _{\\xi \\in \\mathcal {X}_j} \\widetilde{w}_\\xi f(\\xi )$ is exact for all spherical harmonics $f$ of degree $\\le 2^{j+1}$ , $j\\ge 1$ .",
"As before, the number of nodes in $\\mathcal {X}_j$ is $\\le c(d)(\\gamma ^{-1}2^j)^{d-1}$ , since $\\mathcal {X}_j$ is a maximal $\\delta _j$ -net on ${{\\mathbb {S}}^{d-1}}$ .",
"Also the disjoint partition $\\lbrace A_\\xi \\rbrace _{\\xi \\in \\mathcal {X}_j}$ of ${{\\mathbb {S}}^{d-1}}$ exists with $B(\\xi ,\\delta _j/2)\\subset A_\\xi \\subset B(\\xi ,\\delta _j)$ , but the equality $\\widetilde{w}_\\xi =|A_\\xi |$ does not hold in general."
],
[
"Maximal operator",
"The maximal operator is an important technical tool when dealing with Besov and Triebel-Lizorkin spaces.",
"We shall use the following version of the Hardy-Littlewood maximal operator: $\\mathcal {M}_tf(x):=\\sup _{B\\ni x}\\Big (\\frac{1}{|B|}\\int _B |f|^t\\, d\\sigma \\Big )^{1/t},\\quad x\\in {{\\mathbb {S}}^{d-1}}, \\; t>0,$ where the sup is over all spherical caps $B\\subset {{\\mathbb {S}}^{d-1}}$ such that $x\\in B$ .",
"The Fefferman-Stein vector-valued maximal inequality (see [30]) can be written in the form: If $0<p<\\infty , 0<q\\le \\infty $ , and $0<t<\\min \\lbrace p,q\\rbrace $ , then for any sequence of measurable functions $\\lbrace f_\\nu \\rbrace $ on ${{\\mathbb {S}}^{d-1}}$ $\\Big \\Vert \\Big (\\sum _{\\nu }|\\mathcal {M}_tf_\\nu (\\cdot )|^q\\Big )^{1/q} \\Big \\Vert _{L^p}\\le {\\tilde{c}}_1\\Big \\Vert \\Big (\\sum _{\\nu }| f_\\nu (\\cdot )|^q\\Big )^{1/q}\\Big \\Vert _{L^p}.$ From Theorem 2.1 in [12] it follows that the constant ${\\tilde{c}}_1$ above can be written in the form ${\\tilde{c}}_1=\\Big (c^* \\max \\big \\lbrace p/t, (p/t-1)^{-1}\\big \\rbrace \\max \\big \\lbrace 1, (q/t-1)^{-1}\\big \\rbrace \\Big )^{1/t},$ where $c^*>0$ is a constant depending only on $d$ .",
"Note that the area/volume of a spherical cap $B(x,r)$ on ${{\\mathbb {S}}^{d-1}}$ , $d\\ge 2$ , is given by $|B(x,r)|=\\omega _{d-1}\\int _0^r \\sin ^{d-2}v\\,dv.$ Hence $|B(x_1,r_1)|/|B(x_2,r_2)|\\le (r_1/r_2)^{d-1},\\quad 0<r_2\\le r_1\\le \\pi ,\\quad x_1,x_2\\in {{\\mathbb {S}}^{d-1}},$ $1/{\\tilde{c}}_2\\le |B(x,r)|/r^{d-1}\\le {\\tilde{c}}_2,\\quad 0<r\\le \\pi ,\\quad x\\in {{\\mathbb {S}}^{d-1}},$ where ${\\tilde{c}}_2$ is a constant depending only on $d$ ."
],
[
"Inner products of zonal functions",
"A function $f$ on ${{\\mathbb {S}}^{d-1}}$ is zonal if it is invariant under rotation about a fixed axis.",
"If this axis is in the direction of $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , then $f$ can be represented as $f(x)=F(\\eta \\cdot x)$ , $x\\in {{\\mathbb {S}}^{d-1}}$ , for an appropriate function $F:[-1,1]\\rightarrow {\\mathbb {R}}$ .",
"Lemma 2.2 Let $F, G\\in L^\\infty [-1,1]$ .",
"Then there exists $H\\in C[-1,1]$ such that $H(x\\cdot z)=\\int _{{{\\mathbb {S}}^{d-1}}}F(x\\cdot y)G(y\\cdot z)\\,d\\sigma (y),\\quad \\forall x,z\\in {{\\mathbb {S}}^{d-1}}.$ Assume first that $F$ and $G$ are algebraic polynomials of degree $m$ .",
"Then we can expend them in Gegenbauer polynomials to obtain $F=\\sum _{k=0}^m \\hat{F}_kZ_k$ and $G=\\sum _{k=0}^m \\hat{G}_kZ_k$ .",
"Using that $Z_k(x\\cdot y)$ is the kernel of the orthogonal projector onto $\\mathcal {H}_k$ we have $\\int _{{{\\mathbb {S}}^{d-1}}}Z_k(x\\cdot y)Z_k(y\\cdot z)\\,d\\sigma (y)=Z_k(x\\cdot z)$ (see (REF )).",
"Therefore, $\\int _{{{\\mathbb {S}}^{d-1}}}F(x\\cdot y)G(y\\cdot z)\\,d\\sigma (y)=\\sum _{k=0}^m \\hat{F}\\hat{G}Z_k(x\\cdot z)=H(x\\cdot z),$ where $H$ is an algebraic polynomial of degree $m$ .",
"Thus (REF ) holds for polynomials.",
"Finally, a limiting argument implies that (REF ) is valid in general.",
"From Lemma REF it follows that for any $F, G\\in L^\\infty [-1,1]$ $\\int _{{{\\mathbb {S}}^{d-1}}}F(x\\cdot y)G(y\\cdot z)\\,d\\sigma (y)=\\int _{{{\\mathbb {S}}^{d-1}}}F(z\\cdot y)G(y\\cdot x)\\,d\\sigma (y),\\quad \\forall x,z\\in {{\\mathbb {S}}^{d-1}}.$"
],
[
"Inner products of localized functions",
"The estimation of the inner products of well localized functions and functions with small moments on the sphere will play a key role in our further development.",
"The following proposition is an analogue of [10].",
"We replace the vanishing moment condition used in [10] by the weaker “small moments” condition (REF ).",
"Proposition 2.3 Let $K\\in {\\mathbb {N}}$ , $M>K+d-1$ , $N_2\\ge N_1\\ge 1$ $(N_1,N_2\\in {\\mathbb {R}})$ , and $\\kappa _1, \\kappa _2>0$ .",
"Assume $f\\in L^\\infty ({{\\mathbb {S}}^{d-1}})$ and $g\\in W_\\infty ^K({{\\mathbb {S}}^{d-1}})$ , see §REF .",
"Furthermore, assume that for some $x_1,x_2\\in {{\\mathbb {S}}^{d-1}}$ $\\left|\\partial ^\\beta \\breve{g}(y)\\right|&\\le \\frac{\\kappa _1 N_1^{|\\beta |+d-1}}{(1+N_1\\rho (x_1,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}},~0\\le |\\beta |\\le K, \\\\|f(y)|&\\le \\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (x_2,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}}, \\quad \\hbox{and}$ $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}y^\\beta f(y)\\, d\\sigma (y)\\Big |\\le \\kappa _2 N_2^{-K},\\quad 0\\le |\\beta |\\le K-1.$ Then $|\\left\\langle g,f\\right\\rangle |=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}g(y)\\overline{f(y)}\\, d\\sigma (y)\\Big |\\le c_1\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)^{K}N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M},$ where $c_1$ depends only on $d$ , $K$ , and $M$ .",
"Above $\\breve{g}(y):=g(y/|y|)$ for $y\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace $ just as in (REF ).",
"For cases where condition (REF ) may not be satisfied we modify Proposition REF as follows Proposition 2.4 Let $M>d$ , $N_2\\ge N_1\\ge 1$ , and $\\kappa _1, \\kappa _2>0$ .",
"Let $f\\in L^\\infty ({{\\mathbb {S}}^{d-1}})$ and $g\\in W_\\infty ^1({{\\mathbb {S}}^{d-1}})$ .",
"Also, assume that for some $x_1,x_2\\in {{\\mathbb {S}}^{d-1}}$ $\\left|\\partial ^\\alpha \\breve{g}(y)\\right|&\\le \\frac{\\kappa _1 N_1^{d}}{(1+N_1\\rho (x_1,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}},~|\\alpha |=1, \\\\|f(y)|&\\le \\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (x_2,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}}.$ Then $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}g(y)\\overline{f(y)}\\, d\\sigma (y)-g(x_2)\\int _{{{\\mathbb {S}}^{d-1}}}\\overline{f(y)}\\, d\\sigma (y)\\Big |\\le c_2\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M},$ where $c_2$ depends only on $d$ and $M$ .",
"In cases where only the function localizations are known/matter we use Proposition 2.5 Let $M> d-1$ , $N_2\\ge N_1\\ge 1$ , $\\kappa _1, \\kappa _2>0$ .",
"Let $f,g\\in L^\\infty ({{\\mathbb {S}}^{d-1}})$ , and for some $x_1,x_2\\in {{\\mathbb {S}}^{d-1}}$ $\\left|g(y)\\right|\\le \\frac{\\kappa _1 N_1^{d-1}}{(1+N_1\\rho (x_1,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}},$ $|f(y)|\\le \\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (x_2,y))^M},\\quad \\forall y\\in {{\\mathbb {S}}^{d-1}}.$ Then $|\\left\\langle g,f\\right\\rangle |=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}g(y)\\overline{f(y)}\\, d\\sigma (y)\\Big |\\le c_3\\frac{\\kappa _1 \\kappa _2 N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M},$ where $c_3$ depends only on $d$ and $M$ .",
"To streamline our presentation we defer the proofs of Propositions REF , REF , and REF to Section ."
],
[
"Spaces of functions and distributions on the ball and sphere",
"The theory of harmonic Besov and Triebel-Lizorkin spaces on $B^d$ , and their relation with the respective Besov and Triebel-Lizorkin spaces of distributions on ${{\\mathbb {S}}^{d-1}}$ is developed in [15].",
"In this section we review all definitions and results that will be needed from [15].",
"Denote by $\\mathcal {H}(B^d)$ the set of all harmonic functions on the unit ball $B^d$ in ${\\mathbb {R}}^d$ ."
],
[
"Harmonic Besov and Triebel-Lizorkin spaces on $B^d$",
"It is convenient to define the harmonic Besov and Triebel-Lizorkin spaces on $B^d$ by using their expansion in solid spherical harmonics.",
"As in § REF let $\\lbrace Y_{kj}: j=1, \\dots , N(k, d)\\rbrace $ be a real-valued orthonormal basis for $\\mathcal {H}_k$ .",
"The harmonic coefficients of $U\\in \\mathcal {H}(B^d)$ are defined by $b_{k\\nu }(U):= \\frac{1}{a^k}\\int _{{\\mathbb {S}}^{d-1}}U(a\\eta )Y_{k\\nu }(\\eta ) d\\sigma (\\eta )$ for some $0<a<1$ .",
"It is an important observation that the coefficients are independent of $a$ for all $0<a<1$ .",
"This implies the representation $U(r\\xi ) = \\sum _{k=0}^\\infty \\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(U)r^kY_{k\\nu }(\\xi ),\\quad 0\\le r<1, ~\\xi \\in {{\\mathbb {S}}^{d-1}},$ where the convergence is absolute and uniform on every compact subset of $B^d$ .",
"For $U\\in \\mathcal {H}(B^d)$ and $\\beta \\in {\\mathbb {R}}$ we define $J^\\beta U(r\\xi ):= \\sum _{k=0}^\\infty r^k(k+1)^{-\\beta }\\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(U)Y_{k\\nu }(\\xi ),\\quad 0\\le r<1, ~\\xi \\in {{\\mathbb {S}}^{d-1}}.$ The above series converges absolutely and uniformly on every compact subset of $B^d$ and hence $J^\\beta U$ is a well defined harmonic function on $B^d$ .",
"Definition 3.1 Let $s\\in {\\mathbb {R}}$ , $0<q\\le \\infty $ , and $\\beta :=s+1$ .",
"$(a)$ The harmonic Besov space $B^{sq}_p(\\mathcal {H})$ , $0<p\\le \\infty $ , is defined as the set of all $U\\in \\mathcal {H}(B^d)$ such that $\\Vert U\\Vert _{B^{sq}_p(\\mathcal {H})}:= \\Big (\\int _0^1 (1-r)^{(\\beta -s)q}\\Vert J^{-\\beta } U(r\\cdot )\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}^q \\frac{dr}{1-r}\\Big )^{1/q} <\\infty \\quad \\hbox{if}\\; q\\ne \\infty $ and $\\Vert U\\Vert _{B^{s\\infty }_p(\\mathcal {H})} := \\sup _{0<r<1} (1-r)^{\\beta -s}\\Vert J^{-\\beta } U(r\\cdot )\\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty .$ $(b)$ The harmonic Triebel-Lizorkin space $F^{sq}_p(\\mathcal {H})$ , $0<p<\\infty $ , is defined as the set of all $U\\in \\mathcal {H}(B^d)$ such that $\\Vert U\\Vert _{F^{sq}_p(\\mathcal {H})}:= \\Big \\Vert \\Big (\\int _0^1 (1-r)^{(\\beta -s)q}|J^{-\\beta } U(r\\cdot )|^q \\frac{dr}{1-r}\\Big )^{1/q}\\Big \\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty \\quad \\hbox{if}\\; q\\ne \\infty $ and $\\Vert U\\Vert _{F^{s\\infty }_p(\\mathcal {H})} := \\Big \\Vert \\sup _{0<r<1} (1-r)^{\\beta -s}|J^{-\\beta } U(r\\cdot )|\\Big \\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty .$ Choosing an arbitrary $\\beta >s$ above results in equivalent quasi-norms for the spaces $B^{sq}_p(\\mathcal {H})$ and $F^{sq}_p(\\mathcal {H})$ ."
],
[
"Besov and Triebel-Lizorkin spaces on ${{\\mathbb {S}}^{d-1}}$",
"The Besov and Triebel-Lizorkin spaces on ${{\\mathbb {S}}^{d-1}}$ in general are spaces of distributions.",
"As test functions we use the class $\\mathcal {S}:= C^\\infty ({{\\mathbb {S}}^{d-1}})$ of all functions $\\phi $ on ${{\\mathbb {S}}^{d-1}}$ such that $\\Vert Z_k*\\phi \\Vert _2 \\le c(\\phi , m)(1+k)^{-m}, \\quad \\forall k, m\\ge 0.$ Recall that the convolution $Z_k*\\phi $ is defined in (REF ).",
"The topology on $\\mathcal {S}$ is defined by the sequence of norms $P_m(\\phi ):= \\sum _{k=0}^\\infty (k+1)^m \\Vert Z_k*\\phi \\Vert _2= \\sum _{k=0}^\\infty (k+1)^m \\Big (\\sum _{\\nu =1}^{N(k, d)} |\\langle \\phi , Y_{k\\nu }\\rangle |^2\\Big )^{1/2}.$ $\\mathcal {S}$ is complete in this topology.",
"Observe that all $Y_{k\\nu }\\in \\mathcal {S}$ and hence by (REF ) $Z_k(x\\cdot y)\\in \\mathcal {S}$ as a function of $x$ for every fixed $y$ and as function of $y$ for every fixed $x$ .",
"The space $\\mathcal {S}^{\\prime }:=\\mathcal {S}^{\\prime }({{\\mathbb {S}}^{d-1}})$ of distributions on ${{\\mathbb {S}}^{d-1}}$ is defined as the space of all continuous linear functionals on $\\mathcal {S}$ .",
"The pairing of $f\\in \\mathcal {S}^{\\prime }$ and $\\phi \\in \\mathcal {S}$ will be denoted by $\\langle f, \\phi \\rangle := f(\\overline{\\phi })$ , which is consistent with the inner product on $L^2({{\\mathbb {S}}^{d-1}})$ .",
"More precisely, $\\mathcal {S}^{\\prime }$ consists of all linear functionals $f$ on $\\mathcal {S}$ for which there exist constants $c>0$ and $m\\in {\\mathbb {N}}_0$ such that $|\\langle f, \\phi \\rangle | \\le c P_m(\\phi ),\\quad \\forall \\phi \\in \\mathcal {S}.$ For any $f\\in \\mathcal {S}^{\\prime }$ we define $Z_k*f$ by $Z_k*f(x) :=\\langle f, \\overline{Z_k(x\\cdot \\bullet )}\\rangle =\\langle f, Z_k(x\\cdot \\bullet )\\rangle ,$ where on the right $f$ is acting on $\\overline{Z_k(x\\cdot y)}=Z_k(x\\cdot y)$ as a function of $y$ ($Z_k$ is real-valued).",
"Observe that the representation $f=\\sum _{k=0}^\\infty Z_k*f, \\quad \\forall f\\in \\mathcal {S}^{\\prime }$ holds with convergence in distributional sense.",
"Definition 3.2 Let $s\\in {\\mathbb {R}}$ , $0<q\\le \\infty $ , and $\\varphi $ satisfy the conditions: $\\varphi \\in C^\\infty ({\\mathbb {R}}_+)$ , $\\operatorname{supp}\\varphi \\subset [1/2, 2]$ , and $|\\varphi (u)|\\ge c>0$ for $u\\in [3/5, 5/3]$ .",
"For a distribution $f\\in \\mathcal {S}^{\\prime }$ set $\\Phi _0*f = Z_0*f,\\quad \\Phi _j*f = \\sum _{k=0}^\\infty \\varphi \\Big (\\frac{k}{2^{j-1}}\\Big )Z_k*f, ~ j\\ge 1,$ where $Z_k*f$ is defined in (REF ).",
"$(a)$ The Besov space $\\mathcal {B}^{sq}_p:=\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ , $0<p\\le \\infty $ , is defined as the set of all distributions $f\\in \\mathcal {S}^{\\prime }$ such that $\\Vert f\\Vert _{\\mathcal {B}^{s q}_p} :=\\Big (\\sum _{j=0}^\\infty \\Big (2^{sj}\\Vert \\Phi _j*f\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}\\Big )^q\\Big )^{1/q}< \\infty ,$ where the $\\ell ^q$ -norm is replaced by the sup-norm if $q=\\infty $ .",
"$(b)$ The Triebel-Lizorkin space $\\mathcal {F}^{sq}_p:=\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ , $0<p<\\infty $ , is defined as the set of all distributions $f\\in \\mathcal {S}^{\\prime }$ such that $\\Vert f\\Vert _{\\mathcal {F}^{s q}_p} :=\\Big \\Vert \\Big (\\sum _{j=0}^\\infty \\big (2^{sj}|\\Phi _j*f(\\cdot )|\\big )^q\\Big )^{1/q}\\Big \\Vert _{L^p({{\\mathbb {S}}^{d-1}})}< \\infty ,$ where the $\\ell ^q$ -norm is replaced by the sup-norm if $q=\\infty $ .",
"Note that the definitions of the Besov and Triebel-Lizorkin spaces above are independent of the particular selection of the function $\\varphi $ with the required properties, that is, different $\\varphi $ 's produce equivalent quasi-norms."
],
[
"Identification of harmonic Besov and Triebel-Lizorkin spaces",
"We are interested in harmonic functions $U\\in \\mathcal {H}(B^d)$ with coefficients of at most polynomial growth: $|b_{k\\nu }(U)| \\le c(k+1)^\\gamma ,\\quad \\nu =1, \\dots , N(k, d), \\;\\; k=0, 1, \\dots ,$ for some constants $\\gamma , c>0$ .",
"The functions in the harmonic Besov and Triebel-Lizorkin spaces have this property.",
"The relationship between harmonic functions on $B^d$ and distributions on ${{\\mathbb {S}}^{d-1}}$ is clarified by the following Proposition 3.3 $(a)$ To any $U\\in \\mathcal {H}(B^d)$ represented by $(\\ref {rep-U-3a})$ with coefficients satisfying $(\\ref {coef-growth})$ there corresponds a distribution $f\\in \\mathcal {S}^{\\prime }$ , $f=f_U$ , $($ the boundary value function/distribution of $U$$)$ defined by $f:= \\sum _{k=0}^\\infty \\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(U) Y_{k\\nu }\\quad \\hbox{$($convergence in $\\mathcal {S}^{\\prime }$$)$}$ with coefficients $b_{k\\nu }(U)=\\langle f, Y_{k\\nu }\\rangle $ .",
"$(b)$ To any distribution $f\\in \\mathcal {S}^{\\prime }$ with coefficients $b_{k\\nu }(f):=\\langle f, Y_{k\\nu }\\rangle $ there corresponds a harmonic function $U\\in \\mathcal {H}(B^d)$ , $U=U_f$ , $($ the harmonic extension of $f$ to $B^d$$)$ , defined by $U(x) = \\sum _{k=0}^\\infty \\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(f)|x|^kY_{k\\nu }\\Big (\\frac{x}{|x|}\\Big ),\\quad |x|<1,$ with coefficients $b_{k\\nu }(U)=b_{k\\nu }(f)$ obeying $(\\ref {coef-growth})$ , where the series converges uniformly on every compact subset of $B^d$ .",
"$(c)$ For every $U\\in \\mathcal {H}(B^d)$ we have $U_{f_U} = U$ and for every $f\\in \\mathcal {S}^{\\prime }$ we have $f_{U_f} = f$ .",
"The principle results of this subsection are: Theorem 3.4 Let $s\\in {\\mathbb {R}}$ , $0<p < \\infty $ , $0<q\\le \\infty $ .",
"A harmonic function $U\\in F^{s q}_p(\\mathcal {H})$ if and only if its boundary value distribution $f=f_U$ defined by $(\\ref {rep-f-33})$ belongs to $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ , moreover $\\Vert U\\Vert _{F^{s q}_p} \\sim \\Vert f\\Vert _{\\mathcal {F}^{s q}_p}$ .",
"Theorem 3.5 Let $s\\in {\\mathbb {R}}$ , $0<p, q\\le \\infty $ .",
"A harmonic function $U\\in B^{s q}_p(\\mathcal {H})$ if and only if its boundary value distribution $f=f_U$ defined by $(\\ref {rep-f-33})$ belongs to $\\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}})$ , moreover $\\Vert U\\Vert _{B^{s q}_p} \\sim \\Vert f\\Vert _{\\mathcal {B}^{s q}_p}$ ."
],
[
"Harmonic Hardy spaces",
"Here we consider the harmonic Hardy spaces $\\mathcal {H}^p(B^d)$ on the ball (usually denoted by $h^p(B^d)$ ).",
"Definition 3.6 The space $\\mathcal {H}^p :=\\mathcal {H}^p(B^d)$ , $0< p\\le \\infty $ , is defined as the set of all harmonic functions $U\\in \\mathcal {H}(B^d)$ such that $\\Vert U\\Vert _{\\mathcal {H}^p} := \\Vert \\sup _{0\\le r<1} |U(r\\cdot )|\\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty .$ The following identification of harmonic Hardy spaces holds.",
"Theorem 3.7 A harmonic function $U\\in \\mathcal {H}^p(B^d)$ , $0<p<\\infty $ , if and only if its boundary distribution $f_U \\in \\mathcal {F}^{02}_p({{\\mathbb {S}}^{d-1}})$ and $\\Vert U\\Vert _{\\mathcal {H}^p} \\sim \\Vert U\\Vert _{F^{02}_p(\\mathcal {H})} \\sim \\Vert f_U\\Vert _{\\mathcal {F}^{02}_p({{\\mathbb {S}}^{d-1}})}.$ Furthermore, $U\\in \\mathcal {H}^p(B^d)$ , $1<p<\\infty $ , if and only if $f_U \\in L^p({{\\mathbb {S}}^{d-1}})$ and $\\Vert U\\Vert _{\\mathcal {H}^p} \\sim \\Vert f_U\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}.$ In addition, for any $U\\in \\mathcal {H}^p(B^d)$ , $1<p<\\infty $ , $\\Vert U\\Vert _{\\mathcal {H}^p} \\sim \\sup _{0\\le r<1} \\Vert U(r\\cdot )\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}$ and the right-hand side quantity is sometimes used to define $\\Vert U\\Vert _{\\mathcal {H}^p}$ for $p>1$ .",
"To prove this theorem we shall need the following Lemma 3.8 If $U\\in \\mathcal {H}^p(B^d)$ , $0<p\\le \\infty $ , then $|b_{k\\nu }(U)| \\le c(k+1)^\\gamma \\Vert U\\Vert _{\\mathcal {H}^p},\\quad \\nu =1, \\dots , N(k, d), \\;\\; k=0, 1, \\dots ,$ for some constants $\\gamma , c>0$ , depending only on $d$ and $p$ , i.e.",
"inequalities (REF ) are valid.",
"Consequently, there exists a distribution $f_U\\in \\mathcal {S}^{\\prime }$ with spherical harmonic coefficients the same as the coefficients of $U$ , which in turn leads to $U=P*f_U$ with $P(y, x)$ being the Poisson kernel, see (REF ).",
"Here $P*f_U$ is defined by $P*f_U(x):= \\langle f_U, \\overline{P(\\cdot , x)}\\rangle = \\langle f_U, P(\\cdot , x)\\rangle ,$ where $f_U$ acts on $\\overline{P(y, x)} = P(y, x)$ as a function of $y$ $($$P(y, x)$ is real-valued$)$ .",
"To prove (REF ) we invoke Proposition 4.2 from [15] which, in particular, asserts that for any $U\\in B_p^{sq}(\\mathcal {H})$ , $s\\in {\\mathbb {R}}$ , $0<p,q\\le \\infty $ , $|b_{k\\nu }(U)| \\le c(k+1)^\\gamma \\Vert U\\Vert _{B_p^{sq}(\\mathcal {H})},\\quad \\nu =1, \\dots , N(k, d), \\;\\; k=0, 1, \\dots ,$ where the constants $\\gamma , c>0$ depend only on $d,s,p,q$ .",
"If $U\\in \\mathcal {H}^p(B^d)$ then by Definition REF with $\\beta =0$ $\\Vert U\\Vert _{B_p^{-1,1}(\\mathcal {H})}=\\int _0^1\\Vert U(r\\cdot )\\Vert _p dr\\le \\Vert \\sup _{0\\le r<1}|U(r\\cdot )|\\Vert _p= \\Vert U\\Vert _{\\mathcal {H}^p},$ which implies that $\\mathcal {H}^p(B^d)$ is continuously embedded in the harmonic Besov space $B_p^{-1,1}(\\mathcal {H})$ .",
"Now, the above and (REF ) imply (REF ).",
"We set $f_U:= \\sum _{k\\ge 0}\\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(U) Y_{k\\nu }.$ Inequalities (REF ) and Proposition REF lead to the conclusion that the series in (REF ) converge in $\\mathcal {S}^{\\prime }$ and defines a distribution $f_U\\in \\mathcal {S}^{\\prime }$ with coefficients $b_{k\\nu }(f_U):=b_{k\\nu }(U)$ .",
"In turn, this implies that $\\sum _{\\nu =1}^{N(k, d)} b_{k\\nu }(U) Y_{k\\nu }\\Big (\\frac{x}{|x|}\\Big ) = Z_k*f_U\\Big (\\frac{x}{|x|}\\Big ),\\quad |x|<1,$ and hence $U(x)&= \\sum _{k=0}^\\infty |x|^k Z_k*f_U\\Big (\\frac{x}{|x|}\\Big )= \\sum _{k=0}^\\infty |x|^k \\Big \\langle f_U, Z_k\\Big (\\cdot , \\frac{x}{|x|}\\Big )\\Big \\rangle \\\\&=\\lim _{m\\rightarrow \\infty } \\Big \\langle f_U, \\sum _{k=0}^m |x|^k Z_k\\Big (\\cdot , \\frac{x}{|x|}\\Big )\\Big \\rangle = \\langle f_U, P(\\cdot , x)\\rangle = P*f_U(x).$ Here we used the obvious fact that for any $|x|<1$ $P(y, x)=\\lim _{m\\rightarrow \\infty }\\sum _{k=0}^m |x|^k Z_k\\Big (y, \\frac{x}{|x|}\\Big )\\quad \\hbox{(convergence in $\\mathcal {S}$)}.$ The proof is complete.",
"The Hardy space $\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})$ , $0<p<\\infty $ , on the sphere is defined as the set of all distributions $f\\in \\mathcal {S}^{\\prime }$ such that $\\Vert f\\Vert _{\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})} :=\\Vert \\sup _{0\\le r<1}|P*f(r \\cdot )|\\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty .$ A frame characterization of the Triebel-Lizorkin spaces on ${{\\mathbb {S}}^{d-1}}$ has been established in [25] (see Theorem REF (b) below), which along with the same frame characterization of the Hardy spaces $\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})$ from [7] implies that $\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})=\\mathcal {F}_p^{0 2}({{\\mathbb {S}}^{d-1}})$ , $0<p<\\infty $ , with equivalent quasi-norms.",
"By Lemma REF and (REF ) it follows that $U\\in \\mathcal {H}^p(B^d)$ , $0<p<\\infty $ , if and only if $f_U \\in \\mathcal {H}^p({{\\mathbb {S}}^{d-1}})$ and $\\Vert U\\Vert _{\\mathcal {H}^p(B^d)} = \\Vert f_U\\Vert _{\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})}$ .",
"This along with the above observation and Theorem REF implies $\\Vert U\\Vert _{\\mathcal {H}^p(B^d)} = \\Vert f_U\\Vert _{\\mathcal {H}^p({{\\mathbb {S}}^{d-1}})} \\sim \\Vert f_U\\Vert _{\\mathcal {F}_p^{0 2}({{\\mathbb {S}}^{d-1}})} \\sim \\Vert U\\Vert _{\\mathcal {F}_p^{0 2}(\\mathcal {H})},$ which confirms (REF ).",
"The equivalence $\\Vert U\\Vert _{\\mathcal {H}^p} \\sim \\Vert f_U\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}$ , when $1<p<\\infty $ , follows by Lemma REF and the maximal inequality just as in the case of Hardy spaces on ${\\mathbb {R}}^d$ , see [30].",
"For the equivalence $\\sup _{0\\le r<1} \\Vert U(r\\cdot )\\Vert _{L^p({{\\mathbb {S}}^{d-1}})} \\sim \\Vert f_U\\Vert _{L^p({{\\mathbb {S}}^{d-1}})}$ , $1<p<\\infty $ , see [3].",
"The proof is complete."
],
[
"Frame decomposition of distribution spaces on ${{\\mathbb {S}}^{d-1}}$",
"We next recall the construction of the frame (needlets) on ${{\\mathbb {S}}^{d-1}}$ from [25].",
"Note that in dimension $d=2$ the Meyer's periodic wavelets (see [22]) form a basis with the desired properties.",
"The first step in the construction of needlets on ${{\\mathbb {S}}^{d-1}}$ , $d>2$ , is the selection of a real-valued function $\\varphi \\in C^\\infty ({\\mathbb {R}}_+)$ with the properties: $\\operatorname{supp}\\varphi \\subset [1/2, 2]$ , $0\\le \\varphi \\le 1$ , $\\varphi (u)\\ge c>0$ for $u\\in [3/5, 5/3]$ , $\\varphi ^2(u)+\\varphi ^2(u/2)=1$ for $u\\in [1, 2]$ , and hence $\\sum _{\\nu =0}^\\infty \\varphi ^2(2^{-\\nu }u) =1$ for $u\\in [1, \\infty )$ .",
"Set $\\Psi _0:=Z_0,\\quad \\hbox{and}\\quad \\Psi _j:=\\sum _{k=0}^\\infty \\varphi \\Big (\\frac{k}{2^{j-1}}\\Big )Z_k, \\quad j\\ge 1.$ It is easy to see that $f=\\sum _{j=0}^\\infty \\Psi _j*\\Psi _j*f$ for every $f\\in \\mathcal {S}^{\\prime }$ (convergence in $\\mathcal {S}^{\\prime }$ ).",
"The next step is to discretize $\\Psi _j*\\Psi _j$ for $j\\ge 1$ by using the cubature formula on ${{\\mathbb {S}}^{d-1}}$ from (REF ), where $\\mathcal {X}_j$ is a maximal $\\delta _j$ -net with $\\delta _j=\\gamma 2^{-j+1}$ , $0<\\gamma <1$ .",
"In addition, for $j=0$ we set $\\mathcal {X}_0:=\\lbrace e_1\\rbrace $ with $e_1:= (1, 0, \\dots , 0)$ , and $\\widetilde{w}_{e_1}:=\\omega _d$ .",
"Since the cubature formula (REF ) is exact for spherical harmonics of degree $\\le 2^{j+1}$ we have $\\Psi _j*\\Psi _j(x\\cdot y) =\\int _{{\\mathbb {S}}^{d-1}}\\Psi _j(x\\cdot \\eta )\\Psi _j(\\eta \\cdot y)d\\sigma (\\eta )= \\sum _{\\xi \\in \\mathcal {X}_j} \\widetilde{w}_\\xi \\Psi _j(x\\cdot \\xi )\\Psi _j(\\xi \\cdot y),$ which allows to discretize $f=\\sum _{j=0}^\\infty \\Psi _j*\\Psi _j*f$ and obtain $f=\\sum _{j=0}^\\infty \\sum _{\\xi \\in \\mathcal {X}_j}\\langle f,\\psi _\\xi \\rangle \\psi _\\xi ,\\quad \\forall f\\in \\mathcal {S}^{\\prime } \\quad \\hbox{(convergence in $\\mathcal {S}^{\\prime }$)},$ $\\psi _\\xi (x):=\\widetilde{w}_\\xi ^{1/2} \\Psi _j(\\xi \\cdot x), \\quad \\xi \\in \\mathcal {X}_j, \\;j\\ge 0.$ We set $\\mathcal {X}:=\\cup _{j\\ge 0}\\mathcal {X}_j$ assuming that equal points from different sets $\\mathcal {X}_j$ are distinct points in $\\mathcal {X}$ so that $\\mathcal {X}$ can be used as an index set.",
"This completes the construction of the system $\\Psi =\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ .",
"Observe that the frame elements $\\lbrace \\psi _\\xi \\rbrace $ are not only band limited, but also have excellent localization on ${{\\mathbb {S}}^{d-1}}$ .",
"From the properties of $\\varphi $ and Theorem REF and (REF ) it follows that (see also [24], [25]) for any $M>0$ $|\\psi _\\xi (x)| \\le {\\tilde{c}}_4 2^{(j-1)(d-1)/2}(1+2^{j-1}\\rho (x, \\xi ))^{-M},\\quad x\\in {{\\mathbb {S}}^{d-1}},\\; \\xi \\in \\mathcal {X}_j, \\; \\;j\\ge 0,$ where ${\\tilde{c}}_4>0$ is a constant depending only on $d$ , $M$ , $c_7$ and $\\varphi $ .",
"Moreover, the localization of $\\psi _\\xi $ can be improved to sub-exponential as shown in [17].",
"The normalization factor $\\widetilde{w}_\\xi ^{1/2}$ in (REF ) makes all $\\psi _\\xi $ essentially normalized in $L^2({{\\mathbb {S}}^{d-1}})$ , i.e.",
"$\\Vert \\psi _\\xi \\Vert _{L^2({{\\mathbb {S}}^{d-1}})}\\sim 1$ .",
"In what follows we only need the lower bound estimate $\\Vert \\psi _\\xi \\Vert _{L^2({{\\mathbb {S}}^{d-1}})} \\ge {\\tilde{c}}_5, \\quad \\forall \\xi \\in \\mathcal {X}_j, \\;j\\ge 0,$ with a constant ${\\tilde{c}}_5$ depending only on $d$ , $M$ , $c_7$ and $\\varphi $ .",
"Inequality (REF ) follows from (REF ), (REF ), (REF ), the properties of $\\varphi $ , and $\\int _{{\\mathbb {S}}^{d-1}}Z_k^2(\\xi \\cdot x)d\\sigma (x)=Z_k(1)\\sim k^{d-2}$ .",
"We next define the Besov and Triebel-Lizorkin sequence spaces ${\\mathfrak {b}}_p^{sq}$ and ${\\mathfrak {f}}_p^{sq}$ associated to $\\mathcal {X}$ .",
"Definition 3.9 Let $s\\in {\\mathbb {R}}$ , $0<p,q\\le \\infty $ .",
"Then ${\\mathfrak {b}}_p^{sq}:={\\mathfrak {b}}_p^{sq}(\\mathcal {X})$ is defined as the space of all complex-valued sequences $h:=\\lbrace h_{\\xi }\\rbrace _{\\xi \\in \\mathcal {X}}$ such that $\\Vert h\\Vert _{{\\mathfrak {b}}_p^{sq}} :=\\Big (\\sum _{j=0}^\\infty \\Big [2^{j[s+(d-1)(1/2-1/p)]}\\Big (\\sum _{\\xi \\in \\mathcal {X}_j}|h_\\xi |^p\\Big )^{1/p}\\Big ]^q\\Big )^{1/q}<\\infty $ with the usual modification when $p=\\infty $ or $q=\\infty $ .",
"Definition 3.10 Let $s\\in {\\mathbb {R}}$ , $0<p<\\infty $ , and $0<q\\le \\infty $ .",
"Then ${\\mathfrak {f}}_p^{sq}:={\\mathfrak {f}}_p^{sq}(\\mathcal {X})$ is defined as the space of all complex-valued sequences $h:=\\lbrace h_{\\xi }\\rbrace _{\\xi \\in \\mathcal {X}}$ such that $\\Vert h\\Vert _{{\\mathfrak {f}}_p^{sq}} :=\\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\big [|B_\\xi |^{-s/(d-1)-1/2}|h_{\\xi }|{\\mathbb {1}}_{B_\\xi }(\\cdot )\\big ]^q\\Big )^{1/q}\\Big \\Vert _{L^p} <\\infty $ with the usual modification for $q=\\infty $ .",
"Here $B_\\xi :=B(\\xi ,\\gamma 2^{-j+1})$ , $\\xi \\in \\mathcal {X}_j$ , where $\\gamma $ is used in the selection of $\\mathcal {X}_j$ , $|B_\\xi |$ is the measure of $B_\\xi $ and ${\\mathbb {1}}_{B_\\xi }$ is the characteristic function of $B_\\xi $ .",
"Remark 3.11 The replacement of $B_\\xi =B(\\xi ,\\gamma 2^{-j+1})$ in Definition REF with $B(\\xi ,\\gamma 2^{-j})$ or with the disjoint partition sets $A_\\xi $ produces equivalent quasi-norms.",
"This immediately follows from the vector-valued maximal inequality as observed in [10].",
"The main result here asserts that $\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is a self-dual real-valued frame for Besov and Triebel-Lizorkin spaces on the sphere.",
"To state this result we introduce the following analysis and synthesis operators: $S_{\\psi }: f\\mapsto \\lbrace \\langle f, \\psi _\\xi \\rangle \\rbrace _{\\xi \\in \\mathcal {X}},\\quad T_\\psi : \\lbrace h_\\xi \\rbrace _{\\xi \\in \\mathcal {X}} \\mapsto \\sum _{\\xi \\in \\mathcal {X}}h_\\xi \\psi _\\xi .$ Theorem 3.12 Let $s\\in {\\mathbb {R}}$ and $0< p, q< \\infty $ .",
"$(a)$ The operators $S_{\\psi }: \\mathcal {B}_p^{s q} \\rightarrow {\\mathfrak {b}}_p^{s q}$ and $T_\\psi : {\\mathfrak {b}}_p^{s q} \\rightarrow \\mathcal {B}_p^{s q}$ are bounded, and $T_\\psi \\circ S_{\\psi }= I$ on $\\mathcal {B}_p^{s q}$ .",
"Hence, if $f\\in \\mathcal {S}^{\\prime }$ , then $f\\in \\mathcal {B}_p^{sq}$ if and only if $\\lbrace \\langle f,\\psi _\\xi \\rangle \\rbrace _{\\xi \\in \\mathcal {X}}\\in {\\mathfrak {b}}_p^{sq}$ , and $f =\\sum _{\\xi \\in \\mathcal {X}}\\langle f, \\psi _\\xi \\rangle \\psi _\\xi \\quad \\mbox{and}\\quad \\Vert f\\Vert _{\\mathcal {B}_p^{sq}}\\sim \\Vert \\lbrace \\langle f,\\psi _\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}_p^{sq}}.$ $(b)$ The operators $S_{\\psi }: \\mathcal {F}_p^{s q} \\rightarrow {\\mathfrak {f}}_p^{s q}$ and $T_\\psi : {\\mathfrak {f}}_p^{s q} \\rightarrow \\mathcal {F}_p^{s q}$ are bounded, and $T_\\psi \\circ S_{\\psi }= I$ on $\\mathcal {F}_p^{s q}$ .",
"Hence, if $f\\in \\mathcal {S}^{\\prime }$ , then $f\\in \\mathcal {F}_p^{sq}$ if and only if $\\lbrace \\langle f,\\psi _\\xi \\rangle \\rbrace _{\\xi \\in \\mathcal {X}}\\in {\\mathfrak {f}}_p^{sq}$ , and $f =\\sum _{\\xi \\in \\mathcal {X}}\\langle f, \\psi _\\xi \\rangle \\psi _\\xi \\quad \\mbox{and}\\quad \\Vert f\\Vert _{\\mathcal {F}_p^{sq}}\\sim \\Vert \\lbrace \\langle f,\\psi _\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {f}}_p^{sq}}.$ The convergence in $(\\ref {B-disc-calderon})$ and $(\\ref {F-disc-calderon})$ is unconditional in $\\mathcal {B}_p^{sq}$ and $\\mathcal {F}_p^{sq}$ , respectively.",
"For details and proofs, see [25].",
"Remark 3.13 A careful examination of the proofs in [25] shows that the operators $S_{\\psi }$ and $T_{\\psi }$ are uniformly bounded on the respective spaces with parameters $(s, p, q)\\in {\\mathcal {Q}}(A), \\quad \\mbox{for fixed}~A >1,$ where ${\\mathcal {Q}}(A)$ is the index set defined in $(\\ref {indices-1})$ , that is, all constants that appear in the equivalences in Theorem REF depend only on $A$ , $d$ , and $\\varphi $ , if $(s, p, q)\\in {\\mathcal {Q}}(A)$ .",
"In fact, the only nontrivial source of constants is the maximal inequality $(\\ref {max-ineq})$ , however, as seen in $(\\ref {max-const})$ these constant are compatible with the definition of ${\\mathcal {Q}}(A)$ in $(\\ref {indices-1})$ .",
"The above observation will be needed for the construction of new frames below.",
"Remark 3.14 In general, one normally constructs and works with a pair of dual frames $\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\psi }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ on ${{\\mathbb {S}}^{d-1}}$ , see [25].",
"In the construction presented above we consider the case when $\\tilde{\\psi }_\\xi =\\psi _\\xi $ for simplicity.",
"Some embeddings between Besov or Triebel-Lizorkin spaces will be needed.",
"Proposition 3.15 Assume $s,s_0,s_1\\in {\\mathbb {R}}$ and let $0<p,p_0,p_1,q,q_0,q_1\\le \\infty $ in the case of Besov spaces and $0<p,p_0,p_1<\\infty $ , $0<q,q_1,q_2\\le \\infty $ in the case of Triebel-Lizorkin spaces.",
"The following continuous embeddings are valid: $\\mathcal {B}^{s_0q_0}_p\\subset \\mathcal {B}^{s_1q_1}_p,~~\\mathcal {F}^{s_0q_0}_p\\subset \\mathcal {F}^{s_1q_1}_p,\\quad \\hbox{if}\\;\\;s_0=s_1,~q_0\\le q_1~~\\mbox{or}~~s_0>s_1,~\\forall q_0, q_1;$ $\\mathcal {B}^{sq}_{p_0}\\subset \\mathcal {B}^{sq}_{p_1},\\quad \\mathcal {F}^{sq}_{p_0}\\subset \\mathcal {F}^{sq}_{p_1},\\quad \\hbox{if}\\;\\;p_0\\ge p_1;$ $\\mathcal {B}^{s_0q}_{p_0}\\subset \\mathcal {B}^{s_1q}_{p_1},\\quad \\hbox{if}\\;\\;s_0\\ge s_1,~s_0-\\frac{d-1}{p_0}=s_1-\\frac{d-1}{p_1};$ $\\mathcal {F}^{s_0q_0}_{p_0}\\subset \\mathcal {F}^{s_1q_1}_{p_1},\\quad \\hbox{if}\\;\\;s_0> s_1,~s_0-\\frac{d-1}{p_0}=s_1-\\frac{d-1}{p_1},~\\forall q_0, q_1;$ $\\mathcal {B}^{sq}_{p}\\subset \\mathcal {F}^{sq}_{p}\\subset \\mathcal {F}^{sp}_{p}=\\mathcal {B}^{sp}_{p},\\quad \\hbox{if}\\;\\;q<p;$ $\\mathcal {B}^{sp}_{p}=\\mathcal {F}^{sp}_{p}\\subset \\mathcal {F}^{sq}_{p}\\subset \\mathcal {B}^{sq}_{p},\\quad \\hbox{if}\\;\\;p<q.$ The proofs of embeddings (REF ), (REF ), (REF ), and (REF ) are easy and will be omitted.",
"Embedding (REF ) is an immediate consequence of the Nikolski inequality for spherical polynomials.",
"Indeed, by Definition REF it follows that $\\Phi _j*f$ is a spherical polynomial of degree $\\le 2^j$ , i.e.",
"$\\Phi _j*f \\in \\Pi _{2^j}$ .",
"Then by the Nikolski inequality, see e.g.",
"[8], $\\Vert \\Phi _j*f\\Vert _{L^{p_1}}\\le c2^{j(1/p_0-1/p_1)(d-1)}\\Vert \\Phi _j*f\\Vert _{L^{p_0}},\\quad p_0\\le p_1,$ and (REF ) follows readily.",
"The proof of embedding (REF ) relies on the Nikolski inequality (REF ) and can be carried out along the lines of the proof of the same embedding result in the classical case on ${\\mathbb {R}}^n$ from [18], see also [32].",
"We omit the details."
],
[
"Construction of frames by small perturbation",
"Here we present the small perturbation method for construction of frames, developed in [9].",
"Special attention is paid to the dependence of the numerous constants on the parameters of the distribution spaces involved."
],
[
"Setting and conditions on the old frame",
"As in Section REF we denote by $\\mathcal {S}:=C^\\infty ({{\\mathbb {S}}^{d-1}})$ the set of all test functions on ${{\\mathbb {S}}^{d-1}}$ and let $\\mathcal {S}^{\\prime }$ be its dual.",
"We assume that $\\mathcal {Y}$ is a collection of quasi-Banach spaces ${\\mathfrak {B}}={\\mathfrak {B}}({{\\mathbb {S}}^{d-1}})\\subset \\mathcal {S}^{\\prime }$ of distributions on ${{\\mathbb {S}}^{d-1}}$ with quasi-norms $\\Vert \\cdot \\Vert _{\\mathfrak {B}}$ , which are continuously embedded in $\\mathcal {S}^{\\prime }$ , i.e.",
"there exist $m=m({\\mathfrak {B}})\\in {\\mathbb {N}}$ and $C=C({\\mathfrak {B}})>0$ such that $|\\langle f, \\phi \\rangle |\\le C\\Vert f\\Vert _{\\mathfrak {B}}P_m(\\phi )$ for all $f\\in {\\mathfrak {B}}$ , $\\phi \\in \\mathcal {S}$ .",
"Also we assume that $\\mathcal {S}$ is a dense subset of each ${\\mathfrak {B}}\\in \\mathcal {Y}$ .",
"Furthermore, we assume that there exists a collection $\\mathcal {Y}_d$ of quasi-Banach complex-valued sequence spaces ${\\mathfrak {b}}={\\mathfrak {b}}(\\mathcal {X})$ with quasi-norms $\\Vert \\cdot \\Vert _{\\mathfrak {b}}$ , such that every ${\\mathfrak {B}}\\in \\mathcal {Y}$ is associated with a space ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ .",
"We assume that the constants in the quasi-triangle inequalities for the quasi-Banach spaces in $\\mathcal {Y}$ and $\\mathcal {Y}_d$ are uniformly bounded, i.e.",
"there exists a constant $C_1=C_1(\\mathcal {Y},\\mathcal {Y}_d)$ such that $\\begin{split}\\Vert f_1+f_2\\Vert _{\\mathfrak {B}}&\\le C_1(\\Vert f_1\\Vert _{\\mathfrak {B}}+\\Vert f_2\\Vert _{\\mathfrak {B}}), \\quad \\forall f_1,f_2\\in {\\mathfrak {B}},~\\forall {\\mathfrak {B}}\\in \\mathcal {Y};\\\\\\Vert h_1+h_2\\Vert _{\\mathfrak {b}}&\\le C_1(\\Vert h_1\\Vert _{\\mathfrak {b}}+\\Vert h_2\\Vert _{\\mathfrak {b}}), \\quad \\forall h_1,h_2\\in {\\mathfrak {b}},~\\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d.\\end{split}$ A popular version of the Aoki-Rolewicz theorem states (see e.g.",
"[4]) that for any quasi-Banach space ${\\mathfrak {B}}$ with a quasi-norm $\\Vert \\cdot \\Vert _{\\mathfrak {B}}$ satisfying the quasi-triangle inequalities with constant $C_1$ there exists a norm $\\Vert \\cdot \\Vert ^*$ on ${\\mathfrak {B}}$ , such that $\\Vert f\\Vert ^*\\le \\Vert f\\Vert _{\\mathfrak {B}}^\\tau \\le 2\\Vert f\\Vert ^*, \\quad \\forall f\\in {\\mathfrak {B}},\\quad \\mbox{where}~\\tau =\\ln 2/(\\ln 2+\\ln C_1)\\le 1.$ Targeted application of this construction is to the Besov and Triebel-Lizorkin function spaces introduced in Section REF and the corresponding sequence spaces introduced in Section REF .",
"For the Besov spaces the sets $\\mathcal {Y}$ and $\\mathcal {Y}_d$ are given by $\\mathcal {Y}=\\big \\lbrace \\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}}): (s, p, q)\\in {\\mathcal {Q}}(A)\\big \\rbrace \\quad { and }\\quad \\mathcal {Y}_d=\\big \\lbrace {\\mathfrak {b}}^{sq}_p(\\mathcal {X}): (s, p, q)\\in {\\mathcal {Q}}(A)\\big \\rbrace ,$ where $A>1$ is fixed and ${\\mathcal {Q}}(A)$ is introduced in (REF ).",
"A similar observation is valid for the Triebel-Lizorkin spaces $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ and ${\\mathfrak {f}}^{sq}_p(\\mathcal {X})$ .",
"The old frame.",
"We stipulate the existence of a pair of dual frames $\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\psi }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ for all ${\\mathfrak {B}}\\in \\mathcal {Y}$ such that $\\psi _\\xi , \\tilde{\\psi }_\\xi \\in \\mathcal {S}$ , where $\\mathcal {X}$ is a countable index set, with the following properties: A1.",
"The analysis and synthesis operators $S_{\\psi }$ , $S_{\\tilde{\\psi }}$ and $T_{\\psi }$ , $T_{\\tilde{\\psi }}$ from (REF ) have the properties: (a) The operators $S_{\\psi }, S_{\\tilde{\\psi }}: {\\mathfrak {B}}\\mapsto {\\mathfrak {b}}$ are bounded.",
"(b) For any sequence $h=\\lbrace h_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}\\in {\\mathfrak {b}}$ the series $\\sum _{\\xi \\in \\mathcal {X}}h_\\xi \\psi _\\xi $ and $\\sum _{\\xi \\in \\mathcal {X}}h_\\xi \\tilde{\\psi }_\\xi $ converge unconditionally in ${\\mathfrak {B}}$ and $T_\\psi , T_{\\tilde{\\psi }}: {\\mathfrak {b}}\\rightarrow {\\mathfrak {B}}$ are bounded.",
"It is assumed that the norms of the operators $S_{\\psi }$ , $S_{\\tilde{\\psi }}$ and $T_{\\psi }$ , $T_{\\tilde{\\psi }}$ are uniformly bounded relative to ${\\mathfrak {B}}\\in \\mathcal {Y}$ and ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ by a constant $C_2=C_2(\\mathcal {Y},\\mathcal {Y}_d,\\lbrace \\psi _\\xi \\rbrace )>1$ .",
"Thus, for any ${\\mathfrak {B}}\\in \\mathcal {Y}$ and $f\\in {\\mathfrak {B}}$ we have $\\begin{split}C_2^{-1}\\Vert f\\Vert _{{\\mathfrak {B}}} \\le \\Vert S_{\\psi }f\\Vert _{\\mathfrak {b}}=\\Vert \\lbrace \\langle f,\\psi _{\\xi }\\rangle \\rbrace \\Vert _{\\mathfrak {b}}\\le C_2 \\Vert f\\Vert _{{\\mathfrak {B}}},\\\\C_2^{-1}\\Vert f\\Vert _{{\\mathfrak {B}}} \\le \\Vert S_{\\tilde{\\psi }}f\\Vert _{\\mathfrak {b}}=\\Vert \\lbrace \\langle f,\\tilde{\\psi }_{\\xi }\\rangle \\rbrace \\Vert _{\\mathfrak {b}}\\le C_2 \\Vert f\\Vert _{{\\mathfrak {B}}}.\\end{split}$ A2.",
"We have $T_{\\tilde{\\psi }}S_{\\psi }=T_{\\psi }S_{\\tilde{\\psi }}=I$ in $\\mathcal {B}$ , i.e.",
"for any $f\\in {\\mathfrak {B}}$ $f=\\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\tilde{\\psi }_{\\xi }\\rangle \\psi _{\\xi }=\\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\psi _{\\xi }\\rangle \\tilde{\\psi }_{\\xi },$ where the two series converge unconditionally in ${\\mathfrak {B}}$ and hence in $\\mathcal {S}^{\\prime }$ .",
"Note that the compositions $S_{\\psi }T_{\\tilde{\\psi }}$ , $S_{\\tilde{\\psi }}T_{\\psi }$ are projectors due to $(S_{\\psi }T_{\\tilde{\\psi }})^2=S_{\\psi }(T_{\\tilde{\\psi }}S_{\\psi })T_{\\tilde{\\psi }}=S_{\\psi }IT_{\\tilde{\\psi }}=S_{\\psi }T_{\\tilde{\\psi }}.$ A3.",
"In addition, we assume that each ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ obeys the conditions: For any sequence $\\lbrace h_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}\\in {\\mathfrak {b}}$ one has $\\Vert \\lbrace h_\\xi \\rbrace \\Vert _{\\mathfrak {b}}= \\Vert \\lbrace |h_\\xi |\\rbrace \\Vert _{\\mathfrak {b}}$ .",
"If the sequences $\\lbrace h_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}, \\lbrace g_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}\\in {\\mathfrak {b}}$ and $|h_\\xi |\\le |g_\\xi |$ for $\\xi \\in \\mathcal {X}$ , then $\\Vert \\lbrace h_\\xi \\rbrace \\Vert _{\\mathfrak {b}}\\le \\Vert \\lbrace g_\\xi \\rbrace \\Vert _{\\mathfrak {b}}$ .",
"Compactly supported sequences belong to ${\\mathfrak {b}}$ and are dense in ${\\mathfrak {b}}$ .",
"Note that conditions A3 (b)-(c) imply condition A3 (ii) in [9].",
"As a consequence of A1 we obtain that the operator $A:=S_{\\psi }T_{\\psi }$ with matrix ${\\bf A}:=\\lbrace a_{\\xi ,\\eta }\\rbrace _{\\xi , \\eta \\in \\mathcal {X}}, \\quad a_{\\xi , \\eta }:=\\langle \\psi _\\eta ,\\psi _\\xi \\rangle $ is uniformly bounded on the sequence spaces ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ , i.e.",
"$\\Vert A\\Vert _{{\\mathfrak {b}}\\mapsto {\\mathfrak {b}}}\\le C_3:=C_2^2,\\quad \\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d.$"
],
[
"Construction of new frames",
"We next construct a pair of dual frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ for all spaces ${\\mathfrak {B}}\\in \\mathcal {Y}$ , where $\\mathcal {X}$ is the index set from above.",
"For a system $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}\\subset \\cap \\lbrace {\\mathfrak {B}}: {\\mathfrak {B}}\\in \\mathcal {Y}\\rbrace $ of real-valued functions $\\theta _\\xi \\in {\\mathfrak {B}}$ , $\\xi \\in \\mathcal {X}$ , we define the matrices $\\begin{aligned}&{\\bf B}:=\\lbrace b_{\\xi ,\\eta }\\rbrace _{\\xi , \\eta \\in \\mathcal {X}}, \\quad b_{\\xi , \\eta }:=\\langle \\theta _\\eta ,\\psi _\\xi \\rangle ,\\\\&{\\bf D}:=\\lbrace d_{\\xi ,\\eta }\\rbrace _{\\xi ,\\eta \\in \\mathcal {X}}, \\quad d_{\\xi ,\\eta }:=\\langle \\psi _\\eta -\\theta _\\eta ,\\psi _\\xi \\rangle .\\end{aligned}$ The only condition that we require when constructing $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is that the operator $D=S_{\\psi }T_{\\psi }-S_{\\psi }T_{\\theta }, \\;\\; D:{\\mathfrak {b}}\\rightarrow {\\mathfrak {b}},$ with matrix ${\\bf D}$ , defined by $(D h)_\\xi =\\sum _{\\eta \\in \\mathcal {X}} d_{\\xi ,\\eta } h_\\eta $ , has a sufficiently small norm uniformly for all ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ .",
"More precisely we assume that $\\Vert D\\Vert _{{\\mathfrak {b}}\\mapsto {\\mathfrak {b}}}\\le \\epsilon :=\\frac{(1-2^{-\\tau })^{1/\\tau }}{2C_1C_2^4 2^{1/\\tau }},\\quad \\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d,$ with $\\tau $ given in (REF ), where $C_1$ is the constant from (REF ) and $C_2$ is the constant in (REF ).",
"For the operator $B$ with matrix ${\\bf B}$ we have $B=A-D=S_{\\psi }T_{\\theta }$ and hence by (REF ), (REF ), and (REF ) it follows that $B$ is uniformly bounded on $\\mathcal {Y}_d$ , more precisely, $\\Vert B\\Vert _{{\\mathfrak {b}}\\mapsto {\\mathfrak {b}}}\\le C_4, \\quad \\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d,$ with constant $C_4= C_1(C_3+\\epsilon )$ .",
"Condition (REF ) will be sufficient to show that $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is a frame for all spaces ${\\mathfrak {B}}\\in \\mathcal {Y}$ and to construct its dual frame $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ .",
"To this end we introduce the operator: $Tf :=\\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\tilde{\\psi }_\\xi \\rangle \\theta _\\xi ,\\quad f\\in {\\mathfrak {B}}.$ The next three lemmas will be instrumental in the construction of $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ .",
"They are direct adaptations of Lemmas 3.1 - 3.3 in [9]; we omit their proofs.",
"Lemma 4.1 The operators $T_\\theta $ , defined in (REF ) with $\\theta _\\xi $ in the place of $\\psi _\\xi $ , and $T$ are well defined and uniformly bounded, that is, $\\begin{split}\\Vert T_\\theta h\\Vert _{\\mathfrak {B}}\\le C_2C_4\\Vert h\\Vert _{\\mathfrak {b}},\\quad \\forall h\\in {\\mathfrak {b}},~\\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d;\\\\\\Vert Tf\\Vert _{\\mathfrak {B}}\\le C_2^2 C_4\\Vert f\\Vert _{\\mathfrak {B}},\\quad \\forall f\\in {\\mathfrak {B}},~\\forall {\\mathfrak {B}}\\in \\mathcal {Y};\\end{split}$ where $C_2$ is from (REF ) and $C_4$ is from (REF ).",
"Furthermore, the series in (REF ) and (REF ) converge unconditionally in ${\\mathfrak {B}}$ and hence in $\\mathcal {S}^{\\prime }$ .",
"The fact the operator $T$ is invertible plays a key role in this construction.",
"Lemma 4.2 If $(\\ref {oper-eps})$ is satisfied, then $\\Vert I-T\\Vert _{{\\mathfrak {B}}\\mapsto {\\mathfrak {B}}}\\le C_2^2\\epsilon =\\frac{(1-2^{-\\tau })^{1/\\tau }}{2C_1C_2^2 2^{1/\\tau }} \\le \\frac{1}{2},\\quad \\forall {\\mathfrak {B}}\\in \\mathcal {Y},$ and hence $T^{-1}$ exists and $\\Vert T^{-1}\\Vert _{{\\mathfrak {B}}\\mapsto {\\mathfrak {B}}} \\le C_5:=\\frac{2^{1/\\tau }}{(1-2^{-\\tau })^{1/\\tau }},\\quad \\forall {\\mathfrak {B}}\\in \\mathcal {Y},$ where $\\tau $ is from (REF ).",
"Lemma 4.3 Assume $(\\ref {oper-eps})$ holds.",
"Then the operator $H$ with matrix ${\\bf H}:=\\lbrace \\langle T^{-1}\\psi _\\eta , \\tilde{\\psi }_\\xi \\rangle \\rbrace _{\\xi , \\eta \\in \\mathcal {X}}$ is uniformly bounded on ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ , i.e.",
"$\\Vert H\\Vert _{{\\mathfrak {b}}\\mapsto {\\mathfrak {b}}} \\le C_6:=C_2^2C_5,\\quad \\forall {\\mathfrak {b}}\\in \\mathcal {Y}_d.$ The operators from the previous three lemmas can be written as $T_\\theta =T_{\\tilde{\\psi }} B$ , $T=T_{\\theta }S_{\\tilde{\\psi }}=T_{\\tilde{\\psi }}BS_{\\tilde{\\psi }}$ , $I-T=T_{\\tilde{\\psi }}DS_{\\tilde{\\psi }}$ , $H=S_{\\tilde{\\psi }}T^{-1}T_{\\psi }$ .",
"Construction of the dual frame $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ .",
"For any $\\xi \\in \\mathcal {X}$ we define the linear functional $\\tilde{\\theta }_\\xi $ by $\\tilde{\\theta }_\\xi (f)=\\langle f, \\tilde{\\theta }_\\xi \\rangle := \\sum _{\\eta \\in \\mathcal {X}}\\langle T^{-1}\\psi _\\eta , \\tilde{\\psi }_\\xi \\rangle \\langle f, \\tilde{\\psi }_\\eta \\rangle \\quad \\hbox{for}\\;\\; f\\in {\\mathfrak {B}},\\;\\; {\\mathfrak {B}}\\in \\mathcal {Y}.$ Lemma REF and A1 imply that for any $f\\in {\\mathfrak {B}}$ , ${\\mathfrak {B}}\\in \\mathcal {Y}$ , $\\Vert \\lbrace \\langle f, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}}\\le \\Vert H\\Vert _{{\\mathfrak {b}}\\mapsto {\\mathfrak {b}}}\\Vert \\lbrace \\langle f, \\tilde{\\psi }_\\eta \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}}\\le C_6C_2\\Vert f\\Vert _{\\mathfrak {B}}.$ Denote ${\\mathbb {1}}_\\xi :=\\lbrace \\delta _{\\xi \\eta }\\rbrace _{\\eta \\in \\mathcal {X}}$ .",
"Then ${\\mathbb {1}}_\\xi \\in {\\mathfrak {b}}$ by A3 (c) and $\\Vert {\\mathbb {1}}_\\xi \\Vert _{\\mathfrak {b}}>0$ because ${\\mathfrak {b}}$ is a quasi-normed space.",
"Now, condition A3 (b) and inequality (REF ) imply $|\\tilde{\\theta }_\\xi (f)|=|\\langle f, \\tilde{\\theta }_\\xi \\rangle |= \\frac{1}{\\Vert {\\mathbb {1}}_\\xi \\Vert _{\\mathfrak {b}}}\\Vert \\langle f, \\tilde{\\theta }_\\xi \\rangle {\\mathbb {1}}_\\xi \\Vert _{{\\mathfrak {b}}}\\le \\frac{1}{\\Vert {\\mathbb {1}}_\\xi \\Vert _{\\mathfrak {b}}}\\Vert \\lbrace \\langle f, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}}\\le \\frac{C_6C_2}{\\Vert {\\mathbb {1}}_\\xi \\Vert _{\\mathfrak {b}}}\\Vert f\\Vert _{\\mathfrak {B}},$ i.e.",
"$\\tilde{\\theta }_\\xi $ ($\\xi \\in \\mathcal {X}$ ) is a bounded linear functional on every ${\\mathfrak {B}}\\in \\mathcal {Y}$ .",
"Also, for any $f\\in {\\mathfrak {B}}$ by Lemma REF $T^{-1}f\\in {\\mathfrak {B}}$ and using Lemma REF $f=T(T^{-1}f) = \\sum _{\\xi \\in \\mathcal {X}} \\langle T^{-1}f, \\tilde{\\psi }_\\xi \\rangle \\theta _\\xi .$ Furthermore, from the fact that $T^{-1}$ is a bounded operator on ${\\mathfrak {B}}$ and (REF ) it follows that for any $f\\in {\\mathfrak {B}}$ $T^{-1}f = \\sum _{\\eta \\in \\mathcal {X}}\\langle f, \\tilde{\\psi }_\\eta \\rangle T^{-1}\\psi _\\eta ,$ where the series converges unconditionally in ${\\mathfrak {B}}$ and hence in $\\mathcal {S}^{\\prime }$ .",
"This and the fact that $\\tilde{\\psi }_\\xi \\in \\mathcal {S}$ imply $\\langle T^{-1}f, \\tilde{\\psi }_\\xi \\rangle = \\sum _{\\eta \\in \\mathcal {X}}\\langle T^{-1}\\psi _\\eta , \\tilde{\\psi }_\\xi \\rangle \\langle f, \\tilde{\\psi }_\\eta \\rangle = \\langle f, \\tilde{\\theta }_\\xi \\rangle .$ Here the series converges unconditionally and hence absolutely because of the unconditional convergence of the former series.",
"From (REF )–(REF ) it follows that $f = \\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi ,\\quad f\\in {\\mathfrak {B}},$ where $\\langle f, \\tilde{\\theta }_\\xi \\rangle $ is defined in (REF ) and the convergence is unconditional in ${\\mathfrak {B}}$ .",
"The following theorem shows that $\\lbrace \\theta _\\xi \\rbrace $ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ is a pair of dual frames for all spaces ${\\mathfrak {B}}\\in \\mathcal {Y}$ if $\\epsilon $ from (REF ) is sufficiently small.",
"Theorem 4.4 Let $\\lbrace \\psi _\\xi \\rbrace $ , $\\lbrace \\tilde{\\psi }_\\xi \\rbrace $ be a pair of dual old frames for all ${\\mathfrak {B}}\\in \\mathcal {Y}$ satisfying conditions A1–A3 in Subsection REF .",
"Assume $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}\\subset \\cap \\lbrace {\\mathfrak {B}}: {\\mathfrak {B}}\\in \\mathcal {Y}\\rbrace $ satisfies (REF ) and $\\lbrace \\tilde{\\theta }_\\xi \\rbrace $ is defined as in (REF ).",
"Then the analysis operator $S_{\\tilde{\\theta }}:{\\mathfrak {B}}\\rightarrow {\\mathfrak {b}}$ and the synthesis operator $T_\\theta : {\\mathfrak {b}}\\rightarrow {\\mathfrak {B}}$ are uniformly bounded for ${\\mathfrak {B}}\\in \\mathcal {Y}$ , ${\\mathfrak {b}}\\in \\mathcal {Y}_d$ .",
"Furthermore, $T_\\theta S_{\\tilde{\\theta }}=I$ on ${\\mathfrak {B}}$ , i.e.",
"for any $f\\in {\\mathfrak {B}}$ , ${\\mathfrak {B}}\\in \\mathcal {Y}$ , we have $f= \\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi ,$ where the convergence is unconditional in ${\\mathfrak {B}}$ , and $\\Vert f\\Vert _{\\mathfrak {B}}\\le C_7\\Vert \\lbrace \\langle f, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{\\mathfrak {b}},\\quad \\Vert \\lbrace \\langle f, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{\\mathfrak {b}}\\le C_8\\Vert f\\Vert _{\\mathfrak {B}}$ with $C_7=2C_1C_2$ and $C_8=C_2C_5$ .",
"For a proof of Theorem REF see the proof of Theorem 3.5 in [9].",
"The main assumption in constructing the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is the operator norm condition (REF ).",
"A standard tool for evaluating the operator norms in sequence spaces is the following monotonicity lemma.",
"Lemma 4.5 Assume the quasi-norm in ${\\mathfrak {b}}(\\mathcal {X})$ satisfies conditions A3 $(a)$ –$(b)$ in Subsection REF .",
"If the entries of two matrices ${\\bf F}=\\lbrace f_{\\xi ,\\eta }\\rbrace _{\\xi ,\\eta \\in \\mathcal {X}}, {\\bf G}=\\lbrace g_{\\xi ,\\eta }\\rbrace _{\\xi ,\\eta \\in \\mathcal {X}}$ are related by $|f_{\\xi ,\\eta }|\\le g_{\\xi ,\\eta }$ , $\\xi ,\\eta \\in \\mathcal {X}$ , then the respective operators $F$ , $G$ are related by $\\Vert F\\Vert _{{\\mathfrak {b}}\\rightarrow {\\mathfrak {b}}}\\le \\Vert G\\Vert _{{\\mathfrak {b}}\\rightarrow {\\mathfrak {b}}}.$ The relation between ${\\bf F}$ and ${\\bf G}$ implies $|({\\bf F}h)_\\xi |=|\\sum _{\\eta \\in \\mathcal {X}}h_\\eta f_{\\xi ,\\eta }|\\le \\sum _{\\eta \\in \\mathcal {X}}|h_\\eta ||f_{\\xi ,\\eta }|\\le \\sum _{\\eta \\in \\mathcal {X}}|h_\\eta |g_{\\xi ,\\eta } = ({\\bf G}|h|)_\\xi .$ Now, (b) and (a) of A3 imply that for every $h\\in {\\mathfrak {b}}(\\mathcal {X})$ $\\Vert {\\bf F}h\\Vert _{\\mathfrak {b}}\\le \\Vert {\\bf G}|h|\\Vert _{\\mathfrak {b}}\\le \\Vert {\\bf G}\\Vert _{{\\mathfrak {b}}\\rightarrow {\\mathfrak {b}}} \\Vert |h|\\Vert _{\\mathfrak {b}}= \\Vert {\\bf G}\\Vert _{{\\mathfrak {b}}\\rightarrow {\\mathfrak {b}}} \\Vert h\\Vert _{\\mathfrak {b}},$ which implies (REF )."
],
[
"Almost diagonal operators",
"In the next two sections we shall apply the small perturbation method described above for construction of new frames for the Besov spaces from $\\mathcal {Y}=\\big \\lbrace \\mathcal {B}^{sq}_p({{\\mathbb {S}}^{d-1}}): (s, p, q)\\in {\\mathcal {Q}}(A)\\big \\rbrace $ for an arbitrary fixed $A>1$ as well as for the respective collection of Triebel-Lizorkin spaces $\\mathcal {F}^{sq}_p({{\\mathbb {S}}^{d-1}})$ .",
"All spaces from $\\mathcal {Y}$ satisfy (REF ) with $C_1=C_1(A)$ .",
"On account of Theorem REF and Remark REF the frame $\\Psi =\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is real-valued, self-dual, i.e.",
"$\\tilde{\\psi }_\\xi =\\psi _\\xi $ , and conditions A1–A2 in Subsection REF are satisfied with a constant $C_2=C_2(d,A,\\lbrace \\psi _\\xi \\rbrace )$ .",
"Conditions A3 are trivially satisfied for the sequence spaces ${\\mathfrak {b}}^{sq}_p, {\\mathfrak {f}}^{sq}_p$ with $0<p,q<\\infty $ , $s\\in {\\mathbb {R}}$ , and for $\\ell ^p$ , $0<p<\\infty $ , as well.",
"It remains to establish sufficient conditions for verifying the operator norm bound (REF ).",
"To this end, using Lemma REF one can compare the operator matrix elements with the elements of an appropriate almost diagonal matrix (cf.",
"[10], [20]).",
"The almost diagonal matrices we shall use are $\\Omega _{K,M}:=\\lbrace \\omega _{\\xi ,\\eta }^{(K,M)}\\rbrace _{\\xi ,\\eta \\in \\mathcal {X}}$ with entries $\\omega _{\\xi ,\\eta }^{(K,M)}:=\\left(\\frac{\\min \\lbrace N_\\xi ,N_\\eta \\rbrace }{\\max \\lbrace N_\\xi ,N_\\eta \\rbrace }\\right)^{K+(d-1)/2}\\frac{1}{\\left(1+\\min \\lbrace N_\\xi ,N_\\eta \\rbrace \\rho (\\xi ,\\eta )\\right)^M},$ where $N_\\xi := 2^{j-1}$ for $\\xi \\in \\mathcal {X}_j$ , $j\\ge 0$ .",
"Other (non-symmetric) examples of almost diagonal matrices with index set $\\mathcal {X}$ are given in [20].",
"We next show that under appropriate conditions on $K$ and $M$ the operator $\\Omega $ with matrix $\\Omega _{K,M}$ is bounded on ${\\mathfrak {b}}^{sq}_p$ and ${\\mathfrak {f}}^{sq}_p$ .",
"In the following, we shall use the notation $\\mathcal {J}:=(d-1)/\\min \\lbrace 1,p\\rbrace $ in the case of ${\\mathfrak {b}}$ -spaces and $\\mathcal {J}:=(d-1)/\\min \\lbrace 1,p, q\\rbrace $ in the case of ${\\mathfrak {f}}$ -spaces.",
"Theorem 4.6 Let $s\\in {\\mathbb {R}}$ , $0<p, q<\\infty $ .",
"For a fixed $\\delta \\in (0,1]$ assume that $K,M\\in {\\mathbb {N}}$ satisfy $K\\ge \\max \\lbrace s,\\mathcal {J}-s-d+1\\rbrace +\\delta \\quad \\hbox{and}\\quad M\\ge \\mathcal {J}+\\delta .$ Then the operator $\\Omega $ with matrix $\\Omega _{K,M}$ is bounded on ${\\mathfrak {b}}^{sq}_p$ and on ${\\mathfrak {f}}^{sq}_p$ .",
"More precisely, there exists a constant $C_9>0$ such that $\\Vert \\Omega h\\Vert _{{\\mathfrak {b}}^{sq}_p} \\le C_9\\Vert h\\Vert _{{\\mathfrak {b}}^{sq}_p},\\quad \\forall h\\in {\\mathfrak {b}}^{sq}_p;\\qquad \\Vert \\Omega h\\Vert _{{\\mathfrak {f}}^{sq}_p} \\le C_9\\Vert h\\Vert _{{\\mathfrak {f}}^{sq}_p},\\quad \\forall h\\in {\\mathfrak {f}}^{sq}_p.$ Here in the case of ${\\mathfrak {b}}$ -spaces the constant $C_9$ can be written in the form $C_9=\\Big (\\frac{c_{\\dag }}{\\delta ^2}+\\Big (\\frac{c_{\\dag }}{\\delta p}\\Big )^{2/p-1}\\Big )\\Big (\\frac{c_{\\star }}{\\delta }+\\Big (\\frac{c_{\\star }}{\\delta q}\\Big )^{1/q}\\Big ),$ and in the case of ${\\mathfrak {f}}$ -spaces in the form $C_9={c_{\\dag }}^{d/\\mathcal {J}+2|s/(d-1)+1/2|}\\Big (\\frac{c_{\\star }}{\\delta }+\\Big (\\frac{c_{\\star }}{\\delta q}\\Big )^{1/q}\\Big )\\frac{{\\tilde{c}}_1}{\\delta },$ where $c_{\\dag }$ is a constant depending only on $d$ , $c_{\\star }$ is an absolute constant, and ${\\tilde{c}}_1$ is the constant from the maximal inequality (REF ) with $1/t=1/\\min \\lbrace 1,p,q\\rbrace +\\delta /(d-1)$ .",
"Observe that if $(s, p, q)\\in {\\mathcal {Q}}(A)$ for some fixed $A>1$ and $\\delta =1$ then Theorem REF holds with $C_9$ depending only on $d$ and $A$ .",
"To streamline our presentation we defer the long tedious proof of Theorem REF to Section .",
"In light of Theorem REF we next use Proposition REF and the localization property (REF ) to show that the scalar products of the elements of the needlet system $\\lbrace \\psi _\\xi \\rbrace $ from Subsection REF are majorized by the entries of an almost diagonal matrix.",
"Proposition 4.7 For any $K\\in {\\mathbb {N}}_0$ and $M>d-1$ the needlet system $\\lbrace \\psi _\\xi \\rbrace $ from Subsection REF satisfies $|\\langle \\psi _\\eta ,\\psi _\\xi \\rangle |\\le C_{10} \\omega _{\\xi ,\\eta }^{(K,M)},\\quad \\forall \\xi ,\\eta \\in \\mathcal {X},$ where $\\omega _{\\xi ,\\eta }^{(K,M)}$ is defined in (REF ) and $C_{10}= 2^K c_3{\\tilde{c}}_4^2$ with $c_3$ from Proposition REF and ${\\tilde{c}}_4$ from (REF ).",
"Let $\\xi \\in \\mathcal {X}_j$ and $\\eta \\in \\mathcal {X}_k$ .",
"From (REF ) and (REF ) it readily follows that $\\langle \\psi _\\eta ,\\psi _\\xi \\rangle =0$ if $|j-k|\\ge 2$ .",
"The symmetry of $\\omega _{\\xi ,\\eta }^{(K,M)}$ implies that it suffices to consider only the cases $k=j$ and $k=j+1$ .",
"On account of (REF ) condition (REF ) is satisfied for $g=\\psi _\\xi $ with $x_1=\\xi $ , $N_1=N_\\xi $ , and $\\kappa _1={\\tilde{c}}_4 N_\\xi ^{-(d-1)/2}$ and condition (REF ) is satisfied for $f=\\psi _\\eta $ with $x_2=\\eta $ , $N_2=N_\\eta $ , and $\\kappa _2={\\tilde{c}}_4 N_\\eta ^{-(d-1)/2}$ .",
"Then Proposition REF and $N_\\eta =N_\\xi $ for $k=j$ or $N_\\eta =2N_\\xi $ for $k=j+1$ show that (REF ) is satisfied with $C_{10}= c_3{\\tilde{c}}_4^2$ or $C_{10}= 2^K c_3{\\tilde{c}}_4^2$ , respectively.",
"Note that a combination of Proposition REF , Lemma REF , and Theorem REF readily yields another proof of (REF ).",
"Corollary 4.8 Let ${\\mathfrak {b}}={\\mathfrak {b}}_p^{sq}$ or ${\\mathfrak {b}}={\\mathfrak {f}}_p^{sq}$ with $(s, p, q)\\in {\\mathcal {Q}}(A)$ for a fixed $A>1$ .",
"Then $\\lbrace \\psi _\\xi \\rbrace $ satisfies (REF ) with a constant $C_3$ depending only on $d$ , $A$ and $\\varphi $ , namely $C_3=C_9 C_{10}$ , where $C_9$ is from Theorem REF with $K = M = \\left\\lceil Ad\\right\\rceil $ and $C_{10}$ is from Proposition REF with the same $K$ and $M$ .",
"As $K$ and $M$ satisfy assumption (REF ) of Theorem REF with $\\delta =1$ , Proposition REF , Lemma REF , and Theorem REF imply that (REF ) is valid with $C_3=C_9 C_{10}$ .",
"We shall also apply Theorem REF in Section to show that condition (REF ) holds for the constructed new Newtonian kernel frame."
],
[
"Space localization of needlets and Newtonian kernels",
"The basic localization property of the needlets is given in (REF ).",
"In this section we establish some additional localization properties of the needlets introduced in §REF .",
"We also introduce the localized kernels developed in [16].",
"These kernels are linear combinations of shifts of the Newtonian kernel and will be the building blocks in the construction of Newtonian kernel frames in Section ."
],
[
"Properties of the needlets",
"Let $N:=2^{j-1}$ , $j\\in {\\mathbb {N}}$ , and assume that the integer parameter $K$ is even, i.e.",
"$K\\in 2{\\mathbb {N}}$ .",
"Let $\\varphi $ be the $C^\\infty [0,\\infty )$ function introduced in Subsection REF .",
"We define (cf.",
"(REF )) $\\mathcal {K}_N(u):=\\Psi _j(u)=\\sum _{k=0}^{\\infty } \\varphi \\Big (\\frac{k}{N}\\Big )Z_k(u)=\\sum _{N/2<k<2N} \\varphi \\Big (\\frac{k}{N}\\Big )Z_k(u)$ and $\\Lambda _N(u):=(-1)^{K/2} \\sum _{N/2<k<2N} \\varphi \\Big (\\frac{k}{N}\\Big )[k(k+d-2)]^{-K/2}Z_k(u),$ where $Z_k$ is from (REF ).",
"By (REF ) it follows that $-\\Delta _0Z_k(\\eta \\cdot x) =k(k+d-2)Z_k(\\eta \\cdot x),$ implying $\\Delta _0^{K/2}\\Lambda _N(\\eta \\cdot x)= \\mathcal {K}_N(\\eta \\cdot x),\\quad \\eta , x\\in {{\\mathbb {S}}^{d-1}}.$ Here $\\Delta _0$ is the Laplace-Beltrami operator on ${{\\mathbb {S}}^{d-1}}$ (see Subsection REF ).",
"Given $\\eta \\in {{\\mathbb {S}}^{d-1}}$ we extend $\\Lambda _N(\\eta \\cdot x)$ and $\\mathcal {K}_N(\\eta \\cdot x)$ by $\\breve{\\Lambda }_N(\\eta ; x):=\\Lambda _N\\Big (\\frac{\\eta \\cdot x}{|x|}\\Big ),\\quad \\breve{\\mathcal {K}}_N(\\eta ; x):=\\mathcal {K}_N\\Big (\\frac{\\eta \\cdot x}{|x|}\\Big ),\\quad x\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace .$ In light of (REF ) and (REF ) this implies $\\Delta ^{K/2}\\breve{\\Lambda }_N(\\eta ; x)=\\Delta _0^{K/2}\\Lambda _N(\\eta \\cdot x)= \\mathcal {K}_N(\\eta \\cdot x)=\\breve{\\mathcal {K}}_N(\\eta ; x),\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ We shall need the following simple claim.",
"Lemma 5.1 Let $W(x,y):=\\displaystyle {\\frac{x\\cdot y}{|x||y|}}$ for $x,y\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace $ .",
"(a) For any $j=1,2, \\dots , d$ we have $\\frac{\\partial }{\\partial x_j}W(x,y) = P_j(x,y)|x|^{-3}|y|^{-1},\\quad x,y\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace ,$ where $P_j$ is a homogeneous polynomial of degree 2 in $x$ and a homogeneous polynomial of degree 1 in $y$ and $\\Big |\\frac{\\partial }{\\partial x_j}W(x,y)\\Big | \\le 2\\rho (x,y),\\quad \\Big |\\frac{\\partial }{\\partial y_j}W(x,y)\\Big | \\le 2\\rho (x,y),\\quad x,y\\in {{\\mathbb {S}}^{d-1}}.$ (b) For any multi-index $\\beta $ with $|\\beta |\\ge 1$ and any $G\\in C^{|\\beta |}[-1,1]$ we have the representation $\\partial ^\\beta _x G(W(x,y))= \\sum _{1\\le \\nu \\le |\\beta |}G^{(\\nu )}(W(x,y)) R_{\\beta ,\\nu }(x,y)|x|^{-\\nu -2|\\beta |}|y|^{-\\nu }$ with $R_{\\beta ,\\nu }(x,y)=\\!\\sum _{\\begin{array}{c}\\mu \\\\ |\\mu |=2\\nu -|\\beta |\\end{array}}\\prod _{k=1}^d\\Big (|x|^3|y|\\frac{\\partial }{\\partial x_j} W(x,y)\\Big )^{\\mu _k}Q_{\\beta ,\\nu ,\\mu }(x,y),~ \\frac{|\\beta |}{2} < \\nu \\le |\\beta |,$ where $x,y\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace $ , $R_{\\beta ,\\nu }$ , $1\\le \\nu \\le |\\beta |$ , is a homogeneous polynomial of degree $|\\beta |+\\nu $ in $x$ and a homogeneous polynomial of degree $\\nu $ in $y$ , and $Q_{\\beta ,\\nu ,\\mu }$ , $|\\beta |/2 < \\nu \\le |\\beta |$ , is a homogeneous polynomial of degree $3|\\beta |-3\\nu $ in $x$ and a homogeneous polynomial of degree $|\\beta |-\\nu $ in $y$ .",
"The coefficients of $R_{\\beta ,\\nu }$ and $Q_{\\beta ,\\nu ,\\mu }$ are independent of $G$ and some of the polynomials $Q_{\\beta ,\\nu ,\\mu }$ are identically equal to zero.",
"(c) For any multi indices $\\alpha ,\\beta $ with $|\\alpha |=1$ , $|\\beta |\\ge 0$ and any $G\\in C^{|\\beta |+1}[-1,1]$ we have the representation $\\partial ^\\alpha _y\\partial ^\\beta _x G(W(x,y))\\\\= \\sum _{0\\le \\nu \\le |\\beta |}G^{(\\nu +1)}(W(x,y)) (\\partial ^\\alpha _y W(x,y)) R_{\\beta ,\\nu }(x,y)|x|^{-\\nu -2|\\beta |}|y|^{-\\nu }\\\\+ \\sum _{1\\le \\nu \\le |\\beta |}G^{(\\nu )}(W(x,y))\\big [|y|^2\\partial ^\\alpha _yR_{\\beta ,\\nu }(x,y)-\\nu y^\\alpha R_{\\beta ,\\nu }(x,y)\\big ]|x|^{-\\nu -2|\\beta |}|y|^{-\\nu -2},$ where $x,y\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace $ , $R_{0,0}\\equiv 1$ and $R_{\\beta ,\\nu }$ , $1\\le \\nu \\le |\\beta |$ , are from part (b).",
"Clearly, $\\frac{\\partial }{\\partial x_j}W(x,y) = |x|^{-3}|y|^{-1} \\sum _{\\nu \\ne j}x_\\nu (y_j x_\\nu -y_\\nu x_j)\\\\= |x|^{-3}|y|^{-1} \\sum _{\\nu \\ne j}x_\\nu [y_j(x_\\nu -y_\\nu )-y_\\nu (x_j-y_j)]$ and the first inequality of part (a) follows using the Cauchy-Schwarz inequality and that $|x-y|\\le \\rho (x,y)$ for $x,y\\in {{\\mathbb {S}}^{d-1}}$ .",
"The second inequality of part (a) follows from the first one by symmetry.",
"Part (b) follows by induction on $|\\beta |$ .",
"Note that $R_{\\beta ,\\nu }$ is defined recursively by $R_{(0,\\dots ,0),0}(x,y)\\equiv 1$ , $R_{\\beta ,0}(x,y)\\equiv 0$ for $|\\beta |\\ge 1$ , $R_{\\beta ,|\\beta |+1}(x,y)\\equiv 0$ for $|\\beta |\\ge 0$ , and for $|\\alpha |=1$ , $|\\beta |\\ge 0$ $R_{\\beta +\\alpha ,\\nu }(x,y)=(|x|^3|y|\\partial ^\\alpha _x W(x,y))R_{\\beta ,\\nu -1}(x,y)\\\\+|x|^2\\partial ^\\alpha _x R_{\\beta ,\\nu }(x,y)-(\\nu +2|\\beta |)x^\\alpha R_{\\beta ,\\nu }(x,y),\\quad 1\\le \\nu \\le |\\beta |+1.$ Part (c) follows from part (b) by differentiating (REF ) with respect to $y$ for $|\\beta |\\ge 1$ or trivially for $|\\beta |=0$ .",
"From Lemma REF one easily derives localization estimates for zonal functions.",
"Lemma 5.2 Let $K,d\\in {\\mathbb {N}}$ , $G\\in C^K [-1,1]$ .",
"Assume that for some $N\\ge 1$ and $M>0$ $|G^{(\\nu )}(u)|\\le \\frac{\\kappa N^{2\\nu }}{(1+N\\arccos u)^{M+\\nu }}, \\quad \\forall u\\in [-1,1],~0\\le \\nu \\le K,$ where $\\kappa >0$ is a constant depending on $K$ , $d$ , $M$ , and $N$ .",
"Then for all $x,y\\in {{\\mathbb {S}}^{d-1}}$ we have $\\Big |\\partial ^\\beta _x G\\Big (\\frac{y\\cdot x}{|y||x|}\\Big )\\Big |\\le c\\frac{\\kappa N^{|\\beta |}}{(1+N\\rho (y,x))^M}, \\quad 0\\le |\\beta |\\le K,$ $\\Big |\\partial ^\\alpha _y\\partial ^\\beta _x G\\Big (\\frac{y\\cdot x}{|y||x|}\\Big )\\Big |\\le c\\frac{\\kappa N^{|\\beta |+1}}{(1+N\\rho (y,x))^M}, \\quad |\\alpha |=1,~0\\le |\\beta |\\le K-1,$ where $c>0$ is a constant depending only on $K$ and $d$ .",
"For $|\\beta |=0$ (REF ) with $\\nu =0$ coincides with (REF ) with $c=1$ .",
"Let $1\\le |\\beta |\\le K$ .",
"From (REF ) and (REF ) we get $|R_{\\beta ,\\nu }(x,y)|\\le c \\rho (x,y)^{(2\\nu -|\\beta |)_+}$ , $x,y\\in {{\\mathbb {S}}^{d-1}}$ .",
"Using this estimate and (REF ) with $u=y\\cdot x=\\cos \\rho (y, x)$ , $x,y\\in {{\\mathbb {S}}^{d-1}}$ , in (REF ) we get $\\Big |\\partial ^\\beta _x G\\Big (\\frac{y\\cdot x}{|y||x|}\\Big )\\Big |&\\le c\\sum _{1\\le \\nu \\le |\\beta |}|G^{(\\nu )}(y\\cdot x)| \\rho (y, x)^{(2\\nu -|\\beta |)_+}\\\\&\\le c\\sum _{1\\le \\nu \\le |\\beta |}\\kappa N^{2\\nu }(1+N\\rho (y, x))^{-M-\\nu }\\rho (y, x)^{(2\\nu -|\\beta |)_+}\\\\&\\le c\\sum _{1\\le \\nu \\le |\\beta |}\\kappa N^{2\\nu -(2\\nu -|\\beta |)_+}(1+N\\rho (y, x))^{-M-\\nu +(2\\nu -|\\beta |)_+}\\\\&\\le c \\kappa N^{|\\beta |}(1+N\\rho (y, x))^{-M},$ which confirms (REF ).",
"For the proof of (REF ) with the help of (REF ) and (REF ) we estimate the quantities in (REF ) for $x,y\\in {{\\mathbb {S}}^{d-1}}$ as follows $|(\\partial ^\\alpha _y W(x,y))R_{\\beta ,\\nu }(x,y)|\\le c \\rho (x,y)^{1+(2\\nu -|\\beta |)_+},$ $\\big ||y|^2\\partial ^\\alpha _y R_{\\beta ,\\nu }(x,y)-\\nu y^\\alpha R_{\\beta ,\\nu }(x,y)\\big |\\le c \\rho (x,y)^{(2\\nu -|\\beta |-1)_+}.$ Now (REF ) implies for $x,y\\in {{\\mathbb {S}}^{d-1}}$ $\\Big |\\partial ^\\alpha _y\\partial ^\\beta _x G\\Big (\\frac{y\\cdot x}{|y||x|}\\Big )\\Big |&\\le c\\sum _{0\\le \\nu \\le |\\beta |}|G^{(\\nu +1)}(y\\cdot x)| \\rho (y, x)^{1+(2\\nu -|\\beta |)_+}\\\\&+c\\sum _{1\\le \\nu \\le |\\beta |}|G^{(\\nu )}(y\\cdot x)| \\rho (y, x)^{(2\\nu -|\\beta |-1)_+}$ and one completes the proof of (REF ) along the lines of proof of (REF ).",
"For $\\breve{\\Lambda }_N$ , $\\breve{\\mathcal {K}}_N$ and their partial derivatives we have the following estimates: Proposition 5.3 For any $N\\ge 1$ , $K\\in {\\mathbb {N}}$ , $M>0$ and $x,\\eta \\in {{\\mathbb {S}}^{d-1}}$ we have $\\big |\\partial ^\\beta \\breve{\\Lambda }_N(\\eta ; x)\\big |\\le c_4 \\frac{N^{-K+|\\beta |+d-1}}{(1+N\\rho (\\eta ,x))^M},\\quad ~0\\le |\\beta |\\le K+1,$ $\\big |\\partial ^\\alpha _\\eta \\partial ^\\beta _x \\Lambda _N\\Big (\\frac{\\eta \\cdot x}{|\\eta ||x|}\\Big )\\big |\\le c_4\\frac{N^{-K+|\\beta |+d}}{(1+N\\rho (\\eta ,x))^M}, \\quad |\\alpha |=1,~0\\le |\\beta |\\le K,$ and $\\big |\\partial ^\\beta \\breve{\\mathcal {K}}_N(\\eta ; x)\\big |\\le c_5 \\frac{N^{|\\beta |+d-1}}{(1+N\\rho (\\eta ,x))^M},\\quad ~0\\le |\\beta |\\le K+1,$ $\\big |\\partial ^\\alpha _\\eta \\partial ^\\beta _x \\mathcal {K}_N\\Big (\\frac{\\eta \\cdot x}{|\\eta ||x|}\\Big )\\big |\\le c_5\\frac{N^{|\\beta |+d}}{(1+N\\rho (\\eta ,x))^M}, \\quad |\\alpha |=1,~0\\le |\\beta |\\le K.$ where $c_4$ , $c_5$ depend only on $d, K,M$ , and $\\varphi $ .",
"Let $\\lambda (t):=(-1)^{K/2} N^{-K}\\big (t[t+(d-1)/N]\\big )^{-K/2}\\varphi (t)$ , $t\\in [0,\\infty )$ .",
"Clearly, $\\Lambda _N(u) = \\sum _{N/2<k<2N} \\lambda \\Big (\\frac{k}{N}\\Big )Z_k(u),\\quad \\lambda \\in C^\\infty [0, \\infty ),\\quad \\hbox{and}\\quad \\operatorname{supp}\\lambda \\subset [1/2, 2].$ It is readily seen that $\\Vert \\lambda ^{(m)}\\Vert _\\infty \\le cN^{-K}$ for each $m\\ge 0$ with $c=c(d, m, K, \\varphi )$ .",
"Then for any $M>0$ by Theorem REF with $M+K+1$ instead of $M$ we have $|\\Lambda _N^{(\\nu )}(\\cos \\theta )| \\le \\frac{cN^{-K+ d-1}N^{2\\nu }}{(1+N|\\theta |)^{M+K+1}},\\quad |\\theta |\\le \\pi , \\; 0\\le \\nu \\le K+1,$ where $c=c(d, M, K, \\varphi )$ .",
"Applying Lemma REF with $\\kappa =cN^{-K+ d-1}$ , $K$ replaced by $K+1$ , and (REF ) with $|\\theta |=\\rho (\\eta ,x)$ we get (REF ) and (REF ).",
"For the localization of $\\mathcal {K}_N$ we use (REF ) and the fact that $\\Vert \\varphi ^{(m)}\\Vert _\\infty \\le c$ for each $m\\ge 0$ with $c=c(d, m, \\varphi )$ .",
"Thus for any $M>0$ by Theorem REF with $M$ replaced by $M+K+1$ we obtain $|\\mathcal {K}_N^{(\\nu )}(\\cos \\theta )| \\le \\frac{cN^{d-1}N^{2\\nu }}{(1+N|\\theta |)^{M+K+1}},\\quad |\\theta |\\le \\pi , \\; 0\\le \\nu \\le K+1.$ This estimate along with Lemma REF with $\\kappa =cN^{d-1}$ and $K$ replaced by $K+1$ imply (REF ) and (REF ).",
"For $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}_0$ , we set $N_\\xi :=2^{j-1}$ .",
"The elements of the needlet frame $\\Psi =\\lbrace \\psi _\\xi (x): \\xi \\in \\mathcal {X}\\rbrace $ , defined in (REF ), can be represented in terms of the kernels $\\mathcal {K}_N$ as follows $\\psi _\\xi (x):=C^\\diamond _\\xi \\psi ^\\diamond _\\xi (x),\\quad \\psi ^\\diamond _\\xi (x):=\\mathcal {K}_{N_\\xi }(\\xi \\cdot x)=\\Psi _j(\\xi \\cdot x),\\quad C^\\diamond _\\xi :=\\widetilde{w}_\\xi ^{1/2}$ for $x\\in {{\\mathbb {S}}^{d-1}}$ , $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , and the coefficients $C^\\diamond _\\xi $ satisfy $C^\\diamond _\\xi \\le c_9 N_\\xi ^{-(d-1)/2},\\quad \\xi \\in \\mathcal {X},$ with $c_9=2^{(1-d)/2}c_7^{1/2}$ depending only on $d$ (cf.",
"(REF )).",
"In the following sections we assume that the needlet frame $\\Psi $ is fixed; the dependence of some of the constants on $\\varphi $ will not be indicated explicitly."
],
[
"Highly localized kernels in terms of shifts of the Newtonian kernel",
"As already explained in the introduction our tool for approximation of harmonic functions on the ball will consist of linear combinations of shifts of the Newtonian kernel: $\\frac{1}{|x|^{d-2}}\\quad \\hbox{in dimension}\\quad d>2\\quad \\hbox{or}\\quad \\ln \\frac{1}{|x|} \\quad \\hbox{if}\\quad d=2,$ just as in (REF ).",
"The poor localization of the Newtonian kernel, however, creates problems.",
"Its directional derivatives achieve much better localization and are well approximated by finite differences.",
"However, as explained in [16] they do not have either the right localization in the sense of (REF ) or $L^1({{\\mathbb {S}}^{d-1}})$ normalization.",
"We next invoke Theorem 3.1 from [16] to show (see Corollary REF below) the existence of highly localized summability kernels that are linear combinations of finitely many directional derivatives of the Newtonian kernel.",
"Consequently, they will be arbitrarily well approximated by linear combinations of a fixed number of shifts of the Newtonian kernel.",
"Theorem 5.4 (Theorem 3.1 in [16]) Let $d\\ge 2$ , $M>d-2$ , and $0<{\\varepsilon }\\le 1$ .",
"Set $a:=1+{\\varepsilon }$ , $\\delta :=1-a^{-2}$ and $m:=\\left\\lceil (M-d+2)/2\\right\\rceil .$ Consider the function $\\mathcal {F}_{{\\varepsilon }}(u) := {\\varepsilon }^{2m-1} (a^2+1-2au)^{-d/2+1-m},\\quad u\\in [-1,1].$ The function $\\mathcal {F}_{{\\varepsilon }}$ has these properties: $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta ) = {\\varepsilon }^{2m-1} |x-a\\eta |^{-d+2-2m},\\quad \\forall x, \\eta \\in {{\\mathbb {S}}^{d-1}},$ $0< \\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta ) \\le \\frac{c_1^\\#{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^{M}},\\quad \\forall x, \\eta \\in {{\\mathbb {S}}^{d-1}},$ and $\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )d\\sigma (x) \\ge c_2^\\#>0,\\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}},$ where $c_1^\\#, c_2^\\#>0$ are constants depending only on $m$ and $d$ .",
"Furthermore, there exist real numbers $b_0,b_1,\\dots ,b_m$ depending only on ${\\varepsilon }$ , $m$ , and $d$ such that for every $\\eta \\in {{\\mathbb {S}}^{d-1}}$ the function $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )$ is the restriction on ${{\\mathbb {S}}^{d-1}}$ of the harmonic function $\\mathcal {F}_{{\\varepsilon },m}$ defined on ${\\mathbb {R}}^d\\setminus \\lbrace a\\eta \\rbrace $ by $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x):=\\sum _{\\ell =0}^mb_\\ell (\\eta \\cdot \\nabla )^\\ell |x-a\\eta |^{2-d}\\quad \\mbox{if}~d\\ge 3,$ or $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x):=b_0+\\sum _{\\ell =1}^mb_\\ell (\\eta \\cdot \\nabla )^\\ell \\ln \\frac{1}{|x-a\\eta |} \\quad \\mbox{if}~d=2.$ We define the univariate function $F_{\\varepsilon }(u):=\\kappa _{{\\varepsilon },m,d}\\mathcal {F}_{{\\varepsilon }}(u),\\quad u\\in [-1,1],$ where $\\mathcal {F}_{{\\varepsilon }}$ is defined in (REF ) and $\\kappa _{{\\varepsilon },m,d}:=\\Big (\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )\\,d\\sigma (x)\\Big )^{-1},\\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}}.$ Note that $\\kappa _{{\\varepsilon },m,d}$ is independent of $\\eta $ and (REF ) implies $\\kappa _{{\\varepsilon },m,d}\\le 1/c_2^\\#,\\quad \\forall ~ 0<{\\varepsilon }\\le 1.$ Given $\\eta \\in {{\\mathbb {S}}^{d-1}}$ we extend $F_{\\varepsilon }(\\eta \\cdot x)$ just as $\\mathcal {K}_N(\\eta \\cdot x)$ in (REF ) by $\\breve{F}_{\\varepsilon }(\\eta ; x)=F_{\\varepsilon }\\Big (\\frac{\\eta \\cdot x}{|x|}\\Big ),\\quad x\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace .$ In this case (REF ) takes the form $\\Delta \\breve{F}_{\\varepsilon }(\\eta ; x)=\\Delta _0 F_{\\varepsilon }(\\eta \\cdot x),\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ We use $\\breve{F}_{\\varepsilon }$ to bound the derivatives of $F_{\\varepsilon }(\\eta \\cdot x)$ for $x\\in {{\\mathbb {S}}^{d-1}}$ .",
"Corollary 5.5 Let $d\\ge 2$ , $M>d-2$ , $K\\in {\\mathbb {N}}$ .",
"Let $0<{\\varepsilon }\\le 1$ and let $F_{\\varepsilon }$ be defined by (REF )–(REF ).",
"Then for all $x,\\eta \\in {{\\mathbb {S}}^{d-1}}$ we have $F_{\\varepsilon }(\\eta \\cdot x)=\\kappa _{{\\varepsilon },m,d}\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x),$ $\\int _{{{\\mathbb {S}}^{d-1}}} F_{{\\varepsilon }}(\\eta \\cdot y)\\,d\\sigma (y)=1,$ $\\big |\\partial ^\\beta \\breve{F}_{\\varepsilon }(\\eta ; x)\\big |\\le c_8 \\frac{({\\varepsilon }^{-1})^{|\\beta |+d-1}}{(1+{\\varepsilon }^{-1}\\rho (\\eta ,x))^M}, \\quad 0\\le |\\beta |\\le 2K+1,$ $\\big |\\partial ^\\alpha _\\eta \\partial ^\\beta _x F_{\\varepsilon }\\Big (\\frac{\\eta \\cdot x}{|\\eta ||x|}\\Big )\\big |\\le c_8 \\frac{({\\varepsilon }^{-1})^{|\\beta |+d}}{(1+{\\varepsilon }^{-1}\\rho (\\eta ,x))^M}, \\quad |\\alpha |=1,~0\\le |\\beta |\\le 2K,$ where $\\breve{F}_{\\varepsilon }$ is defined by (REF ) and $c_8$ depends only on $d,K,M$ .",
"Identity (REF ) follows from (REF ) and the second part of Theorem REF , while (REF ) follows from (REF ) and (REF ).",
"From (REF ) and (REF ) it follows that for any $\\nu \\in {\\mathbb {N}}_0$ $F_{\\varepsilon }^{(\\nu )}(u)=\\kappa _{{\\varepsilon },m,d}\\,a^{\\nu }{\\varepsilon }^{2m-1} (a^2+1-2au)^{-d/2+1-m-\\nu }\\prod _{k=0}^{\\nu -1} (2m+d-2+2k).$ On the other hand, using that $a=1+{\\varepsilon }$ it is easy to show that $\\frac{1}{5}\\le \\frac{(a^2+1-2au)^{1/2}}{{\\varepsilon }(1+{\\varepsilon }^{-1}\\arccos u)}\\le 2, \\quad u\\in [-1,1],$ see the proof of inequalities (3.7) in [16].",
"The above, (REF ), and (REF ) yield $|F_{\\varepsilon }^{(\\nu )}(u)|\\le \\frac{c {\\varepsilon }^{-(2\\nu +d-1)}}{(1+{\\varepsilon }^{-1}\\arccos u)^{d-2+2m+2\\nu }}\\le \\frac{c {\\varepsilon }^{-(2\\nu +d-1)}}{(1+{\\varepsilon }^{-1}\\arccos u)^{M+\\nu }}$ with $c$ depending on $d$ , $m$ , and $\\nu $ .",
"In turn, this estimate and Lemma REF with $G=F_{\\varepsilon }$ , $N={\\varepsilon }^{-1}$ , $\\kappa =cN^{d-1}$ , and $K$ replaced by $2K+1$ imply (REF ) and (REF ).",
"Note that the extension $\\breve{F}_{\\varepsilon }(\\eta ; x)$ of $F_{\\varepsilon }(\\eta \\cdot x)$ from (REF ) is different from its harmonic extension $\\kappa _{{\\varepsilon },m,d}\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)$ given in (REF ) of the form (REF )–(REF )."
],
[
"Frames in terms of shifts of the Newtonian kernel",
"We now come to the most technical part of our development – the construction of a frame whose elements are finite lineal combinations of shifts of the Newtonian kernel.",
"We shall carry our this construction in several steps."
],
[
"The main technical step in the construction of the new frame on ${{\\mathbb {S}}^{d-1}}$",
"We now focus on the construction of highly localized frame elements $\\lbrace \\theta _\\xi : \\xi \\in \\mathcal {X}\\rbrace $ of the form $\\theta _\\xi (x) =\\sum _{\\nu =1}^{\\tilde{n}} \\frac{a_\\nu }{|x-y_\\nu |^{d-2}},\\quad \\hbox{if}\\quad d>2,$ or $\\theta _\\xi (x) =\\sum _{\\nu =1}^{\\tilde{n}} a_\\nu \\ln \\frac{1}{|x-y_\\nu |},\\quad \\hbox{if}\\quad d=2.$ Here $y_\\nu \\in {\\mathbb {R}}^d$ with $|y_\\nu |>1$ , $a_\\nu \\in {\\mathbb {R}}$ , and $\\lbrace y_\\nu \\rbrace _{\\nu =1}^{\\tilde{n}}$ and $\\lbrace a_\\nu \\rbrace _{\\nu =1}^{\\tilde{n}}$ may vary with $\\xi \\in \\mathcal {X}$ , but $\\tilde{n}$ is fixed.",
"Assume that $\\Psi =\\lbrace \\psi _\\xi : \\xi \\in \\mathcal {X}\\rbrace $ , $\\mathcal {X}=\\cup _{j\\ge 0} \\mathcal {X}_j$ , is the existing frame, described in §REF .",
"For the construction of the new frame elements $\\lbrace \\theta _\\xi \\rbrace $ we utilize the small perturbation method, described in §.",
"In applying this scheme the main step is to construct frame elements $\\theta _\\xi $ , $\\xi \\in \\mathcal {X}$ , of the form (REF )–(REF ) so that $|\\langle \\psi _\\eta -\\theta _\\eta , \\psi _\\xi \\rangle | \\le \\gamma _0\\omega _{\\xi , \\eta },\\quad \\xi ,\\eta \\in \\mathcal {X},$ and $|\\psi _\\xi (x)-\\theta _\\xi (x)| \\le \\frac{\\gamma _0 N_\\xi ^{(d-1)/2}}{(1+N_\\xi \\rho (x, \\xi ))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}},\\;\\xi \\in \\mathcal {X}.$ Here $N_\\xi =2^{j-1}$ for $\\xi \\in \\mathcal {X}_j$ , $\\gamma _0>0$ is a small parameter, $\\omega _{\\xi , \\eta }$ are the entries of an almost diagonal matrix like $\\omega _{\\xi , \\eta }^{(K, M)}$ from (REF ), and $M>0$ is sufficiently large.",
"The result of this construction will be a frame $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ for the Besov and Triebel-Lizorkin spaces of interest.",
"It will be convenient to us to approximate the essentially $L^1$ -normalized frame elements $\\psi _\\xi ^\\diamond (x):=\\mathcal {K}_{N_\\xi }(\\xi \\cdot x)$ defined in (REF ) by essentially $L^1$ -normalized new frame elements $\\theta _\\xi ^\\diamond $ .",
"Then multiplication by constants $C_\\xi ^\\diamond $ (see (REF )) will complete the construction of $L^2$ -normalized frame elements.",
"The construction of the new frame elements $\\theta _\\xi ^\\diamond $ will be carried out in four steps: (a) Approximation of $\\mathcal {K}_{N_\\xi }(\\xi \\cdot x)$ , $\\xi \\in \\mathcal {X}$ , by convolving $\\mathcal {K}_{N_\\xi }$ with the kernel $F_{{\\varepsilon }}(y\\cdot x)$ from (REF ).",
"(b) Discretization of the convolutions by using the cubature formula from (REF ).",
"(c) Truncation of the resulting sums.",
"(d) Approximation of the truncated sums by discrete versions of the operators involved.",
"These approximation steps will be governed by four small parameters (constants): $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ , $\\gamma _4 >0$ .",
"The relations between these parameters and all involved constants will be carefully traced.",
"We next introduce some convenient notation and set up the approximation steps described above.",
"For the only index $\\xi \\in \\mathcal {X}_0$ we set $\\theta _\\xi (x):=\\psi _\\xi (x)\\equiv 1$ .",
"In the remaining part of this subsection we consider $\\xi \\in \\mathcal {X}\\setminus \\mathcal {X}_0=\\cup _{j=1}^\\infty \\mathcal {X}_j$ .",
"Given $0<\\gamma _1\\le 1$ (to be selected), we set ${\\varepsilon }:=\\gamma _1/N_\\xi $ and define $g_1(\\xi ;x):=\\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }(\\xi \\cdot y)F_{{\\varepsilon }}(y\\cdot x)\\, d\\sigma (y),\\quad x\\in {{\\mathbb {S}}^{d-1}},$ where $\\Lambda _{N_\\xi }$ is defined in (REF ), $F_{{\\varepsilon }}(y\\cdot x)$ is the kernel from (REF ) with $\\varepsilon $ from (REF ), and $m$ from (REF ).",
"Given $0<\\gamma _2\\le \\gamma _1$ (to be selected), we let $\\mathcal {Z}_j\\subset {{\\mathbb {S}}^{d-1}}$ be a fixed maximal $\\delta $ -net with $\\delta :=\\gamma _2 2^{-j+1}$ and let $\\lbrace A_\\zeta \\rbrace _{\\zeta \\in \\mathcal {Z}_j}$ be the associated partition of ${{\\mathbb {S}}^{d-1}}$ (see Subsection REF ).",
"Applying cubature formula (REF ) with nodes $\\zeta \\in \\mathcal {Z}_j$ and weights $w_\\zeta =|A_\\zeta |$ to (REF ) we arrive at $g_2(\\xi ;x):=\\sum _{\\zeta \\in \\mathcal {Z}_j} w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )F_{{\\varepsilon }}(\\zeta \\cdot x),\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ Observe that there is no connection between the nodal sets $\\mathcal {X}_j$ and $\\mathcal {Z}_j$ ($j\\in {\\mathbb {N}}$ ).",
"In particular, the cubature $\\sum _{\\zeta \\in \\mathcal {Z}_j}w_\\zeta f(\\zeta )$ from (REF ) has to be exact only for constants, while the cubature $\\sum _{\\xi \\in \\mathcal {X}_j}\\widetilde{w}_\\xi f(\\xi )$ from (REF ) is required to be exact for all spherical harmonics of degree $\\le 2^{j+1}$ .",
"Given $0<\\gamma _3\\le 1$ (to be determined), we truncate the sum in (REF ) by including only the nodes within distance $r_\\xi :=(\\gamma _3 N_\\xi )^{-1}$ from $\\xi $ to obtain $g_3(\\xi ;x):=\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}} w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )F_{{\\varepsilon }}(\\zeta \\cdot x),\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ The functions $g_1(\\xi ;x)$ , $g_2(\\xi ;x)$ , and $g_3(\\xi ;x)$ should be viewed as consecutive approximations of $\\Lambda _{N_\\xi }(\\xi \\cdot x)$ .",
"We obtain consecutive approximations to $\\mathcal {K}_{N_\\xi }(\\xi \\cdot x)$ by applying $\\Delta _0^{K/2}$ to each of the functions $g_1$ , $g_2$ , $g_3$ in (REF ), (REF ), and (REF ).",
"We set $h_1(\\xi ;x):= \\Delta _0^{K/2}g_1(\\xi ;x)=\\Delta _0^{K/2}\\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }(x\\cdot y)F_{{\\varepsilon }}(y\\cdot \\xi )\\, d\\sigma (y) \\\\= \\int _{{{\\mathbb {S}}^{d-1}}}\\Delta _0^{K/2}\\Lambda _{N_\\xi }(x\\cdot y)F_{{\\varepsilon }}(y\\cdot \\xi )\\, d\\sigma (y)= \\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {K}_{N_\\xi }(x\\cdot y)F_{{\\varepsilon }}(y\\cdot \\xi )\\, d\\sigma (y),$ $h_2(\\xi ;x):=\\Delta _0^{K/2}g_2(\\xi ;x)=\\sum _{\\zeta \\in \\mathcal {Z}_j}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\Delta _0^{K/2}F_{{\\varepsilon }}(\\zeta \\cdot x),$ $h_3(\\xi ;x):=\\Delta _0^{K/2}g_3(\\xi ;x)=\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\Delta _0^{K/2}F_{{\\varepsilon }}(\\zeta \\cdot x).$ Above in (REF ) we first used the commutativity of the inner product of zonal functions (REF ) in the definition of $g_1$ followed by (REF ) in the last equality.",
"Observe that $h_3(\\xi ;x)$ is a linear combination of finitely many (independent of $\\xi $ ) terms of the form $\\Delta _0^{K/2}F_{{\\varepsilon }}(\\zeta \\cdot x)=\\kappa _{{\\varepsilon },m,d} \\sum _{\\ell =0}^m b_{\\ell } \\Delta _0^{K/2}(\\zeta \\cdot \\nabla )^\\ell |x-a\\zeta |^{2-d},\\quad \\hbox{if}\\; d\\ge 3,$ see (REF ), (REF ), and (REF ).",
"We have a similar representation of $h_3(\\xi ;x)$ in dimension $d=2$ .",
"Replacing the differential operator $\\Delta _0^{K/2}\\left(\\zeta \\cdot \\nabla \\right)^\\ell $ in (REF ) by its discrete counterpart ${\\mathfrak {L}}_t^{K/2}{\\mathfrak {D}}^\\ell _t(\\zeta )$ with an appropriate small $t=t_j>0$ (to be specified) we arrive at the following definition of $\\theta ^\\diamond _\\xi $ , $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , in dimension $d\\ge 3$ $\\theta ^\\diamond _\\xi (x):=\\kappa _{{\\varepsilon },m,d}\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\sum _{\\ell =0}^m b_{\\ell }{\\mathfrak {L}}_{t_j}^{K/2}{\\mathfrak {D}}^\\ell _{t_j}(\\zeta ) |x-a\\zeta |^{2-d}.$ If $d=2$ we set $\\theta ^\\diamond _\\xi (x):=\\kappa _{{\\varepsilon },m,2}\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\sum _{\\ell =1}^m b_{\\ell } {\\mathfrak {L}}_{t_j}^{K/2}{\\mathfrak {D}}^\\ell _{t_j}(\\zeta ) \\ln \\frac{1}{|x-a\\zeta |}.$ Several remarks are in order: The finite difference operator ${\\mathfrak {D}}^\\ell _t(\\varsigma ):=t^{-\\ell }\\sum _{k=0}^\\ell (-1)^{\\ell -k} \\binom{\\ell }{k} T(\\varsigma ,kt)$ is defined by the translation operator (in ${\\mathbb {R}}^d$ ) in direction $\\varsigma \\in {{\\mathbb {S}}^{d-1}}$ with step $t$ given by $T(\\varsigma ,t)f(x)=f(x+t\\varsigma )$ for $x\\in {\\mathbb {R}}^d$ .",
"Following [8] the rotation $Q_{1,2,t}\\in SO(d)$ is given by $Q_{1,2,t}\\varsigma = Q_{1,2,t}(\\varsigma _1,\\varsigma _2,\\dots ,\\varsigma _d) \\\\:= (\\varsigma _1 \\cos t +\\varsigma _2 \\sin t, -\\varsigma _1 \\sin t +\\varsigma _2 \\cos t,\\varsigma _3,\\dots ,\\varsigma _d),\\quad \\varsigma \\in {{\\mathbb {S}}^{d-1}},$ and $Q_{i,\\ell ,t}\\varsigma $ is defined similarly for any $1\\le i<\\ell \\le d$ .",
"The translation operator corresponding to the rotation $Q_{i,\\ell ,t}$ , $1\\le i<\\ell \\le d$ , is given by $T(Q_{i,\\ell ,t})f(\\varsigma ) := f(Q_{i,\\ell ,t}^{-1}\\varsigma )=f(Q_{i,\\ell ,-t}\\varsigma ).$ The operator ${\\mathfrak {L}}_tf(\\varsigma ):=t^{-2}\\sum _{1\\le i<\\ell \\le d}(T(Q_{i,\\ell ,t}) +T(Q_{i,\\ell ,-t})-2{I})f(\\varsigma ),$ where ${I}$ stands for the identity, approximates well $\\Delta _0 f(\\varsigma )$ for small $t$ ; the powers of ${\\mathfrak {L}}_t$ are defined as usual by ${\\mathfrak {L}}_t^k:={\\mathfrak {L}}_t({\\mathfrak {L}}_t^{k-1})$ .",
"The numbers $a$ , $\\delta $ , $m$ , and $b_\\ell $ , $\\ell =0,1,\\dots ,m$ , are determined in Theorem REF as functions of ${\\varepsilon }$ from (REF ) and $M$ .",
"We now come to the first main assertion in this section.",
"Theorem 6.1 Let $d\\ge 2$ , $K\\in 2{\\mathbb {N}}$ , $M>K+d-1$ , and $\\gamma _0>0$ .",
"Then there exist constants $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ , $\\gamma _4>0$ depending only on $d, K, M, \\gamma _0$ , and for every $j\\in {\\mathbb {N}}$ there exists $t_j>0$ depending only on $d, K, M, \\gamma _0$ , and $j$ such that for every $\\xi \\in \\mathcal {X}$ the element $\\theta ^\\diamond _\\xi $ from (REF ) or (REF ) obeys $\\big |\\partial ^\\beta \\big [\\breve{\\psi }^\\diamond _\\xi (x)-\\breve{\\theta }^\\diamond _\\xi (x)\\big ]\\big |\\le \\frac{\\gamma _0 N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},~ 0\\le |\\beta |\\le K,$ $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[\\psi ^\\diamond _\\xi (y)-\\theta ^\\diamond _\\xi (y)\\right]d\\sigma (y)\\Big |\\le \\gamma _0 N_\\xi ^{-K},~~0\\le |\\beta |\\le K-1.$ Here $\\breve{\\theta }^\\diamond _\\xi (x):=\\theta ^\\diamond _\\xi (x/|x|)$ , $x\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace $ , and $\\breve{\\psi }^\\diamond _\\xi (x):=\\breve{\\mathcal {K}}_{N_\\xi }(\\xi ; x)$ , see (REF ), (REF ).",
"The proof of Theorem REF relies on four lemmas, which establish estimates similar to estimates (REF ) and (REF ) for the differences $\\psi ^\\diamond _\\xi -h_1(\\xi ;\\cdot )$ , $h_1(\\xi ;\\cdot )-h_2(\\xi ;\\cdot )$ , $h_2(\\xi ;\\cdot )-h_3(\\xi ;\\cdot )$ , and $h_3(\\xi ;\\cdot )-\\theta ^\\diamond _\\xi $ .",
"The values of the parameters $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ , $\\gamma _4$ , used in these four lemmas will be selected in the proof of Theorem REF (see (REF )).",
"In light of the last integral in (REF ) we define $\\breve{h}_1(\\xi ;x):=\\int _{{{\\mathbb {S}}^{d-1}}}\\breve{\\mathcal {K}}_{N_\\xi }(y;x)F_{{\\varepsilon }}(y\\cdot \\xi )\\, d\\sigma (y),\\quad x\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ,$ where $\\breve{\\mathcal {K}}_{N_\\xi }(y; x)$ is defined in (REF ) and ${\\varepsilon }$ is from (REF ).",
"Lemma 6.2 Let $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , $K\\in 2{\\mathbb {N}}$ , and $M>K+d-1$ .",
"If $0<\\gamma _1\\le 1$ , ${\\varepsilon }$ is from (REF ) and $F_{{\\varepsilon }}$ is from (REF ), then: $(a)$ For any $\\beta $ , $0\\le |\\beta |\\le K$ , $\\big |\\partial ^\\beta \\big [\\breve{\\mathcal {K}}_{N_\\xi }(\\xi ; x)-\\breve{h}_1(\\xi ;x)\\big ]\\big |\\le c_{10}\\frac{\\gamma _1 N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}};$ $(b)$ For any $\\beta $ , $0\\le |\\beta |\\le K-1$ , $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[\\mathcal {K}_{N_\\xi }(\\xi \\cdot y)-h_1(\\xi ;y)\\right]d\\sigma (y)\\Big |\\le c_{10}\\gamma _1 N_\\xi ^{-K},$ where $c_{10}$ depends only on $d, K, M$ .",
"Let $0\\le |\\beta |\\le K$ .",
"We apply Proposition REF with $g(y)=\\partial ^\\beta _x \\breve{\\mathcal {K}}_{N_\\xi }(y;x)$ , $x_1\\!=\\!x$ , $N_1=N_\\xi $ , $\\kappa _1=c_5 N_\\xi ^{|\\beta |}$ on account of (REF ), and with $f(y)=F_{{\\varepsilon }}(\\xi \\cdot y)$ , $x_2=\\xi $ , $N_2=1/{\\varepsilon }$ , ${\\varepsilon }=\\gamma _1/N_\\xi $ , $\\kappa _2=c_8$ on account of (REF ) with $\\beta =0$ .",
"Observe that $N_2=N_\\xi /\\gamma _1\\ge N_1$ .",
"Hence, because of (REF ), inequality (REF ) implies $\\big |\\partial ^\\beta _x\\breve{h}_1(\\xi ;x)-\\partial ^\\beta _x\\breve{\\mathcal {K}}_{N_\\xi }(\\xi ; x)\\big |\\le c_2c_5c_8 \\frac{\\gamma _1 N_\\xi ^{|\\beta |+d-1} }{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ For the proof of (REF ) we apply Proposition REF with $x_1=\\xi $ , $N_1=N_\\xi $ , $g(y)=\\Lambda _{N_\\xi }(\\xi ; y)$ , $\\kappa _1=c_4 N_\\xi ^{-K}$ in view of (REF ) with $|\\beta |=1$ , $f(y)=F_{{\\varepsilon }}(x\\cdot y)$ with any fixed $x\\in {{\\mathbb {S}}^{d-1}}$ , $N_2=1/{\\varepsilon }$ , ${\\varepsilon }=\\gamma _1/N_\\xi $ , $\\kappa _2=c_8$ in view of (REF ) with $\\beta =0$ .",
"Consequently, because of (REF ), inequality (REF ) implies $\\big |\\Lambda _{N_\\xi }(\\xi \\cdot x)-g_1(\\xi ;x)\\big |\\le c_{11} \\frac{\\gamma _1 N_\\xi ^{-K+d-1} }{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}},$ with $c_{11}:= c_2 c_4 c_8$ .",
"Now, for $0\\le |\\beta |\\le K-1$ we apply consecutively (REF ), (REF ), the fact that the operator $\\Delta _0$ is symmetric, (REF ) with $y$ in place of $x$ , (REF ), and (REF ) to obtain $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[\\mathcal {K}_{N_\\xi }(\\xi \\cdot y)-h_1(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\Delta _0^{K/2}\\left[\\Lambda _{N_\\xi }(\\xi \\cdot y)-g_1(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} \\left[\\Lambda _{N_\\xi }(\\xi \\cdot y)-g_1(\\xi ;y)\\right]\\Delta _0^{K/2}y^\\beta d\\sigma (y)\\Big |\\le c_{11}c_0c_6\\gamma _1 N_\\xi ^{-K}.$ Finally, (REF ) and (REF ) imply (REF ) and (REF ) with $c_{10}:=\\max \\lbrace c_2c_5c_8,c_{11}c_0c_6\\rbrace $ .",
"The proof is complete.",
"The first integral in (REF ), (REF ), and identity (REF ) give another representation of $\\breve{h}_1(\\xi ;x)$ , namely, $\\breve{h}_1(\\xi ;x)= \\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }(\\xi \\cdot y)\\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)\\, d\\sigma (y),\\quad x\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ,$ with $\\breve{F}_{{\\varepsilon }}(y; x)$ defined in (REF ).",
"Using (REF ) and (REF ) we also set for $\\xi \\in \\mathcal {X}_j$ $\\breve{h}_2(\\xi ;x):=\\sum _{\\zeta \\in \\mathcal {Z}_j}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta ; x),\\quad x\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace .$ Lemma 6.3 Assume $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , $K\\in 2{\\mathbb {N}}$ , $M>K+d-1$ , and let $\\gamma _1$ , ${\\varepsilon }$ , $F_{{\\varepsilon }}$ be as in Lemma REF .",
"If $0<\\gamma _2\\le \\gamma _1$ and $\\breve{F}_{{\\varepsilon }}$ is from (REF ), then: $(a)$ For any $\\beta $ , $0\\le |\\beta |\\le K$ , $\\big |\\partial ^\\beta \\big [\\breve{h}_1(\\xi ;x)-\\breve{h}_2(\\xi ;x)\\big ]\\big |\\le c_{20}\\gamma _2 \\gamma _1^{-2K-1} \\frac{ N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}};$ $(b)$ For any $\\beta $ , $0\\le |\\beta |\\le K-1$ , $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[h_1(\\xi ;y)-h_2(\\xi ;y)\\right]d\\sigma (y)\\Big |\\le c_{20}\\gamma _2 \\gamma _1^{-2K-1} N_\\xi ^{-K},$ where $c_{20}$ depends only on $d, K, M$ .",
"Let $0\\le |\\beta |\\le K$ .",
"From (REF ) and (REF ) we get $\\big |\\partial ^\\beta \\big [\\breve{h}_1(\\xi ;x)-\\breve{h}_2(\\xi ;x)\\big ]\\big |\\\\=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\!\\!\\Lambda _{N_\\xi }(\\xi \\cdot y)\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x) d\\sigma (y)-\\!\\sum _{\\zeta \\in \\mathcal {Z}_j} w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta ; x)\\Big |\\\\=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\!\\!\\big [\\Lambda _{N_\\xi }(\\xi \\cdot y)\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\Lambda _{N_\\xi }(\\xi \\cdot \\zeta (y))\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)\\big ] d\\sigma (y)\\Big |\\\\\\le \\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }(\\xi \\cdot y)\\big [\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)\\big ]d\\sigma (y)\\Big |~~~~~~~~~~~~~~~~\\\\+\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\big [\\Lambda _{N_\\xi }(\\xi \\cdot y)-\\Lambda _{N_\\xi }(\\xi \\cdot \\zeta (y))\\big ]\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x) d\\sigma (y)\\Big |,~~~~~~~~~~~~~~~~~~~~$ where $\\zeta (y)$ is defined in §REF .",
"Let $\\eta =\\eta (s)$ , $s\\in [0,\\rho ]$ , $\\rho =\\rho (y,\\zeta (y))$ , be the geodesic line on ${{\\mathbb {S}}^{d-1}}$ such that $\\eta (0)=y$ and $\\eta (\\rho )=\\zeta (y)$ .",
"Then $\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)=\\int _0^\\rho \\left.\\nabla _\\eta \\partial _x^\\beta \\Delta _x^{K/2}F_{{\\varepsilon }}\\Big (\\frac{\\eta \\cdot x}{|\\eta ||x|}\\Big )\\right|_{\\eta =\\eta (s)}\\cdot \\eta ^{\\prime }(s)\\,ds.$ Using in the above representation (REF ), (REF ), $\\rho (y, \\zeta (y))\\le \\gamma _2N_\\xi ^{-1}$ , $\\gamma _2\\le \\gamma _1\\le 1$ we get $\\big |\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)\\big |\\le c_{21} \\frac{\\gamma _2N_\\xi ^{-1}{\\varepsilon }^{-(K+|\\beta |+d)}}{(1+{\\varepsilon }^{-1}\\rho (y, x))^M}\\\\= c_{21} \\frac{\\gamma _2\\gamma _1^{-(K+|\\beta |+1)} N_\\xi ^{K+|\\beta |}{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (y, x))^M},\\quad x,y\\in {{\\mathbb {S}}^{d-1}}.$ Applying Proposition REF with $g(y)=\\Lambda _{N_\\xi }(\\xi \\cdot x)$ , $N_1=N_\\xi $ , $\\kappa _1=c_4 N_\\xi ^{-K}$ (because of (REF ) with $\\beta =0$ ), $f(y)=\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)$ , $N_2=1/{\\varepsilon }\\ge N_1$ , and $\\kappa _2=c_{21}\\gamma _2\\gamma _1^{-2K-1}N_\\xi ^{K+|\\beta |}$ (using the above estimate) we get $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }(\\xi \\cdot y)\\big [\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(y; x)-\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x)\\big ] d\\sigma (y)\\Big |\\\\\\le c_3 c_4 c_{21} \\frac{\\gamma _2 \\gamma _1^{-2K-1}N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi , x))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ Similarly, using (REF ) and (REF ) we obtain $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\big [\\Lambda _{N_\\xi }(\\xi \\cdot y)-\\Lambda _{N_\\xi }(\\xi \\cdot \\zeta (y))\\big ]\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y); x) d\\sigma (y)\\Big |\\\\\\le c_{22} \\frac{\\gamma _2 \\gamma _1^{-2K}N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi , x))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ Substituting the above two estimates in (REF ) we get (REF ) with $c_{20}\\ge c_{23}:= c_{22}+c_3 c_4 c_{21}$ .",
"For the proof of (REF ) we repeat the arguments applied for the proof of (REF ) and get $|g_1(\\xi ;x)-g_2(\\xi ;x)|\\le c_{24} \\frac{\\gamma _2 \\gamma _1^{-1}N_\\xi ^{-K+d-1}}{(1+N_\\xi \\rho (\\xi , x))^M}.$ Now, for $0\\le |\\beta |\\le K-1$ we apply consecutively (REF ), (REF ), the fact that the operator $\\Delta _0$ is symmetric, (REF ), (REF ), and (REF ) to obtain $&\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[h_1(\\xi ;y)-h_2(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\&\\qquad \\qquad =\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\Delta _0^{K/2}\\left[g_1(\\xi ;y)-g_2(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\&\\qquad \\qquad =\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} \\left[g_1(\\xi ;y)-g_2(\\xi ;y)\\right]\\Delta _0^{K/2}y^\\beta d\\sigma (y)\\Big |\\le c_0c_6c_{24} \\gamma _2 \\gamma _1^{-1} N_\\xi ^{-K},$ which yields (REF ) with $c_{20}=\\max \\lbrace c_{23}, c_0c_6c_{24}\\rbrace $ in view of $\\gamma _1\\le 1$ .",
"The estimates on $h_2(\\xi ;\\cdot )-h_3(\\xi ;\\cdot )$ are given in the following lemma, where $\\breve{h}_2(\\xi ;x)$ is as in Lemma REF and for $\\xi \\in \\mathcal {X}_j$ we set $\\breve{h}_3(\\xi ;x)=\\sum _{\\zeta \\in \\mathcal {Z}_j:\\rho (\\zeta ,\\xi )\\le r_\\xi }w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta ; x).$ Lemma 6.4 Let $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , $K\\in 2{\\mathbb {N}}$ , $M>K+d-1$ , and let $\\gamma _1$ , $\\gamma _2$ , ${\\varepsilon }$ , $\\breve{F}_{{\\varepsilon }}$ be as in Lemma REF .",
"If $0<\\gamma _3\\le 1$ and $r_\\xi =(\\gamma _3 N_\\xi )^{-1}$ , then: $(a)$ For any $\\beta $ , $0\\le |\\beta |\\le K$ , we have $\\big |\\partial ^\\beta \\big [\\breve{h}_2(\\xi ;x)-\\breve{h}_3(\\xi ;x)\\big ]\\big |\\le c_{30}\\gamma _3\\gamma _1^{-2K} \\frac{N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},$ $(b)$ For any $\\beta $ , $0\\le |\\beta |\\le K-1$ , we have $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[h_2(\\xi ;y)-h_3(\\xi ;y)\\right]d\\sigma (y)\\Big |\\le c_{30}\\gamma _3\\gamma _1^{-2K} N_\\xi ^{-K},$ where $c_{30}$ depends only on $d, K, M$ .",
"Let $\\xi \\in \\mathcal {X}_j$ .",
"Set $\\Lambda _{N_\\xi }^*(u):=\\Lambda _{N_\\xi }(u){\\mathbb {1}}_{(r_\\xi ,\\pi ]}(u)$ .",
"Then for $0\\le |\\beta |\\le K$ we get from (REF ) and (REF ) $\\partial ^\\beta \\big [\\breve{h}_2(\\xi ;x)-\\breve{h}_3(\\xi ;x)\\big ]&=\\sum _{\\zeta \\in \\mathcal {Z}_j}w_\\zeta \\Lambda _{N_\\xi }^*(\\xi \\cdot \\zeta )\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta ;x)\\\\&= \\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }^*(\\xi \\cdot \\zeta (y))\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y);x)d\\sigma (y),$ where $\\zeta (y)$ is defined in §REF .",
"For $\\Lambda _{N_\\xi }(\\xi \\cdot x)$ estimate (REF ) with $\\beta =0$ and $M$ replaced by $M+1$ yields $\\big |\\Lambda _{N_\\xi }(\\xi \\cdot x)\\big |\\le c_4^* \\frac{N_\\xi ^{-K+d-1}}{ (1+N_\\xi \\rho (\\xi ,x))^{M+1}},\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ This estimate and the inequality $(1+N_\\xi \\rho (\\xi ,x))^{-M-1}\\le \\gamma _3(1+N_\\xi \\rho (\\xi ,x))^{-M}$ , if $\\rho (\\xi ,x)>r_\\xi $ , yield $\\big |\\Lambda _{N_\\xi }^*(\\xi \\cdot x)\\big |\\le c_4^* \\frac{\\gamma _3 N_\\xi ^{-K+d-1}}{ (1+N_\\xi \\rho (\\xi ,x))^M},\\quad x\\in {{\\mathbb {S}}^{d-1}},$ and hence $\\big |\\Lambda _{N_\\xi }^*(\\xi \\cdot \\zeta (y))\\big |\\le c_{31} \\frac{\\gamma _3 N_\\xi ^{-K+d-1}}{ (1+N_\\xi \\rho (\\xi ,y))^M},\\quad y\\in {{\\mathbb {S}}^{d-1}},$ with $c_{31}=2^M c_4^*$ on account of $\\rho (y,\\zeta (y))\\le \\gamma _2 N_\\xi ^{-1}\\le N_\\xi ^{-1}$ and (REF ).",
"Using (REF ) again we get from (REF ) $\\big |\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y);x)\\big |\\le c_{32} \\frac{{\\varepsilon }^{-(K+|\\beta |+d-1)}}{ (1+{\\varepsilon }^{-1}\\rho (y,x))^M}$ with $c_{32}=2^M d^{K/2}c_8$ .",
"We now apply Proposition REF to the integral in (REF ) with $x_1=\\xi $ , $g(y)=\\Lambda _{N_\\xi }^*(\\xi \\cdot \\zeta (y))$ , $N_1=N_\\xi $ , $\\kappa _1=c_{31}\\gamma _3 N_\\xi ^{-K}$ from (REF ), $x_2=x$ , $f(y)=\\partial ^\\beta \\Delta ^{K/2}\\breve{F}_{{\\varepsilon }}(\\zeta (y);x)$ , ${\\varepsilon }=\\gamma _1/N_\\xi $ , $N_2=1/{\\varepsilon }\\ge N_1$ , $\\kappa _2=c_{32}({\\varepsilon }^{-1})^{K+|\\beta |}$ from (REF ), and get (REF ) with $c_{30}\\ge c_3 c_{31} c_{32}$ .",
"For the proof of (REF ) we repeat the argument from the proof of (REF ) and get $|g_2(\\xi ;x)-g_3(\\xi ;x)|\\\\= \\Big |\\int _{{{\\mathbb {S}}^{d-1}}}\\Lambda _{N_\\xi }^*(\\xi \\cdot \\zeta (y))F_{{\\varepsilon }}(\\zeta (y)\\cdot x)d\\sigma (y)\\Big |\\le c_{33} \\frac{\\gamma _3 N_\\xi ^{-K+d-1}}{(1+N_\\xi \\rho (\\xi , x))^M}.$ Now, for $0\\le |\\beta |\\le K-1$ we apply consecutively (REF ), (REF ), the fact that the operator $\\Delta _0$ is symmetric, (REF ), (REF ), and (REF ) to obtain $&\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[h_2(\\xi ;y)-h_3(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\&\\qquad \\qquad =\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\Delta _0^{K/2}\\left[g_2(\\xi ;y)-g_3(\\xi ;y)\\right]d\\sigma (y)\\Big |\\\\&\\qquad \\qquad =\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} \\left[g_2(\\xi ;y)-g_3(\\xi ;y)\\right]\\Delta _0^{K/2}y^\\beta d\\sigma (y)\\Big |\\le c_0 c_6c_{33} \\gamma _3 N_\\xi ^{-K},$ which yields (REF ) with $c_{30}\\ge c_0 c_6c_{33}$ because of $\\gamma _1\\le 1$ .",
"Finally, we set $c_{30}:=\\max \\lbrace c_3 c_{31} c_{32}, c_0 c_6c_{33}\\rbrace $ .",
"Lemma 6.5 Assume $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , $K\\in 2{\\mathbb {N}}$ , $M>K+d-1$ , and let $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ , ${\\varepsilon }$ , $r_\\xi $ , $\\breve{h}_3$ be as in Lemma REF .",
"Let $\\breve{\\theta }^\\diamond _\\xi (x)$ be defined by (REF ) if $d\\ge 3$ or by (REF ) if $d=2$ with $x/|x|$ in the place of $x$ .",
"Then for any $\\gamma _4>0$ there exists $t_j>0$ such that $\\big |\\partial ^\\beta \\big [\\breve{h}_3(\\xi ;x)-\\breve{\\theta }^\\diamond _\\xi (x)\\big ]\\big |\\le \\frac{\\gamma _4 N_\\xi ^{|\\beta |+d-1}}{(1+N_\\xi \\pi )^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},~0\\le |\\beta |\\le K,$ $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\left[h_3(\\xi ;y)-\\theta ^\\diamond _\\xi (y)\\right]d\\sigma (y)\\Big |\\le \\gamma _4 N_\\xi ^{-M},~0\\le |\\beta |\\le K-1.$ Inequalities (REF ) and (REF ) follow from (REF )–(REF ) by approximating the operator $\\Delta _0^{K/2}\\left(\\zeta \\cdot \\nabla \\right)^m$ by ${\\mathfrak {L}}_t^{K/2}{\\mathfrak {D}}^m_t(\\zeta )$ as $t\\rightarrow 0$ and the infinite smoothness of $|(1+{\\varepsilon })\\zeta -x|^{-d+2}$ and $\\log 1/|(1+{\\varepsilon })\\zeta -x|$ on the compact ${{\\mathbb {S}}^{d-1}}$ .",
"This proof follows at once by Lemmas REF , REF , REF , and REF with the following selection of parameters: $\\begin{array}{ll}\\gamma _1:=\\min \\lbrace \\gamma _0/(4c_{10}),1\\rbrace ,&\\gamma _2:=\\min \\lbrace \\gamma _0\\gamma _1^{2K+1}/(4c_{20}),\\gamma _1\\rbrace ,\\\\\\gamma _3:=\\min \\lbrace \\gamma _0\\gamma _1^{2K}/(4c_{30}),1\\rbrace ,&\\gamma _4:=\\gamma _0/4.\\end{array}$"
],
[
"Completion of the construction of new frames on ${{\\mathbb {S}}^{d-1}}$",
"We use the scheme from Section to complete the construction of a pair of dual frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ on ${{\\mathbb {S}}^{d-1}}$ , where each frame element $\\theta _\\xi $ is a linear combination of a fixed number of shifts of the Newtonian kernel.",
"Following the definition $\\psi _\\xi (x):=C^\\diamond _\\xi \\psi ^\\diamond _\\xi (x)$ of the elements of old frame $\\Psi $ given in (REF ), we similarly construct the elements $\\theta _\\xi (x):=C^\\diamond _\\xi \\theta ^\\diamond _\\xi (x), \\quad \\xi \\in \\mathcal {X}_j,~~j\\ge 1,$ of the new frame $\\Theta =\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ .",
"In light of (REF ) and (REF ) we have for $j\\ge 1$ $\\theta _\\xi (x):=C^\\diamond _\\xi \\kappa _{{\\varepsilon },m,d}\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\sum _{\\ell =0}^m b_{\\ell }{\\mathfrak {L}}_{t_j}^{K/2}{\\mathfrak {D}}^\\ell _{t_j}(\\zeta ) |x-a\\zeta |^{2-d},~~~ d\\ge 3,$ $\\theta _\\xi (x):=C^\\diamond _\\xi \\kappa _{{\\varepsilon },m,2}\\sum _{\\begin{array}{c}\\zeta \\in \\mathcal {Z}_j\\\\ \\rho (\\zeta ,\\xi )\\le r_\\xi \\end{array}}w_\\zeta \\Lambda _{N_\\xi }(\\xi \\cdot \\zeta )\\sum _{\\ell =1}^m b_{\\ell }{\\mathfrak {L}}_{t_j}^{K/2}{\\mathfrak {D}}^\\ell _{t_j}(\\zeta ) \\ln \\frac{1}{|x-a\\zeta |},~~ d=2.$ The only frame element excluded from this definition is the constant function corresponding to $\\xi \\in \\mathcal {X}_0$ .",
"For $\\xi \\in \\mathcal {X}_0$ we set $\\theta _\\xi (x):=\\psi _\\xi (x)\\equiv 1$ , $x\\in {{\\mathbb {S}}^{d-1}}$ .",
"In Theorem REF below we collect some important properties of the new frame $\\Theta $ .",
"Its proof is based on Theorem REF and the following lemma.",
"Lemma 6.6 Let $\\eta \\in \\mathcal {X}$ , $K\\in 2{\\mathbb {N}}$ , and $M>K+d-1$ .",
"$(a)$ For any $\\beta $ , $0\\le |\\beta |\\le K$ , we have $\\big |\\partial ^\\beta \\breve{\\mathcal {K}}_{N_\\eta }(\\eta ; x)\\big |\\le c_{40} \\frac{N_\\eta ^{|\\beta |+d-1}}{(1+N_\\eta \\rho (\\eta ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},$ with $\\breve{\\mathcal {K}}_{N_\\eta }(\\eta ; x)$ given by (REF ).",
"$(b)$ For any $\\beta $ , $0\\le |\\beta |\\le K-1$ , we have $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\mathcal {K}_{N_\\eta }(\\eta \\cdot y)d\\sigma (y)\\Big |\\le c_{40}N_\\eta ^{-K}.$ For $\\eta \\in \\mathcal {X}_0$ we have $\\breve{\\mathcal {K}}_{N_\\eta }(\\eta ; x)\\equiv 1/\\omega _d$ , $N_\\eta =2^{-1}$ , and inequalities (REF ), (REF ) are trivial.",
"Let $\\eta \\in \\mathcal {X}\\backslash \\mathcal {X}_0$ .",
"Inequality (REF ) with $N=N_\\eta $ yields (REF ) with $c_{40}\\ge c_5$ .",
"For any multi-index $\\beta $ , $0\\le |\\beta |\\le K-1$ , we get from (REF ), the fact that the operator $\\Delta _0$ is symmetric, (REF ) with $\\beta =0$ and $N=N_\\eta $ , (REF ), and (REF ) that $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\mathcal {K}_{N_\\eta }(\\eta \\cdot y)d\\sigma (y)\\Big |=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} y^\\beta \\Delta _0^{K/2}\\Lambda _{N_\\eta }(\\eta \\cdot y)d\\sigma (y)\\Big |\\\\=\\Big |\\int _{{{\\mathbb {S}}^{d-1}}} \\Lambda _{N_\\eta }(\\eta \\cdot y)\\Delta _0^{K/2}y^\\beta d\\sigma (y)\\Big |\\le c_0c_6c_4N_\\eta ^{-K},$ which proves the lemma with $c_{40}:=\\max \\lbrace c_5,c_0c_6c_4\\rbrace $ .",
"Theorem 6.7 Let $d\\ge 2$ , $K\\in 2{\\mathbb {N}}$ , $M>K+d-1$ , and $0<\\gamma _0\\le 1$ .",
"Then there exist constants $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ , $\\gamma _4>0$ depending only on $d, K, M, \\gamma _0$ , and for every $\\xi \\in \\mathcal {X}_j$ , $j\\in {\\mathbb {N}}$ , there exists $t_j>0$ depending only on $d, K, M, \\gamma _0$ , and $j$ such that: $(a)$ The new frame $\\Theta =\\lbrace \\theta _\\xi \\rbrace _\\xi \\in \\mathcal {X}$ is real-valued and satisfies $\\big |\\partial ^\\beta \\breve{\\theta }_\\xi (x)\\big |\\le c_{41} \\frac{N_\\xi ^{|\\beta |+(d-1)/2}}{(1+N_\\xi \\rho (\\xi ,x))^M},\\quad \\forall x\\in {{\\mathbb {S}}^{d-1}},~\\forall \\xi \\in \\mathcal {X},~ \\forall \\beta ,~ 0\\le |\\beta |\\le K,$ $|\\left\\langle \\psi _\\eta ,\\psi _\\xi -\\theta _\\xi \\right\\rangle | \\le c_{42} \\gamma _0 \\omega _{\\xi ,\\eta }^{(K,M)},\\quad \\forall \\xi ,\\eta \\in \\mathcal {X},$ with $c_{41}=\\max \\lbrace 2^{(d-1)/2}(1+\\pi /2)^M, c_9(c_{40}+\\gamma _0)\\rbrace $ , $c_{42}=c_1 c_{40}c_9^2$ .",
"$(b)$ Every frame element $\\theta _\\xi $ , $\\xi \\in \\mathcal {X}\\backslash \\mathcal {X}_0$ , is a linear combination of at most $\\tilde{n}$ shifts of Newtonian kernels, where $\\tilde{n}\\le c_{43}\\gamma _0^{(4K+3)(1-d)}$ with $c_{43}$ depending only on $d, K, M$ .",
"$(c)$ Moreover, if $\\gamma _0 \\le \\frac{{\\tilde{c}}_5^2}{4 c_{42}},$ where $c_{42}$ is from (REF ) and ${\\tilde{c}}_5$ is from (REF ), then $\\Vert \\theta _\\xi \\Vert _{L^p({{\\mathbb {S}}^{d-1}})} \\sim N_\\xi ^{(d-1)(1/2-1/p)},\\quad \\frac{d-1}{M}<p\\le \\infty , \\quad \\forall \\xi \\in \\mathcal {X},$ with uniformly bounded constants of equivalence for $p\\ge \\frac{d-1+\\delta }{M}$ , $\\delta >0$ .",
"For $\\xi \\in \\mathcal {X}_0$ we have $\\theta _\\xi =1$ , $N_\\xi =2^{-1}$ , and (REF ) is satisfied with $c_{41}\\ge 2^{(d-1)/2}(1+\\pi /2)^M$ .",
"Also $\\psi _\\xi -\\theta _\\xi =0$ and inequality (REF ) is obvious for all $\\eta \\in \\mathcal {X}$ .",
"Let $\\xi \\in \\mathcal {X}\\backslash \\mathcal {X}_0$ .",
"Then (REF ), (REF ) with $\\xi $ in the place of $\\eta $ , and (REF ) imply (REF ) with $c_{41}\\ge c_9(c_{40}+\\gamma _0)$ .",
"For the proof of (REF ) first assume that $N_\\xi \\ge N_\\eta $ .",
"We apply Proposition REF with $g=\\psi _\\eta =C^\\diamond _\\eta \\psi ^\\diamond _\\eta $ , $x_1=\\eta $ , $f=\\psi _\\xi -\\theta _\\xi =C^\\diamond _\\xi (\\psi ^\\diamond _\\xi -\\theta ^\\diamond _\\xi )$ , $x_2=\\xi $ .",
"Lemma REF implies that (REF ) is satisfied with $N_1=N_\\eta $ , $\\kappa _1=c_{40}C^\\diamond _\\eta $ , and Theorem REF implies that (), (REF ) are satisfied with $N_2=N_\\xi $ , $\\kappa _2=\\gamma _0 C^\\diamond _\\xi $ .",
"Now, (REF ) and (REF ) give $|\\left\\langle \\psi _\\eta ,\\psi _\\xi -\\theta _\\xi \\right\\rangle |\\le c_1 \\frac{c_{40}C_9 N_\\eta ^{-\\frac{d-1}{2}}\\gamma _0 C_9 N_\\xi ^{-\\frac{d-1}{2}}(N_\\eta /N_\\xi )^{K}N_\\eta ^{d-1}}{(1+N_\\eta \\rho (\\xi ,\\eta ))^M}\\le c_{42} \\gamma _0 \\omega _{\\xi ,\\eta }^{(K,M)},$ with $c_{42}:=c_1 c_{40}c_9^2$ , which establishes (REF ) in this case.",
"Second, assume that $N_\\xi \\le N_\\eta $ .",
"Here, we apply Proposition REF with $g=\\psi _\\xi -\\theta _\\xi $ , $f=\\psi _\\eta $ , and then (REF ) follows similarly as above.",
"This completes the proof of $(a)$ .",
"The number of $\\zeta \\in \\mathcal {Z}_j$ in (REF ) (or (REF )) can be estimated as follows.",
"From $\\bigcup _{\\zeta \\in \\mathcal {Z}_j:\\rho (\\zeta ,\\xi )\\le r_\\xi }A_\\zeta \\subset B(\\xi ,r_\\xi +\\gamma _2/N_\\xi )$ we find that the total volume covered by $A_\\zeta $ does not exceed $c((\\gamma _3^{-1}+\\gamma _2)/N_\\xi )^{d-1}$ .",
"From this estimate and (REF ) with $\\gamma =\\gamma _2$ we get that the number of $\\zeta \\in \\mathcal {Z}_j$ in (REF ) is at most $c(\\gamma _3\\gamma _2)^{1-d}=c_{44}\\gamma _0^{(4K+3)(1-d)}$ in light of (REF ).",
"Clearly, the number of translation terms in ${\\mathfrak {L}}_{t_j}^{K/2}$ is at most $(d(d-1)/2)^{K/2}+1$ .",
"The number of translation terms in ${\\mathfrak {D}}^\\ell _{t_j}(\\zeta )$ is $\\ell +1$ , $\\ell =0,1,\\dots ,m$ , and every such term is also a term for ${\\mathfrak {D}}^m_{t_j}(\\zeta )$ .",
"This leads to the estimate $\\tilde{n}\\le c_{43}\\gamma _0^{(4K+3)(1-d)}$ with $c_{43}:=c_{44}[(d(d-1)/2)^{K/2}+1](m+1)$ for the number of shifts of Newtonian kernels used in (REF ) or in (REF ).",
"Thus, the proof of $(b)$ is complete.",
"We now establish $(c)$ .",
"Inequality (REF ) with $|\\beta |=0$ along with (REF ) imply $\\Vert \\theta _\\xi \\Vert _{L^p} \\le c_0^{1/p} c_{41} N_\\xi ^{(d-1)(1/2-1/p)}$ with $c_0=c(d)/\\delta $ .",
"To prove the estimate in the other direction: $\\Vert \\theta _\\xi \\Vert _{L^p} \\ge c_{45} N_\\xi ^{(d-1)(1/2-1/p)}$ we first consider the case $p=2$ .",
"From $\\langle \\theta _\\xi ,\\theta _\\xi \\rangle =\\langle \\psi _\\xi ,\\psi _\\xi \\rangle -2\\langle \\psi _\\xi ,\\psi _\\xi -\\theta _\\xi \\rangle +\\langle \\psi _\\xi -\\theta _\\xi ,\\psi _\\xi -\\theta _\\xi \\rangle ,$ (REF ), (REF ), and (REF ), we get $\\langle \\theta _\\xi ,\\theta _\\xi \\rangle \\ge \\langle \\psi _\\xi ,\\psi _\\xi \\rangle -2|\\langle \\psi _\\xi ,\\psi _\\xi -\\theta _\\xi \\rangle |\\ge {\\tilde{c}}_5^2-2\\gamma _0 c_{42}\\omega _{\\xi ,\\xi }^{(K,M)}\\ge {\\tilde{c}}_5^2/2,$ due to $\\omega _{\\xi ,\\xi }^{(K,M)}=1$ (see (REF )).",
"This gives (REF ) for $p=2$ with $c_{45}={\\tilde{c}}_5/\\sqrt{2}$ .",
"Now, consider the case $p<2$ .",
"Using (REF ) with $p=2$ and (REF ) with $p=\\infty $ we obtain ${\\tilde{c}}_5^2/2\\le \\Vert \\theta _\\xi \\Vert _{L^2}^2\\le \\Vert \\theta _\\xi \\Vert _{L^\\infty }^{2-p}\\Vert \\theta _\\xi \\Vert _{L^p}^p\\le \\left(c_{41} N_\\xi ^{(d-1)/2}\\right)^{2-p}\\Vert \\theta _\\xi \\Vert _{L^p}^p,$ which proves (REF ) for $p<2$ with $c_{45}=(c_{41}^{p-2}{\\tilde{c}}_5^2/2)^{1/p}$ .",
"Finally, consider the case $2<p\\le \\infty $ .",
"Using Hölder's inequality, (REF ) with $p=2$ and (REF ) with $1\\le p^{\\prime }<2$ we obtain ${\\tilde{c}}_5^2/2\\le \\Vert \\theta _\\xi \\Vert _{L^2}^2\\le \\Vert \\theta _\\xi \\Vert _{L^{p^{\\prime }}}\\Vert \\theta _\\xi \\Vert _{L^p}\\le c_0^{1/p^{\\prime }}c_{41} N_\\xi ^{(d-1)(1/2-1/p^{\\prime })}\\Vert \\theta _\\xi \\Vert _{L^p},$ which proves (REF ) for $p>2$ with $c_{45}=c_0^{-1+1/p}c_{41}^{-1}{\\tilde{c}}_5^2/2$ .",
"This proves (REF ) for all $p$ and completes the proof of the theorem.",
"Remark 6.8 All poles of the Newtonian kernels in (REF ) and in (REF ) are placed on $m+1$ concentric spheres of radii $1+\\gamma _1 N_\\xi ^{-1}+kt_j$ , $k=0,1,\\dots ,m$ .",
"On every such sphere the poles are located in the spherical cap of radius $(\\gamma _3N_\\xi )^{-1}+t_jK/2$ centred at $(1+\\gamma _1 N_\\xi ^{-1}+kt_j)\\xi $ .",
"Our next step is to show that the above defined system $\\Theta $ coupled with the dual system $\\tilde{\\Theta }=\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ constructed by the scheme from §REF form a pair of frames for all Besov and Triebel-Lizorkin space $\\mathcal {B}^{sq}_p$ , $\\mathcal {F}^{sq}_p$ with parameters $(s, p, q)\\in {\\mathcal {Q}}(A)$ for a fixed $A>1$ with ${\\mathcal {Q}}(A)$ defined in (REF ).",
"Theorem 6.9 Assume $d\\ge 2$ , $A>1$ , and let $\\Theta =\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ be the real-valued system constructed in (REF ) or (REF ), where $K \\ge \\left\\lceil Ad\\right\\rceil , K\\in 2{\\mathbb {N}}, \\quad M = K+d.$ If the constant $\\gamma _0$ in the construction of $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is sufficiently small, namely, $\\gamma _0 \\le \\frac{\\epsilon }{c_{42} C_9},$ where $\\epsilon $ is from (REF ), $c_{42}$ is from (REF ), and $C_9$ is from Theorem REF , then: $(a)$ The synthesis operator $T_\\theta $ defined by $T_\\theta h:= \\sum _{\\xi \\in \\mathcal {X}}h_\\xi \\theta _\\xi $ on sequences of complex numbers $h=\\lbrace h_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is bounded as a map $T_\\theta : {\\mathfrak {b}}_p^{sq} \\mapsto \\mathcal {B}_p^{sq}$ , uniformly with respect to to $(s, p, q)\\in {\\mathcal {Q}}(A)$ .",
"$(b)$ The operator $Tf:=\\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\psi _\\xi \\rangle \\theta _\\xi =T_{\\theta }S_{\\psi }f,$ is invertible on $\\mathcal {B}_p^{sq}$ and $T$ , $T^{-1}$ are bounded on $\\mathcal {B}_p^{sq}$ , uniformly with respect to $(s, p, q)\\in {\\mathcal {Q}}(A)$ .",
"$(c)$ For $(s, p, q)\\in {\\mathcal {Q}}(A)$ the dual system $\\tilde{\\Theta }=\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ consists of bounded linear functionals on $\\mathcal {B}_p^{sq}$ defined by $\\tilde{\\theta }_\\xi (f)=\\langle f, \\tilde{\\theta }_\\xi \\rangle := \\sum _{\\eta \\in \\mathcal {X}}\\langle T^{-1}\\psi _\\eta , \\psi _\\xi \\rangle \\langle f, \\psi _\\eta \\rangle \\quad \\hbox{for}\\;\\; f\\in \\mathcal {B}_p^{sq},$ with the series converging absolutely.",
"Also, the analysis operator $S_{\\tilde{\\theta }}: \\mathcal {B}_p^{sq} \\mapsto {\\mathfrak {b}}_p^{sq},\\;S_{\\tilde{\\theta }} = S_{\\psi }T^{-1}T_{\\psi }S_{\\psi }=S_{\\psi }T^{-1},$ is uniformly bounded with respect to $(s, p, q)\\in {\\mathcal {Q}}(A)$ .",
"Moreover, $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ form a pair of dual frames for $\\mathcal {B}_p^{sq}$ in the following sense: For any $f\\in \\mathcal {B}_p^{sq}$ $f=\\sum _{\\xi \\in \\mathcal {X}} \\langle f, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi \\quad \\hbox{and}\\quad \\Vert f\\Vert _{\\mathcal {B}_p^{sq}} \\sim \\Vert \\lbrace \\langle f, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}_p^{sq}},$ where the convergence is unconditional in $\\mathcal {B}_p^{sq}$ .",
"Furthermore, $(a)$ , $(b)$ , and $(c)$ hold true when $\\mathcal {B}_p^{sq}$ , ${\\mathfrak {b}}_p^{sq}$ are replaced by $\\mathcal {F}_p^{sq}$ , ${\\mathfrak {f}}_p^{sq}$ , respectively.",
"The parameters $K$ and $M$ from (REF ) satisfy (REF ) with $\\delta =1$ for all $(s, p, q)\\in {\\mathcal {Q}}(A)$ and we can apply Theorem REF .",
"From estimate (REF ) in Theorem REF , Lemma REF , and Theorem REF we obtain that $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ satisfies (REF ) due to $\\gamma _0 c_{42} C_9\\le \\epsilon $ .",
"Also, all conditions on the old frame laid in Subsection REF are satisfied as shown in Subsection REF .",
"Now, we apply Lemma REF and Lemma REF to get $(a)$ and $(b)$ .",
"Finally, Theorem REF implies $(c)$ ."
],
[
"Frames on $B^d$ in terms of shifts of the Newtonian kernel",
"Note that by the fact that each frame element $\\theta _\\xi $ , $\\xi \\in \\mathcal {X}$ , is represented as a finite linear combination of shifts of the Newtonian kernel it readily follows that $\\theta _\\xi (x)$ defined in (REF ) or (REF ) as a function of $x\\in B^d$ is harmonic on $B^d$ .",
"This leads immediately to the conclusion that $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ is a pair of dual frames for harmonic Besov and Triebel-Lizorkin spaces in the sense of the following Theorem 6.10 Under the hypothesis of Theorem REF let $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ be the frames from Theorem REF .",
"Then for any $(s, p, q)\\in {\\mathcal {Q}}(A)$ and $U\\in B_p^{sq}(\\mathcal {H})$ $U(x)=\\sum _{\\xi \\in \\mathcal {X}} \\langle f_U, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi (x),~x\\in B^d,\\quad \\hbox{and}\\quad \\Vert U\\Vert _{B_p^{sq}} \\sim \\Vert \\lbrace \\langle f_U, \\tilde{\\theta }_\\xi \\rangle \\rbrace \\Vert _{{\\mathfrak {b}}_p^{sq}}.$ Here $f_U$ is the boundary value of $U$ $($ see Proposition REF $)$ and the series converges uniformly on every compact subset of $B^d$ .",
"Furthermore, the above holds true when $B_p^{sq}(\\mathcal {H})$ , ${\\mathfrak {b}}_p^{sq}$ are replaced by $F_p^{sq}(\\mathcal {H})$ , ${\\mathfrak {f}}_p^{sq}$ , respectively.",
"This theorem follows immediately by Theorems REF , REF , and REF ."
],
[
"Nonlinear approximation from shifts of the Newtonian kernel",
"The primary goal of this article is to establish a Jackson type estimate for nonlinear $n$ -term approximation of harmonic functions on $B^d$ from shifts of the Newtonian kernel in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ .",
"For any $n\\ge 1$ write $\\mathcal {N}_n:=\\Big \\lbrace G: G(x) =a_0+\\sum _{\\nu =1}^{n} \\frac{a_\\nu }{|x-y_\\nu |^{d-2}}, \\; |y_\\nu |>1, a_\\nu \\in {\\mathbb {C}}\\Big \\rbrace ,\\;\\;\\hbox{if $d>2$},$ and $\\mathcal {N}_n:=\\Big \\lbrace G: G(x) =a_0+\\sum _{\\nu =1}^{n} a_\\nu \\ln \\frac{1}{|x-y_\\nu |}, \\; |y_\\nu |>1, a_\\nu \\in {\\mathbb {C}}\\Big \\rbrace ,\\;\\;\\hbox{if $d=2$}.$ Observe that the points $\\lbrace y_\\nu \\rbrace $ above may vary with $G$ and hence $\\mathcal {N}_n$ is nonlinear.",
"Let ${\\mathfrak {B}}$ be one of the spaces $\\mathcal {H}^p(B^d)$ , $B_p^{0q}(\\mathcal {H})$ , or $F_p^{0q}(\\mathcal {H})$ , $0<p,q<\\infty $ .",
"Given $U\\in {\\mathfrak {B}}$ we define $E_n(U)_{{\\mathfrak {B}}}:=\\inf _{G\\in \\mathcal {N}_n}\\Vert U-G\\Vert _{{\\mathfrak {B}}}.$ We call $E_n(U)_{{\\mathfrak {B}}}$ the best nonlinear $n$ -term approximation of $U$ from shifts of the Newtonian kernel in the harmonic space ${\\mathfrak {B}}$ .",
"We now come to the main result in this article.",
"Theorem 7.1 Let $s>0$ , $0<p<\\infty $ , and $1/\\tau =s/(d-1)+1/p$ .",
"If $U\\in B_\\tau ^{s\\tau }(\\mathcal {H})$ , then $U\\in \\mathcal {H}^p(B^d)$ and $E_n(U)_{\\mathcal {H}^p(B^d)} \\le cn^{-s/(d-1)}\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})},\\quad n\\ge 1,$ where the constant $c>0$ depends only on $p, s, d$ .",
"Theorem REF is an immediate consequence of Theorem REF bellow with $q=2$ and Theorem REF .",
"Our approach to approximating a harmonic function $U$ on $B^d$ amounts to first establishing a Jackson estimate for nonlinear $n$ -term approximation of its boundary value function $f_U$ on ${{\\mathbb {S}}^{d-1}}$ from the frame elements $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ constructed in Section and then considering the harmonic extension to $B^d$ of the approximant.",
"The gist of our approximation method is that each frame element $\\theta _\\xi $ is a linear combination of a fixed number of shifts of the Newtonian kernel."
],
[
"Nonlinear $n$ -term frame approximation on {{formula:627eae61-f7a4-4bc7-8b61-8c2efcaf724b}}",
"Let $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ be the frame constructed in Section with parameters to be specified.",
"Denote by $\\Sigma _n$ the set of all functions $g$ on ${{\\mathbb {S}}^{d-1}}$ of the form $g=\\sum _{\\xi \\in Y_n} a_\\xi \\theta _\\xi , \\quad a_\\xi \\in {\\mathbb {C}},$ where $Y_n\\subset \\mathcal {X}$ is an index set such that $\\# Y_n\\le n$ .",
"Define $\\sigma _n(f)_{\\mathfrak {B}}:=\\inf _{g\\in \\Sigma _n}\\Vert f-g\\Vert _{{\\mathfrak {B}}},$ where ${\\mathfrak {B}}$ is one of the spaces $L^p({{\\mathbb {S}}^{d-1}})$ , $\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ , or $\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ , $0<p,q<\\infty $ .",
"As one can expect the smoothness spaces on ${{\\mathbb {S}}^{d-1}}$ governing this kind of approximation should be the Besov spaces $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ with $s$ and $\\tau $ as in Theorem REF .",
"For this to be true, however, $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ have to provide frame decomposition of all spaces involved just as in Theorem REF .",
"Assumptions: The construction of the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ in Section depends on the parameters $A$ , $K$ , $M$ , and $\\gamma _0$ .",
"The main parameter is $A$ .",
"In light of Theorems REF and REF we require that $0<s\\le A, \\;\\; A^{-1}\\le p\\le A, \\;\\; A^{-1}\\le q <\\infty , \\;\\; A^{-1}\\le s/(d-1)+1/p\\le A,\\;\\; A>1,$ which reduces to the following principle conditions: $0<s\\le A, \\;\\; 0<p \\le A, \\;\\; A^{-1}\\le q <\\infty , \\;\\; s/(d-1)+1/p\\le A,\\;\\; A>1.$ The parameters $K$ , $M$ are secondary and are defined as $K:= 2\\left\\lceil \\frac{Ad}{2}\\right\\rceil ,\\quad M:=K+d.$ Further, $\\gamma _0>0$ is a “small” parameter and using the notation from Theorems REF and REF we fix it as $\\gamma _0:=\\min \\left\\lbrace \\frac{\\epsilon }{c_{42} C_9},\\frac{{\\tilde{c}}_5^2}{4 c_{42}}\\right\\rbrace .$ It is easy to see that $\\gamma _0$ depends only on $A$ , $d$ , and the “old\" frame $\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , described in §REF .",
"Conditions (REF ) can be viewed in two ways: (a) Given $A>1$ consider (REF ) as conditions on $s, p, q$ ; (b) Given $s, p, q$ consider (REF ) as conditions on $A$ .",
"For Theorem REF we take $A$ to be the smallest number satisfying (REF ).",
"Either way conditions (REF ) coupled with (REF )–(REF ) imply that the hypotheses of Theorems REF and REF are obeyed and hence the conclusions of these theorems are valid for the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ and the spaces $\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ , $\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ , and $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ .",
"Although we are mainly interested in approximation of functions in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ or their boundary values in $\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})$ , to put it in perspective we shall examine the approximation process at hand in the slightly more general spaces $\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ ."
],
[
"Nonlinear $n$ -term frame approximation in Triebel-Lizorkin spaces on {{formula:de35b824-1464-42f0-a468-02406bca14e9}}",
"Theorem 7.2 Assume $A>1$ and let the parameters $K$ , $M$ , and $\\gamma _0$ be defined as in (REF )–(REF ).",
"Let $s>0$ , $0<p,q<\\infty $ , and $1/\\tau =s/(d-1)+1/p$ , and assume that $s, p, q$ satisfy conditions (REF ).",
"If $f\\in \\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})=\\mathcal {F}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ , then $f\\in \\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and with the notation from (REF ) $\\sigma _n(f)_{\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})} \\le cn^{-s/(d-1)}\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},\\quad n\\ge 1,$ where the constant $c>0$ depends only on $A, d$ .",
"The proof of this theorem depends on the following simple Lemma 7.3 For any $0<p<\\infty $ and any finite subset $Y$ of $\\mathcal {X}$ we have $\\Big \\Vert \\sum _{\\xi \\in Y}|B_\\xi |^{-1/p}{\\mathbb {1}}_{B_\\xi }\\Big \\Vert _{L^{p}}\\le c(\\# Y)^{1/p}$ with $B_\\xi =B(\\xi ,\\gamma N_\\xi ^{-1})$ from Definition REF .",
"First, we observe that for any $j\\in {\\mathbb {N}}$ and any $x\\in {{\\mathbb {S}}^{d-1}}$ the number of points $\\eta \\in \\mathcal {X}_j$ such that $x\\in B_\\eta =B(\\eta ,\\delta _j)$ does not exceed a constant $c(d)$ .",
"Indeed, the spherical caps $\\lbrace B(\\eta ,\\delta _j/2)\\rbrace _{\\eta \\in \\mathcal {X}_j}$ are mutually disjoint and $x\\in B_\\eta $ implies $B(\\eta ,\\delta _j/2)\\subset B(x,3\\delta _j/2)$ , which together with (REF ) justifies the observation.",
"Given $Y\\subset \\mathcal {X}$ we set $\\Omega := \\cup _{\\xi \\in Y}B_\\xi $ and $h(x):= \\min \\big \\lbrace |B_\\xi |: x\\in B_\\xi , \\xi \\in Y\\big \\rbrace $ if $x\\in \\Omega $ .",
"Clearly, if $x\\in B_\\xi $ for some $\\xi \\in Y$ , then the above observation and (REF ) imply $\\sum _{\\begin{array}{c}\\eta \\in Y: x\\in B_\\eta \\\\ |B_\\eta |\\ge |B_\\xi |\\end{array}}(|B_\\xi |/|B_\\eta |)^{1/p} \\le c\\sum _{j=0}^{\\infty } 2^{-j(d-1)/p}=c_\\star <\\infty ,$ yielding $\\sum _{\\xi \\in Y} |B_\\xi |^{-1/p}{\\mathbb {1}}_{B_\\xi }(x)\\le c_\\star h(x)^{-1/p}\\le c_\\star \\Big (\\sum _{\\xi \\in Y} |B_\\xi |^{-1}{\\mathbb {1}}_{B_\\xi }(x)\\Big )^{1/p},\\quad x\\in \\Omega .$ In turn, this readily implies (REF ).",
"Under the hypotheses of Theorem REF assume $f\\!\\in \\!\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ .",
"Embedding (REF ) implies $f\\in \\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ .",
"As we already alluded to above the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ are well defined and the conclusions of Theorems REF and REF are valid for the spaces $\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ .",
"In particular, condition (REF ) implies (REF ) and conditions (REF )–(REF ) implies $(d-1+\\delta )/M\\le p$ with $\\delta =1$ .",
"Hence the assumptions of Theorem REF (c) are fulfilled and from (REF ) and (REF ) we get $\\Vert \\theta _\\xi \\Vert _{L^p}:=\\Vert \\theta _\\xi \\Vert _{L^p({{\\mathbb {S}}^{d-1}})} \\sim N_\\xi ^{(d-1)(1/2-1/p)}\\sim |B_\\xi |^{1/p-1/2},\\quad \\xi \\in \\mathcal {X},$ with constants of equivalence depending only on $d$ and $A$ .",
"Set $a_\\xi :=\\langle f, \\tilde{\\theta }_\\xi \\rangle $ , $\\xi \\in \\mathcal {X}$ .",
"From (REF ) and (REF )–(REF ) we obtain $\\Vert a\\Vert _{{\\mathfrak {b}}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})}\\sim \\Vert a\\Vert _{{\\mathfrak {f}}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})}\\sim \\Big (\\sum _{\\xi \\in \\mathcal {X}} (\\Vert a_\\xi \\theta _\\xi \\Vert _{L^p})^\\tau \\Big )^{1/\\tau }=:N(f).$ We may assume $N(f)>0$ .",
"Denote $\\mathcal {Y}_\\nu :=\\Big \\lbrace \\xi \\in \\mathcal {X}: 2^{-\\nu }N(f)<\\Vert a_\\xi \\theta _\\xi \\Vert _{L^p}\\le 2^{-\\nu +1}N(f)\\Big \\rbrace ,\\quad \\nu \\in {\\mathbb {N}}.$ Then $\\bigcup _{\\nu \\le \\mu }\\mathcal {Y}_\\nu =\\Big \\lbrace \\xi : 2^{-\\mu }N(f)<\\Vert a_\\xi \\theta _\\xi \\Vert _{L^p}\\Big \\rbrace ,\\quad \\mu \\in {\\mathbb {N}},$ and hence $\\#\\mathcal {Y}_\\mu \\le \\sum _{\\nu \\le \\mu }\\#\\mathcal {Y}_\\nu =\\#\\Big (\\bigcup _{\\nu \\le \\mu }\\mathcal {Y}_\\nu \\Big )\\le 2^{\\mu \\tau }.$ Set $F_\\mu :=\\sum _{\\xi \\in \\mathcal {Y}_\\mu }|a_\\xi |^q|B_\\xi |^{-q/2}{\\mathbb {1}}_{B_\\xi }$ .",
"We next show that for $m\\ge 0$ $\\Big \\Vert \\Big (\\sum _{\\mu \\ge m+1}F_\\mu \\Big )^{1/q}\\Big \\Vert _{L^p({{\\mathbb {S}}^{d-1}})}\\le c2^{-m\\tau s/(d-1)}N(f).$ To this end we first estimate $\\Vert F_\\mu \\Vert _{L^{p/q}}$ .",
"Using (REF ), (REF ), Lemma REF with $p$ replaced by $p/q$ , and (REF ) we obtain $\\Vert F_\\mu \\Vert _{L^{p/q}}&= \\Big \\Vert \\sum _{\\xi \\in \\mathcal {Y}_\\mu }\\big (|a_\\xi ||B_\\xi |^{1/p-1/2}\\big )^q|B_\\xi |^{-q/p}{\\mathbb {1}}_{B_\\xi }\\Big \\Vert _{L^{p/q}} \\\\&\\le c 2^{-q(\\mu -1)} N(f)^q\\Big \\Vert \\sum _{\\xi \\in \\mathcal {Y}_\\mu }|B_\\xi |^{-q/p}{\\mathbb {1}}_{B_\\xi }\\Big \\Vert _{L^{p/q}}\\le c 2^{-q\\mu } N(f)^q(\\# \\mathcal {Y}_\\mu )^{q/p}\\\\&\\le c2^{-q\\mu (1-\\tau /p)} N(f)^q= c 2^{-q\\mu \\tau s/(d-1)} N(f)^q.",
"$ To prove (REF ) we consider two cases.",
"If $q\\le p$ , then using (REF ) $\\Big \\Vert \\Big (\\sum _{\\mu \\ge m+1}F_\\mu \\Big )^{1/q}\\Big \\Vert _{L^p}^q= \\Big \\Vert \\sum _{\\mu \\ge m+1}F_\\mu \\Big \\Vert _{L^{p/q}}&\\le \\sum _{\\mu \\ge m+1}\\Vert F_\\mu \\Vert _{L^{p/q}}\\\\\\le c\\sum _{\\mu \\ge m+1} 2^{-q\\mu \\tau s/(d-1)} N(f)^q&\\le c 2^{-qm\\tau s/(d-1)} N(f)^q,$ which implies (REF ).",
"In the case $q>p$ using that $p/q<1$ and (REF ) we have $\\Big \\Vert \\Big (\\sum _{\\mu \\ge m+1}F_\\mu \\Big )^{1/q}\\Big \\Vert _{L^p}^p= \\Big \\Vert \\sum _{\\mu \\ge m+1}F_\\mu \\Big \\Vert _{L^{p/q}}^{p/q}&\\le \\sum _{\\mu \\ge m+1}\\Vert F_\\mu \\Vert _{L^{p/q}}^{p/q}\\\\\\le c\\sum _{\\mu \\ge m+1} 2^{-p\\mu \\tau s/(d-1)} N(f)^p&\\le c 2^{-pm\\tau s/(d-1)} N(f)^p,$ yielding again (REF ).",
"Choose $m\\ge 0$ so that $2^{m\\tau }\\le n<2^{(m+1)\\tau }$ and denote $\\mathcal {Z}_m:=\\bigcup _{\\nu \\le m}\\mathcal {Y}_\\nu $ .",
"Also, set $a^\\star _\\xi := a_\\xi $ if $\\xi \\in \\mathcal {X}\\setminus \\mathcal {Z}_m$ and $a^\\star _\\xi := 0$ if $\\xi \\in \\mathcal {Z}_m$ .",
"By (REF ) it follows that $\\# \\mathcal {Z}_m \\le 2^{m\\tau }\\le n$ .",
"This, the frame representation (REF ) for $f\\in \\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ , and the boundedness of the synthesis operator $T_\\theta $ from Theorem REF (b) yield $\\sigma _n(f)_{\\mathcal {F}_p^{0q}}&\\le \\Big \\Vert f-\\sum _{\\xi \\in \\mathcal {Z}_m}a_\\xi \\theta _\\xi \\Big \\Vert _{\\mathcal {F}_p^{0q}}= \\Big \\Vert \\sum _{\\xi \\in \\mathcal {X}\\setminus \\mathcal {Z}_m}a_\\xi \\theta _\\xi \\Big \\Vert _{\\mathcal {F}_p^{0q}}\\le c\\Vert a^\\star \\Vert _{{\\mathfrak {f}}_p^{0q}}\\\\&= c\\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}\\setminus \\mathcal {Z}_m}|a_\\xi |^q|B_\\xi |^{-q/2}{\\mathbb {1}}_{B_\\xi }\\Big )^{1/q}\\Big \\Vert _{L^p}= c\\Big \\Vert \\Big (\\sum _{\\mu \\ge m+1}F_\\mu \\Big )^{1/q}\\Big \\Vert _{L^p}.$ Finally, we use (REF ), (REF ), and Theorem REF (c) to obtain $\\sigma _n(f)_{\\mathcal {F}_p^{0q}}\\le c2^{-m\\tau s/(d-1)}N(f)\\le cn^{-s/(d-1)}\\Vert a\\Vert _{{\\mathfrak {b}}_\\tau ^{s\\tau }}\\le cn^{-s/(d-1)}\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},$ which confirms (REF ).",
"From Theorem REF and the equivalence $\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})\\sim L^p({{\\mathbb {S}}^{d-1}})$ for $1<p<\\infty $ we immediately get Corollary 7.4 Assume $A>1$ and let the parameters $K$ , $M$ , and $\\gamma _0$ be defined as in (REF )–(REF ).",
"Let $s>0$ , $1<p<\\infty $ , and $1/\\tau =s/(d-1)+1/p$ , and assume that $s, p$ satisfy conditions (REF ) with $q=2$ .",
"If $f\\in \\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ , then $f\\in L^p({{\\mathbb {S}}^{d-1}})$ and $\\sigma _n(f)_{L^p({{\\mathbb {S}}^{d-1}})} \\le cn^{-s/(d-1)}\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},~~n\\ge 1,$ where the constant $c>0$ depends only on $A, d$ ."
],
[
"Nonlinear $n$ -term frame approximation in Besov spaces on {{formula:08e4bfa8-20a7-4f99-a676-7caaadf10446}}",
"Theorem 7.5 Assume $A>1$ and let the parameters $K$ , $M$ , and $\\gamma _0$ be as in (REF )–(REF ).",
"Let $s>0$ , $0<p, q<\\infty $ , $1/\\tau =s/(d-1)+1/p$ , and $q\\ge \\tau $ , and assume that $s, p, q$ satisfy conditions (REF ).",
"If $f\\in \\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ , then $f\\in \\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and for every $n\\ge 1$ we have $\\sigma _n(f)_{\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})} \\le cn^{-s/(d-1)}\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},\\quad p\\le q,$ $\\sigma _n(f)_{\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})} = o(n^{1/q-1/\\tau })\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},\\quad \\tau \\le q<p,$ where the constant $c>0$ depends only on $A, d$ .",
"For the proof of this theorem we shall utilize inequality (6.7) from [25] given in the following Lemma 7.6 Let $0<\\tau < p<\\infty $ and $x_1\\ge x_2\\ge \\dots \\ge 0$ .",
"Then for every $n\\in {\\mathbb {N}}$ we have $\\Big (\\sum _{k=n+1}^\\infty x_k^p\\Big )^{1/p}\\le n^{1/p-1/\\tau }\\Big (\\sum _{k=1}^\\infty x_k^\\tau \\Big )^{1/\\tau }.$ Assume that the hypotheses of Theorem REF are obeyed and let $f\\in \\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ .",
"Embeddings (REF ) and (REF ) imply $f\\in \\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ .",
"As above the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ are well defined and the conclusions of Theorems REF and REF are valid for the spaces $\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ .",
"Denote $a_\\xi :=\\langle f, \\tilde{\\theta }_\\xi \\rangle $ , $\\xi \\in \\mathcal {X}$ .",
"Recall the equivalence (REF ) which holds in this case.",
"Let $\\big \\lbrace \\Vert a_{\\eta _k}\\theta _{\\eta _k}\\Vert _{L^p}\\big \\rbrace _{k=1}^\\infty $ be the non-increasing rearrangement of the sequence $\\big \\lbrace \\Vert a_{\\xi }\\theta _\\xi \\Vert _{L^p}\\big \\rbrace _{\\xi \\in \\mathcal {X}}$ , i.e.",
"$\\Vert a_{\\eta _1}\\theta _{\\eta _1}\\Vert _{L^p}\\ge \\Vert a_{\\eta _2}\\theta _{\\eta _2}\\Vert _{L^p}\\ge \\cdots .$ Consider first the case $p\\le q$ .",
"Fix $n\\ge 1$ and set $a^\\star _{\\eta _k}:= a_{\\eta _k}$ if $k>n$ and $a^\\star _{\\eta _k}:=0$ if $k\\le n$ .",
"Note that from (REF ) and (REF ) it follows that $\\Vert a^\\star \\Vert _{{\\mathfrak {b}}_p^{0p}}\\sim \\Big (\\sum _{\\xi \\in \\mathcal {X}} \\Vert a^\\star _{\\xi }\\theta _{\\xi }\\Vert _{L^p}^p\\Big )^{1/p}=\\Big (\\sum _{k=n+1}^\\infty \\Vert a_{\\eta _k}\\theta _{\\eta _k}\\Vert _{L^p}^p\\Big )^{1/p}.$ Using this, embedding (REF ), and the boundedness of the synthesis operator $T_\\theta $ from Theorem REF (b) we get $\\sigma _n(f)_{\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})}&\\le \\Big \\Vert \\sum _{\\eta \\in \\mathcal {X}}a_\\eta \\theta _\\eta -\\sum _{k=1}^n a_{\\eta _k}\\theta _{\\eta _k}\\Big \\Vert _{\\mathcal {B}_p^{0q}}\\le c\\Big \\Vert \\sum _{k=n+1}^\\infty a_{\\eta _k}\\theta _{\\eta _k}\\Big \\Vert _{\\mathcal {B}_p^{0p}}\\\\& \\le c\\Vert a^\\star \\Vert _{{\\mathfrak {b}}_p^{0p}}\\le c\\Big (\\sum _{k=n+1}^\\infty \\Vert a_{\\eta _k}\\theta _{\\eta _k}\\Vert _{L^p}^p\\Big )^{1/p}.$ Further, we apply the inequality of Lemma REF with $x_k=\\Vert a_{\\eta _k}\\theta _{\\eta _k}\\Vert _{L^p}$ , (REF ) and Theorem REF (c) to obtain $\\sigma _n(f)_{\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})}&\\le c n^{1/p-1/\\tau }\\Big (\\sum _{k=1}^\\infty \\Vert a_{\\eta _k}\\theta _{\\eta _k}\\Vert _{L^p}^\\tau \\Big )^{1/\\tau }\\\\&\\le cn^{-s/(d-1)}\\Vert a\\Vert _{{\\mathfrak {b}}_\\tau ^{s\\tau }}\\le cn^{-s/(d-1)}\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})},$ which confirms (REF ).",
"In the case $q=\\tau $ , we use the embedding $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})\\subset \\mathcal {B}_p^{0\\tau }({{\\mathbb {S}}^{d-1}})$ (see (REF )) to obtain $\\Vert f-g\\Vert _{\\mathcal {B}_p^{0\\tau }} = o(1)\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }}$ with $g=\\sum _{k=1}^n a_{\\eta _k}\\theta _{\\eta _k}$ .",
"Combining this estimate with estimate (REF ) with $q=p$ and applying Hölder's inequality we obtain in the case $\\tau \\le q<p$ $\\big \\Vert f-g\\big \\Vert _{\\mathcal {B}_p^{0q}}\\le \\big \\Vert f-g\\big \\Vert _{\\mathcal {B}_p^{0\\tau }}^{(p-q)\\tau /((p-\\tau )q)}\\big \\Vert f-g\\big \\Vert _{\\mathcal {B}_p^{0p}}^{(q-\\tau )p/((p-\\tau )q)}\\\\=o(1)\\big \\Vert f\\big \\Vert _{\\mathcal {B}_\\tau ^{s\\tau }}^{(p-q)\\tau /((p-\\tau )q)}n^{1/q-1/\\tau }\\big \\Vert f\\big \\Vert _{\\mathcal {B}_\\tau ^{s\\tau }}^{(q-\\tau )p/((p-\\tau )q)}= o(n^{1/q-1/\\tau })\\Vert f\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }}.$ This proves (REF ) and completes the proof of the theorem.",
"Remark 7.7 In comparing Theorem REF and Theorem REF we see that the optimal rate $n^{s/(d-1)}$ holds for approximation in $\\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ for every $q>0$ but only for $q\\ge p$ for approximation in $\\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})$ .",
"Theorem REF cannot be extended for $q<\\tau $ because $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})\\setminus \\mathcal {B}_p^{0q}({{\\mathbb {S}}^{d-1}})\\ne \\emptyset $ if $q<\\tau $ .",
"Remark 7.8 Note that in both Theorem REF and Theorem REF we form the near best approximant as the sum of the $n$ terms $\\langle f, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi $ with the biggest norms $\\Vert \\langle f, \\tilde{\\theta }_\\xi \\rangle \\theta _\\xi \\Vert _{L^p({{\\mathbb {S}}^{d-1}})}$ ."
],
[
"Nonlinear $n$ -term approximation of harmonic functions on {{formula:d156db2a-9cc1-4e83-9c86-d8fe0359551f}}",
"We next use Theorems REF and REF to establish respective Jackson estimates for nonlinear $n$ -term approximation of harmonic functions on $B^d$ from shifts of Newtonian kernel.",
"Theorem 7.9 Let $s>0$ , $0<p,q<\\infty $ , and $1/\\tau =s/(d-1)+1/p$ .",
"If the harmonic function $U\\in B_\\tau ^{s\\tau }(\\mathcal {H})=F_\\tau ^{s\\tau }(\\mathcal {H})$ , then $U\\in F_p^{0q}(\\mathcal {H})$ and $E_n(U)_{F_p^{0q}(\\mathcal {H})} \\le cn^{-s/(d-1)}\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})},\\quad n\\ge 1,$ where the constant $c>0$ depends only on $s, p, q, d$ .",
"By Theorem REF it follows that the boundary value function $f_U$ of $U$ given in Proposition REF belongs to $\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})=\\mathcal {F}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})$ and $\\Vert f_U\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})} \\sim \\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})}$ .",
"Then embedding (REF ) of Proposition REF implies that $f_U\\in \\mathcal {F}_p^{0q}({{\\mathbb {S}}^{d-1}})$ and in turn Theorem REF yields $U\\in F_p^{0q}(\\mathcal {H})$ .",
"With $s, p, q, \\tau $ already fixed, we choose $A:=\\max \\big \\lbrace s, p, q^{-1}, \\tau ^{-1}, 2\\big \\rbrace $ .",
"Then conditions (REF ) are satisfied.",
"Pick the parameters $K$ , $M$ , and $\\gamma _0$ as in (REF )–(REF ).",
"Then the frames $\\lbrace \\theta _\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ , $\\lbrace \\tilde{\\theta }_\\xi \\rbrace _{\\xi \\in \\mathcal {X}}$ are well defined.",
"Appealing to Theorem REF we conclude that for any $n\\ge 1$ there exist $\\xi _1, \\dots , \\xi _n\\in \\mathcal {X}$ and coefficients $a_1, \\dots , a_n\\in {\\mathbb {C}}$ such that $\\Big \\Vert f_U - \\sum _{j=1}^n a_j \\theta _{\\xi _j}\\Big \\Vert _{\\mathcal {F}_p^{02}({{\\mathbb {S}}^{d-1}})}\\le cn^{-s/(d-1)}\\Vert f_U\\Vert _{\\mathcal {B}_\\tau ^{s\\tau }({{\\mathbb {S}}^{d-1}})}\\le cn^{-s/(d-1)}\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})}.$ Write $G_n(x):= \\sum _{j=1}^n a_j \\theta _{\\xi _j}(x)$ , $x\\in B^d$ .",
"From above by harmonic extension using Theorem REF we obtain $\\Vert U - G_n\\Vert _{F_p^{0q}(\\mathcal {H})}\\le cn^{-s/(d-1)}\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})}.$ However, by Theorem REF (b) we know that for every $\\xi \\in \\mathcal {X}\\setminus \\mathcal {X}_0$ the frame element $\\theta _\\xi $ is a linear combination of $\\le \\tilde{n}$ shifts of the Newtonian kernel, where $\\tilde{n}$ is a constant.",
"Therefore, $G_n\\in \\mathcal {N}_{\\tilde{n}n}$ and then estimate (REF ) follows readily by (REF ).",
"Theorem 7.10 Let $s>0$ , $0<p, q<\\infty $ , $1/\\tau =s/(d-1)+1/p$ , and $q\\ge \\tau $ .",
"If the harmonic function $U\\in B_\\tau ^{s\\tau }(\\mathcal {H})=F_\\tau ^{s\\tau }(\\mathcal {H})$ , then $U\\in B_p^{0q}(\\mathcal {H})$ and for every $n\\ge 1$ we have $E_n(U)_{B_p^{0q}(\\mathcal {H})} \\le cn^{-s/(d-1)}\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})},\\quad p\\le q,$ $E_n(U)_{B_p^{0q}(\\mathcal {H})} = o(n^{1/q-1/\\tau })\\Vert U\\Vert _{B_\\tau ^{s\\tau }(\\mathcal {H})},\\quad \\tau \\le q<p,$ where the constant $c>0$ depends only on $A, d$ .",
"The proof of Theorem REF goes along the lines of the proof of Theorem REF with Theorem REF replaced by Theorem REF .",
"We omit it."
],
[
"Approximation of harmonic functions on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and {{formula:b18a6b83-9960-4768-908d-b69d7818a16a}}",
"The results in Section have their analogues for approximation of harmonic functions on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ or ${\\mathbb {R}}_+^d$ .",
"In the following we established the analogue of the main result (Theorem REF ) on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and explain briefly its analogue on ${\\mathbb {R}}_+^d$ .",
"In analogy to the set $\\mathcal {N}_n$ from (REF )–(REF ) we denote by $\\overline{\\mathcal {N}}_n$ the set of all linear combinations of shifts of the Newtonian kernel as in (REF )–(REF ) with the requirement that the poles $y_\\nu \\in B^d$ .",
"The approximation will take place in the harmonic Hardy space $\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})$ .",
"Let $\\mathcal {H}({\\mathbb {R}}^d\\setminus \\overline{B^d})$ denote the set of all harmonic functions $U$ on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ such that $\\lim _{|x|\\rightarrow \\infty } U(x) =0$ if $d>2$ or $\\lim _{|x|\\rightarrow \\infty } U(x) = {\\rm const.",
"}$ if $d=2$ .",
"The harmonic Hardy space $\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})$ , $0<p<\\infty $ , is defined as the set of all harmonic functions $U\\in \\mathcal {H}({\\mathbb {R}}^d\\setminus \\overline{B^d})$ such that $\\Vert U\\Vert _{\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})} := \\Vert \\sup _{r>1}r^{d-2}|U(r\\cdot )|\\Vert _{L^p({{\\mathbb {S}}^{d-1}})} <\\infty .$ Given $U\\in \\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})$ we define $E_n(U)_{\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})}:=\\inf _{G\\in \\overline{\\mathcal {N}}_n}\\Vert U-G\\Vert _{\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})}.$ Denote by $\\bar{B}_p^{sq}(\\mathcal {H})$ the harmonic Besov spaces on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ (see [15]).",
"As one can expect the following Jackson type theorem is valid: Theorem 8.1 Let $s>0$ , $0<p<\\infty $ , and $1/\\tau =s/(d-1)+1/p$ .",
"If $U\\in \\bar{B}_\\tau ^{s\\tau }(\\mathcal {H})$ , then $U\\in \\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})$ and $E_n(U)_{\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})} \\le cn^{-s/(d-1)}\\Vert U\\Vert _{\\bar{B}_\\tau ^{s\\tau }(\\mathcal {H})},\\quad n\\ge 1,$ where the constant $c>0$ depends only on $p, s, d$ .",
"As is well known the Kelvin transform $KU(x):= |x|^{2-d}U(x/|x|^2)$ maps one-to-one $\\mathcal {H}(B^d)$ onto $\\mathcal {H}({\\mathbb {R}}^d\\setminus \\overline{B^d})$ and $K^{-1}=K$ .",
"It is easy to see that the Kelvin transform is an isometric isomorphism of $\\mathcal {H}^p({\\mathbb {R}}^d\\setminus \\overline{B^d})$ onto $\\mathcal {H}^p(B^d)$ .",
"Also, as shown in [15] the Kelvin transform is an isometric isomorphism between the harmonic Besov spaces $\\bar{B}^{sq}_p(\\mathcal {H})$ on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and the harmonic Besov spaces $B^{sq}_p(\\mathcal {H})$ on $B^d$ .",
"Furthermore, it is readily seen by the symmetry lemma that for a fixed $y\\in {\\mathbb {R}}^d$ , $y\\ne 0$ , $K\\Big (\\frac{1}{|x-y|^{d-2}}\\Big )(x) = \\frac{|y|^{2-d}}{|x-y/|y|^2|^{d-2}}, \\quad d>2,$ and $K\\Big (\\ln \\frac{1}{|x-y|}\\Big )(x) = \\ln \\frac{1}{|y|}+ \\ln |x| + \\ln \\frac{1}{|x-y/|y|^2|}, \\quad d=2.$ Assuming that $U\\in \\bar{B}_\\tau ^{s\\tau }(\\mathcal {H})$ we apply estimate (REF ) to $KU$ and use all of the above to conclude that estimate (REF ) holds true.",
"Approximation of harmonic functions on ${\\mathbb {R}}_+^d$ .",
"Closely related to the approximation problem considered above is the problem for nonlinear $n$ -term approximation of functions in the harmonic Hardy spaces $\\mathcal {H}^p({\\mathbb {R}}^d_{+})$ , $0<p<\\infty $ , from linear combinations of shifts of the Newtonian kernel with poles in ${\\mathbb {R}}^d_{-}$ .",
"This problem should be regarded as a limiting case of the same problem on $B(0, R)\\subset {\\mathbb {R}}^d$ as $R\\rightarrow \\infty $ .",
"For lack of space we do not elaborate on this sort of approximation.",
"We would like to observe only that all definitions and statements in this article have analogues in the more common setting on ${\\mathbb {R}}_+^d$ from Harmonic analysis point of view, in particular, our main Jackson estimate (REF ) is valid."
],
[
"Proofs of Propositions ",
"For the proofs of Proposition REF we need the following simple Lemma 9.1 Let $K\\in {\\mathbb {N}}$ , $x_0\\in {{\\mathbb {S}}^{d-1}}$ , $g\\in W_\\infty ^K({{\\mathbb {S}}^{d-1}})$ and $\\breve{g}(x):=g(x/|x|)$ for $x\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace $ .",
"Then for every $x\\in {{\\mathbb {S}}^{d-1}}$ with $\\rho (x,x_0)\\le 1$ we have $\\Big |\\breve{g}(x)-\\sum _{|\\beta |\\le K-1}\\frac{\\partial ^\\beta \\breve{g}(x_0)}{\\beta !",
"}(x-x_0)^\\beta \\Big |\\le c \\rho (x,x_0)^K\\sup _{\\begin{array}{c}z\\in {{\\mathbb {S}}^{d-1}}\\\\ \\rho (z,x_0)\\le \\rho (x,x_0)\\end{array}}\\max _{|\\beta |=K}\\left|\\partial ^\\beta \\breve{g}(z)\\right|$ with $c$ depending only on $d$ and $K$ .",
"Assuming $x\\ne x_0$ , we set $\\eta :=(x-x_0)/|x-x_0|\\in {{\\mathbb {S}}^{d-1}}$ .",
"Then from Taylor's theorem there exists $\\lambda \\in (0,1)$ such that $\\Big |\\breve{g}(x)-\\sum _{|\\beta |\\le K-1}\\frac{\\partial ^\\beta \\breve{g}(x_0)}{\\beta !",
"}(x-x_0)^\\beta \\Big |\\\\=\\frac{|x-x_0|^K}{K!",
"}\\left|(\\eta \\cdot \\nabla )^K\\breve{g}(x_\\lambda )\\right|=\\frac{|x-x_0|^K}{|x_\\lambda |^K K!",
"}\\Big |(\\eta \\cdot \\nabla )^K\\breve{g}\\Big (\\frac{x_\\lambda }{|x_\\lambda |}\\Big )\\Big |,$ where $x_\\lambda :=x_0+\\lambda (x-x_0)$ and the definition of $\\breve{g}$ is used for the last equality.",
"Now, we use that $|x-x_0|\\le \\rho (x,x_0)$ , $|x_\\lambda |\\ge \\cos 1/2$ for $\\lambda \\in (0,1)$ , and $\\left|(\\eta \\cdot \\nabla )^K\\breve{g}(y)\\right|\\le c \\max _{|\\beta |=K}\\left|\\partial ^\\beta \\breve{g}(y)\\right|,\\quad y\\in {{\\mathbb {S}}^{d-1}},$ to complete the proof.",
"We represent $\\left\\langle g,f\\right\\rangle $ in the form $\\begin{split}\\left\\langle g,f\\right\\rangle &=S_1+S_2,\\\\S_1&:=\\int _{{{\\mathbb {S}}^{d-1}}}\\Big (\\breve{g}(y)-\\sum _{|\\beta |\\le K-1}\\frac{\\partial ^\\beta \\breve{g}(x_2)}{\\beta !",
"}(y-x_2)^\\beta \\Big )\\overline{f(y)}\\, d\\sigma (y),\\\\S_2&:=\\sum _{|\\beta |\\le K-1}\\frac{\\partial ^\\beta \\breve{g}(x_2)}{\\beta !",
"}\\int _{{{\\mathbb {S}}^{d-1}}}(y-x_2)^\\beta \\overline{f(y)}\\, d\\sigma (y).\\end{split}$ From (REF ) we get $\\Big |\\int _{{{\\mathbb {S}}^{d-1}}}(y-x_2)^\\beta \\overline{f(y)}\\, d\\sigma (y)\\Big |\\le c\\kappa _2 N_2^{-K},\\quad ~0\\le |\\beta |\\le K-1$ and using (REF ) $|S_2|\\le \\sum _{|\\beta |\\le K-1} \\frac{\\kappa _1 N_1^{|\\beta |+d-1}}{\\beta !",
"(1+N_1\\rho (x_1,x_2))^M} c\\kappa _2 N_2^{-K}\\le c\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)^{K}N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}.$ We bound $S_1$ by $|S_1|\\le \\int _{{{\\mathbb {S}}^{d-1}}}\\Big |\\breve{g}(y)-\\sum _{|\\beta |\\le K-1}\\!\\!\\frac{\\partial ^\\beta \\breve{g}(x_2)}{\\beta !",
"}(y-x_2)^\\beta \\Big ||f(y)|\\, d\\sigma (y)=:\\int _{A_1}+\\int _{A_2}+\\int _{A_3},$ where $A_1&=\\lbrace y\\in {{\\mathbb {S}}^{d-1}}: \\rho (x_2, y)\\le N_1^{-1}\\rbrace ,\\\\A_2&=\\lbrace y\\in {{\\mathbb {S}}^{d-1}}: \\rho (x_2, y)>N_1^{-1}, \\rho (x_1,y)\\le \\rho (x_1,x_2)/2\\rbrace ,\\\\A_3&=\\lbrace y\\in {{\\mathbb {S}}^{d-1}}: \\rho (x_2, y)>N_1^{-1}, \\rho (x_1,y)>\\rho (x_1,x_2)/2\\rbrace .$ For $y\\in A_1$ , Lemma REF and (REF ) imply $\\Big |\\breve{g}(y)-\\sum _{|\\beta |\\le K-1}\\frac{\\partial ^\\beta \\breve{g}(x_2)}{\\beta !",
"}(y-x_2)^\\beta \\Big |\\le c \\rho (y,x_2)^K\\sup _{z\\in A_1}\\max _{|\\beta |= K}\\left|\\partial ^\\beta \\breve{g}(z)\\right|\\\\\\le c \\sup _{z\\in A_1}\\frac{\\kappa _1 N_1^{K+d-1}}{(1+N_1\\rho (x_1,z))^M}\\rho (y,x_2)^K\\le c \\frac{\\kappa _1 N_1^{K+d-1}}{(1+N_1\\rho (x_1,x_2))^M}\\rho (y,x_2)^K$ due to (REF ).",
"Using the above estimate, (), and (REF ) we see that $\\int _{A_1}&\\le c \\frac{\\kappa _1 N_1^{K+d-1}}{(1+N_1\\rho (x_1,x_2))^M}\\int _{A_1}\\rho (y,x_2)^K \\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\le c \\frac{\\kappa _1 \\kappa _2 N_1^{K+d-1}}{(1+N_1\\rho (x_1,x_2))^M} N_2^{-K}\\int _{{\\mathbb {S}}^{d-1}}\\frac{N_2^{d-1}}{(1+N_2\\rho (y,x_2))^{M-K}}\\, d\\sigma (y)\\\\&\\le c\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)^{K}N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}.$ For $y\\in A_2$ we have $\\rho (x_1,x_2)/2\\le \\rho (x_2, y)\\le 3\\rho (x_1,x_2)/2$ , and hence $3N_2\\rho (y,x_2)\\ge (N_2/N_1)(1+N_1\\rho (x_1,x_2))$ , using that $ \\rho (y,x_2)> N_1^{-1}$ .",
"Therefore, $(1+N_2\\rho (y,x_2))^{-M}\\le (N_2\\rho (y,x_2))^{-M}\\le 3^M(N_1/N_2)^M(1+N_1\\rho (x_1,x_2))^{-M}.$ This combined with (REF ) and () implies $\\int _{A_2}&\\le \\int _{A_2}\\frac{\\kappa _1 N_1^{d-1}}{(1+N_1\\rho (x_1,y))^M}\\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\qquad \\qquad \\quad + \\int _{A_2}\\sum _{|\\beta |\\le K-1}\\frac{\\kappa _1 N_1^{|\\beta |+d-1}\\rho (y,x_2)^{|\\beta |}}{\\beta !",
"(1+N_1\\rho (x_1,x_2))^M}\\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\le \\int _{A_2}\\frac{\\kappa _1 N_1^{d-1}}{(1+N_1\\rho (x_1,y))^M}\\, d\\sigma (y)\\frac{3^M\\kappa _2 (N_1/N_2)^MN_2^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}\\\\&+\\sum _{|\\beta |\\le K-1}\\int _{A_2}\\frac{\\kappa _1 N_1^{|\\beta |+d-1}(3/2)^{|\\beta |}\\rho (x_1,x_2)^{|\\beta |}}{\\beta !",
"(1+N_1\\rho (x_1,x_2))^M}\\, d\\sigma (y)\\frac{3^M\\kappa _2 (N_1/N_2)^MN_2^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}\\\\&\\le c\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)^{K}N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}.$ Here we used that $M>K+d-1$ and (REF ) as well as the fact that $N_1/N_2\\le 1$ and $\\sigma (A_2)\\le c\\rho (x_1,x_2)^{d-1}$ .",
"For $y\\in A_3$ , we have $(1+N_1\\rho (x_1,x_2))/2\\le 1+N_1\\rho (x_1,y)$ and $\\rho (x_2, y)>N_1^{-1}$ .",
"Therefore, $\\int _{A_3}&\\le \\int _{A_3}\\frac{\\kappa _1 N_1^{d-1}}{(1+N_1\\rho (x_1,y))^M}\\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\qquad \\qquad \\quad + \\int _{A_3}\\sum _{|\\beta |\\le K-1}\\frac{\\kappa _1 N_1^{|\\beta |+d-1}\\rho (y,x_2)^{|\\beta |}}{\\beta !",
"(1+N_1\\rho (x_1,x_2))^M}\\frac{\\kappa _2 N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\le c \\frac{\\kappa _1 \\kappa _2 N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M}\\int _{A_3}\\frac{(N_1\\rho (y,x_2))^{K}N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\\\&\\le c\\frac{\\kappa _1 \\kappa _2 (N_1/N_2)^{K}N_1^{d-1}}{(1+N_1\\rho (x_1,x_2))^M},$ using that $M>K+d-1$ and $\\int _{{{\\mathbb {S}}^{d-1}}}\\frac{(N_2\\rho (y,x_2))^{K}N_2^{d-1}}{(1+N_2\\rho (y,x_2))^M}\\, d\\sigma (y)\\le \\int _{{{\\mathbb {S}}^{d-1}}}\\frac{N_2^{d-1}}{(1+N_2\\rho (y,x_2))^{M-K}}\\, d\\sigma (y)\\le c$ because of (REF ).",
"This completes the proof.",
"This proof is the same as the proof of Proposition REF for $K=1$ .",
"Instead of estimating $S_2$ in (REF ) we move it to the left-hand side.",
"Only the localization of the first derivatives of $\\breve{g}$ , but not of $\\breve{g}$ itself, is needed here.",
"This proof follows along the lines of the proof of Proposition REF with $K=0$ .",
"Of course, in this case the Taylor series is missing from the definitions of both $S_1$ and $S_2$ in (REF ), i.e.",
"$S_2\\equiv 0$ .",
"Lemma REF is also not used in the proof."
],
[
"Proof of Theorem ",
"This proof depends on the next three lemmas.",
"Lemma 9.2 Let $j,m\\ge 0$ , $0<\\beta \\le 1$ , $x\\in {{\\mathbb {S}}^{d-1}}$ and $\\xi \\in \\mathcal {X}_j$ .",
"Then $\\sum _{\\eta \\in \\mathcal {X}_{j+m}} \\frac{1}{(1+N_\\xi \\rho (x,\\eta ))^{d-1+\\beta }}\\le c^\\star _1 2^{m(d-1)}$ with $c^\\star _1=c(d)\\beta ^{-1}$ .",
"Using that $\\mathcal {X}_{j+m}$ is a maximal $\\gamma 2^{-j-m+1}$ net with a fixed $\\gamma =c(d)\\in (0,1)$ (as stated in §REF ), (REF ), the inequality $(1+\\gamma )(1+N_\\xi \\rho (x,\\eta ))\\ge 1+N_\\xi \\rho (x,y)$ for any $y\\in A_\\eta $ , and (2.6) we obtain $\\sum _{\\eta \\in \\mathcal {X}_{j+m}} \\frac{1}{(1+N_\\xi \\rho (x,\\eta ))^{d-1+\\beta }}\\le c(d)N_\\eta ^{d-1}\\sum _{\\eta \\in \\mathcal {X}_{j+m}} \\frac{|A_\\eta |}{(1+N_\\xi \\rho (x,\\eta ))^{d-1+\\beta }}\\\\\\le c(d)N_\\eta ^{d-1}\\int _{{{\\mathbb {S}}^{d-1}}} \\frac{d\\sigma (y)}{(1+N_\\xi \\rho (x,y))^{d-1+\\beta }}\\le c(d)N_\\eta ^{d-1}c(d)\\beta ^{-1} N_\\xi ^{-d+1},$ which proves (REF ).",
"Lemma 9.3 Let $0<t\\le 1$ and $M\\ge (d-1)/t+\\delta $ , $0<\\delta \\le 1$ .",
"Then for any sequence of complex numbers $\\lbrace h_{\\eta }\\rbrace _{\\eta \\in \\mathcal {X}_{m} }$ , $m\\ge 0$ , and for any $x\\in B_\\xi =B(\\xi ,\\gamma 2^{-j})$ , $\\xi \\in \\mathcal {X}_j$ , $j\\ge 0$ , we have $\\sum _{\\eta \\in \\mathcal {X}_m} \\frac{|h_{\\eta }|}{\\big (1+\\min \\lbrace N_\\xi , N_\\eta \\rbrace \\rho (\\xi ,\\eta )\\big )^{M}}\\\\\\le c^\\star _2 \\max \\big \\lbrace 1,2^{(m-j)(d-1)/t}\\big \\rbrace \\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_m}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big )(x),$ where $c^\\star _2:=(2/\\ln 2) 4^{(d-1)/t}(2/\\gamma )^M \\delta ^{-1}$ with $\\gamma \\in (0,1)$ being the constant from the construction of the old frame $\\Psi $ in §REF .",
"Two cases present themselves here.",
"Case 1: $m\\ge j$ .",
"Set $\\mathcal {Q}_{0}:=\\lbrace \\eta \\in \\mathcal {X}_m: 2^{j-1}\\rho (\\xi ,\\eta )< \\gamma \\rbrace $ and $\\mathcal {Q}_{\\nu }:=\\lbrace \\eta \\in \\mathcal {X}_m: \\gamma 2^{\\nu -1}\\le 2^{j-1}\\rho (\\xi ,\\eta )< \\gamma 2^{\\nu }\\rbrace ,\\quad \\nu \\ge 1.$ Since $0<t\\le 1$ we have for $\\nu \\ge 1$ $\\sum _{\\eta \\in \\mathcal {Q}_\\nu } \\frac{|h_{\\eta }|}{\\bigl (1+2^{j-1}\\rho (\\xi ,\\eta )\\bigr )^{M}}\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 2^{-\\nu M} \\sum _{\\eta \\in \\mathcal {Q}_\\nu } |h_{\\eta }|\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 2^{-\\nu M} \\Big (\\sum _{\\eta \\in \\mathcal {Q}_\\nu } |h_{\\eta }|^t\\Big )^{1/t}.$ The same estimate holds trivially for $\\nu =0$ .",
"Put $\\mathcal {R}_\\nu :=B(\\xi ,2\\gamma (2^{-m}+2^{-j+\\nu })),\\quad \\nu \\ge 0.$ Clearly $\\cup _{\\eta \\in \\mathcal {Q}_\\nu } B_\\eta \\subset \\mathcal {R}_\\nu $ .",
"Using this, the fact that the sets $\\lbrace B_\\eta : \\eta \\in \\mathcal {X}_m\\rbrace $ are disjoint, and (REF ) we obtain for every $x\\in B_\\xi \\subset \\mathcal {R}_\\nu $ $\\sum _{\\eta \\in \\mathcal {Q}_\\nu }|h_{\\eta }|^t&=\\int _{{\\mathbb {S}}^{d-1}}\\biggl (\\sum _{\\eta \\in \\mathcal {Q}_\\nu }|h_\\eta | |B_\\eta |^{-1/t} {\\mathbb {1}}_{B_\\eta }(y)\\biggr )^t \\, d\\sigma (y)\\\\& =\\frac{|\\mathcal {R}_\\nu |}{|B(\\xi , \\gamma 2^{-m})|}\\frac{1}{|\\mathcal {R}_\\nu |}\\int _{\\mathcal {R}_\\nu } \\Big (\\sum _{\\eta \\in \\mathcal {Q}_\\nu }|h_\\eta |{\\mathbb {1}}_{B_\\eta }(y)\\Big )^t d\\sigma (y)\\\\&\\le 4^{d-1} 2^{(m-j+\\nu )(d-1)} \\Big [\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x)\\Big ]^t.$ Therefore, since $M\\ge (d-1)/t+\\delta $ we get for any $x\\in B_\\xi $ $\\sum _{\\eta \\in \\mathcal {X}_m} &\\frac{|h_{\\eta }|}{\\big (1+2^{j-1}\\rho (\\xi ,\\eta )\\big )^{M}}\\\\&\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 4^{(d-1)/t}\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x)\\sum _{\\nu \\ge 0} 2^{-\\nu M}2^{(\\nu -j+m)(d-1)/t}\\\\&\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 4^{(d-1)/t}2^{(m-j)(d-1)/t}\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x)\\sum _{\\nu \\ge 0} 2^{-\\nu \\delta }\\\\&\\le \\frac{2}{\\delta \\ln 2}\\Big (\\frac{2}{\\gamma }\\Big )^M 4^{(d-1)/t}2^{(m-j)(d-1)/t}\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x),$ which confirms (REF ).",
"Case 2: $m<j$ .",
"Set ${\\tilde{\\mathcal {Q}}}_{0}:=\\lbrace \\eta \\in \\mathcal {X}_m: 2^{m-1} \\rho (\\xi ,\\eta )< \\gamma \\rbrace $ and ${\\tilde{\\mathcal {Q}}}_{\\nu }:=\\lbrace \\eta \\in \\mathcal {X}_m: \\gamma 2^{\\nu -1}\\le 2^{m-1}\\rho (\\xi ,\\eta )< \\gamma 2^{\\nu }\\rbrace ,\\quad \\nu \\ge 1.$ Write $\\tilde{\\mathcal {R}}_\\nu :=B(\\xi ,\\gamma 2^{-m+1}(1+2^{\\nu })),\\quad \\nu \\ge 0.$ We use that $0<t\\le 1$ to obtain $\\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu } \\frac{|h_{\\eta }|}{\\bigl (1+2^{m-1}\\rho (\\xi ,\\eta )\\bigr )^{M}}\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 2^{-\\nu M} \\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu } |h_{\\eta }|\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 2^{-\\nu M} \\Big (\\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu } |h_{\\eta }|^t\\Big )^{1/t}.$ Just as in Case 1 we obtain for $x\\in B_\\xi \\subset \\tilde{\\mathcal {R}}_\\nu $ $\\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu }|h_{\\eta }|^t&=\\int _{{\\mathbb {S}}^{d-1}}\\biggl (\\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu }|h_\\eta | |B_\\eta |^{-1/t} {\\mathbb {1}}_{B_\\eta }(y)\\biggr )^t \\, d\\sigma (y)\\\\& =\\frac{|\\tilde{\\mathcal {R}}_\\nu |}{|B(\\xi , \\gamma 2^{-m})|}\\frac{1}{|\\tilde{\\mathcal {R}}_\\nu |}\\int _{\\tilde{\\mathcal {R}}_\\nu } \\Big (\\sum _{\\eta \\in {\\tilde{\\mathcal {Q}}}_\\nu }|h_\\eta |{\\mathbb {1}}_{B_\\eta }(y)\\Big )^t d\\sigma (y)\\\\&\\le 4^{d-1} 2^{\\nu (d-1)} \\Big [\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x)\\Big ]^t.$ As before, since $M\\ge (d-1)/t+\\delta $ we get for any $x\\in B_\\xi $ $\\sum _{\\eta \\in \\mathcal {X}_m} &\\frac{|h_{\\eta }|}{\\big (1+2^{m-1}\\rho (\\xi ,\\eta )\\big )^{M}}\\\\&\\le \\Big (\\frac{2}{\\gamma }\\Big )^M 4^{(d-1)/t}\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x)\\sum _{\\nu \\ge 0} 2^{-\\nu (M-(d-1)/t)}\\\\&\\le \\frac{2}{\\delta \\ln 2}\\Big (\\frac{2}{\\gamma }\\Big )^M 4^{(d-1)/t}\\mathcal {M}_t\\bigl (\\sum _{\\eta \\in \\mathcal {X}_m}|h_\\eta | {\\mathbb {1}}_{B_\\eta }\\bigr )(x),$ which verifies (REF ).",
"The proof of the lemma is complete.",
"In the next lemma we specify the constants in certain well known discrete Hardy inequalities that will be needed.",
"Lemma 9.4 Let $\\beta >0$ , $0<q<\\infty $ , and $a_m \\ge 0$ for $m\\ge 0$ .",
"Then $\\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta }a_m\\Big )^q\\Big )^{1/q}\\le c^\\star _3\\Big (\\sum _{m\\ge 0} a_m^q\\Big )^{1/q}$ and $\\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m=0}^j 2^{-(j-m)\\beta }a_m\\Big )^q\\Big )^{1/q}\\le c^\\star _3\\Big (\\sum _{m\\ge 0} a_m^q\\Big )^{1/q}$ with $c^\\star _3=2^\\beta \\max \\left\\lbrace \\frac{1}{\\beta \\ln 2},\\frac{1}{(\\beta q \\ln 2)^{1/q}}\\right\\rbrace .$ In the case $0<q\\le 1$ inequalities (REF )–(REF ) follow readily by applying the $q$ -inequality and switching the order of summation.",
"More precisely, we have $&\\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta }a_m\\Big )^q\\Big )^{1/q}\\le \\Big (\\sum _{j\\ge 0}\\sum _{m\\ge j} 2^{-(m-j)\\beta q}a_m^q\\Big )^{1/q}\\\\&= \\Big (\\sum _{m\\ge 0}\\sum _{j= 0}^m 2^{-(m-j)\\beta q}a_m^q\\Big )^{1/q}\\le \\Big (\\sum _{\\nu \\ge 0} 2^{-\\nu \\beta q}\\Big )^{1/q}\\Big (\\sum _{m\\ge 0}a_m^q\\Big )^{1/q}.$ Clearly $\\Big (\\sum _{\\nu \\ge 0} 2^{-\\nu \\beta q}\\Big )^{1/q} =\\Big (\\frac{1}{1-2^{-\\beta q}}\\Big )^{1/q}\\le \\frac{2^\\beta }{(\\beta q \\ln 2)^{1/q}}\\le c^\\star _3,$ which implies (REF ).",
"The proof of (REF ) in the case $0<q\\le 1$ is similar and gives the same constant $c^\\star _3$ .",
"In the case $q>1$ using the Hölder's inequality ($1/q^{\\prime }+1/q=1$ ) and switching the order of summation just as above we obtain $&\\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta }a_m\\Big )^q\\Big )^{1/q}= \\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta /q^{\\prime }}2^{-(m-j)\\beta /q}a_m\\Big )^q\\Big )^{1/q}\\\\&\\le \\Big (\\sum _{j\\ge 0}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta }\\Big )^{q/q^{\\prime }}\\Big (\\sum _{m\\ge j} 2^{-(m-j)\\beta }a_m^q\\Big )\\Big )^{1/q}\\\\&\\le \\Big (\\sum _{\\nu \\ge 0} 2^{-\\nu \\beta }\\Big )^{1/q^{\\prime }}\\Big (\\sum _{\\nu \\ge 0} 2^{-\\nu \\beta }\\Big )^{1/q}\\Big (\\sum _{m\\ge 0} a_m^q \\Big )^{1/q}=\\sum _{\\nu \\ge 0} 2^{-\\nu \\beta }\\Big (\\sum _{m\\ge 0} a_m^q \\Big )^{1/q},$ which gives (REF ) with $c^\\star _3\\ge 2^\\beta /(\\beta \\ln 2)$ .",
"The proof of (REF ) in the case $q>1$ is similar and with the same constant.",
"We shall use the abbreviated notation $\\omega _{\\xi ,\\eta }:= \\omega _{\\xi ,\\eta }^{(K,M)}$ for $\\xi ,\\eta \\in \\mathcal {X}$ (see (REF )).",
"We first establish the result for the sequence Besov spaces ${\\mathfrak {b}}^{sq}_p$ , that is, $\\Vert \\Omega h\\Vert _{{\\mathfrak {b}}^{sq}_p}\\le C_9\\Vert h\\Vert _{{\\mathfrak {b}}^{sq}_p}.$ Set $p_*:=\\max \\lbrace p,1\\rbrace $ .",
"We start with the proof of the estimate $\\begin{split}\\Big (\\sum _{\\xi \\in \\mathcal {X}_j}\\Big |\\sum _{\\eta \\in \\mathcal {X}}\\omega _{\\xi ,\\eta }&h_\\eta \\Big |^p\\Big )^{1/p}\\\\&\\le C_{11}\\sum _{m=0}^\\infty 2^{-m(K+(d-1)(1/2-1/p^{\\prime }_*)-\\delta /2)}\\Big (\\sum _{\\eta \\in \\mathcal {X}_{j+m}} |h_\\eta |^p\\Big )^{1/p}\\\\&+C_{11}\\sum _{m=1}^j 2^{-m(K+(d-1)(1/2-1/p)-\\delta /2)}\\Big (\\sum _{\\eta \\in \\mathcal {X}_{j-m}} |h_\\eta |^p\\Big )^{1/p}\\end{split}$ for any $j\\ge 0$ .",
"For $1<p<\\infty $ using (REF ) and the convexity of $u^p$ we obtain $\\sum _{\\xi \\in \\mathcal {X}_j}\\Big |\\sum _{\\eta \\in \\mathcal {X}}\\omega _{\\xi ,\\eta }h_\\eta \\Big |^p\\le 2^{p-1}\\sum _{\\xi \\in \\mathcal {X}_j}\\Big [\\Big (\\sum _{m=0}^\\infty \\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\frac{2^{-m(K+(d-1)/2)}|h_\\eta |}{(1+N_\\xi \\rho (\\xi ,\\eta ))^M}\\Big )^p\\\\+\\Big (\\sum _{m=1}^{j} \\sum _{\\eta \\in \\mathcal {X}_{j-m}}\\frac{2^{-m(K+(d-1)/2)}|h_\\eta |}{(1+N_\\eta \\rho (\\xi ,\\eta ))^M}\\Big )^p\\Big ].$ Applying in the first double sum in the right-hand side of (REF ) twice Hölder's inequality, first in the summation on $m$ and then on $\\eta $ , and Lemma REF with $\\beta =\\delta $ we get with $M_1=(d-1+\\delta )/p^{\\prime }$ and $M_2=M-M_1$ $\\Big (\\sum _{m=0}^\\infty \\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\frac{2^{-m(K+(d-1)/2)}|h_\\eta |}{(1+N_\\xi \\rho (\\xi ,\\eta ))^M}\\Big )^p\\\\\\le \\Big (\\sum _{m=0}^\\infty 2^{-mp^{\\prime }\\delta /2}\\Big )^{p/p^{\\prime }}\\sum _{m=0}^\\infty \\Big (\\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\frac{2^{-m(K+(d-1)/2-\\delta /2)}|h_\\eta |}{(1+N_\\xi \\rho (\\xi ,\\eta ))^M}\\Big )^p\\\\\\le C_{12}^p\\sum _{m=0}^\\infty \\Big (\\sum _{\\eta \\in \\mathcal {X}_{j+m}} \\frac{1}{(1+N_\\xi \\rho (\\xi ,\\eta ))^{M_1p^{\\prime }}}\\Big )^{p/p^{\\prime }}\\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\frac{2^{-m(K+(d-1)/2-\\delta /2)p}|h_\\eta |^p}{(1+N_\\xi \\rho (\\xi ,\\eta ))^{M_2p}}\\\\\\le {c^\\star _1}^{p-1} C_{12}^p\\sum _{m=0}^\\infty \\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\frac{2^{-m(K+(d-1)(1/2-1/p^{\\prime })-\\delta /2)p}|h_\\eta |^p}{(1+N_\\xi \\rho (\\xi ,\\eta ))^{M_2p}},$ where $C_{12}:=\\big (1- 2^{-p^{\\prime }\\delta /2}\\big )^{-1/p^{\\prime }}\\le 5\\delta ^{-1/p^{\\prime }}$ .",
"Applying the same arguments to the second double sum in the right-hand side of (REF ) we obtain $\\Big (\\sum _{m=1}^j \\sum _{\\eta \\in \\mathcal {X}_{j-m}}\\frac{2^{-m(K+(d-1)/2)}|h_\\eta |}{(1+N_\\eta \\rho (\\xi ,\\eta ))^M}\\Big )^p\\\\\\le {c^\\star _1}^{p-1} C_{12}^p\\sum _{m=1}^j\\sum _{\\eta \\in \\mathcal {X}_{j-m}}\\frac{2^{-m(K+(d-1)/2-\\delta /2)p}|h_\\eta |^p}{(1+N_\\eta \\rho (\\xi ,\\eta ))^{M_2p}},$ Note that (REF ) implies $M_2p\\ge d-1+\\delta $ .",
"Substituting (REF ) and (REF ) in (REF ) and using Lemma REF with $\\beta =\\delta $ we get $\\sum _{\\xi \\in \\mathcal {X}_j}\\Big |\\sum _{\\eta \\in \\mathcal {X}}\\omega _{\\xi ,\\eta }h_\\eta \\Big |^p\\\\\\le 2^{p-1} {c^\\star _1}^{p-1} C_{12}^p\\Big (\\sum _{m=0}^\\infty \\sum _{\\eta \\in \\mathcal {X}_{j+m}}\\sum _{\\xi \\in \\mathcal {X}_j}\\frac{2^{-m(K+(d-1)(1/2-1/p^{\\prime })-\\delta /2)p}|h_\\eta |^p}{(1+N_\\xi \\rho (\\xi ,\\eta ))^{d-1+\\delta }}\\\\+\\sum _{m=1}^j \\sum _{\\eta \\in \\mathcal {X}_{j-m}}\\sum _{\\xi \\in \\mathcal {X}_j}\\frac{2^{-m(K+(d-1)/2-\\delta /2)p}|h_\\eta |^p}{(1+N_\\eta \\rho (\\xi ,\\eta ))^{d-1+\\delta }}\\Big )\\\\\\le 2^{p-1} {c^\\star _1}^{p-1} C_{12}^p\\Big (\\sum _{m=0}^\\infty {c^\\star _1} 2^{-m(K+(d-1)(1/2-1/p^{\\prime }_*)-\\delta /2)p}\\sum _{\\eta \\in \\mathcal {X}_{j+m}}|h_\\eta |^p\\\\+\\sum _{m=1}^j {c^\\star _1} 2^{-m(K+(d-1)(1/2-1/p)-\\delta /2)p}\\sum _{\\eta \\in \\mathcal {X}_{j-m}}|h_\\eta |^p\\Big ).$ Now, we raise both sides of (REF ) to the power $1/p<1$ and apply the $1/p$ -inequality to its right-hand side to obtain (REF ) for $1<p<\\infty $ with $C_{11}\\ge C_{13}:=2^{1-1/p} c^\\star _1 C_{12}$ .",
"Let $0<p\\le 1$ .",
"Using the $p$ -inequality, observing that (REF ) implies in this case $Mp\\ge d-1+\\delta p$ , and using Lemma REF with $\\beta =\\delta p\\le 1$ we obtain $\\sum _{\\xi \\in \\mathcal {X}_j}\\Big |\\sum _{\\eta \\in \\mathcal {X}}\\omega _{\\xi ,\\eta }h_\\eta \\Big |^p\\le \\sum _{m=0}^\\infty {c^\\star _1} 2^{-mp\\delta /2}2^{-m(K+(d-1)/2-\\delta /2)p}\\sum _{\\eta \\in \\mathcal {X}_{j+m}}|h_\\eta |^p\\\\+\\sum _{m=1}^j {c^\\star _1} 2^{-mp\\delta /2}2^{-m(K+(d-1)(1/2-1/p)-\\delta /2)p}\\sum _{\\eta \\in \\mathcal {X}_{j-m}}|h_\\eta |^p.$ We now raise both sides of (REF ) to the power $1/p\\ge 1$ , use the convexity of $u^{1/p}$ to break the right-hand side to two terms and apply Hölder's inequality with exponents $r=1/(1-p)$ and $r^{\\prime }=1/p$ in the summations on $m$ in order to get the $1/p$ power inside the sum and to prove (REF ) for $0<p\\le 1$ with $C_{11}:=\\max \\lbrace C_{13}, C_{15}\\rbrace $ , where $C_{15}=2^{1/p-1}{c^\\star _1}^{1/p} C_{14}$ with $c^\\star _1$ is for $\\beta =\\delta p$ and $C_{14}:=\\big (1- 2^{-\\delta p/(2(1-p))}\\big )^{-(1-p)/p}$ .",
"Finally, using (REF ), (REF ), (REF ), and Lemma REF with $\\beta =\\delta /2$ and $a_m=2^{m[s+(d-1)(1/2-1/p)]}\\Big (\\sum _{\\eta \\in \\mathcal {X}_{m}} |h_\\eta |^p\\Big )^{1/p}$ we obtain $\\Vert \\Omega h\\Vert _{{\\mathfrak {b}}_p^{sq}} = \\Big (\\sum _{j=0}^\\infty \\Big [2^{j[s+(d-1)(1/2-1/p)]}\\Big (\\sum _{\\xi \\in \\mathcal {X}_j} \\Big |\\sum _{\\eta \\in \\mathcal {X}}\\omega _{\\xi ,\\eta }h_\\eta \\Big |^p\\Big )^{1/p}\\Big ]^q\\Big )^{1/q}\\\\\\le C_{11}\\Big (\\sum _{j=0}^\\infty \\Big [\\sum _{m=0}^\\infty 2^{j[s+(d-1)(1/2-1/p)]-m(K+(d-1)(1/2-1/p^{\\prime }_*)-\\delta /2)}\\Big (\\sum _{\\eta \\in \\mathcal {X}_{j+m}} |h_\\eta |^p\\Big )^{1/p}\\\\+\\sum _{m=1}^j 2^{j[s+(d-1)(1/2-1/p)]-m(K+(d-1)(1/2-1/p)-\\delta /2)}\\Big (\\sum _{\\eta \\in \\mathcal {X}_{j-m}} |h_\\eta |^p\\Big )^{1/p}\\Big ]^q\\Big )^{1/q}\\\\= \\!C_{11}\\Big (\\sum _{j=0}^\\infty \\!\\Big [\\!\\sum _{m=0}^\\infty \\!2^{-m(K+s+(d-1)(1/p_*-1/p)-\\delta /2)}a_{j+m}+\\sum _{m=1}^j 2^{-m(K-s-\\delta /2)}a_{j-m}\\Big ]^q\\Big )^{1/q}\\\\\\le C_{11}\\Big (\\sum _{j=0}^\\infty \\Big [\\sum _{m\\ge j} 2^{-(m-j)\\delta /2}a_{m}+\\sum _{m=0}^{j-1} 2^{-(j-m)\\delta /2}a_{m}\\Big ]^q\\Big )^{1/q}\\\\\\le 2^{1/q+1}C_{11}c^\\star _3\\Big (\\sum _{m=0}^\\infty a_m^q\\Big )^{1/q}=C_9\\Vert h\\Vert _{{\\mathfrak {b}}_p^{sq}}.$ Thus, (REF ) is established with a constant $C_9=2^{1/q+1}C_{11}c^\\star _3$ of the claimed form.",
"We next prove the result for the sequence Triebel-Lizorkin spaces ${\\mathfrak {f}}^{sq}_p$ , that is, $\\Vert \\Omega h\\Vert _{{\\mathfrak {f}}^{sq}_p}\\le C_9\\Vert h\\Vert _{{\\mathfrak {f}}^{sq}_p}.$ Taking into account Remark REF we chose the quasi-norm of ${\\mathfrak {f}}^{sq}_p$ in Definition REF to be defined with $B_\\xi =B(\\xi ,\\gamma 2^{-j})$ for $\\xi \\in \\mathcal {X}_j$ , $j\\ge 1$ .",
"Thus $B_\\xi \\cap B_\\eta =\\emptyset $ for $\\xi \\ne \\eta \\in \\mathcal {X}_j$ .",
"Let $h\\in {\\mathfrak {f}}^{sq}_p$ .",
"Then $(\\Omega h)_{\\xi }=\\sum _{\\eta \\in \\mathcal {X}} \\omega _{\\xi ,\\eta }h_{\\eta },$ where the series converges absolutely (see proof below).",
"Then by (REF ) $\\Vert \\Omega h\\Vert _{{\\mathfrak {f}}^{sq}_p}:= \\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\big [|B_\\xi |^{-s/(d-1)-1/2}|(\\Omega h)_\\xi |{\\mathbb {1}}_{B_\\xi }\\big ]^q\\Big )^{1/q}\\Big \\Vert _{L^p}\\\\\\le C_{21}\\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\big [N_\\xi ^{s+(d-1)/2}\\sum _{\\eta \\in \\mathcal {X}} \\omega _{\\xi ,\\eta }|h_{\\eta }|{\\mathbb {1}}_{B_\\xi }\\big ]^q\\Big )^{1/q}\\Big \\Vert _{L^p}\\le C_{21} 2^{1/p+1/q} (\\Sigma _1+\\Sigma _2),$ where $C_{21}:={{\\tilde{c}}_2}^{~|s/(d-1)+1/2|}(2/\\gamma )^{s+(d-1)/2}$ , $&\\Sigma _1:=\\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\Big [N_\\xi ^{s+(d-1)/2}\\sum _{\\eta \\in \\mathcal {X}: N_\\eta \\ge N_\\xi }\\omega _{\\xi ,\\eta }|h_{\\eta }| {\\mathbb {1}}_{B_\\xi }\\Big ]^q\\Big )^{1/q}\\Big \\Vert _{L^p},\\quad \\mbox{and}\\\\\\quad &\\Sigma _2:=\\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\Big [N_\\xi ^{s+(d-1)/2}\\sum _{\\eta \\in \\mathcal {X}: N_\\eta < N_\\xi }\\omega _{\\xi ,\\eta }|h_{\\eta }| {\\mathbb {1}}_{B_\\xi }\\Big ]^q\\Big )^{1/q}\\Big \\Vert _{L^p}.$ Write $\\lambda _\\xi := N_\\xi ^{s+(d-1)/2}{\\mathbb {1}}_{B_\\xi }$ , $\\xi \\in \\mathcal {X}$ , and choose $t$ so that $(d-1)/t=\\mathcal {J}+\\delta /2$ .",
"Then $0<t<\\min \\lbrace 1, p, q\\rbrace $ .",
"If $N_\\eta \\ge N_\\xi $ , then $\\omega _{\\xi ,\\eta } = \\Big (\\frac{N_\\xi }{N_\\eta }\\Big )^{K+(d-1)/2}\\big ( 1+ N_\\xi \\rho (\\xi , \\eta )\\big )^{-M}.$ Then we have $&\\Sigma _1 \\le \\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\Big [\\sum _{\\eta \\in \\mathcal {X}: N_\\eta \\ge N_\\xi }\\Big (\\frac{N_\\xi }{N_\\eta }\\Big )^{\\mathcal {J}-s-(d-1)/2+\\delta } (1+N_\\xi \\rho (\\xi , \\eta ))^{-\\mathcal {J}-\\delta }|h_{\\eta }| \\lambda _{\\xi }\\Big ]^q\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}\\\\&=\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\sum _{\\xi \\in \\mathcal {X}_j}\\Big [\\sum _{m\\ge j} 2^{-(m-j)(\\mathcal {J}-s-\\frac{d-1}{2}+ \\delta )}\\sum _{\\eta \\in \\mathcal {X}_m}\\big (1+N_\\xi \\rho (\\xi ,\\eta )\\big )^{-\\mathcal {J}-\\delta }|h_{\\eta }| \\lambda _{\\xi }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}.$ We now apply Lemma REF (with $(d-1)/t$ and $\\delta /2$ in the place of $M$ and $\\delta $ ) and the fact that the sets $\\lbrace B_\\xi : \\xi \\in \\mathcal {X}_j\\rbrace $ are mutually disjoint to obtain $\\Sigma _1 &\\le c^\\star _2 \\Big \\Vert \\Big (\\sum _{j\\ge 0}\\sum _{\\xi \\in \\mathcal {X}_j}\\Big [\\sum _{m\\ge j} 2^{-(m-j)(\\mathcal {J}-s-\\frac{d-1}{2}+ \\delta - \\frac{d-1}{t})}\\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_m}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big ) \\lambda _{\\xi }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}\\\\&\\le c^\\star _2 \\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\sum _{m\\ge j} 2^{-(m-j)\\delta /2}\\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_m}|h_{\\eta }|\\lambda _{\\eta }\\Big )\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}.$ The application of inequality (REF ), the maximal inequality (REF ), the fact that the sets $\\lbrace B_\\eta : \\eta \\in \\mathcal {X}_j\\rbrace $ are mutually disjoint, and (REF ) leads to $\\Sigma _1&\\le c^\\star _2 c^\\star _3\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_j}|h_{\\eta }|\\lambda _{\\eta }\\Big )\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p} \\\\&\\le c^\\star _2 c^\\star _3 {\\tilde{c}}_1\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\sum _{\\eta \\in \\mathcal {X}_j}N_\\eta ^{s+(d-1)/2}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p} \\nonumber \\\\&\\le c^\\star _2 c^\\star _3 {\\tilde{c}}_1 C_{22}\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\sum _{\\eta \\in \\mathcal {X}_j}\\Big [|B_\\eta |^{-s/(d-1)-1/2}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}= c\\Vert f\\Vert _{{\\mathfrak {f}}^{sq}_p} \\nonumber $ with $C_{22}:={\\tilde{c}}_2^{~|s/(d-1)+1/2|}(\\gamma /2)^{s+(d-1)/2}$ .",
"If $N_\\eta < N_\\xi $ , then $\\omega _{\\xi ,\\eta } = \\Big (\\frac{N_\\eta }{N_\\xi }\\Big )^{K+(d-1)/2}\\big (1+ N_\\eta \\rho (\\xi , \\eta )\\big )^{-M}$ and hence $&\\Sigma _2 \\le \\Big \\Vert \\Big (\\sum _{\\xi \\in \\mathcal {X}}\\Big [\\sum _{\\eta \\in \\mathcal {X}: N_\\eta < N_\\xi }\\Big (\\frac{N_\\eta }{N_\\xi }\\Big )^{s+(d-1)/2+\\delta } (1+N_\\eta \\rho (\\xi , \\eta ))^{-\\mathcal {J}-\\delta }|h_{\\eta }| \\lambda _{\\xi }\\Big ]^q\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}\\\\&= \\Big \\Vert \\Big (\\sum _{j\\ge 0}\\sum _{\\xi \\in \\mathcal {X}_j}\\Big [\\sum _{m< j} 2^{-(j-m)(s+\\frac{d-1}{2}+ \\delta )}\\sum _{\\eta \\in \\mathcal {X}_m}\\big (1+N_\\eta \\rho (\\xi ,\\eta )\\big )^{-\\mathcal {J}-\\delta }|h_{\\eta }| \\lambda _{\\xi }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}.$ As above employing Lemma REF , using the fact that the sets $\\lbrace B_\\xi : \\xi \\in \\mathcal {X}_j\\rbrace $ are mutually disjoint, applying (REF ), the maximal inequality (REF ), and (REF ) we obtain $\\Sigma _2&\\le c^\\star _2\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\sum _{m< j} 2^{-(j-m)\\delta }\\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_m}|h_{\\eta }|\\lambda _{\\eta }\\Big )\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p} \\\\&\\le c^\\star _2 c^\\star _3 \\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\mathcal {M}_t\\Big (\\sum _{\\eta \\in \\mathcal {X}_j}|h_{\\eta }|\\lambda _{\\eta }\\Big )\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p} \\nonumber \\\\&\\le c^\\star _2 c^\\star _3 {\\tilde{c}}_1\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\Big [\\sum _{\\eta \\in \\mathcal {X}_j}N_\\eta ^{s+(d-1)/2}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p} \\nonumber \\\\&\\le c^\\star _2 c^\\star _3 {\\tilde{c}}_1 C_{22}\\Big \\Vert \\Big (\\sum _{j\\ge 0}\\sum _{\\eta \\in \\mathcal {X}_j}\\Big [|B_\\eta |^{-s/(d-1)-1/2}|h_{\\eta }|{\\mathbb {1}}_{B_\\eta }\\Big ]^{q}\\Big )^{\\frac{1}{q}}\\Big \\Vert _{L^p}=c\\Vert f\\Vert _{{\\mathfrak {f}}^{sq}_p}, \\nonumber $ where the constants $c^\\star _2, c^\\star _3, {\\tilde{c}}_1, C_{22}$ are as above.",
"Finally, using estimates (REF ) and (REF ) in (REF ) we obtain (REF ) with $C_9=C_{21} 2^{1/p+1/q}c^\\star _2 c^\\star _3 {\\tilde{c}}_1 C_{22}$ , which is of the claimed form.",
"This completes the proof of Theorem REF ."
]
] | 1808.08641 |
[
[
"System of interacting harmonic oscillators in rotationally invariant\n noncommutative phase space"
],
[
"Abstract Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered.",
"A system of $N$ interacting harmonic oscillators in uniform filed and a system of $N$ particles with harmonic oscillator interaction are studied.",
"We analyze effect of noncommutativity on the energy levels of these systems.",
"It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles.",
"The spectrum of $N$ free particles in uniform field in rotationally-invariant noncommutative phase space is also analyzed.",
"It is shown that the spectrum corresponds to the spectrum of a system of $N$ harmonic oscillators with frequency determined by the parameter of momentum noncommutativity."
],
[
"Introduction",
"Recently much attention has been devoted to studies of a quantum space realized on the basis of idea that the spatial coordinates might be noncommutative.",
"The noncommutative space of canonical type has been studied intensively.",
"In the space the coordinates satisfy the following commutation relations $[X_{i},X_{j}]=i\\hbar \\theta _{ij},{}$ where $\\theta _{ij}$ are elements of constant antisymmetric matrix, parameters of coordinate noncommutativity.",
"In noncommutative phase space the momenta are supposed to be noncommutative too.",
"The commutation relations read $[P_{i},P_{j}]=i\\hbar \\eta _{ij}.",
"{}$ Commutation relations for coordinates and momenta are generalized as $[X_{i},P_{j}]=i\\hbar (\\delta _{ij}+\\gamma _{ij}).",
"{}$ where $\\eta _{ij}$ , $\\gamma _{ij}$ are elements of constant matrixes.",
"Coordinates $X_i$ and momenta $P_i$ which satisfy (REF ), (REF ) can be represented as $X_{i}=x_{i}-\\frac{1}{2}\\sum _{j}\\theta _{ij}{p}_{j},\\\\P_{i}=p_{i}+\\frac{1}{2}\\sum _{j}\\eta _{ij}{x}_{j},$ here $x_i$ , $p_i$ are coordinates and momenta which satisfy $[x_{i},x_{j}]=0,\\\\{}[x_{i},p_{j}]=i\\hbar \\delta _{ij},\\\\{}[p_{i},p_{j}]=0.$ On the basis of (REF ), (), the commutation relations for coordinates and momenta read $[X_{i},P_{j}]=i\\hbar \\delta _{ij}+i\\hbar \\sum _k\\theta _{ik}\\eta _{jk}/{4}.$ So, parameters $\\gamma _{ij}$ are considered in the following form $\\gamma _{ij}=\\sum _k \\theta _{ik}\\eta _{jk}/4$ [1].",
"In noncommutative phase space of canonical type (REF )-(REF ) the rotational symmetry is not preserved [2], [3].",
"Algebra (REF )-(REF ) is not rotationally invariant.",
"To preserve this symmetry different types of noncommutative algebras were studied [4], [5], [6], [7]).",
"Much attention has been devoted to studies of position-dependent noncommutativity (see, for example, [8], [9], [10], [11], [12], [13], [14]), noncommutative algebras with spin noncommutativity of coordinates (see, for example, [15], [16], [17]).",
"In our previous paper [18] in order to construct rotationally invariant noncommutative algebra of canonical type we have studied the idea of involving additional coordinates and momenta.",
"The parameters of noncommutativity were considered to be generalized to a tensors constructed with the help of the additional coordinates and momenta.",
"In the present paper a system of interacting harmonic oscillators in uniform field is studied in the rotationally invariant noncommutative phase space.",
"We investigate influence of coordinates noncommutativity and momentum noncommutativity on the spectrum of the system.",
"On the basis of this result energy levels of a system of particles with harmonic oscillator interaction and a system of free particles in uniform field are analyzed in the rotationally invariant noncommutative phase space.",
"Nancommutative harmonic oscillator was studied in papers [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].",
"A system of two coupled harmonic oscillators was examined in two-dimensional noncommutative space [34], [35], in four-dimensional noncommutative phase space [36], [37].",
"In [38], [39] system of free particles in four-dimensional noncommutative phase space of canonical type was studied.",
"In noncommutative space-time classical problems of many particles among them $N$ interacting harmonic oscillators were examined in [40].",
"Studies of many-particle systems in the frame of noncommutative algebra of coordinates and momenta give a possibility to find new effects in the properties of wide class of physical systems caused by space quantization.",
"Considered in the present paper system of interacting harmonic oscillators has various applications.",
"Studies of a system of $N$ interacting harmonic oscillators are important in nuclei physics [41], [42], [43], in quantum chemistry and molecular spectroscopy [44], [45], [46], [47].",
"Recently networks of coupled harmonic oscillators have attracted much attention because of their importance for quantum information processing [48], [49], [50].",
"The studies are also important for searching signatures towards the Planck scale physics which are observable on macroscopic scales.",
"The systems under consideration can be realized on the classical level as a system of oscillators coupled by springs.",
"Our paper is organized as follows.",
"In Section 2 noncommutative algebra which is rotationally invariant and equivalent to noncommutative algebra of canonical type is presented.",
"Section 3 is devoted to studies of the total hamiltonian in rotationally invariant noncommutative phase space.",
"Effect of noncommutativity on the spectrum of a system of N interacting harmonic oscillators in uniform field is examined in the Section 4.",
"Also in this section a system of free particles in uniform filed and a system of particles with harmonic oscillator interaction are studied.",
"Spectrum of a system of two interacting harmonic oscillators and spectrum of three interacting harmonic oscillators are analyzed in Section 5 and Section 6, respectively.",
"Section 7 is devoted to conclusions."
],
[
"Rotationally invariant noncommutative algebra of canonical type",
"In our paper [18] we considered the tensors of noncommutativity to be defined as $\\theta _{ij}=\\frac{c_{\\theta } l^2_{P}}{\\hbar }\\sum _k\\varepsilon _{ijk}\\tilde{a}_{k}, \\\\\\eta _{ij}=\\frac{c_{\\eta }\\hbar }{l^2_{P}}\\sum _k\\varepsilon _{ijk}\\tilde{p}^b_{k}.$ where $c_{\\theta }$ , $c_{\\eta }$ are dimensionless constants, $l_P$ is the Planck length.",
"We use notations $\\tilde{a}_i$ , $\\tilde{b}_i$ $\\tilde{p}^a_i$ , $\\tilde{p}^b_i$ for additional dimensionless coordinates and momenta conjugate to them which are governed by a spherically symmetric systems.",
"For simplicity these systems are considered to be harmonic oscillators $H^a_{osc}=\\hbar \\omega _{osc}\\left(\\frac{(\\tilde{p}^{a})^{2}}{2}+\\frac{\\tilde{a}^{2}}{2}\\right),\\\\H^b_{osc}=\\hbar \\omega _{osc}\\left(\\frac{(\\tilde{p}^{b})^{2}}{2}+\\frac{\\tilde{b}^{2}}{2}\\right).$ The values of parameters of noncommutativity are supposed to be of the order of the Planck scale.",
"So, we put $\\sqrt{{\\hbar }}/\\sqrt{{m_{osc}\\omega _{osc}}}=l_{P}$ .",
"The frequency of the oscillators $\\omega _{osc}$ is considered to be very large which leads to the statement that the oscillators put into the ground states remain in them [18].",
"So, in [18] we proposed the following noncommutative algebra $[X_{i},X_{j}]=ic_{\\theta } l^2_{P} \\sum _k\\varepsilon _{ijk}\\tilde{a}_{k},\\\\{}[X_{i},P_{j}]=i\\hbar \\left(\\delta _{ij}+\\frac{c_{\\theta }c_{\\eta }}{4}({\\bf \\tilde{a}}\\cdot {\\bf \\tilde{p}^{b}})\\delta _{ij}-\\frac{c_{\\theta }c_{\\eta }}{4}{\\tilde{a}}_j{\\tilde{p}}^{b}_i\\right),\\\\{}[P_{i},P_{j}]=\\frac{c_{\\eta }\\hbar ^2}{l_P^2}\\sum _k\\varepsilon _{ijk}\\tilde{p}^{b}_{k}.",
"{}$ Commutation relations for $\\tilde{a}_i$ , $\\tilde{b}_i$ $\\tilde{p}^a_i$ , $\\tilde{p}^b_i$ were considered to be as follows $[\\tilde{a}_{i},\\tilde{a}_{j}]=[\\tilde{b}_{i},\\tilde{b}_{j}]=[\\tilde{a}_{i},\\tilde{b}_{j}]=[\\tilde{p}^{a}_{i},\\tilde{p}^{a}_{j}]=\\nonumber \\\\=[\\tilde{p}^{b}_{i},\\tilde{p}^{b}_{j}]=[\\tilde{p}^{a}_{i},\\tilde{p}^{b}_{j}]=0,\\\\{} [\\tilde{a}_{i},\\tilde{p}^{a}_{j}]=[\\tilde{b}_{i},\\tilde{p}^{b}_{j}]=i\\delta _{ij},\\\\{}[\\tilde{a}_{i},\\tilde{p}^{b}_{j}]=[\\tilde{b}_{i},\\tilde{p}^{a}_{j}]=0 {}\\\\ {} [\\tilde{a}_{i},X_{j}]=[\\tilde{a}_{i},P_{j}]=[\\tilde{p}^{b}_{i},X_{j}]=[\\tilde{p}^{b}_{i},P_{j}]=0.$ So, like in the case of canonical version of noncommutativity with $\\theta _{ij}$ , $\\eta _{ij}$ , $\\gamma _{ij}$ being constants, in the case of $\\theta _{ij}$ , $\\eta _{ij}$ being defined as (REF ), () we can write $[\\theta _{ij}, X_k]=[\\theta _{ij}, P_k]=[\\eta _{ij}, X_k]=[\\eta _{ij}, P_k]=[\\gamma _{ij}, X_k]=[\\gamma _{ij}, P_k]=0$ From this one can state that the proposed algebra $[X_{i},X_{j}]=ic_{\\theta } l^2_{P} \\sum _k\\varepsilon _{ijk}\\tilde{a}_{k},\\\\{}[X_{i},P_{j}]=i\\hbar \\left(\\delta _{ij}+\\frac{c_{\\theta }c_{\\eta }}{4}(\\tilde{\\bf {a}}\\cdot \\tilde{\\bf {p}}^{b})\\delta _{ij}-\\frac{c_{\\theta }c_{\\eta }}{4}{\\tilde{a}}_j{\\tilde{p}}^{b}_i\\right),\\\\{}[P_{i},P_{j}]=\\frac{c_{\\eta }\\hbar ^2}{l_P^2}\\sum _k\\varepsilon _{ijk}\\tilde{p}^{b}_{k},{}$ is equivalent to noncommutative algebra of canonical type at the same time it is rotationally invariant.",
"The noncommutative coordinates and noncommutative momenta which satisfy (REF )-() can be represented as $X_{i}=x_{i}+\\frac{c_{\\theta } l_P^2}{2\\hbar }[\\tilde{\\bf {a}}\\times {\\bf p}]_i=x_{i}+\\frac{1}{2}[{\\mathbf {\\theta }}\\times {\\bf p}]_i,\\\\P_{i}=p_{i}-\\frac{c_{\\eta }\\hbar }{2l_P^2}[{\\bf x}\\times \\tilde{\\bf {p}}^b]_i=p_{i}-\\frac{1}{2}[{\\bf x}\\times {\\mathbf {\\eta }}]_i,$ where coordinates and momenta $x_i$ , $p_i$ satisfy (REF )-() and for convenience the following vectors ${\\mathbf {\\theta }}=(\\theta _1,\\theta _2,\\theta _3), \\ \\ {\\mathbf {\\eta }}=(\\eta _1,\\eta _2,\\eta _3),\\\\\\theta _i=\\frac{1}{2}\\sum _{jk}\\varepsilon _{ijk}{\\theta _{jk}},\\\\\\eta _i=\\frac{1}{2}\\sum _{jk}\\varepsilon _{ijk}{\\eta _{jk}},$ are introduced.",
"After rotation one has $X_{i}^{\\prime }=U(\\varphi )X_{i}U^{+}(\\varphi )$ , $P_{i}^{\\prime }=U(\\varphi )P_{i}U^{+}(\\varphi )$ $a_{i}^{\\prime }=U(\\varphi )a_{i}U^{+}(\\varphi )$ , $p^{b\\prime }_{i}=U(\\varphi )p^b_{i}U^{+}(\\varphi )$ .",
"The rotation operator $U(\\varphi )=\\exp (i\\varphi ({\\bf n}\\cdot {\\bf L^t})/\\hbar )$ , contains the total angular momentum which reads ${\\bf L^t}=[{\\bf x}\\times {\\bf p}]+\\hbar [\\tilde{\\bf {a}}\\times \\tilde{\\bf p}^{a}]+\\hbar [\\tilde{\\bf { b}}\\times \\tilde{\\bf {p}}^{b}]$ [18].",
"The commutation relations for coordinates and momenta remain the same $[X^\\prime _{i},X^\\prime _{j}]=ic_{\\theta } l^2_{P} \\sum _k\\varepsilon _{ijk}\\tilde{a}^\\prime _{k},\\\\{}[X^\\prime _{i},P^\\prime _{j}]=i\\hbar \\left(\\delta _{ij}+\\frac{c_{\\theta }c_{\\eta }}{4}(\\tilde{\\bf {a}}^\\prime \\cdot \\tilde{\\bf {p}}^{b\\prime })\\delta _{ij}-\\frac{c_{\\theta }c_{\\eta }}{4}{\\tilde{a}}^\\prime _j{\\tilde{p}}^{b\\prime }_i\\right),\\\\{}[P^{\\prime }_{i},P^{\\prime }_{j}]= \\frac{c_{\\eta }\\hbar ^2}{l_P^2}\\sum _k\\varepsilon _{ijk}\\tilde{p}^{b\\prime }_{k}.",
"{}$"
],
[
"Hamiltonian of a system of interacting oscillators in noncommutative phase space with rotational symmetry",
"Let us study a system of $N$ interacting harmonic oscillators of masses $m$ and frequencies $\\omega $ in uniform field in noncommutative phase space with rotational symmetry (REF )-().",
"The hamiltonian of the system reads $H_s=\\sum _n\\frac{( {\\bf P}^{(n)})^{2}}{2m}+\\sum _n\\frac{m\\omega ^2( {\\bf X}^{(n)})^{2}}{2}+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}({\\bf X}^{(n)}-{\\bf X}^{(m)})^2+\\nonumber \\\\+\\kappa \\sum _n X^{(n)}_1.$ Here $\\kappa $ and $k$ are constants.",
"The direction of the field for convenience is chosen to coincide with the $X_1$ axis direction.",
"For $\\kappa =0$ , Hamiltonian (REF ) corresponds to nondissipative symmetric network of coupled harmonic oscillators [49].",
"Coordinates and momenta satisfy the following commutation relations $[X^{(n)}_{i},X^{(m)}_{j}]=i\\hbar \\delta _{mn}\\theta ^{(n)}_{ij},\\\\{}[X^{(n)}_{i},P^{(m)}_{j}]=i\\hbar \\delta _{mn}\\left(\\delta _{ij}+\\sum _k\\frac{\\theta ^{(n)}_{ik}\\eta ^{(m)}_{jk}}{4}\\right),\\\\{}[P^{(n)}_{i},P^{(m)}_{j}]=i\\hbar \\delta _{mn}\\eta ^{(n)}_{ij},$ with $\\theta ^{(n)}_{ij}=\\frac{c_{\\theta }^{(n)}l_P^2}{\\hbar }\\sum _k\\varepsilon _{ijk}\\tilde{a}_{k}, \\\\\\eta ^{(n)}_{ij}=\\frac{c_{\\eta }^{(n)}\\hbar }{l_P^2}\\sum _k\\varepsilon _{ijk}\\tilde{p}^b_{k},$ here indexes $m,n=(1...N)$ label the particles.",
"Note that we consider the general case when different particles satisfy noncommutative algebra with different tensors of noncommutativity.",
"The problem of description of composite system in rotationally invariant noncommutative phase space was discussed in our previous paper [54].",
"In the paper we proposed condition on the parameters $c_{\\theta }^{(n)}$ , $c_{\\eta }^{(n)}$ in tensors of noncommutativity on which the list of important results can be obtained (among them the noncommutative coordinates are independent on mass and noncommutative momenta are proportional to mass as it has to be, coordinates and momenta of the center-of-mass commute with the coordinates and momenta of the relative motion [54], the weak equivalence principle is recovered [55]).",
"The conditions read $c^{(n)}_{\\theta }m_n=\\tilde{\\gamma }=const, \\ \\ \\frac{c^{(n)}_{\\eta }}{m_n}=\\tilde{\\alpha }=const.$ Constants $\\tilde{\\gamma }$ , $\\tilde{\\alpha }$ are the same for particles with different masses.",
"We would like also to note that the idea to relate parameters of algebra for coordinates and momenta with mass is also important in deformed space with minimal length [56], [57], [58], two-dimensional noncommutative space of canonical type [59], four-dimensional noncommutative phase space of canonical type [60], [61].",
"In the case of system of harmonic oscillators with masses $m$ taking into account (REF ), (), (REF ) one has $\\theta ^{(n)}_{ij}=\\theta _{ij}=\\frac{c_{\\theta } l_P^2}{\\hbar }\\sum _k\\varepsilon _{ijk}\\tilde{a}_{k}, \\\\\\eta ^{(n)}_{ij}=\\eta _{ij}=\\frac{c_{\\eta }\\hbar }{l_P^2}\\sum _k\\varepsilon _{ijk}\\tilde{p}^b_{k},$ with $c_{\\theta }=\\tilde{\\gamma }/m$ , $c_{\\eta }=\\tilde{\\alpha } m$ .",
"Using representation (REF )-() and (REF ), () the hamiltonian of a system can be written in the following form $H_s=\\sum _n\\left(\\frac{({\\bf p}^{(n)})^{2}}{2m}+\\frac{m\\omega ^2( {\\bf x}^{(n)})^{2}}{2}+\\kappa x^{(n)}_1\\right)+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}({\\bf x}^{(n)}-{\\bf x}^{(m)})^2+\\nonumber \\\\+\\sum _n\\left(-\\frac{({\\mathbf {\\eta }}\\cdot {\\bf L}^{(n)})}{2m}-\\frac{m\\omega ^2({\\mathbf {\\theta }}\\cdot {\\bf L}^{(n)})}{2}+\\frac{\\kappa }{2}[{\\mathbf {\\theta }}\\times {\\bf p}^{(n)}]_1+\\frac{m\\omega ^2}{8}[{\\mathbf {\\theta }}\\times {\\bf p}^{(n)}]^2+\\right.\\nonumber \\\\\\left.+\\frac{[{\\mathbf {\\eta }}\\times {\\bf x}^{(n)}]^2}{8m}\\right)-\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}{\\mathbf {\\theta }}\\cdot [({{\\bf x}}^{(n)}-{{\\bf x}}^{(m)})\\times ({\\bf p}^{(n)}-{\\bf p}^{(m)})]+\\nonumber \\\\+\\mathop {\\sum _{m,n}}\\limits _{m\\ne n} \\frac{k}{8}[{\\mathbf {\\theta }}\\times ({\\bf p}^{(n)}-{\\bf p}^{(m)})]^2,$ where ${\\bf L}^{(n)}=[{\\bf x}^{(n)}\\times {\\bf p}^{(n)}]$ .",
"Because of involving of additional coordinates and additional momenta $\\tilde{a}_i$ , $\\tilde{b}_i$ $\\tilde{p}^a_i$ , $\\tilde{p}^b_i$ we have to consider the total hamiltonian which is the sum of $H_s$ and Hamiltonians of harmonic oscillators $H^a_{osc}$ , $H^b_{osc}$ $H=H_s+H^a_{osc}+H^b_{osc}=H_0+\\Delta H.$ here $H_0=\\langle H_s\\rangle _{ab}+H^a_{osc}+H^b_{osc},\\\\\\Delta H= H-H_0=H_s-\\langle H_s\\rangle _{ab},$ $\\langle ...\\rangle _{ab}$ denotes averaging over degrees of freedom of harmonic oscillators $H^a_{osc}$ $H^b_{osc}$ in the ground states $\\langle ...\\rangle _{ab}=\\langle \\psi ^{a}_{0,0,0}\\psi ^{b}_{0,0,0}|...|\\psi ^{a}_{0,0,0}\\psi ^{b}_{0,0,0}\\rangle $ $\\psi ^{a}_{0,0,0}$ , $\\psi ^{b}_{0,0,0}$ are eigenstates of tree-dimensional harmonic oscillators $H^a_{osc}$ , $H^b_{osc}$ in the ground states in the ordinary space (space with commutative coordinates and commutative momenta).",
"In our previous paper [54] we concluded that up to the second order in $\\Delta H$ one can consider Hamiltonian $H_0$ .",
"For a system of interacting harmonic oscillators using $\\langle \\psi ^{a}_{0,0,0}|\\theta _i|\\psi ^{a}_{0,0,0}\\rangle =\\langle \\psi ^{b}_{0,0,0}|\\eta _i|\\psi ^{b}_{0,0,0}\\rangle =0, \\\\\\langle \\theta _i\\theta _j\\rangle =\\frac{c_{\\theta }^2l_P^4}{\\hbar ^2}\\langle \\psi ^{a}_{0,0,0}| \\tilde{a}_i\\tilde{a}_j|\\psi ^{a}_{0,0,0}\\rangle =\\frac{c_{\\theta }^2l_P^4}{2\\hbar ^2}\\delta _{ij}=\\frac{\\langle \\theta ^2\\rangle \\delta _{ij}}{3},\\\\\\langle \\eta _i\\eta _j\\rangle = \\frac{\\hbar ^2 c_{\\eta }^2}{l_P^4}\\langle \\psi ^{b}_{0,0,0}| \\tilde{p}^{b}_i\\tilde{p}^{b}_j|\\psi ^{b}_{0,0,0}\\rangle =\\frac{\\hbar ^2 c_{\\eta }^2}{2 l_P^4}\\delta _{ij}=\\frac{\\langle \\eta ^2\\rangle \\delta _{ij}}{3},$ and calculating $\\langle [{\\mathbf {\\eta }}\\times {\\bf x}^{(n)}]^2\\rangle _{ab}=\\frac{2}{3}\\langle \\eta ^2\\rangle ({\\bf x}^{(n)})^2, \\ \\ \\langle [{\\mathbf {\\theta }}\\times {\\bf p}^{(n)}]^2\\rangle _{ab}=\\frac{2}{3}\\langle \\theta ^2\\rangle ({\\bf p}^{(n)})^2,\\\\\\langle [{\\mathbf {\\theta }}\\times ({\\bf p}^{(n)}-{\\bf p}^{(m)})]^2\\rangle _{ab}=\\frac{2}{3}\\langle \\theta ^2\\rangle ({\\bf p}^{(n)}-{\\bf p}^{(m)})^2$ the expression for $\\Delta H$ can be written as $\\Delta H=\\sum _n\\left(-\\frac{({\\mathbf {\\eta }}\\cdot {\\bf L}^{(n)})}{2m}-\\frac{m\\omega ^2({\\mathbf {\\theta }}\\cdot {\\bf L}^{(n)})}{2}+\\frac{\\kappa }{2}[{\\mathbf {\\theta }}\\times {\\bf p}^{(n)}]_1+\\frac{m\\omega ^2}{8}[{\\mathbf {\\theta }}\\times {\\bf p}^{(n)}]^2+\\right.\\nonumber \\\\\\left.+\\frac{[{\\mathbf {\\eta }}\\times {\\bf x}^{(n)}]^2}{8m}\\right)-\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}{\\mathbf {\\theta }}\\cdot [({{\\bf x}}^{(n)}-{{\\bf x}}^{(m)})\\times ({\\bf p}^{(n)}-{\\bf p}^{(m)})]+\\nonumber \\\\+\\mathop {\\sum _{m,n}}\\limits _{m\\ne n} \\frac{k}{8}[{\\mathbf {\\theta }}\\times ({\\bf p}^{(n)}-{\\bf p}^{(m)})]^2-\\sum _n\\left(\\frac{\\langle \\eta ^2\\rangle ({\\bf x}^{(n)})^2}{12m}+\\frac{\\langle \\theta ^2\\rangle m\\omega ^2({\\bf p}^{(n)})^2}{12}\\right)-\\nonumber \\\\-\\frac{k}{12}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}\\langle {\\theta }^2\\rangle ({\\bf p}^{(n)}-{\\bf p}^{(m)})^2.",
"\\nonumber \\\\$ So, on the basis of conclusion presented in [54] up to the second order in $\\Delta H$ (or taking into account (REF ), up to the second order in the parameters of noncommutativity) for a system of interacting harmonic oscillators in uniform field we can consider the Hamiltonian as $H_0=\\sum _n\\left(\\frac{({\\bf p}^{(n)})^{2}}{2m}+\\frac{m\\omega ^2( {\\bf x}^{(n)})^{2}}{2}+\\kappa x^{(n)}_1\\right)+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}({\\bf x}^{(n)}-{\\bf x}^{(m)})^2+\\nonumber \\\\+ \\sum _n\\left(\\frac{\\langle \\eta ^2\\rangle ({\\bf x}^{(n)})^2}{12m}+\\frac{\\langle \\theta ^2\\rangle m\\omega ^2({\\bf p}^{(n)})^2}{12}\\right)+\\nonumber \\\\+\\frac{k}{12}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}\\langle {\\theta }^2\\rangle ({\\bf p}^{(n)}-{\\bf p}^{(m)})^2+H^a_{osc}+H^b_{osc}.$"
],
[
"Influence of noncommutativity on the spectrum of a system of $N$ interacting oscillators",
"Let us find energy levels of a system of $N$ interacting harmonic oscillators.",
"It is convenient to introduce effective mass and effective frequency $m_{eff}={m}\\left({1+\\frac{m^2\\omega ^2\\langle \\theta ^2\\rangle }{6}}\\right)^{-1},\\\\\\omega _{eff}=\\left({\\omega ^2+\\frac{\\langle \\eta ^2\\rangle }{6m^2}}\\right)^{\\frac{1}{2}}\\left({1+\\frac{m^2\\omega ^2\\langle \\theta ^2\\rangle }{6}}\\right)^{\\frac{1}{2}}$ and rewrite (REF ) as $H_0=\\sum _n\\left(\\frac{({\\bf p}^{(n)})^{2}}{2m_{eff}}+\\frac{m_{eff}\\omega _{eff}^2( {\\tilde{\\bf x}}^{(n)})^{2}}{2}\\right)-\\frac{N\\kappa ^2}{2m_{eff}\\omega ^2_{eff}}+\\nonumber \\\\+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}({\\tilde{\\bf x}}^{(n)}-{\\tilde{\\bf x}}^{(m)})^2+\\frac{k}{12}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}\\langle {\\theta }^2\\rangle ({\\bf p}^{(n)}-{\\bf p}^{(m)})^2+H^a_{osc}+H^b_{osc}.$ where the vector $\\tilde{{\\bf x}}^{(n)}=\\left(x_1^{(n)}+\\frac{\\kappa }{m_{eff}\\omega ^2_{eff}},x^{(n)}_2,x^{(n)}_3\\right),$ is introduced.",
"Coordinates and momenta ${\\tilde{\\bf x}}^{(n)}$ , ${\\bf p}^{(n)}$ satisfy the ordinary commutation relations $[{\\tilde{x}}^{(n)}_{i},{\\tilde{x}}^{(m)}_{j}]=0,\\\\{}[{\\tilde{x}}^{(n)}_{i},p^{(m)}_{j}]=i\\hbar \\delta _{nm}\\delta _{ij},\\\\{}[p^{(n)}_{i},p^{(m)}_{j}]=0.$ Note also that $[H_0,H^a_{osc}]=[H_0,H^b_{osc}]=0.$ Therefore, the spectrum of $H_0$ reads $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\sum ^{N}_{a=1}\\hbar \\omega _a\\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\\frac{3}{2}\\right)-\\frac{N\\kappa ^2}{2m_{eff}\\omega ^2_{eff}}+\\nonumber \\\\+3\\hbar \\omega _{osc}.$ where $\\omega _1=\\omega _{eff},\\\\\\omega _2=\\omega _3=...=\\omega _N=\\nonumber \\\\=\\left(\\omega ^2_{eff}+\\frac{2kN}{m_{eff}}+\\frac{kN\\langle \\theta ^2\\rangle m_{eff}\\omega ^2_{eff}}{3}+\\frac{2k^2\\langle \\theta ^2\\rangle N^2}{3}\\right)^{\\frac{1}{2}}.$ $n^{(a)}_i$ are quantum numbers ($n^{(a)}_i=0,1,2...$ ).",
"In (REF ) we take into account that the oscillators $H^a_{osc}$ , $H^b_{osc}$ are in the ground states.",
"The first term in (REF ) corresponds to the spectrum of the center-of-mass of the system.",
"The terms with $a=2..N$ corresponds to the spectrum of the relative motion.",
"This can be shown introducing coordinates and momenta of the center-of-mass ${\\bf x}^c=\\sum _n{\\bf x}^{(n)}/N$ , ${\\bf p}^c=\\sum _n {\\bf p}^{(n)}$ , and coordinates and momenta of relative motion $\\Delta {\\bf x}^{(n)}={\\bf x}^{(n)}-{\\bf x}^c$ , $\\Delta {\\bf p}^{(n)}={\\bf p}^{(n)}-{\\bf p}^c/N$ .",
"From (REF ) we can write $H_0=H^{c}+H_{rel}+H^a_{osc}+H^b_{osc},\\\\H^c=\\frac{({\\bf p}^c)^{2}}{2Nm_{eff}}+\\frac{Nm_{eff}\\omega _{eff}^2(\\tilde{\\bf x}^c)^2}{2}-\\frac{N\\kappa ^2}{2m_{eff}\\omega ^2_{eff}},\\\\H_{rel}=\\sum _n\\left(\\frac{(\\Delta {\\bf p}^{(n)})^{2}}{2m_{eff}}+\\frac{m_{eff}\\omega _{eff}^2( \\Delta {\\bf x}^{(n)})^{2}}{2}\\right)+\\nonumber \\\\+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}(\\Delta {\\bf x}^{(n)}-\\Delta {\\bf x}^{(m)})^2+\\frac{k}{12}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}\\langle {\\theta }^2\\rangle (\\Delta {\\bf p}^{(n)}-\\Delta {\\bf p}^{(m)})^2,\\\\{}[H^{c},H_{rel}]=[H^{c},H^a_{osc}+H^b_{osc}]=[H_{rel},H^a_{osc}+H^b_{osc}]=0,{}$ where $\\tilde{\\bf x}^c=\\left(x_1^{c}+{\\kappa }/({m_{eff}\\omega ^2_{eff}}),x^{c}_2,x^{c}_3\\right)$ .",
"So, from (REF ) we have that the noncommutativity of coordinates and noncommutativity of momenta effects on the frequencies in the spectra of the center-of-mass and relative motion of the system.",
"The presents of uniform field shifts the spectrum on a constant.",
"In the limit $\\langle \\theta ^2\\rangle \\rightarrow 0$ , $\\langle \\eta ^2\\rangle \\rightarrow 0$ the expression for $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }$ reduces to known spectrum for system of $N$ interacting harmonic oscillators in uniform field in the ordinary space $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\hbar \\omega \\left(n^{(1)}_1+n^{(1)}_2+n^{(1)}_3+\\frac{3}{2}\\right)+\\nonumber \\\\+\\sum ^{N}_{a=2}\\hbar \\left(\\omega ^2+\\frac{2Nk}{m}\\right)^{\\frac{1}{2}}\\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\\frac{3}{2}\\right)-\\frac{N\\kappa ^2}{2m\\omega ^2}$ From (REF ) setting $\\omega =0$ , one can write the following expression for the spectrum of a system of $N$ particles of mass $m$ with harmonic oscillator interaction.",
"$E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\frac{\\hbar \\langle \\eta ^2\\rangle }{6m^2}\\left(n^{(1)}_1+n^{(1)}_2+n^{(1)}_3+\\frac{3}{2}\\right)+\\nonumber \\\\+\\hbar \\left(\\frac{2kN}{m}+\\frac{\\langle \\eta ^2\\rangle }{6m^2}+\\frac{2k^2\\langle \\theta ^2\\rangle N^2}{3}\\right)^{\\frac{1}{2}}\\sum ^{N}_{a=2}\\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\\frac{3}{2}\\right)-\\nonumber \\\\-\\frac{3N\\kappa ^2m}{\\langle \\eta ^2\\rangle }+3\\hbar \\omega _{osc}.$ The first term in (REF ) corresponds to the spectrum of the center-of-mass of the system.",
"In the contrast to the ordinary space (space with commutative coordinates and commutative momenta) because of momentum noncommutativity the spectrum of the center-of-mass of the system of particles with harmonic oscillator interaction is discreet.",
"The spectrum corresponds to the spectrum of harmonic oscillator with the frequency ${\\hbar \\langle \\eta ^2\\rangle }/{6m^2}$ .",
"The frequency in the spectrum of the relative motion is affected by the noncommutativity of coordinates and noncommutativity of momenta (see second term in (REF )).",
"It is worth mentioning that from (REF ) and (REF ) we have that the effect of coordinates noncommutativity on the spectrum of interacting harmonic oscillators (a system of particles with harmonic oscillator interaction) increases with increasing of the number of particles in the system.",
"For a system of $N$ free particles in uniform field in rotationally invariant noncommutative phase space setting $k=0$ in (REF ) energy levels are as follows $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\sum ^{N}_{a=1}\\frac{\\hbar \\langle \\eta ^2\\rangle }{6m^2}\\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\\frac{3}{2}\\right)-\\frac{3N\\kappa ^2m}{\\langle \\eta ^2\\rangle }+3\\hbar \\omega _{osc}.$ Note that the spectrum of a system of free particles is affected only by momentum noncommutativity.",
"The spectrum corresponds to the spectrum of $N$ oscillators with frequencies determined by the parameter of momentum noncommutativity ${\\hbar \\langle \\eta ^2\\rangle }/{6m^2}$ .",
"We would like also to mention that the presence of uniform field $\\kappa $ shift the spectra (REF ), (REF ), (REF ) by constant."
],
[
"Two interacting oscillators in rotationally invariant noncommutative phase space",
"Let us study particular case when a system consists of two oscillators with masses $m_1$ , $m_2$ and frequencies $\\omega _1$ , $\\omega _2$ and is described by the following Hamiltonian $H_s=\\frac{( {\\bf P}^{(1)})^{2}}{2m_1}+\\frac{({\\bf P}^{(2)})^{2}}{2m_2}+\\frac{m_1\\omega _1^2( {\\bf X}^{(1)})^{2}}{2}+\\frac{m_2\\omega _2^2({\\bf X}^{(2)})^{2}}{2}+k({\\bf X}^{(1)}-{\\bf X}^{(2)})^2.$ where ${\\bf X}^{(n)}$ , ${\\bf P}^{(n)}$ satisfy (REF )-(), ($n=1,2$ ).",
"System of two coupled harmonic oscillators has various applications in physics (see, for example, [34], [51] and references therein).",
"The system is considered as a model in molecular physics [44], [45], used for description of states of light in the framework of two-photon quantum optics [52], [53].",
"Taking into account (REF ) and using representation (REF )-(), we have $H_0=\\left(\\frac{({\\bf p}^{(1)})^{2}}{2m^{(1)}_{eff}}+\\frac{({\\bf p}^{(2)})^{2}}{2m^{(2)}_{eff}}+\\frac{m^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2( {{\\bf x}}^{(1)})^{2}}{2}+\\frac{m^{(2)}_{eff}(\\omega ^{(2)}_{eff})^2( {{\\bf x}}^{(2)})^{2}}{2}\\right)+\\nonumber \\\\+k({{\\bf x}}^{(1)}-{{\\bf x}}^{(2)})^2+\\frac{k}{6}\\left(\\langle ({\\theta ^{(1)}})^2\\rangle ({\\bf p}^{(1)})^2+\\langle (\\theta ^{(2)})^2\\rangle ({\\bf p}^{(2)})^2-\\right.\\nonumber \\\\\\left.-2\\langle {\\theta ^{(1)}\\theta ^{(2)}}\\rangle ({\\bf p}^{(1)}\\cdot {\\bf p}^{(2)})\\right)+H^a_{osc}+H^b_{osc}.$ with $m^{(n)}_{eff}={m_n}\\left({1+\\frac{m_n^2\\omega _n^2\\langle (\\theta ^{(n)})^2\\rangle }{6}}\\right)^{-1},\\\\\\omega ^{(n)}_{eff}=\\left({\\omega _n^2+\\frac{\\langle (\\eta ^{n})^2\\rangle }{6m_n^2}}\\right)^{\\frac{1}{2}}\\left({1+\\frac{m_n^2\\omega _n^2\\langle (\\theta ^{(n)})^2\\rangle }{6}}\\right)^{\\frac{1}{2}},\\\\\\langle \\theta ^{(n)}\\theta ^{(m)}\\rangle =\\frac{c^{(n)}_{\\theta }c^{(m)}_{\\theta }l_P^4}{\\hbar ^2}\\langle \\psi ^{a}_{0,0,0}| \\tilde{a}^2|\\psi ^{a}_{0,0,0}\\rangle =\\frac{3c^{(n)}_{\\theta }c^{(m)}_{\\theta }l_P^4}{2\\hbar ^2},\\\\\\langle (\\eta ^{(n)})^{2}\\rangle =\\frac{\\hbar ^2 (c^{(n)}_{\\eta })^2}{l_P^4}\\langle \\psi ^{b}_{0,0,0}| (\\tilde{p}^{b})^2|\\psi ^{b}_{0,0,0}\\rangle =\\frac{3\\hbar ^2 (c^{(n)}_{\\eta })^2}{2 l_P^4},$ Coordinates $x_i^{(n)}$ and momenta $p_i^{(n)}$ satisfy the ordinary commutation relations.",
"So, the spectrum of $H_0$ is as follows $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\hbar {\\omega }_+\\left(n^{(1)}_1+n^{(1)}_2+n^{(1)}_3+\\frac{3}{2}\\right)+\\nonumber \\\\+\\hbar {\\omega }_{-}\\left(n^{(2)}_1+n^{(2)}_2+n^{(2)}_3+\\frac{3}{2}\\right)+3\\hbar \\omega _{osc}.$ where $\\omega ^2_{\\pm }=\\frac{1}{{2}}\\sum _n\\left((\\omega ^{(n)}_{eff})^2+\\frac{2k}{m^{(n)}_{eff}}+\\frac{km^{(n)}_{eff}(\\omega ^{(n)}_{eff})^2\\langle (\\theta ^{(n)})^{2}\\rangle }{3}+\\right.\\nonumber \\\\\\left.+\\frac{2k^2}{3}\\left(\\langle (\\theta ^{(n)})^{2}\\rangle +\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle \\right)\\right)\\pm \\frac{1}{2}\\sqrt{D},$ $D=\\left(\\sum _n(\\omega ^{(n)}_{eff})^2+\\sum _n\\frac{2k}{m^{(n)}_{eff}}+\\sum _n\\frac{km^{(n)}_{eff}(\\omega ^{(n)}_{eff})^2\\langle (\\theta ^{(n)})^{2}\\rangle }{3}+\\right.\\nonumber \\\\ \\left.+\\sum _n\\frac{2k^2}{3}\\left(\\langle (\\theta ^{(n)})^{2}\\rangle +\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle \\right)\\right)^2-4\\prod _n\\left((\\omega ^{(n)}_{eff})^2+\\frac{2k}{m^{(n)}_{eff}}+\\right.\\nonumber \\\\\\left.+\\frac{km^{(n)}_{eff}(\\omega ^{(n)}_{eff})^2\\langle (\\theta ^{(n)})^{2}\\rangle }{3}+\\frac{2k^2}{3}\\left(\\langle (\\theta ^{(n)})^{2}\\rangle +\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle \\right)\\right)+\\nonumber \\\\+4\\left(\\frac{2k}{m^{(2)}_{eff}}+\\frac{km^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle }{3}+\\frac{2k^2}{3}\\left(\\langle (\\theta ^{(2)})^{2}\\rangle +\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle \\right)\\right)\\times \\nonumber \\\\\\left(\\frac{2k}{m^{(1)}_{eff}}+\\frac{km^{(2)}_{eff}(\\omega ^{(2)}_{eff})^2\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle }{3}+\\frac{2k^2}{3}\\left(\\langle (\\theta ^{(1)})^{2}\\rangle +\\langle \\theta ^{(1)}\\theta ^{(2)}\\rangle \\right)\\right).$ In the case of $m_1=m_2$ we have $m^{(n)}_{eff}=m_{eff}$ , $\\omega ^{(n)}_{eff}=\\omega _{eff}$ and the expressions reduce to $\\omega _-=\\omega _{eff},\\\\\\omega _+=\\left(\\omega ^2_{eff}+\\frac{4k}{m_{eff}}+\\frac{2k\\langle \\theta ^2\\rangle m_{eff}\\omega ^2_{eff}}{3}+\\frac{8k^2\\langle \\theta ^2\\rangle }{3}\\right)^{\\frac{1}{2}}.$ which corresponds to (REF ), () with $N=2$ ."
],
[
"System of three interacting oscillators in rotationally invariant noncommutative phase space",
"Let us study a system of three interacting oscillators with masses $m_1$ , $m_2=m_3=m$ , and frequencies $\\omega _1$ , $\\omega _2=\\omega _3=\\omega $ .",
"The Hamiltonian reads $H_s=\\frac{( {\\bf P}^{(1)})^{2}}{2m_1}+\\frac{( {\\bf P}^{(2)})^{2}}{2m}+\\frac{( {\\bf P}^{(3)})^{2}}{2m}+\\frac{m_1\\omega _1^2( {\\bf X}^{(1)})^{2}}{2}+\\frac{m\\omega ^2({\\bf X}^{(2)})^{2}}{2}+\\nonumber \\\\+\\frac{m\\omega ^2({\\bf X}^{(3)})^{2}}{2}+k({\\bf X}^{(1)}-{\\bf X}^{(2)})^2+k({\\bf X}^{(2)}-{\\bf X}^{(3)})^2+k({\\bf X}^{(3)}-{\\bf X}^{(3)})^2.$ In the case when $\\omega _n=0$ the Hamiltonian (REF ) is used as a model for description of confining forces between quarks [41], [42], [43].",
"Up to the second order in the parameters of noncommutativity one can consider $H_0=\\sum _n\\frac{({\\bf p}^{(n)})^{2}}{2m^{(n)}_{eff}}+\\sum _n\\frac{m^{(n)}_{eff}(\\omega ^{(n)}_{eff})^2( {{\\bf x}}^{(n)})^{2}}{2}+\\nonumber \\\\+\\frac{k}{2}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}({{\\bf x}}^{(n)}-{{\\bf x}}^{(m)})^2+\\frac{k}{12}\\mathop {\\sum _{m,n}}\\limits _{m\\ne n}\\left(\\langle ({\\theta ^{(n)}})^2\\rangle ({\\bf p}^{(n)})^2+\\langle (\\theta ^{(m)})^2\\rangle ({\\bf p}^{(m)})^2-\\right.\\nonumber \\\\\\left.-2\\langle {\\theta ^{(n)}\\theta ^{(m)}}\\rangle ({\\bf p}^{(n)}\\cdot {\\bf p}^{(m)})\\right)+H^a_{osc}+H^b_{osc}.$ where $m^{(n)}_{eff}$ , $\\omega ^{(n)}_{eff}$ , $\\langle {\\theta ^{(n)}\\theta ^{(m)}}\\rangle $ are defined as (REF )-().",
"The spectrum of (REF ) reads $E_{\\lbrace n_1\\rbrace ,\\lbrace n_2\\rbrace ,\\lbrace n_3\\rbrace }=\\sum ^{3}_{a=1}\\hbar \\tilde{\\omega }_a\\left(n^{(a)}_1+n^{(a)}_2+n^{(a)}_3+\\frac{3}{2}\\right)+3\\hbar \\omega _{osc}.\\\\\\tilde{\\omega }_1=\\frac{1}{\\sqrt{2}}\\left(\\omega ^2_{eff}+(\\omega ^{(1)}_{eff})^2+\\frac{2k}{m_{eff}}+\\frac{4k}{m_{eff}^{(1)}}+A_1-\\sqrt{D}\\right)^{\\frac{1}{2}},\\\\\\tilde{\\omega }_2=\\frac{1}{\\sqrt{2}}\\left(\\omega ^2_{eff}+(\\omega ^{(1)}_{eff})^2+\\frac{2k}{m_{eff}}+\\frac{4k}{m^{(1)}_{eff}}+A_1+\\sqrt{D}\\right)^{\\frac{1}{2}},\\\\\\tilde{\\omega }_3=\\left(\\omega _{eff}^2+\\frac{6k}{m_{eff}}\\right)^{\\frac{1}{2}}\\left(1+km_{eff}\\langle {\\theta }^2\\rangle \\right)^{\\frac{1}{2}},$ with $D=\\left(\\omega ^2_{eff}-(\\omega ^{(1)}_{eff})^2+\\frac{4k}{m_{eff}}-\\frac{4k}{m^{(1)}_{eff}}+A_2\\right)^2+\\left(\\frac{2k}{m}+A_3\\right)\\left(2(\\omega ^{(1)}_{eff})^2-\\right.\\nonumber \\\\\\left.-2\\omega ^2_{eff}-\\frac{6k}{m}+\\frac{8k}{m^{(1)}_{eff}}+8\\left(\\frac{2k}{m}+A_4\\right)\\left(\\frac{2k}{m_1}+A_5\\right)\\left(\\frac{2k}{m}+A_3\\right)^{-1}+A_6\\right),\\\\A_1=\\left(\\frac{km_{eff}\\omega ^2_{eff}}{3}+\\frac{2k^2}{3}\\right)\\langle {\\theta }^2\\rangle +\\left(\\frac{2km^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2}{3}+\\frac{8k^2}{3}\\right)\\langle (\\theta ^{(1)})^2\\rangle +\\nonumber \\\\+\\frac{8k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle ,\\\\A_2=\\left(\\frac{2km_{eff}\\omega ^2_{eff}}{3}+\\frac{10k^2}{3}\\right)\\langle {\\theta }^2\\rangle -\\left(\\frac{2km^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2}{3}+\\frac{8k^2}{3}\\right)\\langle (\\theta ^{(1)})^2\\rangle -\\nonumber \\\\-\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle ,\\\\A_3=\\left(\\frac{8k^2}{3}+\\frac{km_{eff}\\omega _{eff}^{2}}{3}\\right)\\langle {\\theta }^2\\rangle -\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle ,\\\\A_4=\\left(\\frac{km^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2}{3}+\\frac{4k^2}{3}\\right)\\langle \\theta \\theta ^{(1)}\\rangle +\\frac{2k^2}{3}\\langle {\\theta }^2\\rangle ,\\\\A_5=\\left(\\frac{km_{eff}(\\omega ^2_{eff})}{3}+\\frac{2k^2}{3}\\right)\\langle \\theta \\theta ^{(1)}\\rangle +\\frac{4k^2}{3}\\langle (\\theta ^{(1)})^2\\rangle ,\\\\A_6=-\\left({km_{eff}\\omega ^2_{eff}}+4k^2\\right)\\langle {\\theta }^2\\rangle +\\left(\\frac{4km^{(1)}_{eff}(\\omega ^{(1)}_{eff})^2}{3}+\\frac{16k^2}{3}\\right)\\langle (\\theta ^{(1)})^2\\rangle +\\nonumber \\\\+\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle .$ here for convenience we use notations $m_{eff}=m^{(2)}_{eff}=m^{(3)}_{eff}$ , $\\omega _{eff}=\\omega ^{(2)}_{eff}=\\omega ^{(3)}_{eff}$ , and $\\theta =\\theta ^{(2)}=\\theta ^{(3)}$ .",
"In the case when the masses and frequencies of the oscillators are equal, $m_1=m$ , $\\omega _1=\\omega $ , the result (REF ) reproduce (REF ) with $N=3$ .",
"We have $\\tilde{\\omega }_1=\\omega _{eff},\\\\\\tilde{\\omega }_2=\\tilde{\\omega }_3=\\left(\\omega ^2_{eff}+\\frac{6k}{m_{eff}}+k\\langle \\theta ^2\\rangle m_{eff}\\omega ^2_{eff}+6k^2\\langle \\theta ^2\\rangle \\right)^{\\frac{1}{2}}.$ For Hamiltonian (REF ) with $\\omega _n=0$ which is considered for description of confining forces between quarks the spectrum is given by (REF ) with (), (), () and $m^{(1)}_{eff}=m_1$ , $m_{eff}=m$ , $\\omega ^{(1)}_{eff}=\\sqrt{\\langle (\\eta ^{1})^2\\rangle }/\\sqrt{6m_1^2}$ , $\\omega _{eff}=\\sqrt{\\langle (\\eta )^2\\rangle }/\\sqrt{6m^2}$ .",
"Note, that because of noncommutativity of coordinates and noncommutativity of momenta the spectrum of the center-of-mass of the system is discrete and corresponds to the spectrum of harmonic oscillator with frequency $\\tilde{\\omega }_1$ ().",
"In a rotationally-invariant space with noncommutativity of coordinates (space which is characterized by (REF ), () and $[P^{(n)}_{i},P^{(m)}_{j}]=0$ ), the spectrum of a system described by Hamiltonian (REF ) with $\\omega _n=0$ has the form (REF ) with frequencies $\\tilde{\\omega }_1=0,\\\\\\tilde{\\omega }_2=\\frac{1}{\\sqrt{2}}\\left(\\frac{2k}{m}+\\frac{4k}{m^{(1)}}+\\frac{2k^2}{3}\\langle {\\theta }^2\\rangle +\\frac{8k^2}{3}\\langle (\\theta ^{(1)})^2\\rangle +\\frac{8k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle +\\sqrt{D}\\right)^{\\frac{1}{2}},\\\\\\tilde{\\omega }_3=\\left(\\frac{6k}{m}+6k^2\\langle {\\theta }^2\\rangle \\right)^{\\frac{1}{2}},$ where $D=\\left(\\frac{4k}{m}-\\frac{4k}{m^{(1)}}+\\frac{10k^2}{3}\\langle {\\theta }^2\\rangle -\\frac{8k^2}{3}\\langle (\\theta ^{(1)})^2\\rangle -\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle \\right)^2+\\left(\\frac{2k}{m}+\\right.\\nonumber \\\\\\left.+\\frac{8k^2}{3}\\langle {\\theta }^2\\rangle -\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle \\right)\\left(-\\frac{6k}{m}+\\frac{8k}{m^{(1)}}+8\\left(\\frac{2k}{m}+\\frac{4k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle +\\frac{2k^2}{3}\\langle {\\theta }^2\\rangle \\right)\\times \\right.\\nonumber \\\\\\times \\left.\\left(\\frac{2k}{m_1}+\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle +\\frac{4k^2}{3}\\langle (\\theta ^{(1)})^2\\rangle \\right)\\left(\\frac{2k}{m}+\\frac{8k^2}{3}\\langle {\\theta }^2\\rangle -\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle \\right)^{-1}-\\right.\\nonumber \\\\\\left.",
"-4k^2\\langle {\\theta }^2\\rangle +\\frac{16k^2}{3}\\langle (\\theta ^{(1)})^2\\rangle +\\frac{2k^2}{3}\\langle \\theta \\theta ^{(1)}\\rangle \\right).\\nonumber \\\\$ The frequencies are obtained putting $\\omega _{eff}=\\omega ^{(1)}_{eff}=0$ , $m^{(1)}_{eff}=m_1$ , $m_{eff}=m$ in (), (), ().",
"Note, that the spectrum of the center-of-mass of the system is not affected by noncommutativity of coordinates (REF ).",
"The noncommutativity has influence on the frequencies in the spectrum of the relative motion (), ()."
],
[
"Conclusions",
"We have considered noncommutative phase space of canonical type with rotational symmetry (REF )-().",
"The corresponding noncommutative algebra is constructed with the help of generalization of parameters of noncommutativity to tensors determined by additional coordinates and additional momenta [18].",
"In the frame of the rotationally invariant noncommutative algebra we have examined a system of $N$ harmonic oscillators with harmonic oscillator interaction in uniform field.",
"The total hamiltonian has been constructed and analyzed (REF ).",
"We have found energy levels of the system up to the second order in the parameters of noncommutativity.",
"We have obtained that noncommutativity affects on the frequencies of the system (REF ).",
"Uniform field causes shift of the spectrum on a constant (REF ).",
"A system of two interacting oscillators and a system of three interacting oscillators have been studied in details and the corresponding spectra have been obtained (REF ), (REF ).",
"As particular cases, a system of particles with harmonic oscillator interaction and a system of free particles have been studied in uniform field in rotationally-invariant noncommutative phase space.",
"We have obtained that the spectrum of free particles in uniform field corresponds to the spectrum of a system of $N$ oscillators with frequencies ${\\hbar \\langle \\eta ^2\\rangle }/{6m^2}$ and is not affected by the coordinates noncommutativity (REF ).",
"For a system of particles with harmonic oscillator interaction in uniform field we have found that the spectrum of the center-of-mass of the system is affected by noncommutativity of momenta and corresponds to the spectrum of harmonic oscillator (see first term in (REF )).",
"The spectrum of the relative motion of the system corresponds to the spectrum of harmonic oscillators with frequencies determined by parameters of momentum noncommutativity and coordinate noncommutativity (see second term in (REF )).",
"We have concluded that effect of coordinates noncommutativity on the spectra of systems with harmonic oscillator interaction (system of interacting harmonic oscillators, system of particles with harmonic oscillator interaction) increases with increasing of the number of particles in the systems (REF ), (REF )."
],
[
"Acknowledgments",
"The author thanks Prof. V. M. Tkachuk for his advices and support during research studies.",
"This work was partly supported by the by the State Found for Fundamental Research under the project F-76."
]
] | 1808.08515 |
[
[
"The Disparate Effects of Strategic Manipulation"
],
[
"Abstract When consequential decisions are informed by algorithmic input, individuals may feel compelled to alter their behavior in order to gain a system's approval.",
"Models of agent responsiveness, termed \"strategic manipulation,\" analyze the interaction between a learner and agents in a world where all agents are equally able to manipulate their features in an attempt to \"trick\" a published classifier.",
"In cases of real world classification, however, an agent's ability to adapt to an algorithm is not simply a function of her personal interest in receiving a positive classification, but is bound up in a complex web of social factors that affect her ability to pursue certain action responses.",
"In this paper, we adapt models of strategic manipulation to capture dynamics that may arise in a setting of social inequality wherein candidate groups face different costs to manipulation.",
"We find that whenever one group's costs are higher than the other's, the learner's equilibrium strategy exhibits an inequality-reinforcing phenomenon wherein the learner erroneously admits some members of the advantaged group, while erroneously excluding some members of the disadvantaged group.",
"We also consider the effects of interventions in which a learner subsidizes members of the disadvantaged group, lowering their costs in order to improve her own classification performance.",
"Here we encounter a paradoxical result: there exist cases in which providing a subsidy improves only the learner's utility while actually making both candidate groups worse-off--even the group receiving the subsidy.",
"Our results reveal the potentially adverse social ramifications of deploying tools that attempt to evaluate an individual's \"quality\" when agents' capacities to adaptively respond differ."
],
[
"Introduction",
"The expanding realm of algorithmic decision-making has not only altered the ways that institutions conduct their day-to-day operations, but has also had a profound impact on how individuals interface with these institutions.",
"It has changed the ways we communicate with each other, receive crucial resources, and are granted important social and economic opportunities.",
"Theoretically, algorithms have great potential to reform existing systems to become both more efficient and equitable, but as exposed by various high-profile investigations [1], [2], [3], [4], prediction-based models that make or assist with consequential decisions are, in practice, highly prone to reproducing past and current patterns of social inequality.",
"While few algorithmic systems are explicitly designed to be discriminatory, there are many underlying forces that drive such socially biased outcomes.",
"For one, since most of the features used in these models are based on proxy, rather than causal, variables, outputs often reflect the various structural factors that bear on a person's life opportunities rather than the individualized characteristics that decision-makers often seek.",
"Much of the previous work in algorithmic fairness has examined a particular undesirable proxy effect in which a classifier's features may be linked to socially significant and legally protected attributes like race and gender, interpreting correlations that have arisen due to centuries of accumulated disadvantage as genuine attributes of a particular category of people [5], [6], [7], [8].",
"But algorithmic models do not only generate outcomes that passively correlate with social advantages or disadvantages.",
"These tools also provoke a certain type of reactivity, in which agents see a classifier as a guide to action and actively change their behavior to accord with the algorithm's preferences.",
"On this view, classifiers both evaluate and animate their subjects, transforming static data into strategic responses.",
"Just as an algorithm's use of certain features differentially advantages some populations over others, the room for strategic response that is inherent in many automated systems also naturally favors social groups of privilege.",
"Admissions procedures that heavily weight SAT scores motivate students who have the means to take advantage of test prep courses and even take the exam multiple times.",
"Loan approval systems that rely on existing lines of credit as an indication of creditworthiness encourage those who can to apply for more credit in their name.",
"Thus an algorithm that scores applicants to determine how a resource should be allocated sets a standard for what an ideal candidate's features ought to be.",
"A responsive subject would look to alter how she appears to a classifier in order to increase her likelihood of gaining the system's approval.",
"But since reactivity typically requires informational and material resources that are not equally accessible to all, even when an algorithm draws on features that seem to arise out of individual effort, these metrics can be skewed to favor those who are more readily able to alter their features.",
"In the machine learning literature, agent reactivity to a classifier is termed “strategic manipulation.” Since previous work in strategic classification has typically depicted agent-classifier interactions as antagonistic, such actions are usually viewed as distortions that aim to undermine a learner's classifier [9], [10].",
"As shown in Hardt et al.",
"[10], a learner who anticipates these responses can, under certain formulations of agent costs, adapt to protect against the misclassification errors that would have resulted from manipulation, recovering an accuracy level that is arbitrarily close to the theoretical maximum.",
"These results are welcome news for a learner who correctly assesses agents' best-responses.",
"Indeed in most strategic manipulation models, agents are depicted as equally able to pursue manipulation, allowing the learner who knows their costs to accurately preempt strategic responses.",
"While there are occasions in which agents do largely face homogenous costs—an even playing field—in many other social use cases of machine learning tools, agents do not encounter the same costs of altering the attributes that are ultimately observed and assessed by the classifier.",
"As such, in this paper we ask, “What are the effects of strategic classification and manipulation in a world of social stratification?” As in previous work in strategic classification, we cast the problem as a Stackelberg game in which the learner moves first and publishes her classifier before candidates best-respond and manipulate their features [9], [10], [11], [12].",
"But in contrast with the models in Brückner & Scheffer [9] and Hardt et al.",
"[10], we formalize the setting of a society comprised of social groups that not only may differ in terms of distributions over unmanipulated features and true labeling functions but also face different costs to manipulation.",
"This extra set of differences brings to light questions that favor an analysis that focuses on the welfares of the candidates who must contend with these classifiers: Do classifiers formulated with strategic behavior in mind impose disparate burdens on different groups?",
"If so, how can a learner mitigate these adverse effects?",
"The altered gameplay and outcomes of strategic classification beg questions of fairness that are intertwined with those of optimality.",
"Though our model is quite general, we obtain technical results that reveal important social ramifications of using classification in systems marked by deep inequalities and a potential for manipulation.",
"Our analysis shows that, under our model, even when the learner knows the costs faced by different groups, her equilibrium classifier will always act to reinforce existing inequalities by mistakenly excluding qualified candidates who are less able to manipulate their features while also mistakenly admitting those candidates for whom manipulation is less costly, perpetuating the relative advantage of the privileged group.",
"We delve into the cost disparities that generate such inevitable classification errors.",
"Next, we consider the impact of providing subsidies to lighten the burden of manipulation for the disadvantaged group.",
"We find that such an intervention can improve the learner's classification performance as well as mitigate the extent to which her errors are inequality-reinforcing.",
"However, we show that there exist cases in which providing subsidies enforces an equilibrium learner strategy that actually makes some individual candidates worse-off without making any better-off.",
"Paradoxically, in these cases, paying a subsidy to the disadvantaged group actually benefits only the learner while both candidate groups experience a welfare decline!",
"Further analysis of these scenarios reveals that, in many cases, all parties would have preferred a world in which manipulation of features was not possible for any candidates.",
"Our paper's agent-centric analysis views data points as representing individuals and classifications as impacting those individuals' welfares.",
"This orientation departs from the dominant perspective in learning theory, which privileges a vendor's predictive accuracy, and instead evaluates classification regimes in light of the social consequences of the outcomes they issue.",
"By incorporating insights and techniques from game theory and economics, domains that consider deeply the effects of various policies on agents' behaviors and outcomes, we hope to broaden the perspective that machine learning takes on socially-oriented tools.",
"Presenting more democratically-inclined analysis has been central to the field of algorithmic fairness, and we hope our work sheds new light on this generic setting of classification with strategic agents."
],
[
"Related Work",
"While many earlier approaches to strategic classification in the machine learning literature have tended to view learner-agent interactions as adversarial [13], [14], our work does not assume inherently antagonistic relationships, and instead, shares the Stackelberg game-theoretic perspective akin to that presented in Brückner & Scheffer [9] and built upon by Hardt et al.",
"[10].",
"Departing from these models' focus on static prediction and homogeneous manipulation costs, Dong et al.",
"[15] propose an online setting of strategic classification in which agents appear sequentially and have individual costs for manipulation that are unknown to the learner.",
"Unlike our work, they take a traditional learner-centric view, whereas our concerns are with the welfare of the candidates.",
"Agent features and potential manipulations in the face of a learner classifier can also be interpreted as serving informational purposes.",
"In the economics literature on signaling theory, agents interact with a principal—the counterpart to our learner—via signals that convey important information relevant to a particular task at hand.",
"Classic works, such as Spence's paper on job-market signaling, focus their analysis on the varying quality of information that signals provide at equilibrium [16].",
"The emphasis in our analysis on different group costs shares features with a recent update to the signaling literature by Frankel & Kartik [17], who also distinguish between natural actions, corresponding to unmanipulated features in our model, and “gaming\" ability, which operate similarly to our cost functions.",
"The connection between gaming capacity and social advantage is also explicitly discussed in work by Esteban & Ray [18] who consider the effects of wealth and lobbying on governmental resource allocation.",
"While most works in the economics signaling literature center on the decay of the informativeness of signals as gaming and natural actions become indistinguishable, some recent work in computer science has also considered the effect of costly signaling on mechanism design [19], [20].",
"In contrast to both of these perspectives, our work highlights the effect of manipulation on a learner's action and as a consequence, on the agents' welfares.",
"In independent, concurrent work appearing at the same conference, Milli et al.",
"[21] also consider the social impacts of strategic classification.",
"Whereas our model highlights the interplay between a learner's Stackelberg equilibrium classifier and agents' best-response manipulations at the feature level, their work traces the relationship between the learner's utility and the social burden, a measure of agents' manipulation costs.",
"They show that an institution must select a point on the outcome curve that trades off its predictive accuracy with the social burden it imposes.",
"In their model, an agent with an unmanipulated feature vector $\\mathbf {x}$ has a likelihood $\\ell (\\mathbf {x})$ of having a positive label and can manipulate to any vector $\\mathbf {y}$ with $\\ell (\\mathbf {y}) \\le \\ell (\\mathbf {x})$ at zero cost, or to $\\mathbf {y}$ with $\\ell (\\mathbf {y}) > \\ell (\\mathbf {x})$ for a positive cost.",
"This assumption, called “outcome monotonicity,\" allows them to reason about manipulations in (one-dimensional) likelihood space rather than feature space, since the optimal learner strategies amount to thresholds on likelihoods.",
"In contrast, we allow features to be differently manipulable (perhaps a student can boost her SAT score via test prep courses, but can do nothing to change her grades from the previous year, and cannot freely obtain a higher SAT score in exchange for a worse record of extracurricular activities), which affects the forms of both the learner's equilibrium classifier and agents' best-response manipulations.",
"Despite these differences in model and focus, their analysis yields results that are qualitatively similar to ours.",
"Highlighting the differential impact of classifiers on social groups, they also find that overcoming stringent thresholds is more burdensome on the disadvantaged group."
],
[
"Model Formalization",
"As in Brückner & Scheffer [9] and Hardt et al.",
"[10], we formalize the Strategic Classification Game as a Stackelberg competition in which the learner moves first by committing to and publishing a binary classifier $f$ .",
"Candidates, who are endowed with “innate” features, best respond by manipulating their feature inputs into the classifier.",
"Formally, a candidate is defined by her $d$ -dimensional feature vector $\\mathbf {x}\\in X = [0,1]^d$ and group membership $A$ or $B$ , with $A$ signifying the advantaged group and $B$ the disadvantaged.",
"Group membership bears on manipulation costs such that a candidate from group $m$ who wishes to move from a feature vector $\\mathbf {x}$ to a feature vector $\\mathbf {y}$ must pay a cost of $c_m(\\mathbf {y}) - c_m(\\mathbf {x})$ .",
"We note that these cost function forms are similar to the class of separable cost functions considered in Hardt et al.",
"[10].",
"We assume that higher feature values indicate higher quality to the learner, and thus restrict our attention to manipulations such that $\\mathbf {y}\\ge \\mathbf {x}$ , where the symbol $\\ge $ signifies a component-wise comparison such that $\\mathbf {y}\\ge \\mathbf {x}$ if and only if $\\forall i \\in [d]$ , $y_i \\ge x_i$ .",
"Throughout this paper, we study non-negative monotone cost functions such that the cost of manipulating from a feature vector $\\mathbf {x}$ to a feature vector $\\mathbf {y}$ increases as $\\mathbf {x}$ and $\\mathbf {y}$ get further apart.",
"To motivate this distinction between features and costs, consider the use of SAT scores as a signal of academic preparedness in the U.S. college admissions process.",
"The high-stakes nature of the SAT has encouraged the growth of a test prep industry dedicated to helping students perform better on the exam.",
"Test preparation books and courses, while also exposing students to content knowledge and skills that are covered on the SAT, promise to “hack\" the exam by training students to internalize test-taking strategies based on the format, structure, and style of its questions.",
"One can view SAT scores as a feature used by a learner building a classifier to select candidates with sufficient academic success according to some chosen standard.",
"The existence of test prep resources then presents an opportunity for some applicants to inflate their scores, which might “trick” the tool into classifying the candidates as more highly qualified than they are in actuality.",
"In this example, a candidate's strategic manipulation move refers to her investment in these resources, which despite improving her exam score, do not confer any genuine benefits to her level of academic preparation for college.",
"Just as access to test prep resources tends to fall along income and race lines, we view candidates' different abilities to manipulate as tied to their group membership.",
"We model these group differences with respect to availability of resources and opportunity by enforcing a cost condition that orders the two groups.",
"We suppose that for all $\\mathbf {x}\\in [0,1]^d$ and $\\mathbf {y}\\ge \\mathbf {x}$ , $ c_A(\\mathbf {y}) - c_A(\\mathbf {x}) \\le c_B(\\mathbf {y}) - c_B(\\mathbf {x}).$ Manipulating from a feature vector $\\mathbf {x}$ to $\\mathbf {y}$ is always at least as costly for a member of group $B$ as it is for a member of group $A$ .",
"We believe our model's inclusion of this cost condition reflects an authentic aspect of our social world wherein one group is systematically disadvantaged with respect to a task in comparison to another.",
"In our setup, we also allow groups to have distinct probability distributions $\\mathcal {D}_A$ and $\\mathcal {D}_B$ over unmanipulated features and to be subject to different true labeling functions $ h_A$ and $h_B$ defined as $h_A(\\mathbf {x}) = {\\left\\lbrace \\begin{array}{ll} 1, & \\forall \\mathbf {x}\\text{ such that } \\sum _{i=1}^dw_{A, i} x_i \\ge \\tau _A , \\\\ 0, & \\forall \\mathbf {x}\\text{ such that } \\sum _{i=1}^dw_{A, i} x_i < \\tau _A , \\end{array}\\right.",
"}$ $h_B(\\mathbf {x}) = {\\left\\lbrace \\begin{array}{ll} 1, & \\forall \\mathbf {x}\\text{ such that } \\sum _{i=1}^dw_{B, i} x_i \\ge \\tau _B , \\\\ 0, & \\forall \\mathbf {x}\\text{ such that } \\sum _{i=1}^dw_{B, i} x_i < \\tau _B .",
"\\end{array}\\right.",
"}$ We assume that $h_A(\\mathbf {x}) = 1 \\Rightarrow h_B(\\mathbf {x}) = 1$ for all $\\mathbf {x}\\in [0, 1]$ .",
"Returning to the SAT example, research has shown that scores are skewed by race even before factoring in additional considerations such as access to manipulation [22].",
"In such cases, the true threshold for the disadvantaged group is lower than that for the advantaged group.",
"We leave this generality in our model to acknowledge and account for the influence that various social and historical factors have on candidates' unmanipulated features and not, we emphasize, as an endorsement of a view that groups are fundamentally different in ability.",
"A formal description of the Strategic Classification Game with Groups is given in the following definition.",
"Definition 1 (Strategic Classification Game with Groups) In the Strategic Classification Game with Groups, candidates with features $\\mathbf {x}\\in [0, 1]^d$ and group memberships $A$ or $B$ are drawn from distributions $\\mathcal {D}_A$ and $\\mathcal {D}_B$ .",
"The population proportion of each group is given by $p_A$ and $p_B$ where $p_A + p_B = 1$ .",
"A candidate from group $m$ pays cost $c_m(\\mathbf {y}) - c_m(\\mathbf {x})$ to move from her original features $\\mathbf {x}$ to $\\mathbf {y}\\ge \\mathbf {x}$ .",
"There exist true binary classifiers $h_A$ and $h_B$ , for candidates of each group.",
"Probability distributions, cost functions, and true binary classifiers are all common knowledge.",
"Gameplay proceeds in the following manner: The learner issues a classifier $f$ generating outcomes $\\lbrace 0, 1\\rbrace $ .",
"Each candidate observes $f$ and manipulates her features $\\mathbf {x}$ to $\\mathbf {y}\\ge \\mathbf {x}$ .",
"A group $m$ candidate with features $\\mathbf {x}$ who moves to $\\mathbf {y}$ earns a payoff $f(\\mathbf {y}) - (c_m(\\mathbf {y}) - c_m(\\mathbf {x})).$ The learner incurs a penalty of $\\begin{split}&C_{FP} \\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}[h_m(\\mathbf {x}) = 0, f(\\mathbf {y}) = 1] \\\\& + C_{FN}\\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}[h_m(\\mathbf {x}) = 1, f(\\mathbf {y}) = 0] ,\\end{split}$ where $C_{FP}$ and $C_{FN}$ denote the cost of a false positive and a false negative respectively.",
"The learner looks to correctly classify candidates with respect to their original features $\\mathbf {x}$ , whereas each candidate hopes to manipulate her features to attain a positive classification, expending as little cost as possible in the process.",
"Under this setup, candidates are only willing to manipulate their features if it flips their classification from 0 to 1 and if the cost of the manipulation is less than 1.",
"We note that defining the utility of a positive classification to be 1 can be considered a scaling and thus is without loss of generality.",
"This learner-candidate interaction is very similar to that studied in Hardt et al.",
"[10].",
"However, our inclusion of groups with distinct manipulation costs leads to an ambiguity regarding a candidate's initial features that does not exist when all candidates have an equal opportunity to manipulate.",
"In very few cases can a vendor distinguish among candidates based on their group membership for the explicit purpose of issuing distinct classification policies, especially if that group category is a protected class attribute.",
"As such, in our setup, we require that a learner publish a classifier that is not adaptive to different agents based on their group identities.",
"It is important to note that the positive results in Hardt et al.",
"'s [10] formulation of the Strategic Classification Game, wherein for separable cost functions, the learner can attain a classification error at test-time that is arbitrarily close to the optimal payoff attainable, do not carry over into this setting of heterogeneous groups and costs.",
"Even when $h_A = h_B$ , the existence of different costs of agent manipulation, even when separable as in our model, introduces a base uncertainty to the learning problem that generates errors that cannot be extricated so long as the learner must publish a classifier that does not distinguish candidates based on their group memberships.",
"Second, an analysis of the learner's strategy and performance, the perspective typically taken in most learning theory papers, contributes only a partial view of the total welfare effect of using classification in strategic settings.",
"The main objective of this paper is to offer a more thorough and holistic inspection of all agents' outcomes, paying special heed to the different outcomes experienced by candidates of the two groups.",
"Insofar as all social behaviors are impelled by goals, interests, and purposes, we should view data that is strategically generated to be the rule rather than the exception in social machine learning settings."
],
[
"Remark on the assumption that $h_A$ and {{formula:a34c465a-5361-42c0-b4ce-a530734946c8}} are known.",
"Our assumption that the learner has knowledge of groups' true labeling functions is not central to our analysis.",
"We make such an assumption to highlight the pure effect of groups' differential costs of manipulation on equilibrium gameplay and consequent welfares rather than the potential side effects due to a learner's noisy estimation of the true classifiers.",
"Our general findings do not substantially rely on this feature of the model, and the overall results carry through into a setting in which the learner optimizes from samples.",
"The differences in costs $c_A$ and $c_B$ encoded by the cost condition is not restricted to referring only to differences in the monetary cost of manipulation.",
"Instead, as is common in information economics and especially signaling theory, “cost” reflects the multiplicity of factors that bear on the effort exertion required by feature manipulation [16], [23], [24], [25].",
"To demonstrate the generality of our formulation of distinct group costs, we show that the cost condition given in (REF ) is equivalent to a more explicit derivation of the choice that an agent faces when deciding whether to manipulate her feature.",
"A rational agent with feature $\\mathbf {x}$ will only pursue manipulation if her value for a positive classification minus her cost of manipulation exceeds her value for a negative classification: $v(f(\\mathbf {x}) = 0) \\le v(f(\\mathbf {y}) = 1) - u(c(\\mathbf {y}) - c(\\mathbf {x})) .$ The monotone function $u$ translates the costs borne by a candidate to manipulate from $\\mathbf {x}$ to $\\mathbf {y}$ into her “utility space,\" i.e., it reflects the value that she places on that expenditure.",
"We can rewrite the previous inequality to be $c(\\mathbf {y}) - c(\\mathbf {x}) \\le u^{-1}\\big (v(f(\\mathbf {y}) = 1)- v(f(\\mathbf {x}) = 0)\\big ) .$ Substituting in $k = u^{-1}\\big (v(f(\\mathbf {y}) = 1)- v(f(\\mathbf {x}) = 0)\\big )$ , we have $c(\\mathbf {y}) - c(\\mathbf {x}) \\le k.$ Since the same cost expenditure is valued more highly by the disadvantaged group than by the advantaged group, the function $u$ is more convex for group $B$ than for group $A$ .",
"Thus all else equal, we have $c_A(\\mathbf {y}) - c_A(\\mathbf {x}) \\le c_B(\\mathbf {y}) - c_B(\\mathbf {x})$ as desired.",
"More generally, the functions $v$ , $c$ , and $u$ may each be different for the groups.",
"As such, the disadvantage encoded in the cost condition can arise due to differences in valuations of classifications ($v$ ), differences in costs ($c$ ), or differences in valuations of those costs ($u$ )."
],
[
"Equilibrium Analysis",
"We begin by studying agents' best-response strategies in the basic Strategic Manipulation Game with Groups in which candidates belong to one of two groups $A$ and $B$ , and the cost condition holds so that group $B$ members face greater costs to manipulation than group $A$ members.",
"To build intuition, we first consider best-response strategies in the one-dimensional case in which candidates have features $x \\in [0,1]$ and group cost functions are of any non-negative monotone form.",
"We then move on to consider the $d$ -dimensional case in which candidate features are given as vectors $\\mathbf {x}\\in [0,1]^d$ and manipulation costs are assumed to be linear."
],
[
"One-dimensional Features",
"In the $d=1$ case, the cost condition given in (REF ) may be written as $c_A^{\\prime }(x) \\le c_B^{\\prime }(x)$ for all $x \\in [0,1]$ .",
"Since the true decision boundaries are linear, in the one-dimensional case, they may be written as threshold functions where thresholds $\\tau _A$ and $\\tau _B$ are constants in $[0,1]$ and for agents in group $m$ , $h_m(x) = 1$ if and only if $x \\ge \\tau _m$ .",
"A university admissions decision based on a single score is an example of such a classifier.",
"Although the SAT does not act as the sole determinant of admissions in the U.S., in countries such as Australia, Brazil, and China, a single exam score is often the only factor of applicant quality that is considered for admissions.",
"When the learner has access to $\\tau _A$ and $\\tau _B$ , and group costs $c_A$ and $c_B$ satisfy the cost condition, the following proposition characterizes the space of undominated strategies for the learner who seeks to minimize any error-penalizing cost function.",
"Proposition 1 (One-D Undominated Learner Strategies) Given group cost functions $c_A$ and $c_B$ and true label thresholds $\\tau _A$ and $\\tau _B$ where $\\tau _B \\le \\tau _A$ , there exists a space of undominated learner threshold strategies $[\\sigma _B, \\sigma _A] \\subset [0, 1]$ where $\\sigma _A = c_A^{-1}(c_A(\\tau _A)+1)$ and $\\sigma _B = c_B^{-1}(c_B(\\tau _B)+1)$ .",
"That is, for any error penalties $C_{FP}$ and $C_{FN}$ , the learner's equilibrium classifier $f$ is based on a threshold $\\sigma \\in [\\sigma _B, \\sigma _A]$ such that for all manipulated features $y$ , $f(y) = {\\left\\lbrace \\begin{array}{ll} 1, & \\forall y \\ge \\sigma , \\\\ 0, & \\forall y < \\sigma .",
"\\end{array}\\right.",
"}$ To understand this result, first notice that if the learner were to face only those candidates from group $A$ , she would achieve perfect classification by labeling as 1 only those candidates with unmanipulated feature $x \\ge \\tau _A$ .",
"This strategy is enacted by considering candidates' best-response manipulations.",
"A rational candidate would only be willing to manipulate her feature if the gain she receives in her classification exceeds her costs of manipulation.",
"The learner would like to guard against manipulations by candidates with $x < \\tau _A$ but still admit candidates with $x \\ge \\tau _A$ , so she considers the maximum manipulated feature $y$ that is attainable by a rational candidate with $x = \\tau _A$ who is willing to spend up to a cost of one in order to secure a better classification, as illustrated in Figure REF .",
"The maximum such $y$ value is $\\sigma _A$ , and thus, the learner sets a threshold at $\\sigma _A$ , admitting all those with $y \\ge \\sigma _A$ and rejecting all those with $y < \\sigma _A$ .",
"The same reasoning applies to a learner facing only group $B$ candidates, and the learner sets a threshold at $\\sigma _B$ , admitting all those candidates with $y \\ge \\sigma _B$ and rejecting all those with $y < \\sigma _B$ .",
"It can be shown that for all valid values of $\\tau _A, \\tau _B, c_A,$ and $c_B$ , necessarily $\\sigma _B \\le \\sigma _A$ .",
"Then all classifiers with threshold $\\sigma < \\sigma _B$ are dominated by $\\sigma _B$ , in the sense that for any arbitrary error penalties $C_{FP}$ and $C_{FN}$ , the learner would suffer higher costs by setting her threshold to be $\\sigma $ rather than $\\sigma _B$ .",
"In the same way, all thresholds $\\sigma > \\sigma _A$ are dominated by $\\sigma _A$ , thus leaving $[\\sigma _B, \\sigma _A]$ to be the space of undominated thresholds.",
"For an account of the full proof of this result (and all omitted proofs), see the appendix.",
"Even without committing to a particular learner cost function, the space of optimal strategies characterized in Proposition REF leads to an important consequence.",
"A rational learner in the Strategic Classification Game always selects a classifier that exhibits the following phenomenon: it mistakenly admits unqualified candidates from the group with lower costs and mistakenly excludes qualified candidates from the group with higher costs.",
"This result is formalized in Proposition REF .",
"To state the proposition, the following definition is instructive.",
"Whereas the true thresholds $\\tau _A$ and $\\tau _B$ are a function of unmanipulated features, the learner only faces candidate features that may have been manipulated.",
"In order to make these observed features commensurable with $\\tau _A$ and $\\tau _B$ , it is helpful for the learner to “translate” a candidate's possibly manipulated feature $y$ to its minimum corresponding original unmanipulated value.",
"Definition 2 (Correspondence with unmanipulated features) For any observed candidate feature $y \\in [0,1]$ , the minimum corresponding unmanipulated feature is defined as $\\begin{split}\\ell _A(y)= \\max \\lbrace 0, c_A^{-1}(c_A(y)-1)\\rbrace ,\\\\\\ell _B(y)= \\max \\lbrace 0, c_B^{-1}(c_B(y)-1)\\rbrace \\end{split}$ for a candidate belonging to group A and group B respectively.",
"The corresponding values $\\ell _A(y)$ and $\\ell _B(y)$ are defined such that a candidate who presents feature $y$ must have as her true unmanipulated feature $x \\ge \\ell _A(y)$ if she is a group A member and $x \\ge \\ell _B(y)$ if she is a group B member.",
"Figure: Group cost functions for a one-dimensional feature xx.",
"τ A \\tau _A and τ B \\tau _B signify true thresholds on unmanipulated features for group AA and BB, but a learner must issue a classifier on manipulated features.",
"The threshold σ A \\sigma _A perfectly classifies group AA candidates; σ B \\sigma _B perfectly classifies group BB candidates.",
"A learner selects an equilibrium threshold σ * ∈[σ B ,σ A ]\\sigma ^* \\in [\\sigma _B, \\sigma _A], committing false positives on group AA (red bracket) and false negatives on group BB (blue bracket).Proposition 2 (Learner's Cost in 1 Dimension) A learner who employs a classifier $f$ based on a threshold strategy $\\sigma \\in [\\sigma _B, \\sigma _A]$ only commits false positives errors on group A and false negatives errors on group B.",
"The cost $C(\\sigma )$ of such a classifier is $ C_{FN} p_B P_{x\\sim \\mathcal {D}_B}\\big [x \\in [\\tau _B, \\ell _B(\\sigma ))\\big ] + C_{FP} p_A P_{x\\sim \\mathcal {D}_A} \\big [x \\in [\\ell _A(\\sigma ), \\tau _A)\\big ], $ where false negative errors entail penalty $C_{FN}$ , and false positive errors entail penalty $C_{FP}$ .",
"A learner who commits to classifying only one of the groups correctly bears costs given by the following corollaries.",
"Corollary 1 A classifier based on $\\sigma _A$ perfectly classifies group A candidates and bears cost $C(\\sigma _A) = C_{FN}p_BP_{x\\sim \\mathcal {D}_B}\\big [x \\in [\\tau _B, \\ell _B(\\sigma ))\\big ].$ Corollary 2 A classifier based on $\\sigma _B$ perfectly classifies group B candidates and bears cost $C(\\sigma _B) = C_{FP} p_AP_{x\\sim \\mathcal {D}_A} \\big [x \\in [\\ell _A(\\sigma ), \\tau _A)\\big ].$ Notice that the learner's errors always cut in the same direction—by unduly benefiting group $A$ candidates and unduly rejecting group $B$ candidates, these errors act to reinforce the existing social inequality that had generated the unequal group cost conditions in the first place.",
"Since these errors arise out of the asymmetric group costs of manipulation, the Strategic Classification Game can be viewed as an interactive model that itself perpetuates the relative advantage of group A over group B candidates.",
"Within the undominated region $[\\sigma _B, \\sigma _A]$ , the equilibrium learner threshold $\\sigma ^*$ is attained as the solution to the optimization problem $\\sigma ^* = \\operatornamewithlimits{arg\\,min}_{\\sigma \\in [\\sigma _B, \\sigma _A]} C(\\sigma ).$ In the game's greatest generality where candidates are drawn from arbitrary probability distributions, groups bear any costs that abide by the cost condition, and the learner has arbitrary error penalties, one cannot specify the equilibrium learner threshold $\\sigma ^*$ any further.",
"However, under some special cases of candidate cost functions and probability distributions, the equilibrium threshold can be characterized more precisely.",
"Specifically, when candidates from both groups are assumed to be drawn from a uniform distribution over unmanipulated features in $[0,1]$ , an error-minimizing learner seeks a threshold value $\\sigma ^*$ that minimizes the length of the interval of errors, given by the following quantity: $ \\sigma ^* = \\operatornamewithlimits{arg\\,min}_{\\sigma \\in [\\sigma _B, \\sigma _A]} \\ell _B(\\sigma ) - \\ell _A(\\sigma ) .$ From here, one natural assumption of candidate cost functions would have that groups $A$ and $B$ bear costs that are proportional to each other.",
"In this case, the curvature of the cost functions is determinative of a learner's equilibrium threshold.",
"Proposition 3 Suppose group cost functions are proportional such that $c_A(x) = q c_B(x)$ for $q \\in (0,1)$ , that $\\mathcal {D}_A$ and $\\mathcal {D}_B$ are uniform on $[0,1]$ , and that $C_{FN} = C_{FP}$ and $p_A = p_B = \\frac{1}{2}$ .",
"Let $\\sigma ^*$ be the learner's equilibrium threshold.",
"When cost functions are strictly concave, $\\sigma ^* = \\sigma _B$ .",
"When cost functions are strictly convex, $\\sigma ^* = \\sigma _A$ .",
"When cost functions are affine, the learner is indifferent between all $\\sigma ^* \\in [\\sigma _B, \\sigma _A]$ ."
],
[
"General $d$ -Dimensional Feature Vectors",
"In the general $d$ -dimensional case of the Strategic Classification Game, candidates are endowed with features that are given by a vector $\\mathbf {x}\\in [0,1]^d$ and can choose to manipulate and present any feature $\\mathbf {y}\\ge \\mathbf {x}$ to the learner.",
"In this section, we consider optimal learner and candidate strategies when group costs are linear such that they may be written as $c_A(\\mathbf {x}) = \\sum _{i=1}^d c_{A, i} x_i;\\hspace{10.0pt} c_B(\\mathbf {x}) = \\sum _{i=1}^d c_{B, i} x_i$ for groups $A$ and $B$ respectively.",
"Now, the cost condition $c_A(\\mathbf {y}) - c_A(\\mathbf {x}) \\le c_B(\\mathbf {y}) - c_B(\\mathbf {x})$ for all $\\mathbf {y}\\ge \\mathbf {x}$ —defined component-wise as before—implies that $\\forall i \\in [d]$ , $c_{A,i} \\le c_{B,i}$ .",
"In $d$ dimensions, the true classifiers $h_A$ and $h_B$ have linear decision boundaries such that for a group $A$ candidate with feature $\\mathbf {x}$ , $h_A(\\mathbf {x}) = {\\left\\lbrace \\begin{array}{ll}1 & \\sum _{i=1}^d w_{A, i}x_i \\ge \\tau _A ,\\\\0 & \\sum _{i=1}^d w_{A, i}x_i <\\tau _A ,\\end{array}\\right.",
"}$ and for a group $B$ candidate with feature $\\mathbf {x}$ , $h_B(\\mathbf {x}) = {\\left\\lbrace \\begin{array}{ll}1 & \\sum _{i=1}^d w_{B, i}x_i \\ge \\tau _B ,\\\\0 & \\sum _{i=1}^d w_{B, i}x_i <\\tau _B.\\end{array}\\right.",
"}$ We assume that all components $x_i$ contribute positively to an agent's likelihood of being classified as 1 so that $w_{A, i}, w_{B, i} \\ge 0$ for all $i$ .",
"To ensure that the cost of manipulation is always non-negative, all cost coefficients are positive: $c_{B,i}, c_{A,i} \\ge 0$ for all $i \\in [d]$ .",
"A candidate may now manipulate any combination of the $d$ components of her initial feature $\\mathbf {x}$ to reach the final feature $\\mathbf {y}$ that she presents to the learner.",
"Despite this increased flexibility on the part of the candidate, we are still able to characterize the performance of undominated learner classifiers, generalizing the result in Proposition REF .",
"All potentially optimal classifiers exhibit the same inequality-reinforcing property inherent within the one-dimensional interval of undominated threshold strategies, trading off false positives on group A candidates with false negatives on group B candidates.",
"Before we formally present this result, we first describe candidates' best-response strategies.",
"Here, a geometric view of the space of potential manipulations is informative.",
"Suppose a candidate endowed with a feature vector $\\mathbf {x}$ faces costs $\\sum _{i=1}^d c_i x_i$ and is willing to expend a total cost of 1 for manipulation.",
"Then she can move to any $\\mathbf {y}\\ge \\mathbf {x}$ contained within the $d$ -simplex with orthogonal corner at $\\mathbf {x}$ and remaining vertices at $\\mathbf {x}+ \\frac{1}{c_i}\\mathbf {e}_i$ where $\\mathbf {e}_i$ is the $i$ th standard basis vector.",
"This region is given by $\\Delta (\\mathbf {x})= \\Big \\lbrace \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i}\\mathbf {e}_i \\in [0,1]^d \\Big | \\sum _{i=1}^d t_i \\le 1\\text{ ; } t_i \\ge 0 \\text{ }\\forall i \\Big \\rbrace .$ $\\Delta (\\mathbf {x})$ , depicted in Figure REF , gives the space of potential movement for a candidate with unmanipulated feature $\\mathbf {x}$ who is willing to expend a total cost of 1.",
"Notice that $t_i$ can be interpreted as the cost that a candidate expends on movement in the $i$ th direction.",
"Thus $\\sum _{i=1}^d t_i$ gives the total cost of manipulation.",
"Moving beyond the range of possible moves, in order to describe how a rational candidate will best-respond to a learner, we must consider the published classifier.",
"Figure: The forward simplex.",
"A candidate in group AA with unmanipulated feature vector 𝐱\\mathbf {x} can manipulate to reach any feature vector 𝐲∈Δ A (𝐱)\\mathbf {y}\\in \\Delta _A(\\mathbf {x}) at a cost of at most 1.Suppose a learner publishes a classifier $f$ based on a hyperplane $\\sum _{i=1}^d g_i y_i = g_0$ , so that $f(\\mathbf {y}) = 1$ if and only if $\\sum _{i=1}^d g_i y_i \\ge g_0$ .",
"A best-response manipulation occurs along the direction that generates the greatest increase in the value $\\sum _{i=1}^d g_i (y_i - x_i)$ for the least cost.",
"As such, a candidate will move in any directions $i \\in \\operatornamewithlimits{arg\\,max}_{i \\in [d]} \\frac{g_i}{c_i}$ .",
"This result is formalized in the following lemma.",
"Lemma 1 ($d$ -D Candidate Best Response) Suppose a learner publishes the classifier $f(\\mathbf {y}) = 1$ if and only if $\\sum _{i=1}^d g_i y_i \\ge g_0$ .",
"Consider a candidate with unmanipulated feature vector $\\mathbf {x}$ and linear costs $\\sum _{i=1}^d c_i x_i$ .",
"If $f(\\mathbf {x}) = 1$ or if for all $i \\in [d]$ , $f(\\mathbf {x}+ \\frac{1}{c_i} \\mathbf {e}_i) = 0$ , the candidate's best response is to set $\\mathbf {y}= \\mathbf {x}$ .",
"Otherwise, letting $K = \\operatornamewithlimits{arg\\,max}_{i \\in [d]} \\frac{g_i}{c_i}$ , her manipulation takes the form $y = \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i} \\mathbf {e}_i$ for any $\\mathbf {t}$ such that $t_i \\ge 0$ for all $i\\in [d]$ , $t_i = 0$ for all $i \\notin K$ , and $\\sum _{i=1}^d g_i( x_i + \\frac{t_i}{c_i}) = g_0$ .",
"While in the $d$ -dimensional case, a candidate has many more choices of manipulation directions to pursue, a best response strategy will always lead her to increase her feature in those components that are most valued by the learner and least costly for manipulation.",
"That is, she behaves according to a “bang for your buck\" principle, in which the optimal manipulations are in the direction or directions where the ratio $\\frac{g_i}{c_i}$ is highest.",
"Despite the fact that the optimal manipulation may not be unique, as in the cases where there are multiple equivalently good directions for a candidate to move in, a learner who knows candidates' costs can still anticipate best-response manipulations and avoid errors on that group.",
"As such, we are once again able to construct a perfect classifier for candidates of group $A$ and a perfect classifier for candidates of group $B$ .",
"Figure: A perfect classifier for group AA.",
"Every candidate with unmanipulated feature vector 𝐱\\mathbf {x} on or above the true decision boundary for group AA is able to manipulate to a point 𝐲∈Δ A (𝐱)\\mathbf {y}\\in \\Delta _A(\\mathbf {x}) on or above the blue decision boundary depicted here.",
"No candidate with an unmanipulated feature vector below the true decision boundary is able to do so.",
"The kink in the blue decision boundary arises due to the restriction of features to [0,1] d [0,1]^d.",
"A perfect classifier for group AA does not need to have this kink; for example, a more lenient perfect classifier can be formed by “straightening” it out.Theorem 1 ($d$ -D Space of Dominant Learner Strategies) In the general $d$ -dimensional Strategic Classification Game with linear costs, there exists a classifier that perfectly classifies group $A$ and a classifier that perfectly classifies group $B$ .",
"All undominated classifiers commit no false negative errors on group $A$ and no false positive errors on group $B$ .",
"A full exposition of the proof appears in the appendix, but here we present an abbreviated explanation of the result.",
"For each group $m$ , the learner computes an optimal boundary that perfectly classifies all of its members by considering the set of simplices $\\lbrace \\Delta _m(\\mathbf {x})\\rbrace $ anchored at the vectors $\\bar{\\mathbf {x}}$ that satisfy $\\mathbf {w}_m^\\intercal \\bar{\\mathbf {x}} = \\tau _m$ and drawing the strictest hyperplane that intersects each simplex.",
"That is for all hyperplanes $g_i: \\sum _{j=1}^d g_{i,j} x_j = g_{i, 0}$ that are constructed to intersect each simplex, then $g_1: \\sum _{j=1}^d g_{1,j} x_j = g_{1, 0}$ is the strictest if for all $\\mathbf {x}\\in [0,1]^d$ , $\\sum _{j=1}^d g_{1,j} x_j = g_{1, 0} \\Rightarrow \\sum _{j=1}^d g_{i,j} x_j = g_{i, 0}\\ge g_{j,0}$ for all $g_i$ .",
"Due to the cost ordering, for any $\\mathbf {x}\\in [0,1]^d$ , $\\Delta _B(\\mathbf {x}) \\subseteq \\Delta _A(\\mathbf {x})$ , and thus wherever a comparison is possible, the group $A$ boundary is at least as strict as the group $B$ boundary.",
"Figure REF gives a visualization of a boundary formed by connecting the simplices $\\Delta (\\bar{\\mathbf {x}})$ ; the corresponding classifier perfectly classifies the group.",
"As in the one-dimensional general costs case, learner strategies necessarily entail inequality-reinforcing classifiers: a rational learner equipped with any error-penalizing cost function will select an equilibrium strategy that trades off undue optimism with respect to group $A$ for undue pessimism with respect to group $B$ .",
"We note that except in the extreme case in which there exists a perfect classifier for all candidates in the population, this result implies that the classifier for group $A$ issues false negatives on group $B$ , and the classifier for group $B$ issues false positives on group $A$ .",
"In order to formalize this result, we would like to generalize the idea behind the minimum correspondence unmanipulated features given by $\\ell _A(\\cdot )$ and $\\ell _B(\\cdot )$ in (REF ) for general $d$ -dimensions and linear costs.",
"A learner who observes a possibly manipulated feature vector $\\mathbf {y}$ must consider the space of unmanipulated feature vectors that the candidate could have had.",
"Thus we can make use of the simplex idea of potential manipulation; however in this case, the learner seeks to project a simplex “backward\" to “undo” the potential candidate manipulation.",
"Since groups are subject to different costs, simplices $\\Delta _A^{-1}(\\mathbf {y})$ and $\\Delta _B^{-1}(\\mathbf {y})$ —a depiction is given in Figure REF —which represent the region from where a candidate could have manipulated, will differ based on the candidate's group membership, with $\\Delta _A^{-1}(\\mathbf {y})= \\Big \\lbrace \\mathbf {y}- \\sum _{i=1}^d \\frac{t_i}{c_{A,i}}\\mathbf {e}_i \\in [0,1]^d \\Big | \\sum _{i=1}^d t_i \\le 1\\text{ ; } t_i \\ge 0 \\text{ }\\forall i \\Big \\rbrace ,\\\\\\Delta _B^{-1}(\\mathbf {y})= \\Big \\lbrace \\mathbf {y}- \\sum _{i=1}^d\\frac{t_i}{c_{B,i}} \\mathbf {e}_i \\in [0,1]^d \\Big | \\sum _{i=1}^d t_i \\le 1\\text{ ; } t_i \\ge 0 \\text{ }\\forall i \\Big \\rbrace .$ We can now use these constructs in order to define $d$ -dimensional generalizations of $\\ell _A(\\mathbf {y})$ and $\\ell _B(\\mathbf {y})$ .",
"Definition 3 (Correspondence with Unmanipulated Features in $d$ -D) For any observed candidate feature $\\mathbf {y}\\in [0,1]^d$ , the minimum corresponding unmanipulated feature vectors are given by $\\ell _A(\\mathbf {y}) = \\big \\lbrace \\mathbf {x}\\in \\Delta _A^{-1}(\\mathbf {y}) \\cap [0,1]^d \\big | \\nexists \\hat{\\mathbf {x}} \\in \\Delta _A^{-1}(\\mathbf {y}) \\text{ such that } \\hat{\\mathbf {x}} < \\mathbf {x}\\big \\rbrace ,\\\\\\ell _B(\\mathbf {y}) = \\big \\lbrace \\mathbf {x}\\in \\Delta _B^{-1}(\\mathbf {y}) \\cap [0,1]^d \\big | \\nexists \\hat{\\mathbf {x}} \\in \\Delta _B^{-1}(\\mathbf {y}) \\text{ such that } \\hat{\\mathbf {x}} < \\mathbf {x}\\big \\rbrace $ for a candidate belonging to group $A$ and group $B$ respectively.",
"The corresponding values $\\ell _A(\\mathbf {y})$ and $\\ell _B(\\mathbf {y})$ are defined such that a candidate who presents feature $\\mathbf {y}$ must have had a true unmanipulated feature vector $\\mathbf {x}\\ge \\bar{\\mathbf {x}}$ for some $\\bar{\\mathbf {x}} \\in \\ell _A(\\mathbf {y})$ if she is a group $A$ member and $\\mathbf {x}\\ge \\bar{\\mathbf {x}}$ for some $\\bar{\\mathbf {x}} \\in \\ell _B(\\mathbf {y})$ if she is a group $B$ member.",
"Figure: The backward simplex.",
"A candidate in group AA with manipulated feature vector 𝐲\\mathbf {y} could have started with any feature vector 𝐱∈Δ A -1 (𝐲)\\mathbf {x}\\in \\Delta _A^{-1}(\\mathbf {y}) and paid a cost of at most 1.For any hyperplane decision boundary $g$ containing vectors $\\mathbf {y}$ , the minimum corresponding feature vectors given by $\\ell _A(\\mathbf {y})$ and $\\ell _B(\\mathbf {y})$ are helpful for determining the effective thresholds that $g$ generates on unmanipulated features for groups $A$ and $B$ .",
"Lemma 2 Suppose a learner classifier $f$ is based on a hyperplane $g: \\sum _{i=1}^d g_i x_i = g_0$ .",
"Construct the set $\\mathcal {L}_m(g) = \\left\\lbrace \\operatornamewithlimits{arg\\,min}_{\\mathbf {x}\\in \\ell _m(\\mathbf {y})} \\sum _{i=1}^d g_i x_i \\Big | \\forall \\mathbf {y}\\text{ s. t. } \\sum _{i=1}^d g_i y_i = g_0 \\right\\rbrace $ Then a group $m$ agent with feature $\\mathbf {x}$ can move to some $\\mathbf {y}$ with $f(\\mathbf {y}) = 1$ and $c_m(\\mathbf {y}) - c_m(\\mathbf {x}) \\le 1$ if and only if $\\mathbf {x}\\ge \\ell $ for some $\\ell \\in \\mathcal {L}_m(g)$ .",
"By definition, for any two $\\ell _1, \\ell _2 \\in \\mathcal {L}_m(g)$ , $\\sum _{i=1}g_i \\ell _{1,i} =\\sum _{i=1}g_i \\ell _{2,i} = g_0 -\\frac{g_{k_m}}{c_{m,k_m}},$ where $ k_m \\in \\operatornamewithlimits{arg\\,max}_{i=[d]} \\frac{g_{i}}{c_{m,i}}$ .",
"Thus a learner who cares only about the true label of presented features, will construct her decision boundary $g$ such that all $\\ell \\in \\mathcal {L}_m(g)$ have the same true label.",
"A cost-minimizing learner who publishes a classifier $f$ based on a hyperplane $g$ on manipulated features will commit errors on those candidates with unmanipulated features $\\mathbf {x}\\in [0,1]^d$ contained within the boundaries given by $\\mathcal {L}_A(g)$ and $\\mathcal {L}_B(g)$ .",
"This space can be understood as the $d$ -dimensional generalization of the $[\\ell _A(\\sigma ), \\ell _B(\\sigma )]$ error interval in one-dimension.",
"Proposition 4 (Learner's Cost in $d$ Dimensions) A learner who publishes an undominated classifier $f$ based on a hyperplane $\\mathbf {g}^\\intercal \\mathbf {x}= g_0$ can only commit false positives on group $A$ candidates and false negatives on group $B$ candidates.",
"The cost of such a classifier is $\\begin{split}&C_{FN} P_{x \\sim \\mathcal {D}_B}\\Big [ \\mathbf {x}\\in \\big (\\mathbf {g}^\\intercal \\mathbf {x}< g_0 - \\frac{g_{k_B}}{c_{k_B}} \\bigcap \\mathbf {w}_B^\\intercal \\mathbf {x}\\ge \\tau _{B}\\ \\big ) \\Big ] \\\\&+ C_{FP} P_{x \\sim \\mathcal {D}_A}\\Big [\\mathbf {x}\\in \\big (\\mathbf {w}_A^\\intercal \\mathbf {x}< \\tau _{A} \\bigcap \\mathbf {g}^\\intercal \\mathbf {x}\\ge g_0 - \\frac{g_{k_A}}{c_{k_A}} \\big ) \\Big ] ,\\end{split}$ where $k_B \\in \\operatornamewithlimits{arg\\,max}_{i\\in [d]} \\frac{g_i}{c_{B,i}}$ and $k_A \\in \\operatornamewithlimits{arg\\,max}_{i\\in [d]}\\frac{g_i}{c_{A,i}}$ ."
],
[
"Learner Subsidy Strategies",
"Since in our setting, the learner's classification errors are directly tied to unequal group costs, we ask whether she would be willing to subsidize group $B$ candidates in order to shrink the manipulation gap between the two groups and as a result, reduce the number of errors she commits.",
"In this section, we formalize subsidies as interventions that a learner can undertake to improve her classification performance.",
"Although in many high-stakes classification settings, the barriers that make manipulation differentially accessible are non-monetary—such as time, information, and social access—in this section, we consider subsidies that are monetary in nature to alleviate the financial burdens of manipulation.",
"We introduce these subsidies for the purpose of analyzing their effects on not only the learner's classification performance but also candidate groups' outcomes.",
"Since subsidies mitigate the inherent disparities in groups' costs and increase access to manipulation, one might expect that their implementation would surely improve group $B$ 's overall welfare.",
"In this section, we show that in some cases, optimal subsidy interventions can surprisingly have the effect of lowering the welfare of candidates from both groups without improving the welfare of even a single candidate."
],
[
"Subsidy Formalization",
"There are different ways in which a learner might choose to subsidize candidates costs.",
"In the main text of this paper, we focus on subsidies that reduce each group $B$ candidate's costs such that the agent need only pay a $\\beta $ fraction of her original manipulation cost.",
"Definition 4 (Proportional subsidy) Under a proportional subsidy plan, the learner pays a proportion $1-\\beta $ of each group B candidate's cost of manipulation for some $\\beta \\in [0, 1]$ .",
"As such, a group B candidate who manipulates from an initial feature vector $\\mathbf {x}$ to a final feature vector $\\mathbf {y}$ bears a cost of $\\beta \\big (c_B(\\mathbf {y}) - c_B(\\mathbf {x})\\big )$ .",
"In the appendix, we also introduce flat subsidies in which the learner absorbs up to a flat $\\alpha $ amount from each group $B$ candidate's costs, leaving the candidate to pay $\\max \\lbrace 0, c_B(\\mathbf {y}) - c_B(\\mathbf {x}) - \\alpha \\rbrace $ .",
"Similar results to those shown in this section hold for flat subsidies.",
"When considering proportional subsidies, the learner's strategy now consists of both a choice of $\\beta $ and a choice of classifier $f$ to issue.",
"The learner's goal is to minimize her penalty $\\begin{split}&C_{FP} \\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}\\big [h_m(\\mathbf {x}) = 0, f(\\mathbf {y}) = 1\\big ] \\\\& + C_{FN}\\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}\\big [h_m(\\mathbf {x}) = 1, f(\\mathbf {y}) = 0\\big ]+ \\lambda cost(f, \\beta ),\\end{split}$ where $ cost(f, \\beta )$ is the monetary cost of the subsidy, $C_{FP}$ and $C_{FN}$ denote the cost of a false positive and a false negative respectively as before, and $\\lambda \\ge 0 $ is some constant that determines the relative weight of misclassification errors and subsidy costs for the learner.",
"For ease of exposition, the remainder of the section is presented in terms of one-dimensional features.",
"In Section REF of the appendix, we show that in many cases, the $d$ -dimensional linear costs setting can be reduced to this one-dimensional setting.",
"As an analog of (REF ), we define $\\ell _B^\\beta (y) = (\\beta c_B)^{-1}(\\beta c_B(y) -1)$ , giving the minimum corresponding unmanipulated feature $x$ for any observed feature $y$ .",
"Under the proportional subsidy, for a given $y$ , the group $B$ candidate must have $x \\ge \\ell ^\\beta _B(y)$ .",
"From this, we define $\\sigma _B^\\beta $ such that $\\ell _{B}^\\beta (\\sigma _B^\\beta ) = \\tau _B$ .",
"In order to compute the cost of a subsidy plan, we must determine the number of group $B$ candidates who will take advantage of a given subsidy benefit.",
"Since manipulation brings no benefit in itself, candidates will only choose to manipulate and use the subsidy if it will lead to a positive classification.",
"For a published classifier $f$ with threshold $\\sigma $ , we then have $cost(f, \\beta ) = \\big (1-\\beta \\big )\\int _{\\ell _B^\\beta (\\sigma )}^\\sigma \\big (c_B(\\sigma ) - c_B(x)\\big ) P_{x\\sim \\mathcal {D}_B}(x) dx.$ Although the learner's optimization problem can be solved analytically for various values of $\\lambda $ , we are primarily interested in taking a welfare-based perspective on the effects of various classification regimes on both the learner and candidate groups.",
"In the following section, we analyze how the implementation of a subsidy plan can alter a learner's classification strategy and consider the potential impacts of such policies on candidate groups."
],
[
"Group Welfare Under Subsidy Plans",
"While a learner would choose to adopt a subsidy strategy primarily in order to reduce her error rate, offering cost subsidies can also be seen as an intervention that might equalize opportunities in an environment that by default favors those who face lower costs.",
"That is, if costs are keeping group $B$ down, then one might believe that reducing costs will surely allow group $B$ a fairer shot at manipulation, and, as a result, a fairer shot at positive classification.",
"Alas we find that mitigating cost disparities by way of subsidies does not necessarily lead to better outcomes for group $B$ candidates.",
"In fact, an optimal subsidy plan can actually reduce the welfares of both groups.",
"Paradoxically, in some cases, the subsidy plan boosts only the learner's utility, whereas every individual candidate from both groups would have preferred that she offer no subsidies at all.",
"The following theorem captures the surprising result that subsidies can be harmful to all candidates, even those from the group that would appear to benefit.",
"Theorem 2 (Subsidies can harm both groups) There exist cost functions $c_A$ and $c_B$ satisfying the cost conditions, learner distributions $\\mathcal {D}_A$ and $\\mathcal {D}_B$ , true classifiers with threshold $\\tau _A$ and $\\tau _B$ , population proportions $p_A$ and $p_B$ , and learner penalty parameters $C_{FN}$ , $C_{FP}$ , and $\\lambda $ , such that no candidate in either group has higher payoff at the equilibrium of the Strategic Classification Game with proportional subsidies compared with the equilibrium of the Strategic Classification Game with no subsidies, and some candidates from both group $A$ and group $B$ are strictly worse off.",
"We note that a slightly weaker version of the theorem holds for flat subsidies.",
"In particular, there exist cases in which some individual candidates have higher payoff at the equilibrium of the Strategic Classification Game with flat subsidies compared with the equilibrium with no subsidies, but both group $A$ and group $B$ candidates have lower payoffs on average with the subsidies.",
"To prove the theorem, it suffices to give a single case in which both candidate groups are harmed by the use of subsidies.",
"However, to illustrate that this phenomenon does not arise only as a rare corner case, we provide one such example here plus two in the appendix, and discuss general conditions under which this occurs.",
"In each example, we consider a particular instance of the Strategic Classification Game and compare the welfares of candidates at equilibrium when the learner is able to select a proportional subsidy with their welfares at equilibrium when no subsidy is allowed.",
"Example 1 Suppose that a learner is error-minimizing such that $C_{FN} = C_{FP} = 1$ and $\\lambda = \\frac{3}{4}$ .",
"Suppose that unmanipulated features for both groups are uniformly distributed with $p_A = p_B = \\frac{1}{2}$ .",
"Let group cost functions be given by $c_A(x) = 8\\sqrt{x} + x$ and $c_B(x) = 12\\sqrt{x}$ ; note that the cost condition $c^{\\prime }_A(x) < c^{\\prime }_B(x)$ holds for $x \\in [0,1]$ .",
"Let the true group thresholds be given by $\\tau _A = 0.4$ and $\\tau _B = 0.3$ .",
"When subsidies are not allowed, the learner chooses a classifier with threshold $\\sigma ^* = \\sigma _B \\approx 0.398$ at equilibrium.",
"This threshold perfectly classifies all candidates from group $B$ , while permitting false positives on candidates from group $A$ with features $x \\in [0.272, 0.4)$ .",
"If the learner decides to implement a proportional subsidies plan, at equilibrium the learner chooses a classifier with threshold $\\sigma ^*_{prop} = \\sigma _A \\approx 0.546$ and a subsidy parameter $\\beta ^* = 0.558$ .",
"Her new threshold now correctly classifies all members of group $A$ , while committing false negatives on group $B$ members with features $x \\in [0.3, 0.348)$ .",
"Some candidates in group $B$ are thus strictly worse-off, while none improve.",
"Without the subsidy offering, group B members had been perfectly classified, but now there exist some candidates who are mistakenly excluded.",
"Further, one can show that candidates who are positively classified must pay more to manipulate to the new threshold in spite of receiving the subsidy benefit.",
"This increased cost is due to the fact that the higher classification threshold imposes greater burdens on manipulation than the $\\beta $ subsidy alleviates.",
"Group $A$ candidates are also strictly worse-off since the threshold increase eliminates false positive benefits that some members had previously been granted in the no-subsidy regime.",
"Moreover, all candidates who manipulate must expend more to do so, since these candidates do not receive a subsidy payment.",
"Only the learner is strictly better off with the implementation of this subsidy plan.",
"Additional examples in the appendix show cases in which both groups experience diminished welfare when they bear linear costs.",
"Even when the learner has an error function that penalizes false negatives twice as harshly as false positives and thus is explicitly concerned with mistakenly excluding group B candidates, an equilibrium subsidy strategy can still make both groups worse-off.",
"We thus highlight two consequences of subsidy interventions: On the one hand, with reduced cost burdens, more candidates from the disadvantaged group should be able to manipulate to reach a positive classification.",
"However, subsidy payments also allow a learner to select a classifier that is at least as strict as the one issued without offering subsidies.",
"These are opposing forces, and these examples show that without needing to distort underlying group probability distributions or the learner's penalty function in extreme ways, the effect of mitigating manipulation costs may be outweighed by the overall impact of a stricter classifier.",
"This result can also be extended to show that a setup in which candidates are unable to manipulate their features at all can be preferred by all three parties—groups $A$ and $B$ as well as the learner—to both the manipulation and subsidy regimes.",
"We provide an informal statement of this proposition below and defer the interested reader to its formal statement and demonstration in the appendix.",
"Proposition 5 There exist general cost functions such that the outcomes issued by a learner's equilibrium classifier under a non-manipulation regime is preferred by all parties—the learner, group $A$ , and group $B$ —to outcomes that arise both under her equilibrium manipulation classifier and under her equilibrium subsidy strategy."
],
[
"Discussion",
"Social stratification is constituted by forms of privilege that exist along many different axes, weaving and overlapping to create an elaborate mesh of power relations.",
"While our model of strategic manipulation does not attempt to capture this irreducible complexity, we believe this work highlights a likely consequence of the expansion of algorithmic decision-making in a world that is marked by deep social inequalities.",
"We demonstrate that the design of classification systems can grant undue rewards to those who appear more meritorious under a particular conception of merit while justifying exclusions of those who have failed to meet those standards.",
"These consequences serve to exacerbate existing inequalities.",
"Our work also shows that attempts to resolve these negative social repercussions of classification, such as implementing policies that help disadvantaged populations manipulate their features more easily, may actually have the opposite effect.",
"A learner who has offered to mitigate the costs facing these candidates may be encouraged to set a higher classification standard, underestimating the deeper disadvantages that a group encounters, and thus serving to further exclude these populations.",
"However, it is important to note that these unintended consequences do not always arise.",
"A conscientious learner who offers subsidies to equalize the playing field can guard against such paradoxes by making sure to classify agents in the same way even when offering to mitigate costs.",
"Other research in signaling and strategic classification has considered models in which manipulation is desirable from the learner's point of view [17], [26].",
"Though this perspective diverges from the one we consider here, we acknowledge that there do exist cases in which manipulation serves to improve a candidate's quality and thus leads a learner to encourage such behaviors.",
"It is important to note, however, that although this account may accurately represent some social classification scenarios, differential group access to manipulation remains an issue, and in fact, cases in which manipulation genuinely improves candidate quality may present even more problematic scenarios for machine learning systems.",
"As work in algorithmic fairness has shown, feedback effects of classification can lead to deepening inequalities that become “justified\" on the basis of features both manipulated and “natural\" [27].",
"The rapid adoption of algorithmic tools in social spheres calls for a range of perspectives and approaches that can address a variety of domain-specific concerns.",
"Expertise from other disciplines ought to be imported into machine learning, informing and infusing our research in motivation, application, and technical content.",
"As such, our work seeks to investigate, from a theoretical learning perspective, some of the potential adverse effects of what sociology has called “quantification,\" a world increasingly governed by metrics.",
"In doing so, we bring in techniques from game theory and information economics to model the interaction between a classifier and its subjects.",
"This paper adopts a framework that tries to capture the genuine unfair aspects of our social reality by modeling group inequality in a population of agents.",
"Although this perspective deviates from standard idealized settings of learner-agent interaction, we believe that so long as machine learning tools are designed for deployment in the imperfect social world, pursuing algorithmic fairness will require us to explicitly build models and theory to address critical issues such as social stratification and unequal access."
],
[
"Acknowledgements",
"We thank Alex Frankel, Rupert Freeman, Manish Raghavan, Hanna Wallach, and Glen Weyl for constructive input and discussion on this project and related topics."
],
[
"Proof of Proposition ",
"We first construct the optimal learner classifier when facing only candidates of a single group.",
"Suppose the learner encounters only group $A$ candidates.",
"Then using her knowledge that the true classifier $h_A$ is based on a threshold $\\tau _A \\in [0,1]$ , she can construct a classifier that admits those candidates with scores $x\\ge \\tau _A$ and rejects candidates $x < \\tau _A$ .",
"Since the maximal manipulation cost that any candidate would be willing to undertake is 1, for all $x \\in [0,1]$ , $c_A(y) - c_A(x) \\le 1$ and therefore $y \\le c_A^{-1}(c_A(x)+1)$ Thus a candidate with feature $x = \\tau _A$ would be able to move to any feature $y \\le \\sigma _A$ where $\\sigma _A = c_A^{-1}(c_A(\\tau _A)+1)$ .",
"Repeating the same reasoning for group $B$ , a candidate with feature $x = \\tau _B$ would be willing to move to any feature $y \\le \\sigma _B$ where $\\sigma _B = c_B^{-1}(c_B(\\tau _B) + 1)$ .",
"Now we want to show that $[\\sigma _B, \\sigma _A]$ marks an interval of undominated strategies.",
"First we prove the ordering that $\\sigma _B \\le \\sigma _A$ for all cost functions $c_B$ and $c_A$ and all thresholds $\\tau _B \\le \\tau _A$ .",
"Recall that since $h_A(x) = 1 \\Rightarrow h_B(x) = 1$ , we have $\\tau _B \\le \\tau _A$ .",
"Although we cannot order $c_B(\\tau _B)$ and $c_A(\\tau _A)$ , we have, by monotonicity of $c_B$ $ c_B(\\tau _B) \\le c_B(\\tau _A).$ Let $\\Delta = c_B(\\tau _A) - c_B(\\tau _B)$ .",
"Notice that if $\\Delta \\ge 1$ , $c_B(\\tau _B) + 1 \\le c_B(\\tau _A)$ , and so $\\sigma _B = c_B^{-1}(c_B(\\tau _B) + 1 ) \\le \\tau _A < \\sigma _A , $ where the last inequality is due to monotonicity of $c_A$ .",
"Let us consider the $\\Delta \\in (0,1)$ case.",
"By the cost condition, we can write $c_B^{\\prime } (\\tau _A) \\ge c_A^{\\prime }(\\tau _A)$ .",
"This implies that $c_B^{-1}(c_B(\\tau _A) + 1 ) \\le c_A^{-1}(c_A(\\tau _A) + 1 )$ Substituting in $c_B(\\tau _A) = c_B(\\tau _B) + \\Delta $ , we have $c_B^{-1}( c_B(\\tau _B) + \\Delta + 1 ) \\le c_A^{-1}(c_A(\\tau _A) + 1 ) = \\sigma _A.$ By monotonicity of $c_B$ , the left hand side is $\\ge \\sigma _B$ , and we have that $\\sigma _B \\le \\sigma _A$ as desired.",
"Notice that for all $\\sigma < \\sigma _B$ , the learner commits false positive errors on candidates from group $B$ , since $\\sigma _B$ is optimal for group $B$ classification.",
"She commits more false positives on group $A$ candidates as well and does not commit any fewer false negatives because of the monotonicity of $c_B$ and $c_A$ .",
"Thus for any error function with $C_{FP} > 0$ , the threshold classifier $\\sigma _B$ dominates $\\sigma $ .",
"Similarly, for all $\\sigma > \\sigma _A$ , the learner commits false negative errors on candidates from group $A$ , since $\\sigma _A$ is optimal for group $A$ classification.",
"She also commits more false negatives on group $B$ while committing no fewer false positives.",
"Thus for any error function with $C_{FN} > 0$ , the threshold classifier $\\sigma _A$ dominates $\\sigma $ .",
"For all $\\sigma \\in [\\sigma _B, \\sigma _A]$ , the learner trades off false negatives on group $B$ for false positives on group $A$ , and we call this range of threshold strategies undominated.",
"$\\Box $"
],
[
"Proof of Proposition ",
"We compute the cost of a learner's threshold strategy $\\sigma \\in [\\sigma _B, \\sigma _A]$ by first examining its performance on each group individually.",
"Recall from Proposition REF that the optimal learner threshold that perfectly classifies all $B$ candidates is $\\sigma _B$ .",
"Thus for all threshold strategies based on $\\sigma \\in (\\sigma _B, \\sigma _A]$ , the learner commits false negative errors on group $B$ .",
"To compute which members of group $B$ are subject to these errors, consider a learner classifier $f$ based on a threshold $\\sigma $ .",
"In order to manipulate to reach the feature threshold $\\sigma $ , a group $B$ candidate must have an unmanipulated $x$ such that $ c_B(\\sigma ) - c_B(x) \\le 1, $ $ x \\ge c_B^{-1}(c_B(\\sigma ) + 1 ) = \\ell _B(\\sigma ).$ We know that $\\tau _B \\le \\ell _B(\\sigma )$ by monotonicity of $c_B$ , and thus for all group $B$ candidates with feature $x \\in [ \\tau _B, \\ell _B(\\sigma ) )$ , the learner issues classification $f(x) = 0$ , even though $h_B(x) = 1$ .",
"These are the false negative errors issued on group $B$ for which the learner bears cost $C_{FN} p_B P_{x \\sim \\mathcal {D}_B}\\big [x \\in [ \\tau _B, \\ell _B(\\sigma )) \\big ]$ Following the same reasoning, notice that since $\\sigma _A$ is the optimal threshold policy for a learner facing only group $A$ candidates, a classifier $f$ based on any $\\sigma \\in [\\sigma _B, \\sigma _A)$ commits false positive errors on some group $A$ candidates.",
"Then repeating the steps that we carried out for group $B$ , we see that for all group $A$ candidates with $x$ such that $x \\ge c_A^{-1}(c_A(\\sigma )+1 = \\ell _A(\\sigma )$ the classifier $f$ issues a positive classification; $f(x) = 1$ .",
"Since $\\ell _A(\\sigma ) \\le \\tau _A$ , candidates with features $x \\in [\\ell _A(\\sigma ), \\tau _A)$ , have true label $h_A(x) = 0$ , and the learner commits false positive errors that bear cost $C_{FP} p_A P_{x \\sim \\mathcal {D}_A}\\big [x \\in [\\ell _A(\\sigma ), \\tau _A) \\big ]$ Combining (REF ) and (REF ), the total cost of any classifier $f$ based on a threshold $\\sigma \\in [\\sigma _B, \\sigma _A]$ , we obtain our desired result.",
"$\\Box $"
],
[
"Proofs of Corollaries ",
"These results follow by considering strategies $\\sigma _B$ , which commits no errors on group $B$ and thus only bears the cost given in (REF ), and $\\sigma _A$ , which commits no errors on group $A$ and thus only bears the cost given in (REF ).",
"$\\Box $"
],
[
"Proof of Proposition ",
"Under the assumption of uniform feature distributions for both groups, minimizing a classifier's probability of error amounts to choosing the threshold $\\sigma $ as $\\operatornamewithlimits{arg\\,min}_{\\sigma \\in [\\sigma _B, \\sigma _A]} \\ell _B(\\sigma ) - \\ell _A(\\sigma ).$ With proportional group costs $c_A(x) = q c_B(x)$ for $q\\in (0,1)$ , we have that $\\ell _B^{\\prime }(\\sigma ) &= \\frac{(c_B)^{\\prime }(\\sigma )}{\\Big (c_B\\Big )^{\\prime }\\Big (\\big (c_B\\big )^{-1}\\big (c_B(\\sigma )-1\\big )\\Big )}\\\\&=\\frac{(c_B)^{\\prime }(\\sigma )}{\\Big (c_B\\Big )^{\\prime }\\Big (\\ell _B(\\sigma )\\Big )}$ and $\\ell _A^{\\prime }(\\sigma ) &= \\frac{(c_A)^{\\prime }(\\sigma )}{\\Big (c_A\\Big )^{\\prime }\\Big (\\big (c_A\\big )^{-1}\\big (c_A (\\sigma )-1\\big )\\Big )} \\\\&= \\frac{(q c_B)^{\\prime }(\\sigma )}{\\Big (q c_B\\Big )^{\\prime }\\Big (\\big (c_A\\big )^{-1}\\big (c_A (\\sigma )-1\\big )\\Big )} \\\\&= \\frac{(c_B)^{\\prime }(\\sigma )}{\\Big (c_B\\Big )^{\\prime }\\Big (\\big (c_A\\big )^{-1}\\big (c_A (\\sigma )-1\\big )\\Big )} \\\\&= \\frac{(c_B)^{\\prime }(\\sigma )}{\\Big (c_B\\Big )^{\\prime }\\Big (\\ell _A(\\sigma )\\Big )} .$ When $c_A$ and $c_B$ are strictly concave, since $\\ell _B(\\sigma ) > \\ell _A(\\sigma )$ , $(c_B)^{\\prime }(\\ell _A(\\sigma )) > (c_B)^{\\prime }(\\ell _B(\\sigma ))$ and therefore $\\ell _A^{\\prime }(\\sigma ) < \\ell _B^{\\prime }(\\sigma )$ for all $\\sigma \\in [\\sigma _B, \\sigma _A]$ , and the quantity $\\ell _B(\\sigma ) - \\ell _A(\\sigma )$ is monotonically increasing in $\\sigma $ .",
"Thus the optimal classifier threshold is $\\sigma ^* = \\sigma _B$ .",
"Similarly, when $c_A$ and $c_B$ are strictly convex, $\\ell _A^{\\prime }(\\sigma ) > \\ell _B^{\\prime }(\\sigma )$ for all $\\sigma \\in [\\sigma _B, \\sigma _A]$ , and the quantity $\\ell _B(\\sigma ) - \\ell _A(\\sigma )$ is monotonically decreasing in $\\sigma $ .",
"Thus the optimal classifier threshold is $\\sigma ^* = \\sigma _A$ .",
"Thus the optimal classifier threshold is $\\sigma ^* = \\sigma _A$ .",
"Finally, when $c_A$ and $c_B$ are affine, $\\ell _A^{\\prime }(\\sigma ) = \\ell _B^{\\prime }(\\sigma )$ for all $\\sigma \\in [\\sigma _B, \\sigma _A]$ , and the quantity $\\ell _B(\\sigma ) - \\ell _A(\\sigma )$ is constant for all $\\sigma \\in [\\sigma _B, \\sigma _A]$ .",
"Thus the learner is indifferent between all thresholds $\\sigma \\in [\\sigma _B, \\sigma _A]$ .",
"$\\Box $"
],
[
"Proof of Lemma ",
"Consider a candidate with unmanipulated feature $\\mathbf {x}\\in [0,1]^d$ and manipulation cost $\\sum _{i=1}^d c_i x_i$ who faces a classifier $f(\\mathbf {y})$ with linear decision boundary given by $\\sum _{i=1}^d g_i y_i = g_0$ .",
"Recall that the utility a candidate receives for presenting feature $\\mathbf {y}\\ge \\mathbf {x}$ is given by $f(\\mathbf {y}) - c(\\mathbf {x}, \\mathbf {y})$ .",
"When $f(\\mathbf {x}) = 1$ , it is trivial that the candidate's best response to select $\\mathbf {y}=\\mathbf {x}$ .",
"Notice that if for all $i \\in [d]$ , $f(\\mathbf {x}+ \\frac{1}{c_i}\\mathbf {e}_i) = 0$ , then we have that $\\mathbf {g}^\\intercal \\mathbf {x}+ \\frac{g_k}{c_k} < g_0$ , so $\\frac{c_k(g_0 - \\mathbf {g}^\\intercal \\mathbf {x})}{g_k} > 1$ The manipulation from $\\mathbf {x}$ to $\\mathbf {y}= \\mathbf {x}+ \\sum _{i\\in K} \\frac{t_i}{c_i}\\mathbf {e}_i$ such that $\\mathbf {g}^\\intercal \\mathbf {y}= g_0$ entails cost $c(\\mathbf {y}) - c(\\mathbf {x}) = \\sum _{i \\in K} t_i = \\frac{c_k(g_0 - \\mathbf {g}^\\intercal \\mathbf {x})}{g_k} > 1$ and manipulating to achieve a positive classification using only components in $K$ would require a cost $>$ 1.",
"By definition, keeping the sum $\\sum _{i\\in K} t_i$ , but selecting different $t_i$ such that some $i\\notin K$ , $t_i > 0 $ would yield an even lower value $\\mathbf {g}^\\intercal \\mathbf {x}+ \\sum _{i=1}^d \\frac{g_i t_i}{c_i}$ .",
"Thus manipulating from $\\mathbf {x}$ to $\\mathbf {y}$ such that $f(\\mathbf {y}) = 1$ entails a cost $c(\\mathbf {y}) - c(\\mathbf {x}) > 1$ , and the candidate would not move at all, since the utility for moving $1 - (c(\\mathbf {y}) - c(\\mathbf {x})) < 0$ makes her worse-off than being subject to a negative classification without expending any cost on feature manipulation.",
"Thus she selects $\\mathbf {y}= \\mathbf {x}$ .",
"Now we consider the case where $f(\\mathbf {x}) = 0$ and there exists $i \\in [d]$ such that $f(\\mathbf {x}+ \\frac{1}{c_i}\\mathbf {e}_i) = 1$ .",
"Let $k \\in K = \\operatornamewithlimits{arg\\,max}_{i\\in [d]} \\frac{g_i}{c_i}$ .",
"We prove that the best-response manipulation for candidates with these $\\mathbf {x}$ moves to $\\mathbf {y}= \\mathbf {x}+ \\sum _{i =1}^d \\frac{t_i}{c_i}\\mathbf {e}_i$ where $t_i \\ge 0$ , $t_j = 0$ for all $j\\notin K$ , and $\\mathbf {g}^\\intercal (\\mathbf {x}+ \\sum _{i \\in K}\\frac{t_i}{c_i}\\mathbf {e}_i) = g_0$ .",
"Note that such a $\\mathbf {y}$ may not be unique—there may be multiple best-response manipulated features that achieve the same candidate utility, since they all result in the same candidate cost, and thus regardless of choices $i \\in K$ , we have that $\\sum _{i \\in K} t_i = \\frac{c_k(g_0 - \\mathbf {g}^\\intercal \\mathbf {x})}{g_k}$ The utility of any move to $\\mathbf {y}$ satisfying (REF ) is given by $f(\\mathbf {y}^*) - c(\\mathbf {x}, {\\mathbf {y}}^*) = 1- \\sum _{i=1}t_i$ Let us pick any such $\\mathbf {y}$ and call it $\\mathbf {y}^*$ since we will show that all other manipulations that are not of the form given in (REF ) generate lower utility for the candidate than $\\mathbf {y}^*$ .",
"We now show that for any manipulation to $\\mathbf {y}$ , $\\sum _{i=1}^d t_i \\le 1$ .",
"By assumption, for some $i$ , we have $f(\\mathbf {x}+ \\frac{1}{c_i}\\mathbf {e}_i) = 1 \\Rightarrow \\mathbf {g}^\\intercal \\mathbf {x}+ \\frac{g_i}{c_i} \\ge g_0$ Thus by (REF ), we have that $ \\sum _{i \\in K} t_i \\le \\frac{c_k \\frac{g_i}{c_i}}{g_k}$ .",
"By definition of $k,$ this is at most one since $\\frac{g_k}{c_k} \\ge \\frac{g_i}{c_i}$ for all $i\\in [d]$ .",
"Suppose on the contrary that there exists another manipulated feature $\\hat{\\mathbf {y}} \\ne \\mathbf {y}^*$ that is optimal and is not of the form (REF ): $ f(\\hat{\\mathbf {y}}) - (c(\\hat{\\mathbf {y}}) -c(\\mathbf {x})) \\ge 1 - \\frac{c_k(g_0 - \\mathbf {g}^\\intercal \\mathbf {x})}{g_k} \\ge 0$ Then it must be the case that moving to $\\hat{\\mathbf {y}}$ achieves a positive classification with a lower cost burden.",
"We write $\\hat{\\mathbf {y}} =\\mathbf {x}+ \\sum _{i=1}\\hat{t}_i\\mathbf {e}_i$ where $\\mathbf {e}_i$ is the $i^{\\text{th}}$ standard basis vector, and $\\hat{t}_j = \\hat{y}_j-x_j$ to highlight the components that have been manipulated from $\\mathbf {x}$ to $\\hat{\\mathbf {y}}$ .",
"First, we suppose that $\\hat{\\mathbf {y}}$ is such that there exists some component $\\hat{\\mathbf {y}}_j > 0$ where $j \\notin K = \\operatornamewithlimits{arg\\,max}_{i\\in [d]} \\frac{g_i}{c_i}$ .",
"Now we construct a feature $\\hat{\\mathbf {y}}^{\\prime }$ by selecting this component, and decreasing $\\hat{t}_j = 0$ and increasing a component $k \\in K$ by $\\frac{c_j \\hat{t}_j}{c_k}$ .",
"That is $ \\hat{\\mathbf {y}} ^{\\prime } = \\hat{\\mathbf {y}} - \\hat{t}_j\\mathbf {e}_j +\\frac{c_j \\hat{t}_j}{c_k} \\mathbf {e}_k$ The cost of manipulation from $\\mathbf {x}$ to $\\hat{\\mathbf {y}} ^{\\prime }$ is the same as that for manipulation to $\\hat{\\mathbf {y}}$ : $c(\\hat{\\mathbf {y}}^{\\prime }) - c(\\mathbf {x}) = \\sum _{i=1}^d c_i \\hat{y}_i - \\hat{t}_j c_j + c_k \\frac{c_j \\hat{t}_j}{c_k} = \\sum _{i=1}^d c_i \\hat{y}_i $ Notice that now we have $\\sum _{i=1}^d g_i \\hat{y}^{\\prime }_i = \\sum _{i=1}^d g_i \\hat{y}_i - g_j \\hat{t}_j + \\frac{g_k c_j \\hat{t}_j}{c_k}> \\sum _{i=1}^d g_i \\hat{y}_i \\ge g_0.$ Thus the candidate can manipulate to $\\hat{\\mathbf {y}}^{\\prime }$ by expending the same cost with $\\sum _{i=1}^d g_i \\hat{y}^{\\prime }_i > g_0$ Then by continuity of $g$ , there must exist some $\\bar{\\mathbf {y}} \\le \\hat{\\mathbf {y}}^{\\prime }$ such that $\\sum _{i=1}^d g_i \\bar{y}_i \\in [g_0, \\sum _{i=1}^d g_i \\hat{y}^{\\prime }_i )$ .",
"Thus since costs are monotonically increasing, $c(\\mathbf {x}, \\bar{\\mathbf {y}}) < c(\\mathbf {x}, \\hat{\\mathbf {y}})$ and since $\\bar{\\mathbf {y}}$ reaches the same classification, and we have shown that $\\hat{\\mathbf {y}}$ could not have been optimal, which is a contradiction.",
"Now we consider the case where $\\hat{\\mathbf {y}} = \\mathbf {x}+ \\sum _{i=1}^d {\\hat{t}_i}\\mathbf {e}_i$ is such that $\\hat{t}_j = 0$ for all $j\\notin K$ , but $\\mathbf {g}^\\intercal \\hat{\\mathbf {y}} \\ne g_0$ .",
"If $\\mathbf {g}^\\intercal \\hat{\\mathbf {y}} < g_0$ , then $\\hat{\\mathbf {y}}$ is negatively classified and thus trivially receives a lower utility than manipulating to any feature $\\mathbf {y}$ that is positively classified and associated with total cost $\\sum _i t_i \\le 1$ .",
"If $\\mathbf {g}^\\intercal \\hat{\\mathbf {y}} > g_0$ , then there are two possibilities: If $c(\\hat{\\mathbf {y}}) - c(\\mathbf {x}) \\ge 1$ , then once again, she receives at most a utility of 0, and thus manipulating to $\\hat{\\mathbf {y}}$ is a suboptimal move.",
"If $c(\\hat{\\mathbf {y}}) - c(\\mathbf {x}) <1$ , then we show the optimal manipulation is the one that moves from $\\mathbf {x}$ to $\\mathbf {y}= \\mathbf {x}+ \\sum _{i=1} t_i \\mathbf {e}_i$ where $\\mathbf {g}^\\intercal \\mathbf {y}= g_0$ and $t_j = 0,$ $\\forall j \\notin K$ —the move dictated by (REF ).",
"This feature $\\mathbf {y}$ also achieves a positive classification, but we argue that it does so at a lower cost than $\\hat{\\mathbf {y}}$ .",
"Since $\\mathbf {g}^\\intercal \\hat{\\mathbf {y}} > g_0$ , we can define $\\Delta =\\mathbf {g}^\\intercal \\hat{\\mathbf {y}} - g_0 > 0 $ The manipulation from $\\mathbf {x}$ to $\\hat{\\mathbf {y}} - \\frac{\\Delta }{g_k}\\mathbf {e}_k$ for any choice of $k$ attains a higher utility since it receives the same classification since $\\mathbf {g}^\\intercal (\\hat{\\mathbf {y}} - \\frac{\\Delta }{c_k}\\mathbf {e}_k) = g_0$ but does so at a cost $c({\\mathbf {y}}) - c(\\mathbf {x})= c(\\hat{\\mathbf {y}}) - c(\\mathbf {x})- {\\Delta }$ Since we already showed that all manipulations to $\\mathbf {y}$ of the form given in (REF ) bear the same cost, then we have shown that all such $\\mathbf {y}$ are preferable to $\\hat{\\mathbf {y}}$ .",
"By monotonicity of $c({\\mathbf {y}}) - c(\\mathbf {x})$ and $\\sum _{i=1}^d g_i x_i$ , all manipulations with lower cost entail a negative classification and thus a lower utility, and such only those manipulations to $\\mathbf {y}$ are optimal.",
"$\\Box $"
],
[
"Proof of Theorem ",
"We first prove that a learner who has access to the linear decision boundary for the true classifier can construct a classifier that commits no errors on any candidates from a single group; thus, in our setting, perfect classifiers exist for groups $A$ and $B$ .",
"We then prove that all undominated classifiers commit no false positives on group $B$ and no false negatives on group $A$ .",
"Suppose true classifiers are given by $h_A$ and $h_B$ based on decision boundaries $\\sum _{i=1}^d w_{A, i} x_i = \\tau _A$ and $\\sum _{i=1}^d w_{B, i} x_i = \\tau _B$ , costs are $c_A(\\mathbf {x}) = \\sum _{i=1}^d c_{A, i} x_i$ and $c_B(\\mathbf {x}) = \\sum _{i=1}^d c_{B, i} x_i$ .",
"Claim 1: When facing candidates from a single group, a learner who has access to true decision boundary $\\sum _{i=1}^d w_{i} x_i = \\tau $ and manipulation costs $\\sum _{i=1}^d c_i x_i$ can construct a perfect classifier.",
"Consider those features $\\bar{\\mathbf {x}} \\in [0,1]^d$ that lie on the true decision boundary $\\sum _{i=1}^d w_{i} x_i = \\tau $ and thus have true labels 1.",
"For each of these $\\bar{\\mathbf {x}}$ , we construct $\\Delta (\\bar{\\mathbf {x}})$ as defined in (REF ) to represent the candidate's space of potential manipulation to form the set $\\lbrace \\Delta (\\bar{\\mathbf {x}})\\rbrace $ for all $\\bar{\\mathbf {x}}$ on the boundary.",
"Notice that when all candidates face the same cost, the set of $j^{\\text{th}}$ vertices of each of the simplices $\\Delta (\\bar{\\mathbf {x}})$ , given by $\\mathbf {v}_j(\\bar{\\mathbf {x}}) = \\bar{\\mathbf {x}} + \\frac{1}{c_j}\\mathbf {e}_j$ , are coplanar.",
"Each of these hyperplanes can be described as a set $\\left\\lbrace \\mathbf {y}: \\sum _{i=1}^d w_{i} y_i = \\tau + \\frac{w_j}{c_j}\\right\\rbrace .$ Let $k \\in \\operatornamewithlimits{arg\\,max}_j \\frac{w_j}{c_j}$ .",
"We define $g_1$ to be a notational shortcut for the hyperplane corresponding to feature $k$ , so $g_1 = \\left\\lbrace \\mathbf {y}: \\sum _{j=1}^d g_{1,j} y_j = g_{1,0} \\right\\rbrace ,$ where $g_{1,0} = \\tau + \\frac{w_k}{c_k}$ and $g_{1,i} = w_i$ for all $i \\in \\lbrace 1,...,d\\rbrace $ .",
"We define a classifier $f_1$ based on the hyperplane $g_1$ : $f_1(\\mathbf {y}) = {\\left\\lbrace \\begin{array}{ll}1 & \\sum _{j=1}^d g_{1,j} y_j \\ge g_{1,0} ,\\\\0 & \\sum _{j=1}^d g_{1,j} y_j < g_{1,0} .\\end{array}\\right.",
"}$ To show that $f_1$ is a perfect classifier of all candidates with these generic costs, we show that it commits no false positive errors and no false negative errors.",
"Notice that since $g_1$ was constructed to be precisely the hyperplane that contains all vertices $\\mathbf {v}_k(\\bar{\\mathbf {x}}) = \\bar{\\mathbf {x}} + \\frac{1}{c_k}$ of the simplices $\\Delta (\\bar{\\mathbf {x}})$ where $k \\in \\operatornamewithlimits{arg\\,max}_{j\\in [d]} \\frac{w_j}{c_j}$ , then all $\\bar{\\mathbf {x}}$ on the true decision boundary $\\sum _{i=1}^d w_{i} x_i = \\tau $ can indeed manipulate to $\\mathbf {v}_k(\\bar{\\mathbf {x}})$ and reach $g_1$ to gain a positive classification.",
"Similarly, all candidates with features $\\mathbf {x}$ such that $\\sum _{i=1}^d w_{i} x_i > \\tau $ , can move to the $k^{\\text{th}}$ vertex of the simplex $\\Delta (\\mathbf {x})$ given by $\\mathbf {v}_k({\\mathbf {x}}) = \\mathbf {x}+ \\frac{1}{c_k}\\mathbf {e}_k$ in order to be classified positively since $ \\sum _{i=1}^d w_{i}v_{k, i}({\\mathbf {x}}) > \\tau + \\frac{w_k}{c_k} \\Rightarrow \\sum _{i=1}^d g_{1, j} v_{k,i}({\\mathbf {x}})> g_{1,0} .$ Thus $f_1$ correctly classifies all these candidates positively and permits no false negatives.",
"Consider the optimal manipulation for all true negative candidates $\\mathbf {x}$ .",
"By Lemma REF , the optimal manipulation would be either to not move at all, guaranteeing a negative classification, or to move $\\mathbf {x}$ to some point $\\mathbf {y}= \\mathbf {x}+ \\sum _{i=1}^d\\frac{t_i}{c_i}\\mathbf {e}_i$ where $t_j=0$ for all $j \\notin \\operatornamewithlimits{arg\\,max}_{j\\in [d]}\\frac{g_{1,j}}{c_j}$ .",
"But since $\\sum _{i=1}^d w_{i} x_i < \\tau $ , then for all such $\\mathbf {y}$ , $ \\sum _{i=1}^d w_{i} y_i \\le \\sum _{i=1}^d w_{i} x_i + \\frac{w_k}{c_k} < \\tau + \\frac{w_k}{c_k} \\Rightarrow \\sum _{i=1}^d g_{1, j} y_j < g_{1,0}$ and thus the classifier based on the hyperplane $g_1$ also issues a classification $f_1(\\mathbf {x}) = 0$ and admits no false positives.",
"Thus we have shown that the hyperplane $g_1$ supports a perfect classifier $f_1$ as defined in (REF ).",
"Now we move on to group-specific claims, where groups have distinct costs and potentially distinct true decision boundaries, but we continue to use the constructions of $f_1$ and $g_1$ from Claim 1.",
"Claim 2: Let $f^A_1$ be the classifier based on boundary $g_1$ for group $A$ , and let $f^B_1$ be the classifier based on boundary $g_1$ for group $B$ , as in (REF ), but with group-specific costs and true decision boundary parameters.",
"Then $\\forall \\mathbf {y}\\in [0,1]^d$ , $f^A_1(\\mathbf {y}) =1 \\Rightarrow f^B_1(\\mathbf {y}) = 1.$ We first prove the claim for the case in which $h_A = h_B$ with decision bounday $\\sum _{i=1}^d w_i x_i = \\tau $ .",
"We then show that it also holds when the two are not equal.",
"By the cost condition that $c_A(\\mathbf {y}) - c_A(\\mathbf {x}) \\le c_B(\\mathbf {y}) - c_B(\\mathbf {x})$ for all $\\mathbf {x}\\in [0,1]^d$ and $\\mathbf {y}\\ge \\mathbf {x}$ , we know that for any given $\\mathbf {x}$ , $\\Delta _B(\\mathbf {x}) \\subseteq \\Delta _A(\\mathbf {x}).$ Let $k_A \\in \\operatornamewithlimits{arg\\,max}_{j\\in [d]} \\frac{w_j}{c_{A,j}}$ and $k_B \\in \\operatornamewithlimits{arg\\,max}_{j\\in [d]} \\frac{w_j}{c_{B,j}}$ , so that $g^A_1$ and $g^B_1$ are defined as $\\sum _{i=1}^d w_i y_i = \\tau + \\frac{w_{k_A}}{c_{A,k_A}} \\iff g^A_1: \\sum _{j=1}^d g^A_{1,j} y_j = g^A_{1,0}, $ $\\sum _{i=1}^d w_i y_i = \\tau + \\frac{w_{k_B}}{c_{B, k_B}} \\iff g^B_1: \\sum _{j=1}^d g^B_{1,j} y_j = g^B_{1,0} .$ Then since for all $i \\in [d]$ , $c_{A, i} \\le c_{B, i}$ , we must have that $\\tau + \\frac{w_{k_A}}{c_{A,k_A}} \\ge \\tau + \\frac{w_{k_B}}{c_{A,k_B}} \\ge \\tau + \\frac{w_{k_B}}{c_{B, k_B}},$ and thus $g^A_{1, 0} \\ge g^B_{1,0}$ .",
"Since $f^A_1$ is the classifier based on $g^A_1$ and $f^B_1$ is based on $g^B_1$ , we have that $\\forall \\mathbf {y}\\in [0,1]^d$ , $f^A_1(\\mathbf {y}) = 1 \\Rightarrow f^B_1(\\mathbf {y}) = 1.$ Now consider the case in which $h_A$ and $h_B$ differ.",
"Recall the assumption $h_A(\\mathbf {x}) = 1 \\Rightarrow h_B(\\mathbf {x}) = 1$ for all $\\mathbf {x}\\in [0,1]^d$ .",
"Thus for all $\\mathbf {x}\\in [0,1]^d$ , $\\sum _{i=1} w_{A, i} x_i \\ge \\tau _A \\Rightarrow \\sum _{i=1} w_{B, i}x_i\\ge \\tau _B.$ Recall that the hyperplanes ${g}_1^A$ , ${g}_1^B$ are constructed as shifts of $\\sum _{i=1} w_{A, i} x_i \\ge \\tau _A$ and $\\sum _{i=1} w_{B, i}x_i\\ge \\tau _B$ by the set of simplices $\\lbrace \\Delta _A(\\bar{\\mathbf {x}}_A)\\rbrace $ and $\\lbrace \\Delta _B(\\bar{\\mathbf {x}}_B)\\rbrace $ for $\\bar{\\mathbf {x}}_A$ such that $\\sum _{i=1} w_{A, i} \\bar{x}_{A,i} = \\tau _A $ and $\\bar{\\mathbf {x}}_B$ such that $\\sum _{i=1} w_{B, i} \\bar{x}_{B,i} = \\tau _B$ .",
"Since $\\Delta _B(\\mathbf {x}) \\subseteq \\Delta _A(\\mathbf {x})$ , ${g}_1^A$ and ${g}_1^B$ support classifiers $f_1^A$ and $f_1^B$ such that $f_1^A(\\mathbf {y}) = 1\\Rightarrow f_1^B(\\mathbf {y}) = 1.$ Claim 3: All undominated classifiers commit no false negative errors on group $A$ members and no false positive errors on group $B$ members when candidates best respond.",
"Fix a classifier $f$ and consider a group $A$ candidate with true feature vector $\\bar{\\mathbf {x}}$ who manipulates to best response $\\bar{\\mathbf {y}}$ such that $h_A(\\bar{\\mathbf {x}}) = 1$ but $f(\\bar{\\mathbf {y}}) = 0$ .",
"Thus the classifier $f$ makes a false negative error on this candidate.",
"We show that we can construct another classifier $\\hat{f}$ that correctly classifies $\\bar{\\mathbf {x}}$ under its optimal manipulation with respect to $\\hat{f}$ .",
"We prove that $\\hat{f}$ commits no more errors than does $f$ and commits strictly fewer errors since it commits no false negatives on group $A$ candidates.",
"Construct the classifier $\\hat{f}$ such that $\\hat{f}(\\mathbf {y}) = {\\left\\lbrace \\begin{array}{ll}1 & f(\\mathbf {y}) = 1 \\text{ or } f^A_1(\\mathbf {y}) = 1, \\\\0 & \\text{ otherwise,}\\end{array}\\right.",
"}$ where $f^A_1(\\mathbf {y})$ is based on the boundary $\\sum _{j=1}g^A_{1,j}y_j = g^A_{1,0}$ .",
"We first argue that $f$ and $\\hat{f}$ make exactly the same set of false positive errors.",
"Consider a potential false positive error that $\\hat{f}$ issues on a candidate with feature $\\mathbf {x}$ from group $A$ .",
"Such a candidate cannot manipulate to a feature $\\mathbf {y}$ to “trick” classifier $f_1^A$ , since we have shown in Claim 1 that $f_1^A$ perfectly classifies all group $A$ candidates, and thus does not admit false positives.",
"Thus any potential false positive error must be due to $f(\\mathbf {y}) = 1$ , in which case $\\hat{f}$ and $f$ issue the same false positive error.",
"Now we consider a potential false positive error that $\\hat{f}$ issues on a candidate with feature $\\mathbf {x}$ from group $B$ .",
"By Claim 2, $f_1^A(\\mathbf {y}) = 1 \\Rightarrow f_1^B(\\mathbf {y}) = 1$ , and thus we would have that the candidate with feature $\\mathbf {x}$ was able to manipulate to some feature $\\mathbf {y}$ such that $f_1^B(\\mathbf {y}) = 1$ .",
"But this is a contradiction, since we know that $f_1^B$ commits no false positives on group $B$ members, and thus $f_1^A(\\mathbf {y})$ does not commit false positives on group $B$ .",
"Thus if $\\hat{f}$ commits a false positive, then it must be the case that $f$ committed the same false positive.",
"Consider a potential false negative error that $\\hat{f}$ issues on a candidate with feature $\\mathbf {x}$ from group $B$ .",
"Then it must be the case that $\\mathbf {x}$ can manipulate to some $\\mathbf {y}$ such that both $f(\\mathbf {y}) = 0$ and $f_1^A(\\mathbf {y})=0$ , and thus it be the case that ${f}$ commits the same false negative.",
"Lastly, consider a potential false negative error on a candidate from group $A$ .",
"By claim 1, this candidate must have been able to manipulate to some feature vector $\\mathbf {y}$ such that $f_1^A(\\mathbf {y}) = 1$ , since $f_1^A$ commits no errors on group $A$ members.",
"Thus when a candidate with unmanipulated feature $\\mathbf {x}$ can manipulate to some $\\mathbf {y}$ such that $f_1^A(\\mathbf {y}) = 1$ yet can only present a (possibly different) feature $\\mathbf {y}$ such that $f(\\mathbf {y}) =0$ , then $\\hat{f}$ correctly classifies this candidate positively, even when $f$ does not.",
"Thus $\\hat{f}$ makes no false negative errors on group $B$ .",
"Thus $\\hat{f}$ commits strictly fewer errors than $f$ —none of which are false negatives on group $A$ members—and $f$ is dominated by $\\hat{f}$ .",
"The second half of the claim can be proved through an analogous argument.",
"Combining Claims 1 and 3, we conclude that we can construct perfect classifiers for group $A$ that commit only false negative errors on group $B$ and perfect classifiers for group $B$ that commit only false positive errors on group $A$ .",
"$f_1^A$ and $f_1^B$ are examples of such classifiers, though they are not unique.",
"$\\Box $"
],
[
"Proof of Lemma ",
"$\\Rightarrow $ direction: Assume a group $m$ candidate with feature $\\mathbf {x}$ can move to $\\mathbf {y}$ such that $f(\\mathbf {y}) = 1$ and $c_m(\\mathbf {y}) - c_m(\\mathbf {x}) \\le 1$ , we show that necessarily $\\mathbf {x}\\ge \\ell $ for some $\\ell \\in \\mathcal {L}_m(g)$ .",
"If $\\mathbf {x}$ can move to $\\mathbf {y}$ , then $\\mathbf {x}\\in \\Delta ^{-1}(\\mathbf {y})$ .",
"By the definition of $\\ell _m(\\mathbf {y}), \\mathbf {x}\\ge \\bar{\\mathbf {x}}$ for some $\\bar{\\mathbf {x}} \\in \\ell _m(\\mathbf {y})$ .",
"Then by monotonicity of $g$ , we have that $\\sum _{i=1}^d g_i x_i \\ge \\sum _{i=1}^dg_i \\bar{x}_i \\ge \\min _{x\\in \\ell _m(\\mathbf {y})} \\sum _{i=1}^dg_i {x}_i $ Thus $\\mathbf {x}\\ge \\ell $ for some $\\ell \\in \\mathcal {L}_m(g)$ .",
"$$ direction: Assume some group $m$ candidate has feature $\\mathbf {x}\\ge \\ell $ for some $\\ell \\in \\mathcal {L}_m(g)$ .",
"Then she can move to some $\\mathbf {y}$ such that $f(\\mathbf {y}) = 1$ and $c_m(\\mathbf {y}) - c_m(\\mathbf {x}) \\le 1$ .",
"If $\\mathbf {x}\\ge \\ell $ for some $\\ell \\in \\mathcal {L}_m(g)$ , then $\\sum _{i=1}^d g_i x_i \\ge \\sum _{i=1}^d g_i \\ell _i ,$ where $\\ell \\in \\Delta ^{-1}_m(\\mathbf {y})$ for some $\\mathbf {y}$ such that $\\sum _{i=1}g_iy_i = g_0$ and $f(\\mathbf {y}) = 1$ .",
"Since $\\ell $ is defined as $\\operatornamewithlimits{arg\\,min}_{x \\in \\ell _m(\\mathbf {y})}\\sum _{i=1}g_ix_i$ , then we have $\\sum _{i=1}^d g_i \\ell _i = \\sum _{i=1}^d \\big (g_i y_i - \\max _{t_i} \\sum _{i=1}^d\\frac{g_i t_i}{c_{m,i}}\\big ) ,$ where $t_i \\ge 0$ and $\\sum _{i=1}t_i = 1$ as shown before.",
"Then substituting $\\sum _{i=1}^d g_i y_i = g_0$ , we have that $\\sum _{i=1} g_i \\ell _i + \\frac{g_{k_m}}{c_{m,k_m}} = g_0 ,$ where $ k_m \\in \\operatornamewithlimits{arg\\,max}_{i=[d]} \\frac{g_{i}}{c_i}$ .",
"Since $\\mathbf {x}\\ge \\ell $ , $\\mathbf {x}$ can also manipulate to some $\\mathbf {y}$ with $f(\\mathbf {y}) = 1$ , bearing a cost $\\le 1$ .",
"$\\Box $"
],
[
"Proof of Proposition ",
"If a learner publishes an undominated classifier $f$ , then by Theorem REF , the hyperplane $g: \\mathbf {g}^\\intercal \\mathbf {x}= g_0$ that supports this classifier can only commit inequality-reinforcing errors: only false positives on group $A$ members and only false negatives on group $B$ members.",
"As proved in Lemma REF , the set $\\mathcal {L}_m(g)$ determines the effective threshold on unmanipulated features $\\mathbf {x}$ for a candidate of group $m$ .",
"We have already shown that for any two $\\ell _1, \\ell _2 \\in \\mathcal {L}_m(g)$ , $\\sum _{i=1}g_i \\ell _{1,i} =\\sum _{i=1}g_i \\ell _{2,i} = g_0 -\\frac{g_{k_m}}{c_{k_m}}$ where $ k_m \\in \\operatornamewithlimits{arg\\,max}_{i=[d]} \\frac{g_{i}}{c_{m,i}}$ .",
"For any $\\ell \\in \\mathcal {L}_B(g)$ , we have $\\sum _{i=1}^dg_i\\ell _i + \\frac{g_{k_B}}{c_{B, k_B}}= g_0$ Thus combining these results, those group $B$ candidates with features $\\mathbf {x}\\in [0,1]^d$ in the intersection $ \\mathbf {g}^\\intercal \\mathbf {x}< g_0 - \\frac{g_{k_B}}{c_{B, k_B}} \\bigcap \\mathbf {w}_B^\\intercal \\mathbf {x}\\ge \\tau _B $ are classified as false negatives.",
"For group $A$ , we consider $\\ell \\in \\mathcal {L}_A(g)$ : $\\sum _{i=1}^dg_i\\ell _i + \\frac{g_{k_A}}{c_{A, k_A}}= g_0$ and thus group $A$ candidates with features $\\mathbf {x}\\in [0,1]^d$ in the intersection $\\mathbf {w}_A^\\intercal \\mathbf {x}< \\tau _A \\bigcap \\mathbf {g}^\\intercal \\mathbf {x}\\ge g_0 - \\frac{g_{k_A}}{c_{A, k_A}} $ are classified as false positives.",
"Thus the cost publishing $g$ is $&C_{FN} P_{x \\sim \\mathcal {D}_B}\\big [ \\mathbf {x}\\in \\big (\\mathbf {g}^\\intercal \\mathbf {x}< g_0 - \\frac{g_{k_B}}{c_{k_B}} \\bigcap \\mathbf {w}_B^\\intercal \\mathbf {x}\\ge \\tau _{B}\\ \\big ) \\big ] \\\\&+ C_{FP} P_{x \\sim \\mathcal {D}_A}\\big [\\mathbf {x}\\in \\big (\\mathbf {w}_A^\\intercal \\mathbf {x}< \\tau _{A} \\bigcap \\mathbf {g}^\\intercal \\mathbf {x}\\ge g_0 - \\frac{g_{k_A}}{c_{k_A}} \\big ) \\big ]$ $\\Box $"
],
[
"Reduction from the $d$ -dimensional setting to the one-dimensional setting",
"We first show that under certain conditions of a learner's equilibrium classifier strategy, a $d$ -dimensional subsidy analysis is equivalent to a one-dimensional subsidy analysis.",
"In general $d$ -dimensions, those features $\\mathbf {y}$ attainable from an unmanipulated feature $\\mathbf {x}\\in [0,1]^d$ , where $f(\\mathbf {x}) = 0$ , is given by $ \\mathbf {y}\\le \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i}\\mathbf {e}_i \\text{ where } \\sum _{i=1}^d t_i = 1$ where the right hand side gives the simplex $\\Delta (\\mathbf {x})$ of potential manipulation.",
"By Lemma REF , if a candidate moves from $\\mathbf {x}$ to $\\mathbf {y}\\ne \\mathbf {x}$ , then she selects $\\mathbf {t}$ such that $t_j=0$ for all $j\\notin K = \\operatornamewithlimits{arg\\,max}_{i=[d]}\\frac{g_i}{c_i}$ .",
"Staying within the simplex implies $\\sum _{i=1}^d t_i \\le 1$ .",
"Increasing the candidate's available cost to expend from 1 to $n$ increases her range of motion such that now she can move to any $ \\mathbf {y}\\le \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i}\\mathbf {e}_i \\text{ where } \\sum _{i=1}^d t_i = n$ She continues to manipulate in the spirit of Lemma REF —optimal moves entail choices of $\\mathbf {t}$ such that $t_j = 0$ for all $j\\notin K$ —however now, she is willing to manipulate if $\\exists i \\in [d]$ such that $f(\\mathbf {x}+ \\frac{n}{c_i}\\mathbf {e}_i) = 1$ and thus chooses $\\mathbf {t}$ such that $\\sum _{i=1}t_i \\le n$ .",
"Since offering a subsidy does not change the form of the group $B$ cost function, a candidate from group $B$ will pursue the same manipulation strategy given by the vector $\\mathbf {t}$ under subsidy regimes as long as the classifier's decision boundaries stay the same.",
"By definition, all such choices of $\\mathbf {y}$ resulting from a manipulation via $\\mathbf {t}$ have equivalent values $\\mathbf {g}^\\intercal \\mathbf {y}$ .",
"When costs are subsidized through a flat $\\alpha $ or a proportional $\\beta $ subsidy, a candidate with feature $\\mathbf {x}$ can manipulate to any $\\mathbf {y}_\\alpha , \\mathbf {y}_\\beta \\ge \\mathbf {x}$ that satisfies $\\mathbf {y}_\\alpha &\\in [\\mathbf {x}, \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i}\\mathbf {e}_i] \\text{ where } \\sum _{i=1}^d t_i = 1 + \\alpha \\\\\\mathbf {y}_\\beta &\\in [\\mathbf {x}, \\mathbf {x}+ \\sum _{i=1}^d \\frac{t_i}{c_i}\\mathbf {e}_i] \\text{ where } \\sum _{i=1}^d t_i = \\frac{1}{\\beta }$ We can pursue a dimensionality reduction by mapping each feature $\\mathbf {x}\\in [0,1]^d$ to $\\mathbf {g}^\\intercal \\mathbf {x}\\in \\mathbb {R}_+$ .",
"Rather than considering an optimal manipulation in $d$ -dimensions from $\\mathbf {x}$ to $\\mathbf {y}$ , we instead consider the relationship between the cost of the manipulation and the change from $\\mathbf {g}^\\intercal \\mathbf {x}$ to $\\mathbf {g}^\\intercal \\mathbf {y}$ : $\\sum _{i=1}^d c_i (y_i - x_i) \\iff \\sum _{i=1}^d {g_i}(y_i-x_i) $ where $g_i$ gives the coefficients of the linear decision boundary that supports $f$ , and $\\mathbf {x}$ optimally manipulates to $\\mathbf {y}$ .",
"We want to show that such a relationship is linear.",
"Consider optimal manipulations: If a candidate chooses not to manipulate at all, she will incur a cost of 0 and will also move from $\\sum _{i=1}^d g_i (y_i - x_i) = 0$ .",
"Since optimal manipulations (under any “budget\" constraint) only are along $k^{\\text{th}}$ components, a move from $\\mathbf {x}$ to $\\mathbf {y}$ always entails a total cost of $\\sum _{i\\in K}^d c_i(y_i - x_i) $ accompanied with $ \\sum _{i\\in K}^d g_i(y_i - x_i)=\\mathbf {g}^\\intercal (\\mathbf {y}- \\mathbf {x}) $ Thus we can write her total cost $c$ for a move from $\\mathbf {x}$ to $\\mathbf {y}$ as $\\frac{c_k}{g_k}(\\mathbf {g}^\\intercal \\mathbf {y}- \\mathbf {g}^\\intercal \\mathbf {x})$ for any $k \\in K$ .",
"Recall that by Lemma REF , optimal non-stationary manipulations move from $\\mathbf {x}$ to $\\mathbf {y}>\\mathbf {x}$ such that $\\sum _{i=1}^dg_iy_i= g_0$ , so in these cases, we can also write the above as $\\frac{c_k}{g_k} (g_0 - \\mathbf {g}^\\intercal \\mathbf {x}) $ Thus we can consider candidates' unmanipulated $d$ -dimensional features $\\mathbf {x}$ as one-dimensional features $\\mathbf {g}^\\intercal \\mathbf {x}$ and classifiers $f$ based on $d$ -dimensional hyperplanes $g: \\sum _{i=1}^d g_ix_i = g_0$ as imposing one-dimensional thresholds $g_0$ .",
"However a learner may also choose a different optimal subsidy strategy, thus publishing a classifier that now admits candidates differently.",
"Formally, suppose a learner first publishes a classifier $f_1$ based on a decision boundary $g_1: \\sum _{i=1}^d g_{1,i}x_i = g_{1,0}$ to which a candidate's optimal response follows the form given in Lemma REF with $k_1 \\in \\operatornamewithlimits{arg\\,max}_{i\\in [d]} \\frac{g_{1,i}}{c_i}$ .",
"If a learner then chooses to change her strategy when implementing a subsidy, thus publishing a different classifier $f_2$ based on decision boundary $g_2: \\sum _{i=1}^d g_{2,i} x_i = g_{2,0}$ , a candidate's optimal manipulation strategy will continue to adhere to Lemma REF , however, now, $k_2 \\in \\operatornamewithlimits{arg\\,max}_{i\\in [d]} \\frac{g_{2,i}}{c_i}$ .",
"Whereas the corresponding one-dimensional cost function $c(\\mathbf {y}) - c(\\mathbf {x})$ for best-response manipulations when facing classifier $f_1$ was given by $\\frac{c_{k_1}}{g_{1,k_1}} ( \\mathbf {g}_1^\\intercal (\\mathbf {y}- \\mathbf {x})) $ Her corresponding cost function when facing classifier $f_2$ is $\\frac{c_{k_2}}{g_{2,k_2}} (\\mathbf {g}_2^\\intercal (\\mathbf {y}-\\mathbf {x})) $ When these cost functions are the same, as when the coefficients $g_{1,i} = g_{2,i}$ for all $i$ , the agent's strategies when facing $f_1$ and $f_2$ are identical when reduced to one-dimension.",
"This case arises, for example, when the learner continues to perfectly classify a single group in both the non-subsidy regime and the subsidy regime.",
"In these cases, we can transition to considering just one-dimensional manipulations from $\\mathbf {g}^\\intercal \\mathbf {y}$ to $\\mathbf {g}^\\intercal \\mathbf {x}$ , where candidates bear linear costs of manipulation given in (REF )."
],
[
"Proof of Proposition ",
"Working from the subsidy and no-subsidy comparisons given in Proposition REF , we show that all three parties would have preferred the outcomes of a non-manipulation world to those in both of the manipulation cases.",
"To facilitate comparisons of welfare across classification regimes, we formalize group-wide utilities in the following definition.",
"Definition 5 (Group welfare under a proportional subsidy) The average welfare of group $B$ under classifier $f_{prop}$ and a proportional subsidy with parameter $\\beta $ is given by $\\begin{split}W_B(f_{prop}, \\beta ) = & \\int _{R_1} P_{x\\sim \\mathcal {D}_B}(x)dx\\\\&+ \\int _{R_2} \\big (1 - \\beta (c_B(y(x)) - c_B(x)) \\big ) P_{x\\sim \\mathcal {D}_B}(x)dx,\\end{split}$ $W_A(f_{prop}, 1) = & \\int _{R_1} P_{x\\sim \\mathcal {D}_A}(x)dx\\\\&+ \\int _{R_2} \\big (1 - (c_A(y(x)) - c_A(x)) \\big ) P_{x\\sim \\mathcal {D}_A}(x)dx,$ where $y(x)$ is the best response of a candidate with unmanipulated feature $x$ , $R_1$ sums over those candidates who are positively classified by $f_{prop}$ without expending any cost, and $R_2$ sums over those candidates who are positively classified after manipulating their features.",
"Since group $A$ members do not receive subsidy benefits, their welfare form is the same across no-subsidy and subsidy regimes.",
"We use $W_A(f_{prop})$ to denote $W_A(f_{prop}, 1)$ , the average welfare for group $A$ under classifier $f_{prop}$ with no subsidy.",
"Definition 6 (Group welfare in a non-manipulation setting) The average welfare of group $m$ under classifier $f_0$ in a non-manipulation setting is given by $W_{m}(f_0) = & \\int _{R} P_{x\\sim \\mathcal {D}_m}(x)dx$ where $R$ sums over candidates who are positively classified by $f_0$ .",
"Proposition 6 There exist cost functions $c_A$ and $c_B$ satisfying the cost conditions, learner distributions $\\mathcal {D}_A$ and $\\mathcal {D}_B$ , true classifiers with threshold $\\tau _A$ and $\\tau _B$ , population proportions $p_A$ and $p_B$ , and learner penalty parameters $C_{FN}$ , $C_{FP}$ , and $\\lambda $ , such that $W_A (f_{prop}^*) < W_A (f_0^*), \\qquad W_B(f_{prop}^*, \\beta ^*) < W_B (f_0^*),$ $W_A (f_1^*) < W_A (f_0^*), \\qquad W_B(f_1^*) < W_B (f_0^*),$ $C(f_{prop}^*, \\beta ^*) > C(f_0^*), \\qquad C(f_1^*) > C(f_0^*)$ where $f_0$ is the equilibrium classifier in the non-manipulation regime, $f_1^*$ is the equilibrium classifier in the manipulation regime, and ($f_{prop}^*, \\beta ^*$ ) is the equilibrium classifier in the subsidy regime.",
"The average welfare of each group, $W_m(\\cdot )$ , as well as the learner, $1-C(\\cdot )$ , is higher at the equilibrium of the non-manipulation game compared with the equilibria of the Strategic Classification Game with proportional subsidies and compared with the equilibrium of the Strategic Classification Game with no subsidies.",
"Example 2 Now we consider a case in which candidates have linear cost functions $c_A(x) = 3x$ and $c_B(x) = 4x$ .",
"To show that diminished welfare for both candidate groups can occur without requiring distortions of probability distributions or cost functions, we consider a learner who seeks to avoid errors on group $B$ in both the subsidy and the non-subsidy regimes by penalizing false negatives twice as much as false positives, with $C_{FN} = \\frac{2}{3}$ , $C_{FP} = \\frac{1}{3}$ , and $\\lambda = \\frac{3}{4}$ .",
"As in the previous example, we assume that the underlying unmanipulated features for both groups are uniformly distributed with $p_A = p_B = \\frac{1}{2}$ , and that $\\tau _A = 0.4$ and $\\tau _B = 0.3$ .",
"Now the equilibrium learner classifier without subsidies is based on threshold $\\sigma _1^* = \\sigma _B = 0.55$ , which perfectly classifies all candidates from group $B$ , while permitting false positives on candidates from group $A$ with features $x \\in [0.217, 0.4)$ .",
"Under a proportional subsidy intervention, the learner's equilibrium action is to choose threshold $\\sigma _{prop}^* = \\sigma ^\\beta _B \\approx 0.552$ and $\\beta ^* = 0.994$ , which again perfectly classifies $B$ candidates.",
"Notice that now her optimal threshold commits fewer false positive errors on group $A$ members, while still committing false positives on those members with features $x \\in [0.219, 0.4)$ .",
"Here, even when the learner has a cost penalty that is explicitly concerned with mistakenly excluding group $B$ candidates and then seeks to offer a subsidy benefit to further alleviate their costs, group $B$ members are still no better off.",
"They receive the same classifications as before and it can be shown that all candidates who manipulate must spend more to reach the higher threshold, even while accounting for the subsidy benefit!",
"Some group $A$ candidates are also worse off since the threshold has increased, and they receive no subsidy benefits.",
"As before, only the learner gains from the intervention.",
"Example 3 This example is based on Example REF .",
"Now we consider the case a learner seeks $\\sigma _1^* \\in [\\sigma _B, \\sigma _A]$ where $\\sigma _A = 0.733$ and $\\sigma _B = 0.55$ .",
"Suppose she seeks to equalize the number of false positives she commits on group $A$ and the number of false negatives for group $B$ and thus chooses $\\sigma _1^* = 0.64$ such that $\\ell _A(\\sigma _1^*) = 0.31$ $\\ell _B(\\sigma _1^*) = 0.39$ Thus group $B$ candidates with features $x \\in [0.3, 0.39)$ are mistakenly excluded, and group $A$ candidates with features $x\\in [0.31, 0.4)$ are mistakenly admitted.",
"Upon implementing a subsidy and minimizing the same error penalty as in Example 1, the learner selects an optimal proportional $\\beta $ subsidy such that $\\sigma _{prop}^* = \\sigma _A = 0.733; \\beta = 0.806$ Under this regime, group $B$ members are worse-off because many more candidates now receive false negative classifications $x \\in [0.3, 0.423) $ Others who do secure positive classifications must pay more to do so.",
"Candidates in group $A$ are now perfectly classified, though this actually entails a welfare decline, since some candidates lose their false positive benefits.",
"The learner is also strictly better off with a total penalty decline $C(\\sigma _0^*) = 0.183 \\rightarrow C(\\sigma _{prop}^*, \\beta ^*) = 0.128$ Recall that the learner's utility is given by $1 - C(\\cdot )$ .",
"Thus we have that $W_A(\\sigma _{prop}^*, \\beta ^*) < W_A(\\sigma _1^*)$ $W_B(\\sigma _{prop}^*, \\beta ^*) < W_B(\\sigma _1^*)$ $C(\\sigma _1^*) > C(\\sigma _{prop}^*, \\beta ^*)$ Now consider a non-manipulation regime, in which the learner selects to equalize the number of false negatives for group $B$ and the number of false positives for group $A$ , she now chooses a threshold on unmanipulated features $\\sigma _0^* = 0.35$ Some group $A$ candidates lose false positive benefits in the manipulation regime, though on the whole, the group fares better off because all those candidates with features $x \\in [0.39, 0.64)$ need not expend any costs in order to receive a positive classification.",
"Group $B$ candidates are strictly better off since they both receive fewer false negatives and need not pay to manipulate.",
"The learner is also better off here because she reduces her error down to $C(\\tau ^*) = 0.1$ .",
"Thus comparing the non-manipulation regime, the no-subsidy manipulation regime, and the subsidy regime, we have that utility comparisons for all three parties is given by $W_A(\\sigma _0^*)>W_A(\\sigma _1^*)>W_A(\\sigma _{prop}^*, \\beta ^*)$ $W_B(\\sigma _0^*)>W_B(\\sigma _1^*)>W_B(\\sigma _{prop}^*, \\beta ^*) $ $ 1-C(\\sigma _{0}^*)>1-C(\\sigma _{prop}^*, \\beta ^*) > 1- C(\\sigma _{1}^*) $"
],
[
"Flat Subsidies",
"Here we give analogous definitions and results for flat subsidies in which the learner absorbs up to a flat $\\alpha $ amount from each group $B$ candidate's costs and show that qualitatively similar results hold.",
"Definition 7 (Flat subsidy) Under a flat subsidy plan, the learner pays an $\\alpha > 0$ benefit to all members of group $B$ .",
"As such, a group B candidate who manipulates from an initial score $\\mathbf {x}$ to a final score $\\mathbf {y}\\ge \\mathbf {x}$ bears a cost of $\\max \\lbrace 0, c_B(\\mathbf {y}) - c_B(\\mathbf {x}) - \\alpha \\rbrace $ .",
"A learner's strategy now consists of both a choice of $\\alpha $ and a choice of classifier $f$ to issue.",
"The learner's goal is to minimize her penalty $\\begin{split}&C_{FP} \\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}[h_m(\\mathbf {x}) = 0, f(\\mathbf {y}) = 1] \\\\& + C_{FN}\\sum _{m \\in \\lbrace A, B\\rbrace } p_m P_{\\mathbf {x}\\sim \\mathcal {D}_m}[h_m(\\mathbf {x}) = 1, f(\\mathbf {y}) = 0] \\\\& + \\lambda cost(f, \\alpha ),\\end{split}$ We can define $\\ell _B^\\alpha (y) = c_B^{-1}\\Big (c_B(y) - (1+\\alpha )\\Big ).$ Under the $\\alpha $ subsidy, for an observed feature $y$ , the group $B$ candidate must have unmanipulated feature $x \\ge \\ell ^\\alpha _B(y)$ .",
"From these functions, we define $\\sigma _B^\\alpha $ and $\\sigma _B^\\beta $ such that $\\ell _B^\\alpha (\\sigma _B^\\alpha ) = \\tau _B$ , and $\\ell _{B}^\\beta (\\sigma _B^\\beta ) = \\tau _B$ .",
"Under a flat $\\alpha $ subsidy, setting a threshold at $\\sigma ^\\alpha _B$ correctly classifies all group $B$ members; under a proportional $\\beta $ subsidy, a threshold at $\\sigma ^\\beta _B$ correctly classifies all group $B$ members.",
"From this, we define $\\sigma _B^\\alpha $ such that $\\ell _{B}^\\alpha (\\sigma _B^\\alpha ) = \\tau _B$ .",
"Under a flat $\\alpha $ subsidy, setting a threshold at $\\sigma ^\\alpha _B$ correctly classifies all group $B$ members.",
"In order to compute the cost of a subsidy plan, we must determine the number of group $B$ candidates who will take advantage of a given subsidy benefit.",
"Since manipulation brings no benefit in itself, candidates will still only choose to manipulate and use the subsidy if it will lead to a positive classification.",
"For the flat $\\alpha $ subsidy, $cost(f, \\alpha )$ is given by $\\int _{c_B^{-1}(c_B(\\sigma ) - \\alpha )}^\\sigma \\!\\!\\!\\!\\!\\!\\!\\!",
"[c_B(\\sigma ) - c_B(x)] P_{D_B}(x) dx + \\alpha \\int _{\\ell _B^\\alpha (\\sigma )}^{c_B^{-1}(c_B(\\sigma ) - \\alpha )} P_{D_B}(x)dx,$ where $\\sigma $ is the threshold for classifier $f$ .",
"The first integral refers to the benefits paid out to candidates with manipulation costs less than the $\\alpha $ amount offered.",
"The latter refers to the total sum of full $\\alpha $ payments offered to those with costs greater than $\\alpha $ .",
"Definition 8 (Group welfare under a flat subsidy) The average welfare of group B under classifier $f$ and a flat subsidy with parameter $\\alpha $ is given by $W_B(f, \\alpha ) = &\\int _{R_1} P_{x\\sim \\mathcal {D}_B}(x) dx \\\\&+ \\int _{R_2} (1-c_B(y(x) - c_B(x))) P_{x\\sim \\mathcal {D}_B}(x) dx$ where $y(x)$ is the best response of a candidate with unmanipulated feature $x$ , $R_1$ sums over those candidates who are positively classified without expending any cost, and $R_2$ sums over those candidates who are positively classified after manipulating their features.",
"Note that under the flat subsidy, group $B$ costs have the form $\\max \\lbrace 0,c_B(y) - c_B(x) - \\alpha \\rbrace $ The formulation of average group $A$ welfare is the same in this setting and follows the same form given in Definition 5.",
"Theorem 3 (Subsidies can harm both groups) There exist cost functions $c_A$ and $c_B$ satisfying the cost conditions, learner distributions $\\mathcal {D}_A$ and $\\mathcal {D}_B$ , true classifiers with threshold $\\tau _A$ and $\\tau _B$ , population proportions $p_A$ and $p_B$ , and learner penalty parameters $C_{FN}$ , $C_{FP}$ , and $\\lambda $ , such that $W_A (f_{prop}^*) < W_A (f_0^*), \\qquad W_B(f_{prop}^*, \\alpha ^*) < W_B (f_0^*),$ where $f_{prop}^*$ and $\\alpha ^*$ are the learner's equilibrium classifier and subsidy choice in the Strategic Classification Game with flat subsidies and $f_0^*$ is the learner's equilibrium classifier in the Strategic Classification Game with no subsidies."
]
] | 1808.08646 |
[
[
"Spectral Pruning: Compressing Deep Neural Networks via Spectral Analysis\n and its Generalization Error"
],
[
"Abstract Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices.",
"The concept of model compression is also important for analyzing the generalization error of deep learning, known as the compression-based error bound.",
"However, there is still huge gap between a practically effective compression method and its rigorous background of statistical learning theory.",
"To resolve this issue, we develop a new theoretical framework for model compression and propose a new pruning method called {\\it spectral pruning} based on this framework.",
"We define the ``degrees of freedom'' to quantify the intrinsic dimensionality of a model by using the eigenvalue distribution of the covariance matrix across the internal nodes and show that the compression ability is essentially controlled by this quantity.",
"Moreover, we present a sharp generalization error bound of the compressed model and characterize the bias--variance tradeoff induced by the compression procedure.",
"We apply our method to several datasets to justify our theoretical analyses and show the superiority of the the proposed method."
],
[
"Introduction",
"Currently, deep learning is the most promising approach adopted by various machine learning applications such as computer vision, natural language processing, and audio processing.",
"Along with the rapid development of the deep learning techniques, its network structure is becoming considerably complicated.",
"In addition to the model structure, the model size is also becoming larger, which prevents the implementation of deep neural network models in edge-computing devices for applications such as smartphone services, autonomous vehicle driving, and drone control.",
"To overcome this problem, model compression techniques such as pruning, factorization [7], [8], and quantization [13] have been extensively studied in the literature.",
"Among these techniques, pruning is a typical approach that discards redundant nodes, e.g., by explicit regularization such as $\\ell _1$ and $\\ell _2$ penalization during training [21], [33], [16].",
"It has been implemented as ThiNet [24], Net-Trim [1], NISP [34], and so on [7].",
"A similar effect can be realized by implicit randomized regularization such as DropConnect [32], which randomly removes connections during the training phase.",
"However, only few of these techniques (e.g., Net-Trim [1]) are supported by statistical learning theory.",
"In particular, it unclear which type of quantity controls the compression ability.",
"On the theoretical side, compression-based generalization analysis is a promising approach for measuring the redundancy of a network [2], [36].",
"However, despite their theoretical novelty, the connection of these generalization error analyses to practically useful compression methods is not obvious.",
"In this paper, we develop a new compression based generalization error bound and propose a new simple pruning method that is compatible with the generalization error analysis.",
"Our method aims to minimize the information loss induced by compression; in particular, it minimizes the redundancy among nodes instead of merely looking at the amount of information of each individual node.",
"It can be executed by simply observing the covariance matrix in the internal layers and is easy to implement.",
"The proposed method is supported by a comprehensive theoretical analysis.",
"Notably, the approximation error induced by compression is characterized by the notion of the statistical degrees of freedom [26], [5].",
"It represents the intrinsic dimensionality of a model and is determined by the eigenvalues of the covariance matrix between each node in each layer.",
"Usually, we observe that the eigenvalue rapidly decreases (Fig.",
"REF ) for several reasons such as explicit regularization (Dropout [31], weight decay [20]), and implicit regularization [14], [12], which means that the amount of important information processed in each layer is not large.",
"In particular, the rapid decay in eigenvalues leads to a low number of degrees of freedom.",
"Then, we can effectively compress a trained network into a smaller one that has fewer parameters than the original.",
"Behind the theory, there is essentially a connection to the random feature technique for kernel methods [3].",
"Compression error analysis is directly connected to generalization error analysis.",
"The derived bound is actually much tighter than the naive VC-theory bound on the uncompressed network [4] and even tighter than recent compression-based bounds [2].",
"Further, there is a tradeoff between the bias and the variance, where the bias is induced by the network compression and the variance is induced by the variation in the training data.",
"In addition, we show the superiority of our method and experimentally verify our theory with extensive numerical experiments.",
"Our contributions are summarized as follows: We give a theoretical compression bound which is compatible with a practically useful pruning method, and propose a new simple pruning method called spectral pruning for compressing deep neural networks.",
"We characterize the model compression ability by utilizing the notion of the degrees of freedom, which represents the intrinsic dimensionality of the model.",
"We also give a generalization error bound when a trained network is compressed by our method and show that the bias–variance tradeoff induced by model compression appears.",
"The obtained bound is fairly tight compared with existing compression-based bounds and much tighter than the naive VC-dimension bound."
],
[
"Model Compression Problem and its Algorithm",
"Suppose that the training data $D_{\\mathrm {tr}}= \\lbrace (x_i,y_i)\\rbrace _{i=1}^{n}$ are observed, where $x_i \\in \\mathbb {R}^{d_x}$ is an input and $y_i$ is an output that could be a real number ($y_i \\in \\mathbb {R}$ ), a binary label ($y_i \\in \\lbrace \\pm 1\\rbrace $ ), and so on.",
"The training data are independently and identically distributed.",
"To train the appropriate relationship between $x$ and $y$ , we construct a deep neural network model as $f(x) =(W^{(L)} \\eta ( \\cdot ) + b^{(L)}) \\circ \\dots \\circ (W^{(1)} x + b^{(1)}),$ where $W^{(\\ell )} \\in \\mathbb {R}^{m_{\\ell + 1} \\times m_{\\ell }}$ , $b^{(\\ell )} \\in \\mathbb {R}^{m_{\\ell + 1}}$ ($\\ell = 1,\\dots ,L$ ), and $\\eta : \\mathbb {R}\\rightarrow \\mathbb {R}$ is an activation function (here, the activation function is applied in an element-wise manner; for a vector $x \\in \\mathbb {R}^d$ , $\\eta (x)=(\\eta (x_1),\\dots ,\\eta (x_d))^\\top $ ).",
"Here, $m_\\ell $ is the width of the $\\ell $ -th layer such that $m_{L+1} = 1$ (output) and $m_1 = d_x$ (input).",
"Let $\\widehat{f}$ be a trained network obtained from the training data $D_{\\mathrm {tr}}= \\lbrace (x_i,y_i)\\rbrace _{i=1}^{n}$ where its parameters are denoted by $(\\hat{W}^{(\\ell )},\\hat{b}^{(\\ell )})_{\\ell =1}^L$ , i.e., $\\widehat{f}(x) = (\\hat{W}^{(L)}\\eta (\\cdot ) + \\hat{b}^{(L)}) \\circ \\dots \\circ (\\hat{W}^{(1)} x + \\hat{b}^{(1)})$ .",
"The input to the $\\ell $ -th layer (after activation) is denoted by $\\phi ^{(\\ell )}(x) =\\eta \\circ (\\hat{W}^{(\\ell -1)}\\eta (\\cdot ) + \\hat{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\hat{W}^{(1)} x + \\hat{b}^{(1)}).$ We do not specify how to train the network $\\widehat{f}$ , and the following argument can be applied to any learning method such as the empirical risk minimizer, the Bayes estimator, or another estimator.",
"We want to compress the trained network $\\widehat{f}$ to another smaller one $f^{\\sharp }$ having widths $(m^{\\sharp }_{\\ell })_{\\ell =1}^L$ with keeping the test accuracy as high as possible.",
"To compress the trained network $\\widehat{f}$ to a smaller one $f^{\\sharp }$ , we propose a simple strategy called spectral pruning.",
"The main idea of the method is to find the most informative subset of the nodes.",
"The amount of information of the subset is measured by how well the selected nodes can explain the other nodes in the layer and recover the output to the next layer.",
"For example, if some nodes are heavily correlated with each other, then only one of them will be selected by our method.",
"The information redundancy can be computed by a covariance matrix between nodes and a simple regression problem.",
"We do not need to solve a specific nonlinear optimization problem unlike the methods in [21], [33], [1]."
],
[
"Algorithm Description",
"Our method basically simultaneously minimizes the input information loss and output information loss, which will be defined as follows.",
"(i) Input information loss.",
"First, we explain the input information loss.",
"Denote $\\phi (x) = \\phi ^{(\\ell )}(x)$ for simplicity, and let $\\phi _J(x) = (\\phi _j(x))_{j \\in J} \\in \\mathbb {R}^{m^{\\sharp }_{\\ell }}$ be a subvector of $\\phi (x)$ corresponding to an index set $J\\in [m_\\ell ]^{m^{\\sharp }_{\\ell }}$ , where $[m] := \\lbrace 1,\\dots ,m\\rbrace $ (here, duplication of the index is allowed).",
"The basic strategy is to solve the following optimization problem so that we can recover $\\phi (x)$ from $\\phi _J(x)$ as accurately as possible: $& \\hat{A}_J := (\\hat{A}^{(\\ell )}_J=) \\mathop {\\mathrm {argmin}}_{A \\in \\mathbb {R}^{m_\\ell \\times |J|}} \\widehat{\\mathrm {E}}[\\Vert \\phi - A \\phi _J\\Vert ^2] + \\Vert A\\Vert _\\tau ^2, $ where $\\widehat{\\mathrm {E}}[\\cdot ]$ is the expectation with respect to the empirical distribution ($\\widehat{\\mathrm {E}}[f] = \\frac{1}{n}\\sum _{i=1}^{n} f(x_i)$ ) and $\\Vert A\\Vert _\\tau ^2 = \\mathrm {Tr}[A \\mathrm {I}_\\tau A^\\top ]$ for a regularization parameter $\\tau \\in \\mathbb {R}_+^{|J|}$ and $\\mathrm {I}_\\tau := \\mathrm {diag}\\left(\\tau \\right)$ (how to set the regularization parameter will be given in Theorem REF ).",
"The optimal solution $\\hat{A}_J$ can be explicitly expressed by utilizing the (noncentered) covariance matrix in the $\\ell $ -th layer of the trained network $\\widehat{f}$ , which is defined as $\\widehat{\\Sigma }:= \\widehat{\\Sigma }^{(\\ell )}= \\frac{1}{n} \\sum _{i=1}^n \\phi (x_i) \\phi (x_i)^\\top , $ defined on the empirical distribution (here, we omit the layer index $\\ell $ for notational simplicity).",
"Let $\\widehat{\\Sigma }_{I,I^{\\prime }} \\in \\mathbb {R}^{K \\times H}$ for $K,H \\in \\mathbb {N}$ be the submatrix of $\\widehat{\\Sigma }$ for the index sets $I \\in [m_\\ell ]^{K}$ and $I^{\\prime } \\in [m_\\ell ]^{H}$ such that $\\widehat{\\Sigma }_{I,I^{\\prime }} = (\\widehat{\\Sigma }_{i,j})_{i\\in I,j\\in I^{\\prime }}$ .",
"Let $F = \\lbrace 1,\\dots ,m_\\ell \\rbrace $ be the full index set.",
"Then, we can easily see that $\\hat{A}_J = \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1}.$ Hence, the full vector $\\phi (x)$ can be decoded from $\\phi _J(x)$ as $\\phi (x) \\approx \\hat{A}_J \\phi _J(x) = \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1} \\phi _J(x).$ To measure the approximation error, we define $L^{(\\mathrm {A})}_{\\tau }(J) = \\min _{A \\in \\mathbb {R}^{m_\\ell }\\times |J|} \\widehat{\\mathrm {E}}[\\Vert \\phi -A \\phi _J\\Vert ^2] + \\Vert A \\Vert _\\tau ^2.$ By substituting the explicit formula $\\hat{A}_J$ into the objective, this is reformulated as $L^{(\\mathrm {A})}_{\\tau }(J) = &\\mathrm {Tr}[\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1}\\widehat{\\Sigma }_{J,F} ].$ (ii) Output information loss.",
"Next, we explain the output information loss.",
"Suppose that we aim to directly approximate the outputs $Z^{(\\ell )} \\phi $ for a weight matrix $Z^{(\\ell )} \\in \\mathbb {R}^{m \\times m_\\ell }$ with an output size $m \\in \\mathbb {N}$ .",
"A typical situation is that $Z^{(\\ell )} = \\hat{W}^{(\\ell )}$ so that we approximate the output $\\hat{W}^{(\\ell )} \\phi $ (the concrete setting of $Z^{(\\ell )}$ will be specified in Theorem REF ).",
"Then, we consider the objective $L^{(\\mathrm {B})}_{\\tau } (J)& := \\sum _{j=1}^m \\min \\limits _{\\alpha \\in \\mathbb {R}^{m_\\ell }}\\left\\lbrace \\widehat{\\mathrm {E}}[( Z_{j,:}^{(\\ell )} \\phi -\\alpha ^\\top \\phi _J)^2] \\!+\\!",
"\\Vert \\alpha ^\\top \\Vert _\\tau ^2 \\right\\rbrace \\\\&=\\mathrm {Tr}\\lbrace Z^{(\\ell )} [\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1}\\widehat{\\Sigma }_{J,F} ] Z^{(\\ell )\\top }\\rbrace ,$ where $Z^{(\\ell )}_{j,:}$ means the $j$ -th raw of the matrix $Z^{(\\ell )}$ .",
"It can be easily checked that the optimal solution $\\hat{\\alpha }_J$ of the minimum in the definition of $L^{(\\mathrm {B})}_{\\tau }$ is given as $\\hat{\\alpha }_J = \\hat{A}_J^\\top Z_{j,:}^{(\\ell ) \\top }$ for each $j=1,\\dots ,m$ .",
"(iii) Combination of the input and output information losses.",
"Finally, we combine the input and output information losses and aim to minimize this combination.",
"To do so, we propose to the use of the convex combination of both criteria for a parameter $0 \\le \\theta \\le 1$ and optimize it with respect to $J$ under a cardinality constraint $|J| = m^{\\sharp }_{\\ell }$ for a prespecified width $m^{\\sharp }_{\\ell }$ of the compressed network: $& \\min _J ~~L^{(\\theta )}_{\\tau }(J) ~= \\theta L^{(\\mathrm {A})}_\\tau (J) + (1- \\theta )L^{(\\mathrm {B})}_{\\tau }(J) \\\\ &~~~\\mathrm {s.t.",
"}~~~J \\in [m_\\ell ]^{m^{\\sharp }_{\\ell }}.",
"$ We call this method spectral pruning.",
"There are the hyperparameter $\\theta $ and regularization parameter $\\tau $ .",
"However, we see that it is robust against the choice of hyperparameter in experiments (Sec.",
").",
"Let $J^{\\sharp }_{\\ell }$ be the optimal $J$ that minimizes the objective.",
"This optimization problem is NP-hard, but an approximate solution is obtained by the greedy algorithm since it is reduced to maximization of a monotonic submodular function [19].",
"That is, we start from $J = \\emptyset $ , sequentially choose an element $j^* \\in [m_\\ell ]$ that maximally reduces the objective $L^{(\\theta )}_{\\tau }$ , and add this element $j^*$ to $J$ ($J \\leftarrow J \\cup \\lbrace j^*\\rbrace $ ) until $|J| = m^{\\sharp }_{\\ell }$ is satisfied.",
"After we chose an index $J^{\\sharp }_{\\ell }$ ($\\ell = 2,\\dots ,L$ ) for each layer, we construct the compressed network $f^{\\sharp }$ as $f^{\\sharp }(x) =(W^{\\sharp (L)} \\eta ( \\cdot ) + b^{\\sharp (L)}) \\circ \\dots \\circ (W^{\\sharp (1)} x + b^{\\sharp (1)})$ , where $W^{\\sharp (\\ell )} = W^{(\\ell )}_{J^{\\sharp }_{\\ell +1},F} \\hat{A}_{J^{\\sharp }_{\\ell }}^{(\\ell )}$ and $b^{\\sharp (\\ell )} = b^{(\\ell )}_{J^{\\sharp }_{\\ell +1}}$ .",
"An application to a CNN is given in Appendix .",
"The method can be executed in a layer-wise manner, thus it can be applied to networks with complicated structures such as ResNet."
],
[
"Compression accuracy Analysis and Generalization Error Bound",
"In this section, we give a theoretical guarantee of our method.",
"First, we give the approximation error induced by our pruning procedure in Theorem REF .",
"Next, we evaluate the generalization error of the compressed network in Theorem REF .",
"More specifically, we introduce a quantity called the degrees of freedom [26], [5], [30], [29] that represents the intrinsic dimensionality of the model and determines the approximation accuracy.",
"For the theoretical analysis, we define a neural network model with norm constraints on the parameters $W^{(\\ell )}$ and $b^{(\\ell )}$ ($\\ell = 1,\\dots ,L$ ).",
"Let $R >0$ and $R_b>0$ be the upper bounds of the parameters, and define the norm-constrained model as ${\\mathcal {F}}& :=\\textstyle \\lbrace (W^{(L)} \\eta ( \\cdot ) + b^{(L)}) \\circ \\dots \\circ (W^{(1)} x + b^{(1)}) \\mid \\\\ &\\textstyle \\max _{j} \\Vert W^{(\\ell )}_{j,:}\\Vert \\le \\frac{ R}{\\sqrt{m_{\\ell +1}}},~\\Vert b^{(\\ell )}\\Vert _\\infty \\le \\frac{R_b}{\\sqrt{m_{\\ell + 1}}}~\\rbrace ,$ where $W^{(\\ell )}_{j,:}$ means the $j$ -th raw of the matrix $W^{(\\ell )}$ , $\\Vert \\cdot \\Vert $ is the Euclidean norm, and $\\Vert \\cdot \\Vert _\\infty $ is the $\\ell _\\infty $ -normWe are implicitly supposing $R,R_b\\simeq 1$ so that $\\Vert W^{(\\ell )}\\Vert _{\\mathrm {F}},\\Vert b^{(\\ell )}\\Vert =O(1)$ .. We make the following assumption for the activation function, which is satisfied by ReLU and leaky ReLU [25].",
"Assumption 1 We assume that the activation function $\\eta $ satisfies (1) scale invariance: $\\eta (a x) = a\\eta (x)$ for all $a >0$ and $x\\in \\mathbb {R}^d$ and (2) 1-Lipschitz continuity: $|\\eta (x) - \\eta (x^{\\prime })| \\le \\Vert x - x^{\\prime }\\Vert $ for all $x,x^{\\prime }\\in \\mathbb {R}^d$ , where $d$ is arbitrary."
],
[
"Approximation Error Analysis",
"Here, we evaluate the approximation error derived by our pruning procedure.",
"Let $(m^{\\sharp }_{\\ell })_{\\ell = 1}^L$ denote the width of each layer of the compressed network $f^{\\sharp }$ .",
"We characterize the approximation error between $f^{\\sharp }$ and $\\widehat{f}$ on the basis of the degrees of freedom with respect to the empirical $L_2$ -norm $\\Vert g\\Vert _n^2 := \\frac{1}{n}\\sum _{i=1}^{n} \\Vert g(x_i)\\Vert ^2$ , which is defined for a vector-valued function $g$ .",
"Recall that the empirical covariance matrix in the $\\ell $ -th layer is denoted by $\\widehat{\\Sigma }^{(\\ell )}$ .",
"We define the degrees of freedom as $\\textstyle \\hat{N}_{\\ell }(\\lambda ):= \\mathrm {Tr}[ \\widehat{\\Sigma }^{(\\ell )}( \\widehat{\\Sigma }^{(\\ell )} + \\lambda \\mathrm {I})^{-1}]=\\sum \\nolimits _{j=1}^{m_\\ell }\\hat{\\mu }_j^{(\\ell )}/(\\hat{\\mu }_j^{(\\ell )} + \\lambda ),$ where $(\\hat{\\mu }_j^{(\\ell )})_{j=1}^{m_\\ell }$ are the eigenvalues of $\\widehat{\\Sigma }^{(\\ell )}$ sorted in decreasing order.",
"Roughly speaking, this quantity quantifies the number of eigenvalues above $\\lambda $ , and thus it is monotonically decreasing w.r.t.",
"$\\lambda $ .",
"The degrees of freedom play an essential role in investigating the predictive accuracy of ridge regression [26], [5], [3].",
"To characterize the output information loss, we also define the output aware degrees of freedom with respect to a matrix $Z^{(\\ell )}$ as $\\hat{N}^{\\prime }_{\\ell }(\\lambda ;Z^{(\\ell )}) := \\mathrm {Tr}[Z^{(\\ell )} \\widehat{\\Sigma }^{(\\ell )}( \\widehat{\\Sigma }^{(\\ell )} + \\lambda \\mathrm {I})^{-1}Z^{(\\ell )\\top }].$ This quantity measures the intrinsic dimensionality of the output from the $\\ell $ -th layer for a weight matrix $Z^{(\\ell )}$ .",
"If the covariance $\\widehat{\\Sigma }^{(\\ell )}$ and the matrix $Z^{(\\ell )}$ are near low rank, $\\hat{N}^{\\prime }_{\\ell }(\\lambda ;Z^{(\\ell )})$ becomes much smaller than $\\hat{N}_{\\ell }(\\lambda )$ .",
"Finally, we define $N_\\ell ^\\theta (\\lambda ) := \\theta \\hat{N}_\\ell (\\lambda ) + (1-\\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda ;Z^{(\\ell )})$ .",
"To evaluate the approximation error induced by compression, we define $\\lambda _\\ell > 0$ as $\\textstyle \\lambda _\\ell = \\inf \\lbrace \\lambda \\ge 0 \\mid m^{\\sharp }_{\\ell }\\ge 5 \\hat{N}_{\\ell }(\\lambda ) \\log (80 \\hat{N}_{\\ell }(\\lambda )) \\rbrace .$ Conversely, we may determine $m^{\\sharp }_{\\ell }$ from $\\lambda _\\ell $ to obtain the theorems we will mention below.",
"Along with the degrees of freedom, we define the leverage score ${\\tilde{\\tau }^{(\\ell )}} \\in \\mathbb {R}^{m_\\ell }$ as ${\\tilde{\\tau }^{(\\ell )}}_j :=\\frac{1}{\\hat{N}_{\\ell }(\\lambda _\\ell )} [\\widehat{\\Sigma }^{(\\ell )} (\\widehat{\\Sigma }^{(\\ell )} + \\lambda _\\ell \\mathrm {I})^{-1}]_{j,j}~(j\\in [m_\\ell ]).$ Note that $\\sum _{j=1}^{m_\\ell } {\\tilde{\\tau }^{(\\ell )}}_j = 1$ originates from the definition of the degrees of freedom.",
"The leverage score can be seen as the amount of contribution of node $j \\in [m_\\ell ]$ to the degrees of freedom.",
"For simplicity, we assume that ${\\tilde{\\tau }^{(\\ell )}}_j > 0$ for all $\\ell ,j$ (otherwise, we just need to neglect such a node with ${\\tilde{\\tau }^{(\\ell )}}_j=0$ ).",
"For the approximation error bound, we consider two situations: (i) (Backward procedure) spectral pruning is applied from $\\ell =L$ to $\\ell = 2$ in order, and for pruning the $\\ell $ -th layer, we may utilize the selected index $J^{\\sharp }_{\\ell + 1}$ in the $\\ell +1$ -th layer and (ii) (Simultaneous procedure) spectral pruning is simultaneously applied for all $\\ell = 2,\\dots ,L$ .",
"We provide a statement for only the backward procedure.",
"The simultaneous procedure also achieves a similar bound with some modifications.",
"The complete statement will be given as Theorem REF in Appendix .",
"As for $Z^{(\\ell )}$ for the output information loss, we set $Z^{(\\ell )}_{k,:}= {\\scriptstyle \\sqrt{m_{\\ell } q_{j_k}^{(\\ell )}} (\\max _{j^{\\prime }} \\Vert \\hat{W}^{(\\ell )}_{j^{\\prime },:}\\Vert )^{-1}} \\hat{W}^{(\\ell )}_{j_k,:}~(k=1,\\dots , m^{\\sharp }_{\\ell +1})$ where we let $J^{\\sharp }_{\\ell +1}=\\lbrace j_1,\\dots ,j_{m^{\\sharp }_{\\ell +1}}\\rbrace $ , and $q_j^{(\\ell )} := \\frac{( {\\tilde{\\tau }^{(\\ell +1)}}_j)^{-1}}{\\sum _{j^{\\prime } \\in J^{\\sharp }_{\\ell + 1}} ({\\tilde{\\tau }^{(\\ell +1)}}_{j^{\\prime }})^{-1}}~(j \\in J^{\\sharp }_{\\ell + 1})$ and $q_j^{(\\ell )} = 0~(\\text{otherwise})$ .",
"Finally, we set the regularization parameter $\\tau $ as $\\tau \\leftarrow m^{\\sharp }_{\\ell }\\lambda _\\ell {\\tilde{\\tau }^{(\\ell )}}$ .",
"Theorem 1 (Compression rate via the degrees of freedom) If we solve the optimization problem (REF ) with the additional constraint $\\sum _{j \\in J} ({\\tilde{\\tau }^{(\\ell )}}_j)^{-1} \\le \\frac{5}{3}m_\\ell m^{\\sharp }_{\\ell }$ for the index set $J$ , then the optimization problem is feasible, and the overall approximation error of $f^{\\sharp }$ is bounded by $\\Vert \\widehat{f}- f^{\\sharp }\\Vert _n\\le \\sum \\nolimits _{\\ell = 2}^L\\left( \\bar{R}^{L-\\ell + 1} \\sqrt{ {\\textstyle \\prod _{\\ell ^{\\prime }=\\ell }^L}\\zeta _{\\ell ^{\\prime },\\theta } } \\right) \\sqrt{\\lambda _\\ell }$ for $\\bar{R}= \\sqrt{\\hat{c}} R$ , where $\\hat{c}$ is a universal constant, and $\\zeta _{\\ell ,\\theta }:=N_\\ell ^\\theta (\\lambda _\\ell ) \\left( \\theta \\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{ \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}}\\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} } + (1-\\theta )m_{\\ell }\\right)^{-1}$$\\Vert \\cdot \\Vert _{\\mathrm {op}}$ represents the operator norm of a matrix (the largest absolute singular value)..",
"The proof is given in Appendix .",
"To prove the theorem, we essentially need to use theories of random features in kernel methods [3], [30].",
"The main message from the theorem is that the approximation error induced by compression is directly controlled by the degrees of freedom.",
"Since the degrees of freedom $\\hat{N}_{\\ell }(\\lambda )$ are a monotonically decreasing function with respect to $\\lambda $ , they become large as $\\lambda $ decreases to 0.",
"The behavior of the eigenvalues determines how rapidly $\\hat{N}_{\\ell }(\\lambda )$ increases as $\\lambda \\rightarrow 0$ .",
"We can see that if the eigenvalues $\\hat{\\mu }_1^{(\\ell )} \\ge \\hat{\\mu }_2^{(\\ell )} \\ge \\dots $ rapidly decrease, then the approximation error $\\lambda _\\ell $ can be much smaller for a given model size $m^{\\sharp }_{\\ell }$ .",
"In other words, $f^{\\sharp }$ can be much closer to the original network $\\widehat{f}$ if there are only a few large eigenvalues.",
"The quantity $\\zeta _{\\ell ,\\theta }$ characterizes how well the approximation error $\\lambda _{\\ell ^{\\prime }}$ of the lower layers $\\ell ^{\\prime } \\le \\ell $ propagates to the final output.",
"We can see that a tradeoff between $\\zeta _{\\ell ,\\theta }$ and $\\theta $ appears.",
"By a simple evaluation, $N_\\ell ^\\theta $ in the numerator of $\\zeta _{\\ell ,\\theta }$ is bounded by $m_\\ell $ ; thus, $\\theta =1$ gives $\\zeta _{\\ell ,\\theta }\\le 1$ .",
"On the other hand, the term $\\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{ \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}}\\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }$ takes a value between $m_{\\ell + 1}$ and 1; thus, $\\theta = 1$ is not necessarily the best choice to maximize the denominator.",
"From this consideration, we can see that the value of $\\theta $ that best minimizes $\\zeta _{\\ell ,\\theta }$ exists between 0 and 1, which supports our numerical result (Fig.",
"REF ).",
"In any situation, small degrees of freedom give a small $\\zeta _{\\ell ,\\theta }$ , leading to a sharper bound."
],
[
"Generalization Error Analysis",
"Here, we derive the generalization error bound of the compressed network with respect to the population risk.",
"We will see that a bias–variance tradeoff induced by network compression appears.",
"As usual, we train a network through the training error $\\widehat{\\Psi }(f) := \\frac{1}{n} \\sum _{i=1}^{n} \\psi (y_i,f(x_i))$ , where $\\psi :\\mathbb {R}\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ is a loss function.",
"Correspondingly, the expected error is denoted by $\\Psi (f) := \\mathrm {E}[\\psi (Y,f(X))]$ , where the expectation is taken with respect to $(X,Y) \\sim P$ .",
"Our aim here is to bound the generalization error $\\Psi (f^{\\sharp })$ of the compressed network.",
"Let the marginal distribution of $X$ be $P_{{\\mathcal {X}}}$ and that of $y$ be $P_{{\\mathcal {Y}}}$ .",
"First, we assume the Lipschitz continuity for the loss function $\\psi $ .",
"Assumption 2 The loss function $\\psi $ is $\\rho $ -Lipschitz continuous: $|\\psi (y,f) - \\psi (y,f^{\\prime })| \\le \\rho |f - f^{\\prime }|~(\\forall y \\in \\mathrm {supp}(P_{{\\mathcal {Y}}}),~~\\forall f,f^{\\prime } \\in \\mathbb {R})$ .",
"The support of $P_{{\\mathcal {X}}}$ is bounded: $\\Vert x\\Vert \\le D_x~~(\\forall x \\in \\mathrm {supp}(P_{{\\mathcal {X}}})).$ For a technical reason, we assume the following condition for the spectral pruning algorithm.",
"Assumption 3 We assume that $0 \\le \\theta \\le 1$ is appropriately chosen so that $\\zeta _{\\ell ,\\theta }$ in Theorem REF satisfies $\\zeta _{\\ell ,\\theta }\\le 1$ almost surely, and spectral pruning is solved under the condition $\\sum _{j \\in J} ({\\tilde{\\tau }^{(\\ell )}}_j)^{-1} \\le \\frac{5}{3}m_\\ell m^{\\sharp }_{\\ell }$ on the index set $J$ .",
"As for the choice of $\\theta $ , this assumption is always satisfied at least by the backward procedure.",
"The condition on the linear constraint on $J$ is merely to ensure the leverage scores are balanced for the chosen index.",
"Note that the bounds in Theorem REF can be achieved even with this condition.",
"If $L_\\infty $ -norm of networks is loosely evaluated, the generalization error bound of deep learning can be unrealistically large because there appears $L_\\infty $ -norm in its evaluation.",
"However, we may consider a truncated estimator $[\\!",
"[ \\widehat{f}(x)]\\!]",
":= \\max \\lbrace -M,\\min \\lbrace M,\\widehat{f}(x)\\rbrace \\rbrace $ for sufficiently large $0 < M \\le \\infty $ to moderate the $L_\\infty $ -norm (if $M = \\infty $ , this does not affect anything).",
"Note that the truncation procedure does not affect the classification error for a classification task.",
"To bound the generalization error, we define $\\delta _{1}$ and $\\delta _{2}$ for $(m^{\\sharp }_{2},\\dots ,m^{\\sharp }_{L})$ and $(\\lambda _2,\\dots ,\\lambda _L)$ satisfying relation (REF ) as $\\log _+(x) = \\max \\lbrace 1,\\log (x)\\rbrace $ .",
"$&\\textstyle \\delta _{1}= \\sum _{\\ell = 2}^L\\big ( \\bar{R}^{L-\\ell + 1}\\text{\\small $\\sqrt{\\prod _{\\ell ^{\\prime } = \\ell }^L \\zeta _{\\ell ^{\\prime },\\theta }} \\big )$ } \\sqrt{ \\lambda _\\ell },~~\\\\ &\\delta _{2}^2 = \\frac{1}{n} \\sum _{\\ell =1}^L m^{\\sharp }_{\\ell } m^{\\sharp }_{\\ell + 1}\\log _+\\!",
"\\left({\\textstyle 1 +\\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\hat{R}_{\\infty }}}\\right), $ where $\\hat{R}_{\\infty }:= \\min \\lbrace \\bar{R}^L D_x + \\sum _{\\ell = 1}^L \\bar{R}^{L - \\ell } \\bar{R}_b,M\\rbrace , ~\\hat{G} := L \\bar{R}^{L-1} D_x + \\sum _{\\ell = 1}^L \\bar{R}^{L - \\ell }$ for $\\bar{R}= \\sqrt{\\hat{c}} R$ and $\\bar{R}_b= \\sqrt{\\hat{c}} R_b$ with the constants $\\hat{c}$ introduced in Theorem REF .",
"Let $R_{n,t}:= \\frac{ 1 }{n}\\left(t+\\sum \\nolimits _{\\ell =2}^L \\log (m_\\ell )\\right)$ for $t > 0$ .",
"Then, we obtain the following generalization error bound for the compressed network $f^{\\sharp }$ .",
"Theorem 2 (Generalization error bound of the compressed network) Suppose that Assumptions REF , REF , and REF are satisfied.",
"Then, the spectral pruning method presented in Theorem REF satisfies the following generalization error bound.",
"There exists a universal constant $C_1 >0$ such that for any $t >0$ , it holds that $\\Psi ([\\!",
"[f^{\\sharp }]\\!",
"])\\le &\\hat{\\Psi }([\\![\\widehat{f}]\\!",
"])+\\rho \\Big \\lbrace \\delta _{1}+ C_1 \\hat{R}_{\\infty }(\\delta _{2}+ \\delta _{2}^2+ \\sqrt{R_{n,t}}) \\Big \\rbrace \\\\\\lesssim &\\hat{\\Psi }([\\![\\widehat{f}]\\!])",
"\\!+\\!",
"\\sum \\nolimits _{\\ell =2}^L \\!\\!",
"\\sqrt{\\lambda _\\ell }\\!+\\!",
"{\\textstyle \\sqrt{\\frac{\\sum _{\\ell =1}^L m^{\\sharp }_{\\ell +1} m^{\\sharp }_{\\ell }}{n}{\\textstyle \\log _+(\\hat{G})}}},$ uniformly over all choices of $m^{\\sharp }_{} = (m^{\\sharp }_{1},\\dots ,m^{\\sharp }_{L})$ with probability $1 - 2 e^{-t}$ .",
"The proof is given in Appendix .",
"From this theorem, the generalization error of $f^{\\sharp }$ is upper-bounded by the training error of the original network $\\widehat{f}$ (which is usually small) and an additional term.",
"By Theorem REF , $\\delta _{1}$ represents the approximation error between $\\widehat{f}$ and $f^{\\sharp }$ ; hence, it can be regarded as a bias.",
"The second term $\\delta _{2}$ is the variance term induced by the sample deviation.",
"It is noted that the variance term $\\delta _{2}$ only depends on the size of the compressed network rather than the original network size.",
"On the other hand, a naive application of the theorem implies $\\Psi ([\\![\\widehat{f}]\\!])",
"- \\hat{\\Psi }([\\![\\widehat{f}]\\!])",
"\\le \\mathrm {\\tilde{O}}\\big ( \\sqrt{\\frac{1}{n}\\sum _{\\ell =1}^L m_{\\ell +1} m_{\\ell }} \\big )$ for the original network $\\widehat{f}$ , which coincides with the VC-dimension based bound [4] but is much larger than $\\delta _{2}$ when $m^{\\sharp }_{\\ell } \\ll m_{\\ell }$ .",
"Therefore, the variance is significantly reduced by model compression, resulting in a much improved generalization error.",
"Note that the relation between $\\delta _{1}$ and $\\delta _{2}$ is a tradeoff due to the monotonicity of the degrees of freedom.",
"When $m^{\\sharp }_{\\ell }$ is large, the bias $\\delta _{1}$ becomes small owing to the monotonicity of the degrees of freedom, but the variance $\\delta _{2}(m^{\\sharp }_{})$ will be large.",
"Hence, we need to tune the size $(m^{\\sharp }_{\\ell })_{\\ell =1}^L$ to obtain the best generalization error by balancing the bias ($\\delta _{1}$ ) and variance ($\\delta _{2}$ ).",
"The generalization error bound is uniformly valid over the choice of $m^{\\sharp }_{} $ (to ensure this, the term $R_{n,t}$ appears).",
"Thus, $m^{\\sharp }_{} $ can be arbitrary and chosen in a data-dependent manner.",
"This means that the bound is a posteriori, and the best choice of $m^{\\sharp }_{} $ can depend on the trained network."
],
[
"Relation to Existing Work",
"A seminal work [2] showed a generalization error bound based on how the network can be compressed.",
"Although the theoretical insights provided by their analysis are quite instructive, the theory does not give a practical compression method.",
"In fact, a random projection is proposed in the analysis, but it is not intended for practical use.",
"The most difference is that their analysis exploits the near low rankness of the weight matrix $W^{(\\ell )}$ , while ours exploits the near low rankness of the covariance matrix $\\widehat{\\Sigma }^{(\\ell )}$ .",
"They are not directly comparable; thus, we numerically compare the intrinsic dimensionality of both with a VGG-19 network trained on CIFAR-10.",
"Table REF summarizes a comparison of the intrinsic dimensionalities.",
"For our analysis, we used $\\hat{N}_{\\ell }(\\lambda _\\ell ) \\hat{N}_{\\ell +1}(\\lambda _{\\ell + 1}) k^2 $ for the intrinsic dimensionality of the $\\ell $ -th layer, where $k$ is the kernel sizeWe omitted quantities related to the depth $L$ and $\\log $ term, but the intrinsic dimensionality of [2] also omits these factors..",
"This is the number of parameters in the $\\ell $ -th layer for the width $m^{\\sharp }_{\\ell } \\simeq \\hat{N}_{\\ell }(\\lambda _\\ell )$ where $\\lambda _\\ell $ was set as $\\lambda _\\ell = 10^{-3} \\times \\mathrm {Tr}[\\widehat{\\Sigma }_{(\\ell )}]$ , which is sufficiently small.",
"We can see that the quantity based on our degrees of freedom give significantly small values in almost all layers.",
"Table: Comparison of the intrinsic dimensionality ofour degrees of freedom and existing one.",
"They are computed for a VGG-19 network trained on CIFAR-10.The PAC-Bayes bound [9], [36] is also a promising approach for obtaining the nonvacuous generalization error bound of a compressed network.",
"However, these studies “assume” the existence of effective compression methods and do not provide any specific algorithm.",
"[30], [29] also pointed out the importance of the degrees of freedom for analyzing the generalization error of deep learning but did not give a practical algorithm.",
"Figure: Relation between accuracy and the hyper parameters λ ℓ \\lambda _\\ell and θ\\theta .The best λ\\lambda and θ\\theta are indicated by the star symbol."
],
[
"Numerical Experiments",
"In this section, we conduct numerical experiments to show the validity of our theory and the effectiveness of the proposed method."
],
[
"Eigenvalue Distribution and Compression Ability",
"We show how the rate of decrease in the eigenvalues affects the compression accuracy to justify our theoretical analysis.",
"We constructed a network (namely, NN3) consisting of three hidden fully connected layers with widths $(300,1000,300)$ following the settings in [1] and trained it with 60,000 images in MNIST and 50,000 images in CIFAR-10.",
"Figure REF shows the magnitudes of the eigenvalues of the 3rd hidden layers of the networks trained for each dataset (plotted on a semilog scale).",
"The eigenvalues are sorted in decreasing order, and they are normalized by division by the maximum eigenvalue.",
"We see that eigenvalues for MNIST decrease much more rapidly than those for CIFAR-10.",
"This indicates that MINST is “easier” than CIFAR-10 because the degrees of freedom (an intrinsic dimensionality) of the network trained on MNIST are relatively smaller than those trained on CIFAR-10.",
"Figure REF presents the (relative) compression error $\\Vert \\widehat{f}- f^{\\sharp }\\Vert _{n}/\\Vert \\widehat{f}\\Vert _n$ versus the width $m^{\\sharp }_{3}$ of the compressed network where we compressed only the 3rd layer and $\\lambda _3$ was fixed to a constant $10^{-6} \\times \\mathrm {Tr}[\\widehat{\\Sigma }_{(\\ell )}]$ and $\\theta = 0.5$ .",
"It shows a rapid decrease in the compression error for MNIST than CIFAR-10 (about 100 times smaller).",
"This is because MNIST has faster eigenvalue decay than CIFAR-10.",
"Figure REF shows the relation between the test classification accuracy and $\\lambda _\\ell $ .",
"It is plotted for a VGG-13 network trained on CIFAR-10.",
"We chose the width $m^{\\sharp }_{\\ell }$ that gave the best accuracy for each $\\lambda _\\ell $ under the constraint of the compression rate (relative number of parameters).",
"We see that as the compression rate increases, the best $\\lambda _\\ell $ goes down.",
"Our theorem tells that $\\lambda _\\ell $ is related to the compression error through (REF ), that is, as the width goes up, $\\lambda _\\ell $ must goes down.",
"This experiment supports the theoretical evaluation.",
"Figure REF shows the relation between the test classification accuracy and the hyperparameter $\\theta $ .",
"We can see that the best accuracy is achieved around $\\theta =0.3$ for all compression rates, which indicates the superiority of the “combination” of input- and output-information loss and supports our theoretical bound.",
"For low compression rate, the choice of $\\lambda _\\ell $ and $\\theta $ does not affect the result so much, which indicates the robustness of the hyper-parameter choice."
],
[
"Compression on ImageNet Dataset",
"We applied our method to the ImageNet (ILSVRC2012) dataset [6].",
"We compared our method using the ResNet-50 network [15] (experiments for VGG-16 network [28] are also shown in Appendix REF ).",
"Our method was compared with the following pruning methods: ThiNet [24], NISP [34], and sparse regularization [16] (which we call Sparse-reg).",
"As the initial ResNet network, we used two types of networks: ResNet-50-1 and ResNet-50-2.",
"For training ResNet-50-1, we followed the experimental settings in [24] and [34].",
"During training, images were resized as in [24].",
"to 256 $\\times $ 256; then, a 224 $\\times $ 224 random crop was fed into the network.",
"In the inference stage, we center-cropped the resized images to 224 $\\times $ 224.",
"For training ResNet-50-2, we followed the same settings as in [16].",
"In particular, images were resized such that the shorter side was 256, and a center crop of 224 $\\times $ 224 pixels was used for testing.",
"The augmentation for fine tuning was a 224 $\\times $ 224 random crop and its mirror.",
"We compared ThiNet and NISP for ResNet-50-1 (we call our model for this situation “Spec-ResA”) and Sparse-reg for ResNet-50-2 (we call our model for this situation “Spec-ResB”) for fair comparison.",
"The size of compressed network $f^{\\sharp }$ was determined to be as close to the compared network as possible (except, for ResNet-50-2, we did not adopt the “channel sampler” proposed by [16] in the first layer of the residual block; hence, our model became slightly larger).",
"The accuracies are borrowed from the scores presented in each paper, and thus we used different models because the original papers of each model reported for each different model.",
"We employed the simultaneous procedure for compression.",
"After pruning, we carried out fine tuning over 10 epochs, where the learning rate was $10^{-3}$ for the first four epochs, $10^{-4}$ for the next four epochs, and $10^{-5}$ for the last two epochs.",
"We employed $\\lambda _\\ell = 10^{-6}\\times \\mathrm {Tr}[\\widehat{\\Sigma }_{(\\ell )}]$ and $\\theta = 0.5$ .",
"Table: Performance comparison of our method and existing ones for ResNet-50 on ImageNet.“ft” indicates fine tuning after compression.Table REF summarizes the performance comparison for ResNet-50.",
"We can see that for both settings, our method outperforms the others for about $1\\%$ accuracy.",
"This is an interesting result because ResNet-50 is already compact [24] and thus there is less room to produce better performance.",
"Moreover, we remark that all layers were simultaneously trained in our method, while other methods were trained one layer after another.",
"Since our method did not adopt the channel sampler proposed by [16], our model was a bit larger.",
"However, we could obtain better performance by combining it with our method."
],
[
"Conclusion",
"In this paper, we proposed a simple pruning algorithm for compressing a network and gave its approximation and generalization error bounds using the degrees of freedom.",
"Unlike the existing compression based generalization error analysis, our analysis is compatible with a practically useful method and further gives a tighter intrinsic dimensionality bound.",
"The proposed algorithm is easily implemented and only requires linear algebraic operations.",
"The numerical experiments showed that the compression ability is related to the eigenvalue distribution, and our algorithm has favorable performance compared to existing methods."
],
[
"Acknowledgements",
"TS was partially supported by MEXT Kakenhi (18K19793, 18H03201 and 20H00576) and JST-CREST, Japan.",
"—Appendix—"
],
[
"Extension to convolutional neural network",
"An extension of our method to convolutional layers is a bit tricky.",
"There are several options, but to perform channel-wise pruning, we used the following “covariance matrix” between channels in the experiments.",
"Suppose that a channel $k$ receives the input $\\phi _{k;u,v}(x)$ where $1 \\le u \\le I_\\tau ,~1\\le v \\le I_h$ indicate the spacial index, then “covariance” between the channels $k$ and $k^{\\prime }$ can be formulated as $\\widehat{\\Sigma }_{k,k^{\\prime }} =\\frac{1}{n}\\sum _{i=1}^{n} (\\frac{1}{I_\\tau I_h} \\sum _{u,v} \\phi _{k;u,v}(x_{i}) \\phi _{k^{\\prime };u,v}(x_{i}))$ .",
"As for the covariance between an output channel $k^{\\prime }$ and an input channel $k$ (which corresponds to the $(k^{\\prime },k)$ -th element of $Z^{(\\ell )} \\widehat{\\Sigma }_{F,J} = \\mathrm {Cov}(Z^{(\\ell )} \\phi (X),\\phi _J(X))$ for the fully connected situation), it can be calculated as $\\widehat{\\Sigma }_{k^{\\prime },k} =\\frac{1}{n}\\sum _{i=1}^{n} (\\frac{1}{I_\\tau I_h} \\sum _{u,v}\\frac{1}{I^{\\prime }_{(u,v)}} \\sum _{u^{\\prime },v^{\\prime }: (u,v) \\in \\mathrm {Res}(u^{\\prime },v^{\\prime })}(Z^{(\\ell )} \\phi (x_i))_{k^{\\prime };u^{\\prime },v^{\\prime }}(x_{i}) \\phi _{k;u,v}(x_{i}))$ , where $\\mathrm {Res}(u^{\\prime },v^{\\prime })$ is the receptive field of the location $u^{\\prime },v^{\\prime }$ in the output channel $k^{\\prime }$ , and $I^{\\prime }_{(u,v)}$ are the number of locations $(u^{\\prime },v^{\\prime })$ that contain $(u,v)$ in their receptive fields."
],
[
"Proof of Theorem ",
"The output of its internal layer (before activation) is denoted by $\\hat{F}_{\\ell }(x) = (\\hat{W}^{(\\ell )}\\eta (\\cdot ) + \\hat{b}^{(\\ell )}) \\circ \\dots \\circ (\\hat{W}^{(1)} x + \\hat{b}^{(1)}).$ We denote the set of row vectors of $Z^{(\\ell )}$ by $\\mathcal {Z}_\\ell $ , i.e., $\\mathcal {Z}_\\ell =\\lbrace Z^{(\\ell )\\top }_{1,:},\\dots , Z^{(\\ell )\\top }_{m,:} \\rbrace $ .",
"Conversely, we may define $Z^{(\\ell )}$ by specifying ${\\mathcal {Z}}_\\ell $ .",
"Here, we restate Theorem REF in a complete form that contains both of backward procedure and simultaneous procedure.",
"Theorem 3 (Restated) Assume that the regularization parameter $\\tau $ in the pruning procedure (REF ) is defined by the leverage score $\\tau \\leftarrow \\tau ^{(\\ell )} := m^{\\sharp }_{\\ell }\\lambda _\\ell {\\tilde{\\tau }^{(\\ell )}}$ .",
"(i) Backward-procedure: Let $\\mathcal {Z}_\\ell $ for the output information loss be $\\mathcal {Z}_\\ell = \\left\\lbrace \\frac{ \\sqrt{m_{\\ell } q_j^{(\\ell )}}}{\\max _{j^{\\prime }} \\Vert \\hat{W}^{(\\ell )}_{j^{\\prime },:}\\Vert } \\hat{W}^{(\\ell )}_{j,:} \\mid j \\in J^{\\sharp }_{\\ell +1}\\right\\rbrace $ where $q_j^{(\\ell )} := \\frac{( {\\tilde{\\tau }^{(\\ell +1)}}_j)^{-1}}{\\sum _{j^{\\prime } \\in J^{\\sharp }_{\\ell + 1}} ({\\tilde{\\tau }^{(\\ell +1)}}_{j^{\\prime }})^{-1}}~(j \\in J^{\\sharp }_{\\ell + 1})$ and $q_j^{(\\ell )} = 0~(\\text{otherwise})$ , and define $\\zeta _{\\ell ,\\theta }=N_\\ell ^\\theta (\\lambda _\\ell ) \\left( \\theta \\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{ \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}}\\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} } + (1-\\theta )m_{\\ell }\\right)^{-1}$$\\Vert \\cdot \\Vert _{\\mathrm {op}}$ represents the operator norm of a matrix (the largest absolute singular value).. Then, if we solve the optimization problem (REF ) with an additional constraint $\\sum _{j \\in J} ({\\tilde{\\tau }^{(\\ell )}}_j)^{-1} \\le \\frac{5}{3}m_\\ell m^{\\sharp }_{\\ell }$ for the index set $J$ , then the optimization problem is feasible, and the overall approximation error of $f^{\\sharp }$ is bounded by $\\Vert \\widehat{f}- f^{\\sharp }\\Vert _n\\le \\sum \\nolimits _{\\ell = 2}^L\\left( \\bar{R}^{L-\\ell + 1} \\sqrt{ {\\textstyle \\prod _{\\ell ^{\\prime }=\\ell }^L}\\zeta _{\\ell ^{\\prime },\\theta } } \\right) \\sqrt{\\lambda _\\ell },$ for $\\bar{R}= \\sqrt{\\hat{c}} R$ where $\\hat{c}$ is a universal constant.",
"(ii) Simultaneous-procedure: Suppose that there exists $c_{\\mathrm {scale}}> 0$ such that $\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 \\le c_{\\mathrm {scale}}R^2 {\\tilde{\\tau }^{(\\ell +1)}}_{j},$ and we employ ${\\mathcal {Z}}_\\ell = \\lbrace \\hat{W}^{(\\ell )}_{j,:}/\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert \\mid j \\in [m_{\\ell +1}]\\rbrace $ for the output aware objective.",
"Then, we have the same bound as (REF ) for $q_j^{(\\ell )} = ({\\tilde{\\tau }^{(\\ell +1)}}_{j})^{-1}/\\sum _{j^{\\prime } \\in [m_{\\ell + 1}]} ({\\tilde{\\tau }^{(\\ell +1)}}_{j^{\\prime }})^{-1}~(\\forall j \\in [m_{\\ell + 1}])$ and $\\zeta _{\\ell , \\theta } =c_{\\mathrm {scale}}N_\\ell ^\\theta (\\lambda _\\ell ) \\left( \\theta {\\textstyle \\frac{m^{\\sharp }_{\\ell +1} \\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta ) m^{\\sharp }_{\\ell +1} \\right)^{-1}.$ The assumption (REF ) is rather strong, but we see that it is always satisfied by $c_{\\mathrm {scale}}= 1$ when $\\lambda _\\ell = 0$ and by $c_{\\mathrm {scale}}= \\mathrm {Tr}[\\widehat{\\Sigma }_{(\\ell +1)}]/(m_{\\ell +1}\\min _{j} \\widehat{\\Sigma }_{(\\ell +1),(j,j)})$ when $\\lambda _\\ell = \\infty $ .",
"Thus, it is satisfied if the variances of the nodes in the $\\ell +1$ -th layer is balanced, which is ensured if we are applying batch normalization."
],
[
"Preparation of lemmas",
"To derive the approximation error bound.",
"we utilize the following proposition that was essentially proven by [3].",
"This proposition states the connection between the degrees of freedom and the compression error, that is, it characterize the sufficient width $m^{\\sharp }_{\\ell }$ to obtain a pre-specified compression error $\\lambda _\\ell $ .",
"Actually, we will see that the eigenvalue essentially controls this relation through the degrees of freedom.",
"Proposition 1 There exists a probability measure $q_\\ell $ on $\\lbrace 1,\\dots ,m_\\ell \\rbrace $ such that for any $\\delta \\in (0,1)$ and $\\lambda > 0$ , i.i.d.",
"sample $v_1,\\dots ,v_m \\in \\lbrace 1,\\dots ,m_\\ell \\rbrace $ from $q_\\ell $ satisfies, with probability $1-\\delta $ , that $& \\inf _{\\beta \\in \\mathbb {R}^m}\\left\\lbrace \\left\\Vert \\alpha ^\\top \\eta (\\hat{F}_{\\ell -1}(\\cdot )) - \\sum _{j=1}^m \\beta _j q_\\ell (v_j)^{-1/2} \\eta (\\hat{F}_{\\ell -1}(\\cdot ))_{v_j} \\right\\Vert _n^2+ m \\lambda \\Vert \\beta \\Vert ^2 \\right\\rbrace \\\\&\\le 4 \\lambda \\alpha ^\\top \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda \\mathrm {I})^{-1} \\alpha ,$ for every $\\alpha \\in \\mathbb {R}^{m_\\ell }$ , if $m \\ge 5 \\hat{N}_{\\ell }(\\lambda ) \\log (16 \\hat{N}_{\\ell }(\\lambda )/\\delta ).$ Moreover, the optimal solution $\\hat{\\beta }$ satisfies $\\Vert \\hat{\\beta }\\Vert _2^2 \\le \\frac{4 \\Vert \\alpha \\Vert ^2}{m}$ .",
"This is basically a direct consequence from Proposition 1 in [3] and its discussions.",
"The original statement does not include the regularization term $m \\lambda \\Vert \\beta \\Vert ^2$ in the LHS and $\\alpha ^\\top \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda \\mathrm {I})^{-1} \\alpha $ in the right hand side.",
"However, by carefully following the proof, the bound including these additional factors is indeed proven.",
"The norm bond of $\\hat{\\beta }$ is guaranteed by the following relation: $m \\lambda \\Vert \\hat{\\beta }\\Vert ^2 \\le 4 \\lambda \\alpha ^\\top \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda \\mathrm {I})^{-1} \\alpha \\le 4 \\lambda \\Vert \\alpha \\Vert ^2.$ Proposition REF indicates the following lemma by the the scale invariance of $\\eta $ , $\\eta (a x) = a \\eta (x)~(a >0)$ .",
"Lemma 1 Suppose that $\\tau ^{\\prime }_j = \\frac{1}{\\hat{N}_{\\ell }(\\lambda )}\\sum _{l=1}^{m_\\ell }U_{j,l}^2\\frac{\\hat{\\mu }_l^{(\\ell )}}{\\hat{\\mu }_l^{(\\ell )} + \\lambda }= \\frac{1}{\\hat{N}_{\\ell }(\\lambda )} [\\widehat{\\Sigma }^{(\\ell )} (\\widehat{\\Sigma }^{(\\ell )} + \\lambda \\mathrm {I})^{-1}]_{j,j}~~(j \\in \\lbrace 1,\\dots ,m_\\ell \\rbrace ),$ where $U = (U_{j,l})_{j,l}$ is the orthogonal matrix that diagonalizes $\\widehat{\\Sigma }^{(\\ell )}$ , that is, $\\widehat{\\Sigma }^{(\\ell )}= U\\mathrm {diag}\\left(\\hat{\\mu }_1^{(\\ell )},\\dots ,\\hat{\\mu }_{m_\\ell }^{(\\ell )}\\right) U^\\top $ .",
"For $\\lambda >0$ , and any $1/2 > \\delta >0$ , if $m \\ge 5 \\hat{N}_{\\ell }(\\lambda ) \\log (16 \\hat{N}_{\\ell }(\\lambda )/\\delta ),$ then there exist $v_1,\\dots , v_m \\in \\lbrace 1,\\dots ,m_\\ell \\rbrace $ such that, for every $\\alpha \\in \\mathbb {R}^{m_\\ell }$ , $& \\inf _{\\beta \\in \\mathbb {R}^m}\\left\\lbrace \\left\\Vert \\alpha ^\\top \\eta (\\hat{F}_{\\ell -1}(\\cdot )) - \\sum _{j=1}^m \\beta _j {\\tau ^{\\prime }_j}^{-1/2} \\eta (\\hat{F}_{\\ell -1}(\\cdot ))_{v_j} \\right\\Vert _n^2+ m \\lambda \\Vert \\beta \\Vert ^2 \\right\\rbrace \\\\& \\le 4 \\lambda \\alpha ^\\top \\widehat{\\Sigma }^{(\\ell )} (\\widehat{\\Sigma }^{(\\ell )} + \\lambda \\mathrm {I})^{-1} \\alpha ,$ and $\\sum _{j=1}^m {\\tau _j^{\\prime }}^{-1} \\le (1-2\\delta )^{-1} m \\times m_\\ell .$ Suppose that the measure $Q_\\ell $ is the counting measure, $Q_\\ell (J) = |J|$ for $J \\subset \\lbrace 1,\\dots ,m_\\ell \\rbrace $ , and $q_\\ell $ is a density given by $q_\\ell (j) = \\tau ^{\\prime }_j~(j \\in \\lbrace 1,\\dots ,m_\\ell \\rbrace )$ with respect to the base measure $Q_\\ell $ .",
"Suppose that $v_1,\\dots ,v_m \\in \\lbrace 1,\\dots ,m_\\ell \\rbrace $ is an i.i.d.",
"sequence distributed from $q_\\ell \\mathrm {d}Q_\\ell $ , then [3] showed that this sequence satisfies the assertion given in Proposition REF .",
"Notice that $\\mathrm {E}_v[\\frac{1}{m} \\sum _{j=1}^m q_\\ell (v_j)^{-1}] =\\mathrm {E}_v[q_\\ell (v)^{-1}] = \\int _{[m_\\ell ]} q_\\ell (v)^{-1} q_\\ell (v) \\mathrm {d}Q_\\ell (v) = \\int _{[m_\\ell ]} 1 \\mathrm {d}Q_\\ell (v) = m_\\ell $ , thus an i.i.d.",
"sequence $\\lbrace v_1,\\dots ,v_m\\rbrace $ satisfies $\\frac{1}{m} \\sum _{j=1}^m q_\\ell (v_j)^{-1} \\le m_\\ell /(1-2 \\delta )$ with probability $2\\delta $ by the Markov's inequality.",
"Combining this with Proposition REF , the i.i.d.",
"sequence $\\lbrace v_1,\\dots ,v_m\\rbrace $ and $\\tau ^{\\prime }_j = q_\\ell (v_j)$ satisfies the condition in the statement with probability $1-(\\delta + 1- 2\\delta ) = \\delta >0$ .",
"This ensures the existence of sequences $\\lbrace v_j\\rbrace _{j=1}^m$ and $\\lbrace \\tau ^{\\prime }_j\\rbrace _{j=1}^m$ that satisfy the assertion."
],
[
"General fact",
"Since Lemma REF with $\\delta = 1/5$ states that if $m^{\\sharp }_{\\ell } \\ge 5 \\hat{N}_{\\ell }(\\lambda _\\ell ) \\log (80 \\hat{N}_{\\ell }(\\lambda _\\ell ))$ , then there exists $J \\subset [m_\\ell ]^{m^{\\sharp }_{\\ell } }$ such that $\\inf _{\\alpha \\in \\mathbb {R}^{|J|}}\\Vert z^\\top \\phi - \\alpha ^\\top \\phi _{J}\\Vert _n^2 +\\lambda _\\ell |J| \\Vert \\alpha \\Vert _{\\tau ^{\\prime }}^2 \\le 4 \\lambda _\\ell z^\\top \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda \\mathrm {I})^{-1} z~~~(\\forall z \\in \\mathbb {R}^{m_\\ell }),$ and $\\sum _{j \\in J^{\\sharp }_{\\ell }} ({\\tilde{\\tau }^{(\\ell )}}_j)^{-1} \\le \\frac{5}{3} m_\\ell \\times m^{\\sharp }_{\\ell }$ is satisfied (here, note that $\\tau ^{\\prime }$ given in Eq.",
"(REF ) is equivalent to ${\\tilde{\\tau }^{(\\ell )}}$ ).",
"Evaluation of $L^{(\\mathrm {A})}_{\\tau }(J)$ : By setting $z = e_j~(j=1,\\dots ,m_\\ell )$ where $e_j$ is an indicator vector which has 1 at its $j$ -th component and 0 in other components, and summing up them for $j=1,\\dots ,m_\\ell $ , it holds that $L^{(\\mathrm {A})}_{\\tau }(J) = \\inf _{A \\in \\mathbb {R}^{m_\\ell \\times |J|}} \\Vert \\phi - A\\phi _J \\Vert _n^2 + \\lambda _\\ell |J| \\Vert A\\Vert _{\\tau ^{\\prime }}^2\\le 4 \\lambda _\\ell \\hat{N}_{\\ell }(\\lambda _\\ell ),$ for the same $J$ as above.",
"Here, the optimal $A$ , which is denoted by $\\hat{A}_J$ , is given by $\\hat{A}_J = \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1}.$ Evaluation of $L^{(\\mathrm {B})}_{\\tau }(J)$ : By letting $z \\in \\mathcal {Z}_\\ell $ and summing up them, we also have $L^{(\\mathrm {B})}_{\\tau }(J)=\\inf _{B \\in \\mathbb {R}^{m^{\\sharp }_{\\ell + 1} \\times |J|}} \\Vert Z^{(\\ell )} \\phi - B \\phi _J \\Vert _n^2 + \\lambda _\\ell |J| \\Vert B\\Vert _{\\tau ^{\\prime }}^2\\le 4 \\lambda _\\ell \\mathrm {Tr}[Z^{(\\ell )} \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda _\\ell \\mathrm {I})^{-1} Z^{(\\ell )\\top }].$ for the same $J$ as above.",
"Remind again that the optimal $B$ , which is denoted by $\\hat{B}_J$ , is given by $\\hat{B}_J = Z^{(\\ell )} \\widehat{\\Sigma }_{F,J}(\\widehat{\\Sigma }_{J,J} + \\mathrm {I}_\\tau )^{-1} = Z^{(\\ell )} \\hat{A}_J.$ Combining the bounds for $L^{(\\mathrm {A})}_{\\tau }(J)$ and $L^{(\\mathrm {B})}_{\\tau }(J)$ : By combining the above evaluation, we have that $\\theta L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{}) + (1-\\theta ) L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{})\\le 4 \\lambda _\\ell \\lbrace \\theta \\hat{N}_{\\ell }(\\lambda _\\ell )+(1-\\theta ) \\mathrm {Tr}[Z^{(\\ell )} \\widehat{\\Sigma }^{(\\ell )} (\\widehat{\\Sigma }^{(\\ell )} + \\lambda _\\ell \\mathrm {I})^{-1} Z^{(\\ell )\\top }]\\rbrace ,$ where $J^{\\sharp }_{\\ell }$ is the minimizer of $\\theta L^{(\\mathrm {A})}_{\\tau }(J) + (1-\\theta ) L^{(\\mathrm {B})}_{\\tau }(J)$ with respect to $J$ .",
"From now on, we let $\\tau = \\lambda _\\ell m^{\\sharp }_{\\ell } \\tau ^{\\prime }(=\\lambda _\\ell |J^{\\sharp }_{\\ell }| \\tau ^{\\prime } )$ as defined in the main text."
],
[
" (i) Backward-procedure",
"From now on, we give the bound corresponding to the backward-procedure.",
"The proof consists of three parts: (i) evaluation of the compression error in each layer, (ii) evaluation of the norm of the weight matrix for the compressed network, and (iii) overall compression error of whole layer.",
"In (i), we use Lemma REF to evaluate the compression error based on the eigenvalue distribution of the covariance matrix.",
"In (ii), we again use Lemma REF to bound the norm of the compressed network.",
"This is important to evaluate the overall compression error because the norm controls how the compression error in each layer propagates to the final output.",
"In (iii), we combine the results in (i) and (ii) to obtain the overall compression error.",
"First, note that, for the choice of ${\\mathcal {Z}}_\\ell = \\left\\lbrace \\frac{ \\sqrt{m_{\\ell } q_j^{(\\ell )}}}{\\max _{j^{\\prime }} \\Vert \\hat{W}^{(\\ell )}_{j^{\\prime },:}\\Vert } \\hat{W}^{(\\ell )}_{j,:} \\mid j \\in J^{\\sharp }_{\\ell + 1} \\right\\rbrace $ , it holds that $L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{\\ell }) \\le 4 \\lambda _\\ell \\sum _{z \\in {\\mathcal {Z}}_\\ell }z^\\top \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda _\\ell \\mathrm {I})^{-1} z\\le 4 \\lambda _\\ell \\sum _{z \\in {\\mathcal {Z}}_\\ell } \\Vert z\\Vert ^2\\le 4 \\lambda _\\ell m_{\\ell } \\sum _{j\\in J^{\\sharp }_{\\ell +1}} q_j^{(\\ell )} = 4 \\lambda _\\ell m_{\\ell }.$ Compression error bound: Here, we give the compression error bound of the backward procedure.",
"For the optimal $J^{\\sharp }_{\\ell }$ , we have that $\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{}}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 & =\\hat{W}^{(\\ell )}_{j,:}[\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{}}(\\widehat{\\Sigma }_{J^{\\sharp },J^{\\sharp }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp },F} ](\\hat{W}^{(\\ell )}_{j,:} )^\\top \\\\& = \\mathrm {Tr}\\lbrace [\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }}(\\widehat{\\Sigma }_{J^{\\sharp },J^{\\sharp }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp },F} ](\\hat{W}^{(\\ell )}_{j,:})^\\top \\hat{W}^{(\\ell )}_{j,:} \\rbrace ,$ and the optimal $\\alpha $ in the left hand side is given by $\\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }_{\\ell }}$ .",
"Hence, it holds that $&\\sum _{j\\in J^{\\sharp }_{\\ell +1}}\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\lbrace \\Vert {q_j^{(\\ell )}}^{1/2} \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace =\\sum _{j\\in J^{\\sharp }_{\\ell +1} } {q_j^{(\\ell )}}\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace \\\\& =\\sum _{j\\in J^{\\sharp }_{\\ell +1}} {q_j^{(\\ell )}}\\mathrm {Tr}\\lbrace [\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{\\ell }}(\\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },J^{\\sharp }_{\\ell }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },F} ](\\hat{W}^{(\\ell )}_{j,:})^\\top \\hat{W}^{(\\ell )}_{j,:} \\rbrace \\\\&=\\mathrm {Tr}\\lbrace [\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{\\ell }}(\\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },J^{\\sharp }_{\\ell }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },F} ]( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )} \\rbrace \\\\& \\le \\mathrm {Tr}[ \\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{\\ell }}(\\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },J^{\\sharp }_{\\ell }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },F} ]\\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} \\\\& = L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{\\ell }) \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}}\\le L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{\\ell })\\frac{ \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}\\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 \\\\&\\le L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{\\ell })\\frac{m_{\\ell } \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}\\frac{R^2}{m_{\\ell +1}m_\\ell },$ where where we used the assumption $\\max _{j} \\Vert W^{(\\ell )}_{j,:}\\Vert \\le R/\\sqrt{m_{\\ell +1}}$ .",
"In the same manner, we also have that $&\\sum _{j\\in J^{\\sharp }_{\\ell +1} }{q_j^{(\\ell )}}\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}}\\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace \\\\& =\\frac{\\max _{j^{\\prime }} \\Vert \\hat{W}^{(\\ell )}_{j^{\\prime },:}\\Vert ^2 }{m_{\\ell }}\\sum _{j\\in {J^{\\sharp }_{\\ell +1}}}\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\Vert { \\textstyle \\frac{ \\sqrt{m_{\\ell } q_j^{(\\ell )}}}{\\max _{j^{\\prime }} \\Vert \\hat{W}^{(\\ell )}_{j^{\\prime },:}\\Vert } }\\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }} \\right\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2=\\\\&\\le L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{\\ell }) \\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{m_{\\ell }}\\le L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{\\ell }) \\frac{R^2}{m_\\ell m_{\\ell +1}}.$ These inequalities imply that $& \\sum _{j\\in J^{\\sharp }_{\\ell +1}}q_j^{(\\ell )}\\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\phi _{J^{\\sharp }}\\Vert _n^2+ \\Vert \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\Vert _\\tau ^2 \\right\\rbrace \\\\&\\le 4 \\lambda _\\ell \\frac{\\lbrace \\theta \\hat{N}_{\\ell }(\\lambda _\\ell )+(1-\\theta ) \\mathrm {Tr}[Z^{(\\ell )} \\widehat{\\Sigma }_\\ell (\\widehat{\\Sigma }_\\ell + \\lambda _\\ell \\mathrm {I})^{-1} Z^{(\\ell )\\top }]\\rbrace }{\\theta \\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{ m_{\\ell } \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}}\\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} } + (1-\\theta )}\\frac{R^2}{m_\\ell m_{\\ell +1}} \\\\&\\le 4 \\lambda _\\ell \\frac{\\lbrace \\theta \\hat{N}_{\\ell }(\\lambda _\\ell )+(1-\\theta ) m_\\ell \\rbrace }{\\left[\\theta \\frac{ \\max _{j \\in [m_{\\ell +1}]} \\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2}{ m_{\\ell } \\Vert ( \\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}}\\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} } + (1-\\theta ) \\right]m_\\ell }\\frac{R^2}{ m_{\\ell +1}} \\\\& \\le 4 \\lambda _\\ell \\zeta _{\\ell ,\\theta } \\frac{R^2 }{m_{\\ell +1}}.$ Norm bound of the coefficients: Here, we give an upper bound of the norm of the weight matrices for the compressed network.",
"From (REF ) and the definition that $\\tau ^{(\\ell )} = \\lambda _\\ell m^{\\sharp }_{\\ell }{\\tilde{\\tau }^{(\\ell )}}$ , we have that $\\sum _{j\\in J^{\\sharp }_{\\ell +1} }q_j^{(\\ell )} \\Vert \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\Vert _{{\\tilde{\\tau }^{(\\ell )}}}^2\\le \\frac{1}{ \\lambda _\\ell m^{\\sharp }_{\\ell }} 4 \\lambda _\\ell \\zeta _{\\ell ,\\theta } \\frac{R^2 }{m_{\\ell +1}}= 4 \\zeta _{\\ell ,\\theta } \\frac{R^2 }{m_{\\ell +1}m^{\\sharp }_{\\ell }}.$ Here, by Eq.",
"(REF ), the condition $\\sum _{j \\in J^{\\sharp }_{\\ell + 1}} ({\\tilde{\\tau }^{(\\ell +1)}}_j)^{-1} \\le \\frac{5}{3} m_{\\ell +1} m^{\\sharp }_{\\ell +1}$ is feasible, and under this condition, we also have that $\\sum _{j\\in J^{\\sharp }_{\\ell + 1}}({\\tilde{\\tau }^{(\\ell + 1)}}_j)^{-1} \\Vert \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\Vert _{{\\tilde{\\tau }^{(\\ell )}}}^2\\le 4 \\zeta _{\\ell ,\\theta } \\frac{R^2 }{m_{\\ell +1}m^{\\sharp }_{\\ell }} \\times \\frac{5}{3} m_{\\ell +1} m^{\\sharp }_{\\ell +1}= \\frac{20}{3} \\zeta _{\\ell ,\\theta }\\frac{m^{\\sharp }_{\\ell +1}}{m^{\\sharp }_{\\ell }} R^2,$ where we used the definition $q$ Similarly, the approximation error bound Eq.",
"(REF ) can be rewritten as $& \\sum _{j\\in J^{\\sharp }_{\\ell + 1}}({\\tilde{\\tau }^{(\\ell + 1)}}_j)^{-1}\\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\phi _{J^{\\sharp }}\\Vert _n^2\\le \\frac{20}{3} \\lambda _\\ell \\zeta _{\\ell ,\\theta } m^{\\sharp }_{\\ell +1} R^2.",
"$ For $\\ell = L$ , the same inequality holds for $m_{L+1}= m^{\\sharp }_{L+1} = 1$ and ${\\tilde{\\tau }^{(L+1)}}_j =1~(j=1)$ .",
"Overall approximation error bound: Given these inequalities, we bound the overall approximation error bound.",
"Let $J^{\\sharp }_{\\ell }$ be the optimal index set chosen by Spectral Pruning for the $\\ell $ -th layer, and the parameters of compressed network be denoted by $W^{\\sharp (\\ell )} = \\hat{W}^{(\\ell )}_{J^{\\sharp }_{\\ell +1},[m_\\ell ]}\\hat{A}_{J^{\\sharp }_{\\ell }}\\in \\mathbb {R}^{m^{\\sharp }_{\\ell +1} \\times m^{\\sharp }_{\\ell }}, ~~b^{\\sharp (\\ell )} = \\hat{b}^{(\\ell )}_{J^{\\sharp }_{\\ell + 1}}\\in \\mathbb {R}^{m^{\\sharp }_{\\ell +1}}.$ Then, it holds that $f^{\\sharp }(x) & = (W^{\\sharp (L)} \\eta (\\cdot ) + b^{\\sharp (L)}) \\circ \\dots \\circ (W^{\\sharp (1)} x + b^{\\sharp (1)}).$ Then, due to the scale invariance of $\\eta $ , we also have $f^{\\sharp }(x) & = (W^{\\sharp (L)} \\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{\\frac{1}{2}}} \\eta (\\cdot ) + b^{\\sharp (L)}) \\circ (\\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{-\\frac{1}{2}}} W^{\\sharp (L-1)} \\mathrm {I}_{({\\tilde{\\tau }^{(L-1)}})^{\\frac{1}{2}}} \\eta (\\cdot ) + \\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{-\\frac{1}{2}}} b^{\\sharp (L-1)})\\dots \\circ (\\mathrm {I}_{({\\tilde{\\tau }^{(2)}})^{-\\frac{1}{2}}} W^{\\sharp (1)} x + \\mathrm {I}_{({\\tilde{\\tau }^{(2)}})^{-\\frac{1}{2}}} b^{\\sharp (1)}).$ Then, if we define as $\\widetilde{W}^{(\\ell )} = \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell +1)}})^{-\\frac{1}{2}}} W^{\\sharp (\\ell )} \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell )}})^{\\frac{1}{2}} }$ and $\\tilde{b}^{(\\ell )} = \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell +1)}})^{-\\frac{1}{2}}} b^{\\sharp (\\ell )}$ , then we also have another representation of $f^{\\sharp }$ as $f^{\\sharp }(x) & = (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(1)} x + \\tilde{b}^{(1)}).$ In the same manner, the original trained network $\\widehat{f}$ is also rewritten as $\\widehat{f}(x)& = (\\hat{W}^{(L)} \\eta (\\cdot ) + \\hat{b}^{(L)}) \\circ \\dots \\circ (\\hat{W}^{(1)} x + \\hat{b}^{(1)}) \\\\& = (\\hat{W}^{(L)} \\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{\\frac{1}{2}} } \\eta (\\cdot ) + \\hat{b}^{(L)}) \\circ (\\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{-\\frac{1}{2}}} \\hat{W}^{(L-1)} \\mathrm {I}_{({\\tilde{\\tau }^{(L-1)}})^{\\frac{1}{2}}} \\eta (\\cdot ) + \\mathrm {I}_{({\\tilde{\\tau }^{(L)}})^{-\\frac{1}{2}}} \\hat{b}^{(L-1)})\\circ \\dots \\circ (\\mathrm {I}_{({\\tilde{\\tau }^{(2)}})^{-\\frac{1}{2}}} \\hat{W}^{(1)} x + \\mathrm {I}_{({\\tilde{\\tau }^{(2)}})^{-\\frac{1}{2}}} \\hat{b}^{(1)}) \\\\& =: (\\ddot{W}^{(L)} \\eta (\\cdot ) + \\ddot{b}^{(L)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}),$ where we defined $\\ddot{W}^{(\\ell )} := \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell +1)}})^{-\\frac{1}{2}}} \\hat{W}^{(\\ell )} \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell )}})^{\\frac{1}{2}}}$ and $\\ddot{b}^{(\\ell )} := \\mathrm {I}_{({\\tilde{\\tau }^{(\\ell +1)}})^{-\\frac{1}{2}}} \\hat{b}^{(\\ell )}$ .",
"Then, the difference between $f^{\\sharp }$ and $\\widehat{f}$ can be decomposed into $& f^{\\sharp }(x) - \\widehat{f}(x) = (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(1)} x + \\tilde{b}^{(1)})- (\\ddot{W}^{(L)} \\eta (\\cdot ) + \\ddot{b}^{(L)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\\\=&\\sum _{\\ell =2}^L\\Big \\lbrace (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\widetilde{W}^{(\\ell )} \\eta (\\cdot ) + \\tilde{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\\\& - (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\ddot{W}^{(\\ell )} \\eta (\\cdot ) + \\ddot{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\Big \\rbrace .$ We evaluate the $\\Vert \\cdot \\Vert _n$ -norm of this difference.",
"First, notice that Eq.",
"(REF ) is equivalent to the following inequality: $& \\Vert (\\widetilde{W}^{(\\ell )} \\eta (\\cdot ) + \\tilde{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} \\cdot + \\ddot{b}^{(1)})\\\\&- (\\ddot{W}^{(\\ell )}_{J^{\\sharp }_{\\ell +1},[m_\\ell ]} \\eta (\\cdot ) + \\ddot{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} \\cdot + \\ddot{b}^{(1)})\\Vert _n^2\\le \\hat{c}\\lambda _\\ell \\zeta _{\\ell ,\\theta } m^{\\sharp }_{\\ell +1} R^2.$ (We can check that, even for $\\ell = 2$ , this inequality is correct.)",
"Next, by evaluating the Lipschitz continuity of the $\\ell $ -th layer of $f^{\\sharp }$ as $\\Vert \\widetilde{W}^{(\\ell )} g - \\widetilde{W}^{(\\ell )} g^{\\prime } \\Vert _n^2& = \\frac{1}{n} \\sum _{i=1}^n \\Vert \\widetilde{W}^{(\\ell )} g(x_i) - \\widetilde{W}^{(\\ell )} g^{\\prime }(x_i) \\Vert ^2 \\\\& = \\frac{1}{n} \\sum _{i=1}^n (g(x_i) - g^{\\prime }(x_i))^\\top (\\widetilde{W}^{(\\ell )})^\\top \\widetilde{W}^{(\\ell )} (g(x_i) - g^{\\prime }(x_i)) \\\\& \\le \\frac{1}{n} \\sum _{i=1}^n \\Vert g(x_i) - g^{\\prime }(x_i)\\Vert ^2 \\mathrm {Tr}[ (\\widetilde{W}^{(\\ell )})^\\top \\widetilde{W}^{(\\ell )}] \\\\& \\le \\hat{c}\\zeta _{\\ell ,\\theta } \\frac{m^{\\sharp }_{\\ell +1}}{m^{\\sharp }_{\\ell }} R^2 \\Vert g - g^{\\prime }\\Vert _n^2,$ for $g, g^{\\prime }: \\mathbb {R}^d \\rightarrow \\mathbb {R}^{m^{\\sharp }_{\\ell }}$ , then it holds that $& \\Vert (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\widetilde{W}^{(\\ell )} \\eta (\\cdot ) + \\tilde{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\\\& - (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\ddot{W}^{(\\ell )} \\eta (\\cdot ) + \\ddot{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\Vert _n^2 \\\\\\le &\\prod _{\\ell ^{\\prime } = \\ell +1}^L \\hat{c}\\zeta _{\\ell ^{\\prime },\\theta }\\frac{m^{\\sharp }_{\\ell ^{\\prime }+1}}{m^{\\sharp }_{\\ell ^{\\prime }}} R^2 \\cdot \\hat{c}\\lambda _\\ell \\zeta _{\\ell ,\\theta } m^{\\sharp }_{\\ell +1} R^2\\le \\lambda _\\ell \\prod _{\\ell ^{\\prime } = \\ell }^L (\\hat{c}\\zeta _{\\ell ^{\\prime },\\theta } R^2)= \\lambda _\\ell \\bar{R}^{2(L - \\ell + 1)} \\prod _{\\ell ^{\\prime }=\\ell }^L \\zeta _{\\ell ^{\\prime },\\theta }.$ Then, by summing up the square root of this for $\\ell =2,\\dots ,L$ , then we have the whole approximation error bound."
],
[
"Simultaneous procedure",
"Here, we give bounds corresponding to the simultaneous-procedure.",
"The proof techniques are quite similar to the forward procedure.",
"However, instead of the $\\ell _2$ -norm bound derived in the backward-procedure, we derive $\\ell _\\infty $ -norm bound for both of the approximation error and the norm bounds.",
"We let $q_j^{(\\ell )} = ({\\tilde{\\tau }^{(\\ell )}}_j)^{-1}$ for $j=1,\\dots ,m_{\\ell + 1}$ .",
"As for the input aware quantity $L^{(\\mathrm {A})}_{\\tau }$ , for any $j \\in [m_{\\ell + 1}]$ , it holds that $&\\sum _{j=1}^{m_{\\ell + 1}} \\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\lbrace \\Vert {q_j^{(\\ell )}}^{1/2} \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace =\\sum _{j=1}^{m_{\\ell + 1}} \\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace \\\\& =\\sum _{j=1}^{m_{\\ell + 1}} {q_j^{(\\ell )}} \\hat{W}^{(\\ell )}_{j,:} [\\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{\\ell }}(\\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },J^{\\sharp }_{\\ell }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },F} ](\\hat{W}^{(\\ell )}_{j,:})^\\top \\\\& \\le \\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} \\mathrm {Tr}\\lbrace \\widehat{\\Sigma }_{F,F} - \\widehat{\\Sigma }_{F,J^{\\sharp }_{\\ell }}(\\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },J^{\\sharp }_{\\ell }} + \\mathrm {I}_\\tau )^{-1} \\widehat{\\Sigma }_{J^{\\sharp }_{\\ell },F} \\rbrace \\\\& \\le c_{\\mathrm {scale}}R^2 \\frac{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 } L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{\\ell }).$ Moreover, as for the output aware quantity $L^{(\\mathrm {B})}_{\\tau }$ , we have that $& \\sum _{j =1}^{m_{\\ell + 1}}{q_j^{(\\ell )}}\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}}\\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace \\\\& =\\sum _{j =1}^{m_{\\ell + 1}}{q_j^{(\\ell )}}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2\\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}} \\left\\Vert { \\textstyle \\frac{1}{\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert } }\\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }} \\right\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\\\&\\le c_{\\mathrm {scale}}R^2 L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{\\ell }).",
"$ By combining these inequalities, it holds that $\\sum _{1 \\le j \\le m_{\\ell + 1}}{q_j^{(\\ell )}} \\inf _{\\alpha \\in \\mathbb {R}^{|J^{\\sharp }_{\\ell }|}}\\left\\lbrace \\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\alpha ^\\top \\phi _{J^{\\sharp }_{\\ell }}\\Vert _n^2 +\\Vert \\alpha \\Vert _{\\tau }^2 \\right\\rbrace & \\le \\frac{[\\theta L^{(\\mathrm {A})}_{\\tau }(J^{\\sharp }_{\\ell }) + (1 - \\theta ) L^{(\\mathrm {B})}_{\\tau }(J^{\\sharp }_{\\ell })] }{\\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta )} c_{\\mathrm {scale}}R^2 \\\\& \\le 4 c_{\\mathrm {scale}}\\lambda _\\ell \\frac{\\theta \\hat{N}_{\\ell }(\\lambda _\\ell ) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda _\\ell ; N_\\ell )}{\\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta )}R^2$ Therefore, by the definition of $q_j^{(\\ell )}$ and $\\tau $ , it holds that $\\sum _{1 \\le j \\le m_{\\ell + 1}}({\\tilde{\\tau }^{(\\ell + 1)}}_j)^{-1} \\Vert \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\Vert _{{\\tilde{\\tau }^{(\\ell )}}}^2\\le 4 c_{\\mathrm {scale}}\\frac{\\theta \\hat{N}_{\\ell }(\\lambda _\\ell ) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda _\\ell ; N_\\ell )}{m^{\\sharp }_{\\ell }\\left[ \\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta )\\right]}R^2= 4 \\zeta _{\\ell ,\\theta }\\frac{m^{\\sharp }_{\\ell + 1}}{m^{\\sharp }_{\\ell }} R^2.$ Similarly, the approximation error bound can be evaluated as $& \\sum _{j \\in [m_{\\ell + 1}] }({\\tilde{\\tau }^{(\\ell + 1)}}_j)^{-1}\\Vert \\hat{W}^{(\\ell )}_{j,:} \\phi - \\hat{W}^{(\\ell )}_{j,:} \\hat{A}_{J^{\\sharp }} \\phi _{J^{\\sharp }}\\Vert _n^2\\le 4 c_{\\mathrm {scale}}\\lambda _\\ell \\frac{\\theta \\hat{N}_{\\ell }(\\lambda _\\ell ) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda _\\ell ; N_\\ell )}{\\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta )}R^2= 4 \\lambda _\\ell \\zeta _{\\ell ,\\theta }m^{\\sharp }_{\\ell + 1} R^2.$ This gives the following equivalent inequality: $& \\sum _{j \\in [m_{\\ell }]} \\Vert (\\widetilde{W}^{(\\ell )}_{j,:} \\eta (\\cdot ) + \\tilde{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} \\cdot + \\ddot{b}^{(1)})\\\\&~~~~- (\\ddot{W}^{(\\ell )}_{j,:} \\eta (\\cdot ) + \\ddot{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} \\cdot + \\ddot{b}^{(1)})\\Vert _n^2 \\\\&\\le 4 c_{\\mathrm {scale}}\\lambda _\\ell [\\theta \\hat{N}_{\\ell }(\\lambda _\\ell ) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda _\\ell ; N_\\ell )] R^2.$ Moreover, the norm bound (REF ) gives the following Lipschitz continuity bound of each layer: $\\sum _{j \\in [m_{\\ell + 1}]} \\Vert \\widetilde{W}^{(\\ell )}_{j,:} g - \\widetilde{W}^{(\\ell )}_{j,:} g^{\\prime } \\Vert _n^2& = \\sum _{j \\in [m_{\\ell + 1}]} \\frac{1}{n} \\sum _{i=1}^n ( \\widetilde{W}^{(\\ell )}_{j,:} g(x_i) - \\widetilde{W}^{(\\ell )}_{j,:} g^{\\prime }(x_i) )^2 \\\\& = \\sum _{j \\in [m_{\\ell + 1}]} \\Vert \\widetilde{W}^{(\\ell )}_{j,:}\\Vert ^2\\sum _{j^{\\prime } \\in [m_\\ell ]} \\frac{1}{n} \\sum _{i=1}^n (g_{j^{\\prime }}(x_i) - g_{j^{\\prime }}^{\\prime }(x_i))^2 \\\\& \\le 4 c_{\\mathrm {scale}}\\frac{\\theta \\hat{N}_{\\ell }(\\lambda _\\ell ) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell }(\\lambda _\\ell ; N_\\ell )}{ m^{\\sharp }_{\\ell } \\left[ \\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta ) \\right]}R^2\\Vert g - g^{\\prime }\\Vert _n^2$ for $g, g^{\\prime }: \\mathbb {R}^d \\rightarrow \\mathbb {R}^{m^{\\sharp }_{\\ell }}$ .",
"Combining these inequalities, it holds that $& \\Vert (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\widetilde{W}^{(\\ell )} \\eta (\\cdot ) + \\tilde{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\\\& - (\\widetilde{W}^{(L)} \\eta (\\cdot ) + \\tilde{b}^{(L)}) \\circ \\dots \\circ (\\widetilde{W}^{(\\ell +1)} \\eta (\\cdot ) + \\tilde{b}^{(\\ell +1)}) \\circ (\\ddot{W}^{(\\ell )} \\eta (\\cdot ) + \\ddot{b}^{(\\ell )}) \\circ (\\ddot{W}^{(\\ell -1)} \\eta (\\cdot ) + \\ddot{b}^{(\\ell -1)}) \\circ \\dots \\circ (\\ddot{W}^{(1)} x + \\ddot{b}^{(1)}) \\Vert _n^2 \\\\\\le &\\lambda _\\ell \\bar{R}^{2(L - \\ell + 1)}\\prod _{\\ell ^{\\prime } = \\ell }^L c_{\\mathrm {scale}}\\frac{[\\theta \\hat{N}_{\\ell ^{\\prime }}(\\lambda _{\\ell ^{\\prime }}) + (1 - \\theta ) \\hat{N}^{\\prime }_{\\ell ^{\\prime }}(\\lambda _{\\ell ^{\\prime }}; N_{\\ell ^{\\prime }})]}{\\theta {\\textstyle \\frac{\\max _{j} q_j^{(\\ell )}\\Vert \\hat{W}^{(\\ell )}_{j,:}\\Vert ^2 }{\\Vert (\\hat{W}^{(\\ell )})^\\top \\mathrm {I}_{q^{(\\ell )}} \\hat{W}^{(\\ell )}\\Vert _{\\mathrm {op}} }} + (1- \\theta )}\\frac{1}{\\prod _{\\ell ^{\\prime } = \\ell +1}^L m^{\\sharp }_{\\ell ^{\\prime }}}.$ By summing up the square root of this for $\\ell =2,\\dots ,L$ , we obtain the assertion."
],
[
"Notations",
"For a sequence of the width $m^{\\prime } = (m^{\\prime }_{2},\\dots ,m^{\\prime }_{L})$ , let $\\hat{\\mathcal {F}}_{m^{\\prime }}:= & \\lbrace f(x) = (W^{(L)} \\eta ( \\cdot ) + b^{(L)}) \\circ \\dots \\circ (W^{(1)} x + b^{(1)}) \\\\& \\mid \\Vert W^{(\\ell )}\\Vert _{\\mathrm {F}}^2\\le \\bar{R}^2,~\\Vert b^{(\\ell )}\\Vert _2\\le \\bar{R}_b,~W^{(\\ell )} \\in \\mathbb {R}^{m^{\\prime }_{\\ell +1} \\times m^{\\prime }_{\\ell }},~b^{(\\ell )} \\in \\mathbb {R}^{m^{\\prime }_{\\ell + 1}}~(1 \\le \\ell \\le L)\\rbrace .$ Proposition 2 Under Assumptions REF and REF , the $\\ell _\\infty $ -norm of $f \\in \\hat{\\mathcal {F}}_{m^{\\prime }}$ is bounded as $\\Vert f\\Vert _\\infty &\\le \\bar{R}^{L} D_x + \\sum _{\\ell = 1}^L \\bar{R}^{L-\\ell } \\bar{R}_b.$ The proof is easy to see the Lipschitz continuity of the network with respect to $\\Vert \\cdot \\Vert $ -norm is bounded by $\\Vert W^{(\\ell )}\\Vert _{\\mathrm {F}}$ .",
"By the scale invariance of the activation function $\\eta $ , $\\hat{\\mathcal {F}}_{m^{\\sharp }_{}}$ can be rewritten as $\\hat{\\mathcal {F}}_{m^{\\sharp }_{}}= & \\lbrace f(x) = (W^{(L)} \\eta ( \\cdot ) + b^{(L)}) \\circ \\dots \\circ (W^{(1)} x + b^{(1)}) \\\\& \\mid \\Vert W^{(\\ell )}\\Vert _{\\mathrm {F}}^2\\le \\frac{m^{\\sharp }_{\\ell +1}}{m^{\\sharp }_{\\ell }} \\bar{R}^2,~\\Vert b^{(\\ell )}\\Vert _2\\le \\sqrt{m^{\\sharp }_{\\ell + 1}} \\bar{R}_b,~W^{(\\ell )} \\in \\mathbb {R}^{m^{\\sharp }_{\\ell +1} \\times m^{\\sharp }_{\\ell }},~b^{(\\ell )} \\in \\mathbb {R}^{m^{\\sharp }_{\\ell + 1}}~(1 \\le \\ell \\le L)\\rbrace .$ Hence, from Theorem REF and the argument in Appendix , we can see that under Assumption REF , it holds that $f^{\\sharp }\\in \\hat{\\mathcal {F}}_{m^{\\sharp }_{}}.$ for both of the backward-procedure and the simultaneous-procedure.",
"Therefore, the compressed network $f^{\\sharp }$ of both procedures with the constraint has $\\ell _\\infty $ -bound such as $\\Vert [\\!",
"[f^{\\sharp }]\\!",
"]\\Vert _\\infty \\le \\hat{R}_{\\infty }.$"
],
[
"Proof of Theorem ",
"Remember that the $\\epsilon $ -internal covering number of a (semi)-metric space $(T,d)$ is the minimum cardinality of a finite set such that every element in $T$ is in distance $\\epsilon $ from the finite set with respect to the metric $d$ .",
"We denote by $N(\\epsilon ,T,d)$ the $\\epsilon $ -internal covering number of $(T,d)$ .",
"The covering number of the neural network model $\\hat{\\mathcal {F}}_{m^{\\prime }}$ can be evaluated as follows (see for example [30]): Proposition 3 The covering number of $\\hat{\\mathcal {F}}_{m^{\\prime }}$ is bounded by $\\log N(\\epsilon , \\hat{\\mathcal {F}}_{m^{\\prime }}, \\Vert \\cdot \\Vert _\\infty ) \\le C \\frac{\\sum _{\\ell =1}^L m^{\\prime }_\\ell m^{\\prime }_{\\ell +1}}{n}\\log _+ \\left(1 + \\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\delta } \\right)$ for a universal constant $C > 0$ .",
"We define $\\mathcal {G}_{m^{\\prime }} = \\lbrace g(x_i,y_i) = \\psi (y, f(x)) \\mid f \\in \\hat{\\mathcal {F}}_{m^{\\prime }}\\rbrace ,$ for $m^{\\prime } = (m_2,\\dots ,m_L)$ .",
"Then, its Rademacher complexity can be bounded as follows.",
"Lemma 2 Let $(\\epsilon _i)_{i=1}^n$ be i.i.d.",
"Rademacher sequence, that is, $P(\\epsilon _i = 1) = P(\\epsilon _i = -1) = \\frac{1}{2}$ .",
"There exists a universal constant $C > 0$ such that, for all $\\delta > 0$ , $\\mathrm {E}\\left[\\sup _{f \\in \\mathcal {G}_{m^{\\prime }} } \\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i g(x_i,y_i) \\right|\\right]&\\le C\\rho \\Bigg [ \\hat{R}_{\\infty }\\sqrt{\\frac{\\sum _{\\ell =1}^L m^{\\prime }_\\ell m^{\\prime }_{\\ell +1}}{n}\\log _+ \\left(1 + \\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\hat{R}_{\\infty }} \\right)} \\\\&~~\\vee \\hat{R}_{\\infty }\\frac{\\sum _{\\ell =1}^L m^{\\prime }_\\ell m^{\\prime }_{\\ell +1}}{n}\\log _+ \\left(1 + \\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\delta } \\right) \\Bigg ],$ where the expectation is taken with respect to $\\epsilon _i,x_i,y_i$ .",
"Since $\\psi $ is $\\rho $ -Lipschitz continuous, the contraction inequality Theorem 4.12 of [22] gives an upper bound of the RHS as $& \\mathrm {E}\\left[\\sup _{g \\in \\mathcal {G}_{m^{\\prime }}} \\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i g(x_i,y_i) \\right| \\right] \\le 2 \\rho \\mathrm {E}\\left[\\sup _{g \\in \\hat{\\mathcal {F}}_{m^{\\prime }}} \\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i f(x_i) \\right|\\right].$ We further bound the RHS.",
"By Theorem 3.1 in [10] or Lemma 2.3 of [27] with the covering number bound (Proposition REF ), there exists a universal constant $C^{\\prime }$ such that $\\mathrm {E}\\left[\\sup _{f \\in \\hat{\\mathcal {F}}_{m^{\\prime }}}\\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i f(x_i) \\right| \\right] \\le &C^{\\prime }\\Bigg [ \\hat{R}_{\\infty }\\sqrt{\\frac{\\sum _{\\ell =1}^L m^{\\prime }_\\ell m^{\\prime }_{\\ell +1}}{n}\\log _+ \\left(1 + \\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\hat{R}_{\\infty }} \\right)} \\\\&\\vee \\hat{R}_{\\infty }\\frac{\\sum _{\\ell =1}^L m^{\\prime }_\\ell m^{\\prime }_{\\ell +1}}{n}\\log _+ \\left(1 + \\frac{4 \\hat{G}\\max \\lbrace \\bar{R},\\bar{R}_b\\rbrace }{\\hat{R}_{\\infty }} \\right)\\Bigg ].$ This concludes the proof.",
"Now we are ready to probe the theorem.",
"[Proof of Theorem REF ] Since $\\mathcal {G}_{m^{\\prime }}$ is separable with respect to $\\Vert \\cdot \\Vert _\\infty $ -norm, by the standard symmetrization argument, we have that $P\\left\\lbrace \\sup _{g \\in \\mathcal {G}_{m^{\\prime }}} \\left| \\frac{1}{n} \\sum _{i=1}^n g(x_i,y_i) - \\mathrm {E}_{X,Y}[g]\\right| \\ge 2 \\mathrm {E}\\left[\\sup _{f \\in \\mathcal {G}_{m^{\\prime }} } \\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i g(x_i,y_i) \\right|\\right]+ 3 \\hat{R}_{\\infty }\\sqrt{\\frac{2 t }{n}} \\right\\rbrace \\le 2 e^{-t}$ for all $t > 0$ (see, for example, Theorem 3.4.5 of [11]).",
"Taking uniform bound with respect to the choice of $m^{\\prime } \\in [m_2] \\times [m_3] \\times \\dots \\times [m_L]$ , we have that $P\\Bigg \\lbrace & \\sup _{g \\in \\mathcal {G}_{m^{\\prime }}} \\left| \\frac{1}{n} \\sum _{i=1}^n g(x_i,y_i) - \\mathrm {E}_{X,Y}[g]\\right|\\ge 2 \\mathrm {E}\\left[\\sup _{f \\in \\mathcal {G}_{m^{\\prime }} } \\left| \\frac{1}{n} \\sum _{i=1}^n \\epsilon _i g(x_i,y_i) \\right|\\right]+ 3 \\hat{R}_{\\infty }\\sqrt{\\frac{2 (t + \\sum _{\\ell =2}^L \\log (m_{\\ell }) )}{n}} \\\\&\\text{for all $m^{\\prime } \\in [m_2] \\times [m_3] \\times \\dots \\times [m_L]$ uniformly}\\Bigg \\rbrace \\le 2 e^{-t}.$ Now, the generalization error of $f^{\\sharp }$ can decomposed into $\\Psi ([\\!",
"[f^{\\sharp }]\\!])",
"= \\underbrace{\\Psi ([\\!",
"[f^{\\sharp }]\\!])",
"- \\hat{\\Psi }([\\!",
"[f^{\\sharp }]\\!",
"])}_{\\clubsuit } +\\underbrace{\\hat{\\Psi }([\\!",
"[f^{\\sharp }]\\!])",
"- \\hat{\\Psi }([\\![\\widehat{f}]\\!",
"])}_{\\diamondsuit } + \\hat{\\Psi }([\\![\\widehat{f}]\\!",
"]).$ Since the truncation operation $[\\!",
"[\\cdot ]\\!",
"]$ does not increase the $\\Vert \\cdot \\Vert _\\infty $ -norm of two functions, we can apply the inequality (REF ) and Lemma REF also for $[\\!",
"[f^{\\sharp }]\\!",
"]$ to bound the term $\\clubsuit $ .",
"The term $\\diamondsuit $ can be bounded as $\\hat{\\Psi }([\\!",
"[f^{\\sharp }]\\!])",
"- \\hat{\\Psi }([\\![\\widehat{f}]\\!",
"])& \\le \\frac{1}{n} \\sum _{i=1}^n | \\psi (y_i,[\\!",
"[f^{\\sharp }(x_i)]\\!])",
"- \\psi (y_i,[\\![\\widehat{f}(x_i)]\\!])",
"|\\le \\frac{1}{n} \\sum _{i=1}^n \\rho | [\\!",
"[f^{\\sharp }(x_i) ]\\!",
"]- [\\![\\widehat{f}(x_i)]\\!",
"]| \\\\& \\le \\rho \\sqrt{\\frac{1}{n} \\sum _{i=1}^n ( [\\!",
"[f^{\\sharp }(x_i) ]\\!]-[\\!",
"[\\widehat{f}(x_i) ]\\!",
"])^2 } =\\rho \\Vert [\\!",
"[f^{\\sharp }]\\!",
"]- [\\![\\widehat{f}]\\!",
"]\\Vert _n\\le \\rho \\Vert f^{\\sharp }- \\widehat{f}\\Vert _n\\le \\rho \\delta _{1}.$ Combining these inequalities, we obtain the assertion."
],
[
"Additional numerical experiments",
"This section gives additional numerical experiments for compressing the network."
],
[
"Compressing VGG-16 on ImageNet",
"Here, we also applied our method to compress a publicly available VGG-16 network [28] on the ImageNet dataset.",
"We apply our method to the ImageNet dataset [6].",
"We used the ILSVRC2012 dataset of the ImageNet dataset, which consists of 1.3M training data and 50,000 validation data.",
"Each image is annotated into one of 1,000 categories.",
"We applied our method to this network and compared it with existing methods, namely APoZ [17], SqueezeNet [18], and ThiNet [24].",
"All of them are applied to the same VGG-16 network.",
"For fair comparison, we followed the same experimental settings as [24]; the way of training data generation, data augmentation, performance evaluation schemes and so on.",
"The results are summarized in Table REF .",
"It summarizes the Top-1/Top-5 classification accuracies, the number of parameters (#Param), and the float point operations (FLOPs) to classify a single image.",
"Our method is indicated by “Spec-(type).\"",
"We employed the simultaneous procedure for compression.",
"In Spec-Conv, we applied our method only to the convolutional layers (it is not applied to the fully connected layers (FC)).",
"The size of compressed network $f^{\\sharp }$ was set to be the same as that of ThiNet-Conv.",
"Spec-GAP is a method that replaces the FC layers of Spec-Conv with a global average pooling (GAP) layer [23], [35].",
"Here, we again set the number of channels in each layer of Spec-GAP to be same as that of ThiNet-GAP.",
"We employed $\\lambda _\\ell = 10^{-6}\\times \\mathrm {Tr}[\\widehat{\\Sigma }_{(\\ell )}]$ and $\\theta = 0.5$ for our method.",
"We see that in both situations, out method outperforms ThiNet in terms of accuracy.",
"This shows effectiveness of our method while our method is supported by theories.",
"Table: Performance comparison on ImageNet dataset.Our proposed method is compared with APoZ-2, and ThiNet.",
"Our method is indicated as “Spec-(type).\""
]
] | 1808.08558 |
[
[
"Considerations on Quantum-Based Methods for Communication Security"
],
[
"Abstract In this paper we provide an intuitive-level discussion of the challenges and opportunities offered by quantum-based methods for supporting secure communications, e.g., over a network.",
"The goal is to distill down to the most fundamental issues and concepts in order to provide a clear foundation for assessing the potential value of quantum-based technologies relative to classical alternatives.",
"It is hoped that this form of exposition can provide greater clarity of perspective than is typically offered by mathematically-focused treatments of the topic.",
"It is also hoped that this clarity extends to more general applications of quantum information science such as quantum computing and quantum sensing."
],
[
"Introduction",
"Quantum-based technologies exploit physical phenomena that cannot be efficiently exhibited or simulated using technologies that exploit purely classical physics.",
"For example, a quantum sensor may use quantum phenomena to probe a system to discern classical and/or quantum properties of the system that cannot be directly measured by classical sensing technologies.",
"Quantum computing, by contrast, generalizes the classical unit of information, the bit, in the form of a quantum bit, or qubit, and exploits quantum computational operators that cannot be efficiently simulated using classical Boolean-based operators.",
"Secure quantum-based communication protocols have emerged as among the first practical technologies for which advantages over classical alternatives have been rigorously demonstrated.",
"As will be discussed, however, these advantages rely on a set of assumptions about the capabilities of potential adversaries (hackers) as well as those of the communicating parties.",
"Because the quantum advantage can be lost if these assumptions are relaxed, the utility of quantum-based communication must be assessed based on the assumed scenario in which it will be applied.",
"In the next section we discuss scenarios in which classical cryptography can facilitate unconditionally secure communications.",
"We then discuss a more general class of communication scenarios in which classical methods cannot provide unconditional guarantees of security but may offer practically sufficient ones.",
"We then provide a high-level description of how special properties of quantum systems can be exploited to enlarge the range of scenarios for which unconditional communication security can be achieved.",
"This provides context for realistically examining how the tantalizing theoretical features of quantum-based approaches to communication security may translate to practical advantages over classical alternatives."
],
[
"Secure Communications",
"Suppose two parties, Alice and Bob, know they will have need for unconditionally secure communications at various times in the future.",
"If they determine that they are unlikely to communicate more than a total of $n$ bits before the next time they meet then they can create a sequence of random bits, referred to as a one-time pad (OTP), and each keep a copy for use to mask their messages.",
"For example, a week later Alice can contact Bob using whatever unsecure communication medium she chooses, e.g., phone or email, and then send her $k$ -bit private message encrypted by performing an exclusive-orThe exclusive-or function of two bits $a$ and $b$ is 0 if are the same and 1 if they differ.",
"(XOR) of it with the first $k$ bits of the OTP.",
"Upon receipt of the encrypted k-bit message, Bob will simply invert the mask by applying the same XOR operation using the first $k$ bits of the OTP.",
"Even if an eavesdropper, Eve, is able to monitor all communications between Alice and Bob, she will not be able to access the private information (i.e., original plain-text messages) without a copy of their OTP.",
"Thus, the OTP protocol offers unconditional security against eavesdropping, but its use is limited to parties who have previously established a shared OTP.",
"The question is whether a secure protocol can be established between two parties who have never communicated before."
],
[
"Public Key Encryption",
"At first glance it appears that there is no way for Alice and Bob to communicate for the first time in a way that is secure against an eavesdropper who has access to every bit of information they exchange.",
"However, a commonly-used analogy can quickly convey how this might be done.",
"Suppose Alice wishes to mail a piece of paper containing a secret message to Bob.",
"To ensure security during transport she places the paper in a box and applies a lock before sending.",
"When Bob receives the box he of course can't open it because of the lock, but he can apply his own lock and send the box back to Alice.",
"Upon receipt, Alice removes her lock and sends the box back to Bob, who can now open it and read the message.",
"If it is assumed that the box and locks can't be compromised then this protocol is secure even if Eve is able to gain physical access to the box during transport.",
"An analogous protocol can be applied to digital information if it is possible for Alice and Bob to sequentially encrypt a given message and then sequentially decrypt it.",
"To do so, however, Alice must be able to remove her encryption mask after Bob has applied his.",
"In other words, their respective encryption operations must commute and not be invertible by Eve.",
"It turns out that no classical protocol can satisfy the necessary properties for unconditional security.",
"However, a practical equivalent of unconditional security can potentially be achieved in the sense that Eve may be able to invert the encryption – but only if she expends thousand years of computation time.",
"Under the assumption that security of the message will be irrelevant at that point in the distant future, the protocol can be regarded as unconditionally secure for all practical purposes.",
"At present there are technically no protocols that provably require such large amounts of computational effort, but some do if certain widely-believed conjectures (relating to one-way functions) are true.",
"Assuming that these conjectures are in fact true, classical public-key protocols would seem to offer practically the same level of security as a one-time pad but without the limitation of prior communication.",
"On the other hand... estimating the expected amount of time necessary to break a classical public-key protocol is very difficult.",
"Even if it is assumed that the amount of work required by Eve grows exponentially with the length of a critical parameter, a particular value for that parameter must be chosen.",
"For all existing protocols the value of this parameter introduces an overhead coefficient (both in computational time and space) which may not be exponential but may grow such that the protocol becomes impractical in most real-world contexts.",
"Suppose the parameter is selected based on a tradeoff between practical constraints and a minimum acceptable level of security, e.g., that it would take Eve 500 years to break the encryption using the fastest existing supercomputer.",
"What if Eve can apply 1000 supercomputers and break it in six months?",
"Or what if she develops an optimized implementation of the algorithm that is 1000 times faster?",
"Breaking the code may still require time that is exponential in the value of the parameter, but the real question is how to estimate the range of parameter values that are at risk if Eve applies all available resources to crack a given message.",
"As an example, in 1977 it was estimated that the time required to break a message encrypted with the RSA public-key protocol using a particular parameter value would be on the order of many quadrillion years.",
"However, improved algorithms and computing resources permitted messages of this kind to be broken only four years later, and by 2005 it was demonstrated that the same could be done in only a day.",
"The difficulty of making predictions, especially about the future [3], raises significant doubts about the extent to which any particular classical public-key scheme truly provides a desired level of security for all practical purposes, and it is this nagging concern that motivates interest in quantum-based protocols that offer true unconditional security, at least in theory."
],
[
"Quantum Key Distribution (QKD)",
"Quantum-based public-key protocols have been developed that provide unconditional security guaranteed by the laws of physics.",
"In the case of Quantum Key Distribution (QKD) [2], its security is achieved by exploiting properties that only hold for qubits.",
"The first is the no-cloning theorem, which says that the complete quantum state of a qubit cannot be copied.",
"The second is that a pair of qubits can be generated with entangled states such that the classical binary value measured for one by a particular measurement process using parameter value $\\Theta $ will be identical to what is measured for the other using the same parameter value, but not necessarily if the second measurement is performed with a different parameter value $\\Theta ^{\\prime }\\ne \\Theta $ .",
"The no-cloning theorem is clearly non-classical in the sense that a qubit stored in one variable can't be copied into a different variable the way the content of a classical binary variable can be copied into another variable or to many other variables.",
"For example, if the state of a given qubit is somehow placed into a different qubit then the state of the original qubit will essentially be erased in the processThe theoretical physics explaining why quantum states can't be cloned, and the details of how qubits are prepared and manipulated, are not important in the present context for the same reason that details of how classical bits are implemented in semiconductor devices are not relevant to discussions of algorithmic issues..",
"In other words, the state of the qubit should not be viewed as having been copied but rather teleported from the first qubit to the second qubit.",
"If it is simply measured, however, then its state collapses to a classical bit and all subsequent measurements will obtain the same result.",
"Based on these properties, the following simple quantum communication protocolThis toy protocol is intended only to illuminate the key concepts in a way that links to classical one-time pad (variations can be found in [7]).",
"Much more complete expositions of the general theory and practice of quantum cryptography can be found in [1], [4], [5], [6].",
"can be defined: Alice and Bob begin by agreeing on a set of $k$ distinct measurement parameter values $\\Theta =\\lbrace \\Theta _1,\\Theta _2,...,\\Theta _k\\rbrace $ .",
"This is done openly without encryption, i.e., Eve sees everything.",
"Alice and Bob each separately choose one of the $k$ parameter values but do not communicate their choices, thus Eve has no knowledge of them.",
"Alice generates a pair of entangled qubits.",
"She measures one and sends the other to Bob.",
"Bob reports his measured value.",
"If Alice sees that it is not the same as hers then she chooses a different parameter and repeats the process.",
"She does this for each parameter value until only one is found that always (for a sufficiently large number of cases) yields the same measured value as Bob but does not give results expected for different $\\Theta $ values.",
"At this point Alice and Bob have established a shared parameter value that is unknown to Eve.",
"The process can now be repeated to create a shared sequence of random bits that can be used like an ordinary one-time pad.",
"In fact, subsequent communications can be conducted securely using classical bits.",
"The security of the above protocol derives from the fact that Eve cannot clone $k$ copies of a given qubit to measure with each $\\Theta _k$ , and simply measuring transmitted qubits will prevent Alice and Bob from identifying a unique shared measurement parameter.",
"In other words, Eve may corrupt the communication channel but cannot compromise its information.",
"At this point Alice and Bob can create a shared OTP (which they can verify are identical by using a checksum or other indicator) and communicate with a level of security beyond what is possible for any classical public-key protocol."
],
[
"The Authentication Challenge",
"For research purposes it is natural to introduce simplifying assumptions to make a challenging problem more tractable.",
"The hope is that a solution to the simplified problem will provide insights for solving the more complex variants that arise in real-world applications.",
"This was true of the lockbox example in which it was assumed that Eve might obtain physical access to the locked box but is not able to dismantle and reassemble the box, or pick the lock, to access the message inside.",
"The secure digital communication problem as posed in this paper also has such assumptions.",
"Up to now it has been assumed that Eve has enormous computational resources at her disposal sufficient to overcome the exponential computational complexity demanded to break classical protocols.",
"Despite these resources, it has also been assumed that she is only able to passively monitor the channel between Alice and Bob.",
"This is necessary because otherwise she could insert herself and pretend to be Alice when communicating with Bob and pretend to be Bob when communicating with Alice.",
"This is referred to as a Man-In-The-Middle (MITM) attack, which exploits what is known as the authentication problem.",
"To appreciate why there can be no general countermeasure to MITM attacks, consider the case of Eve monitoring all of Alice's outgoing communications.",
"At some point Eve sees that Alice is trying to achieve first-time communication with a guy named Bob.",
"Eve can intercept the messages intended for Bob and pretend to be Bob as the two initiate a secure quantum-based protocol.",
"Pretending to be Alice, Eve does the same with Bob.",
"Now all unconditionally secure communications involve Eve as a hidden go-between agent.",
"In many respects it might seem easier to actively tap into a physical channel (e.g., optical fiber or copper wire) than to passively extract information from a bundle of fibers or wires within an encased conduit, but of course it's possible to add physical countermeasures to limit Eve's ability to penetrate that conduit.",
"On the other hand, if that can be done then it might seem possible to do something similar to thwart passive monitoring.",
"Ultimately no quantum public-key protocol can be unconditionally secure without a solution to the authentication problem.",
"Many schemes have been developed in this regard, but ultimately they all rely on additional assumptions and/or restrictions or else involve mechanisms that potentially could facilitate a comparable level of security using purely classical protocols.",
"As an example, suppose a company called Amasoft Lexicon (AL) creates a service in which customers can login and communicate with other registered customers such that AL serves as a trusted intermediary to manage all issues relating to authentication.",
"This may involve use of passwords, confirmation emails or text messages to phones, etc., but ultimately it must rely on information that was privately established at some point between itself and each of its customers, e.g., Alice and Bob.",
"Suppose each customer is required to set up a strong password.",
"Initially, how is that information exchanged securely with AL?",
"One option might be to require the customer to physically visit a local provider so that the person's identity can be verified, and a secure password can be established, without having to go through an unsecure channel.",
"Okay, but how long must the password be?",
"If it is to be repeatedly used then it would become increasingly vulnerable as Eve monitors more and more messages.",
"To avoid repeated use of a short password, AL could give Alice a drive containing 4TB of random bits for an OTP that would be shared only by her and AL.",
"The same would be done using a different OTP when Bob registers.",
"Now Alice can initiate unconditionally secure communicates with AL, and AL can do the same with Bob, and therefore Alice and Bob can communicate with unconditional security via AL.",
"Regardless of whether communications through AL involve a quantum component, the security of the overall system depends on the trusted security of AL – and on the security practices of its customers in maintaining the integrity of their individual OTPs.",
"The situation can be viewed as one of replacing one point of vulnerability with a different one.",
"For example, what prevents Eve from seeking employment at AL?",
"Are there sufficient internal safeguards to protect against nefarious actions of AL employees?"
],
[
"The Complexity Challenge",
"Complexity is a double-edged sword in the context of communication security.",
"On the one hand it can be used to increase the computational burden on Eve.",
"On the other hand, it can introduce more points of vulnerability for her to exploit as the scale of the implementation (amount of needed software and hardware) increases.",
"In the case of quantum-based protocols there is need for highly complex infrastructure to support the transmission of qubits and the preservation of entangled states.",
"The details are beyond the scope of this paper, but it is safe to say that as implementation details become more concretely specified the number of identified practical vulnerabilities grows.",
"An argument can be made that as long as the theory is solid the engineering challenges will eventually be surmounted.",
"This may be verified at some point in the indefinite future, but it is worthwhile to consider the number of practical security challenges that still exist in current web browsers, operating systems, etc., despite the recognized commercial and regulatory interests in addressing them.",
"The critical question is whether the investment in quantum-based infrastructure to support quantum-based secure communication protocols is analogous to a homeowner wanting to improve his security by installing a titanium front door with sophisticated intruder detection sensors but not making any changes to windows and other doors.",
"The natural response to the titanium door analogy is to agree that quantum-based technologies represent only one part of the overall security solution and that of course there are many other vulnerabilities which also must be addressed.",
"However, this raises a new question: Is it possible that a complete solution can be developed that doesn't require any quantum-based components?",
"It may turn out that it is only feasible to guarantee practically sufficient levels of security (as opposed to unconditional) and only for specialized infrastructure and protocols tailored to specific use-cases.",
"If the scope of a given use-case is sufficiently narrow (e.g., communications of financial information among a fixed number of banks) then the prospects for confidently establishing a desired level of security are greatly improved.",
"In other words, relative simplicity tends to enhance trust in the properties of a system because it is difficult to be fully confident about anything that is too complicated to be fully understood."
],
[
"Discussion",
"The foregoing considerations on the status of quantum-based approaches for secure communications have leaned strongly toward a sober, devil's-advocate perspectiveSee the appendices for more succinct expressions of arguments considered in this paper..",
"This was intentional to firmly temper some of the overly-enthusiastic depictions found in the popular media.",
"For example, the following is from media coverage of an announcement in May of 2017 about the launch of a quantum-based “unhackable” fiber network in China: “The particles cannot be destroyed or duplicated.",
"Any eavesdropper will disrupt the entanglement and alert the authorities,” a researcher at the Chinese Academy of Sciences is quoted as saying.",
"Hopefully our discussion thus far clarifies the extent to which there is a factual basis for this quote and how the implicit conclusion (i.e., that the network is “unhackable”) goes somewhat beyond that basis.",
"One conclusion that cannot be doubted is that remarkable progress has been made toward implementing practical systems based on theoretically-proposed quantum techniques.",
"Another equally-important conclusion that can be drawn is that China is presently leading this progress.",
"In many respects the situation is similar to the early days of radar when it was touted as a sensing modality that could not be evaded by any aircraft or missile because it had the means “to see through clouds and darkness.” While this claimed capability was not inaccurate, that power motivated the development of increasingly sophisticated countermeasures to mask the visibility of aircraft to enemy radar, thus motivating the development of increasingly more sophisticated technologies to counter those countermeasures.",
"The lesson from this is that every powerful technology will demand continuing research and development to meet new challenges and to support new applications.",
"It is likely that the real value of future quantum fiber networks will not be communication security but rather to support the needs of distributed quantum sensing applications.",
"More specifically, quantum information from quantum-based sensors and related technologies can only be transmitted via special channels that are implemented to preserve entangled quantum states.",
"The future is quantum, so the development of infrastructure to manage and transmit quantum information has to be among the highest of priorities."
],
[
"Conclusion",
"In retrospect it seems almost ludicrous to suggest that any technology could ever offer something as unequivocally absolute as “unconditional guaranteed security,” but that doesn't mean quantum-based technologies don't represent the future state-of-the-art for maximizing network communication security.",
"More importantly, surmounting the theoretical and practical challenges required to realize this state-of-the-art will have much more profound implications than simply supporting the privacy concerns of Alice and Bob."
],
[
"Devil's Advocate Arguments",
" “The theoretical guarantees provided by QKD are only satisfied under certain assumptions.",
"It may be that those assumptions can't be satisfied in any practical implementation and thus QKD provides no theoretical advantages over classical alternatives.” “If it's possible to implement the highly-complex infrastructure needed to support QKD, and to provide physical security against MITM attacks, then it should also be possible to implement physical security against passive monitoring.",
"If that can be achieved then there is no need for QKD.” “The complexity associated with QKD may make it less secure than simpler classical alternatives.",
"Just consider the number of security challenges that still exist in current web browsers, operating systems, etc., despite the recognized commercial and regulatory interests in addressing them.” “Progress on the development of classical protocols (e.g., based on elliptic curve cryptography) may very well lead to rigorous guarantees about the asymmetric computational burden imposed on Eve.",
"If so, this would provide essentially unconditional security for all practical purposes.” “The need for provable unconditional security may be limited to only a few relatively narrow contexts in which classical alternatives are sufficient.",
"For example, communications of financial information among a fixed number of banks could potentially be supported using classical one-time pads that are jointly established at regular intervals.” “QKD assumptions on what the physical infrastructure is required to support, and on what Eve is and is not able to do, seem to evolve over time purely to conform to the limits of what the theoretical approach can accommodate.",
"This raises further doubts about QKD's true scope of practical applicability.” “Implementing quantum infrastructure to support QKD is analogous to a homeowner wanting to improve security by installing a titanium front door but not making any changes to windows and other doors.",
"In the case of Alice and Bob, for example, it's probably much easier for Eve to place malware on their computers, or place sensors at their homes, than to identify and compromise a network link somewhere between them.”"
],
[
"Replies to the Devil's advocate:",
" “If demands are set too high at the outset then no progress can ever be made to improve the status quo.” “Even if it is true that most security-critical applications will demand specially-tailored solutions, the availability of quantum-based tools will offer greater flexibility in producing those solutions.” “People can assume responsibility for their local security but have no choice but to trust the security of infrastructure outside their control.” “A network that supports quantum information is unquestionably more powerful than one that does not.",
"It is impossible to foresee the many ways this power will be exploited down the road, but it is hard to imagine that enhanced security will not be included.”"
]
] | 1808.08591 |
[
[
"Experimental demonstration of high-rate measurement-device-independent\n quantum key distribution over asymmetric channels"
],
[
"Abstract Measurement-device-independent quantum key distribution (MDI-QKD) can eliminate all detector side channels and it is practical with current technology.",
"Previous implementations of MDI-QKD all use two symmetric channels with similar losses.",
"However, the secret key rate is severely limited when different channels have different losses.",
"Here we report the results of the first high-rate MDI-QKD experiment over $asymmetric$ channels.",
"By using the recent 7-intensity optimization approach, we demonstrate $>$10x higher key rate than previous best-known protocols for MDI-QKD in the situation of large channel asymmetry, and extend the secure transmission distance by more than 20-50 km in standard telecom fiber.",
"The results have moved MDI-QKD towards widespread applications in practical network settings, where the channel losses are asymmetric and user nodes could be dynamically added or deleted."
],
[
"Optimization Algorithm",
"Here we briefly describe the optimization algorithm, as proposed in Ref.",
"[34], for the intensities and the probabilities of sending them in the 7-intensity optimization method.",
"As described in the main text, there are a total of 12 parameters that need to be optimized: $[s_A,\\mu _A,\\nu _A,p_{s_A},p_{\\mu _A},p_{\\mu _A},s_B,\\mu _B,\\nu _B,p_{s_B},p_{\\mu _B},p_{\\mu _B}]$ To navigate such a large parameter space, a local search algorithm is necessary.",
"However, the key rate function versus these parameters is in fact non-smooth, hence local search does not work well in this case.",
"To address this, we need to convert the parameters to polar coordinate: $[r_s,\\theta _s, r_\\mu , r_\\nu , \\theta _{\\mu \\nu }, p_{s_A},p_{\\mu _A},p_{\\mu _A},p_{s_B},p_{\\mu _B},p_{\\mu _B}]$ where the conversion follows that: $\\begin{aligned}r_{s}&=\\sqrt{{s_A}^2+{s_B}^2}\\\\\\theta _s&=arctan({s_A\\over s_B})\\end{aligned}$ and the same applies for $r_\\mu ,\\theta _{\\mu \\nu }$ and $r_\\nu ,\\theta _{\\mu \\nu }$ .",
"Note that here we have locked the polar angle of $\\mu $ and $\\nu $ to be always equal.",
"This is because, the optimal values of the decoy intensities for Alice and Bob always satisfy [34] ${\\mu _A \\over \\nu _A}={\\mu _B \\over \\nu _B}$ for the 7-intensity optimization method.",
"The intensity probabilities are not involved in non-smoothness and therefore do not need to be changed.",
"With the 11 parameters now, we can perform a local search algorithm, such as coordinate descent [34], [27], to efficiently find the set of optimal intensities.",
"Coordinate descent algorithm alternatively optimizes each variable at a time while keeping others constant.",
"When all variables are searched, the algorithm starts over again with the first variable.",
"With enough iterations, this algorithm can reach a local maximum point.",
"For asymmetric MDI-QKD, the algorithm can find the global maximum point (which is the only local maximum)."
],
[
"The Detail of MDI-QKD Systems",
"Alice and Bob each possess an internally modulated laser which emits phase-randomized laser pulses at a clock rate of 75 MHz.",
"The pulse width is about 3.4 ns, and the center wavelength is at 1550.12 nm with a full width at half maximum (FWHM) of about 15 pm.",
"AM4, is used to modulate the vacuum state, where the extinction ratio between the signal state and the vacuum state is larger than 23 dB.",
"The MZI divides each incoming pulse into two time-bins with 6.4 ns time interval.",
"The superconducting nanowire single-photon detectors (SNSPD1 and SNSPD2) have detection efficiency 70% and dark count rate 30 cps.",
"Due to an extra insertion loss 1.2 dB in Charlie and 15% non-overlap between laser pulse and detection time window, the total system detection efficiency is 46%.",
"To get a high visibility of two-photon interference , we adopt several feedback systems [18] to calibrate the polarization, time and spectrum modes of the signal pulses generated by two independent laser sources, as shown in Fig.",
"2 in main text.",
"First, to make sure that the polarization of two pulses are indistinguishable, we plug a EPC and a PBS before the BS in the BSM.",
"Intensities of the reflection port of the PBS are monitored by a SNSPD, which outputs a feedback signal to control the EPC to minimize the intensities.",
"Besides, to precisely overlap timing mode of the two signal pulses, there are two calibration processes in the experiments.",
"First, Charlie generates two synchronization lasers(Synls, 1570nm) pulses which are synchronized by a crystal oscillator circuit (COC).",
"The pulses are respectively sent to Alice and Bob, who detected it by a PD.",
"The output signals of the PDs are used to synchronize lasers in Alice and Bob.",
"Second, Alice and Bob respectively send their signal laser pulses to Charlie, who use the SNSPD to measure the arriving time of the pulses.",
"Then, by using a programmable delay chip, Charlie adjusts the time delay between the two SynL according to the arriving time difference.",
"The total timing calibration precision is below 10ps and the arriving time jitter of SNSPD is below 50ps, both them are far smaller that the pulse width of 2.4 ns.",
"For the spectrum mode, we first use an optical spectrum analyzer (OSA,YOKOGAWA AQ6370B) to calibrate Alice's signal pulses and phase feedback pulses wavelength at 1550.12nm.",
"Then we observe the Hong-Ou-Mandel (HOM) interference of two pulses in Charlie and adjust the operating temperature of the laser in Bob to the value where the HOM dip are found.",
"Alice and Bob need to have a shared phase reference frame which fluctuates with temperature and stress.",
"Using a PSL, Alice sends laser pulses from her AMZI to Bob's AMZI via an additional link between Alice and Bob.",
"Bob monitors the power at one of outputs of his interferometer with a SPAPD and then minimizes the counts of SPAPD by using a PS in Bob's AMZI."
],
[
"Application to asymmetric quantum digital signature",
"In MDI quantum digital signature (QDS) [46], [39], [40], there exists a sufficiently large signature length which makes the protocol secure, if the condition $Q_{Z}^{0,0}+Q_{Z}^{1,1}[1-h(e_{X}^{1,1})]-h(E_Z)>0$ holds.",
"Here, $Q_{Z}^{i,i}$ is the lower bound on the count rate when Alice and Bob sent $Z$ -basis pulses containing $i$ photons, $e_X^{1,1}$ is the upper bound for the single-photon phase error rate and $E_Z$ is the quantum bit error rate (QBER) in $Z$ basis.",
"Apparently, Eq.",
"(REF ) is the same as the key rate formula of MDI-QKD [13] except for that MDI-QDS omits the inefficient factor of error correction.",
"Hence, we can directly apply the proposed asymmetric method to improve the performance of MDI-QDS over asymmetric channels."
],
[
"Application to asymmetric twin-field QKD",
"Unlike MDI-QKD, in the case of TF-QKD [29], the key bits are generated from singles $\\mathinner {|{01}\\rangle }+\\mathinner {|{10}\\rangle }$ rather than coincidences $\\mathinner {|{11}\\rangle }$ .",
"Alice and Bob send signals whose phases are announced and post-selected to be in the same \"phase-slice\", and the signals are received by a third party, Eve, who announces the detector events at detectors $C$ and $D$ .",
"The key rate can be written as [29], $R={d\\over M}\\lbrace Q_1[1-h_2(e_1)]-fQ_\\mu h_2(E_\\mu )\\rbrace $ where $M$ is the number of phase slices, $d$ is a phase slice post-selection factor, $Q_1,e_1$ are the estimated gain and phase-error rate of single photons, $Q_\\mu ,E_\\mu $ are the observed gain and QBER for the signal states, and $f$ is the error correction efficiency.",
"Conceptually, for single photons, the asymmetry between transmittances in the two channels decreases the visibility of single-photon interference, resulting in higher QBER.",
"In reality, Alice and Bob both use WCP sources.",
"Consider Alice and Bob sending intensities $\\mu _A$ and $\\mu _B$ .",
"Let us define $\\begin{aligned}\\gamma _A&=\\sqrt{\\mu _A\\eta _A\\eta _d}\\\\\\gamma _B&=\\sqrt{\\mu _B\\eta _B\\eta _d}\\end{aligned}$ where $\\eta _A,\\eta _B$ are the channel transmittances between Alice (Bob) and Charlie, and $\\eta _d$ is the detector efficiency.",
"Following [25], the received intensities $D_C,D_D$ at the detectors (here for simplicity we ignore the dark counts) are: $\\begin{aligned}D_C&=(\\gamma _A^2+\\gamma _B^2-2\\gamma _A\\gamma _Bcos\\phi )/2\\\\D_D&=(\\gamma _A^2+\\gamma _B^2+2\\gamma _A\\gamma _Bcos\\phi )/2\\\\\\end{aligned}$ where $\\phi $ is the relative phase between Alice's and Bob's signals.",
"Consider the detector $C$ , the visibility can be written as: $\\begin{aligned}v_1={{I_{max}-I_{min}}\\over {I_{max}+I_{min}}}={{2\\gamma _A\\gamma _B}\\over {\\gamma _A^2+\\gamma _B^2}}={{2}\\over {{k}+{1\\over k}}} \\\\\\end{aligned}$ where $k={\\sqrt{{\\mu _A\\eta _A}\\over {\\mu _B\\eta _B}}}$ is the ratio between arriving intensities at Eve's beam splitter.",
"$v_1$ is a function that reaches maximum 1 when $k=1$ , and monotonically decreases with $k$ when $k$ deviates from 1.",
"In fact, we can compare this with the two-photon interference visibility from two WCP sources as used in MDI-QKD: $\\begin{aligned}v_2=1-{{P_{coin}}\\over {P_C}{P_D}}={{2\\mu _A\\eta _A\\mu _B\\eta _B}\\over {(\\mu _A\\eta _A+\\mu _B\\eta _B)^2}}={{2}\\over {2+k^2+{1 \\over k^2}}} \\\\\\end{aligned}$ where $P_C,P_D,P_{coin}$ are respectively the count probability of detector $C$ , detector $D$ , and the coincidence count probability.",
"We have plotted both visibilities versus ratio of arriving intensities in Fig.",
"REF .",
"We can see that just like for MDI-QKD, the single-photon interference visibility in TF-QKD heavily depends on the balance of arriving intensities.",
"The visibility of the interference between the two pulses directly affects the observed QBER for both the signal and decoy states.",
"Therefore, we can also apply our method to compensate for channel asymmetry (such as using $\\eta _A\\mu _A=\\eta _B\\mu _B$ ) and obtain higher single-photon interference visibility and lower QBER.",
"Note that the single-photon gain $Q_1$ also depends on the intensity values [41].",
"Hence an asymmetric choice of intensities, i.e., $\\eta _A\\mu _A=\\eta _B\\mu _B$ , may not be the optimal method.",
"A careful optimization of the parameters, following the approach in [34], are likely required to produce the optimal key rate in asymmetric TF-QKD.",
"Note also that the decoupling of basis might not work in TF-QKD.",
"The reason is that, unlike MDI-QKD where the two bases are inherently asymmetric (Charles only measures in $Z$ basis [34]), for TF-QKD the two bases are symmetric, and both will depend on the visibility of interference.",
"Some extensions of the initial TF-QKD protocol use only one basis $X$ for encoding and another basis $Z$ for decoy-state analysis, where decoupling of bases and using different intensity choices potentially might work, but a rigorous study of this will be the subject of future studies.",
"Figure: The visibility for a single-photon interference (used in TF-QKD) and for a two-photon interference (used in MDI-QKD) versus the ratio of intensities used by Alice and Bob, using WCP sources.",
"Here we consider two channels with 50km difference in standard fiber (i.e.",
"with η A /η B =0.1\\eta _A/\\eta _B=0.1).",
"The visibility is the highest in both cases when Alice and Bob use intensities that satisfy μ A η A =μ B η B \\mu _A\\eta _A=\\mu _B\\eta _B, and the visibility drops as ratio of arriving intensities at Charles becomes imbalanced.",
"Note that a two-photon interference with WCP sources can only reach a maximum of 50% visibility while single-photon interference can reach 100%.",
"Therefore, just like for MDI-QKD, the visibility of single-photon interference in TF-QKD (and as a result its QBER) heavily depends on the balance of intensities.",
"List of main parameters and experimental results.",
"Here, $s_{11}^Z$ is the yield of single photon states in the $Z$ basis and $e_{11}^X$ is the phase-flip error rate of single photon states.",
"$Q_{s_As_B}^Z$ and $E_{s_As_B}^Z$ are the observed yield and QBER for source $s_As_B$ , respectively.",
"R is the final key rate.",
"Table: NO_CAPTION[ht!]",
"List of the total gains and error gains of bell state $\\psi ^-$ in the cases of $L_A=10km$ .",
"The first row indicates the channel distance length and attenuation from Bob to Charlie.",
"The notation $\\alpha \\beta $ shown in the second column denotes the pulse pair from Alice source $\\alpha $ and Bob source $\\beta $ , respectively.",
"Table: NO_CAPTION[ht!]",
"List of the total gains and error gains of bell state $\\psi ^-$ in the cases of $L_A=0km$ .",
"The first row indicates the channel distance length and attenuation from Bob to Charlie.",
"The notation $\\alpha \\beta $ shown in the second column denotes the pulse pair from Alice source $\\alpha $ and Bob source $\\beta $ , respectively.",
"Table: NO_CAPTION"
]
] | 1808.08584 |
[
[
"Fermion spectral function in hot strongly interacting matter from the\n functional renormalization group"
],
[
"Abstract We present the first calculation of fermion spectral function at finite temperature in quark-meson model in the framework of the functional renormalization group (FRG).",
"We compare the results in two truncations, after first evolving flow equation of effective potential, we investigate the spectral function either by taking the IR values as input to calculate one-loop self-energy or by taking the scale-dependent values as input to evolve the flow equation of the fermion two-point function.",
"The latter one is a self-consistent procedure in the framework of FRG.",
"In both truncations, we find a multi-peak structure in the spectral function, indicating quark collective excitations realized in terms of the Landau damping.",
"However, in contrast to fermion zero-mode in the one-loop truncation, we find a fermion soft-mode in the self-consistent truncation, which approaches the zero-mode as temperature increases."
],
[
"Introduction",
"The Quantum Chromodynamics (QCD) phase transitions at finite temperature and density provide a deep insight into the strong interacting matter created in high energy nuclear collisions and compact stars.",
"The properties of the QGP phase near the critical temperature ($T_c$ ) acquire much interest, the heavy-ion collisions has suggested that the QGP matter is an ideal fluid [2], [1], indicating that the created matter is a strongly coupled system.",
"The spectral properties of quark and hadron in this strongly interacting matter, are of fundamental importance for identifying the relevant degrees of freedom in the equation of state and respective transport properties.",
"Whether quark can be described by well-defined quasi-particles has long been investigated, quasi-particles correspond to peaks with a small width in the spectral function with relevant quantum numbers.",
"Quenched lattice QCD simulation indicates the existence of the quasi-particles of quarks with small decay width [3], [4], [5].",
"Finite temperature gauge theory with Dyson-Schwinger equation also predicts quasi-particle properties of quarks [6], [7].",
"At high temperature, where the hard thermal loop (HTL) approximation applies, quarks still have some collective excitation known by the normal quasi-quark and the plasmino branches in the spectral function [8], [9], [10].",
"It has also been investigated that, quarks in the QGP phase can be described within a quasi-particle picture with a multi-peak spectral function [11], [12], [13], whenever the interaction is mediated by scalar, pseudo-scalar, vector and axial-vector meson, which may exist as bosonic excitations in the QGP phase [13].",
"In the vicinity of $T_c$ of the chiral phase transition, non-perturbative effects are important, and one may expect, that the quark spectral functions will possess novel properties, when non-perturbative methods are adopted.",
"In this paper, we employ the functional renormalization group (FRG) [14], [15], [16], [17], [18], [19] approach to the quark-meson model.",
"As a non-perturbative method, FRG enables us to incorporate fluctuation effects beyond mean field theory, see Refs. [14].",
"The self-consistent treatment of fluctuations is important towards the understanding of physics near a phase transition.",
"Since the FRG allows a description of scale transformation, it provides a deep insight into the system where scale dependence plays a crucial role.",
"To calculate the spectral function in the usually used imaginary time formalism with FRG, an analytical continuation is required to bring the imaginary time in the Euclidean two point function at finite temperature to the real time in the Minkowski space [21], [22], [20], [23], [24], [25], [26].",
"This method has been applied to the study of real time observables such as shear viscosity [26] and soft modes [27] near the QCD critical point.",
"The consistent investigation of spectral function in the framework of the FRG has been applied to various systems, including meson spectral function in a chiral phase transition [23], [24], quark spectral function in vacuum [28], meson spectral function in a pion superfluid system [29].",
"When the FRG is put to use to investigate the quark spectral function, the most crucial difference is that one takes into account the scale dependence of the meson masses, and hence the thresholds of each decay, creation and scattering channel.",
"The scale dependence is a high order effect and brings about difference in spectral function mainly in the following three aspects.",
"First, it gives arise to novel structures in the imaginary part and real part of the self-energy.",
"Second, the Landau damping which is the dominant effect at high temperature, is forbidden at low energy when considering the scale dependence of the meson masses, and leads to more peaks at low energy region at high temperature.",
"Third, it is found that a fermion zero mode starts to appear when temperature is comparable to meson mass [13], which also originates from the Landau damping effect.",
"However, when the FRG is adopted, this zero mode becomes a soft mode, and approaches the origin when temperature increases.",
"We organize the paper as follows.",
"The FRG flow equations for the effective potential and the two truncations to calculate the quark self-energy are derived in Section .",
"The procedure to solve the flow equations and the numerical results are presented in Section .",
"We summarize in Section ."
],
[
"Flow equations and Truncation",
"As an low energy effective model, the quark-meson model comes from the partial bosonization of the four-fermion interaction model and exhibits many of the global symmetries of QCD.",
"It is widely used as an effective chiral model to demonstrate the spontaneous chiral symmetry breaking in vacuum and its restoration at finite temperature and density [30], [31], [32].",
"Here we take the two-flavor version of the model with pseudo-scalar mesons ${\\bf \\pi }$ and scalar meson $\\sigma $ as the dominant meson degrees of freedom at energy scale up to $\\Lambda \\approx 1$ GeV.",
"The Euclidean effective action of the model at finite temperature $T$ and density $\\mu $ is given as $\\Gamma = \\int _x&\\Big [&\\bar{\\psi }\\left(\\partial \\!\\!\\!/-\\gamma _0\\mu \\right)\\psi +g\\bar{\\psi }\\left(\\sigma +i\\gamma _5{\\vec{\\tau }}\\cdot {\\vec{\\pi }}\\right)\\psi \\nonumber \\\\&+&\\frac{1}{2}(\\partial _\\mu \\phi )^2+U(\\phi ^2)-c\\sigma \\Big ],$ where the abbreviation $\\int _x$ stands for $\\int _0^\\beta dx_0\\int d^3x$ with the inverse of temperature $\\beta =1/T$ , and ${\\vec{\\tau }}$ are the Pauli matrices in flavor space.",
"The Yukawa coupling is chosen as $g=3.2$ to fit the quark mass in vacuum.",
"The fermion field $\\psi $ and meson field $\\phi $ are defined as $\\psi =(u, d)$ and $\\phi =(\\sigma ,\\pi _1,\\pi _2,\\pi _0)$ .",
"The explicit chiral symmetry breaking term $-c\\sigma $ corresponds to a finite current quark mass $m_0$ .",
"Quantum and thermal fluctuations are of particular importance in the vicinity of a phase transition and are conveniently included within the framework of FRG.",
"The core quantity in this approach is the averaged effective action $\\Gamma _k$ at the RG scale $k$ in Euclidean space, its scale dependence is described by the flow equation [14], [15], [16], [17], [18], [19] $\\dot{\\Gamma }_k =\\text{Tr}\\int _p\\left[\\frac{1}{2}G_{\\phi ,k}(p)\\dot{R}_{\\phi ,k}(p)-G_{\\psi ,k}(p)\\dot{R}_{\\psi ,k}(p)\\right],$ where $\\dot{\\Gamma }_k=\\partial _k\\Gamma _k$ and so as for $\\dot{R}_k$ .",
"The symbol '$\\text{Tr}$ ' represents the summation over all inner degrees of freedom of mesons and quarks.",
"$G_{\\phi ,k}(q) &=& \\left(\\Gamma _k^{(2)}[\\phi ]+R_{\\phi ,k}(q)\\right)^{-1},\\nonumber \\\\G_{\\psi ,k}(q) &=& \\left(\\Gamma _k^{(2)}[\\psi ]+R_{\\psi ,k}(q)\\right)^{-1}$ are the FRG modified meson and quark propagators with the two-point functions $\\Gamma _k^{(2)}[\\phi ]=\\delta ^2\\Gamma _k/\\delta \\phi ^2$ and $\\Gamma _k^{(2)}[\\psi ]=\\delta ^2\\Gamma _k/\\delta \\psi \\delta \\bar{\\psi }$ and the two regulators $R_{\\phi ,k}$ and $R_{\\psi ,k}$ .",
"The evolution of the flow from the ultraviolet limit $k=\\Lambda $ to infrared limit $k=0$ encodes in principle all the quantum and thermal fluctuations in the action.",
"To suppress the fluctuations with momentum smaller than the scale $k$ during the evolution, an infrared regulator $R$ is introduced in the flow equation.",
"At finite temperature and density where the Lorentz symmetry is broken, we employ the optimized regulator function which is the three dimensional analogue of the 4-momentum regulator [33], [34].",
"The bosonic and fermionic regulators are chosen to be $R_{\\phi ,k}(p) &=& \\vec{p}^2 r_B(y),\\nonumber \\\\R_{\\psi ,k}(p) &=& \\vec{\\gamma }\\cdot \\vec{p} r_F(y)$ in momentum space with $y=\\vec{p}^2/k^2$ and $r_B(y)=(1/y-1)\\Theta (1-y)$ and $r_F(y)=(1/\\sqrt{y}-1)\\Theta (1-y)$ .",
"The regulators $R_{\\phi ,k}$ and $R_{\\psi ,k}$ in the propagators $G_\\phi $ and $G_\\psi $ amount to having regularized three-momenta $\\vec{p}_r^2=\\vec{p}^2(1+r_B(y))$ and $\\vec{p}_r=\\vec{p}(1+r_F(y))$ for bosons and fermions respectively.",
"The three dimensional regulators break down the Lorentz symmetry in vacuum.",
"However, physical quantities are measured in the ground state at $k=0$ , where the regulators vanish and the Lorentz symmetry is guaranteed.",
"In order to derive the meson and quark propagators $G_{\\phi ,k}$ and $G_{\\psi ,k}$ , we expand the effective potential around the mean field $\\langle \\sigma \\rangle _k$ , which describes the chiral symmetry breaking, and introduce the chiral invariant $\\rho _k=\\langle \\sigma \\rangle _k^2$ .",
"The RG-modified propagators of mesons and quark are $G_{\\phi ,k}^{-1}&=&p_0^2+\\vec{p}_r^2+m_{\\phi ,k}^2,\\nonumber \\\\G_{\\psi ,k}^{-1}&=&-\\gamma _0(ip_0+\\mu )+\\vec{\\gamma }\\cdot \\vec{p}+m_{f,k},$ with $m_{\\sigma ,k}^2=2U^{\\prime }+4\\rho _k U^{\\prime \\prime }$ and $m_{\\pi ,k}^2=2U^{\\prime }$ , $U^{\\prime }$ and $U^{\\prime \\prime }$ are first and second order derivatives of effective potential with respect to $\\rho $ , and quark mass $m_{f,k}=g\\langle \\sigma \\rangle _k$ .",
"Assuming uniform field configurations, the integral over space and imaginary time becomes trivial, and the effective action $\\Gamma _k=\\beta V U_k$ is fully controlled by the potential $U_k$ with $V$ and $\\beta $ being the space and time regions of the system.",
"The flow equation $U_k$ hence comes directly from that of $\\Gamma _k$ , namely $\\partial _kU_k=(T/V)\\partial _k\\Gamma _k$ .",
"The flow equation of the effective potential is then calculated by $\\partial _kU_k~=~\\frac{1}{2}J_\\phi (E_{\\sigma ,k})+\\frac{3}{2}J_\\phi (E_{\\pi ,k})-N_cN_fJ_\\psi (E_{\\psi ,k}),$ with $J_\\phi $ and $J_\\psi $ are one-loop threshold functions, the explicit expressions are presented in Appendix by Eq.",
"(REF ), and the energies are given by $E_{\\phi ,k}=\\sqrt{k^2+m_{\\phi ,k}^2}$ and $E_{\\psi ,k}=\\sqrt{k^2+m_{\\psi ,k}^2}$ .",
"In the following, we are going to present two truncations to calculate the self-energy and spectral function.",
"In both truncations, we first evolve the flow of the effective potential, and then take the masses as input to calculate the self-energy.",
"In truncation A, we take scale-dependent masses $m_{\\sigma ,k}$ and $m_{\\pi ,k}$ at IR-minimum $\\rho _{k=0}$ as input to integrate the flow of the two point function; while in truncation B, we take the IR masses $m_{\\sigma ,k=0}$ and $m_{\\pi ,k=0}$ to directly calculate the one-loop self-energy of quark.",
"The diagrammatic description is presented in FIG.",
"REF Figure: The diagrammatic presentation of self-energy in two truncations, ∂ k Γ ψ,k (2) \\partial _k\\Gamma _{\\psi ,k}^{(2)} is the flow equation for fermion two-point function in truncation A, Σ\\Sigma is the one-loop self-energy in truncation B."
],
[
"Truncation A",
"In truncation A, we first evolve the flow of effective potential and prepare the scale-dependent meson masses as input.",
"We then integrate the flow equation of the two-point function to obtain the self-energy in the infrared limit.",
"The flow equation of fermion two-point function has Dirac structure, namely is a $4\\times 4$ matrix in Dirac space.",
"For u quark of a certain color, the flow equation of the two-point function $\\partial _k{\\Gamma }^{(2)}_{\\bar{u}u}(p)~=~-g^2\\widetilde{\\partial }_k\\int _q&\\Big [&G_\\sigma (q-p)G_u(q)\\\\&+&3G_\\pi (q-p)(i\\gamma _5)G_u(q)(i\\gamma _5)\\Big ],\\nonumber $ where $\\widetilde{\\partial }_k$ is the derivative of RG-scale k that only acts on the regulator $R_{\\phi ,k}$ and $R_{\\psi ,k}$ in the propagators.",
"In this work, we consider only the spectral function at zero external momentum $\\vec{p}=0$ .",
"In the ultraviolet, the Euclidean inverse quark propagator at $\\vec{p}=0$ at the IR expansion point is $G^{-1}_{E,\\Lambda }(ip_0)=\\Gamma ^{(2)}_{\\bar{u} u,\\Lambda }(ip_0)=-\\gamma ^0(ip_0+\\mu )+g\\langle \\sigma \\rangle .$ The inverse propagator at scale $k$ is then an integral of the flow equation two-point function from the ultraviolet $k=\\Lambda $ down to $k$ $G^{-1}_{E,k}(ip_0)&=&\\Gamma ^{(2)}_{\\bar{u} u,k}(ip_0)\\\\&=&-\\gamma ^0(ip_0+\\mu )+g\\langle \\sigma \\rangle +\\int _\\Lambda ^k\\partial _k{\\Gamma }^{(2)}_{\\bar{u}u}(ip_0) dk.\\nonumber $ Hence, the Euclidean inverse propagator can be written in terms of k-dependent self energy $G^{-1}_{E,k}(ip_0)=\\Gamma ^{(2)}_{\\bar{u} u,k}(ip_0)=-\\gamma ^0(ip_0+\\mu )+g\\langle \\sigma \\rangle +\\Sigma _k(ip_0)$ , with the initial condition $\\Sigma _\\Lambda (ip_0)=0$ .",
"The scale-dependent self-energy in Euclidean space is thus $\\Sigma _k(ip_0)-\\Sigma _\\Lambda (ip_0)=\\int _\\Lambda ^k\\partial _k{\\Gamma }^{(2)}_{\\bar{u}u}(ip_0) dk.$ At ultraviolet limit $k=\\Lambda $ , no fluctuation is included, $\\Sigma _\\Lambda (ip_0)=0$ agrees with the bare propagator.",
"As the scale is lowered, quantum fluctuation are included, contributing to the self-energy of the quark.",
"The spectral function is a real-time quantity which requires analytical continuation to bring imaginary time to real time $G^{-1}_{R}(\\omega )=-G^{-1}_{E}(ip_0\\rightarrow \\omega +i\\eta ).$ The inverse retarded propagator is $G^{-1}_{R,k}(\\omega )=\\gamma ^0(\\omega +\\mu +i\\eta )-g\\langle \\sigma \\rangle -\\Sigma _{R,k}(\\omega )$ and the corresponding retarded self-energy is $\\Sigma _{R,k}(\\omega )~=~\\int _\\Lambda ^k\\partial _k{\\Gamma }^{(2)}_{\\bar{u}u}(\\omega +i\\eta )dk.$ The quark propagator at zero momentum $\\vec{p}=0$ can be decomposed to the positive energy part and negative energy part $G_{R,k}(\\omega )~=~G_{+,k}(\\omega )\\Lambda _+\\gamma ^0+G_{-,k}(\\omega )\\Lambda _-\\gamma ^0,$ with projection operators $\\Lambda _\\pm \\equiv (1\\pm \\gamma ^0)/2$ acting onto spinors whose chirality is equal(+) or opposite(-) to the helicity.",
"We call +(-)-sector as 'quark' ('anti-quark') sector.",
"Hence, propagator for positive and negative energy parts are $G_{\\pm ,k}(\\omega )&=&\\frac{1}{2}\\text{Tr}\\big [G_{R,k}(\\omega )\\gamma ^0\\Lambda _\\pm \\big ]\\nonumber \\\\&=&\\big [\\omega +i\\eta +\\mu \\mp m_f-\\Sigma _{\\pm ,k}(\\omega )\\big ]^{-1},$ with $\\Sigma _{\\pm ,k}(\\omega )&=&\\frac{1}{2}\\text{Tr}\\big [\\Sigma _{R,k}(\\omega )\\gamma ^0\\Lambda _\\pm \\big ].$ We here focus on the self-energy of the 'quark' sector, the flow equation is given by $\\partial _k \\Sigma _{\\pm ,k}(\\omega )=\\frac{1}{2}\\text{Tr}\\Big [\\partial _k{\\Gamma }^{(2)}_{\\bar{u}u}(\\omega )\\gamma ^0\\Lambda _\\pm \\Big ],$ which, after integral over the three momentum, Matsubara sum and analytical continuation, has the following structure, $\\partial _k\\Sigma _{+,k}(\\omega )=g^2\\Big (J^S_{\\psi \\sigma }(\\omega )+J^S_{\\sigma \\psi }(\\omega )+3J^{PS}_{\\psi \\pi }(\\omega )+3J^{PS}_{\\pi \\psi }(\\omega )\\Big ),\\nonumber \\\\$ with $J^S_{\\psi \\sigma }, J^S_{\\sigma \\psi }, J^{PS}_{\\psi \\pi }, J^{PS}_{\\pi \\psi }$ are the threshold functions presented in the appendix.",
"At vanishing quark number, from the charge conjugation symmetry, we have a relation between $G_\\pm $ , $G_+(\\omega )=-G_-^*(-\\omega ),$ but finite chemical potential breaks the charge conjugation symmetry.",
"We limit our study to the case with $\\mu =0$ and focus on the spectral function of quark sector only.",
"The spectral function is defined through the imaginary part of the retarded propagator $\\rho _k(\\omega )=-(1/\\pi )\\text{Im} G_{R,k}(\\omega )$ .",
"This can decomposed similarly, $\\rho _k(\\omega )=\\rho _{+,k}(\\omega )\\Lambda _+\\gamma _0+\\rho _{-,k}(\\omega )\\Lambda _-\\gamma _0 $ , with $\\rho _{\\pm ,k}(\\omega )&=&-\\frac{1}{\\pi }\\text{Im} G_{\\pm ,k}\\\\&=&-\\frac{1}{\\pi }\\frac{\\text{Im}\\Sigma _{\\pm ,k}(\\omega )}{\\big (\\omega \\mp m_f-\\text{Re}\\Sigma _{\\pm ,k}(\\omega )\\big )^2+\\text{Im}\\Sigma _{\\pm ,k}(\\omega )^2}.\\nonumber $"
],
[
"Truncation B",
"In another truncation, we first evolve the flow equation of the effective potential, and find the infrared quark mass $m_{f,k=0}(T,\\mu )$ and meson mass $m_{\\sigma ,k=0}(T,\\mu ),~m_{\\pi ,k=0}(T,\\mu )$ at certain temperature and density.",
"We then take these quantities as input to calculate the self-energy of quark at one-loop order.",
"This method has been discussed in Ref.",
"[11], [12], [13], here we take the FRG result as an input.",
"In Euclidean space, the fermion self-energy $\\Sigma _E(ip_0)=&-&g^2\\int _q\\big [G_\\psi (q)G_\\sigma (p+q)\\nonumber \\\\&+&3G_\\psi (q)(i\\gamma ^5)G_\\pi (p+q)(i\\gamma ^5)\\big ].$ The analytical continuation is taken by $\\Sigma _R(\\omega )=\\Sigma _E(ip_0)\\big |_{ip_0\\rightarrow \\omega +i\\eta }$ .",
"Taking the energy projection as before, we have the self-energy of 'quark' and 'anti-quark' sector $\\Sigma _{\\pm }(\\omega )~=~\\frac{1}{2}\\text{Tr}[\\Sigma _{R}(\\omega )\\gamma ^0\\Lambda _\\pm ]$ .",
"The self-energy has contribution from the scalar channel and the pseudo-scalar channel, where the interaction mediated by sigma meson and pion respectively, $\\Sigma _{\\pm }(\\omega )=\\Sigma ^S_{\\pm }(\\omega )+3\\Sigma ^{PS}_{\\pm }(\\omega ).$ One may first deal with the imaginary part of the scalar channel $&&\\text{Im}\\Sigma ^S_+(\\omega )~~=~~-\\frac{g^2}{2\\pi }\\int _0^{+\\infty }\\frac{p^2dp}{E_\\sigma }\\Big \\lbrace \\\\&+&\\delta (\\omega +E_\\sigma +E_f)~\\Big (1+\\frac{m_f}{E_f}\\Big )~\\Big [1+n_b(E_\\sigma )-n_f(E_f)\\Big ]\\nonumber \\\\&+&\\delta (\\omega +E_\\sigma -E_f)~\\Big (1-\\frac{m_f}{E_f}\\Big )~\\Big [n_b(E_\\sigma )+n_f(E_f)\\Big ]\\nonumber \\\\&+&\\delta (\\omega -E_\\sigma +E_f)~\\Big (1+\\frac{m_f}{E_f}\\Big )~\\Big [n_b(E_\\sigma )+n_f(E_f)\\Big ]\\nonumber \\\\&+&\\delta (\\omega -E_\\sigma -E_f)~\\Big (1-\\frac{m_f}{E_f}\\Big )~\\Big [1+n_b(E_\\sigma )-n_f(E_f)\\Big ]\\Big \\rbrace .\\nonumber $ The momentum integral can be carried out analytically, giving [12] $\\text{Im}\\Sigma ^S_+(\\omega )=&-&\\frac{g^2}{64\\pi }\\frac{(\\omega +M_+)(\\omega -M_-)}{\\omega ^3}\\nonumber \\\\&\\times &\\sqrt{(\\omega ^2-M_+^2)(\\omega ^2-M_-^2)}\\nonumber \\\\&\\times &\\left[\\coth \\frac{\\omega ^2+M_+M_-}{4\\omega T}+\\tanh \\frac{\\omega ^2-M_-M_+}{4\\omega T}\\right]\\nonumber \\\\&\\times &\\left(\\Theta (\\omega ^2-M_+^2)-\\Theta (M_+^2-\\omega ^2)\\right),$ where $M_+=m_\\sigma +m_f$ and $M_-=|m_\\sigma -m_f|$ and $\\Theta (x)$ is the step function.",
"The self-energy of the pseudo-scalar channel can be obtain from that of the scalar channel by taking the substitution $m_f\\rightarrow -m_f$ and $m_\\sigma \\rightarrow m_\\pi $ .",
"The self-energy $\\Sigma ^R(\\omega )$ has an ultraviolet divergence which originates from the T-independent part $\\Sigma ^R_{T=0}(\\omega )\\equiv \\lim _{T\\rightarrow 0^+}\\Sigma ^R(\\omega )$ .",
"The divergence can be removed by imposing the on-shell renormalization condition on the T-independent part of the quark propagator.",
"The T-dependent part, $\\Sigma ^R_{T\\ne 0}(\\omega )=\\Sigma ^R(\\omega )-\\Sigma ^R_{T=0}(\\omega )$ is free from divergence.",
"The real part and imaginary part are related by the Krames-Kronig relation, $\\text{Re} \\Sigma _{R,T\\ne 0}(\\omega )=\\mathcal {P}\\int _{-\\infty }^{+\\infty }\\frac{dz}{\\pi }\\frac{\\text{Im} \\Sigma _{R,T\\ne 0}(z) }{z-\\omega }.$ The spectral function of quark and anti-quark are then given by $\\rho _{\\pm }(\\omega )&=&-\\frac{1}{\\pi }\\text{Im} G_{\\pm }\\\\&=&-\\frac{1}{\\pi }\\frac{\\text{Im}\\Sigma _{\\pm }(\\omega )}{\\big (\\omega \\mp m_f-\\text{Re}\\Sigma _{\\pm }(\\omega )\\big )^2+\\text{Im}\\Sigma _{\\pm }(\\omega )^2}.\\nonumber $"
],
[
"Numerical method and Result",
"To numerically solve the flow equations for the effective potential and the two-point functions, we adopt the grid method, and assume the initial condition at the ultraviolet limit, $U_\\Lambda (\\rho )=\\frac{1}{2}m_\\Lambda ^2\\rho +\\frac{1}{4}\\lambda _\\Lambda \\rho ^2,$ for one-dimensional grid, and for the quark self-energy $\\Sigma _{+,\\Lambda }(\\omega )=0.$ During the process of integrating the flow equations from the ultraviolet limit to the infrared limit, the condensates $\\langle \\sigma \\rangle _k$ is obtained by locating the minimum of the $k$ -dependent effective potential $U_k$ .",
"Choosing the quark mass $m_q=300$ MeV, pion mass $m_\\pi =135$ MeV and pion decay constant $f_\\pi =93$ MeV in vacuum, the corresponding initial parameters are $m_\\Lambda ^2/\\Lambda ^2=0.618$ , $\\lambda _\\Lambda =1$ and $c/\\Lambda ^3=0.0025$ with the cutoff $\\Lambda =900$ MeV.",
"In the numerical calculation, the infrared limit $k=0$ cannot be reached.",
"Instead, the evolution of the flow equation is stopped at $k_{\\text{IR}}<10$ MeV, where the condensate and coupling have reached a stable structure.",
"By solving the flow equation of the effective potential Eq.",
"(REF ), we obtain the dependence of chiral condensate and various masses on temperature, with critical temperature $T_c\\sim 170$ MeV.",
"The result is presented in Fig.REF .",
"Figure: Temperature dependence of chiral condensate and various masses.The imaginary part of the self-energy is proportional to the difference between the decay and creation rates of the quasi-quark.",
"Each of the four channels has a Dirac delta function, indicating the energy threshold for each process, see Eq.",
"(REF ,REF ,REF ,REF ).",
"One can give a physical interpretation to each of the channels, (I) $\\delta (\\omega +E_\\phi +E_\\psi )$ describes a annihilation of a quasi-quark with an on-shell free anti-quark and boson, (IV) $\\delta (\\omega -E_\\phi -E_\\psi )$ is a decay of a quasi-quark to on-shell free quark and a boson, (II) $\\delta (\\omega +E_\\phi -E_\\psi )$ is the decay process of a quasi-quark state Q, into an on-shell quark via a collision with a thermally excited boson, and its inverse process $Q+b\\leftrightarrow q$ .",
"(III) $\\delta (\\omega -E_\\phi +E_\\psi )$ corresponds to a pair annihilation process between the quasi-quark and a thermally excited anti-quark with an emission of a boson into the thermal bath, and its inverse process $Q+\\bar{q}\\leftrightarrow b$ .",
"The latter two processes are called Landau damping, which vanishes at $T=0$ , as it involves thermally excited particles in the initial states.",
"The Landau damping plays an important role in the spectral function as temperature rises and is closely related to the three peak structure at high temperature.",
"Process (II) and (III) both cause a mixing between the quark and anti-quark hole through coupling to thermally excited boson as discussed in Ref.",
"[12].",
"In truncation A, the real part and imaginary part of the self-energy are calculated separately, the real part is given by the principle value integral.",
"While for the imaginary part of the two-point function, the integral over RG-scale $k$ only has contribution from a few scales $k_i$ due to the appearance of aforementioned Dirac delta functions.",
"The following structures are encountered in the integration of the imaginary part of the flow equation, where $g(k)=\\pm E_{\\phi ,k}\\pm (\\mp ) E_{\\psi ,k}$ , and $k_i$ are zero points of the delta-function at a certain energy $\\omega $ , with $\\omega +g(k_i)=0$ , $&&\\int _\\Lambda ^0f(k)\\delta (\\omega +g(k))dk=-\\sum _i\\frac{f(k_i)}{|g^{\\prime }(k_i)|},\\\\&&\\int _\\Lambda ^0f(k)\\delta ^{\\prime }(\\omega +g(k))dk\\nonumber \\\\&&\\qquad =\\sum _i\\left(\\frac{f^{\\prime }(k_i)}{g^{\\prime }(k_i)}-\\frac{f(k_i)g^{\\prime \\prime }(k_i)}{g^{\\prime }(k_i)^2}\\right)\\frac{1}{|g^{\\prime }(k_i)|}.\\nonumber $ It is required that $g(k)$ is a continuously differentiable function with $g^{\\prime }$ nowhere zero.",
"In the integral of the flow equation $g(k)$ has certain points where the derivatives are zero, the domain must be broken up to exclude the $g^{\\prime }= 0$ point.",
"These $g^{\\prime }(k_i)= 0$ points are similar to the van-Hove singularities in the density of states in condensed matter physics[35], $g(\\omega )=\\sum _n\\int \\frac{d^3k}{(2\\pi )^3}\\delta (\\omega -\\omega _n(\\vec{k}))=\\sum _n\\int \\frac{dS_\\omega }{(2\\pi )^3}\\frac{1}{|\\nabla \\omega _n(\\vec{k})|}$ .",
"The group velocity $\\nabla \\omega _n(\\vec{k})$ vanishes at certain momenta, resulting a divergent integrand.",
"The divergence is integrable in 3-dimensions, in lower dimensions, the van-Hove singularity appears.",
"The van Hove singularities have also attracted interests in high energy physics [36], [37], [38], [39].",
"In our case, the integral over $k$ is one-dimensional and gives divergence in the imaginary part at finite temperature.",
"This divergence exists only in truncation A, when scale dependence of various masses are taken into consideration, and appears only when two conditions are satisfied at the same time, that $\\omega +\\delta E(k^*)=0$ and $\\delta E^{\\prime }(k^*)=0$ .",
"The scale dependence of meson mass also causes divergence in the real part of the self-energy.",
"In FIG.REF , we present the RG-scale dependence of threshold for channels (I) to (IV), $\\pm E_{\\phi ,k}\\pm E_{\\psi ,k}$ , ($\\phi =\\sigma , \\pi $ ) for three temperature $T=50,~170,~300$ MeV, with the dashed lines for pions and solid line for sigma.",
"For terms related to Landau damping (II) and (III), at certain $\\omega $ , the k-integral runs into points that $\\delta E^{\\prime }(k^*)=0$ , leading to the divergence in imaginary part.",
"At low temperature, when the Landau damping is suppressed, the diverge does not appear.",
"In contrast, in truncation B, we take $m_{\\phi ,k=0}$ as input, the imaginary part and real part are always finite after remove the zero-temperature part.",
"Figure: The RG-scale dependence of threshold for channels (I) to (IV), ±E φ,k ±E ψ,k \\pm E_{\\phi ,k}\\pm E_{\\psi ,k}, (φ=σ,π\\phi =\\sigma , \\pi ) for three temperature T=50,170,300T=50,~170,~300MeV.We first present the spectral function of quark sector at relatively low temperature, In FIG.REF , from top to bottom, are figures for the spectral function, the real part and imaginary part of the self-energy, with the black solid line represents truncation A, and red dashed line for truncation B.",
"The zero-temperature result can be found in Ref.",
"[28], to which, we have also found the same result.",
"At T=50 MeV, the system is still in the chiral symmetry breaking phase, with quark mass $m_f=296$ MeV and meson mass $m_\\sigma =477$ MeV and $m_\\pi =140$ MeV.",
"In both truncations we have delta-peaks around fermion mass, $\\omega =315$ MeV for truncation A, and $\\omega =296$ MeV for truncation B.",
"The peak structure in $\\rho _+(\\omega )$ emerges at the \"quasi-pole\" [12] which is defined as zero of the real part of the inverse propagator $\\omega -m_f-\\text{Re}\\Sigma _+(\\omega )=0$ , providing that the imaginary part is small enough at that point.",
"In both truncations, the real part of the inverse propagator only has one \"quasi-pole\", which is the cross point of $\\omega -m_f$ (the blue dotted line) with $\\text{Re}\\Sigma _+(\\omega )$ in the figure.",
"The difference in the position of the peak in both truncations comes the inclusion and subtraction of different fluctuation.",
"The Landau damping is still well suppressed, giving small imaginary part and flat structures in the spectral function at low energy.",
"At large energy, when $|\\omega |>m_\\psi +m_\\phi $ the decay processes (I) and (IV) take place, giving finite imaginary part and the continuous spectrum in the spectral function in truncation A.",
"While in truncation B, this continuous spectrum is subtracted when performing the renormalization.",
"Figure: Quark spectral function at T=50T=50MeV, from top to bottom are spectral function, real and imaginary part of the quark self-energy.",
"The black solid line stands for truncation A, while red dashed line for truncation B.",
"The blue dotted line represent ω-m f \\omega -m_f, with m f =296m_f=296MeV.The chiral phase transition takes place at the temperature about $T=170$ MeV, pion and sigma meson have not been degenerate yet, with $m_f=130$ MeV, $m_\\sigma =259$ MeV, $m_\\pi =212$ MeV.",
"For truncation A, the threshold of various channels is presented in FIG.REF .",
"In the scattering channel with the thermally excited boson (II) and the annihilation channel with the thermally excited anti-quark (III), $\\pm E_{\\phi ,k}\\mp E_{\\psi ,k}$ has points where the derivative with RG-scale vanished.",
"This brings about divergence in the imaginary part, and oscillation in the real part, see black lines in Fig.REF .",
"The imaginary part is discontinuous at the four van-Hove singularities, and is zero when $\\omega \\le |\\pm E_{\\phi ,k}\\mp E_{\\psi ,k}|_{\\text{min}}$ , where process (II) (III) are forbidden.",
"$\\omega -m_f-\\text{Re}\\Sigma _+(\\omega )$ has several quasi-poles in truncation A, yet the spectral function has only one peak, when the imaginary part is small $\\omega =21$ MeV indicating a quasi-particle mode here.",
"For other quasi-poles, the imaginary parts are too large to form a peak, giving several bumps instead.",
"In truncation B, however, we have a quite different case, the imaginary part is non-zero but finite when process (II) and (III) are allowed.",
"The real part has five quasi-poles, at three of them, the imaginary part is small enough to give a peak in the spectral function.",
"When process (II) and (III) are allowed, the imaginary part is large and gives only small bumps in the spectral function, see the red dashed lines in FIG.REF .",
"The spectral function presents a three peak structure in truncation B, with one peak at the origin, and two quasi-pole where the imaginary part is small.",
"This is the typical structure at $T\\sim m_b$ and has been discussed in-depth in Ref.[12].",
"The Landau damping, which causes peaks in imaginary part, is essential in the three peak structure in the spectral function.",
"In truncation A, the spectral function also presents a peak at small $\\omega $ but not at at the exact origin, namely, instead of a zero mode, we have a soft-mode in truncation A, which also arises from the Landau damping effect.",
"Figure: Quark spectral function at T=170T=170MeV, from top to bottom are spectral function, real and imaginary part of the quark self-energy.",
"The black solid line stands for truncation A, while red dashed line for truncation B.",
"The blue dotted line represent ω-m f \\omega -m_f, with m f =130m_f=130MeV.Finally, we analyze the spectral function in both truncations at $T=300$ MeV, where the system has reached the chiral restored phase, with $m_f=24$ MeV, and degenerate meson mass $m_\\sigma \\approx m_\\pi =490$ MeV.",
"For truncation A, the threshold of each channel is presented in the last figure in FIG.REF .",
"There is a large area where channel (II) (III) are forbidden, leading to $\\text{Im}\\Sigma _+=0$ at small energy.",
"Channel (II) (III) both have points where $\\pm E^{\\prime }_{\\phi ,k^*}\\mp E^{\\prime }_{\\psi ,k^*}=0$ , where the imaginary part diverges.",
"The real part of the inverse propagator has 7 quasi-poles, at two of which, the imaginary parts are too large to give a peak at $\\omega =10,\\pm 160$ MeV.",
"For the three quasi-poles at low energy, the imaginary part is small and gives three peaks.",
"For the two quasi-poles at large $\\omega $ , the Landau damping effect gives two bumps in the spectral function.",
"While in truncation B, the Landau damping effect again gives two peaks in the imaginary part of the self-energy, and an oscillation in real part.",
"$\\omega -m_f-\\text{Re}\\Sigma _+(\\omega )=0$ has five quasi-poles, $\\text{Im}\\Sigma _+$ has relative large values at four of the quasi-poles, leading to two peaks with finite width, and a delta-peak at the origin.",
"Figure: Quark spectral function at T=300T=300MeV, from top to bottom are spectral function, real and imaginary part of the quark self-energy.",
"The black solid line stands for truncation A, while red dashed line for truncation B.",
"The blue dotted line represent ω-m f \\omega -m_f, with m f =24m_f=24MeV.Two scales are of crucial importance in the structure of the fermion spectral function, $m_f/m_b$ and $T/m_b$ .",
"The three peak structure is most obvious when $m_f/m_b\\ll 1$ and $T/m_b\\sim 1$ .",
"When these two factors are approached, the peaks become higher and sharper.",
"For the chiral crossover, $T/m_b\\sim 1$ takes place around the critical temperature, where we observe three sharp peaks in truncation B.",
"For high temperature $T/m_b\\approx 0.61$ , $m_f/m_b\\ll 1$ is satisfied, thus fermion is almost massless, the three peak structure is also obvious but with finite width.",
"In truncation B, the appearance of zero-mode can be demonstrated by analyzing the real and imaginary part of self-energy at $\\omega =0$ .",
"In the case of finite fermion mass, one always has $\\text{Im}\\Sigma _+(\\omega =0)=0$ , $\\text{Re}\\Sigma _+(\\omega )$ can be exactly calculated to give a quasi-pole very close to the origin, thus a peak will appear at the origin.",
"In truncation A, the multi-peak structure is also observed as $T/m_b$ approaches unit.",
"However, the zero-mode becomes a soft-mode, where the quasi-pole is slightly away from the origin.",
"The disappearance of this zero-mode results from the limited scale of momentum in the propagator and the scale dependence of the meson masses.",
"One may expect that with larger $\\Lambda $ , the soft-mode would be closer to the origin."
],
[
"Summary",
"We investigate the spectral function of quark in two truncations in a quark-meson model with functional renormalization group.",
"In both truncations, we first solve the flow equation of the effective potential and find out the temperature and scale dependence of fermion and meson masses.",
"In truncation A, we take the scale-dependent masses in step one as the input, and evolve the flow equation of the two-point function; in truncation B, we take the masses in the infrared as input and calculate the one-loop self-energy.",
"After the analytical continuation, we have the spectral function.",
"When one consistently integrates the flow equation of the two-point function, the RG-scale dependence of the energy thresholds of decay and creation channels leads to van Hove singularities at finite temperature, where Landau damping plays an important role.",
"This singularity leads to divergence in both the real and imaginary part.",
"Another feature is that, at high temperature, the Landau damping is forbidden at low energy, leading to zero imaginary part and several peaks in the low energy area.",
"In comparison, when directly calculating the one-loop self-energy, one gets a three-peak structure when temperature rises and becomes comparable to meson mass.",
"The spectral function has a peak at the origin, namely a fermion zero-mode, and the other peaks comes from the Landau damping effect.",
"Our work presents a first calculation of finite temperature quark spectral function, and supports quasi-particle picture of quarks around the critical temperature.",
"Note Added: During the writing of this manuscript, we became aware of the work by R. A. Tripolt et al.",
"[28], where the fermion spectral function at zero temperature was investigated.",
"Acknowledgement: The work is supported by the National Natural Science Foundation of China (Grant Nos.",
"11335005, 11575093, and 11775123), MOST (Grant Nos.",
"2013CB922000 and 2014CB845400), and Tsinghua University Initiative Scientific Research Program."
],
[
"Loop functions",
"With the boson and fermion occupation numbers and their derivatives, $&& n_B(x)=\\frac{1}{e^{x/T}-1},\\quad n_F(x)=\\frac{1}{e^{x/T}+1},\\nonumber \\\\&& n^{\\prime }_B(x)=\\frac{dn_B(x)}{d x},\\quad ~~ n^{\\prime }_F(x)=\\frac{dn_F(x)}{d x},$ the loop functions $J_{\\phi }$ and $J_\\psi $ in the flow equation for effective potential are explicitly expressed as $J_\\phi &=& \\frac{k^4}{3\\pi ^2} \\frac{1+2n_B(E_\\phi )}{2E_\\phi },\\nonumber \\\\J_\\psi &=& \\frac{k^4}{3\\pi ^2} \\frac{1-n_F(E_\\psi -\\mu )-n_F(E_\\psi +\\mu )}{E_\\psi }.$ The threshold function $J^S_{\\psi \\sigma }, J^S_{\\sigma \\psi }, J^{PS}_{\\psi \\pi }, J^{PS}_{\\pi \\psi }$ in the two-point function can be obtained taking derivatives with respect to the corresponding energy, $J^S_{\\psi \\sigma }(ip_0)&=&-\\frac{1}{2E_\\psi }\\frac{\\partial }{\\partial E_\\psi }J^S(ip_0),\\qquad ~~J^S_{\\sigma \\psi }(ip_0)~=~-\\frac{1}{2E_\\sigma }\\frac{\\partial }{\\partial E_\\sigma }J^S(ip_0),\\nonumber \\\\J^{PS}_{\\psi \\pi }(ip_0)&=&-\\frac{1}{2E_\\psi }\\frac{\\partial }{\\partial E_\\psi }J^{PS}(ip_0),\\qquad J^{PS}_{\\pi \\psi }(ip_0)~=~-\\frac{1}{2E_\\pi }\\frac{\\partial }{\\partial E_\\pi }J^{PS}(ip_0).$ After the analytical continuation, $J^S$ in Minkovski space is $J^S(\\omega )=-\\frac{k^4}{3\\pi ^2}\\frac{1}{4E_\\phi }\\bigg \\lbrace &&\\frac{1}{\\omega +\\mu +E_\\phi +E_\\psi +i\\eta }\\left(1-\\frac{m_f}{E_\\psi }\\right)(1+n_B(E_\\phi )-n_F(E_\\psi +\\mu ))\\nonumber \\\\&+&\\frac{1}{\\omega +\\mu +E_\\phi -E_\\psi +i\\eta }\\left(1+\\frac{m_f}{E_\\psi }\\right)(n_B(E_\\phi )+n_F(E_\\psi -\\mu ))\\nonumber \\\\&+&\\frac{1}{\\omega +\\mu -E_\\phi +E_\\psi +i\\eta }\\left(1-\\frac{m_f}{E_\\psi }\\right)(n_B(E_\\phi )+n_F(E_\\psi +\\mu ))\\nonumber \\\\&+&\\frac{1}{\\omega +\\mu -E_\\phi -E_\\psi +i\\eta }\\left(1+\\frac{m_f}{E_\\psi }\\right)(1+n_B(E_\\phi )-n_F(E_\\psi -\\mu ))\\bigg \\rbrace ,$ Making substitution by $m_f\\rightarrow -m_f$ and $E_\\sigma \\rightarrow E_\\pi $ then we can get threshold for pseudoscalar channel $J^{PS}(\\omega )$ .",
"The real part is given by the principle value, while the imaginary part of the threshold function is then, $\\text{Im}J^S(\\omega )~=~\\frac{k^4}{3\\pi }\\frac{1}{4E_\\phi }~\\Big \\lbrace & &\\delta (\\omega +\\mu +E_\\phi +E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu +E_\\phi -E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi -\\mu )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu -E_\\phi +E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu -E_\\phi -E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi -\\mu )\\Big )\\Big \\rbrace .$ Following Eq.",
"(REF ), one have the imaginary part of the threshold functions $\\text{Im}J^S_{\\psi \\phi }$ , $\\text{Im}J^S_{\\phi \\psi }$ for the scalar channel.",
"$\\text{Im}J^S_{\\psi \\phi }(\\omega )~=~-\\frac{k^4}{3\\pi }\\frac{1}{8E_\\phi E_\\psi }~\\bigg \\lbrace &&\\delta ^{\\prime }(\\omega +\\mu +E_\\phi +E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&-&\\delta ^{\\prime }(\\omega +\\mu +E_\\phi -E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi -\\mu )\\Big )\\nonumber \\\\&+&\\delta ^{\\prime }(\\omega +\\mu -E_\\phi +E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&-&\\delta ^{\\prime }(\\omega +\\mu -E_\\phi -E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi -\\mu )\\Big )\\bigg \\rbrace \\\\-\\frac{k^4}{3\\pi }\\frac{m_f}{8E_\\phi E_\\psi ^3}~\\bigg \\lbrace &&\\delta (\\omega +\\mu +E_\\phi +E_\\psi )\\left[1+n_B(E_\\phi )-n_F(E_\\psi +\\mu )-E_\\psi \\left(\\frac{E_\\psi }{m_f}-1\\right)n^{\\prime }_F(E_\\psi +\\mu )\\right]\\nonumber \\\\&-&\\delta (\\omega +\\mu +E_\\phi -E_\\psi )\\left[n_B(E_\\phi )+n_F(E_\\psi -\\mu )-E_\\psi \\left(\\frac{E_\\psi }{m_f}+1\\right)n^{\\prime }_F(E_\\psi -\\mu )\\right]\\nonumber \\\\&+&\\delta (\\omega +\\mu -E_\\phi +E_\\psi )\\left[n_B(E_\\phi )+n_F(E_\\psi +\\mu )+E_\\psi \\left(\\frac{E_\\psi }{m_f}-1\\right)n^{\\prime }_F(E_\\psi +\\mu )\\right]\\nonumber \\\\&-&\\delta (\\omega +\\mu -E_\\phi -E_\\psi )\\left[1+n_B(E_\\phi )-n_F(E_\\psi -\\mu )+E_\\psi \\left(\\frac{E_\\psi }{m_f}+1\\right)n^{\\prime }_F(E_\\psi -\\mu )\\right]\\bigg \\rbrace \\nonumber $ $\\text{Im}J^S_{\\phi \\psi }(\\omega )~=~-\\frac{k^4}{3\\pi }\\frac{1}{8E_\\phi ^2}~\\Big \\lbrace &&\\delta ^{\\prime }(\\omega +\\mu +E_\\phi +E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&+&\\delta ^{\\prime }(\\omega +\\mu +E_\\phi -E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi -\\mu )\\Big )\\nonumber \\\\&-&\\delta ^{\\prime }(\\omega +\\mu -E_\\phi +E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi +\\mu )\\Big )\\nonumber \\\\&-&\\delta ^{\\prime }(\\omega +\\mu -E_\\phi -E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi -\\mu )\\Big )\\Big \\rbrace \\\\+\\frac{k^4}{3\\pi }\\frac{1}{8E_\\phi ^3}~\\Big \\lbrace &&\\delta (\\omega +\\mu +E_\\phi +E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi +\\mu )-E_\\phi n^{\\prime }_B(E_\\phi )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu +E_\\phi -E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi -\\mu )-E_\\phi n^{\\prime }_B(E_\\phi )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu -E_\\phi +E_\\psi )\\left(1+\\frac{m_f}{E_\\psi }\\right)\\Big (n_B(E_\\phi )+n_F(E_\\psi +\\mu )-E_\\phi n^{\\prime }_B(E_\\phi )\\Big )\\nonumber \\\\&+&\\delta (\\omega +\\mu -E_\\phi -E_\\psi )\\left(1-\\frac{m_f}{E_\\psi }\\right)\\Big (1+n_B(E_\\phi )-n_F(E_\\psi -\\mu )-E_\\phi n^{\\prime }_B(E_\\phi )\\Big )\\Big \\rbrace \\nonumber $"
]
] | 1808.08535 |
[
[
"Primeness and dynamics of some classes of entire functions"
],
[
"Abstract In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member f of this class with respect to a transcendental entire function g that permutes with f. In particular we show that the Julia sets of f and g are identical."
],
[
"Introduction",
"A meromorphic function $F$ is said to be factorizable with $f$ and $g$ as left and right factors respectively, if it can be expressed as $F:=f\\circ g$ where $f$ is meromorphic and $g$ is entire ($g$ may be meromorphic when $f$ is rational).",
"$F$ is said to be prime (left-prime and right prime), if every factorization of $F$ of the above form implies either $f$ is bilinear or $g$ is linear ($f$ is bilinear whenever $g$ is transcendental and $g$ is linear whenever $f$ is transcendental).",
"When the factors $f$ and $g$ of $F$ are restricted to entire functions, the factorization is said to be in the entire sense.",
"Accordingly, we may use the term $F$ is prime (left-prime and right prime) in the entire sense.",
"For factorization of meromorphic functions one may refer to Gross [5] and Chuang and Yang [2].",
"Suppose $H$ and $\\alpha $ are entire functions satisfying $\\alpha ^{^{\\prime }}$ has at least one zero, $T(r,\\alpha )=o(T(r,H)) \\mbox{ as } r\\rightarrow \\infty $ , $H^{^{\\prime }}$ and $\\alpha ^{^{\\prime }}$ have no common zeros.",
"Now consider the class $\\mathcal {F}:= \\lbrace F_a(z)=H(z)+a\\alpha (z): a\\in \\mathbb {C} \\rbrace .$ $\\mathcal {F}$ is a most general class containing both periodic as well as non-periodic prime entire functions.",
"Urabe [15] proved that the periodic entire functions of the form $F_a(z)=H(z)+\\frac{a}{2}\\sin 2z,$ where $H(z)=\\sin z h(\\cos z)$ in which $ h(w)=\\textrm {exp}\\left\\lbrace \\psi \\left((2w^{2}-1)^{2}\\right) \\right\\rbrace \\mbox{ for some entire function } \\psi ,$ are prime for most values of $a\\in \\mathbb {C}$ .",
"We shall denote by $\\mathcal {U}$ , the class of all functions of the form (REF ).",
"Also, Qiao [12] proved that the periodic entire functions of the form $G_a(z):=\\cos z\\left(H(\\sin z)+2a\\right),$ where $H$ is an odd transcendental entire function such that order of $H(\\sin z)$ is finite, is prime for most values of $a\\in \\mathbb {C}.$ Let us denote by $\\mathcal {Q}$ , the class of functions of the form (REF ).",
"It is quite simple to note that the classes $\\mathcal {U}$ and $\\mathcal {Q}$ are subclasses of $\\mathcal {F}$ .",
"Also $\\mathcal {F}$ contains the class of non-periodic prime entire functions due to Singh and Charak [14].",
"Now it is natural to investigate the primeness of members of $\\mathcal {F}$ .",
"The importance of this investigation lies in the fact that if $F_a\\in \\mathcal {F}$ happens to be prime in entire sense and $g$ is any entire function permutable with $F_a$ , then Julia sets of $g$ and $F_a$ are identical for most values of $a$ ; for example see [8], [9], [16].",
"Let $f$ be an entire function.",
"We denote by $f^n$ the $nth$ iterate of $f$ .",
"By a Julia set $J(f)$ of an entire function $f$ we mean the complement $\\mathbb {C}\\setminus F(f)$ of Fatou set $F(f) $ of $f$ defined as $F(f):=\\left\\lbrace z\\in \\mathbb {C}:\\lbrace f^n\\rbrace \\mbox{ is normal in a neighborhood of }z\\right\\rbrace .$ In this paper, we shall find out some more subclasses of $\\mathcal {F}$ which are prime and study the dynamics of such subclasses with respect to any non-linear entire function permuting with a given member of these subclasses.",
"We shall adopt the following notations in our discussions throughout: $\\mathcal {H}(D): \\mbox{ the class of all holomorphic functions on a domain } D\\subset \\mathbb {C}.$ $\\mathbb {C}^{*}:= \\mathbb {C}\\setminus \\left\\lbrace 0\\right\\rbrace ,$ the punctured plane.",
"$\\mathcal {E}_{T}: \\mbox{ the class of all transcendental entire functions.",
"}$"
],
[
"Factorization of some periodic subclasses of $\\mathcal {F}$",
"The subclasses $\\mathcal {U}\\mbox{ and }\\mathcal {Q}$ of $\\mathcal {F}$ consist of prime periodic entire functions.",
"We shall prove the primeness of a more general subclass of $\\mathcal {F}$ containing $\\mathcal {U}.$ Also, we shall investigate some interesting properties of the functions in this general subclass as well as $\\mathcal {Q}.$ Actually, the purpose of this investigation is to study the dynamics of these subclasses of $\\mathcal {F}$ in the next section.",
"Lemma 2.1 Let $H$ and $\\alpha $ be entire functions such that $H^{\\prime }$ and $\\alpha ^{\\prime }$ have no common zeros.",
"Let $H_a(z):= H(z)+a\\alpha (z)$ , where $a\\in \\mathbb {C}$ .",
"Then there exists a countable set $E\\subset \\mathbb {C}$ such that for each $c\\in \\mathbb {C}$ and for any $ z, w \\in \\lbrace z\\in \\mathbb {C}: H_{a}(z)=c, H_{a}^{^{\\prime }}(z)=0\\rbrace $ , $\\alpha (z)=\\alpha (w)$ for each $a\\notin E$ .",
"For any critical point $z_0$ of $H_a$ , we have $ a = -\\frac{H^{^{\\prime }}(z_0)}{\\alpha ^{^{\\prime }}(z_0)}$ Define $m(z):=-\\frac{H^{^{\\prime }}(z)}{\\alpha ^{^{\\prime }}(z)}, ~~~~~ z\\in \\mathbb {C}.$ Set $A:= \\mathbb {C} \\setminus \\lbrace z\\in \\mathbb {C}: m^{\\prime }(z)=0 \\mbox{ or } \\infty \\rbrace .$ Then $A$ is an open subset of $\\mathbb {C}$ .",
"Let $\\left\\lbrace G_i: i\\in \\mathbb {N}\\right\\rbrace $ be an open covering of $A$ such that $m|_{G_{i}} $ is univalent and each $D_{i}=m(G_{i})$ is an open disk.",
"Now consider $M(z)&:=H(z)+m(z)\\alpha (z)\\\\x_{i}(w)&:=(m|_{G_{i}})^{-1}(w),~ w \\in D_{i}~ (i=1,2,3,\\cdots )\\\\y_{i}(w)&:=M(x_{i}(w)),~ w \\in D_{i} ~(i=1,2,3,\\cdots ) \\\\I&:=\\lbrace (i,j)\\in \\mathbb {N}\\times \\mathbb {N}: D_{i}\\cap D_{j}\\ne \\phi , y_{i}\\lnot \\equiv y_{j}\\mbox{ on } D_{i}\\cap D_{j}\\rbrace \\\\S_{ij}&:= \\lbrace w\\in D_{i}\\cap D_{j}: y_{i}(w)=y_{j}(w)\\rbrace , ~ (i,j)\\in I \\\\E_{0}&:=\\left(\\bigcup _{i=1}^{\\infty }D_{i}\\right)\\setminus \\left(\\left\\lbrace m(p): m^{^{\\prime }}(p)=0;p\\in \\mathbb {C} \\right\\rbrace \\cup \\left(\\cup _{(i,j)\\in I}S_{ij}\\right)\\right).$ Then $E=\\mathbb {C}\\setminus E_{0}$ is an at most countable subset of $\\mathbb {C}$ .",
"Using $\\left(\\ref {eq:20}\\right)$ in $\\left(\\ref {eq:21}\\right)$ , we get $M^{^{\\prime }}(z)=H^{^{\\prime }}(z)+m^{^{\\prime }}(z)\\alpha (z)+m(z)\\alpha ^{^{\\prime }}(z)=m^{^{\\prime }}(z)\\alpha (z).$ By $\\left(\\ref {eq:23}\\right)$ with the help of $\\left(\\ref {eq:21} \\right)$ and $\\left(\\ref {eq:22}\\right)$ , we obtain $y_{i}(w)&=H(x_i(w))+m(x_i(w))\\alpha (x_i(w)) \\nonumber \\\\&=H(x_i(w))+w\\alpha (x_i(w))$ Also by $\\left(\\ref {eq:23}\\right)$ together with $\\left(\\ref {eq:22}\\right)$ and $ \\left(\\ref {eq:27}\\right)$ , we have $ y_{i}^{^{\\prime }}(w) &= M^{^{\\prime }}(x_i(w))\\cdot x_{i}^{^{\\prime }}(w) \\nonumber \\\\&=m^{^{\\prime }}(x_{i}(w))\\cdot \\alpha (x_i(w))\\cdot x_{i}^{^{\\prime }}(w) \\nonumber \\\\&= \\alpha (x_{i}(w)).$ Since $a\\in E_0$ , we have $\\lbrace z\\in \\mathbb {C}: H^{\\prime }_{a}(z)=0\\rbrace =\\bigcup _{i=1}^{\\infty }\\left\\lbrace x_{i}(a)\\right\\rbrace $ .",
"This can be verified as follows.",
"By assumption, $H^{^{\\prime }}$ and $\\alpha ^{^{\\prime }}$ have no common zeros.",
"Then, since by () $m(x_{i}(a))=a$ for $a\\in D_{i}$ , $\\alpha ^{^{\\prime }}(x_{i}(a))\\ne 0$ and hence $H^{^{\\prime }}_{a}(x_i(a))=0.$ Conversely, let $H^{^{\\prime }}_a(z_0)=0$ .",
"Then $H^{^{\\prime }}(z_0)+a\\alpha ^{^{\\prime }}(z_0)=0$ which implies that $m(z_0)=a$ .",
"Since $a\\in E_0$ , $m^{^{\\prime }}(z_0)\\ne 0$ and so $x_i(a)=z_0$ for $a\\in D_{i}$ for some $i$ .",
"Moreover, if $y_{i}(a)=y_{j}(a)$ for some $a\\in E_0$ , then by (), () and (), it follows that $y^{^{\\prime }}_{i}(a)=y^{^{\\prime }}_{j}(a)$ .",
"From (REF ), we have $ \\alpha (x_{i}(a))=\\alpha (x_{j}(a)).$ Hence, there exists a countable set $E$ in $\\mathbb {C}$ such that for each $c\\in \\mathbb {C}$ , for any $z, w \\in \\lbrace z\\in \\mathbb {C}: H_{a}(z)=c, H_{a}^{^{\\prime }}(z)=0\\rbrace $ such that $\\alpha (z)=\\alpha (w)$ , provided $a\\notin E$ .",
"Considering $H_a(z):=\\cos z\\cdot h(\\sin {z})+a\\sin {z}$ , where $h\\in \\mathcal {E}_{T}$ , in Lemma REF and redefine $E_0$ in () as follows: $ E_{0}=\\left(\\bigcup _{i=1}^{\\infty }D_{i}\\right)\\setminus \\left(\\left\\lbrace m(p): m^{^{\\prime }}(p)=0;p\\in \\mathbb {C} \\right\\rbrace \\cup \\left(\\cup _{(i,j)\\in I}S_{ij}\\right)\\cup \\left(\\cup _{i=1}^{\\infty }\\left\\lbrace p: h(\\sin (x_i(p)))=0\\right\\rbrace \\right)\\right).$ By (REF ) , we have $\\sin x_i(a)=\\sin x_j(a), \\mbox{ and hence } \\cos x_i(a)h(\\sin x_i(a))=\\cos x_j(a)h(\\sin x_j(a)).$ Again by (REF ), we obtain $\\cos x_i(a)=\\cos x_j(a)$ and hence we obtain: Lemma 2.2 Let $h\\in \\mathcal {E}_{T}$ such that $h(\\pm 1)\\ne 0.$ Put $H_a(z):=\\cos z\\cdot h(\\sin {z})+a\\sin {z}$ , where $a\\in \\mathbb {C}$ .",
"Then, there exists a countable set $E$ of complex numbers such that any two roots $u$ and $v$ of the simultaneous equations $H_a(z)=c,\\ H^{^{\\prime }}_{a}(z)=0,$ $\\cos u=\\cos v$ and $\\sin u=\\sin v$ for any constant $c\\in \\mathbb {C}$ , provided $a\\notin E.$ Let $f(z)$ be periodic entire function of period $\\lambda \\ne 0$ .",
"Then for $z\\in \\mathbb {C}$ , we shall denote by $[z]$ , the set $\\lbrace z+n\\lambda :n\\in \\mathbb {Z}\\rbrace $ and for $A\\subset \\mathbb {C}$ , by$[A]$ , the set $\\lbrace [z]:z\\in A\\rbrace .$ Theorem 2.3 Let $H\\in \\mathcal {E}_{T}$ such that $H(-z)=-H(z), H^{^{\\prime }}(0)\\ne 0,$ and the order of $H(\\sin z)$ is finite.",
"Put $H_a(z):=\\cos z(H(\\sin z)+2a)$ , where $a\\in \\mathbb {C}$ .",
"Then there exists a countable set $E\\subset \\mathbb {C}$ such that $H_a$ satisfies the following properties for each $a\\notin E$ .",
"(i) $H_a$ is prime.",
"(ii) $\\#\\left\\lbrace [z]: H^{^{\\prime }}_a(z)=0\\right\\rbrace =\\infty $ (iii) for any $z_1, z_2 \\in \\left\\lbrace z\\in \\mathbb {C}: H_a(z)=c, H^{^{\\prime }}_a(z)=0 \\right\\rbrace ,$ $\\cos z_1=\\cos z_2$ for any $c\\in \\mathbb {C}$ .",
"(iv) $H^{^{\\prime }}_a$ has only simple zeros.",
"$(i)$ and $(iii)$ follow from [[12], Theorem 1] and Lemma REF respectively.",
"Define $k(w)=\\frac{w^2+1}{2w}H\\left(\\frac{w^{2}-1}{2iw}\\right).$ Then $k\\in \\mathcal {H}(\\mathbb {C}^{*})$ with essential singularities at 0 and $\\infty $ , and $H_a(z)=\\left(k(w)+a\\left(\\frac{w^2+1}{w}\\right)\\right)\\circ e^{iz}.$ Put $H_{a}(z)=k_{a}(e^{iz}),$ where $k_a(w)=k(w)+a(\\frac{w^2+1}{w})$ .",
"Since $H^{^{\\prime }}_{a}(z)=k^{^{\\prime }}_a(e^{iz})\\cdot i e^{iz}$ , we have $H^{^{\\prime }}_a(z)=0 \\Leftrightarrow k^{^{\\prime }}_a(e^{iz})=0.$ Thus $k^{^{\\prime }}_a(w)=0$ if and only if $\\frac{w^{2}k^{^{\\prime }}(w)}{w^{2}-1}=-a.$ By Picard's big theorem, we have $\\#\\left\\lbrace w\\in \\mathbb {C}^{*}: k^{^{\\prime }}_{a}(w)=0\\right\\rbrace =\\infty $ for every $a\\in \\mathbb {C}$ with at most two exceptions.",
"By (REF ) and (REF ), we get $\\#\\left\\lbrace [z]: H^{^{\\prime }}_a(z)=0\\right\\rbrace =\\infty ,$ for every $a\\in \\mathbb {C}$ with at most two exceptions.",
"This proves $(ii).$ To establish $(iv)$ it is enough to prove that $k^{^{\\prime }}_a$ has simple zeros (see (REF )).",
"Suppose that $k_{a}^{^{\\prime }}(w_0)=0$ , $ k_{a}^{^{\\prime \\prime }}(w_0)=0$ .",
"Then $w_0(w^{2}_0-1)k^{^{\\prime \\prime }}(w_0)-2k^{^{\\prime }}(w_0)=0,~~~ a=\\frac{-w^{2}_0 k^{^{\\prime }}(w_0)}{w^{2}_0-1}$ $\\it {Claim}$ : $w(w^2-1)k^{^{\\prime \\prime }}(w)-2k^{^{\\prime }}(w)\\lnot \\equiv 0$ on $\\mathbb {C}^{*}.$ Suppose that $w(w^2-1)k^{^{\\prime \\prime }}(w)-2k^{^{\\prime }}(w)\\equiv 0$ on $\\mathbb {C}^{*}$ .",
"Then $k^{^{\\prime \\prime }}(w)=\\frac{2}{w(w^2-1)}k^{^{\\prime }}(w),~~~w\\in \\mathbb {C}^{*}.$ Since $k^{^{\\prime }}(w)$ has no zeros at $w=\\pm 1$ , $k^{^{\\prime \\prime }}(w)$ has poles at $w=\\pm 1$ and this is a contradiction.",
"This establishes the claim.",
"Set $E=\\left\\lbrace a=\\frac{-w^2k^{^{\\prime }}(w)}{w^2-1}: k^{^{\\prime \\prime }}_a(w)=0, w\\ne 0\\right\\rbrace .$ If $a\\notin E$ , then $\\left\\lbrace w\\in \\mathbb {C}^{*}:k_{a}^{^{\\prime }}(w)=0, k_{a}^{^{\\prime \\prime }}(w)=0 \\right\\rbrace =\\phi .$ Therefore, $k^{^{\\prime }}_{a}$ has only simple zeros for $a\\notin E$ .",
"This proves $(iv).$ Adjoining the exceptions obtained in $(i), \\ (ii)$ and $(iii)$ with $E$ , the theorem holds for each $a\\notin E$ .",
"Theorem 2.4 Let $h\\in \\mathcal {E}_{T}$ such that $h(\\pm 1)\\ne 0.$ Put $H_a(z):=\\cos {z}\\cdot h(\\sin {z})+a\\sin {z}$ , where $a\\in \\mathbb {C}$ .",
"Then the set $\\lbrace a\\in \\mathbb {C}: H_{a} \\mbox{ is not prime in entire sense }\\rbrace $ is at most countable.",
"Remark 2.5 Theorem REF generalizes Theorem 1 of Urabe [16].",
"Proof of theorem REF.",
"Define $h_a(w):=\\left(\\frac{w^2+1}{2w}\\right)h\\left(\\frac{w^2-1}{2iw}\\right)+a\\left(\\frac{w^2-1}{2iw}\\right).$ Then $h_a\\in \\mathcal {H}(\\mathbb {C}^{*})$ with essential singularities at 0 and $\\infty .$ Let $H_a(z)=h_{a}(e^{iz})$ .",
"We can choose a countable subset $E$ of complex plane for which the assertion of Lemma REF holds with respect to $H_{a}$ .",
"We may assume $0\\in E$ .",
"By Second fundamental theorem of Nevanlinna [[6], Theorem 2.3, p.43], we can choose $t\\in (0,1)$ such that the inequalities $\\overline{N}(r,0,H^{^{\\prime }}_{a})\\ge tm(r,h^{^{\\prime }}_a(e^{iz}))$ and $\\overline{N}(r,c,H_{a})\\ge tm(r,H_a(z))$ hold on a set of $r$ of infinite measure for any $c\\in \\mathbb {C}$ .",
"Suppose $H_{a}(z)=f(g(z))$ , we consider the following cases one by one: Case (i): When $f,\\ g \\in \\mathcal {E}_{T}$ .",
"Since $H_{a}^{^{\\prime }}(z)=f^{^{\\prime }}(g(z))g^{^{\\prime }}(z)$ , by (REF ) we find that $f^{^{\\prime }}$ has infinitely many zeros $\\lbrace t_{k}\\rbrace $ , say.",
"Since every solution of $g(z)=t_k$ is also a solution of the simultaneous equations $H_{a}(z)=f(t_k),\\ H_{a}^{^{\\prime }}(z)=0,$ by Lemma REF it follows that all the roots of $g(z)=t_{k}$ lie on a single straight line $l_n$ .",
"The set $\\left\\lbrace l_{n}:n\\in \\mathbb {N}\\right\\rbrace $ is infinite otherwise by Edrei's theorem [4], $g$ would reduce to a polynomial (of degree 2) which is not in reason.",
"Therefore, by Theorem 3 of Kobayashi [7], we have $ g(z)=P(e^{Az}),$ where $P$ is a quadratic polynomial and $A$ is a non-zero constant.",
"Let $\\left\\lbrace z_{k,j}\\right\\rbrace _{j=1}^{\\infty }$ be the roots of $g(z)=t_k$ .",
"Then $\\left\\lbrace z_{k,j}\\right\\rbrace _{j=1}^{\\infty }$ are also the common roots of the simultaneous equations $H_{a}(z)=f(t_{k}), \\ H^{^{\\prime }}_{a}(z)=0.$ By Lemma REF , we have $\\cos z_{k,i}=\\cos z_{k,j} \\mbox{ and } \\sin z_{k,i}=\\sin z_{k,j}.$ This implies that $z_{k,i}=z_{k,j}+2\\pi m_0,\\mbox{ for some }m_0\\in \\mathbb {Z}.$ By (REF ), $g$ is a periodic function with period $2\\pi i/lA$ , where $l=1 $ or 2.",
"Thus, we have $A=i/N$ for the integer $N=lm_0 $ , and $H_a(z)=h_a(e^{iz})=f(P(e^{iz/N})).$ Put $w=e^{iz/N}$ .",
"Then $h_a(w^{N})=f(P(w))\\mbox{ for all } w\\ne 0.$ Note that the left hand side of (REF ) has an essential singularity at 0 but the right hand side is holomorphic at $0.$ This contradiction shows that Case(i) can't occur.",
"Case (ii): When $f\\in \\mathcal {E}_{T}$ and $g$ is a polynomial of degree at least two.",
"By (REF ), $f^{^{\\prime }}$ has infinitely many zeros $\\left\\lbrace t_k\\right\\rbrace $ , say.",
"Let $p_k$ and $q_k$ be two roots of $g(z)=t_k$ .",
"Then $p_k$ and $q_k$ are also common roots of the simultaneous equations $H_a(z)=f(t_k), \\ H^{^{\\prime }}_a(z)=0.$ By Lemma REF , it follows that $p_k-q_k=2m_k\\pi ,$ for some integer $m_k$ .",
"Further, by Renyi's theorem [13], $g(z)=bz^2+cz+d$ for some $b\\ne 0, c, d\\in \\mathbb {C}$ and hence $p_k+q_k=-\\frac{c}{b}.$ Now by (REF ) and (REF ), it follows that all $p_k$ and $q_k$ lie on a single straight line (independent of $t_k$ , $k\\in \\mathbb {N}$ ).",
"Since $H_{a}^{^{\\prime }}$ is periodic, we have $N(r,0,H^{^{\\prime }}_{a})\\le N(r,0,f^{^{\\prime }}(g))+N(r,0,g^{^{\\prime }})$ $=N(r,0,f^{^{\\prime }}(g))+O(\\log r)=o(m(r,h^{^{\\prime }}(e^{iz}))).$ This contradicts (REF ) showing that Case(ii) is not possible.",
"Case (iii): When $f$ is polynomial of degree $d$ $(\\ge 2)$ and $g\\in \\mathcal {E}_{T}$ .",
"By Renyi's Theorem [13], $g$ is periodic and therefore, we can express $g$ as $g(z)=G(e^{Bz}),$ $G\\in \\mathcal {H}(\\mathbb {C}^{*})$ with an essential singularity at 0 or $\\infty $ and $B$ is a non-zero constant.",
"Let $w_0$ be a zero of $f^{^{\\prime }}$ .",
"Then $G(z)=w_0$ has at most finitely many roots and so by Picard's big theorem it follows that $f^{^{\\prime }}$ has exactly one zero, say $w_0.$ Thus we can express $f^{\\prime }$ as $f^{^{\\prime }}(w)=b(w-w_0)^{d-1} \\mbox{ and hence } f(w)=\\alpha (w-w_0)^{d}+c$ for some constants $\\alpha (\\ne 0)$ and $c$ .",
"Therefore, $H_a(z)=\\alpha \\left(g(z)-w_0\\right)^d+c.$ Hence $N(r,c,H_a)=dN(r,w_0,g).$ Since $G(w)=w_0$ has at most finitely many roots, $N(r,w_0,g)=o(m(r,H_a(z)))$ which is contrary to (REF ) showing that Case(iii) also fails to occur.",
"Hence $H_a(z)$ is prime in entire sense.",
"$\\Box $ Using Theorem REF and Lemma REF and following the proofs of $(ii)$ and $(iv)$ in Theorem REF , we obtain: Theorem 2.6 Let $h\\in \\mathcal {E}_{T}$ such that $h(\\pm 1)\\ne 0$ .",
"Put $H_a(z):=\\cos {z} \\cdot h(\\sin {z})+a\\sin {z}$ , where $a\\in \\mathbb {C}$ .",
"Then there exists a countable set $E\\subset \\mathbb {C}$ such that $H_a$ possesses the following properties for each $a\\notin E$ : (i) $H_a$ is prime in entire sense; (ii) $\\#\\left\\lbrace [z]: H^{^{\\prime }}_a(z)=0\\right\\rbrace =\\infty $ ; (iii) for any $z_1, z_2 \\in \\left\\lbrace z\\in \\mathbb {C}: H_a(z)=c, H^{^{\\prime }}_a(z)=0 \\right\\rbrace $ , $~~\\cos z_1=\\cos z_2 \\mbox{ and }~~\\sin z_1=\\sin z_2$ for any $c\\in \\mathbb {C}$ ; and (iv) $H^{^{\\prime }}_a$ has only simple zeros."
],
[
"Dynamics of non-linear entire functions permutable with members of subclasses of $\\mathcal {F}$",
"The main result in this section is obtained by utilizing the argument due to Y. Noda [9], faithfully, with certain modifications.",
"Let $D_0:=D\\setminus \\left\\lbrace z: f^{^{\\prime }}(z)g^{^{\\prime }}(z)=0 \\right\\rbrace $ .",
"For $f, g \\in \\mathcal {H}(D)$ , define a relation on $D_0$ with respect to $f$ and $g$ , denoted by $\\sim _{\\left(f,g\\right)} $ as follows: Let $z, w \\in D_0$ .",
"We write $z \\sim _{\\left(f,g\\right)} w$ if and only if $f(z)=f(w), g(z)=g(w)$ and there are neighborhoods $U_z$ and $U_w$ of $z,$ and $w$ respectively such that $f(U_z)=f(U_w)$ , $g(U_z)=g(U_w)$ and $\\left(f_{|U_w}\\right)^{-1}of_{|U_z}=\\left(g_{|U_w}\\right)^{-1}\\circ g_{|U_z}$ in $U_z$ .",
"Then $\\sim _{\\left(f,g\\right)} $ is an equivalence relation on $D_0.$ Y. Noda [[9], Lemma 2.1] proved that for $f, \\ g \\in \\mathcal {H}(D)$ and $z_0 \\in \\mathbb {C}$ , there exist a neighborhood $N_{z_0}$ of $z_0, \\ h\\in \\mathcal {H}(N_{z_0})$ and $\\phi , \\ \\psi \\in \\mathcal {H}(h(N_{z_0}))$ satisfying $f^{^{\\prime }}(z)\\ne 0, g^{^{\\prime }}(z)\\ne 0, h^{^{\\prime }}(z)\\ne 0$ $z\\in N_{z_0}\\setminus \\lbrace z_0\\rbrace ;$ $z\\sim _{\\left(f,g\\right)}w$ if and only if $h(z)=h(w)$ $z,w \\in N_{z_0}\\setminus \\lbrace z_0\\rbrace ;$ and $f=\\phi \\circ h, g=\\psi \\circ h.$ This information lead Y. Noda [9] to extend the above equivalence relation to $D$ as follows: Let $z, w\\in D$ .",
"We write $z\\sim _{\\left(f,g\\right)}w$ if and only if $f(z)=f(w), g(z)=g(w)$ and there exists a conformal map $\\phi $ defined in a neighborhood of $h_1(z)$ such that $\\phi (h_1(z))=h_2(w)$ , $\\phi _1=\\phi _2\\circ \\phi $ , $\\psi _1=\\psi _2\\circ \\phi ,$ where $h_j, \\phi _j, \\psi _j (j=1,2)$ satisfy the conclusions of Lemma $2.1$ of Noda[9]( mentioned in the preceding discussion).",
"Using this equivalence relation, Y. Noda[9] proved the existence of the greatest common right factor of entire functions: Lemma 3.1 ([9], p.5) Let $f, \\ g \\in \\mathcal {H}(\\mathbb {C}).$ Then there exits $F \\in \\mathcal {H}(\\mathbb {C})$ and $\\phi , \\ \\psi \\in \\mathcal {H}(F(\\mathbb {C}))$ such that $f=\\phi \\circ F$ and $g=\\psi \\circ F.$ $F(z)=F(w)$ if and only if $z\\sim _{\\left(f,g\\right)} w.$ The entire function $F$ in Lemma REF is called the greatest common right factor of $f$ and $g$ (for a more general definition one may refer to [[9], p.2]).",
"We also require the following key lemmas for proving our result in this section: Lemma 3.2 ([1], Satz 6) Let $f\\in \\mathcal {E}_{T} $ such that $f$ permutes with a polynomial $g$ .",
"Then $g(z)=\\omega z+\\beta $ $\\left(\\omega =\\textrm {exp}\\left\\lbrace 2\\pi ik / p \\right\\rbrace , \\ k,p\\in \\mathbb {N}, (k,p)=1\\right)$ .",
"Further, if $\\omega \\ne 1$ , then $f(z)=c+(z-c)F_0\\left((z-c)^p\\right),$ where $c=\\beta /{(1-\\omega )}$ and $F_0$ is an entire function.",
"Lemma 3.3 ([1], Satz 7) Let $f$ and $g$ be permutable entire functions.",
"Then there exist a positive integer $n$ and $R_0>0$ such that $M\\left(r,g\\right)<M\\left(r,f^n\\right)$ for all $r>R_0$ .",
"Lemma 3.4 ([3],Theorem 1) Let $f,\\ g\\in \\mathcal {E}_{T}$ .",
"Then $\\limsup _{r\\rightarrow \\infty }\\frac{\\log {M\\left(r,f\\circ g\\right)}}{\\log {M\\left(r,g\\right)}}=\\infty .$ Lemma 3.5 ([9], Lemma 2.5) Suppose that $f, \\ g\\in \\mathcal {H}(\\mathbb {C})$ are permutable and $\\left(F,S\\right)$ be a greatest common right factor of $f$ and $g$ .",
"Let there be a subset $A\\subset \\mathbb {C}$ such that $\\#f(A)=1$ and $\\#g\\left(A\\right)=1$ and $r$ be the order of $f$ at some point of $g\\left(A\\right)$ .",
"Then there exists a subset $A^{\\prime }\\subset A$ such that $\\#F(A^{\\prime })=1$ and $\\#A^{\\prime }\\ge \\# A/{r}$ .",
"Lemma 3.6 ([9], Lemma 3.1) Let $f\\in \\mathcal {E}_{T}$ and $A$ be a discrete subset of $\\mathbb {C}$ such that $\\#f^{-1}\\left(A\\right)=\\infty $ .",
"Then $\\sup _{w\\in A}\\#\\left(f^{-1}\\left(\\lbrace w\\rbrace \\right)\\cap A^c\\right)=\\infty $ .",
"Lemma 3.7 ([9],Lemma 5.3) Let $f$ be a periodic entire function and $A$ be a discrete subset of $\\mathbb {C}$ such that $\\#\\left[f^{-1}\\left(A\\right)\\right]=\\infty $ .",
"Then $\\sup _{w\\in A}\\#\\left[f^{-1}\\left(\\lbrace w\\rbrace \\right)\\cap A^c\\right]=\\infty $ .",
"Lemma 3.8 ([9], Lemma 5.4) Let $h\\in \\mathcal {H}(\\mathbb {C}^{*})$ satisfying that $\\#\\left\\lbrace w:h^{^{\\prime }}(w)=0\\right\\rbrace =\\infty $ .",
"Put $f(z)=h(e^{z})$ .",
"Let $g\\in \\mathcal {E}_{T}$ such that $g$ permutes with $f$ .",
"Then for each $N\\in \\mathbb {N}$ , there exists $c$ such that $g^{^{\\prime }}(c)=0, \\#\\left\\lbrace [z]: f(z)=c, f^{^{\\prime }}(g(z))=0\\right\\rbrace \\ge N.$ Lemma 3.9 ([11], Lemma 2.1) Suppose that $f,\\ g\\in \\mathcal {E}_{T}$ such that $g(z)=af(z)+b$ , where $a,b\\in \\mathbb {C}$ .",
"If $g$ permutes with $f$ , then $J(f)=J(g)$ .",
"Lemma 5.1 of [9] can be straight way extended to: Lemma 3.10 Suppose that $f$ is periodic entire function with a period $\\lambda \\ne 0$ and $g$ be entire function permutable with $f$ and $\\left(F,S\\right)$ be a greatest common right factor of $f$ and $\\textrm {exp}\\left\\lbrace (2\\pi i/ \\lambda ) g \\right\\rbrace $ .",
"Let there be a subset $A\\subset \\mathbb {C}$ such that $\\#f(A)=1$ and $\\#\\textrm {exp}\\left\\lbrace (2\\pi i/\\lambda )g(A) \\right\\rbrace =1$ and $r$ be the order of $f$ at some point of $g\\left(A\\right)$ .",
"Then there exists a subset $A^{\\prime }\\subset A$ such that $\\#F(A^{\\prime })=1$ and $\\#A^{\\prime }\\ge \\# A/{r}$ .",
"Now we state and prove the main result of this section: Theorem 3.11 Let $f$ be a periodic entire function satisfying the following properties: (i) $f$ is prime in entire sense; (ii) $\\#\\left\\lbrace [z]: f^{^{\\prime }}(z)=0\\right\\rbrace =\\infty ;$ (iii) the set $\\left\\lbrace z\\in \\mathbb {C}: f(z)=c, f^{^{\\prime }}(z)=0 \\right\\rbrace $ is distributed over a finite number of distinct straight lines for any $c\\in \\mathbb {C}$ ; and (iv) the multiplicities of zeros of $f^{^{\\prime }}$ are uniformly bounded.",
"If $g$ is any non-linear entire function permutable with $f$ , then $J(g)=J(f)$ .",
"Suppose that $\\lambda \\ne 0$ is the period of $f$ and assume that $g\\in \\mathcal {E}_{T}$ such that $g$ permutes with $f$ .",
"By $(ii)$ and Lemma REF , for each positive integer $N$ there exists $c$ such that $g^{^{\\prime }}(c)=0\\mbox{ and }\\# \\left\\lbrace [z]: f(z)=c, f^{^{\\prime }}(g(z))=0\\right\\rbrace \\ge N.$ Let $\\mathcal {A}$ be a subset of $\\mathbb {C}$ such that $[z]\\ne [w]$ for $z,w\\in \\mathcal {A}, z\\ne w$ and $[\\mathcal {A}]=\\left\\lbrace [z]: f(z)=c, f^{^{\\prime }}(g(z))=0\\right\\rbrace .$ Since $f\\circ g(\\mathcal {A})=g\\circ f(\\mathcal {A})=\\left\\lbrace g(c)\\right\\rbrace ,$ we have $ g(\\mathcal {A})\\subset \\left\\lbrace z:f(z)=g(c), f^{^{\\prime }}(z)=0 \\right\\rbrace .$ In $(iii)$ , let us assume that the solutions of the simultaneous equations are distributed on $t$ straight lines, then there exists a subset $\\mathcal {B}\\subset \\mathcal {A}$ such that $g(\\mathcal {B})$ lie on a single straight line(which is parallel to the line passing through the origin and $\\lambda $ ) with $\\#\\mathcal {B}\\ge N/t$ .",
"Claim 1: $[g{\\mathcal {(}B)}]$ is finite.",
"Suppose that $[g(\\mathcal {B})]$ is infinite.",
"Let $X$ be the set of distinct points such that $[X]=[g(\\mathcal {B})]$ and $[z]\\ne [w],~~z,w\\in X, z\\ne w.$ We can choose the set $X$ such that all points of $X$ lie on a small line segment.",
"Since $X$ is infinite, $X$ has an accumulation point which contradicts (REF ).",
"This establishes the claim.",
"Let $z_1, \\cdots , z_p \\in \\mathbb {C},(p\\ge 1)$ such that $g(\\mathcal {B})\\subset \\bigcup _{i=1}^{p}\\left\\lbrace z_i+n\\lambda : n\\in \\mathbb {Z}\\right\\rbrace .$ Therefore, we have a subset $\\mathcal {C}\\subset \\mathcal {B} $ and $z_i, $ for some $i\\in \\left\\lbrace 1,\\cdots ,p\\right\\rbrace $ such that $g(\\mathcal {C})\\subset \\left\\lbrace z_i+n\\lambda : n\\in \\mathbb {Z}\\right\\rbrace \\mbox{ and }\\#\\mathcal {C}\\ge N/pt.$ This implies that $\\#\\left(\\textrm {exp}\\left\\lbrace \\left(2\\pi i/\\lambda \\right) g(\\mathcal {C}) \\right\\rbrace \\right)=1$ .",
"On the other hand, $f(\\mathcal {C})=\\left\\lbrace c\\right\\rbrace $ and thus $\\#f(\\mathcal {C})=1$ .",
"By Lemma REF , Lemma REF and $(iv),$ there exist $F\\in \\mathcal {H}(\\mathbb {C}), \\ \\phi , \\psi \\in \\mathcal {H}(F(\\mathbb {C}))$ and a subset $\\mathcal {D}\\subset \\mathcal {C}$ such that $f=\\phi \\circ F, \\ \\textrm {exp}\\left\\lbrace \\left(2\\pi i/\\lambda \\right) g \\right\\rbrace =\\psi \\circ F$ , $\\#F(\\mathcal {D})=1$ , $\\# \\mathcal {D}\\ge N/(s+1)pt $ , where $s$ denotes the maximum multiplicity of zeros of $f^{^{\\prime }}$ .",
"Since we can choose $N$ arbitrarily large, $F\\in \\mathcal {E}_{T}$ .",
"We claim that $F(\\mathbb {C})=\\mathbb {C}.$ For, it is enough to show that $F$ has no exceptional value on $\\mathbb {C}$ .",
"Suppose on the contrary that there exists $c\\in \\mathbb {C}$ such that $F=c+e^{Q}$ for some entire function $Q$ .",
"Then $f(z)=\\phi (c+e^{w})\\circ Q(z)$ .",
"Since $f$ is prime in entire sense, $Q(z)=a_1 z+a_2(a_1\\ne 0).$ Thus $f$ has a period $2\\pi i/a_1.$ Since $\\lambda $ is the fundamental period of $f$ , $2\\pi i/a_1= \\lambda p$ for some integer $p$ .",
"Therefore, $a_1= 2\\pi i/{\\lambda p}$ , $F(z)=c+\\textrm {exp}\\left\\lbrace (2\\pi i/{\\lambda p})z+a_2 \\right\\rbrace .$ Since $\\#F(\\mathcal {D})=1$ , we have $(z-w)\\in \\lambda \\mathbb {Z}$ for all $z,w \\in \\mathcal {D}$ .",
"Thus $[z]=[w]$ for all $z,w \\in \\mathcal {D}$ , a contradiction, and therefore, $F(\\mathbb {C})=\\mathbb {C}.$ Hence, $\\phi , \\ \\psi \\in \\mathcal {H}(\\mathbb {C}).$ Since $\\textrm {exp}\\left\\lbrace \\left(2\\pi i/\\lambda \\right) g \\right\\rbrace =\\psi \\circ F$ , we have $\\psi (z)\\ne 0$ for $z\\in \\mathbb {C}.$ Therefore, $\\psi =\\textrm {exp}\\left\\lbrace G \\right\\rbrace $ with $G\\in \\mathcal {H}(\\mathbb {C})$ and so $\\textrm {exp}\\left\\lbrace \\left(2\\pi i/\\lambda \\right) g \\right\\rbrace =\\textrm {exp}\\left\\lbrace G \\right\\rbrace \\circ F.$ Hence $g=\\left(\\lambda /2\\pi i\\right)G\\circ F+q\\lambda $ for some $q \\in \\mathbb {Z}.$ Put $K=\\left(\\lambda /2\\pi i\\right)G+q\\lambda $ .",
"Then $g=K\\circ F$ .",
"Since $F$ is transcendental and $f$ is prime in entire sense, we see that $\\phi $ is linear.",
"Therefore, $g=K\\circ \\phi ^{-1}\\circ f$ .",
"Put $g_1=K\\circ \\phi ^{-1}$ .",
"Then $g=g_1\\circ f.$ Note that $f\\circ g_1=g_1\\circ f$ .",
"If $g_1$ is transcendental, then by the same argument there exists $g_2\\in \\mathcal {H}(\\mathbb {C})$ such that $g=g_1\\circ f=g_2\\circ f{}^{2}$ .",
"Similarly we have $g=g_n\\circ f^{n}(n=1,2,\\cdots )$ , whenever all $g_n(n=1,2,\\cdots )$ are transcendental.",
"This gives a contradiction by Lemma REF and Lemma REF .",
"Thus, $g_n$ is a polynomial for some $n$ .",
"By Lemma REF , we find that $g_n(z)=lz+m(l\\ne 0)$ and so $g=lf^{n}+m$ .",
"Hence by Lemma REF $J(g)=J(f)$ .",
"Remark 3.12 Condition $(iii)$ in Theorem REF as well as Theorem REF implies that the set $\\left\\lbrace z\\in \\mathbb {C}: H_a(z)=c, H^{^{\\prime }}_a(z)=0 \\right\\rbrace $ is distributed over two distinct straight lines and single straight line respectively.",
"Remark 3.13 Let $f$ be a periodic entire function satisfying the conclusion of Theorem REF (or Theorem REF ).",
"Let $g\\in \\mathcal {E}_{T}$ such that $g$ permutes with $f$ .",
"Then by Theorem REF and Remark REF , $J(f)=J(g).$"
],
[
"A non-periodic subclass of $\\mathcal {F}$",
"Let $H(z)=P(z)\\cdot F(\\alpha (z)),$ where $P(z)$ is polynomial of degree $n$ , and $F, \\alpha \\in \\mathcal {H}(\\mathbb {C})$ such that $F_{a}(z):=H(z)-a \\alpha (z)\\in \\mathcal {F}$ for any $a\\in \\mathbb {C}$ .",
"On the similar lines of Lemma REF we get: Lemma 4.1 Let $H(z)=P(z)\\cdot F(\\alpha (z)),$ where $P(z)$ is polynomial of degree $n$ , and $F, \\alpha \\in \\mathcal {H}(\\mathbb {C}).$ Put $F_{a}(z)=H(z)-a\\alpha (z)$ , where $a\\in \\mathbb {C}$ .",
"Suppose that $H^{^{\\prime }}$ and $\\alpha ^{^{\\prime }}$ have no common zeros.",
"Then $\\# \\lbrace z\\in \\mathbb {C}:F_{a}(z)=c, F_{a}^{^{\\prime }}(z)=0\\rbrace \\le n$ , for all $c \\in \\mathbb {C} $ , provided that $a \\notin E.$ Lemma 4.2 [10] Let $F\\in \\mathcal {E}_{T}$ satisfy $N\\left(r,0,F^{^{\\prime }}\\right)> k T\\left( r,F^{^{\\prime }}\\right)$ on a set of $r$ of infinite linear measure for some $k>0$ .",
"Assume that the simultaneous equations $F(z)=c, F^{^{\\prime }}(z)=0$ have finitely many solutions only for any constant $c$ .",
"Then $F$ is left-prime in the entire sense.",
"Theorem 4.3 Let $F_a(z)= H(z)-a\\cdot \\alpha (z)\\in \\mathcal {F}$ , where $a\\in \\mathbb {C}, \\ H(z)=P(z)\\cdot F(\\alpha (z))$ and $P(z)$ is polynomial of degree $n$ .",
"Then there exists a countable set $E \\subset \\mathbb {C}$ such that $F_{a}$ satisfies the following properties for each $a\\notin E$ .",
"$(a)$ $F_{a}$ is left prime in entire sense.",
"$F_a$ happens to be prime if $P(z)$ is a polynomial of degree 1.",
"$(b)$ $\\#\\lbrace z\\in \\mathbb {C}: F_{a}(z)=c, F^{^{\\prime }}_{a}(z)=0\\rbrace \\le n$ for all $c\\in \\mathbb {C}$ .",
"$(c)$ $F_{a}$ has infinitely many critical points and each is of multiplicity $1.$ Remark 4.4 By (b) and the first half of (c) of Theorem REF , it follows that for each $a\\notin E$ , $F_a$ is non-periodic.",
"From $(b)$ and first half of $(c)$ in Theorem REF we observe that $F_{a}$ can not be of the form $f\\circ q$ for some periodic entire function $f$ and polynomial $q$ .",
"Suppose on the contrary that $F_{a}(z)=f\\circ q(z),$ for some periodic entire function $f$ with period $\\lambda $ and for some polynomial $q$ .",
"Then $F^{^{\\prime }}_{a}(z)=f^{^{\\prime }}(q(z))\\cdot q^{^{\\prime }}(z).$ By first half of (c) in Theorem REF , $f^{^{\\prime }}$ has infinitely many zeros.",
"Let $f^{^{\\prime }}(z_0)=0$ for some $z_0 \\in \\mathbb {C}$ .",
"Then $f(z_{0}+n\\lambda )=f(z_0)$ and $f^{^{\\prime }}(z_{0}+n\\lambda )=0$ for all $n\\in \\mathbb {Z}$ .",
"Let $w_{n}\\in \\mathbb {C}$ be such that $q(w_n)=z_{0}+n\\lambda $ for $n\\in \\mathbb {Z}$ .",
"Then $F_{a}(w_n)=f(z_0)$ and $F_{a}^{^{\\prime }}(w_n)=0$ for all $n \\in \\mathbb {Z}$ , which violates (b) of Theorem REF .",
"In fact an entire function satisfying $(b)$ and $(c)$ of Theorem REF is not of the form $g\\circ Q$ , where $g$ is periodic entire function and $Q$ is a polynomial.",
"Thus by Ng[[8], Theorem 1] it follows that if $f\\in \\mathcal {E}_{T}$ satisfying $(a), \\ (b),$ and $(c)$ of Theorem REF and permuting with a non-linear entire function $g,$ then $J(f)=J(g);$ Theorem REF provides an illustration of this conclusion.",
"Proof of Theorem REF Let $E_0$ be a countable subset of $\\mathbb {C}$ such that the assertions of Lemma REF hold for $F_{a}(z)=H(z)-a\\alpha (z)$ as soon as $a\\in \\mathbb {C}\\setminus E_{0}$ .",
"Clearly, $N(r,0,F_{a}^{^{\\prime }})=N\\left(r,a,\\frac{H^{^{\\prime }}}{\\alpha ^{^{\\prime }}}\\right).$ By the second fundamental theorem of Nevanlinna [[6], Theorem 2.3, p.43], it follows that for any $k\\in (0,1)$ , $N\\left(r,a,\\frac{H^{^{\\prime }}}{\\alpha ^{^{\\prime }}}\\right)> k T \\left(r,\\frac{H^{^{\\prime }}}{\\alpha ^{^{\\prime }}}\\right)$ hold for every $r$ outside a set of finite linear measure and for every $a$ outside $E_1$ , where $E_1$ is at most countable subset of $\\mathbb {C}$ .",
"Let $E_2=E_{0}\\cup E_{1}$ .",
"Then $E_2$ is an at most countable subset of $\\mathbb {C}$ and (REF ) holds for every $a\\in \\mathbb {C}\\setminus E_2$ , and thus from (REF ), we have $N(r,0,F_{a}^{^{\\prime }})>k T\\left(r, \\frac{H^{^{\\prime }}}{\\alpha ^{^{\\prime }}}\\right).$ Since $T(r,\\alpha )=o\\left(T(r,H)\\right)$ , (REF ) gives $N(r,0,F_{a}^{^{\\prime }})>k T\\left(r, H^{^{\\prime }}\\right).$ and hence $N(r,0,F_{a}^{^{\\prime }})>k T\\left(r, F_{a}^{^{\\prime }}\\right)$ holds for all $r$ outside a set of finite linear measure and for all $a\\in \\mathbb {C}\\setminus E_2$ .",
"Thus, combining Lemma REF with Lemma REF , it follows that $F_{a}(z)$ is left-prime in entire sense, for all $a\\in \\mathbb {C}\\setminus E_2$ .",
"To show that $F_a$ is right prime in entire sense, when $P(z)$ is linear polynomial.",
"Let $F_a=g\\circ h$ , where $g\\in \\mathcal {E}_{T}$ and $h$ is a polynomial of degree atleast two.",
"Then from (REF ), $g^{\\prime }$ has infinitely many zeros $\\left\\lbrace z_n\\right\\rbrace $ .",
"For all $n$ sufficiently large, $h(z)=z_n$ admits atleast two distinct roots which comes out to be the solutions of the simultaneous equations $F_a(z)=g(z_n), F_a^{\\prime }(z)=0,$ contradicting the fact that these simultaneous equations have atmost one solution.",
"Thus $h$ is linear and hence $F_a$ is right prime in entire sense.",
"This shows that $F_a$ is prime in entire sense.",
"By Remark REF and Lemma 3.1 of [2], $F_a$ is prime.",
"(b) follows from Lemma REF whereas the first half of (c) follows from equation (REF ) .",
"To prove the second half of (c), suppose there is $z_0\\in \\mathbb {C}$ such that $F_{a}^{^{\\prime }}(z_0)=0, F_{a}^{^{\\prime \\prime }}(z_0)=0$ .",
"Then $a= \\frac{H^{^{\\prime }}(z_0)}{\\alpha ^{^{\\prime }}(z_0)},~~ \\alpha ^{^{\\prime }}(z_0)H^{^{\\prime \\prime }}(z_0)-\\alpha ^{^{\\prime \\prime }}(z_0)H^{^{\\prime }}(z_0)=0$ Claim 1 : $\\alpha ^{^{\\prime }}(z)H^{^{\\prime \\prime }}(z)-\\alpha ^{^{\\prime \\prime }}(z)H^{^{\\prime }}(z)\\lnot \\equiv 0$.",
"Suppose on the contrary that $\\alpha ^{^{\\prime }}(z) H^{^{\\prime \\prime }}(z)-\\alpha ^{^{\\prime \\prime }}(z)H^{^{\\prime }}(z)\\equiv 0.$ Then $H^{^{\\prime \\prime }}=\\frac{\\alpha ^{^{\\prime \\prime }}}{\\alpha ^{^{\\prime }}}H^{^{\\prime }}.$ Since $\\alpha ^{^{\\prime }} $ has at least one zero and $ \\alpha ^{^{\\prime }}$ and $H^{^{\\prime }}$ have no common zero which implies that $H^{^{\\prime \\prime }}$ has a pole, a contradiction and this proves the claim.",
"Now it follows that the set $E_{3}:=\\left\\lbrace t=\\frac{H^{^{\\prime }}(z)}{\\alpha ^{^{\\prime }}(z)}:F^{^{\\prime \\prime }}_{t}(z)=0 \\right\\rbrace $ is at most countable and for any $a\\notin E_3, \\left\\lbrace z:\\alpha ^{^{\\prime }}(z)H^{^{\\prime \\prime }}(z)-\\alpha ^{^{\\prime \\prime }}(z)H^{^{\\prime }}=0 \\right\\rbrace =\\phi $ .",
"Therefore, $F^{^{\\prime }}_{a}(z)$ has only simple zeros for $a\\notin E_3$ .",
"The set $E:=E_2\\cup E_3$ is at most countable and the above conclusions hold for each $a \\notin E$ .",
"$\\Box $"
]
] | 1808.08394 |
[
[
"Lepton masses and mixing in a two-Higgs-doublet model"
],
[
"Abstract Within the framework of the two-Higgs Doublet Model (2HDM), we attempt to find some discrete, non-abelian flavour symmetry which could provide an explanation for the masses and mixing matrix elements of leptons.",
"Unlike the Standard Model, currently there is no need for the flavour symmetry to be broken.",
"With the GAP program we investigate all finite subgroups of the U3 group up to the order of 1025.",
"Up to such an order there is no group for which it is possible to select free model parameters in order to match the masses of charged leptons, masses of neutrinos, and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix elements in a satisfactory manner."
],
[
"I. Introduction",
"Despite the success of the Standard Model (SM) in providing a description of the current experimental data, there is a widespread belief that sooner or later an increase in available energy or accuracy of measurements will lead to the detection of a discrepancy between experimental results and theoretical predictions.",
"The SM cannot be considered a complete theory because it does not provide answers to many pressing questions.",
"One of the most important issues to be resolved concerns the masses of the fundamental constituents of matter, quarks and leptons.",
"At the moment we still cannot theoretically predict their masses; we are only able to obtain their values from experimental data.",
"The discovery of the Higgs particle offers a partial solution to the problem, yet it does not resolve it completely.",
"Particles acquire their mass by means of interaction with the Higgs field, and rather than examine the numerical values of the masses, we are currently more interested in the question of why particles interact so differently with the Higgs field, or in other words, why the Yukawa couplings cannot be theoretically predicted.",
"The solution to the problem of the elementary particles' masses is important in itself, because it would reduce the amount of unknown free parameters in the SM.",
"Another important reason for conducting such investigations is that they create a great opportunity to understand the origin of the masses of physical bodies.",
"The main part of the mass of each physical body comes from the interaction of the ingredients contained in it.",
"But it is not the entire mass; the remaining part (though small) being the masses of individual fermions, which are still undetermined.",
"Several proposals to solve this problem, at least partially, can be found in the literature (see e.g.",
"[1], [2], [3]).",
"Although the problem concerns all matter constituents, here we will concentrate on an attempt to explain the masses and mixing angles of leptons.",
"One of the most common approaches consists in the imposition of a flavour symmetry on the leptonic part of Yukawa Lagrangian (for review see e.g.",
"[4], [5], [6]).",
"This approach was particularly popular and successful before 2011 when it was discovered that the reactor-mixing angle $\\theta _{13}$ is non-vanishing [7], [8].",
"The models with an additional flavour symmetry are very popular, but they are by no means the only ones – some papers have been published in which the very existence of such a symmetry is denied [9], [10].",
"Attempts at solving the problem of lepton masses by a horizontal symmetry are dependent on the manner of the introduction of neutrino masses as well as on the Higgs sector for spontaneous symmetry breaking.",
"In the simplest case of the conventional SM with one Higgs doublet in which neutrinos are massless, only three additional right-handed neutrinos are introduced.",
"In such an extension of the SM, without introducing the Majorana term, neutrinos are Dirac particles [11].",
"It is not necessary to introduce the right-handed neutrinos to obtain their masses.",
"Instead, it is possible to use the existing left-handed neutrinos to form the Majorana masses [11].",
"Both cases will be considered in this paper.",
"Within the framework of the Standard Model with one Higgs boson, a discrete symmetry for Yukawa couplings provides the relations for the three-dimensional mass matrices of charged leptons $(M_l)$ and neutrinos $(M_\\nu )$ [12], [13]: $A^{i\\dag }_L \\left( M_l M_l^\\dag \\right) A_L^i &=& \\left( M_l M_l^\\dag \\right),\\\\A^{i\\dag }_L \\left( M_\\nu M_\\nu ^\\dag \\right) A_L^i &=& \\left( M_\\nu M_\\nu ^\\dag \\right),$ where $A^i_L=A_L(g_i), \\;i=1,2,\\ldots , N$ are three dimensional representation matrices for the left-handed lepton doublets for some N-order flavour symmetry group $\\mathcal {G}$ .",
"In such a case, the I-st Schur’s lemma implies that $M_{l}M_{l}^{\\dag }$ and $M_{\\nu }M_{\\nu }^{\\dag }$ are proportional to the identity matrices, which clearly entails the trivial lepton mixing matrix (known in the literature as Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [14], [15], [16]).",
"In order to avoid the relations given in () and predict the non-trivial lepton masses and their mixing in this case, the family symmetry has to be broken.",
"As a rule, the flavour symmetry $\\mathcal {G}$ is spontaneously broken by scalar singlet Higgs fields called flavons (see e.g.",
"[17], [5], [18]).",
"However, it can also be broken by introducing a bigger number of normal Higgs multiplets (e.g.",
"[18], [19]).",
"The latter way is indeed more economical, since the spontaneous gauge symmetry breaking in this case gives rise to the particles’ masses, simultaneously leading to the break of a family symmetry.",
"Additional flavon scalar fields in this framework are therefore redundant.",
"Such models were considered many times in the literature, but mostly in the frame of supersymmetric models where Higgs doublets were singlets of a flavour group [20], [21], or, in a more general approach, where only one selected flavour group was tested [22], [23], [24].",
"In the present work we attempt to explore how the Two Higgs Doublet Model works in the context discussed above.",
"As distinct from the previous works, we do not consider a few selected discrete groups, but instead we try to find a flavour symmetry in all groups up to 1025 order with one restriction, i.e.",
"each of our groups must have at least one faithful, three dimensional irreducible representation.",
"In the next Section, we briefly introduce the flavour symmetry in the 2HDM model and show how the symmetry transformation between two Higgs doublets provides an opportunity to avoid the consequences of Schur's lemma.",
"We also present all the formulas needed to conduct the computations in the case of Dirac and Majorana neutrinos.",
"In Sec.",
"III the results of the final scan of the Yukawa matrices, the lepton masses and the PMNS mixing matrix elements are presented, and finally in Sec.",
"IV we draw our conclusions.",
"To begin, the leptonic part of the Yukawa Lagrangian with Dirac neutrinos will be considered.",
"In contrast to the Standard Model, two Higgs doublets $\\Phi _{i}$ contribute to the lepton masses (the so-called Two-Higgs-Doublet-Model of type III [25]) as follows [26]Note that, in comparison to our notation, in the paper [26]: $\\Phi _{i}=\\left( \\phi _{i}^{+}, \\phi _{i}^{0}/\\sqrt{2}\\right)^T,\\;i=1,2.$ $\\mathcal {L}_{Y}=&-&\\sum \\limits _{i=1,2}\\sum \\limits _{\\alpha ,\\beta =e,\\mu ,\\tau }\\left( (h_{i}^{(l)})_{\\alpha ,\\beta }\\left[\\overline{L}_{\\alpha L}\\tilde{\\Phi }_{i}l_{\\beta R}\\right] \\right.\\nonumber \\\\&+& \\left.",
"(h^{(\\nu )}_{i})_{\\alpha \\beta } \\Big [\\overline{L}_{\\alpha L}\\Phi _{i}\\nu _{\\beta R}\\Big ]\\right)+h.c.$ where: $L_{\\alpha L}=\\left(\\begin{array}{c}\\nu _{\\alpha L}\\\\ ł_{\\alpha L}\\end{array}\\right),\\quad \\Phi _{i}=\\left(\\begin{array}{c} \\phi _{i}^{0}\\\\\\phi _{i}^{-}\\end{array}\\right),\\;i=1,2$ are gauge doublets for the left-handed lepton and Higgs fields and the fields $l_{\\beta R},\\;\\nu _{\\beta R}$ stand for the right-handed lepton and neutrino fields, respectively.",
"The couplings $ h^{(l)}_{i}$ and $h^{(\\nu )}_{i}$ create the 3-dimensional Yukawa matrices.",
"The spontaneous gauge symmetry breaking gives non-zero vacuum expectation values (VEVs) $v_{i}$ for the Higgs doublets: $\\left<\\Phi _{i}\\right>=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}v_{i}\\\\0 \\end{array}\\right),$ and the mass matrices read as follows [26]: $M^{l}&=&-\\frac{1}{\\sqrt{2}}\\left(v_{1}^{*}h^{(l)}_{1}+v_{2}^{*}h^{(l)}_{2}\\right), \\\\M^{\\nu }&=&\\frac{1}{\\sqrt{2}}\\left(v_{1}h^{(\\nu )}_{1}+v_{2}h^{(\\nu )}_{2}\\right).$ In general the vacuum expectation values can be complex, $ v_{i}= \\vert v_{i} \\vert e^{i\\varphi _{i}} $ , but they are restricted by the Fermi coupling constant: $\\sqrt{\\vert v_{1} \\vert ^2 + \\vert v_{2} \\vert ^2} =\\left( \\sqrt{2} G_{F}\\right)^{-1/2} \\simeq 246 \\ GeV.$ Family symmetry of our theory implies, that after the transformation of fields occurring in the 2HDM Lagrangian by the 3 dimensional ($A_{L},\\;A^{R}_{l},\\;A^{R}_{\\nu }$ ) and 2 dimensional ($A_{\\Phi }$ ) representations of a flavour group $\\mathcal {G}$ : $ L_{\\alpha L} \\rightarrow L^{\\prime }_{\\alpha L}&=& \\left(A_{L}\\right)_{\\alpha , \\chi } L_{\\chi L},\\ l_{\\beta R}\\rightarrow l^{\\prime }_{\\beta R}=\\left(A^{R}_{l}\\right)_{\\beta , \\delta } l_{\\delta R} \\nonumber \\\\\\nu _{\\beta R}\\rightarrow \\nu ^{\\prime }_{\\beta R}&=&\\left(A^{R}_{\\nu }\\right)_{\\beta , \\delta } \\nu _{\\delta R},\\ \\Phi _{i}\\rightarrow \\Phi ^{\\prime }_{i}=\\left(A_{\\Phi }\\right)_{ik}\\Phi _{k},$ the full 2HDM Lagrangian does not change: $\\mathcal {L}\\left(L_{\\alpha L},l_{\\beta R},\\nu _{\\gamma R},\\Phi _{i}\\right) =\\mathcal {L}\\left(L^{\\prime }_{\\alpha L},l^{\\prime }_{\\beta R},\\nu ^{\\prime }_{\\gamma R},\\Phi ^{\\prime }_{i}\\right).$ Given that all the transformation matrices in Eq.",
"(REF ) are unitary, the only parts of the total 2HDM Lagrangian for which the aforementioned relations are not automatically fulfilled are the Yukawa Lagrangian and the Higgs potential.",
"Imposition of symmetry on these terms of the model severely restricts their forms.",
"The invariance of the Yukawa Lagrangian is expressed as follows: $\\mathcal {L}^{^{\\prime }}_{Y}\\equiv &&-\\sum _{i=1,2}\\sum _{\\alpha ,\\beta =e,\\mu ,\\tau }\\left( \\left(h^{(l)}_{i}\\right)_{\\alpha ,\\beta }\\left[\\overline{L^{\\prime }}_{\\alpha L}\\tilde{\\Phi ^{\\prime }}_{i}l^{\\prime }_{\\beta R}\\right] \\right.",
"\\nonumber \\\\&& \\left.+\\left(h^{(\\nu )}_{i}\\right)_{\\alpha \\beta } \\Big [\\overline{L^{\\prime }}_{\\alpha L}\\Phi ^{\\prime }_{i}\\nu ^{\\prime }_{\\beta R}\\Big ]\\right)+h.c.=\\mathcal {L}_{Y}.$ With regard to the Higgs potential, there appear to be two possibilities.",
"The first one assumes that: $V\\left(\\Phi ^{\\prime }_{1},\\Phi ^{\\prime }_{2}\\right)=V\\left(\\Phi _{1},\\Phi _{2}\\right),$ which implies that before and after the transformation for Higgs fields (Eq.",
"(REF )) the coefficients in the potential remain exactly the same and the vacuum expectation values are equal, $v_i^\\prime =v_i$ .",
"There is also a second possibility, useful for phenomenological reasons, we allow for the modification of VEVs, which transform in the same way as the Higgs fields: $v^{\\prime }_{i}=\\left(A_{\\Phi }\\right)_{ik}v_{k}.$ In this case, the form of the Higgs potential does not change, the terms in the potential do not vary, while only the potential coefficients undergo change.",
"This kind of invariance is known in the literature as the form-invariance (see [26], [27]).",
"After the unitary transformation (Eq.",
"(REF )), the condition given in Eq.",
"(REF ) is unchanged, and hence also: $\\sqrt{\\vert v_{1}^{^{\\prime }} \\vert ^2 + \\vert v_{2}^{^{\\prime }} \\vert ^2} \\simeq 246 \\ GeV.$ The vacuum expectation values, whose sum of squares is constant, will need adjustments to meet the experimental requirements.",
"Thus, from the point of view of flavour symmetry, the type of Higgs potential is irrelevant, so in our approach, the issue of what symmetry for Higgs Lagrangian is chosen becomes insignificant.",
"With reference to the Eq.",
"(REF ), in order to find symmetric Yukawa matrices $h^{(l)}_{i},\\;h^{(\\nu )}_{i}$ , $ i=1,2$ , one can readily express the symmetry conditions as the eigenequation for a direct product of unitary group representations to the eigenvalue 1 (see e.g.",
"[28]): $\\left((A_{\\Phi })^{\\dag } \\otimes (A_{L})^{\\dag } \\otimes (A^{R}_{l})^{T}\\right)_{k,\\alpha ,\\delta ;i,\\beta ,\\gamma } (h^{l}_{i})_{\\beta ,\\gamma }&=& (h^{l}_{k})_{\\alpha ,\\delta }, \\nonumber \\\\\\left((A_{\\Phi })^{T} \\otimes (A_{L})^{\\dag } \\otimes (A^{R}_{\\nu })^{T}\\right)_{k,\\alpha ,\\delta ;i,\\beta ,\\gamma } (h^{\\nu }_{i})_{\\beta ,\\gamma }&=& (h^{\\nu }_{k})_{\\alpha ,\\delta },$ for the charged leptons and neutrinos, respectively.",
"Both relations (Eq.",
"(REF )) need to be satisfied for any group's element $g\\in \\mathcal {G}$ .",
"It is however sufficient that they are fulfilled only for the group generators [28], which considerably reduces the time of the computation.",
"In such a model, the invariance equations for the mass matrices are not trivial.",
"For the symmetric Higgs potential (Eq.",
"(REF )) : $A_{L}M^{l(\\nu )}\\left(A^{R}_{l (\\nu )}\\right)^{\\dag }=\\frac{1}{\\sqrt{2}}\\sum _{i,k=1}^{2} h^{l(\\nu )}_{i} \\left(A_{\\Phi }\\right)_{i,k}v_{k} \\ne M^{l(\\nu )}.$ then Eq.",
"(REF -) are not satisfied and we avoid the consequences of the Schur's Lemma.",
"The same can be shown for the form-invariant Higgs potential, where Eq.",
"(REF ) is satisfied.",
"In this context we can obtain the non-trivial mass matrices without the introduction of additional flavon fields."
],
[
" B. Majorana neutrinos",
"For Majorana neutrinos the Yukawa term has to be changed.",
"In 2HDM, the simplest Yukawa Lagrangian can be taken as the non-renormalizable Weinberg term in the form: $\\mathcal {L}_{Y}^{\\nu } \\equiv &&-\\frac{g}{M} \\sum \\limits _{i,k=1}^{2}\\sum \\limits _{\\alpha ,\\beta =e,\\mu ,\\tau } h^{(i,k)}_{\\alpha ,\\beta }\\left(\\overline{L}_{\\alpha L}{\\Phi }_{i}\\right)\\left({\\Phi }_{k} {L}_{\\beta R}^{c}\\right) \\nonumber \\\\&&+ h.c,$ where ${L}_{\\beta R}^{c}=C\\overline{L}_{\\beta L}^{T}$ is the charge conjugated lepton doublet fields.",
"After the spontaneous symmetry breaking, the neutrino mass matrix is obtained: $M^{\\nu }_{\\alpha ,\\beta }=\\frac{g}{M}\\sum _{i,k=1}^{2}v_{i}v_{k}h^{(i,k)}_{\\alpha ,\\beta }.$ As in the preceding case, in compliance with the requirement of flavour symmetry for the Yukawa Lagrangian (Eq.",
"(REF )), the neutrino Yukawa matrices must satisfy the eigenvalue equation: $\\left((A_{\\Phi })^{T} \\otimes \\left(A_{\\Phi }\\right)^{T}\\otimes \\left(A_{L}\\right)^{\\dag } \\otimes \\left(A_{L}\\right)^{\\dag }\\right)_{k,m,\\chi ,\\eta ;\\ i,j,\\alpha ,\\beta } \\left(h^{(i,j)}_{\\alpha ,\\beta }\\right) = \\left(h^{(k,m)}_{\\chi ,\\eta }\\right).$ Such flavour symmetric Yukawa couplings restrict the neutrino mass matrix from Eq.",
"(REF )."
],
[
" A. The candidates for the flavour group $\\mathcal {G}_{F}$",
"The flavour group $\\mathcal {G}_{F}$ , which is imposed on the 2HDM Lagrangian, cannot be arbitrary.",
"Due to the fields' transformations defined in Eq.",
"(REF ), the group must possess at least one 2-dimensional (for $A_{\\Phi }$ ) and at least one 3-dimensional irreducible representation (for $A_{L}$ , $A^{R}_{\\nu }$ and $A^{R}_{l}$ ).",
"The sole application of irreducible representations is justified given the fact that were any of the representations ($A_{\\Phi }, A_{L}$ or $A^{R}_{\\nu }(A^{R}_{l})$ ) to be reducible, invariant Yukawa couplings would split up into independent sets for irreducible representations (see e.g.[28]).",
"In the selection of flavour symmetry groups, we have limited ourselves to finite-dimensional groups of the order of at most 1025, which are furthermore subgroups of the $U(3)$ group (at least one of the 3-dimensional irreducible or reducible representations must be faithful) [29], [30].",
"This additional condition is not necessary [31] but it significantly reduces the number of groups to be processed.",
"Using the GAP (version 4.7.6) [32] system for computational discrete algebra with the included Small Groups Library [33] and REPSN [34] packages, we have found in total 10862 groups with at least one 2-dimensional and at least one 3-dimensional irreducible representation, but only 413 of these groups are subgroups of the $U(3)$ group.",
"They split into two disjoint sets.",
"Each group has either at least one faithful 3-dimensional irreducible representation (there are 173 such groups), or at least one faithful 1+2 reducible representation (there are 240 such groups).",
"Some groups are also subgroups of the $U(2)$ group.",
"They have at least one faithful 2-dimensional irreducible representation (none of them have any faithful 3-dimensional irreducible representation).",
"None of the groups have any faithful 1+1+1 reducible, faithful 1+1 reducible or faithful 1-dimensional irreducible representation.",
"All the groups which delivered any solutions belong to polycyclic groups that use the polycyclic presentation for element arithmetic (so called $PC$ -groups).",
"Table: Groups of the order of at most 100 subject to consideration (all the groups are listed in the ancillary file).",
"Here: “[o,i] [\\ o\\ ,\\ i\\ ] ” the ii-th group of the order oo in the Small Groups Library catalogue, “StructureDescription” a short string which provides some insight into the structure of the group under consideration, “2-D” the number of 2-dimensional irreducible representations, “3-D” the number of 3-dimensional irreducible representations, “U(2)U(2)” an indicator why the group is classified as a subgroup of the U(2)U(2) group (at least one 2-dimensional irreducible is faithful), “U(3)U(3)” an indicator why the group is classified as a subgroup of the U(3)U(3) group (either at least one 3-dimensional irreducible or one 1+2 reducible representation is faithful), “L” the number of different combinations of representations for charged leptons, “DN” the number of different combinations of representations for Dirac neutrinos, “MN” the number of different combinations of representations for Majorana neutrinos, “L+DN” the number of pairs of different combinations of representations for charged leptons and Dirac neutrinos, “L+MN” the number of pairs of different combinations of representations for charged leptons and Majorana neutrinos.",
"Note that the “L” and the “DN” are always equal and that the “L+DN” is twice that number.",
"All zero values are suppressed.All the detected groups of the order of at most 1025 are listed in the ancillary file, while Table REF shows groups of the order of at most 100 only which underwent investigation."
],
[
"B. Yukawa couplings matrices in the model with Dirac neutrinos",
"In the case of all the groups subject to consideration, there exist 267 groups that gave in total 748672 different combinations of 2- and 3-dimensional irreducible representations that give 1-dimensional degeneration subspace for all generators, which is the solution to the equations in Eq.",
"(REF ).",
"This common vector gives Yukawa matrices for charged leptons ($h^{(l)}$ ) and for neutrinos ($h^{(\\nu )}$ ), which are interrelated.",
"All the possible solutions for Yukawa matrices for charged leptons and for Dirac neutrinos can be expressed through 7 base forms ($\\omega =e^{2 \\pi i/3}$ ): $h_{1}^{(1)} = \\left(\\begin{array}{ccc}0 & 0 & 1 \\\\1 & 0 & 0 \\\\0 & 1 & 0 \\\\\\end{array}\\right), \\ \\quad h_{2}^{(1)} = \\left(\\begin{array}{ccc}0 & 1 & 0 \\\\0 & 0 & 1 \\\\1 & 0 & 0 \\\\\\end{array}\\right),$ $h_{1}^{(2)} = \\left(\\begin{array}{ccc}0 & 0 & 1 \\\\\\omega ^{2} & 0 & 0 \\\\0 & \\omega & 0 \\\\\\end{array}\\right), \\quad h_{2}^{(2)} = \\left(\\begin{array}{ccc}0 & 1 & 0 \\\\0 & 0 & \\omega \\\\\\omega ^{2} & 0 & 0 \\\\\\end{array}\\right),$ $h_{1}^{(3)} = \\left(\\begin{array}{ccc}0 & 0 & 1 \\\\\\omega & 0 & 0 \\\\0 & \\omega ^{2} & 0 \\\\\\end{array}\\right), \\quad h_{2}^{(3)} = \\left(\\begin{array}{ccc}0 & 1 & 0 \\\\0 & 0 & \\omega ^{2} \\\\\\omega & 0 & 0 \\\\\\end{array}\\right),$ the next three $h_{1}^{(i)}$ and $h_{2}^{(i)}$ for i = 4,5,6, are obtained from those given in Eq.",
"(REF )-Eq.",
"(REF ) by interchange, using the rule: $h_{1}^{(3+i)}=h_{2}^{(i)}, \\ \\ h_{2}^{(3+i)}=h_{1}^{(i)}$ for i= 1,2,3, and finally$U(3)$ subgroups which have no faithful 3-dimensional irreducible representation give only these diagonal solutions.",
": $h_{1}^{(7)} = \\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & \\omega ^{2} & 0 \\\\0 & 0 & \\omega \\\\\\end{array}\\right), \\quad h_{2}^{(7)} = \\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & \\omega & 0 \\\\0 & 0 & \\omega ^{2} \\\\\\end{array}\\right).$ Symmetric Yukawa matrices for any considered group and for any irreducible representation within the group can be expressed now by seven (i=1,2,..,7) basic matrix forms as follows: a) for Dirac neutrinos: $\\lbrace h_{1}^{(\\nu )}, h_{2}^{(\\nu )}\\rbrace = \\lbrace h_{1}^{(i)}, e^{i\\varphi } h_{2}^{(i)}\\rbrace ,$ b) for charged leptons: $\\lbrace h_{1}^{(l)}, h_{2}^{(l)}\\rbrace = \\lbrace h_{2}^{(i)}, e^{-i(\\delta _{l} +\\varphi )} h_{1}^{(i)}\\rbrace ,$ where $\\varphi $ is a phase distinctive for a group and for irreducible representations and $\\delta _{l}=0,\\pi .$ In order to find the lepton masses and mixing matrix we constructed the hermitian matrices as in Eq.",
"(REF ) $ M^{l}M^{l\\dagger }$ and $ M^{\\nu }M^{\\nu \\dagger }$ .",
"For all the possible Yukawa matrices we have obtained only three different forms $(x=l, \\nu )$ : $M_{x}M_{x}^{\\dagger } = \\vert c_{x}|^2\\left(\\begin{array}{ccc}1+\\kappa ^{2} & \\kappa e^{-i(\\eta _{x} + 2 k \\pi /3)} & \\kappa e^{i(\\eta _{x} - 2 k \\pi /3)} \\\\\\kappa e^{i(\\eta _{x} + 2 k \\pi /3)} & 1+\\kappa ^{2} & \\kappa e^{-i \\eta _{x}} \\\\\\kappa e^{-i(\\eta _{x} - 2 k \\pi /3)} & \\kappa e^{i\\eta _{x}}& 1+\\kappa ^{2} \\\\\\end{array} \\right),$ with $k=-1,0,+1$ and $\\kappa = \\vert v_{2}\\vert /\\vert v_{1}\\vert $ , the same for neutrinos and for charged leptons.",
"The only difference lies in the phase $\\eta _x$ .",
"For Dirac neutrinos, $\\eta _{\\nu }= \\varphi +\\varphi _{2} -\\varphi _{1},$ and for charged leptons, $\\eta _{l} = \\delta _{l}+\\varphi +\\varphi _{2}-\\varphi _{1},$ where $\\varphi _i (i=1,2)$ are phases of the VEVs $v_i$ .",
"After diagonalization of Eq.",
"(REF ) $ U^{\\dagger } \\left(M_{x} M_{x}^{\\dagger }\\right) U= diag\\left(m_{x1}^{2},m_{x2}^{2},m_{x3}^{2}\\right),$ we obtain: $&m_{x1}^{2}& = |c_{x}|^2 \\left(1+\\kappa ^{2} + 2 \\kappa cos\\left(\\eta _{x}\\right)\\right), \\\\&m_{x2}^{2}& = |c_{x}|^2 \\left(1+\\kappa ^{2} + 2 \\kappa sin\\left(\\eta _{x}-\\frac{\\pi }{6}\\right)\\right),\\\\&m_{x3}^{2}& = |c_{x}|^2 \\left(1+\\kappa ^{2} - 2 \\kappa sin\\left(\\eta _{x}+\\frac{\\pi }{6}\\right)\\right),$ and the diagonalization matrix U: $U = \\frac{1}{\\sqrt{3}}\\left(\\begin{array}{ccc}e^{- \\frac{2}{3} \\pi i k} & \\omega e^{- \\frac{2}{3} \\pi i k} & \\omega ^{2} e^{- \\frac{2}{3} \\pi i k} \\\\1 & \\omega ^{2} & \\omega \\\\1 & 1 & 1 \\\\\\end{array} \\right).$ This matrix does not depend on the phase $\\eta _{x}$ , so it is identical for charged leptons and for the neutrino.",
"Therefore, it is not possible to reconstruct the correct mixing matrix.",
"For groups and their irreducible representations for which $\\delta _{l}= 0$ , neutrinos and lepton masses are proportional and the mixing matrix is 3x3 identity matrix.",
"For groups and for representations where $\\delta _{l}=\\pi $ , masses of charged leptons and neutrinos are not proportional.",
"If the formulas in Eqs.",
"(REF -) describe the masses of neutrinos, then for the charged leptons there is: $&m_{l1}^{2} &= |c_{l}|^2 \\left(1+\\kappa ^{2} - 2 \\kappa cos\\left(\\eta _{\\nu }\\right)\\right),\\\\&m_{l2}^{2} &= |c_{l}|^2 \\left(1+\\kappa ^{2} - 2 \\kappa sin\\left(\\eta _{\\nu }-\\frac{\\pi }{6}\\right)\\right),\\\\&m_{l3}^{2} &= |c_{l}|^2 \\left(1+\\kappa ^{2} + 2 \\kappa sin\\left(\\eta _{\\nu }+\\frac{\\pi }{6}\\right)\\right).$ Similarly in this case we cannot reconstruct the PMNS mixing matrix for which the 3x3 anti-diagonal identity matrix is obtained.",
"Regardless of whether the masses of charged leptons are described by formula Eqs.",
"(REF -)) (groups with $\\delta _{l} =0$ ) or by Eq.",
"(REF -) (groups with $\\delta _{l} = \\pi $ ), three parameters $(c_{l}, \\kappa , \\eta _{x}) $ cannot be selected in such a way as to obtain the physical masses of the electron, muon and tau (in any case, one obtains: $0 \\leqslant m_{e} / m_{\\mu } \\leqslant 1$ and $1 \\leqslant m_{\\tau } / m_{\\mu } \\leqslant 2$ ).",
"The SM extended by one additional doublet of Higgs particles (2HDM) does not posses a discrete family symmetry (in the groups under examination) that can explain the masses of charged leptons, masses of neutrinos having the nature of Dirac particles and the PMNS matrix."
],
[
"C. Yukawa couplings matrices in the model with Majorana neutrinos",
"In the previous subsection, we did not receive masses of charged leptons consistent with the experiment.",
"This was the case when the neutrinos were Dirac particles.",
"In the case of Majorana neutrinos, we must once again look for a possible symmetry, since a symmetric solution for the Weinberg component may also deliver other groups of symmetry for charged leptons.",
"We are currently looking for symmetries satisfying the Eg.",
"(REF ) for the charged leptons and Eq.",
"(REF ) for neutrinos.",
"Out of all the groups under consideration, there exist 195 groups that gave in total 20888 solutions.",
"All the found symmetries, as a solution of Eq.",
"(REF ), give a two dimensional space which is common to all generators of the groups in question.",
"Each time we find two 36-dimensional vectors p and r and any linear combination of p and r, gives symmetric Yukawa matrices for Majorana neutrinos.",
"Thus these Yukawa matrices are given by: $h^{(i,k)} = x p_{i,k} + y r_{i,k},$ where x and y are two free complex numbers.",
"A detailed analysis of the solution for Majorana neutrinos is as follows: $&h^{(1,1)}= x h_{2}^{(7)},\\quad &h^{(1,2)}= y I_{3}, \\nonumber \\\\&h^{(2,1)}= y e^{i \\delta } I_{3},\\quad &h^{(2,2)}= x e^{i(\\delta + 2\\varphi )} h_{1}^{(7)},$ and for the charged leptons Yukawa matrices: $\\lbrace h_{1}^{(l)}, h_{2}^{(l)}\\rbrace = \\lbrace h_{2}^{(7)}, e^{-i( \\delta _{l} +\\varphi )} h_{1}^{(7)}\\rbrace ,$ where $h_{1}^{(7)}$ , $h_{2}^{(7)}$ are given by Eq.",
"(REF ) and $I_{3}$ is 3x3 identity matrix.",
"As previously observed, the phases $\\delta =(0,\\pi ), \\delta _{l}$ and $\\varphi $ depend on the group and its representations.The resulting neutrino mass matrix Eq.",
"(REF ) has the form; $M^{\\nu } =&&\\frac{g}{2 M}\\left(x \\vert v_{1}\\vert ^{2} e^{2 i \\varphi _{1}}h_{2}^{(7)}+ y \\vert v_{1} v_{2}\\vert e^{i(\\varphi _{1} + \\varphi _{2})} I_{3} \\left(1+e^{i\\delta }\\right) \\right.",
"\\nonumber \\\\&&+ \\left.",
"x \\vert v_{2}\\vert ^{2} e^{i(\\delta +2(\\varphi _{2}+\\varphi ))}h_{1}^{(7)}\\right),$ which gives the squares of neutrino masses: $m_{1}^{2}=&&\\vert c_{\\nu }\\vert ^{2} \\left[ 1 +\\kappa ^4 +2\\beta ^2 \\kappa ^2 + 2 \\beta ^2 \\kappa ^2 \\cos (\\delta )+4 \\beta \\kappa ^3\\cos \\left(\\frac{\\delta }{2}\\right)\\cos \\left(\\frac{\\delta }{2}+\\tau +2\\varphi \\right) \\right.\\nonumber \\\\&& \\left.",
"+4 \\beta \\kappa \\cos \\left(\\frac{\\delta }{2}\\right) \\cos \\left(\\frac{\\delta }{2}+\\tau \\right)+2\\kappa ^2 \\cos (\\delta +2 \\tau +2 \\varphi ) \\right],$ $m_{2}^{2}=&&\\vert c_{\\nu }\\vert ^{2} \\left[ 1 +\\kappa ^4 +2\\beta ^2 \\kappa ^2+ 2 \\beta ^2 \\kappa ^2 \\cos (\\delta )+4 \\beta \\kappa ^3\\cos \\left(\\frac{\\delta }{2}\\right)\\sin \\left(\\frac{\\delta }{2}+\\tau +2\\varphi -\\frac{\\pi }{6}\\right) \\right.",
"\\nonumber \\\\&&\\left.",
"+4 \\beta \\kappa \\cos \\left(\\frac{\\delta }{2}\\right) \\sin \\left(\\frac{\\delta }{2}+\\tau -\\frac{\\pi }{6}\\right)-2\\kappa ^2 \\sin \\left(\\delta +2 \\tau +2\\varphi +\\frac{\\pi }{6}\\right) \\right],$ $m_{3}^{2}=&&\\vert c_{\\nu }\\vert ^{2} \\left[1+ \\kappa ^4+2 \\beta ^2 \\kappa ^2+2 \\beta ^2 \\kappa ^2\\cos (\\delta )-4 \\beta \\kappa ^3 \\cos \\left(\\frac{\\delta }{2}\\right)\\sin \\left(\\frac{\\delta }{2}+\\tau +2\\varphi +\\frac{\\pi }{6}\\right) \\right.",
"\\nonumber \\\\&& \\left.-4 \\beta \\kappa \\cos \\left(\\frac{\\delta }{2}\\right) \\sin \\left(\\frac{\\delta }{2}+\\tau +\\frac{\\pi }{6}\\right)+2\\kappa ^2 \\sin \\left(\\delta +2 \\tau +2\\varphi -\\frac{\\pi }{6}\\right) \\right],$ where $\\beta =y/x$ , $\\tau =\\varphi _{2} - \\varphi _{1}$ and $c_{\\nu }= g x \\vert v_{1}\\vert ^{2} e^{2 i \\varphi _{1}}/(2 M)$ .",
"The masses of charged leptons are given by the same formulae as given above Eqs.",
"(REF -) and it is impossible to fit electron, muon and tau lepton masses.",
"Our mass matrices for neutrinos and charged leptons are diagonal, so we also do not have the ability to fit the mixing matrix.",
"We also see that the symmetry condition for Majorana neutrinos (Eq.REF ) does not give any new flavour symmetry group with new three dimensional representation $A_L$ which would not be present in the symmetry equations for charged leptons (Eq.REF ).",
"The Yukawa matrices $h_1^{(l)}$ and $h_2^{(l)}$ in Eq.REF for charged leptons are exactly the same as in the case of Dirac neutrinos (Eq.REF ).",
"This is not a good conclusion.",
"In the set of groups we are consider, regardless of the adopted neutrino sector, we will not find a symmetry that gives acceptable solutions for mass of charged leptons."
],
[
"IV. Conclusions",
"We have explored the possibility of using some discrete flavour symmetry to explain the masses and mixing matrix elements of leptons in the Standard Model with the Higgs particle sector extended by one additional Higgs doublet - 2HDM.",
"In general, we have assumed that the total Lagrangian model has a full flavour symmetry with one exception.",
"We also admit that the Higgs potential in our model is only invariant in form, which is often used in the model description of experimental data.",
"In such a model, we have avoided having to break the family symmetry and introduce flavon fields.",
"We have investigated discrete groups that are subgroups of the continuous $U(3)$ group up to the order of 1025.",
"Models in which neutrinos have the nature of Dirac particles and models with Majorana neutrinos have been considered.",
"Following a close analysis of these 413 groups and all their possible combinations of 2- and 3-dimensional irreducible representations, it is established that none of them can reproduce the current experimental data.",
"Thus, in the 2HDM model with the symmetrical or only form-invariant Higgs potential, in the class of the groups under consideration, there is no discrete family symmetry that would fully clarify the masses and parameters of the mixing matrix for leptons.",
"In addition, we have observed that the set of symmetric Yukawa matrices for charged leptons is independent from the nature of the neutrinos.",
"This can serve as a guidance for further search for family symmetry.",
"In the models with two Higgs particles, regardless of the adopted neutrino sector and the set of groups that we consider, we do not find a symmetry that gives real masses of charged leptons, even approximately.",
"This work has been supported by the Polish National Science Centre (NCN) under grant no.",
"UMO - 2013/09/B/ST2/03382"
]
] | 1808.08384 |
[
[
"An Experimental Comparison of SONC and SOS Certificates for\n Unconstrained Optimization"
],
[
"Abstract Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications.",
"We present a polynomial time algorithm to approximate such lower bounds via sums of nonnegative circuit polynomials (SONC).",
"As a main result, we carry out the first large-scale comparison of SONC, using this algorithm and different geometric programming (GP) solvers, with the classical sums of squares (SOS) approach, using several of the most common semidefinite programming (SDP) solvers.",
"SONC yields bounds competitive to SOS in several cases, but using significantly less time and memory.",
"In particular, SONC/GP can handle much larger problem instances than SOS/SDP."
],
[
"Introduction",
"Minimizing a given real, multivariate polynomial $f \\in [{x}]$ is the fundamental challenge of polynomial optimization.",
"This problem has countless applications, see e.g., [24].",
"Since it is -hard, e.g., [26], one commonly guarantees nonnegativity via a certificate of nonnegativity, a sufficient, but not necessary condition, which is easier to check than nonnegativity itself.",
"The standard certificates of nonnegativity are sums of squares (SOS), which have a long history dating back to Hilbert; see [19] and see [32] for a historical overview.",
"SOS certificates can be detected via semidefinite programming (SDP) and were tremendously successful in recent years; see e.g., [5], [26], [24], [25] for an overview.",
"The SOS approach has, however, certain limits: In 2006, Blekherman proved that for fixed degree $2d \\ge 4$ and $n \\rightarrow \\infty $ almost every nonnegative polynomial is not SOS [4].",
"Moreover, the SDP, which is equivalent to detecting an $n$ -variate degree $d$ SOS, has a matrix-variable of size $\\binom{n+d}{d}$ .",
"Thus, the corresponding SDPs are too large to be solved even for moderate size of $n$ and $d$ , often even if further preprocessing tools are exploited.",
"In the recent past, building on work by Reznick [31], Iliman and the second author [21] suggested a certificate based on sums of nonnegative circuit polynomials (SONC), a particular class of sparse polynomials, as an alternative to SOS for hard polynomial optimization problems; see section:preliminaries for details.",
"Generalizing an approach by Ghasemi and Marshall [16], [18], they showed in [22] that SONC certificates can be detected via geometric programming (GP) if the investigated polynomial has a simplex Newton polytope.",
"Joint with Dressler and Iliman, the second author provided a first heuristic for computing SONC certificates for polynomials with non-simplex Newton polytope [10].",
"While investigated examples suggested significant shorter runtimes for SONC/GP compared to SOS/SDP, the approach had shortcomings so far: Due to the non-existence of a software for the SONC approach, examples could only be computed with severe effort.",
"As a consequence, no large scale comparison of both approaches had been carried out; the existing examples had to be considered as preliminary data.",
"There existed no deterministic algorithm to compute SONC certificates for polynomials with non-simplex Newton polytope.",
"In this article, we resolve these shortcomings.",
"We present an algorithm for computing SONC certificates for sparse polynomials with arbitrary Newton polytope; see algorithm:bound.",
"The key idea is to generate a covering of the Newton polytope of a given $f$ via solving several linear optimization problems and then considering the distribution of coefficients as part of the GP we have to solve.",
"We show that this algorithm is not only systematic, but also outperforms the heuristic approach in [10] with respect to the quality of the computed bounds; see section:describingthesoftware for further details.",
"Moreover, our algorithm:bound runs in polynomial time, see theorem:fullalgopolytime.",
"Our main contribution is an experimental large-scale comparison of SONC/GP versus SOS/SDP using these algorithms, which are implemented in our software POEM (Effective Methods in Polynomial Optimization; developed since July 2017) [33].",
"The software is available at https://www3.math.tu-berlin.de/combi/RAAGConOpt/poem.html.",
"We let it compete with a broad range of the existing standard SDP solvers, explicitly allowing them to exploit sparsity.",
"We generated a database of 16921 test cases covering polynomials with various sorts of Newton polytopes, number of variables in the range of 2 to 40, degree in the range of 6 to 60, and a cardinality of terms in the range of 6 to 500.",
"Among these are 501 trivial cases, which are sums of monomial squares.",
"The full database of instances is available at the software homepage; see subsection:generatingpolynomials for further details on its generation.",
"The complete database of computations (around 23 GB) is available on inquiry.",
"We discuss all computational results in detail in subsection:evaluation.",
"In summary, our software executing algorithm:bound, combined with the solver ECOS [9], solved most instances within a few seconds.",
"Even for the largest of our instances it required less than 5 minutes.",
"Over all, we obtained a SONC bound in 16240 (98.9 %) of the 16420 investigated non-trivial cases.",
"In contrast, we obtained an SOS bound only in 5161 (31.4 %) out of the 16420 investigated non-trivial cases.",
"In 5732 (34.9 %) cases, where the SOS solvers did not find a bound, they did not terminate due to numerical issues, or ran out of memory.",
"The remaining 6028 cases led to SDPs with matrices of sizes beyond $1000 \\times 1000$ , which we established experimentally as a reasonable threshold for 16 GB of RAM.",
"In comparison, we achieved a SONC but no SOS bound in 11760 (71.6 %) of the cases and an SOS but no SONC bound in only 44 (0.0027 %) of the cases.",
"Among the 4479 cases (27.3 % of the entire nontrivial cases) where we achieved both SOS and SONC bounds, usually the SOS bound is better (by more than $10^{-3}$ ), namely in 3693 cases (82.5 %).",
"In 782 of these cases, however, the Newton polytope was a scaled standard simplex, where SOS is at least as good as SONC by construction.",
"Moreover, only in 1625 (36.28 %) of the cases the difference of the bounds was greater than 1.",
"We visualize the main outcome of our computations in figure:plotbardifference: In the left plot, we depict the number of instances such that the optimality bounds ${}$ and ${}$ , which we obtain using SOS and the SONC certificates, satisfy: $- $ is the interval on the lower axis.",
"In the right plot, we present the ratio of the runtime of the fastest SOS method compared to the runtime of our SONC algorithm with ECOS and depict for how many instances the ratio lies in the given interval.",
"Figure: Number of instances where (left) -- lies in the interval given on the lower axis; (right) the ratio SOS/SONC\\operatorname{SOS}/\\operatorname{SONC} of the running time lies in the interval given on the lower axis.The upper red bar represents instances, where the problem size for SOS lies above our threshold.The article is organized as follows: In section:preliminaries we introduce our notation and previous results about SONCs.",
"In section:describingthesoftware we introduce develop algorithm:bound and prove that it runs in polynomial time theorem:fullalgopolytime.",
"In section:runningtestcases we describe the setup of our experiment and present its outcome.",
"Finally, we draw the conclusions from our experiments in section:conclusion."
],
[
"Acknowledgements",
"We thank Mareike Dressler, Meike Hatzel, Sadik Iliman and Michael Joswig for their helpful suggestions.",
"We thank Benjamin Lorenz for his invaluable technical support.",
"Both authors of this article are supported by the DFG grant WO 2206/1-1."
],
[
"Preliminaries",
"In this section we introduce our basic notation, sums of squares, sums of nonnegative circuit polynomials, and geometric programs."
],
[
"Representing Sparse Polynomials",
"Throughout the paper, we use bold letters for vectors (small) and matrices (capital), e.g., ${{x}}=(x_1,\\ldots ,x_n) \\in ^n$ .",
"Let ${_{\\ge 0}}$ and ${_{> 0}}$ be the set of nonnegative and positive real numbers, respectively.",
"Furthermore, let $[{x}] = [x_1,\\ldots ,x_n]$ be the ring of real $n$ -variate polynomials.",
"We denote the set of all $n$ -variate polynomials of degree less than or equal to $2d$ by ${[{x}]_{n,2d}}$ .",
"For $p \\in [{x}]$ we denote the total degree of $p$ by ${\\deg (p)}$ .",
"We mostly regard sparse polynomials $p \\in [{x}]$ supported on a finite set ${A} \\subset ^n$ ; we write ${{p}}$ if a clarification is necessary.",
"Thus, $p$ is of the form ${p({x})} = \\sum _{{\\alpha } \\in A}^{} b_{{\\alpha }}{x}^{{\\alpha }}$ with ${b_{{\\alpha }}} \\in \\setminus \\lbrace 0\\rbrace $ and ${{x}^{{\\alpha }}} = x_1^{\\alpha _1} \\cdots x_n^{\\alpha _n}$ .",
"Unless stated differently, we follow the convention ${t} = \\# A$ .",
"The support of $p$ can be expressed as an $n \\times t$ matrix, which we denote by ${{A}}$ , such that the $j$ -th column of ${A}$ is ${\\alpha (j)}$ .",
"Hence, $p$ is uniquely described by the pair $({A},{b})$ , written $p = {{{A}}{{b}}}$ .",
"We denote by ${(p)} = \\left(\\lbrace {\\alpha } \\in ^n : b_{{\\alpha }} \\ne 0\\rbrace \\right) = \\left({{p}}\\right)$ the Newton polytope of $p$ with vertices ${{p}} = \\left\\lbrace \\alpha \\in {p} : {\\alpha } \\text{ is vertex of } {p}\\right\\rbrace $ .",
"A lattice point is called even if it is in $(2)^n$ and a term $b_{{\\alpha }}{x}^{{\\alpha }}$ is called a monomial square if $b_{{\\alpha }} > 0$ and ${\\alpha }$ even.",
"We define ${{p}} = \\left\\lbrace {\\alpha } \\in {p} : {\\alpha } \\in (2)^n, b_{{\\alpha }} > 0\\right\\rbrace $ as the set of monomial squares in the support of $p$ .",
"Moreover, we use the notation ${{p}} = {p} \\setminus {p}$ for all elements of the support of $p$ , which are not a monomial square.",
"We indicate the elements of the support which are in the interior of $(p)$ by ${{p}} = {p} \\setminus \\partial (p)$ ."
],
[
"Formulation of the SOS-problem",
"Let $p \\in [{x}]$ .",
"In unconstrained polynomial optimization we are interested in computing the infimum of $p$ given by $\\begin{aligned}{p^*} & = & \\inf _{{x} \\in ^n} p({x}) \\ = \\ \\sup \\lbrace \\gamma \\in \\ : \\ p({x}) - \\gamma \\ge 0 \\text{ for all } {x} \\in ^n\\rbrace .\\end{aligned}$ Note that a polynomial is bounded below if and only if its homogenization is nonnegative.",
"Since deciding nonnegativity is -hard [26], deciding, whether $p^*$ is finite, also is -hard.",
"A polynomial $p$ is a sum of squares (SOS) if there exist $s_1,\\ldots ,s_r \\in [{x}]$ such that $p = \\sum _{j = 1}^r s_j^2$ .",
"Writing a polynomial as a sum of squares provides a certificate for nonnegativity.",
"We denote $\\begin{aligned}{} & = & \\sup \\lbrace \\gamma \\in \\ : \\ p({x}) - \\gamma \\text{ is a sum of squares}\\rbrace ,\\end{aligned}$ and have $\\le p^*$ .",
"Let $2d = \\deg (p)$ .",
"The bound is the optimal value of the following semidefinite program (SDP) SDP-SOS = X0,0= b0 + || d X,-= b1||2d X0 where the last conditions means ${X}$ is positive semidefinite, and we write ${\\alpha }\\le {\\beta }$ if and only if $\\alpha _i\\le \\beta _i$ for all $i=1,\\ldots ,n$ .",
"For background on SOS and SDPs see e.g., [5], [26].",
"Given a feasible starting point and an accuracy $\\varepsilon > 0$ , an SDP can be solved up to accuracy $\\varepsilon $ in time polynomial in the problem size.",
"The size of the problem is, however, the major drawback of this approach: ${X}$ is a $\\binom{n+d}{d}\\times \\binom{n+d}{d}$ -matrix and we have $\\binom{n+2d}{2d}$ constraints.",
"Thus, if both the number of variables and the degree grow larger, the problem requires an exponentially large amount of RAM, even to state it.",
"Therefore, in this case, solving the problem becomes practically infeasible, even for moderate sizes of $d$ and $n$ ."
],
[
"Sums of Nonnegative Circuit Polynomials",
"We introduce the fundamental facts of SONC polynomials, which we use in this article.",
"SONCs are constructed by circuit polynomials; which were first introduced in [21]: A circuit polynomial $p = ({A},{b}) \\in [{x}]$ is of the form ${p({x})} & = & \\sum _{j=0}^r b_{{\\alpha }(j)} {x}^{{\\alpha }(j)} + b_{{\\beta }} {x}^{{\\beta }}, $ with $0 \\le {r} \\le n$ , coefficients ${b_{{\\alpha }(j)}} \\in _{> 0}$ , ${b_{{\\beta }}} \\in $ , exponents ${{\\alpha }(j)} \\in (2)^n$ , ${{\\beta }} \\in ^n$ , such that the following condition holds: There exist unique, positive barycentric coordinates $\\lambda _j$ relative to the ${\\alpha }(j)$ with $j=0,\\ldots ,r$ satisfying $& & {\\beta } \\ = \\ \\sum _{j=0}^r \\lambda _j {\\alpha }(j) \\ \\text{ with } \\ \\lambda _j \\ > \\ 0 \\ \\text{ and } \\ \\sum _{j=0}^r \\lambda _j \\ = \\ 1.$ For every circuit polynomial $p$ we define the corresponding circuit number as ${\\Theta _p} \\ = \\ \\prod _{j = 0}^r \\left(\\frac{b_{{\\alpha }(j)}}{\\lambda _j}\\right)^{\\lambda _j}.$ Condition (REF ) implies that ${A}(p)$ forms a minimal affine dependent set.",
"Those sets are called circuits, see e.g., [29].",
"More specifically, Condition (REF ) yields that $(p)$ is a simplex with even vertices ${\\alpha }(0), {\\alpha }(1),\\ldots ,{\\alpha }(r)$ and that the exponent ${\\beta }$ is in the strict interior of $(p)$ if $\\dim ((p)) \\ge 1$ .",
"Therefore, we call the terms $p_{{\\alpha }(0)} {x}^{{\\alpha }(0)},\\ldots ,p_{{\\alpha }(r)} {x}^{{\\alpha }(r)}$ the outer terms and $p_{{\\beta }} {x}^{{\\beta }}$ the inner term of $p$ .",
"Circuit polynomials are proper building blocks for nonnegativity certificates since the circuit number alone determines whether they are nonnegative.",
"[[21], Theorem 3.8] Let $p$ be a circuit polynomial of the form (REF ).",
"Then $p$ is nonnegative if and only if: $p$ is a sum of monomial squares, or the coefficient $b_{{\\beta }}$ of the inner term of $p$ satisfies $|b_{{\\beta }}| \\le \\Theta _p$ .",
"Note that $\\Theta _p$ can be computed by solving a system of linear equations if it is known that ${\\beta }$ is in the relative interior of $(p)$ and by solving an LP otherwise.",
"We define for every $n,d \\in $ the set of sums of nonnegative circuit polynomials (SONC) in $n$ variables of degree $2d$ as ${C_{n,2d}} \\ = \\ \\left\\lbrace f \\in [{x}]_{n,2d} \\ :\\ f = \\sum _{\\rm finite} p_i, \\quad p_i \\text{ is a nonnegative circuit polynomial} \\right\\rbrace .$ We denote by SONC both the set of SONC polynomials and the property of a polynomial to be a sum of nonnegative circuit polynomials.",
"In what follows let ${P_{n,2d}}$ be the cone of nonnegative $n$ -variate polynomials of degree at most $2d$ and ${\\Sigma _{n,2d}}$ be the corresponding cone of sums of squares respectively.",
"An important observation is, that SONC polynomials form a convex cone independent of the SOS cone: [[21], Proposition 7.2] $C_{n,2d}$ is a convex cone satisfying: $C_{n,2d} \\subseteq P_{n,2d}$ for all $n,d \\in $ , $C_{n,2d} \\subseteq \\Sigma _{n,2d}$ if and only if $(n,2d)\\in \\lbrace (1,2d),(n,2),(2,4)\\rbrace $ , $\\Sigma _{n,2d} \\lnot \\subseteq C_{n,2d}$ for all $(n,2d)$ with $2d \\ge 6$ .",
"For further details about the SONC cone see [13], [21], [11]."
],
[
"Previous Methods for the Computation of SONC Decompositions",
"Geometric programming was introduced in [12].",
"It is equivalent to a convex optimization problem.",
"Applications include problems in circuit design problems, nonlinear network flow, and optimal control; for an overview see [3], [6] A function $p : _{>0}^n\\rightarrow $ of the form ${p({z})} = p(z_1,\\ldots ,z_n) = cz_1^{\\alpha _1}\\cdots z_n^{\\alpha _n}$ with $c > 0$ and $\\alpha _i \\in $ is called a monomial (function).",
"A sum ${\\sum _{i=0}^k c_iz_1^{\\alpha _{1}(i)}\\cdots z_n^{\\alpha _{n}(i)}}$ of monomials with $c_i > 0$ is called a posynomial (function).",
"A geometric program (GP) has the following form: z>0n p0(z) pi(z)1i=1,...,N qj(z)= 1j=1,...,M where $p_0,\\dots ,p_m$ are posynomials and $q_1,\\dots ,q_l$ are monomial functions.",
"Geometric programs are convex and can hence be solved with interior point methods.",
"In [28], the authors prove worst-case polynomial time complexity of this method; see also [3].",
"In [22] Iliman and the second author used GP and SONC to compute lower bounds for polynomials if their Newton polytope is a simplex.",
"For the case of an non-simplex Newton polytope, Dressler, Iliman and the second author propose the following heuristic approach to obtain lower bound for polynomials via SONC in [10].",
"Choose a triangulation $T_1,\\ldots ,T_\\ell $ of the points corresponding to monomial squares.",
"Choose weights $f_{{\\beta },i}$ for all $({\\beta },i)\\in {p}\\times \\lbrace 1,\\ldots ,\\ell \\rbrace $ with ${\\beta }\\in T_i$ , satisfying $\\sum _{i=1}^\\ell f_{{\\beta },i}=f_{{\\beta }}$ .",
"Define polynomials $g_i \\ = \\ \\sum _{\\beta \\in T_i\\cap {p}} f_{{\\beta },i} {x}^{{\\beta }} \\text{ for all $1\\le i\\le \\ell $}$ Then each ${g_i}$ is a simplex, so we can apply the previous approach.",
"However, they do not provide methods to obtain either a good triangulation or a good distribution of the weights, see example:DIdW165.5."
],
[
"A Polynomial Time Algorithm for Computing SONC Certificates",
"Let $p={{A}}{{b}}$ be some polynomial with ${A}\\in ^{n\\times t}$ and ${b}\\in ^t$ .",
"In this section we describe an algorithm to find a lower bound for $p$ .",
"We begin with recalling the special case where the Newton polytope forms a simplex and every point, which is not a vertex, lies in the interior.",
"Afterwards, we describe two ways for handling arbitrary polynomials.",
"The first approach splits the problem into several instances of the simplex case.",
"In the second approach we have to solve a single larger optimization problem, which is similar to the problem from the simplex case.",
"The first approach can be parallelized easily, while the second yields better results.",
"Our initial hypothesis was that the first approach would run significantly faster, which turned out not to be the case.",
"Regarding a comparison of run times, see table:splitstrategytime.",
"As a first relaxation, every coefficient of a monomial non-square gets a negative sign.",
"Their exponents form the negative points, while the exponents of the monomial squares form the positive points.",
"This corresponds to the worst case where every possibly negative term is negative for the same argument.",
"Additionally, we can thus focus on the positive orthant.",
"For each negative point we select positive points, which form a simplex, that contains the negative point in its interior.",
"Next, we identify all other negative points, which lie in this simplex.",
"We continue this procedure until each negative point is covered by at least one simplex.",
"Then we fix a distribution of the negative coefficients and compute an optimal distribution of the positive coefficients via GP, such that each simplex is split into nonnegative circuit polynomials.",
"The adjustment we have to make for the constant term thus yields a lower bound for the polynomial."
],
[
"Computing the Newton polytope",
"Recall that a point in a set $A \\subset ^n$ is extremal if and only if it cannot be written as a convex combination of the other points in $A$ .",
"We need an efficient algorithm for deciding whether the Newton polytope corresponding to a given polynomial $p$ is a simplex.",
"This means, we have to find the extremal points of the set of exponents.",
"So, for every ${v} \\in A(p)$ we ask for the feasibility of the following LP: $\\begin{aligned}\\sum _{u\\in {p}\\setminus \\lbrace v\\rbrace } \\lambda _{{u}} \\cdot {u} &= {v}\\\\\\sum _{u\\in {p}\\setminus \\lbrace v\\rbrace } \\lambda _{{u}} &= 1\\\\\\lambda _{{u}} &\\ge 0 \\qquad \\text{for all } {u}\\in {p}\\setminus \\lbrace {v}\\rbrace \\end{aligned}$ which can be decided in polynomial time [23].",
"Solving this LP for every ${v}\\in {p}$ , computes ${p}$ in polynomial time.",
"Furthermore, the solutions of these LPs yield convex combinations of the interior points with respect to the vertices.",
"For later we need the following result, which is slightly stronger.",
"The lemma is folklore.",
"We, however, provide a short proof for convenience of the reader.",
"For every non-extremal point ${v}\\in {p}\\setminus {p}$ , we can efficiently compute affinely independent ${v}_0,...,{v}_m \\in {p}$ with $m\\le n$ such that and ${v}\\in \\left(\\lbrace {v}_0,...,{v}_m\\rbrace \\right)$ .",
"Solve LP:extremal to obtain a convex combination ${v} = \\sum _{i=0}^m\\lambda _i {v}_i$ .",
"Assume $m>n$ , and let $\\mu $ be a non-trivial element in $\\operatorname{ker}\\begin{pmatrix}1 & \\cdots & 1 \\\\{v}_0 & \\cdots & {v}_m\\end{pmatrix}$ Assign $\\eta = \\min \\left\\lbrace \\frac{\\lambda _i}{\\mu _i}: i\\le m, \\mu _i>0\\right\\rbrace $ and choose coefficients $\\lambda _i^{\\prime } = \\lambda _i - \\eta \\mu _i$ .",
"Then ${v} = \\sum _{i=0}^m (\\lambda _i^{\\prime }-\\eta \\mu _i){v}_i$ is a new convex combination but one of the coefficients vanished, so we obtain a shorter convex combination of ${v}$ .",
"We delete the corresponding column in the matrix (REF ) and iterate the process until we obtain a matrix with trivial kernel.",
"Once this is the case the vectors ${v}_i-{v}_0$ are affinely independent.",
"Note that this can only occur if $m\\le n$ ,"
],
[
"The SONC-Problem for Simplex Newton polytope",
"In this part, we assume ${p} = {p} \\cup {p}$ and that ${p}$ is a simplex of dimension ${h}\\le n$ .",
"We recall how to compute SONC certificates using GPs in this case.",
"This was first shown by Iliman and the second author in [22] generalizing an idea of Ghasemi and Marshall [17], where the latter regard the problem only for the case that ${p}$ is the standard simplex using theoretical results by Fidalgo and Kovacec [14].",
"Let ${\\lambda }$ be the barycentric coordinates of ${p}$ with respect to ${p}$ , i.e.",
"${p}\\cdot {\\lambda }= {p}$ in matrix notation.",
"Note that ${\\lambda }$ is unique, since ${p}$ forms a simplex.",
"This means we write each interior point as a convex combination of the vertices of $(p)$ .",
"In particular, we have $0< \\lambda _{i,j}< 1$ for all $i,j$ .",
"We define as the solution of the following GP.",
"j=0t-h-1 X0,j SONC-simplex = j=0t-h-1Xi,jbi1i<h i=0h-1(Xi,ji,j)i,j=-bh+jj<t-h Following thm:CircuitPolynomialNonnegativity, the second set of constraints would have a “$\\ge $ ”, but for the optimum we always have equality.",
"Hence, we can use the above form, so we have a GP.",
"Assume that is attained at ${}$ and let $\\gamma =-b_0$ .",
"Then ${}$ is a nonnegativity certificate for $p+\\gamma $ in the following way.",
"For $0\\le j<t-h$ let $f_j \\ = \\ \\sum _{i=0}^{h-1} _{i,j} \\cdot {x}^{{\\alpha }(i)} + b_{h+j}{x}^{{\\alpha }(h+j)}.$ Then each $f_j$ is nonnegative due to the second constraint and thm:CircuitPolynomialNonnegativity.",
"Therefore, $p+\\gamma \\ = \\ \\sum _{j=0}^{t-h-1} f_i \\ge 0$ is a sum of nonnegative circuit polynomials.",
"Thus, $-\\gamma $ is a lower bound for $p$ .",
"Regarding the feasibility of the problem problem:sonc we observe: The problem problem:sonc is always feasible.",
"Assume we have some assignment for ${X}$ .",
"Then we get a feasible solution ${X^{\\prime }}$ via $X^{\\prime }_{i,j} &= \\frac{X_{i,j}}{1 - b_i +\\sum _{j=0}^h X_{i,j}} &&\\text{for $i<h$}\\\\X^{\\prime }_{0,j} &= \\left(-b_{h+j}\\cdot \\prod _{i=1}^{h} \\left(\\frac{X^{\\prime }_{i,j}}{\\lambda _i}\\right)^{\\lambda _i}\\right)^{\\frac{1}{\\lambda _0}} &&\\text{for $j<t-h$}$"
],
[
"The SONC-problem for Arbitrary Polynomials",
"In order to compute a lower bound for an arbitrary polynomial $p$ , we cover $(p)$ by simplices and solve a variation of the problem previously discussed.",
"For each ${\\alpha }\\in {p}$ we have to find a simplex spanned by elements of ${p}$ such that ${\\alpha }$ is contained in the interior of this simplex.",
"In this way we cover all elements of ${p}$ with simplices.",
"Note that this covering is not a triangulation in general.",
"Since all vertices of the Newton polytope are monomial squares by assumption (otherwise we have $p^* = -\\infty $ trivially) such a cover exists.",
"Figure: Covering negative terms (red dots) with simplices (colored triangles) whose vertices are monomial squares (black squares).If a vertex occurs in more than one simplex, then its coefficient is split evenly.To compute the cover, we define the following auxiliary program 0 LPHull LPHull(u, p) = v p vv= u v p v= 1 v0vp The cover can be computed in polynomial time using the following algorithm.",
"The idea is illustrated in figure:decomposition.",
"${p}, {p}$ : sets of points in $^n$ $(A^1,\\ldots ,A^)$ : sequence of sets of points, such that each $A^i\\subseteq {p}$ forms the vertices of a simplex; and each ${u}\\in {p}$ appears in the strict interior of at least one $A^i$ .",
"[1] Coverp, p $U := {p}$ uncovered points $k := 0$ $U \\ne \\emptyset $ Choose ${u}\\in U$ $k := k + 1$ ${\\lambda }^* := \\operatorname{LPHull}({u},{p})$ solution vector via simplex method $S := \\lbrace {v} \\in {p}: {\\lambda }^*_{{v}} > 0\\rbrace $ This is a simplex.",
"$A^k := S \\cup ({p} \\cap {(S)})$ $U := U \\setminus A^k$ $(A^1, \\ldots , A^k)$ For the running time, we observe: Solving problem:cover with the simplex method ensures that ${\\lambda }^*$ has at most $n+1$ non-zero entries, so $S$ is a simplex.",
"In theory, however, this algorithm may need exponential time.",
"Alternatively, we can solve the problem:cover using some method with ensured polynomial running time.",
"Then our solution ${\\lambda }^*$ can contain more than $n+1$ non-zero entries and the resulting set $S^{\\prime }=\\lbrace {v} \\in {p}: {\\lambda }^*_{{v}} > 0\\rbrace $ is affinely dependent.",
"In this case, we apply lem:coverreduction, which reduces $S^{\\prime }$ to an affinely independent set $S$ in polynomial time.",
"To check whether some point lies in ${(S)}$ , we have to solve a linear equation system.",
"With QR-factorisation, we can thus compute $A^k$ in $\\mathcal {O}\\left(n^2\\cdot (n+|{p}|)\\right)$ steps.",
"In each iteration of the while-loop we have ${u} \\in {(S)}$ , so at the end ${u} \\in A^k$ holds.",
"Therefore, at least one element is removed from $U$ in every iteration of the loop.",
"Hence, the while-loop has at most $|{p}|$ iterations.",
"Each $A^k$ is a simplex with some interior points.",
"Their vertices correspond to monomial squares of $p$ , and each non-square is contained in at least one simplex.",
"Thus, we have a polynomial time algorithm to cover all non-squares by simplices.",
"algorithm:cover yields a list of sets $A^1,\\ldots ,A^$ , where each $A^k$ represents exponents of a polynomial whose Newton polytope is a simplex.",
"Next we investigate how to split the coefficients in order to obtain nonnegative circuit polynomials.",
"We describe two possible strategies."
],
[
"Even splitting of the coefficients",
"For each $i$ where $b_i x^{{\\alpha }(i)}$ is a monomial square, we count how many simplices contain ${\\alpha }(i)$ and evenly distribute $b_i$ .",
"So we define $b_i^k & \\ = \\ \\frac{b_i}{\\#\\lbrace j:{\\alpha }(i) \\in A^j\\rbrace } &p^k & \\ = {A^k}{{b}^k}$ restricting ${b}^k$ to the exponents occurring in $A^k$ .",
"For each $p^k$ we solve the corresponding GP problem:sonc from sec:simplex.",
"If a polynomial $p^k$ does not contain a constant term, then we adapt the GP problem:sonc and check for feasibility of $\\begin{aligned}\\sum _{j=0}^{t-h-1} X_{i,j}^k & \\ \\le \\ b_i^k && \\text{for $i < h$}\\\\\\prod _{i=0}^{h-1} \\left(\\frac{X_{i,j}^k}{\\lambda _{i,j}^k}\\right)^{\\lambda _{i,j}} & \\ = \\ -b_{h+j}^k && \\text{for $j<t-h$}\\end{aligned}$ with optimum $^k = 0$ if it is feasible and $^k = \\infty $ otherwise.",
"So with $= \\sum _{k=1}^^k$ we get $-$ as a lower bound for the polynomial."
],
[
"Variable splitting of the coefficients",
"For ${\\alpha }(i)\\in {p}$ we compute $b_i^k$ as in eq:split.",
"Then we solve the following GP k=1j=0t-h-1 X0,jk GP-SONC k=1j=0t-h-1 Xi,jk bi1i< h, i=0h-1 (Xi,jki,jk)i,jk = -bh+jkj<t-h, 1k .",
"The meaning of the variable is, that we use $X_{i,j}^k$ weight from $b_i$ to balance the negative weight $b_{h+j}^k$ in the simplex given by $A^k$ .",
"So we fix a cover and a distribution of the negative weights and then search for the optimal distribution of the positive weights.",
"Fixing $b_{h+j}^k$ is necessary in this approach, since if we additionally searched for an optimal distribution of the negative coefficients, the resulting problem would no longer be a GP.",
"The full algorithm to compute a lower bound for a polynomial via SONC is as follows.",
"$p$ : polynomial ${opt}$ : float, such that $p+{opt}$ is a SONC [1] Bound$p$ compute ${p}$ and ${p}$ ${p} \\cap {p} \\ne \\emptyset $ 'unbounded' ${p}$ is simplex $X := $ solution of problem:sonc $(A^1,\\ldots ,A^) = \\operatorname{Cover}({p}, {p})$ $b_i^k := \\frac{b_i}{\\#\\lbrace j:{\\alpha }(i) \\in A^j\\rbrace }$ according to split choice solve problem:soncvarsplit or several of problem:sonc ${opt}:= \\sum _i X_{0,i} - b_{{0}}$ ${opt}$ algorithm:bound runs in polynomial time.",
"As shown in algorithm:cover, we have $\\le t$ .",
"Hence, the size of the GP is polynomially bounded in the input size.",
"Furthermore, computing the convex hull and solving a GP can be done in polynomial time, see subsection:convexhull and [28].",
"Therefore, this algorithm computes a lower bound for a polynomial and requires time polynomial in the input size.",
"Additionally, the algorithm shows that SONC is in particular well-suited for polynomials of high degree.",
"The running time for SONC is independent of the degree in the sense that the following numbers are independent of the degree: the sizes of the LPs to compute the Newton polytope, the sizes of the LPs in algorithm:cover, the number of arithmetic operations, to compute the barycentric coordinates, the problem size of each type of GP we have.",
"The size of both LP:extremal andproblem:cover is in $\\mathcal {O}(nt)$ .",
"We have to compute $(t-h)\\cdot \\le t^2$ tuples of barycentric coordinates, each of size $n+1$ .",
"The largest GP problem:soncvarsplit has size $\\mathcal {O}(h(t-h))\\subseteq \\mathcal {O}(t^3)$ .",
"So none of these depends on $d$ ."
],
[
"Limits of the Algorithm",
"Here, we address cases, where algorithm:bound may fail to find a bound, although the polynomial is bounded.",
"The key concept is a degenerate term, which is defined as a term $b_{{\\alpha }}{x}^{{\\alpha }}\\in {p}$ , such that ${\\alpha }\\in \\partial {p}$ , and in the covering there is a simplex $A^k$ with ${\\alpha }\\in A^k$ but ${0}\\notin A^k$ .",
"Then ${\\alpha }$ is called a degenerate point.",
"The main issue is, that adding weight to the constant term does not influence nonnegativity of the circuit polynomial with support $A^k$ .",
"Hence, we can use at most the given weights of the other positive points in $A^k$ .",
"If some degenerate term forces us to use the entire weight for every positive term, then there might not be enough weight left to even out other negative terms.",
"As example take $p = x^2-2xy + y^2 -2x-2y+1 = (x+y-1)^2\\ge 0$ The only covering is $\\left(\\lbrace (0,2),(1,1),(2,0)\\rbrace , \\lbrace (2,0),(1,0),(0,0)\\rbrace ,\\lbrace (0,2),(0,1),(0,0)\\rbrace \\right)$ .",
"To make the first circuit nonnegative, we have to use the entire coefficients of $x^2$ and $y^2$ , to get the nonnegative circuit polynomial $x^2-2xy+y^2$ .",
"Then however, we have to decompose $-2x-2y+1$ as a SONC, which is not possible, since the expression is unbounded."
],
[
"Experimental Results",
"We discuss the actual physical running time of the algorithms described above.",
"First, we describe the setup of our experiment and explain how our random instances were created.",
"Afterwards, we discuss a few selected examples, which exhibit well the differences of the methods.",
"In the end, we present how the program behaved on a large set of examples."
],
[
"Experimental Setup",
"We give an overview about the experimental setup.",
"Software The entire experiment was steered by our Python based software POEM (Effective Methods in Polynomial Optimization), [33], which we develop since July 2017.",
"POEM is open source, under GNU public license, and freely available at: https://www3.math.tu-berlin.de/combi/RAAGConOpt/poem.html For our experiment, POEM calls a range of further software and solvers for computing both SONC and SOS certificates.",
"For SONC, on the one hand, we use CVXPY [8] with the solver ECOS [9].",
"On the other hand, we solve the problems with Matlab, using CVX [15], calling the solvers SDPT3 [35] and SeDuMi [34].",
"For SOS, we use the established packages YALMIP [27], Gloptipoly [20] and SOSTOOLS [30], the latter both with and without the “sparse” option.",
"Furthermore, we provide our own implementation to construct the SDP for SOS.",
"For Matlab we use CVX with SDPT3 and SeDuMi, and for Python we use CVXPY with CVXOPT [2].",
"Investigated Data The experiment was carried out on a database containing 16921 polynomials with a wide range of variables, terms, degrees, and Newton polytopes.",
"We created the database randomly using POEM.",
"Further details can be found in subsection:generatingpolynomials; the full database of instances is available at the homepage cited above.",
"Hardware and System We used a Intel(R) Core(TM) i7-6700 CPU with 3.4 GHz, 4 cores, 8 threads and 16 GB of RAM under openSUSE Leap 42.3 for our computations.",
"Stopping Criteria For our own methods in Python we used a tolerance of $\\varepsilon =10^{-7}$ .",
"For all computations in Matlab we used the default accuracy $\\varepsilon =1.49\\text{e-8}$ and $1.22\\text{e-4}$ for inaccurate solutions.",
"To avoid memory errors, we call SOS only if the size of the matrix lies below a certain bound.",
"For Python we check $\\binom{n+d}{d}\\le 120$ and for Matlab $\\binom{n+d}{d}\\le 400$ .",
"If $400<\\binom{n+d}{d}\\le 1000$ , we only call SOSTOOLS with the “sparse” option.",
"To avoid excessive run times, we call call SONC with variable split in Matlab only if $t(n+1)<3000$ .",
"We obtained all of these thresholds experimentally.",
"Runtime and Memory The overall running time for all our instances was 28 days and created about 23 GB of data, mainly consisting of the certificates of nonnegativity."
],
[
"Generating Polynomials",
"In what follows we describe how we generated our examples.",
"As parameters for the input size we define by $n$ the number of variables, taking values $n=2,3,4,8,10,20,30,40$ , $d$ the degree, taking values $d=6,8,10,20,30,40,50,60$ , $t$ the number of monomials, taking values $t=6,9,12,20,24,30,50,100,200,300,500$ , ${inner}$ a lower bound for the number of summands whose exponent is not a vertex of the Newton polytope (only for the case “arbitrary” below).",
"We created random polynomials of the following three classes.",
"standard simplex We have ${p} = \\lbrace 0, d\\mathbf {e}_1,\\ldots ,d\\mathbf {e}_n\\rbrace $ and $t-n-1$ many elements in ${p}$ , which we choose randomly using an even distribution over the lattice points of the interior of the scaled standard simplex.",
"simplex We put ${v}_0=0$ and randomly choose by even distribution $n$ points from the scaled standard simplex ${d/2}{n}$ and double their entries.",
"These points form the vertex set ${p} = \\lbrace {v}_0,{v}_1,\\ldots ,{v}_n\\rbrace $ .",
"Afterwards we choose a random ${\\lambda }\\in [0,1]^{n+1}$ and normalize it to $| {\\lambda }|_1 = \\sum _{i = 0}^n \\lambda _i = 1$ .",
"Then we compute ${v}$ as element wise rounding of $\\left(\\sum _{i=0}^n \\lambda _i v_i\\right)$ and check whether ${v}\\in \\operatorname{int}(\\operatorname{conv}\\lbrace v_0,\\ldots ,v_n\\rbrace )$ .",
"If that is the case, then we add ${v}$ to the set ${p}$ .",
"We iterate this process until we reach $\\#{p}=t$ or some threshold of iterations (in which case the generation fails).",
"arbitrary We put ${v}_0=0$ and randomly choose by even distribution $t-{inner}-1$ points from the scaled standard simplex ${d/2}{n}$ and double their entries.",
"All further vertices are chosen as in the simplex case, only as convex combination of $\\lbrace {v}_0,{v}_1,\\ldots ,{v}_{t-{inner}-1}\\rbrace $ (and thus may fail as well, if does not finish after a certain number of iterations).",
"This way, we have ensured, that we have at least ${inner}$ points in ${p}\\setminus {p}$ .. We compute the convex hull of the set of chosen points (trivial in both simplex cases).",
"Let $h$ be the number of vertices of the hull.",
"Finally, we create an array of length $h$ , normally distributed from $\\mathcal {N}\\left(0,\\left(\\frac{t}{n}\\right)^2\\right)$ , take the absolute value of each entry, and concatenate this with an array of length $t-h$ normally distributed from $\\mathcal {N}\\left(0,1\\right)$ .",
"This is the coefficient vector ${b}$ .",
"We chose standard deviation $\\frac{t}{n}$ to keep the lower bounds in a reasonable range.",
"For each combination of parameters $(n,d,t)$ we ran the procedure with 10 different seeds and all three of the above shapes.",
"For the case “arbitrary”, we additionally take ${inner}=\\frac{k}{5}(t-n-1)$ for $k=1,2,3,4$ .",
"In the end, we created 16921 instances.",
"The limit of our implementation lies in the range, the random number generator can handle.",
"In our setup, the methods “simplex” and “arbitrary” fail for $n=40$ and $d=60$ .",
"To pick evenly distributed vectors, we choose random numbers in the range of 0 to $\\binom{40+60}{40}$ , which becomes too large for the underlying C data type to handle."
],
[
"Discussion of Selected Examples",
"Before we present the outcome of the entire experiment, we discuss a couple of chosen examples.",
"In each table, “opt” denotes the optimal value or , according to the chosen strategy SONC or SOS respectively.",
"Since the polynomials discussed in the example:standardsimplex,example:generalnewton,example:dwarfedcube are very huge, we do not explicitly state them in the article.",
"They are available online via https://www3.math.tu-berlin.de/combi/RAAGConOpt/comparison_paper/ [Improved bounds with algorithm:bound] We start with Example 5.5 from [10].",
"$p = 1 + 3\\cdot x_0^{2} x_1^{6} + 2\\cdot x_0^{6} x_1^{2} + 6\\cdot x_0^{2} x_1^{2} - 1\\cdot x_0^{1} x_1^{2} - 2\\cdot x_0^{2} x_1^{1} - 3\\cdot x_0^{3} x_1^{3}$ In this paper, the authors achieve a lower bound $0.5732$ for $p$ using SONC with even split of coefficients and a bound $0.6583$ with a different, arbitrarily chosen split.",
"Using the same cover, as given in the paper, our algorithm:bound yields the optimum ${opt} = -0.693158$ , so we have $0.693158$ as an improved lower bound for $p$ given by SONC.",
"[Range of runtimes of solvers] We consider a polynomial whose Newton polytope is a standard simplex with $n=10$ , $d=30$ , $t=200$ .",
"Running time and results are shown in table:examplestandardsimplex.",
"All three solvers arrive at the same optimum, and ECOS is by far the fastest method.",
"Due to the problem size, we did not attempt to compute a bound using SOS.",
"Table: Standard simplex Newton polytope, n=10n=10, d=30d=30, t=200t=200[Numerically stable via SONC/GP but unstable via SOS/SDP] We consider the following randomly generated polynomial, whose Newton polytope is a non-standard simplex, and which satisfies $n=5$ , $d=8$ , $t=10$ .",
"$p \\ = \\ &1.9450715782850738 + 4.267736494409075\\cdot x_3^{2} x_4^{2} + 0.8128309873524124\\cdot x_2^{8} \\\\&+ 0.3863534030996798\\cdot x_1^{4} x_2^{2} x_3^{2} + 1.5114805777890852\\cdot x_1^{6} x_4^{2} + 1.07826527350598\\cdot x_0^{6} x_1^{2} \\\\&- 0.7496903447028966\\cdot x_0^{1} x_1^{2} x_2^{1} x_3^{1} x_4^{1} + 0.0328087476137118\\cdot x_0^{1} x_1^{3} x_2^{1} x_3^{1} x_4^{1} \\\\&- 2.5827966329699446\\cdot x_0^{1} x_1^{1} x_2^{2} x_3^{1} x_4^{1} - 1.1539503636520094\\cdot x_0^{2} x_1^{1} x_2^{1} x_3^{1} x_4^{1}$ The results of our computation are shown in table:examplesimplex.",
"For SOS, numerical issues occur.",
"Only Sedumi, Gloptipoly, and SOSTOOLS returned a solution.",
"Furthermore, Gloptipoly's solution violates the constraints by more than $\\varepsilon =10^{-7}$ , as denoted by $-1$ in the column “verify”.",
"SOSTOOLS returned an answer, which is far higher than the other two SOS bounds, likely caused by a numerical instability.",
"But the main observation is that SONC does not have any numerical issues and (therefore) yields a dramatically better bound than SOS, which we could obtain with three different solvers.",
"Again, ECOS is by far the fastest solver.",
"Table: example:simplex: The Newton polytope is a simplex; parameters: n=5n=5, d=8d=8, t=10t=10.",
"[A huge polynomial with non-simplex Newton polytope] As in example:standardsimplex we consider a polynomial with $n=10$ , $d=30$ , $t=200$ .",
"Here, however, the Newton polytope is no simplex but has an arbitrary shape.",
"We have the additional parameter ${inner} = 100$ ; i.e., we have 100 monomial squares and 100 possibly negative terms.",
"This increases the difficulty of the problem, since we first have to compute a cover of the Newton polytope with simplices, as described in algorithm:cover.",
"We present our results in table:examplegeneral.",
"In Matlab, splitting the coefficients evenly and solving many instances of the simplex case greatly reduces the running time.",
"For Python the difference is much smaller.",
"As expected, for both solvers the variable split yields a better result.",
"Table: Example : Polynomial with general Newton polytope, n=10n=10, d=30d=30, t=200t=200, inner=100{inner} = 100[A polynomial with combinatorially challenging Newton polytope] The dwarfed cube is a polytope, that exhibits problematic behavior from a combinatorial point of view, in the sense, that it causes many convex hull algorithms to perform badly [1].",
"So we want to see, how our algorithms behave on a polynomial whose Newton polytope is the dwarfed cube.",
"We choose a full support for the polynomial in the sense that every lattice point of the dwarfed cube occurs in the support.",
"We show the results in table:example:dwarfedcube.",
"SOS and SONC differ by less than $1\\%$ , but again SONC with ECOS is the fastest method, this time by a factor 4 compared to the fastest SOS-method YALMIP.",
"Table: Example : Polynomial supported over the full 7-dimensional dwarfed cube, scaled by a factor 4, as support.",
"Corresponding parameters: n=7n=7, d=6d=6, t=113t=113, inner=63{inner} = 63"
],
[
"Evaluation of the Experiment",
"In this section we present and evaluate the results of our experiment.",
"The running time of SONC is independent of the degree This fact was observed in examples in [10] (see also further reference there in).",
"We proved it more formally in theorem:degreeindependent, but it left open the possibility that the degree influences the number of iterations in the LPs or GPs.",
"Here, we confirm experimentally that the running time for SONC is practically not affected by the degree.",
"We investigate our test cases of arbitrary Newton polytope with $n=4$ , $t=20$ , using problem:soncvarsplit solved by ECOS.",
"The results are shown in table:degreeindependent.",
"Table: Timing for n=4n=4, t=20t=20, arbitrary Newton polytope, using ECOS, ordered by time.We see that the running time is independent of the degree.Hence, for the following observations, we mainly regard dependencies on the number of variables and terms, and the shape of the Newton polytope.",
"SONC behavior on the standard simplex We show how SONC behaves on the scaled standard simplex, which is the corresponding Newton polytope in the common approach to investigate (dense) polynomials in $n$ variables of degree $d$ .",
"The results are in presented table:standardsimplex.",
"Even for instances as large as 40 variables and 500 terms, we can solve the problem in a few seconds.",
"Note that we only consider instances with $d > n$ , since otherwise the interior of the Newton polytope is empty.",
"Table: Average running time for SONC with ECOS, where the Newton polytope is the standard simplex; depending on number of terms tt and number of variables nn.A “-” indicates that no such instance exists.A comparison of splitting strategies for SONC We compare the strategy to split coefficients variably as described in subsubsection:variablesplit with the strategy to split coefficients evenly as described in subsubsection:evensplit and as it was done in [10].",
"First note, whenever both strategies found a solution, then the variable split is at least as good as the even split.",
"However, there are 421 cases, where the even split found a solution, but the variable split failed; probably the increased problem size leads to numerical issues.",
"Using ECOS, we cannot verify our initial hypothesis regarding a speed improvement gained by using even split.",
"The quotients of the running time of even splitting divided by the running time of variable splitting range from 0.03 to 3.07, but with an average 1.284.",
"A closer inspection shows that only 1042 instances are solved faster by even splitting, whereas for 7893 instances the variable split is faster.",
"If we distinguish this behavior by the number of terms and the number of variables, then we get the results from table:splitstrategytime.",
"Only for polynomials with many terms, but few variables, we get ratios below 1.",
"So only in these cases, the even splitting is faster on average.",
"Since the bound obtained by the even splitting can be as large as to have an overflow, i.e.",
"numerically infinite, we therefore suggest to use the variable splitting always as first choice.",
"Table: Median quotient of the time “even” over “variable” splitting.Handling degenerate terms In our whole data set, 2413 instances contain degenerate terms.",
"For 37 of these, we could certify, that they are unbounded.",
"In 2279 cases, which means about 94%, we could compute a bound via SONC, despite the presence of degenerate points.",
"So we conclude experimentally, that degenerate terms might cause problems for SONC, but usually we still can compute a bound.",
"Qualitative runtime comparison of SOS and SONC In table:timeSOS, in the first lines of each cell, we show the average running time of the fastest one among the SOS methods which we ran on each of our instances.",
"Already for 8 variables in degree 6 we observe a drastic increase in the running time.",
"Furthermore, in each entry marked with a “$\\times $ ”, the dimension of the psd-matrix in the corresponding SDP would have exceeded $1000\\times 1000$ .",
"During our experiments we observed that problems beyond this bound regularly lead to memory errors, because they require more than 16GB RAM.",
"Even if they do not, we easily have run times of several hours, despite using the “sparse” option of SOSTOOLS.",
"Thus, we did not attempt solving these problems systematically.",
"Table: Average running time of the fastest SOS method (normal font) compared to ECOS with variable split (bold), depending on degree dd and number of variables nn.Note that for higher degree, we generally have more terms.A “×\\times ” indicates, that solving was not attempted, to avoid memory errors.A “-” indicates, we have no solved instance with these parameters.A “ * ^*” means, we only attempted SOSTOOLS with the sparse option.We already observed that the running time of SONC depends on the number of terms, but not on the degree.",
"The running time of SOS, however, depends on the degree primarily, since, in contrast to SONC, computing an SOS certificate does not keep the support invariant.",
"In order to compare both methods we hence proceed as follows: For each pair $(n,d)$ we consider the largest number of terms $t$ , where we created instances, and compute the average running time of SONC only for those instances.",
"Thus, we have a worst-case scenario for SONC in our setting.",
"The results are presented in table:timeSONC in the second lines.",
"Note that for large degrees we can also expect a large Newton polytopes and hence can have more terms in the instances.",
"Thus, we consider our setup to be disadvantageous for SONC especially in these cases.",
"The “-” in the table mean, we do not have instances with arbitrary Newton polytope for these parameters.",
"For $(n,d)=(40,60)$ , we met the limit of our method for generating polynomials, which we explained in subsection:generatingpolynomials.",
"In the other cases, our generation algorithm failed to find enough terms.",
"We make the following two main observations: We can solve far larger and hence far more instances with SONC, than we could solve with SOS.",
"Even in the parameter range, where SOS found a solution, SONC usually is faster (despite the disadvantage for SONC in the setup).",
"To exemplify this difference, we take a close look at the instances satisfying $n=3$ and $d=20$ .",
"We evaluate the average runtimes for varying numbers of terms, as shown in figure:plotn3d20.",
"The running time of SONC starts at 61ms for $t=6$ and reaches 3.56s for $t=500$ .",
"The growth of the runtime is roughly linear in $t$ .",
"On contrast, SOS requires more than 5s, even for few terms, going up to around 25s.",
"Additionally, we observe, that SOS already exploited sparsity to some extent; for fewer terms, we can expect a smaller Newton polytope, which results in smaller SDPs.",
"Figure: Average running time of SONC and SOS for instances with n=3n=3 and d=20d=20, depending on the number of terms.Quantitative comparison of SONC and SOS For every single instance, we could solve with SONC, ECOS was the fastest method, unless it failed to find a solution at all.",
"The latter happened in only 195 instances.",
"Since such a failure, however, only takes few seconds or even milliseconds, a user can quickly switch to another method like Matlab/SeDuMi.",
"Overall, among our 16921 instances, we found a SONC bound for 16240 of them.",
"Only in 4 cases, the fastest SOS method found a bound faster than SONC.",
"In all of these, ECOS failed, so we found the SONC bound with Matlab/SeDuMi, which is considerably slower.",
"Even if we restrict ourselves to SONC with variable splitting of the coefficients, there are still only 15 instances where SOS was quicker in finding a solution.",
"Since these were of different sizes, we could not find a pattern among these 15 instances.",
"The largest difference and the best ratio for SOS regarding the running times was 5.6 seconds to 33.3 seconds.",
"An overview of the ratios of the running times between SOS and SONC is shown in the right graphic of figure:plotbardifference.",
"In about half of the cases, where both algorithms found a solution, SONC was faster by a factor of at least 4.",
"However, there are instances with a speedup of more than 1000, and the largest ratio is around 20,000.",
"In the left graphic of figure:plotbardifference we list the differences between the optima, obtained by SONC and SOS.",
"When both algorithms found a bound, then in most cases SOS found the better bound, but sometimes the bound of SONC was better.",
"But the main observation is that there were only 44 instances, where SOS found a bound and SONC failed.",
"Contrary to this, we have 11760 polynomial where SONC succeeded but SOS could not find a bound, or would have exceeded our thresholds, see subsection:experimental:setup.",
"Of the 3693 instances, where SOS found a better bound than SONC, 782 instances (21.2%) have the scaled standard simplex as Newton polytope.",
"Since the scaled standard simplex is an $H$ -simplex, we have by construction that $\\le $ in these cases, see [21].",
"In addition, there are 501 instances, which are sums of monomial squares and 136 instances, where both algorithms failed.",
"Out of these at least 37 instances are unbounded.",
"For the remaining 99 the status is unknown, since checking whether a polynomial is bounded is -hard on its own, see subsection:SOS."
],
[
"Resume and Outlook",
"This article contains of two main contributions.",
"First, we present a new algorithm, including implementation, to compute lower bounds for multivariate polynomials using SONC certificates.",
"This is the first implementation of this kind and it is the first algorithm to compute such a bound based on nonnegative circuit polynomials, where after giving the polynomial, even for arbitrary Newton polytope, no further human interaction is required.",
"Furthermore, this algorithm computes a lower bound for a sparse polynomial in polynomial time.",
"Second, we provide an experimental comparison of SONC (using our algorithm and GP) solvers and SOS (using various standard SDP solvers).",
"Based on these experiments, we draw the following key conclusions: The runtime for SONC certificates depends on the number of variables and the cardinality of the support only.",
"It does not depend on the degree.",
"Among the two strategies for splitting coefficients introduced in subsection:arbitrarynewton, the variable splitting strategy yields always better bounds and is, to our surprise, mostly faster than the even splitting strategy.",
"Among the tested GP solvers, ECOS is by far the best one for computing SONC certificates.",
"We remark, however, that Mosek announced to add an exponential cone solver in its next release, which we have not tested yet.",
"SONC certificates can be computed extremely fast.",
"Even for very large instances, the runtime remains in the realm of seconds.",
"In a direct comparison of those cases where both SONC and SOS yield a bound, SONC is faster than SOS in over $\\mathbf {99.5\\%}$ of all cases.",
"If both strategies yield a bound, then SOS yields better bounds than the current SONC algorithm in the clear majority ($82.5\\%$ ) of the cases.",
"However, often the bounds do not differ very much, and there is a nontrivial amount of cases, when the SONC bound is better.",
"See figure:plotbardifference for details.",
"SONC is extremely robust.",
"Even for the majority of instances with degenerate terms, a potential source of problems for SONC, we were able to compute bounds.",
"Moreover it uses vastly less memory than SOS.",
"While in our experiments (for 16 GB RAM) SOS certificates can be computed up to corresponding SDPs with matrices of size roughly $1000 \\times 1000$ (these are e.g., instances of type $n=4$ , $d=20$ ), SONC easily handles cases of 40 variables and 500 terms.",
"For unconstrained polynomial optimization, we summarize our findings as follows: In a standard setting, it is, due to the short runtime and the little memory requirements, almost always best to compute a SONC certificate first, and then, depending on the size of the problem, try to compute an SOS certificate afterwards.",
"If an SOS certificate is computable, then often, but not always, SOS yields better bounds than SONC.",
"Especially for large, sparse instances, SONC is the certificate of choice.",
"Large instances with many terms in the boundary are problematic for both SOS and SONC.",
"In its current stage, SONC certificates are not ready for most applications, yet.",
"The main problems to overcome are, in our opinion: Improve the SONC theory of constrained optimization introduced in [10], [11], relate it to Chandrasekaran's and Shah's SAGE certificates [7], and implement constrained optimization in POEM.",
"Develop and implement a good way to compute minimizers exploiting a given SONC decomposition in order to obtain upper bounds and to compute the duality gap.",
"Improve our algorithms (e.g., with respect to finding optimal coverings or carrying out preprocessing steps) to achieve better lower bounds via SONC.",
"We will tackle these issues in future publications.",
"The main challenge here is how this approach can be refined without losing polynomial running time in theory, and without vastly increasing the runtime and the required memory in practice."
]
] | 1808.08431 |
[
[
"A fractional notion of length and an associated nonlocal curvature"
],
[
"Abstract Here a new notion of fractional length of a smooth curve, which depends on a parameter $\\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years.",
"It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant.",
"Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve.",
"This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature."
],
[
"Background",
"The origins of fractional perimeter and nonlocal curvature began with the work of Caffarelli, Roquejoffre, and Savin [4] who defined, up to a multiplicative constant, the $\\sigma $ -perimeter, for $0<\\sigma <1$ , of a measurable set $E\\subseteq \\mathbb {R}^n$ relative to an open, bounded set $\\Omega \\subseteq \\mathbb {R}^n$ by $\\text{Per}_\\sigma (E,\\Omega ):=\\mathcal {I}(E\\cap \\Omega ,E^c\\cap \\Omega )+{\\cal I}(E\\cap \\Omega ,E^c\\cap \\Omega ^c)+{\\cal I}(E\\cap \\Omega ^c,E^c\\cap \\Omega ),$ where ${\\cal I}(A,B):=\\frac{1}{\\alpha _{n-1}}\\int _A\\int _B |x-y|^{-n-\\sigma }dxdy,\\qquad A\\cap B=\\emptyset ,$ and $\\alpha _{n-1}$ is the volume of the unit ball in $\\mathbb {R}^{n-1}$ .",
"In the case where $E$ is contained in $\\Omega $ , the $\\sigma $ -perimeter is related to the $H^{\\sigma /2}$ -norm of the characteristic function of $E$ .",
"It is known [5] that if the boundary of $E$ is smooth, then $\\lim _{\\sigma \\uparrow 1} (1-\\sigma )\\text{Per}_\\sigma (E,B_r)={\\cal H}^{n-1}(\\partial E\\cap B_r)$ for almost every $r>0$ , where $B_r$ is the ball centered at the origin of radius $r$ .",
"A set $E\\subseteq \\mathbb {R}^n$ is a minimizer of the $\\sigma $ -perimeter relative to $\\Omega $ if over all measurable sets $F\\subseteq \\mathbb {R}^n$ such that $E\\setminus \\Omega =F\\setminus \\Omega $ we have $\\text{Per}_\\sigma (E,\\Omega )\\le \\text{Per}_\\sigma (F,\\Omega ).$ Besides the relation (REF ), it is known that the $\\sigma $ -perimeter functional $\\Gamma $ -converges to the classical notion of perimeter [3].",
"If the boundary of a minimizer $E$ is sufficiently regular, then it must satisfy $\\int _{\\mathbb {R}^n}\\frac{\\tilde{\\chi }_E(x)}{|z-x|^{n+\\sigma }} dx=0\\qquad \\text{for all}\\ z\\in \\partial E,$ where $\\tilde{\\chi }_E:=\\chi _E-\\chi _{E^c}$ , $\\chi _E$ is the characteristic function for the set $E$ , and this integral is taken in the principle-value sense.",
"Because of the connection between the $\\sigma $ -perimeter and the areal measure (REF ), and the fact that surfaces that minimize their area subject to a fixed boundary condition must have zero mean curvature, it is reasonable to define a nonlocal mean-curvature by $H_\\sigma (z):=\\frac{1}{\\omega _{n-2}}\\int _{\\mathbb {R}^n}\\frac{\\tilde{\\chi }_E(x)}{|z-x|^{n+\\sigma }} dx\\qquad \\text{for all}\\ z\\in \\partial E,$ where $\\omega _{n-2}$ is the area of the unit sphere in $\\mathbb {R}^{n-2}$ .",
"Notice that this quantity is independent of $\\Omega $ and, hence, well-defined for any point on the surface that is the boundary of the set $E$ .",
"Assuming that $\\partial E$ is smooth, this curvature converges to the classical mean-curvature [1] in the following sense: $\\lim _{\\sigma \\uparrow 1}(1-\\sigma )H_\\sigma (z)=H(z).$ The minimizers of the $\\sigma $ -perimeter functional, called $\\sigma $ -minimal surfaces, have been studied in great detail in recent years.",
"The regularity of $\\sigma $ -minimal surfaces has been investigated by Valdinoci and collaborators [6], [18], [12], [16].",
"Among other things, it is known that $\\sigma $ -minimal surfaces are smooth off of a singular set of dimension at most $n-8$ for $\\sigma $ sufficiently close to 1.",
"While this is in agreement with a well-known result for classical minimal surfaces [13], $\\sigma $ -minimal surfaces may have features different from their classical counterparts, in that they may stick to the boundary instead of being orthogonal to it [10], [18].",
"The motion of surfaces by nonlocal mean-curvature has been investigated using level set methods [7], [8], [9], [14]."
],
[
"Extension and motivation",
"The above discussion of nonlocal mean-curvature applies to surfaces that are the boundary of a set.",
"However, Paroni, Podio-Guidugli, and Seguin discovered that it is possible to define these concepts for any smooth (hyper)surface [15].",
"The main idea is to define a fractional notion of area and find a condition similar to (REF ) that a minimizer of this functional must satisfy.",
"Towards this end, they first showed that for a bounded set $E$ with smooth boundary and bounded, open $\\Omega $ containing $E$ one can write $\\text{Per}_\\sigma (E,\\Omega )= \\frac{1}{\\alpha _{n-1}} \\int _E\\int _{E^c} |x-y|^{-n-\\sigma }dxdy = \\frac{1}{2\\alpha _{n-1}}\\int _{{\\cal X}(\\partial E)}|x-y|^{-n-\\sigma }dxdy,$ where ${\\cal X}(\\partial E)$ is the set of all pairs $(x,y)\\in \\mathbb {R}^n\\times \\mathbb {R}^n$ such that the oriented line segment connecting $x$ to $y$ crosses $\\partial E$ an odd number of times.",
"The validity of (REF ) follows from the fact that ${\\cal X}(\\partial E)$ and $(E\\times E^c)\\cup (E^c\\times E)$ agree up to a set of ${\\cal H}^{2n}$ -measure zero.",
"As the far right-hand side of (REF ) only depends on the surface $\\partial E$ , and not the set $E$ , this motivates the following definition of $\\sigma $ -area for a smooth, compact surface ${\\cal S}$ : $\\text{Area}_\\sigma ({\\cal S},\\Omega ):=\\frac{1}{2\\alpha _{n-1}}\\int _{{\\cal X}({\\cal S})}|x-y|^{-n-\\sigma }\\max \\lbrace \\chi _\\Omega (x),\\chi _\\Omega (y)\\rbrace dxdy,$ where it is assumed that ${\\cal S}$ is contained in $\\Omega $ .",
"The presence of $\\max \\lbrace \\chi _\\Omega (x),\\chi _\\Omega (y)\\rbrace $ in the integrand is necessary to ensure the integral converges.",
"In this way, it is similar to the role $\\Omega $ plays in the definition of the $\\sigma $ -perimeter.",
"The $\\sigma $ -area satisfies a limit relationship analogous to (REF ).",
"It was shown [15] that if ${\\cal S}$ minimizes the $\\sigma $ -area relative to all smooth, compact, oriented surfaces in $\\Omega $ that have the same boundary as ${\\cal S}$ , then ${\\cal S}$ must satisfy $\\int _{{\\cal A}_i(z)}|z-y|^{-n-\\sigma }dy-\\int _{{\\cal A}_e(z)}|z-y|^{-n-\\sigma }dy=0\\quad {\\rm for \\ all}\\ z\\in {\\cal S},$ where ${\\cal A}_e(z)&:=\\big \\lbrace y\\in \\mathbb {R}^n\\ |\\ \\big ((z,y)\\in {\\cal X}({\\cal S})\\ \\text{and}\\ (z-y)\\cdot {\\bf n}(z)> 0\\big )\\\\&\\hspace{72.26999pt}\\text{or } \\big ((z,y)\\in {\\cal X}({\\cal S})^c\\ \\text{and}\\ (z-y)\\cdot {\\bf n}(z)< 0\\big )\\big \\rbrace ,\\\\{\\cal A}_i(z)&:=\\big \\lbrace y\\in \\mathbb {R}^n\\ |\\ \\big ((z,y)\\in {\\cal X}({\\cal S})^c\\ \\text{and}\\ (z-y)\\cdot {\\bf n}(z)> 0\\big )\\\\&\\hspace{72.26999pt}\\text{or } \\big ((z,y)\\in {\\cal X}({\\cal S})\\ \\text{and}\\ (z-y)\\cdot {\\bf n}(z)< 0\\big )\\big \\rbrace .$ See Figure REF for a depiction of these sets.",
"Figure: The solid line depicts 𝒮{\\cal S}.",
"The set of points of density 1 for 𝒜 e (z){\\cal A}_e(z) is shown in light grey, and the set of points of density 1 for 𝒜 i (z){\\cal A}_i(z) is in dark grey.",
"The dashed lines depict the part of the essential boundary between these sets that is not part of 𝒮{\\cal S}.This motivates defining the nonlocal mean-curvature of ${\\cal S}$ , a smooth, oriented surface that need not be compact, at $z$ using the left-hand side of (REF )—that is, $H_s(z):=\\frac{1}{\\omega _{n-2}} \\int _{\\mathbb {R}^n} \\frac{\\hat{\\chi }_{\\cal S}(z,y)}{|z-y|^{n+\\sigma }}dy\\quad {\\rm for \\ all}\\ z\\in {\\cal S},$ where $\\hat{\\chi }(z,y):={\\left\\lbrace \\begin{array}{ll}1 & y\\in {\\cal A}_i(z),\\\\0 & y\\notin {\\cal A}_i(z)\\cup {\\cal A}_e(z),\\\\-1 & y\\in {\\cal A}_e(z).\\end{array}\\right.",
"}$ Notice that $H_\\sigma $ does not depend on $\\Omega $ .",
"Unsurprisingly, this curvature satisfies the limit relation (REF ).",
"To motivate a definition of fractional length we will consider the $\\sigma $ -area in two dimensions, where a hypersurface is a one-dimensional curve.",
"When $n=2$ , the $\\sigma $ -area becomes $\\text{Area}_\\sigma ({\\cal S},\\Omega )=\\frac{1}{4} \\int _{{\\cal X}({\\cal S})}\\frac{\\max \\lbrace \\chi _\\Omega (x),\\chi _\\Omega (y)\\rbrace }{|x-y|^{2+\\sigma }}dxdy.$ The domain of integration here consists of line segments that are described by their end points.",
"A given line segment connecting $x$ to $y$ can be viewed as a one-dimensional disc and hence can be described by its midpoint $p$ , a unit vector ${\\bf u}$ normal to the disc, and a radius $r$ so that $(x,y)=(p-r{\\bf u}^{\\prime },p+r{\\bf u}^{\\prime }),$ where ${\\bf u}^{\\prime }$ is obtained by rotating ${\\bf u}$ clockwise by $90^\\circ $ .",
"Utilizing this change of variables, (REF ) can be rewritten in the $n=2$ case as $\\text{Area}_s({\\cal S},\\Omega )=\\frac{1}{4} \\int _{{\\cal D}({\\cal S})}(2r)^{-1-\\sigma }\\max \\lbrace \\chi _\\Omega (p-r{\\bf u}^{\\prime }),\\chi _\\Omega (p+r{\\bf u}^{\\prime })\\rbrace d{\\cal H}^4(p,{\\bf u},r),$ where ${\\cal D}({\\cal S})$ consists of all triples $(p,{\\bf u},r)$ describing those one-dimensional discs that intersect ${\\cal S}$ an odd number of times.",
"It is this formula for the fractional length that can be generalized to a curve in $n$ dimensions.",
"Before this generalization is done, we first study the measure theoretic properties of the set of all discs that intersect a curve an odd number of times and other related sets of discs in Section .",
"In Section the fractional length is defined and it is shown that it converges, in an appropriate limit, to the classical notion of length up to a multiplicative constant.",
"Next, Section is dedicated to computing the Euler–Lagrange equation associated with the fractional length and the result is used to motivate a definition of nonlocal curvature for a curve.",
"The Appendix contains several change of variables formulas based on the area formula that are useful in established the desired results."
],
[
"Sets of discs",
"Here the set of all $(n-1)$ -dimensional discs, and various subsets of it, are studied in $\\mathbb {R}^n$ , with $n>2$ .",
"The results established here make precise which discs are integrated over in the definition of the nonlocal length.",
"Moreover, they will be crucial in computing the variation of the nonlocal length.",
"We use ${\\cal U}_n$ to denote the set of unit vectors in $\\mathbb {R}^n$ , and set ${\\cal U}_\\perp ^2:=\\lbrace ({\\bf a},{\\bf b})\\in {\\cal U}_n\\times {\\cal U}_n\\ |\\ {\\bf a}\\cdot {\\bf b}=0\\rbrace .$ The $(n-1)$ -dimensional disc with center $p$ , normal unit vector ${\\bf u}$ , and a radius $r$ is denoted by $D(p,{\\bf u},r):=\\lbrace p+\\xi {\\bf v}\\ |\\ ({\\bf u},{\\bf v})\\in {\\cal U}_\\perp ^2,\\ \\xi \\in [0,r)\\rbrace .$ By the boundary $\\partial D(p,{\\bf u},r)$ of one of these discs we mean the $(n-2)$ -dimensional manifold $\\lbrace p+r{\\bf v}\\in \\mathbb {R}^n\\ |\\ {\\bf v}\\in {\\cal U}_n\\cap \\lbrace {\\bf u}\\rbrace ^\\perp \\rbrace ,$ where $\\lbrace {\\bf u}\\rbrace ^\\perp $ is the set of all vectors orthogonal to ${\\bf u}$ .",
"Given a $C^1$ , compact one-dimensional manifold ${\\cal C}$ , let ${\\bf t}(z)$ denote a unit tangent vector to ${\\cal C}$ at $z$ .",
"Consider the following subsets of the set of all discs ${\\cal D}:=\\mathbb {R}^n\\times {\\cal U}_n\\times \\mathbb {R}^+$ : ${\\cal D}_{\\text{tan}}&:= \\lbrace (p,{\\bf u},r)\\in {\\cal D}\\ |\\ \\text{there is a } z\\in \\bar{D}(p,{\\bf u},r)\\cap {\\cal C}\\ \\text{such that}\\ {\\bf t}(z)\\cdot {\\bf u}=0\\rbrace ,\\\\{\\cal D}_\\infty &:= \\lbrace (p,{\\bf u},r)\\in {\\cal D}\\ |\\ {\\cal H}^0(\\bar{D}(p,{\\bf u},r)\\cap {\\cal C})=\\infty \\rbrace ,\\\\{\\cal D}_{\\partial 1}&:= \\lbrace (p,{\\bf u},r)\\in {\\cal D}\\backslash {\\cal D}_\\text{tan}\\ |\\ {\\cal H}^0(\\partial D(p,{\\bf u},r)\\cap {\\cal C})=1\\rbrace ,\\\\{\\cal D}_{\\partial 2}&:= \\lbrace (p,{\\bf u},r)\\in {\\cal D}\\backslash {\\cal D}_\\text{tan}\\ |\\ {\\cal H}^0(\\partial D(p,{\\bf u},r)\\cap {\\cal C})\\ge 2\\rbrace ,\\\\{\\cal D}_{\\partial }&:= {\\cal D}_{\\partial 1}\\cup {\\cal D}_{\\partial 2},\\\\{\\cal D}_\\text{odd}&:=\\lbrace (p,{\\bf u},r)\\in {\\cal D}\\backslash ({\\cal D}_{\\rm tan}\\cup {\\cal D}_\\partial )\\ |\\ {\\cal H}^0(D(p,{\\bf u},r)\\cap {\\cal C}) \\text{ is an odd number}\\rbrace ,\\\\{\\cal D}_\\text{even}&:=\\lbrace (p,{\\bf u},r)\\in {\\cal D}\\backslash ({\\cal D}_{\\rm tan}\\cup {\\cal D}_\\partial )\\ |\\ {\\cal H}^0(D(p,{\\bf u},r)\\cap {\\cal C}) \\text{ is an even number}\\rbrace .$ The following lemma discusses the measure theoretic properties of these sets.",
"Lemma 2.1 The following facts are true: ${\\cal D}_\\infty \\subseteq {\\cal D}_\\text{\\rm tan}$ , ${\\cal H}^{2n-1}({\\cal D}_\\partial \\cap {\\cal E})<\\infty $ for any bounded, open set ${\\cal E}\\subseteq {\\cal D}$ , ${\\cal H}^{n+2}({\\cal D}_\\text{\\rm tan})=0$ , ${\\cal H}^{2n-1}({\\cal D}_{\\partial 2})=0$ , ${\\cal D}_\\text{\\rm even}$ and ${\\cal D}_\\text{\\rm odd}$ are open subsets of ${\\cal D}$ , ${\\cal D}={\\cal D}_\\text{\\rm odd}\\cup {\\cal D}_\\text{\\rm even}\\cup {\\cal D}_\\partial \\cup {\\cal D}_{\\rm tan}$ , where this is a disjoint union.",
"Item 1) Consider $(p,{\\bf u},r)\\in {\\cal D}_\\infty $ , so that there are an infinite number of points in $\\bar{D}(p,{\\bf u},r)\\cap {\\cal C}$ .",
"Since ${\\cal C}$ is compact it follows that this intersection has a cluster point, say $z\\in \\bar{D}(p,{\\bf u},r) \\cap {\\cal C}$ .",
"Suppose that $(p,{\\bf u},r)\\notin {\\cal D}_\\text{tan}$ so that ${\\bf t}(z)\\cdot {\\bf u}\\ne 0$ .",
"It follows that there is a neighborhood of $z$ such that in this neighborhood the curve ${\\cal C}$ can be approximated by a straight line through $z$ with direction ${\\bf t}(z)$ .",
"This implies that there are no points in this neighborhood besides $z$ in the intersection $\\bar{D}(p,{\\bf u},r)\\cap {\\cal C}$ .",
"This would contradict the fact that $z$ is a cluster point of $\\bar{D}(p,{\\bf u},r) \\cap {\\cal C}$ .",
"Thus, we must have $(p,{\\bf u},r)\\in {\\cal D}_\\text{tan}$ .",
"Item 2) Let ${\\cal E}$ be a bounded, open subset of ${\\cal D}$ .",
"Find $R>0$ such that if $(p,{\\bf u},r)\\in {\\cal E}$ , then $r\\in (0,R]$ .",
"Consider the set ${\\cal A}:={\\cal C}\\times {\\cal U}_\\perp ^2\\times (0,R]$ and the function $\\Psi :{\\cal A}\\rightarrow {\\cal D}$ defined by (REF ) in the Appendix.",
"Notice that ${\\cal D}_\\partial \\cap {\\cal E}\\subseteq \\Psi ({\\cal A})$ .",
"Since $\\Psi $ is Lipschitz on ${\\cal A}$ and ${\\cal H}^{2n-1}({\\cal A})<\\infty $ , it follows that ${\\cal H}^{2n-1}({\\cal D}_\\partial \\cap {\\cal E})<\\infty $ .",
"Item 3) Similar to the proof of Item 2, consider the set ${\\cal A}:=\\bigcup _{z\\in {\\cal C}} \\lbrace z\\rbrace \\times \\lbrace {\\bf t}(z),-{\\bf t}(z)\\rbrace \\times \\big ({\\cal U}_n\\cap \\lbrace {\\bf t}(z)\\rbrace ^\\perp \\big )\\times \\mathbb {R}^+\\times \\mathbb {R}^+.$ One can show that ${\\cal D}_{\\text{tan}}\\subseteq \\Xi ({\\cal A})$ , where $\\Xi $ is defined in (REF ) of the Appendix.",
"Since ${\\cal H}^{n-2}({\\cal U}_n\\cap \\lbrace {\\bf t}(z)\\rbrace ^\\perp )<\\infty $ , we have ${\\cal H}^{n+2}({\\cal A})=0$ .",
"Thus, since $\\Xi $ is locally Lipschitz, ${\\cal H}^{n+2}({\\cal D}_\\text{tan})=0$ .",
"Item 4) Consider the set ${\\cal G}:=\\lbrace (z_1,z_2,{\\bf a},{\\bf b})\\in {\\cal C}\\times {\\cal C}\\times {\\cal U}^2_\\perp \\ |\\ {\\bf a}\\cdot (z_2-z_1)>0,\\, {\\bf b}\\cdot (z_2-z_1)=0\\rbrace .$ and define the function $H:{\\cal G}\\rightarrow {\\cal D}$ by $H(z_1,z_2,{\\bf a},{\\bf b}):=(z_1+\\frac{|z_2-z_1|^2}{2{\\bf a}\\cdot (z_2-z_1)}{\\bf a},{\\bf b},\\frac{|z_2-z_1|^2}{2{\\bf a}\\cdot (z_2-z_1)}).$ One can check that the boundary of the disc $D(F(z_1,z_2,{\\bf a},{\\bf b}))$ intersects ${\\cal C}$ at $z_1$ and $z_2$ .",
"Thus, ${\\cal D}_{\\partial 2}\\subseteq H({\\cal G})$ .",
"Moreover, $H$ is locally Lipschitz on ${\\cal G}$ .",
"It follows that since ${\\cal H}^{2n-1}({\\cal G})=0$ , we must have ${\\cal H}^{2n-1}({\\cal D}_{\\partial 2})=0$ .",
"Item 5) This is clear from the definitions of ${\\cal D}_\\text{even}$ and ${\\cal D}_\\text{odd}$ .",
"Item 6) This follows from Item 1 and the definition of the various sets involved.",
"The previous result yields enough information to obtain the properties of ${\\cal D}_\\text{odd}$ we require.",
"Proposition 2.2 The set ${\\cal D}_\\text{\\rm odd}$ is locally of finite perimeter.",
"Moreover, the essential boundaryFor the definition of sets of finite perimeter and essential boundary see, for example, Ambrosio, Fusco, and Pallara [2].",
"$\\partial ^*{\\cal D}_\\text{\\rm odd}$ of this set coincides with the set ${\\cal D}_{\\partial 1}$ up to a set of ${\\cal H}^{2n-1}$ -measure zero.",
"From Items 5 and 6 of Lemma REF , we see that $\\partial ^* {\\cal D}_\\text{odd}\\subseteq {\\cal D}_\\partial \\cup {\\cal D}_\\text{tan}$ .",
"Thus, from Items 2–4 of the same lemma we have for $n>2$ ${\\cal H}^{2n-1}(\\partial ^*{\\cal D}_\\text{odd}\\cap {\\cal E})\\le {\\cal H}^{2n-1}(({\\cal D}_\\partial \\cup {\\cal D}_\\text{tan})\\cap {\\cal E})\\le {\\cal H}^{2n-1}({\\cal D}_{\\partial 1}\\cap {\\cal E})<\\infty $ whenever ${\\cal E}\\subseteq {\\cal D}$ is a bounded, open set.",
"By a result of Federer, see 4.5.11 of [11], we can conclude that ${\\cal D}_\\text{odd}$ has finite perimeter in ${\\cal E}$ .",
"Thus, ${\\cal D}_\\text{odd}$ is locally of finite perimeter.",
"Moreover, it is known, see Ambrosio, Fusco, and Pallara [2] Theorem 3.61, that it follows that ${\\cal D}_\\text{odd}$ has density either 0, $1/2$ , or 1 at ${\\cal H}^{2n-1}$ -a.e.",
"point of ${\\cal E}$ , and $\\partial ^*{\\cal D}_\\text{odd}\\cap {\\cal E}$ consists of those points with density $1/2$ up to a set of ${\\cal H}^{2n-1}$ -measure zero.",
"Thus, using Items 5 and 6 from the lemma again, we conclude that $\\partial ^*{\\cal D}_\\text{odd}\\cap {\\cal E}$ and $({\\cal D}_\\partial \\cup {\\cal D}_\\text{tan})\\cap {\\cal E}$ agree up to a set of ${\\cal H}^{2n-1}$ -measure zero.",
"Finally, since $n>2$ , Items 3 and 4 of the lemma yields ${\\cal H}^{2n-1}({\\cal D}_\\text{tan}\\cup {\\cal D}_{\\partial 2})=0$ , and so $\\partial ^*{\\cal D}_\\text{odd}\\cap {\\cal E}$ and ${\\cal D}_{\\partial 1}\\cap {\\cal E}$ agree up to a set of ${\\cal H}^{2n-1}$ -measure zero.",
"Since this holds for any open bounded ${\\cal E}$ , this establishes the result.",
"Since ${\\cal D}_\\text{odd}$ is locally a set of finite perimeter, it has an exterior unit normal at ${\\cal H}^{2n-1}$ -a.e.",
"point of its essential boundary.",
"The next result describes this normal vector.",
"Proposition 2.3 For ${\\cal H}^{2n-1}$ -a.e.",
"$(p,{\\bf u},r)\\in \\partial ^*{\\cal D}_\\text{\\rm odd}$ there exists a unique $z\\in \\partial D(p,{\\bf u},r)\\cap {\\cal C}$ .",
"Moreover, for such $(p,{\\bf u},r)$ the exterior unit-normal $\\nu (p,{\\bf u},r)\\in \\mathbb {R}^n\\times \\lbrace {\\bf u}\\rbrace ^\\perp \\times \\mathbb {R}$ is given by $\\nu (p,{\\bf u},r):={\\left\\lbrace \\begin{array}{ll}\\mathbf {(}p,{\\bf u},r) & \\text{if } {\\cal H}^0(D(p,{\\bf u},r)\\cap {\\cal C}) \\text{ is an odd number}\\\\-\\mathbf {(}p,{\\bf u},r) & \\text{if } {\\cal H}^0(D(p,{\\bf u},r)\\cap {\\cal C}) \\text{ is an even number},\\end{array}\\right.",
"}$ where $\\mathbf {(}p,{\\bf u},r):=\\frac{\\Big ( z-p+\\frac{(p-z)\\cdot {\\bf t}}{{\\bf u}\\cdot {\\bf t}}{\\bf u}, \\frac{(p-z)\\cdot {\\bf t}}{{\\bf u}\\cdot {\\bf t}}(p-z),r\\Big )}{\\sqrt{ |p-z|^2 + \\Big (\\frac{(p-z)\\cdot {\\bf t}}{{\\bf u}\\cdot {\\bf t}}\\Big )^2|z-p|^2 + \\Big (\\frac{(p-z)\\cdot {\\bf t}}{{\\bf u}\\cdot {\\bf t}}\\Big )^2 + r^2}}$ and $z$ is the unique point in $\\partial D(p,{\\bf u},r)\\cap {\\cal C}$ and ${\\bf t}$ is tangent to ${\\cal C}$ at $z$ .",
"Begin by noticing that ${\\cal D}_\\partial =\\Psi ({\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+)$ , where $\\Psi $ is the function defined in (REF ) of the Appendix.",
"This means that ${\\cal D}_\\partial $ is an immersed submanifold of ${\\cal D}$ .",
"Moreover, the function $\\Psi $ is an embedding on the preimage of ${\\cal D}_{\\partial 1}$ under $\\Psi $ , and so ${\\cal D}_{\\partial 1}$ is a submanifold of ${\\cal D}$ .",
"From Proposition REF , $\\partial ^*{\\cal D}_\\text{\\rm odd}$ coincides with the set ${\\cal D}_{\\partial 1}$ up to a set of ${\\cal H}^{2n-1}$ -measure zero and hence is ${\\cal H}^{2n-1}$ -rectifiable.",
"It follows that the approximate tangent space to $\\partial ^*{\\cal D}_\\text{\\rm odd}$ , where it exists, coincides with the tangent space of ${\\cal D}_{\\partial 1}$ .",
"Thus, to calculate the exterior unit-normal on $\\partial ^*{\\cal D}_\\text{odd}$ , we first find this tangent space.",
"Let $(p,{\\bf u},r)\\in {\\cal D}_{\\partial 1}$ and find the unique $(z,{\\bf a},{\\bf b},r)\\in {\\cal C}\\times {\\cal U}_\\perp ^2 \\times \\mathbb {R}^+$ that gets mapped to $(p,{\\bf u},r)$ under $\\Psi $ .",
"Denote by ${\\bf t}$ a tangent vector to ${\\cal C}$ at $z$ .",
"Since $(p,{\\bf u},r)\\notin {\\cal D}_\\text{tan}$ , we must have ${\\bf b}\\cdot {\\bf t}\\ne 0$ .",
"A curve in ${\\cal C}\\times {\\cal U}_\\perp ^2 \\times \\mathbb {R}^+$ that passes through $(z,{\\bf a},{\\bf b},r)$ induces, via the mapping $\\Psi $ , a curve in ${\\cal D}_{\\partial 1}$ that passes through $(p,{\\bf u},r)$ .",
"By differentiating such curves we can generate vectors in the tangent space of ${\\cal D}_{\\partial 1}$ at $(p,{\\bf u},r)$ .",
"In particular, one can find that the following vectors are in the tangent space: $({\\bf t},{\\bf 0},0),\\quad ({\\bf c},{\\bf 0},0),\\quad ({\\bf 0},{\\bf d},0),\\quad ({\\bf a},0,1),\\quad (r{\\bf b},-{\\bf a},0),$ where ${\\bf c}$ and ${\\bf d}$ are any vectors orthogonal to both ${\\bf a}$ and ${\\bf b}$ .",
"This generates a list of $2n-1$ linearly independent vectors since ${\\bf b}\\cdot {\\bf t}\\ne 0$ .",
"Thus, these vectors span the tangent space at $(p,{\\bf u},r)$ .",
"A vector in $\\mathbb {R}^n\\times \\lbrace {\\bf u}\\rbrace ^\\perp \\times \\mathbb {R}$ , which is the tangent space to ${\\cal D}$ at $(p,{\\bf u},r)$ , that is orthogonal to the list of vectors in (REF ) is $\\Big (-{\\bf a}+\\frac{{\\bf a}\\cdot {\\bf t}}{{\\bf b}\\cdot {\\bf t}}{\\bf b},\\ r\\frac{{\\bf a}\\cdot {\\bf t}}{{\\bf b}\\cdot {\\bf t}}{\\bf a},\\ 1\\Big ).$ Since $(p,{\\bf u},r)=\\Psi (z,{\\bf a},{\\bf b},r)=(z+r{\\bf a},{\\bf b},r)$ , we can replace ${\\bf a}$ with $(p-z)/r$ and ${\\bf b}$ with ${\\bf u}$ .",
"Doing so and normalizing this vector results in the vector $\\mathbf {$ } defined in (REF ).",
"The vector $\\mathbf {$ } at $(p,{\\bf u},r)$ is pointing outward from ${\\cal D}_\\text{odd}$ if the interior of the disc associated with $(p,{\\bf u},r)$ crosses ${\\cal C}$ an odd number of times.",
"To see this, let $\\gamma $ be a smooth curve in ${\\cal D}$ defined on an interval of $\\mathbb {R}$ containing zero such that $\\gamma (0)=(p,{\\bf u},r)$ and $\\gamma ^{\\prime }(0)=\\mathbf {$ }.",
"For small negative values of $t$ , we have $\\gamma (t) \\in {\\cal D}_\\text{odd}$ since the last component of $\\mathbf {$ } is positive and so the disc $D(\\gamma (t))$ will intersect the curve an odd number of times and its boundary will not intersect the curve.",
"Moreover, $\\gamma (t)\\notin {\\cal D}_\\text{odd}$ for small positive $t$ because the curve ${\\cal C}$ will cross the disc $D(\\gamma (t))$ one more time than the disc $D(p,{\\bf u},r)$ since the boundary of this disc intersects the curve.",
"Using similar logic, one can see that $-\\mathbf {$ } is the outward normal if $D(p,{\\bf u},r)\\cap {\\cal C}$ has an even number of points.",
"In the next section we will use the notation ${\\cal D}({\\cal C}):={\\cal D}_\\text{odd}$ to highlight that this set of discs depends on the curve ${\\cal C}$ .",
"The previous two results characterized $\\partial ^*{\\cal D}({\\cal C})$ , but we have to go one step further and understand the boundaries of the sets ${\\cal D}_\\Omega $ and ${\\cal D}_\\Omega ({\\cal C})$ , which are defined by ${\\cal D}_\\Omega :=\\lbrace (p,{\\bf u},r)\\in \\ {\\cal D}\\ |\\ \\partial D(p,{\\bf u},r)\\cap \\Omega \\ne \\emptyset \\rbrace \\quad \\text{and}\\quad {\\cal D}_\\Omega ({\\cal C}):={\\cal D}({\\cal C})\\cap {\\cal D}_\\Omega ,$ where $\\Omega \\subseteq \\mathbb {R}^n$ is an open, bounded set with smooth boundary.",
"First, notice that ${\\cal D}_\\Omega $ is an open subset of ${\\cal D}$ .",
"Moreover, it is not difficult to see that $\\partial {\\cal D}_\\Omega $ is a smooth $(2n-1)$ -dimensional submanifold of ${\\cal D}$ since $\\Omega $ has smooth boundary.",
"Finally, it follows from Proposition REF that for ${\\cal H}^{2n-1}$ -a.e.",
"$(p,{\\bf u},r)\\in \\partial ^*{\\cal D}({\\cal C})$ , we have $(p,{\\bf u},r)\\in {\\cal D}_{\\Omega }$ since ${\\cal C}$ is contained in $\\Omega $ .",
"Thus, we have $\\partial ^*{\\cal D}_\\Omega ({\\cal C})=\\partial ^*{\\cal D}({\\cal C})\\cup \\partial {\\cal D}_\\Omega ,$ where this equality holds up to a set of ${\\cal H}^{2n-1}$ -measure zero and the union is disjoint."
],
[
"Fractional length",
"In this section we define a fractional notion of length and show that in an appropriate limit as $\\sigma $ goes to 1, this converges to the ${\\cal H}^1$ measure up to a multiplicative constant.",
"Let $\\Omega $ be an open, bounded set with smooth boundary.",
"Motivated by (REF ), given a $C^1$ compact curve ${\\cal C}$ in $\\mathbb {R}^n$ , define the $\\sigma $ -length of ${\\cal C}$ relative to $\\Omega $ by $\\text{Len}_\\sigma ({\\cal C},\\Omega ):=\\int _{{\\cal D}({\\cal C})} r^{1-n-\\sigma }\\sup _{{\\bf v}\\in {\\cal U}_n\\cap \\lbrace {\\bf u}\\rbrace ^\\perp } \\chi _\\Omega (p+r{\\bf v}) d{\\cal H}^{2n}(p,{\\bf u},r),$ where ${\\cal D}({\\cal C}):={\\cal D}_\\text{odd}$ .",
"For simplicity, assume that ${\\cal C}$ is contained in $\\Omega $ .",
"Using the definitions in (REF ), the fractional length can be rewritten as $\\text{Len}_\\sigma ({\\cal C},\\Omega )=\\int _{{\\cal D}_\\Omega ({\\cal C})} r^{1-n-\\sigma } d{\\cal H}^{2n}(p,{\\bf u},r).$ To show that this definition yields a finite number, first notice that ${\\cal D}_\\Omega ({\\cal C})\\subseteq \\Xi \\Big (\\bigcup _{\\xi \\in \\mathbb {R}^+} {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\lbrace \\xi \\rbrace \\times [\\xi ,\\xi + d(\\Omega )]\\Big ),$ where $d(\\Omega )$ is the diameter of $\\Omega $ and $\\Xi $ is defined in (REF ) of the Appendix.",
"To see this, consider $(p,{\\bf u},r)\\in {\\cal D}_\\Omega ({\\cal C})$ .",
"Since $D(p,{\\bf u},r)$ intersects ${\\cal C}$ a finite number of times, we can find $z\\in {\\cal C}\\cap D(p,{\\bf u},r)$ with minimal distance to $p$ such that $(p-z)\\cdot {\\bf u}=0$ .",
"Set $\\xi =|p-z|$ , ${\\bf a}=(p-z)/\\xi $ , and ${\\bf b}={\\bf u}$ .",
"It follows that $\\Xi (z,\\xi ,{\\bf a},{\\bf b},r)=(p,{\\bf u},r)$ .",
"Since $z$ is the closest point on ${\\cal C}$ to $p$ in $D(p,{\\bf u},r)$ , we must have $\\xi \\le r$ otherwise $D(p,{\\bf u},r)$ would not intersect ${\\cal C}$ .",
"Moreover, $r\\le \\xi +d(\\Omega )$ since if this were not true then $\\partial D(p,{\\bf u},r)\\cap \\Omega =\\emptyset $ .",
"It follows that $(p,{\\bf u},r)$ is an element of the set on the right-hand side of (REF ), so (REF ) holds.",
"Thus, we can utilize the change of variables formula (REF ) in the Appendix to find that $\\text{Len}_\\sigma ({\\cal C},\\Omega )&\\le \\int _{\\cal C}\\int _0^\\infty \\int _{{\\cal U}_\\perp ^2}\\int _\\xi ^{\\xi +d(\\Omega )} r^{1-n-\\sigma }\\xi ^{n-2} |{\\bf b}\\cdot {\\bf t}(z)|drd{\\cal H}^{2n-3}({\\bf a},{\\bf b}) d\\xi dz\\\\&\\le {\\cal H}^1({\\cal C}) {\\cal H}^{2n-3}({\\cal U}_\\perp ^2) \\int _0^\\infty \\int _\\xi ^{\\xi +d(\\Omega )} r^{1-n-\\sigma }\\xi ^{n-2} drd\\xi \\\\&=\\frac{1}{2-n-\\sigma }{\\cal H}^1({\\cal C}) {\\cal H}^{2n-3}({\\cal U}_\\perp ^2) \\int _0^\\infty \\Big ( \\frac{\\xi ^{n-2}}{(\\xi +d(\\Omega ))^{n-2+\\sigma }}-\\xi ^{-\\sigma }\\Big ) d\\xi $ and the remaining integral involving $\\xi $ is finite.",
"The next goal is to show that the fractional length converges in an appropriate limit to the classical notion of length up to some multiplicative constant.",
"Doing so will require the following result.",
"Lemma 3.1 For any ${\\bf c}\\in {\\cal U}_n$ , we have $\\int _{{\\cal U}^2_\\perp } |{\\bf b}\\cdot {\\bf c}| d{\\cal H}^{2n-3}({\\bf a},{\\bf b})=4 \\alpha _{n-1}\\alpha _{n-2},$ where $\\alpha _n$ is the volume of the unit ball in $\\mathbb {R}^n$ .",
"First notice that for any vector ${\\bf v}\\in \\mathbb {R}^{n-1}$ , we have $\\int _{{\\cal U}_{n-1}} |{\\bf b}\\cdot {\\bf v}| d{\\bf b}=2\\int _{{\\cal B}_{n-2}}\\int _0^{\\pi /2} |{\\bf v}|\\cos \\theta d\\theta d{\\cal H}^{n-2}=2|{\\bf v}|\\alpha _{n-2},$ where ${\\cal B}_{n-2}$ is the unit ball in $\\mathbb {R}^{n-2}$ .",
"Letting $P_{\\bf a}$ denote the projection onto the plane orthogonal to ${\\bf a}$ , we can compute $\\int _{{\\cal U}^2_\\perp } |{\\bf b}\\cdot {\\bf c}| d{\\cal H}^{2n-3}({\\bf a},{\\bf b})&=\\int _{{\\cal U}_n}\\int _{{\\cal U}_n\\cap \\lbrace {\\bf a}\\rbrace ^\\perp } |{\\bf b}\\cdot P_{\\bf a}{\\bf c}| d{\\bf b}d{\\bf a}\\\\& = 2\\alpha _{n-2} \\int _{{\\cal U}_n} |P_{\\bf a}{\\bf c}| d{\\bf a}\\\\& = 4\\alpha _{n-2} \\int _{{\\cal B}_{n-1}}\\int _0^{\\pi /2} \\sin \\theta d\\theta d{\\cal H}^{n-1}\\\\& = 4\\alpha _{n-2}\\alpha _{n-1}.$ Theorem 3.2 If ${\\cal C}$ is a $C^1$ , compact, one-dimensional manifold and $\\Omega \\subseteq \\mathbb {R}^n$ is an open, bounded set such that ${\\cal C}\\subseteq \\Omega $ , then $\\lim _{\\sigma \\uparrow 1} (1-\\sigma ){\\rm Len}_\\sigma ({\\cal C},\\Omega )=\\frac{4\\alpha _{n-1}\\alpha _{n-2}}{(n-1)}{\\cal H}^1({\\cal C}).$ Begin by setting $\\varepsilon :=(1-\\sigma )^{1/n}$ and ${\\cal D}_\\varepsilon ({\\cal C}):=\\lbrace (p,{\\bf u},r)\\in {\\cal D}({\\cal C})\\ |\\ r\\le \\varepsilon \\rbrace .$ One can show that ${\\cal D}_\\Omega ({\\cal C})\\setminus {\\cal D}_\\varepsilon ({\\cal C})\\subseteq \\Xi \\Big (\\bigcup _{\\xi \\in \\mathbb {R}^+} {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\lbrace \\xi \\rbrace \\times [\\max \\lbrace \\xi ,\\varepsilon \\rbrace ,\\xi + d(\\Omega )]\\Big ),$ using an argument similar to that justifying (REF ).",
"Thus, using the change of variables (REF ) there is a constant $C_n$ depending on ${\\cal C}$ and $n$ such that $\\int _{{\\cal D}_\\Omega ({\\cal C})\\setminus {\\cal D}_\\varepsilon ({\\cal C})} r^{1-n-\\sigma }d{\\cal H}^{2n}(p,{\\bf u},r)& \\le C_n \\int _0^\\infty \\int _{\\max \\lbrace \\xi ,\\varepsilon \\rbrace }^{\\xi + d(\\Omega )} \\xi ^{n-2} r^{1-n-\\sigma } dr d\\xi \\\\&=\\frac{C_n}{n+\\sigma -2} \\Big [ \\int _0^\\varepsilon \\xi ^{n-2}[\\varepsilon ^{2-n-\\sigma }-(\\xi +d(\\Omega ))^{n-2-\\sigma }] d\\xi \\\\&\\qquad + \\int _\\varepsilon ^\\infty [\\xi ^{-\\sigma }-\\xi ^{n-2}(\\xi +d(\\Omega ))^{2-n-\\sigma }]d\\xi \\Big ].$ Since $\\int _0^\\varepsilon \\xi ^{n-2}[\\varepsilon ^{2-n-\\sigma }-(\\xi +d(\\Omega ))^{n-2-\\sigma }] d\\xi \\le \\frac{\\varepsilon ^{1-\\sigma }}{n-1}\\\\$ and $\\int _\\varepsilon ^\\infty [\\xi ^{-\\sigma }-\\xi ^{n-2}(\\xi +d(\\Omega ))^{2-n-\\sigma }]d\\xi \\le \\frac{(2-n-\\sigma )\\varepsilon ^{-\\sigma }}{\\sigma },$ it follows that $\\lim _{\\sigma \\uparrow 1} (1-\\sigma )\\int _{{\\cal D}_\\Omega ({\\cal C})\\setminus {\\cal D}_\\varepsilon ({\\cal C})} r^{1-n-\\sigma }d{\\cal H}^{2n}(p,{\\bf u},r) = 0.$ Thus, $\\lim _{\\sigma \\uparrow 1} (1-\\sigma )\\text{Len}_\\sigma ({\\cal C},\\Omega )=\\lim _{\\sigma \\uparrow 1}\\int _{{\\cal D}_\\varepsilon (\\Omega )}(1-\\sigma )r^{1-n-\\sigma } d{\\cal H}^{2n}(p,{\\bf u},r).$ Each $(p,{\\bf u},r)\\in {\\cal D}_\\varepsilon ({\\cal C})$ may intersect ${\\cal C}$ multiple times, however we know it intersects ${\\cal C}$ at least once.",
"Thus, we can arbitrarily associate each $(p,{\\bf u},r)$ with some point $z\\in {\\cal C}$ .",
"Let $c(p,{\\bf u},r)$ denote the selected point in ${\\cal C}$ .",
"We can think of $c$ as a mapping from ${\\cal D}_\\varepsilon ({\\cal C})$ to ${\\cal C}$ .",
"Many such mappings exist, but here we select one.",
"For $z\\in {\\cal C}$ and $({\\bf a},{\\bf b})\\in {\\cal U}_\\perp ^2$ set $C_\\varepsilon (z,{\\bf a},{\\bf b}):=\\lbrace (\\xi ,r)\\in \\mathbb {R}^+\\times \\mathbb {R}^+\\ |\\ (z+\\xi {\\bf a},{\\bf b},r)\\in {\\cal D}_\\varepsilon ({\\cal C})\\ \\text{and}\\ c(z+\\xi {\\bf a},{\\bf b},r)=z\\rbrace .$ It follows from the definition of $c$ that the function $\\Xi $ defined in (REF ) is injective on the set $\\bigcup _{(z,{\\bf a},{\\bf b})\\in {\\cal C}\\times {\\cal U}_\\perp ^2} \\lbrace z\\rbrace \\times \\lbrace {\\bf a}\\rbrace \\times \\lbrace {\\bf b}\\rbrace \\times C_\\varepsilon (z,{\\bf a},{\\bf b}).$ Thus, by the change of variables (REF ) we have $\\int _{{\\cal D}_\\varepsilon (\\Omega )} r^{1-n-\\sigma }d{\\cal H}^{2n}(p,{\\bf u},r) \\\\= \\int _{\\cal C}\\int _{{\\cal U}_\\perp ^2} \\int _{C_\\varepsilon (z,{\\bf a},{\\bf b})} r^{1-n-\\sigma } \\xi ^{n-2} |{\\bf b}\\cdot {\\bf t}(z)| d{\\cal H}^2(\\xi ,r) d{\\cal H}^{2n-3}({\\bf a},{\\bf b}) dz.$ Since ${\\cal C}$ is a smooth curve, for all $(z,{\\bf a},{\\bf b})$ such that ${\\bf a}$ is not parallel to ${\\bf t}(z)$ there is a $\\varepsilon _0$ such that if $\\varepsilon \\le \\varepsilon _0$ we have $C_\\varepsilon (z,{\\bf a},{\\bf b})=\\lbrace (\\xi ,r)\\in \\mathbb {R}^+\\times \\mathbb {R}^+\\ |\\ \\xi \\in [0,\\varepsilon ]\\ \\text{and}\\ r\\in [\\xi ,\\varepsilon ]\\rbrace .$ Thus, $\\lim _{\\sigma \\uparrow 1}(1-\\sigma ) \\int _{C_\\varepsilon (z,{\\bf a},{\\bf b})} r^{1-n-\\sigma } \\xi ^{n-2} d{\\cal H}^2(\\xi ,r) & = \\lim _{\\sigma \\uparrow 1}(1-\\sigma ) \\int _{0}^\\varepsilon \\int _\\xi ^\\varepsilon r^{1-n-\\sigma } \\xi ^{n-2} drd\\xi \\\\&= \\frac{1}{n-1}.$ Putting this together with (REF ) and (REF ) we find $\\lim _{\\sigma \\uparrow 1} (1-\\sigma )\\text{Len}_\\sigma ({\\cal C},\\Omega ) = \\frac{1}{n-1} \\int _{\\cal C}\\int _{{\\cal U}_\\perp ^2} |{\\bf b}\\cdot {\\bf t}(z)| d{\\cal H}^{2n-3}({\\bf a},{\\bf b}) dz.$ With the help of Lemma REF , we obtain (REF )."
],
[
"Variation of Len$_\\sigma $ and nonlocal curvature",
"This section is dedicated to computing the Euler–Lagrange equation associated with the functional Len$_\\sigma $ .",
"To compute this, we will use what is known as a transport theorem.",
"The version of this transport theorem applicable here was established by Seguin [17].",
"For convenience, the statement of this theorem is formulated below in the form it will be applied here.",
"Theorem 4.1 Let ${\\cal I}$ be an open interval of $\\mathbb {R}$ containing zero.",
"For each $\\epsilon \\in {\\cal I}$ , let ${\\cal O}_\\varepsilon $ be an open subset of ${\\cal D}$ that is locally of finite perimeter such that there exists a $(2n-1)$ -dimensional Riemannian manifold ${\\cal N}$ and a function $\\Lambda \\in C^1({\\cal I},W^{1,\\infty }({\\cal N},\\mathbb {R}^{n}))$ such that the following conditions hold: for all $\\varepsilon \\in {\\cal I}$ , the differential of $\\Lambda _\\varepsilon :=\\Lambda (\\epsilon ,\\cdot ):{\\cal N}\\rightarrow \\mathbb {R}^n$ is injective, where it exists, $\\partial ^*{\\cal O}_\\varepsilon $ and $\\Lambda _\\varepsilon ({\\cal N})$ differ by a set of ${\\cal H}^{2n-1}$ -measure zero, and ${\\cal H}^{2n-1}(\\lbrace (p,{\\bf u},r)\\in \\partial ^*{\\cal O}_\\varepsilon \\ |\\ {\\cal H}^0(\\Lambda _\\varepsilon ^{-1}(\\lbrace (p,{\\bf u},r)\\rbrace )>1)$ .",
"It follows that for any $f\\in W^{1,1}({\\cal D},\\mathbb {R})$ we have $\\frac{d}{d\\varepsilon }\\int _{{\\cal O}_\\varepsilon } f\\, d{\\cal H}^{2n} \\Big |_{\\varepsilon =0} = \\int _{\\partial ^*{\\cal O}_0} f {\\bf v}\\cdot {\\bf n}\\, d{\\cal H}^{2n-1},$ where ${\\bf n}$ is the exterior unit-normal to ${\\cal O}_t$ and ${\\bf v}$ is the “velocity\" associated with $\\Lambda $ at $\\varepsilon =0$ , which is defined by ${\\bf v}(p,{\\bf u},r):=\\frac{\\partial \\Lambda }{\\partial \\varepsilon }(\\varepsilon ,\\Lambda _\\varepsilon ^{-1}(p,{\\bf u},r))|_{\\varepsilon =0}.$ Theorem 4.2 Let ${\\cal C}$ be a one-dimensional, compact $C^1$ manifold in $\\mathbb {R}^n$ , with $n>2$ .",
"A necessary and sufficient condition for the vanishing of the first variation with respect to curves with the same boundary as ${\\cal C}$ of the $\\sigma $ -length relative to $\\Omega $ is that for all $z\\in {\\cal C}$ we have $\\int _{{\\cal A}_\\text{\\rm odd}(z)}\\frac{\\big [-{\\bf a}+\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}{\\bf b}\\big ]\\sqrt{({\\bf a}\\cdot {\\bf t}(z))^2+({\\bf b}\\cdot {\\bf t}(z))^2}}{(2r)^{1+\\sigma }\\sqrt{2+(1+r^2)\\big (\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}\\big )^2}}{\\cal H}^{2n-2}({\\bf a},{\\bf b},r)\\\\=\\int _{{\\cal A}_\\text{\\rm even}(z)}\\frac{\\big [-{\\bf a}+\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}{\\bf b}\\big ]\\sqrt{({\\bf a}\\cdot {\\bf t}(z))^2+({\\bf b}\\cdot {\\bf t}(z))^2}}{(2r)^{1+\\sigma }\\sqrt{2+(1+r^2)\\big (\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}\\big )^2}}{\\cal H}^{2n-2}({\\bf a},{\\bf b},r),$ where ${\\cal A}_\\text{\\rm odd}(z)&:=\\lbrace ({\\bf a},{\\bf b},r)\\in {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\ |\\ {\\cal H}^0(D(z+r{\\bf a},{\\bf b},r)\\cap {\\cal C})\\ \\text{is an odd number}\\rbrace ,\\\\{\\cal A}_\\text{\\rm even}(z)&:=\\lbrace ({\\bf a},{\\bf b},r)\\in {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\ |\\ {\\cal H}^0(D(z+r{\\bf a},{\\bf b},r)\\cap {\\cal C})\\ \\text{is an even number}\\rbrace .$ Let ${\\bf w}:{\\cal C}\\rightarrow \\mathbb {R}^n$ be a $C^1$ function that satisfies two conditions: (i) ${\\bf w}$ is zero at the endpoints of ${\\cal C}$ and (ii) ${\\bf w}(z)$ is orthogonal to the tangent space of ${\\cal C}$ at $z\\in {\\cal C}$ .",
"For each $\\varepsilon >0$ define the set ${\\cal C}_\\varepsilon :=\\lbrace z+\\varepsilon {\\bf w}(z)\\ |\\ z\\in {\\cal C}\\rbrace .$ For sufficiently small $\\varepsilon $ , ${\\cal C}_\\varepsilon $ is a one-dimensional $C^1$ manifold that is contained in $\\Omega $ .",
"We seek a necessary and sufficient condition for $0&=\\lim _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\big (\\text{Len}_\\sigma ({\\cal C}_\\varepsilon ,\\Omega )-\\text{Len}_\\sigma ({\\cal C},\\Omega )\\big )\\\\&=\\lim _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\Big (\\int _{{\\cal D}_\\Omega ({\\cal C}_\\varepsilon )} r^{1-n-\\sigma } d{\\cal H}^{2n}(p,{\\bf u},r)-\\int _{{\\cal D}_\\Omega ({\\cal C})} r^{1-n-\\sigma } d{\\cal H}^{2n}(p,{\\bf u},r) \\Big ).$ To compute this limit, which is a derivative with respect to $\\varepsilon $ , we will use the Transport Theorem REF .",
"To applying this result, we must show that the set ${\\cal D}_\\Omega ({\\cal C}_\\varepsilon )$ , as it evolves with $\\varepsilon $ , the properties listed in the theorem.",
"Towards this end, let ${\\cal N}$ be the $(2n-1)$ -dimensional manifold defined as the disjoint union of ${\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+$ and $\\partial {\\cal D}_\\Omega $ , and define a function $\\Lambda :{\\cal I}\\times {\\cal M}\\rightarrow {\\cal D}$ , where ${\\cal I}$ is the interval of admissible $\\varepsilon $ values being considered, by $\\Lambda (\\varepsilon ,m):={\\left\\lbrace \\begin{array}{ll}(z+\\varepsilon {\\bf w}(z)+r{\\bf a},{\\bf b},r) & \\text{if} \\ m=(z,{\\bf a},{\\bf b},r)\\in {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+,\\\\(p,{\\bf u},r) & \\text{if}\\ m=(p,{\\bf u},r)\\in \\partial {\\cal D}_\\Omega .\\end{array}\\right.",
"}$ Notice that $\\Lambda $ is $C^1$ and that for all $\\varepsilon $ , $\\Lambda (\\varepsilon ,{\\cal M})$ agrees with $\\partial ^*{\\cal D}_\\Omega ({\\cal C}_\\varepsilon )$ up to a set of ${\\cal H}^{2n-1}$ -measure zero by Proposition REF and the remarks at the end of Section .",
"Setting $\\Lambda _\\varepsilon :=\\Lambda (\\cdot ,\\varepsilon )$ , since for each $\\varepsilon $ we have $h_\\varepsilon ({\\cal M})={\\cal D}_\\partial \\cup \\partial {\\cal D}_\\Omega $ , see the beginning of the proof of Proposition REF , it follows from Lemma REF Item 4 that ${\\cal H}^{2n-1}(\\lbrace (p,{\\bf u},r)\\in \\partial ^*{\\cal D}_\\Omega ({\\cal C}_\\varepsilon )\\ |\\ {\\cal H}^0(\\Lambda _\\varepsilon ^{-1}(\\lbrace (p,{\\bf u},r)\\rbrace )>1)\\le {\\cal H}^{2n-1}({\\cal D}_{\\partial 2})=0.$ These properties of $\\Lambda $ ensure that Theorem REF can be applied.",
"To apply this, we need the velocity associated with $\\Lambda $ .",
"In this case, one can compute ${\\bf v}(p,{\\bf u},r):={\\left\\lbrace \\begin{array}{ll}({\\bf w}(z),\\textbf {0},0) & \\text{if} \\ (p,{\\bf u},r)\\in \\partial ^*{\\cal D}({\\cal C}),\\\\(\\textbf {0},\\textbf {0},0) & \\text{if}\\ (p,{\\bf u},r)\\in \\partial {\\cal D}_\\Omega .\\end{array}\\right.",
"}$ where $z\\in {\\cal C}$ is the unique point corresponding to $(p,{\\bf u},r)\\in \\partial ^*{\\cal D}({\\cal C})$ as described by Proposition REF .",
"Using (REF ), the limit in (REF ) can be computed to obtain $0=\\int _{\\partial ^* {\\cal D}({\\cal C})} r^{1-n-\\sigma } {\\bf v}(p,{\\bf u},r)\\cdot \\nu (p,{\\bf u},r) d{\\cal H}^{2n-1}(p,{\\bf u},r).$ The next step is to use the change variables formula (REF ) in the Appendix to find that $0=\\int _{\\cal C}\\Big (\\int _{{\\cal A}_\\text{odd}(z)}-\\int _{{\\cal A}_\\text{even}(z)}\\Big ) \\frac{{\\bf w}(z)\\cdot \\big (-{\\bf a}+\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}{\\bf b}\\big )\\sqrt{({\\bf a}\\cdot {\\bf t}(z))^2+({\\bf b}\\cdot {\\bf t}(z))^2}}{(2r)^{1+\\sigma }\\sqrt{2+(1+r^2)\\big (\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}\\big )^2}}{\\cal H}^{2n-2}({\\bf a},{\\bf b},r)dz.$ Standard arguments show that this condition must hold for all normal variations ${\\bf w}$ if and only if (REF ) holds.",
"Since a curve connecting two points of minimal length has zero curvature, the preceding result motivates the following definition.",
"The nonlocal curvature vector $\\kappa _\\sigma $ at $z\\in {\\cal C}$ is defined by $\\kappa _\\sigma (z):=\\Big (\\int _{{\\cal A}_\\text{odd}(z)}-\\int _{{\\cal A}_\\text{even}(z)}\\Big ) \\frac{\\big (-{\\bf a}+\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}{\\bf b}\\big )\\sqrt{({\\bf a}\\cdot {\\bf t}(z))^2+({\\bf b}\\cdot {\\bf t}(z))^2}}{r^{1+\\sigma }\\sqrt{2+(1+r^2)\\big (\\frac{{\\bf a}\\cdot {\\bf t}(z)}{{\\bf b}\\cdot {\\bf t}(z)}\\big )^2}}{\\cal H}^{2n-2}({\\bf a},{\\bf b},r)dz.$ Notice that this vector is orthogonal to the curve at $z$ , however there is no reason to believe that this vector is parallel to the classical normal to the curve.",
"The nonlocal scalar curvature can be define as the magnitude of this vector: $\\kappa _\\sigma (z):=|\\kappa _\\sigma (z)|$ .",
"Note that the nonlocal scalar curvature for a curve does not agree with the nonlocal mean curvature given in (REF ) in the $n=2$ case.",
"This is not a contradiction as the formula for the nonlocal curvature of a curve was derived using the results in Section , which required $n>2$ .",
"In particular, the fact that ${\\cal H}^{2n-1}({\\cal D}_\\text{tan})=0$ only follows from Item 3 of Lemma REF when $n>2$ .",
"This result was needed in determining the structure of $\\partial ^*{\\cal D}({\\cal C})$ ."
],
[
"Appendix: some change of variables",
"Let ${\\cal C}$ be a one-dimensional, compact $C^1$ manifold in $\\mathbb {R}^n$ .",
"Lemma 5.1 Consider the function $\\Phi :{\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\rightarrow \\mathbb {R}^n\\times \\mathbb {R}^n$ defined by $\\Phi (z,{\\bf a},{\\bf b},\\xi ):=(z+\\xi {\\bf a},{\\bf b})\\quad \\text{for all}\\ (z,{\\bf a},{\\bf b},\\xi )\\in {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+.$ If ${\\cal A}$ is a subset of ${\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+$ and $f:\\Phi ({\\cal A})\\rightarrow \\mathbb {R}$ is an integrable function, then $\\int _{\\Phi ({\\cal A})}\\Big [ \\sum _{(z,{\\bf a},{\\bf b},\\xi )\\in \\Phi ^{-1}(p,{\\bf u})} f(p,{\\bf u})\\Big ]d{\\cal H}^{2n-1}(p,{\\bf u}) \\\\= \\int _{\\cal A}f(z+\\xi {\\bf a},{\\bf b})\\xi ^{n-2} |{\\bf b}\\cdot {\\bf t}(z)| d{\\cal H}^{2n-1}(z,{\\bf a},{\\bf b},\\xi ),$ where ${\\bf t}(z)$ is a unit-vector tangent to the curve ${\\cal C}$ at the point $z$ .",
"It suffices to prove the result for a set of the form ${\\cal A}={\\cal C}_{\\cal A}\\times {\\cal U}_{\\cal A}\\times {\\cal I}_{\\cal A}$ , where ${\\cal C}_{\\cal A}\\subseteq {\\cal C}$ , ${\\cal U}_{\\cal A}\\subseteq {\\cal U}_\\perp ^2$ , and ${\\cal I}_{\\cal A}\\subseteq \\mathbb {R}^+$ .",
"Moreover, by employing a partition of unity, we can reduce the problem to the case where ${\\cal U}_{\\cal A}$ is covered by one chart.",
"Thus, there is a set $U_{\\cal A}\\subseteq \\mathbb {R}^{2n-3}$ and a diffeomorphism $\\chi :U_{{\\cal A}}\\rightarrow {\\cal U}_{\\cal A}$ .",
"Since ${\\cal U}_{\\cal A}\\subseteq \\mathbb {R}^n\\times \\mathbb {R}^n$ , we can view this function as $\\chi =(\\chi _1,\\chi _2)$ , where $\\chi _1,\\chi _2:\\mathbb {R}^{2n-3}\\rightarrow \\mathbb {R}^n$ .",
"Recall that if $g$ is an integrable function defined on ${\\cal U}_{\\cal A}$ , then $\\int _{{\\cal U}_{\\cal A}} g({\\bf a},{\\bf b})\\, d{\\cal H}^{2n-3}({\\bf a},{\\bf b})=\\int _{U_{\\cal A}} g(\\chi _1(w),\\chi _2(w)) J_{\\chi }(w)\\, dw,$ where $J_{\\chi }=\\sqrt{\\text{det} (\\nabla \\chi ^\\top \\nabla \\chi )}$ is the Jacobin of $\\chi $ .",
"Also, if $h$ is an integrable function defined on ${\\cal C}_{\\cal A}$ and $\\phi $ is a parameterization of ${\\cal C}$ then $\\int _{{\\cal C}_{\\cal A}} h(z)\\, dz=\\int _{C_{\\cal A}} h(\\phi (s)) |\\phi ^{\\prime }(s)|\\, ds,$ where $C_{\\cal A}$ is the subset of $\\mathbb {R}$ such that $\\phi (C_{\\cal A})={\\cal C}_{{\\cal A}}$ .",
"Set $A:=C_{\\cal A}\\times U_{\\cal A}\\times {\\cal I}_{\\cal A}$ and define $\\tilde{\\Phi }:A\\rightarrow \\Phi ({\\cal A})$ by $\\tilde{\\Phi }(s,w,\\xi ):=\\Phi (\\phi (s) ,\\chi _1(w),\\chi _2(w),\\xi )=(\\phi (s)+\\xi \\chi _1(w), \\chi _1(w))$ and $\\tilde{f}:A\\rightarrow \\mathbb {R}$ by $\\tilde{f}(s,w,\\xi )=f(\\phi (s)+\\xi \\chi _1(w),\\chi _1(w)).$ By the Area Formula (see Theorem 2.71 of Ambrosio, Fusco, and Pallara [2]), it follows that $\\int _{\\Phi ({\\cal A})}\\Big [ \\sum _{(s,w,\\xi )\\in \\Phi ^{-1}(p,{\\bf u})} \\tilde{f}(s,w,\\xi )\\Big ] d{\\cal H}^{2n-1}(p,{\\bf u})=\\int \\tilde{f}(s,w,\\xi )J_{\\tilde{\\Phi }}(s,w,\\xi ) d{\\cal H}^{2n-1}(s,w,\\xi ).$ Thus, by (REF ) and (REF ), the result will follow once it is shown that $J_{\\tilde{\\Phi }}=\\xi ^{n-2}|\\chi _2\\cdot {\\bf t}| |\\phi ^{\\prime }| J_{\\chi }.$ To establish this, first notice that $\\nabla \\tilde{\\Phi }=\\begin{blockarray}{cccc}1 & 2n-3 & 1 & \\\\\\begin{block}{(c|c|c)c}\\phi ^{\\prime } & \\xi \\nabla \\chi _1 & \\chi _1 & n \\\\\\cline {1-3}\\textbf {0} & \\nabla \\chi _2 & \\textbf {0} & n \\\\\\end{block}\\end{blockarray}.$ and hence, noting that $|\\chi _1|^2=1$ and $\\nabla \\chi _1^\\top \\chi _1={\\bf 0}$ , we have $\\nabla \\tilde{\\Phi }^\\top \\nabla \\tilde{\\Phi }=\\begin{blockarray}{ccc}\\begin{block}{(c|c|c)}|\\phi ^{\\prime }|^2 & \\xi \\phi ^{\\prime \\top }\\nabla \\chi _1& \\chi _1\\cdot \\phi ^{\\prime } \\rule {0pt}{2.6ex}\\\\\\cline {1-3}\\xi \\nabla \\chi _1^\\top \\phi ^{\\prime } & \\xi ^2\\nabla \\chi _1^\\top \\nabla \\chi _1+\\nabla \\chi _2^\\top \\nabla \\chi _2 & 0 \\rule {0pt}{2.6ex}\\\\\\cline {1-3}\\chi _1\\cdot \\phi ^{\\prime } & 0 & 1 \\rule {0pt}{2.6ex}\\\\\\end{block}\\end{blockarray}\\, .$ Since switching rows or columns of a matrix does not change the absolute value of its determinant, we have $|\\text{det}(\\nabla \\tilde{\\Phi }^\\top \\nabla \\tilde{\\Phi })| = \\left|\\text{det}\\,\\begin{blockarray}{ccc}\\begin{block}{(c|c|c)}|\\phi ^{\\prime }|^2 & \\chi _1\\cdot \\phi ^{\\prime } & \\xi \\phi ^{\\prime \\top }\\nabla \\chi _1 \\rule {0pt}{2.6ex}\\\\\\cline {1-3}\\chi _1\\cdot \\phi ^{\\prime } & 1 & 0 \\rule {0pt}{2.6ex}\\\\\\cline {1-3}\\xi \\nabla \\chi _1^\\top \\phi ^{\\prime } & 0 & \\xi ^2\\nabla \\chi _1^\\top \\nabla \\chi _1+\\nabla \\chi _2^\\top \\nabla \\chi _2 \\rule {0pt}{2.6ex}\\\\\\end{block}\\end{blockarray}\\, \\right| .$ Recall that the determinant of a block matrix can be computed using $\\det \\begin{blockarray}{cc}& \\\\\\begin{block}{(c|c)}{\\bf A}& {\\bf B}\\\\\\cline {1-2}{\\bf C}& {\\bf D}\\rule {0pt}{2.6ex}\\\\\\end{block}\\end{blockarray}&=\\det ({\\bf A})\\det ({\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}).$ This identity will be used with ${\\bf D}=\\xi ^2\\nabla \\chi _1^\\top \\nabla \\chi _1+\\nabla \\chi _2^\\top \\nabla \\chi _2$ .",
"Let $P_{\\chi _1}$ denote the projection onto the plane orthogonal to the vector $\\chi _1$ .",
"After some computation, using the identity $|P_{\\chi _1} \\phi ^{\\prime }|^2=|\\phi ^{\\prime }|^2-(\\chi _1\\cdot \\phi ^{\\prime })^2$ , one finds that $|\\text{det}(\\nabla \\tilde{\\Phi }^\\top \\nabla \\tilde{\\Phi })|=|P_{\\chi _1}\\phi ^{\\prime }|^2 \\left|\\text{det}\\Big [ \\nabla \\chi _1^\\top (\\xi ^2\\textbf {1}_n-\\frac{\\xi ^2}{|P_{\\chi _1}\\phi ^{\\prime }|^2} P_{\\chi _1}\\phi ^{\\prime }\\otimes P_{\\chi _1}\\phi ^{\\prime })\\nabla \\chi _1+\\nabla \\chi _2^\\top \\nabla \\chi _2\\Big ]\\right|.$ where $\\textbf {1}_n$ is the identity function on $\\mathbb {R}^n$ .",
"It follows that $|\\text{det}(\\nabla \\tilde{\\Phi }^\\top \\nabla \\tilde{\\Phi })|=|P_{\\chi _1}\\phi ^{\\prime }|^2 \\Big | \\text{det}\\Big [ \\nabla \\chi ^\\top \\begin{blockarray}{cc}& \\\\\\begin{block}{(c|c)}\\xi ^2\\textbf {1}_n-\\xi ^2 \\tilde{{\\bf t}}\\otimes \\tilde{{\\bf t}}& \\textbf {0} \\\\\\cline {1-2}\\textbf {0} & \\textbf {1}_n \\rule {0pt}{2.6ex}\\\\\\end{block}\\end{blockarray}\\nabla \\chi \\Big ]\\Big |,$ where $\\tilde{{\\bf t}}=P_{\\chi _1}\\phi ^{\\prime }/|P_{\\chi _1}\\phi ^{\\prime }|$ .",
"To simplify the right-hand side of the previous equation, set ${\\bf L}$ equal to the square $2n\\times 2n$ matrix in the previous equation between $\\nabla \\chi ^\\top $ and $\\nabla \\chi $ and recall the fact that $\\text{det}(\\nabla \\chi ^\\top {\\bf L}\\nabla \\chi )=J_{\\chi }^2\\text{det}({\\bf I}^\\top {\\bf L}{\\bf I}),$ where ${\\bf I}$ is the natural injection of the range of $\\nabla \\chi $ into $\\mathbb {R}^{2n}$ .",
"One can find an orthonormal basis for $\\mathbb {R}^n$ of the form $({\\bf e}_1,{\\bf e}_2,\\dots ,{\\bf e}_{n-2},\\chi _1,\\chi _2)$ such that $\\tilde{{\\bf t}}$ can be written as a linear combination of ${\\bf e}_{n-2}$ and $\\chi _2$ .",
"Notice that $\\lbrace ({\\bf e}_1,{\\bf 0}),\\dots , ({\\bf e}_{n-2},{\\bf 0}),({\\bf 0},{\\bf e}_1),\\dots ,({\\bf 0},{\\bf e}_{n-2}),\\frac{1}{\\sqrt{2}}(\\chi _2,-\\chi _1)\\rbrace $ is an orthonormal basis for the tangent space of ${\\cal U}_\\perp ^2$ at $(\\chi _1,\\chi _2)$ .",
"By writing the the matrix ${\\bf I}^\\top \\begin{blockarray}{cc}& \\\\\\begin{block}{(c|c)}\\xi ^2\\textbf {1}_n-\\xi ^2 \\tilde{{\\bf t}}\\otimes \\tilde{{\\bf t}}& \\textbf {0} \\\\\\cline {1-2}\\textbf {0} & \\textbf {1}_n \\rule {0pt}{2.6ex}\\\\\\end{block}\\end{blockarray}\\,{\\bf I}$ relative to the basis (REF ), one can compute its determinant, and hence, putting together (REF ) and (REF ), we have that $|\\text{det}(\\nabla \\chi ^\\top {\\bf L}\\nabla \\chi )|=\\xi ^{2n-4}|P_{\\chi _1}\\phi ^{\\prime }|^2 |\\chi _2\\cdot \\tilde{{\\bf t}}|^2J^2_{\\chi }.$ Since $|P_{\\chi _1}\\phi ^{\\prime }|^2 |\\chi _2\\cdot \\tilde{{\\bf t}}|^2=|\\chi _2 \\cdot P_{\\chi _1}{\\bf t}|^2|\\phi ^{\\prime }|^2=|\\chi _2\\cdot {\\bf t}|^2|\\phi ^{\\prime }|^2$ , this proves (REF ).",
"Figure: A depiction of the disc associated with Ξ(z,𝐚,𝐛,ξ,r)\\Xi (z,{\\bf a},{\\bf b},\\xi ,r).A slight variation on the change of variables formula (REF ) will be needed.",
"Namely, we will require a change of variables according to the function $\\Xi :{\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\times \\mathbb {R}^+\\rightarrow \\mathbb {R}^n\\times \\mathbb {R}^n\\times \\mathbb {R}$ defined by $\\Xi (z,{\\bf a},{\\bf b},\\xi ,r):=(z+\\xi {\\bf a},{\\bf b},r)\\quad \\text{for all}\\ (z,{\\bf a},{\\bf b},\\xi ,r)\\in {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\times \\mathbb {R}^+.$ The function $\\Xi $ allows us to describe discs using points on the curve ${\\cal C}$ ; see Figure REF .",
"Since $\\Xi $ acts like the identity on the last variable $r$ , the previous lemma immediately implies that if ${\\cal A}$ is a subset of ${\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\times \\mathbb {R}^+$ and $f:\\Xi ({\\cal A})\\rightarrow \\mathbb {R}$ is an integrable function, then $\\int _{\\Xi ({\\cal A})}\\Big [ \\sum _{(z,{\\bf a},{\\bf b},\\xi ,r)\\in \\Xi ^{-1}(p,{\\bf u},r)} f(p,{\\bf u},r)\\Big ]d{\\cal H}^{2n}(p,{\\bf u}, r) \\\\= \\int _{\\cal A}f(z+\\xi {\\bf a},{\\bf b},r)\\xi ^{n-2} |{\\bf b}\\cdot {\\bf t}(z)| d{\\cal H}^{2n}(z, {\\bf a},{\\bf b},\\xi ,r).$ Figure: A depiction of the disc associated with Ψ(z,𝐚,𝐛,r)\\Psi (z,{\\bf a},{\\bf b},r).Another useful change of variables will be needed.",
"This time the function, which is closely related to $\\Xi $ , will allow us to describe all discs whose boundary touches the curve ${\\cal C}$ ; see Figure REF .",
"Define the function $\\Psi :{\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+\\rightarrow \\mathbb {R}^n\\times \\mathbb {R}^n\\times \\mathbb {R}$ by $\\Psi (z,{\\bf a},{\\bf b},r):=(z+r{\\bf a},{\\bf b},r)\\quad \\text{for all}\\ (z,{\\bf a},{\\bf b},r)\\in {\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}.$ It is possible to show that if ${\\cal A}$ is a subset of ${\\cal C}\\times {\\cal U}_\\perp ^2\\times \\mathbb {R}^+$ and $f:\\Psi ({\\cal A})\\rightarrow \\mathbb {R}$ is an integrable function, then $\\int _{\\Psi ({\\cal A})}\\Big [ \\sum _{(z,{\\bf a},{\\bf b},r)\\in \\Psi ^{-1}(p,{\\bf u},r)} f(p,{\\bf u},r)\\Big ]d{\\cal H}^{2n}(p,{\\bf u}, r) \\\\= \\int _{\\cal A}f(z+r{\\bf a},{\\bf b},r)(2r)^{n-2} \\sqrt{({\\bf a}\\cdot {\\bf t}(z))^2+({\\bf b}\\cdot {\\bf t}(z))^2} d{\\cal H}^{2n}(z,{\\bf a},{\\bf b},r).$ The proof of this result is similar to that of Lemma REF and thus will not be presented."
]
] | 1808.08654 |
[
[
"Unique Solutions of Contractions, CCS, and their HOL Formalisation"
],
[
"Abstract The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's \"unique solution of equations\" technique.",
"The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions.",
"(The formalisation consists of about 20,000 lines of proof scripts in Standard ML.)",
"Some refinements of the theory itself are obtained.",
"In particular we remove the constraints on summation, which must be weakly-guarded, by moving to rooted contraction, that is, the coarsest precongruence contained in the contraction preorder."
],
[
"Introduction",
"A prominent proof method for bisimulation, put forward by Robin Milner and widely used in his landmark CCS book [22] is the unique solution of equations, whereby two tuples of processes are componentwise bisimilar if they are solutions of the same system of equations.",
"This method is important in verification techniques and tools based on algebraic reasoning [2], [29], [30].",
"In the versions of Milner's unique solution theorems for proving that all solutions are weakly (or rooted) bisimilar (in practice these are the most relevant cases), however, Milner's proof method has severe syntactical limitations, such that the equations must be “guarded and sequential,” that is, the variables of the equations may only be used underneath a visible prefix and proceed, in the syntax tree, only by the sum and prefix operators.",
"One way of overcoming such limitations is to replace equations with special inequations called contractions [32], [33].",
"Contraction is a preorder that, roughly, places some efficiency constraints on processes.",
"The uniqueness of solutions of a system of contractions is defined as with systems of equations: any two solutions must be bisimilar.",
"The difference with equations is in the meaning of a solution: in the case of contractions the solution is evaluated with respect to the contraction preorder, rather than bisimilarity.",
"With contractions, most syntactic limitations of the unique-solution theorem can be removed.",
"One constraint that still remains in [33] (in which the issue is bypassed using a more restrictive CCS syntax) is the occurrences of direct sums, due to the failure of the substitutivity of contraction under direct sums.",
"The main goal of the work described in this paper is a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus on the theory of unique solutions of contractions.",
"The formalisation, however, is not confined to the theory of unique solutions of equations, but embraces a significant portion the theory of CCS [22] (mostly because the theory of unique solutions relies on a large number of more fundamental results).",
"Indeed the formalisation encompasses the basic properties of strong and weak bisimilarity (e.g.",
"the fixed-point and substitutivity properties), the basic properties of rooted bisimilarity (the congruence induced by weak bisimilarity, also called observation congruence), and their algebraic laws.",
"Further extensions (beyond Nesi [24]) include four versions of “bisimulation up to” techniques (e.g., bisimulation up-to bisimilarity) [22], [34], and the expansion and contraction preorder (two efficiency-like refinements of weak bisimilarity).",
"Concerning rooted bisimilarity, the formalisation includes Hennessy Lemma and Deng Lemma (Lemma 4.1 and 4.2 of [13]), and two long proofs saying the rooted bisimilarity is the coarsest (largest) congruence contained in (weak) bisimilarity: one following Milner's book [22], with the hypothesis that no processes can use up all labels; the other without such hypothesis, essentially formalising van Glabbeek's paper [10].",
"Similar theorems are proved for the rooted contraction preorder.",
"In this respect, the work is an extensive experiment with the use of the HOL theorem prover and its most recent developments, including a package for expressing coinductive definitions.",
"From the view of CCS theory, this formalisation has offered us the possibility of further refining the theory of unique solutions of equations, as formally proving a previously known result gives us a chance to see what's really needed for establishing that result.",
"In particular, the existing theory [33] has placed limitations on the body of the contractions due to the substitutivity problems of weak bisimilarity and other behavioural relations with respect to the sum operator.",
"We have thus refined the contraction-based proof technique, by moving to rooted contraction, that is, the coarsest precongruence contained in the contraction preorder.",
"The resulting unique-solution theorem is now valid for rooted bisimilarity (hence also for bisimilarity itself), and places no constraints on the occurrences of sums.",
"Another advantage of the formalisation is that we can take advantage of results about different equivalences or preorders that share a similar proof structure.",
"Examples are: the results that rooted bisimilarity and rooted contraction are, respectively, the coarsest congruence contained in weak bisimilarity and the coarsest precongruence contained in the contraction preorder; the result about unique solution of equations for weak bisimilarity that uses the contraction preorder as an auxiliary relation, and other unique solution results (e.g., the one for rooted in which the auxiliary relation is rooted contraction); various forms of enhancements of the bisimulation proof method (the `up-to' techniques).",
"In these cases, moving between proofs there are only a few places in which the HOL proof scripts have to be modified.",
"Then the successful termination of the proof gives us a guarantee that the proof is complete and trustworthy, removing the risk of overlooking or missing details as in hand-written proofs.",
"Section presents basic background materials on CCS, including its syntax, operational semantics, bisimilarity and rooted bisimilarity.",
"Section discussed equations and contractions.",
"Section REF presents rooted contraction and the related unique-solution result for rooted bisimilarity.",
"Section highlights our formalisation in HOL4.",
"Finally, Section and discuss related work, conclusions, and a few directions for future work."
],
[
"CCS",
"We assume a possibly infinite set of names ${L} = \\lbrace a, b,\\ldots \\rbrace $ forming input and $\\overline{\\mbox{output}}$ actions, plus a special invisible action $\\tau \\notin {L}$ , and a set of variables $A,B,\\ldots $ for defining recursive behaviours.",
"Given a deadlock 0, the class of CCS processes is then inductively defined from 0 by the operators of prefixing, parallel composition, summation (binary choice), restriction, recursion and relabeling: $\\begin{array}{ccl}\\mu & := & \\tau \\hspace{0.3pt} \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; a \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; \\overline{a} \\\\P & := & {0}\\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; \\mu .",
"P \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; P_1 | P_2 \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\;P_1 + P_2 \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;({\\nu } a\\:\\!",
")\\, P \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; A \\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; {\\tt rec}\\: A .",
"P\\; \\; \\; \\mbox{\\Large {$\\mid $}}\\;\\;\\; P\\; [r\\!\\!f] \\end{array}$ The operational semantics of CCS is given by means of a Labeled Transition System (LTS), shown in Fig.",
"REF as SOS rules (the symmetric version of the two rules for parallel composition and the rule for sum are omitted).",
"A CCS expression uses only weakly-guarded sums if all occurrences of the sum operator are of the form $\\mu _1.P_1 + \\mu _2.P_2 + \\ldots + \\mu _n.P_n$ , for some $n \\ge 2$ .",
"The immediate derivatives of a process $P$ are the elements of the set $\\lbrace P^{\\prime } \\; \\mid \\;P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime } \\mbox{for some $\\mu $}\\rbrace $ .",
"Figure: Structural Operational Semantics of CCSSome standard notations for transitions: $\\mathrel {\\stackrel{{\\;\\;\\epsilon \\;\\;}}{\\mbox{}}}$ is the reflexive and transitive closure of $\\mathrel {\\stackrel{{\\;\\;\\tau \\;\\;}}{\\mbox{}}}$ , and $\\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}$ is $\\mathrel {\\stackrel{{\\;\\;\\epsilon \\;\\;}}{\\mbox{}}}\\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}\\mathrel {\\stackrel{{\\;\\;\\epsilon \\;\\;}}{\\mbox{}}}$ (the composition of the three relations).",
"Moreover, $ P \\mathrel {\\stackrel{{\\;\\; {\\widehat{\\mu }} \\;\\;}}{\\mbox{}}}P^{\\prime }$ holds if $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ or ($\\mu =\\tau $ and $P=P^{\\prime }$ ); similarly $ P \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}P^{\\prime }$ holds if $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ or ($\\mu =\\tau $ and $P=P^{\\prime }$ ).",
"We write $P \\:(\\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}})^n P^{\\prime }$ if $P$ can become $P^{\\prime }$ after performing $n$ $\\mu $ -transitions.",
"Finally, $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}$ holds if there is $P^{\\prime }$ with $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ , and similarly for other forms of transitions.",
"Letters ${\\cal R}$ , ${\\cal S}$ range over relations.",
"We use infix notation for relations, e.g., $P \\mathrel {\\cal R}Q$ means that $(P,Q) \\in {\\cal R}$ .",
"We use a tilde to denote a tuple, with countably many elements; thus the tuple may also be infinite.",
"All notations are extended to tuples componentwise; e.g., $\\widetilde{P} \\mathrel {\\cal R}\\widetilde{Q}$ means that $P_i \\mathrel {\\cal R}Q_i$ , for each component $i$ of the tuples $\\widetilde{P}$ and $\\widetilde{Q}$ .",
"And $ C [\\widetilde{P}] $ is the process obtained by replacing each hole $[\\cdot ]_{i}$ of the context $ C $ with $P_i$ .",
"We write ${\\cal R}^{\\rm {c}}$ for the closure of a relation under contexts.",
"Thus $P\\: {\\cal R}^{\\rm {c}}\\: Q$ means that there are context $ C $ and tuples $\\widetilde{P},\\widetilde{Q}$ with $P = C [\\widetilde{P}] , Q = C [\\widetilde{Q}] $ and $\\widetilde{P} \\mathrel {\\cal R}\\widetilde{Q}$ .",
"We use the symbol $\\stackrel{\\mbox{\\scriptsize def}}{=}$ for abbreviations.",
"For instance, $P \\stackrel{\\mbox{\\scriptsize def}}{=}G $ , where $G$ is some expression, means that $P$ stands for the expression $G$ .",
"If $\\le $ is a preorder, then $\\ge $ is its inverse (and conversely)."
],
[
"Bisimilarity and rooted bisimilarity",
"The equivalences we consider here are mainly weak ones, in that they abstract from the number of internal steps being performed: Definition 2.1 A process relation ${{\\cal R}}$ is a bisimulation if, whenever $P\\mathrel {\\cal R}Q$ , we have: $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ implies that there is $Q^{\\prime }$ such that $Q \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}Q^{\\prime }$ and $P^{\\prime } \\mathrel {\\cal R}Q^{\\prime }$ ; $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ ,implies that there is $P^{\\prime }$ such that $P \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}P^{\\prime }$ and $P^{\\prime } \\mathrel {\\cal R}Q^{\\prime }$ .",
"$P$ and $Q$ are bisimilar, written as $P \\approx Q$ , if $P \\mathrel {\\cal R}Q$ for some bisimulation ${\\cal R}$ .",
"We sometimes call bisimilarity the weak one, to distinguish it from strong bisimilarity ($\\sim $ ), obtained by replacing in the above definition the weak answer $Q\\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}Q^{\\prime }$ with the strong $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ .",
"Weak bisimilarity is not preserved by the sum operator (except for guarded sums).",
"For this, Milner introduced observational congruence, also called rooted bisimilarity [13], [31]: Definition 2.2 Two processes $P$ and $Q$ are rooted bisimilar, written as $P\\mathrel {\\approx ^{\\rm {c}}}Q$ , if we have: $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ implies that there is $Q^{\\prime }$ such that $Q\\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ and $P^{\\prime } \\approx Q^{\\prime }$ ; $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ implies that there is $P^{\\prime }$ such that $P\\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ and $P^{\\prime } \\approx Q^{\\prime }$ .",
"Theorem 2.3 $\\mathrel {\\approx ^{\\rm {c}}}$ is a congruence in CCS, and it is the coarsest congruence contained in $\\approx $ .",
"Uniqueness of solutions of equations [22] intuitively says that if a context $ C $ obeys certain conditions, then all processes $P$ that satisfy the equation $ P \\approx C [P] $ are bisimilar with each other.",
"We need variables to write equations.",
"We use capital letters $X,Y,Z$ for these variables and call them equation variables.",
"The body of an equation is a CCS expression possibly containing equation variables.",
"Thus such expressions, ranged over by $E$ , live in the CCS grammar extended with equation variables.",
"Definition 3.1 Assume that, for each $i$ of a countable indexing set $I$ , we have variables $X_i$ , and expressions $E_i$ possibly containing such variables.",
"Then $\\lbrace X_i = E_i\\rbrace _{i\\in I}$ is a system of equations.",
"(There is one equation for each variable $X_i$ .)",
"We write $E[\\widetilde{P}]$ for the expression resulting from $E$ by replacing each variable $X_i$ with the process $P_i$ , assuming $\\widetilde{P}$ and $\\widetilde{X}$ have the same length.",
"(This is syntactic replacement.)",
"Definition 3.2 Suppose $\\lbrace X_i = E_i\\rbrace _{i\\in I}$ is a system of equations: $\\widetilde{P}$ is a solution of the system of equations for $\\approx $ if for each $i$ it holds that $P_i \\approx E_i [\\widetilde{P}]$ ; it has a unique solution for $\\approx $ if whenever $\\widetilde{P}$ and $\\widetilde{Q}$ are both solutions for $\\approx $ , then $\\widetilde{P} \\approx \\widetilde{Q}$ .",
"For instance, the solution of the equation $ X = a. X$ is the process $R \\stackrel{\\mbox{\\scriptsize def}}{=}{\\tt rec}\\: A .",
"\\, (a.",
"A)$ , and for any other solution $P$ we have $P \\approx R$ .",
"In contrast, the equation $X = a| X$ has solutions that may be quite different, for instance, $K$ and $K | b$ , for $K \\stackrel{\\mbox{\\scriptsize def}}{=}{\\tt rec}\\: K .",
"\\, (a. K)$ .",
"(Actually any process capable of continuously performing $a$ actions (while behaves arbitrarily on other actions) is a solution for $X = a| X$ .)",
"Definition 3.3 (guardedness of equations) A system of equations $\\lbrace X_i = E_i\\rbrace _{i\\in I}$ is weakly guarded if, in each $E_i$ , each occurrence of an equation variable is underneath a prefix; (strongly) guarded if, in each $E_i$ , each occurrence of an equation variable is underneath a visible prefix; sequential if, in each $E_i$ , each of its subexpressions with occurrence of an equation variable, apart from the variable itself, is in forms of prefixes or sums.",
"Theorem 3.4 (unique solution of equations, [22]) A system of guarded and sequential equations (without direct sums) $\\lbrace X_i = E_i\\rbrace _{i\\in I}$ has a unique solution for $\\approx $ .",
"To see the need of the sequentiality condition, consider the equation (from [22]) $X = {\\nu } a\\: (a. X | \\overline{a})$ where $X$ is guarded but not sequential.",
"Any process that does not use $a$ is a solution."
],
[
"Contractions",
"The constraints on the unique-solution Theorem REF can be weakened if we move from equations to a special kind of inequations called contractions.",
"Intuitively, the bisimilarity contraction $\\mathrel {\\succeq _{\\rm {bis}}}$ is a preorder in which $P \\mathrel {\\succeq _{\\rm {bis}}}Q $ holds if $P \\approx Q$ and, in addition, $Q$ has the possibility of being at least as efficient as $P$ (as far as $\\tau $ -actions performed).",
"Process $Q$ , however, may be nondeterministic and may have other ways of doing the same work, and these could be slow (i.e., involving more $\\tau $ -steps than those performed by $P$ ).",
"Definition 3.5 A process relation ${{\\cal R}}$ is a (bisimulation) contraction if, whenever $P\\mathrel {\\cal R}Q$ , $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ implies there is $Q^{\\prime }$ such that $Q \\mathrel {\\stackrel{{\\;\\; {\\widehat{\\mu }} \\;\\;}}{\\mbox{}}}Q^{\\prime }$ and $P^{\\prime } \\mathrel {\\cal R}Q^{\\prime }$ ; $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ implies there is $P^{\\prime }$ such that $P \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}P^{\\prime }$ and $P^{\\prime } \\approx Q^{\\prime }$ .",
"Bisimilarity contraction, written as $P \\mathrel {\\succeq _{\\rm {bis}}}Q$ ($P$ contracts to $Q$ ), if $P\\ {\\cal R}\\ Q$ for some contraction ${\\cal R}$ .",
"In the first clause $Q$ is required to match $P$ 's challenge transition with at most one transition.",
"This makes sure that $Q$ is capable of mimicking $P$ 's work at least as efficiently as $P$ .",
"In contrast, the second clause of Definition REF , on the challenges from $Q$ , entirely ignores efficiency: it is the same clause of weak bisimulation — the final derivatives are even required to be related by $\\approx $ , rather than by ${\\cal R}$ .",
"Bisimilarity contraction is coarser than the expansion relation $\\mathrel {\\succeq _{\\rm {e}}}$ [1], [32].",
"This is a preorder widely used in proof techniques for bisimilarity and that intuitively refines bisimilarity by formalising the idea of `efficiency' between processes.",
"Clause (1) is the same in the two preorders.",
"But in clause (2) expansion uses $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ , rather than $P \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}P^{\\prime }$ ; moreover with contraction the final derivatives are simply required to be bisimilar.",
"An expansion $P \\mathrel {\\succeq _{\\rm {e}}}Q$ tells us that $Q$ is always at least as efficient as $P$ , whereas the contraction $P \\mathrel {\\succeq _{\\rm {bis}}}Q$ just says that $Q$ has the possibility of being at least as efficient as $P$ .",
"Example 3.6 We have $ a \\lnot \\mathrel {\\succeq _{\\rm {bis}}}\\tau .",
"a$ .",
"However, $a+ \\tau .",
"a \\mathrel {\\succeq _{\\rm {bis}}}a$ , as well as its converse, $ a \\mathrel {\\succeq _{\\rm {bis}}}a +\\tau .",
"a $ .",
"Indeed, if $P \\approx Q$ then $ P \\mathrel {\\succeq _{\\rm {bis}}}P +Q$ .",
"The last two relations do not hold with $\\mathrel {\\succeq _{\\rm {e}}}$ , which explains the strictness of the inclusion ${\\mathrel {\\succeq _{\\rm {e}}}} \\subset {\\mathrel {\\succeq _{\\rm {bis}}}}$ .",
"Like (weak) bisimilarity and expansion, contraction is preserved by all operators but (direct) sum."
],
[
"Systems of contractions",
"A system of contractions is defined as a system of equations, except that the contraction symbol $\\mathrel {\\succeq }$ is used in the place of the equality symbol $=$ .",
"Thus a system of contractions is a set $\\lbrace X_i \\mathrel {\\succeq }E_i\\rbrace _{i\\in I}$ where $I$ is an indexing set and expressions $E_i$ may contain the contraction variables $\\lbrace X_i\\rbrace _{i\\in I}$ .",
"Definition 3.7 Given a system of contractions $\\lbrace X_i \\mathrel {\\succeq }E_i\\rbrace _{i\\in I}$ , we say that: $\\widetilde{P}$ is a solution (for $\\mathrel {\\succeq _{\\rm {bis}}}$ ) of the system of contractions if $\\widetilde{P} \\mathrel {\\succeq _{\\rm {bis}}}\\widetilde{E} [\\widetilde{P}]$ ; the system has a unique solution (for $\\approx $ ) if $\\widetilde{P} \\approx \\widetilde{Q}$ whenever $\\widetilde{P}$ and $\\widetilde{Q}$ are both solutions.",
"The guardedness of contractions follows Def.",
"REF (for equations).",
"Lemma 3.8 Suppose $\\widetilde{P}$ and $\\widetilde{Q}$ are solutions for $\\mathrel {\\succeq _{\\rm {bis}}}$ of a system of weakly-guarded contractions that uses weakly-guarded sums.",
"For any context $ C $ that uses weakly-guarded sums, if $ C [\\widetilde{P}] \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} R$ , then there is a context $ C^{\\prime } $ that uses weakly-guarded sums such that $R \\mathrel {\\succeq _{\\rm {bis}}} C^{\\prime } [\\widetilde{P}] $ and $ C [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}\\approx C^{\\prime } [\\widetilde{Q}] $ .There's no typo here: $ C [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} \\approx C^{\\prime } [\\widetilde{Q}] $ means $\\exists {\\widetilde{R}}.\\; C [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} {\\widetilde{R}}\\approx C^{\\prime } [\\widetilde{Q}] $ .",
"Same as in Lemma REF .",
"(sketch from [33]) Let $n$ be the length of the transition $ C [\\widetilde{P}] \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}R$ (the number of `strong steps' of which it is composed), and let $ C^{\\prime \\prime } [\\widetilde{P}] $ and $ C^{\\prime \\prime } [\\widetilde{Q}] $ be the processes obtained from $ C [\\widetilde{P}] $ and $ C [\\widetilde{Q}] $ by unfolding the definitions of the contractions $n$ times.",
"Thus in $ C^{\\prime \\prime } $ each hole is underneath at least $n$ prefixes, and cannot contribute to an action in the first $n$ transitions; moreover all the contexts have only weakly-guarded sums.",
"We have $ C [\\widetilde{P}] \\mathrel {\\succeq _{\\rm {bis}}} C^{\\prime \\prime } [\\widetilde{P}] $ , and $ C [\\widetilde{Q}] \\mathrel {\\succeq _{\\rm {bis}}} C^{\\prime \\prime } [\\widetilde{Q}] $ , by the substitutivity properties of $\\mathrel {\\succeq _{\\rm {bis}}}$ (we exploit here the syntactic constraints on sums).",
"Moreover, since each hole of the context $ C^{\\prime \\prime } $ is underneath at least $n$ prefixes, applying the definition of $ \\mathrel {\\succeq _{\\rm {bis}}}$ on the transition $ C [\\widetilde{P}] \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} R$ , we infer the existence of $ C^{\\prime } $ such that $ C^{\\prime \\prime } [\\widetilde{P}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} C^{\\prime } [\\widetilde{P}] \\mathrel {\\preceq _{\\rm {bis}}}R$ and $ C^{\\prime \\prime } [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} C^{\\prime } [\\widetilde{Q}] .",
"$ Finally, again applying the definition of $\\mathrel {\\succeq _{\\rm {bis}}}$ on $ C [\\widetilde{Q}] \\mathrel {\\succeq _{\\rm {bis}}} C^{\\prime \\prime } [\\widetilde{Q}] $ , we derive $ C [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} \\approx C^{\\prime } [\\widetilde{Q}] .$ Theorem 3.9 (unique solution of contractions for $\\approx $ ) A system of weakly-guarded contractions having only weakly-guarded sums, has a unique solution for $\\approx $ .",
"(sketch from [33]) Suppose $\\widetilde{P}$ and $\\widetilde{Q}$ are two such solutions (for $\\approx $ ) and consider the relation ${\\cal R}\\stackrel{\\mbox{\\scriptsize def}}{=}\\lbrace (R,S) \\; \\mid \\;R \\approx C [\\widetilde{P}] , S \\approx C [\\widetilde{Q}] \\mbox{ for some context$ C $ (weakly-guarded sum only)} \\rbrace \\hspace{5.0pt}.$ We show that ${\\cal R}$ is a bisimulation.",
"Suppose $R\\ {\\cal R}\\ S$ vis the context $C$ , and $R \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} R^{\\prime }$ .",
"We have to find $S^{\\prime }$ with $S \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}}S^{\\prime }$ and $R^{\\prime }\\ {\\cal R}\\ S^{\\prime }$ .",
"From $R \\approx C[{\\widetilde{P}}]$ , we derive $C[{\\widetilde{P}}]\\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} R^{\\prime \\prime } \\approx R^{\\prime }$ for some $R^{\\prime \\prime }$ .",
"By Lemma REF , there is $C^{\\prime }$ with $R^{\\prime \\prime } \\mathrel {\\succeq _{\\rm {bis}}}C^{\\prime }[{\\widetilde{P}}]$ and $C[{\\widetilde{Q}}]\\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} \\approx C^{\\prime }[{\\widetilde{Q}}]$ .",
"Hence, by definition of $\\approx $ , there is also $S^{\\prime }$ with $S \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} S^{\\prime } \\approx C^{\\prime }[{\\widetilde{Q}}]$ .",
"This closes the proof, as we have $R^{\\prime } \\approx C^{\\prime }[{\\widetilde{P}}]$ and $S^{\\prime } \\approx C^{\\prime }[{\\widetilde{Q}}]$ ."
],
[
"Rooted contraction",
"The unique solution theorem of Section REF requires a constrained syntax for sums, due to the congruence and precongruence problems of bisimilarity and contraction with such operator.",
"We show here that the constraints can be removed by moving to the induced congruence and precongruence, the latter called rooted contraction: Definition 3.10 Two processes $P$ and $Q$ are in rooted contraction, written as $P\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}Q$ , if $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ implies that there is $Q^{\\prime }$ with $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ and $P^{\\prime }\\mathrel {\\succeq _{\\rm {bis}}}Q^{\\prime }$ ; $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}Q^{\\prime }$ implies that there is $P^{\\prime }$ with $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}P^{\\prime }$ and $P^{\\prime } \\approx Q^{\\prime }$ .",
"The precise formulation of this definition was guided by the HOL theorem prover and the following two principles: (1) the definition should not be recursive, along the lines of rooted bisimilarity $\\mathrel {\\approx ^{\\rm {c}}}$ in Def.",
"REF ; (2) the definition should be built on top of existing contraction relation $\\mathrel {\\succeq _{\\rm {bis}}}$ (because of its completeness).",
"A few other candidates were quickly tested and rejected, e.g., because of precongruence issue.",
"The proof of the precongruence result below is along the lines of the analogous result for rooted bisimilarity with respect to bisimilarity.",
"Theorem 3.11 $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ is a precongruence in CCS, and it is the coarsest precongruence contained in $\\mathrel {\\succeq _{\\rm {bis}}}$ .",
"For a system of rooted contractions, the meaning of “solution for $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ ” and of a unique solution for $\\mathrel {\\approx ^{\\rm {c}}}$ is the expected one — just replace in Def.",
"REF the preorder $\\mathrel {\\succeq _{\\rm {bis}}}$ with $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ , and the equivalence $\\approx $ with $\\mathrel {\\approx ^{\\rm {c}}}$ .",
"For this new relation, the analogous of Lemma REF and of Theorem REF can now be stated without constraints on the sum operator.",
"The schema of the proofs is almost identical, because all properties of $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ needed in this proof is its precongruence, which is indeed true on unrestricted contexts including direct sums: Lemma 3.12 Suppose $\\widetilde{P}$ and $\\widetilde{Q}$ are solutions for $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ of a system of weakly-guarded contractions.",
"For any context $ C $ , if $ C [\\widetilde{P}] \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} R$ , then there is a context $ C^{\\prime } $ such that $R \\mathrel {\\succeq _{\\rm {bis}}} C^{\\prime } [\\widetilde{P}] $ and $ C [\\widetilde{Q}] \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}}\\approx C^{\\prime } [\\widetilde{Q}] $ .",
"Theorem 3.13 (unique solution of contractions for $\\mathrel {\\approx ^{\\rm {c}}}$ ) A system of weakly-guarded contractions has a unique solution for $\\mathrel {\\approx ^{\\rm {c}}}$ .",
"(thus also for $\\approx $ ) We first follow the same steps as in the proof of Theorem REF to show the relation ${\\cal R}$ (now with $\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ and unrestricted context $C$ ) in (REF ) is bisimulation, exploting Lemma REF .",
"Then it remains to show that, for any two process $P$ and $Q$ with action $\\mu $ , if $P \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} P^{\\prime }$ then there is $Q^{\\prime }$ such that $Q \\mathrel {\\stackrel{{\\;\\;\\mu \\;\\;}}{\\mbox{}}} Q^{\\prime }$ (not $Q \\mathrel {\\stackrel{{\\;\\;{\\widehat{\\mu }}\\;\\;}}{\\mbox{}}} Q^{\\prime }$ !)",
"and $P^{\\prime }\\ {\\cal R}\\ Q^{\\prime }$ , and also for the converse direction, exploting Lemma 4.13 of [22] (surprisingly).",
"By definition of bisimulation (not $\\approx $ !)",
"and $\\approx ^c$ , we actually proved $P\\approx ^c Q$ instead of $P \\approx Q$ .",
"We highlight here a formalisation of CCS in the HOL theorem prover (HOL4) [36], including the new concepts and theorems proposed in the first half of this paper.",
"The whole formalisation (apart from minor fixes and extensions in this paper) is described in [37], and the proof scripts are in HOL4 official exampleshttps://github.com/HOL-Theorem-Prover/HOL/tree/master/examples/CCS.",
"The current work consists of about 20,000 lines of proof scripts in Standard ML.",
"Higher Order Logic (or HOL Logic) [16], which traces its roots back to LCF [11], [21] by Robin Milner and others since 1972, is a variant of Church’s simple theory of types (STT) [6], plus a higher order version of Hilbert's choice operator $\\varepsilon $ , Axiom of Infinity, and Rank-1 (prenex) polymorphism.",
"HOL4 has implemented the original HOL Logic, while some other theorem provers in HOL family (e.g.",
"Isabelle/HOL) have certain extensions.",
"Indeed the HOL Logic has considerable simpler logical foundations than most other theorem provers.",
"As a consequence, formal theories built in HOL is easily convincible and can also be easily ported to other proof systems, sometimes automatically [18].",
"HOL4 is written in Standard ML, a single programming language which plays three different roles: It serves as the underlying implementation language for the core HOL engine; it is used to implement tactics (and tacticals) for writing proofs; it is used as the command language of the HOL interactive shell.",
"Moreover, using the same language HOL4 users can write complex automatic verification tools by calling HOL's theorem proving facilities.",
"(The formal proofs of theorems in CCS theory are mostly done by an interactive process closely following their informal proofs, with minimal automatic proof searching.)",
"In this formalisation we consider only single-variable equations/contractions.",
"This considerably simplifies the required proofs in HOL, also enhances the readability of proof scripts without loss of generality.",
"(For paper proofs, the multi-variable case is just a routine adaptation.)"
],
[
"CCS and its transitions by SOS rules",
"In our CCS formalisation, the type “$\\beta $ Label” ('b or $\\beta $ is the type variable for actions) accounts for visible actions, divided into input and output actions, defined by HOL's Datatype package: val _ = Datatype `Label = name 'b | coname 'b`; The type “$\\beta $ Action” is the union of all visible actions, plus invisible action $\\tau $ (now based on HOL's option theory).",
"The cardinality of “$\\beta $ Action” (and therefore of all CCS types built on top of it) depends on the choice (or type-instantiation) of $\\beta $ .",
"The type “($\\alpha $ , $\\beta $ ) CCS”, accounting for the CCS syntaxThe order of type variables $\\alpha $ and $\\beta $ is irrelevant.",
"Our choice is aligned with other CCS literals.",
"$\\mathrm {CCS}(h,k)$ is the CCS subcalculus which can use at most $h$ constants and $k$ actions.",
"[12] Thus, to formalize theorems on such a CCS subcalculus, the needed CCS type can be retrieved by instantiating the type variables $\\alpha $ and $\\beta $ in “($\\alpha $ , $\\beta $ ) CCS” with types having the corresponding cardinalities $h$ and $k$ .",
"Monica Nesi goes too far by adding another type variable $\\gamma $ for value-passing CCS [25]., is then defined inductively: ('a or $\\alpha $ is the type variable for recursion variables, “$\\beta $ Relabeling” is the type of all relabeling functions, ` is for backquotes of HOL terms): val _ = Datatype `CCS = nil \t\t | var 'a \t\t | prefix ('b Action) CCS \t\t | sum CCS CCS \t\t | par CCS CCS \t\t | restr (('b Label) set) CCS \t\t | relab CCS ('b Relabeling) \t\t | rec 'a CCS`; We have added some grammar support, using HOL's powerful pretty printer, to represent CCS processes in more readable forms (c.f.",
"the column HOL (abbrev.)",
"in Table REF , which summarizes the main syntactic notations of CCS).",
"For the restriction operator, we have chosen to allow a set of names as a parameter, rather than a single name as in the ordinary CCS syntax; this simplifies the manipulation of processes with different orders of nested restrictions.",
"Table: Syntax of CCS operators, constant and actionsThe transition semantics of CCS processes follows Structural Operational Semantics (SOS) in Fig.",
"REF : u..P $-$ u$\\rightarrow $ P[PREFIX] P $-$ u$\\rightarrow $ P P $+$ Q $-$ u$\\rightarrow $ P[SUM1] P $-$ u$\\rightarrow $ P Q $+$ P $-$ u$\\rightarrow $ P[SUM2] P $-$ u$\\rightarrow $ P P $\\parallel $ Q $-$ u$\\rightarrow $ P $\\parallel $ Q[PAR1] P $-$ u$\\rightarrow $ P Q $\\parallel $ P $-$ u$\\rightarrow $ Q $\\parallel $ P[PAR2] P $-$ label l$\\rightarrow $ P Q $-$ label (COMPL l)$\\rightarrow $ Q P $\\parallel $ Q $-$$\\tau $$\\rightarrow $ P $\\parallel $ Q[PAR3] P $-$ u$\\rightarrow $ Q ((u = $\\tau $ ) (u = label l) l L COMPL l L) $\\nu $ L P $-$ u$\\rightarrow $ $\\nu $ L Q[RESTR] P $-$ u$\\rightarrow $ Q relab P rf $-$ relabel rf u$\\rightarrow $ relab Q rf[RELABELING] CCS_Subst P (rec A P) A $-$ u$\\rightarrow $ P rec A P $-$ u$\\rightarrow $ P[REC] The rule REC (Recursion) says that if we substitute all appearances of variable $A$ in $P$ to $({\\tt rec}\\: A .",
"P)$ and the resulting process has a transition to $P^{\\prime }$ with action $u$ , then $({\\tt rec}\\: A .",
"P)$ has the same transition.",
"From HOL's viewpoint, these SOS rules are inductive definitions on the tenary relation TRANS of type “($\\alpha $ , $\\beta $ ) CCS $\\rightarrow $$\\beta $ Action $\\rightarrow $ ($\\alpha $ , $\\beta $ ) CCS $\\rightarrow $ bool”, generated by HOL's Holreln function.",
"A useful function that we have defined, exploiting the interplay between HOL4 and Standard ML (and following an idea by Nesi [24]) is a complex Standard ML function taking a CCS process and returning a theorem indicating all its direct transitions.If the input process could yield something infinite branching, due to the use of recursion or relabeling operators, the program will loop forever without outputting a theorem.",
"For instance, we know that the process $(a.0 | \\bar{a}.0)$ has three possible transitions: $(a.0 | \\bar{a}.0) \\overset{a}{\\longrightarrow }(0 | \\bar{a}.0)$ , $(a.0 | \\bar{a}.0)\\overset{\\bar{a}}{\\longrightarrow } (a.0 | 0)$ and $(a.0 | \\bar{a}.0)\\overset{\\tau }{\\longrightarrow } (0 | 0)$ .",
"To completely describe all possible transitions of a process, if done manually, the following facts should be proved: (1) there exists transitions from $(a.0 | \\bar{a}.0)$ (optional); (2) the correctness for each of the transitions; and (3) the non-existence of other transitions.",
"For large processes it may be surprisingly hard to manually prove the non-existence of transitions.",
"Hence the usefulness of appealing to the new function CCS_TRANS_CONV.",
"For instance this function is called on the process $(a.0 | \\bar{a}.0)$ thus: (“ is for double-backquotes of HOL terms, > is HOL's prompt) > CCS_TRANS_CONV ``par (prefix (label (name \"a\")) nil) (prefix (label (coname \"a\")) nil)`` This returns the following theorem, indeed describing all immediate transitions of the process: In a..nil $\\parallel $ Out a..nil $-$ u$\\rightarrow $ E ((u = In a) (E = nil $\\parallel $ Out a..nil) (u = Out a) (E = In a..nil $\\parallel $ nil)) (u = $\\tau $ ) (E = nil $\\parallel $ nil)[Example.ex_A]"
],
[
"Bisimulation and Bisimilarity",
"To define (weak) bisimilarity, we first need to define weak transitions of CCS processes.",
"Following the name adopted by Nesi [24], we define a (possibly empty) sequence of $\\tau $ -transitions between two processes as a new relation called EPS ($\\overset{\\epsilon }{\\Rightarrow }$ ), which is the RTC (reflexive transitive closure, denoted by $^*$ in HOL4) of ordinary $\\tau $ -transitions of CCS processes: EPS = (E E. E $-$$\\tau $$\\rightarrow $ E)[EPS_def] Then we can define a weak transition as an ordinary transition wrapped by two $\\epsilon $ -transitions: E $=$ u$\\Rightarrow $ E E1 E2.",
"E $\\overset{\\epsilon }{\\Rightarrow }$ E1 E1 $-$ u$\\rightarrow $ E2 E2 $\\overset{\\epsilon }{\\Rightarrow }$ E[WEAK_TRANS] For the definition of bisimilarity and the associated coinduction principle [35], we have taken advantage of HOL's coinductive relation package (Holcoreln [15]), a new tool since its Kananaskis-11 release (March 3, 2017).https://hol-theorem-prover.org/kananaskis-11.release.html#new-tools This essentially amounts to defining bisimilarity as the greatest fixed-point of the appropriate functional on relations.",
"Precisely we call the Holcoreln command as follows: (here WB is meant to be WEAKEQUIV ($\\approx $ ) in the rest of this paper; !",
"and ?",
"stand for universal and existential quantifiers.)",
"val (WB_rules, WB_coind, WB_cases) = Hol_coreln ` (!(P :('a, 'b) CCS) (Q :('a, 'b) CCS).",
"(!l.",
"(!P'.",
"TRANS P (label l) P' ==> \t (?Q'.",
"WEAK_TRANS Q (label l) Q' /\\ WB P' Q')) /\\ \t (!Q'.",
"TRANS Q (label l) Q' ==> \t (?P'.",
"WEAK_TRANS P (label l) P' /\\ WB P' Q'))) /\\ (!P'. TRANS P tau P' ==> (?Q'.",
"EPS Q Q' /\\ WB P' Q')) /\\ (!Q'. TRANS Q tau Q' ==> (?P'.",
"EPS P P' /\\ WB P' Q')) ==> WB P Q)`; Holcoreln returns 3 theorems, of the first being always the same as input termOur mixing of HOL notation and mathematical notation in this paper is not arbitrary.",
"We have to paste here the original proof scripts, which is written in HOL's ASCII term notation (c.f.",
"[15] for more details).",
"HOL4 also supports writing Unicode symbols directly in proof scripts but we did not make use of them.",
"However, all formal definitions and theorems in the paper are automatically generated from HOL4 in which we have made an effort for generating Unicode and TeX outputs as natural as possible.",
"What is really arbitrary is the presense/absense of outermost universal quantifiers in all generated theorems.",
"(now proved automatically as a theorem).",
"The second and third theorems, namely WBcoind and WBcases, express the coinduction proof method for bisimilarity (i.e.",
"any bisimulation is contained in bisimilarity) and the fixed-point property of bisimilarity (bisimilarity itself is a bisimulation, thus the largest bisimulation): WB.",
"(a0 a1.",
"WB a0 a1 (l. (P. a0 $-$ label l$\\rightarrow $ P Q. a1 $=$ label l$\\Rightarrow $ Q WB P Q) Q. a1 $-$ label l$\\rightarrow $ Q P. a0 $=$ label l$\\Rightarrow $ P WB P Q) (P. a0 $-$$\\tau $$\\rightarrow $ P Q. a1 $\\overset{\\epsilon }{\\Rightarrow }$ Q WB P Q) Q. a1 $-$$\\tau $$\\rightarrow $ Q P. a0 $\\overset{\\epsilon }{\\Rightarrow }$ P WB P Q) a0 a1.",
"WB a0 a1 WB a0 a1[WB_coind, WEAK_EQUIV_coind] a0 a1.",
"WB a0 a1 (l. (P. a0 $-$ label l$\\rightarrow $ P Q. a1 $=$ label l$\\Rightarrow $ Q WB P Q) Q. a1 $-$ label l$\\rightarrow $ Q P. a0 $=$ label l$\\Rightarrow $ P WB P Q) (P. a0 $-$$\\tau $$\\rightarrow $ P Q. a1 $\\overset{\\epsilon }{\\Rightarrow }$ Q WB P Q) Q. a1 $-$$\\tau $$\\rightarrow $ Q P. a0 $\\overset{\\epsilon }{\\Rightarrow }$ P WB P Q[WB_cases, WEAK_EQUIV_cases] The coinduction principle WBcoind says that any bisimulation is contained in the resulting relation (i.e.",
"it is largest), but it didn't constrain the resulting relation in the set of fixed points (e.g.",
"even the universal relation — the set of all pairs — would fit with this theorem); the purpose of WBcases is to further assert that the resulting relation is indeed a fixed point.",
"Thus WBcoind and WBcases together make sure that bisimilarity is the greatest fixed point, as the former contributes to “greatest” while the latter contributes to “fixed point”.",
"Without HOL's coinductive relation package, bisimilarity would have to be defined by following literally Def.",
"REF ; then other properties of bisimilarity, such as the fixed-point property in WBcases, would have to be derived manually (which is quite hard; indeed it was one of the main results in Nesi's formalisation work in HOL88 [24])."
],
[
"Context, guardedness and (pre)congruence",
"We have chosen to use $\\lambda $ -expressions (with the type “($\\alpha $ , $\\beta $ ) CCS $\\rightarrow $ ($\\alpha $ , $\\beta $ ) CCS”) to represent multi-hole contexts.",
"This choice has a significant advantage over one-hole contexts, as each hole corresponds to one appearance of the same variable in single-variable expressions (or equations).",
"Thus contexts can be directly used in formulating the unique solution of equations theorems in single-variable cases.",
"The precise definition is given inductively: CONTEXT (t. t) CONTEXT (t. p) CONTEXT e CONTEXT (t. a..e t) CONTEXT e1 CONTEXT e2 CONTEXT (t. e1 t $+$ e2 t) CONTEXT e1 CONTEXT e2 CONTEXT (t. e1 t $\\parallel $ e2 t) CONTEXT e CONTEXT (t. $\\nu $ L (e t)) CONTEXT e CONTEXT (t. relab (e t) rf)[CONTEXT_rules] A context is weakly guarded (WG) if each hole is underneath a prefix: WG (t. p) CONTEXT e WG (t. a..e t) WG e1 WG e2 WG (t. e1 t $+$ e2 t) WG e1 WG e2 WG (t. e1 t $\\parallel $ e2 t) WG e WG (t. $\\nu $ L (e t)) WG e WG (t. relab (e t) rf)[WG_rules] A context is (strongly) guarded (SG) if each hole is underneath a visible prefix: SG (t. p) CONTEXT e SG (t. label l..e t) SG e SG (t. a..e t) SG e1 SG e2 SG (t. e1 t $+$ e2 t) SG e1 SG e2 SG (t. e1 t $\\parallel $ e2 t) SG e SG (t. $\\nu $ L (e t)) SG e SG (t. relab (e t) rf)[SG_rules] A context is sequential (SEQ) if each of its subcontexts with a hole, apart from the hole itself, is in forms of prefixes or sums: (c.f.",
"Def.",
"REF and p.101,157 of [22] for the informal definitions.)",
"SEQ (t. t) SEQ (t. p) SEQ e SEQ (t. a..e t) SEQ e1 SEQ e2 SEQ (t. e1 t $+$ e2 t)[SEQ_rules] In the same manner, we can also define variants of contexts (GCONTEXT) and weakly guarded contexts (WGS) in which only guarded sums are allowed (i.e.",
"arbitrary sums are forbidden): GCONTEXT (t. t) GCONTEXT (t. p) GCONTEXT e GCONTEXT (t. a..e t) GCONTEXT e1 GCONTEXT e2 GCONTEXT (t. a1..e1 t $+$ a2..e2 t) GCONTEXT e1 GCONTEXT e2 GCONTEXT (t. e1 t $\\parallel $ e2 t) GCONTEXT e GCONTEXT (t. $\\nu $ L (e t)) GCONTEXT e GCONTEXT (t. relab (e t) rf)[GCONTEXT_rules] WGS (t. p) GCONTEXT e WGS (t. a..e t) GCONTEXT e1 GCONTEXT e2 WGS (t. a1..e1 t $+$ a2..e2 t) WGS e1 WGS e2 WGS (t. e1 t $\\parallel $ e2 t) WGS e WGS (t. $\\nu $ L (e t)) WGS e WGS (t. relab (e t) rf)[WGS_rules] A (pre)congruence is a relation on CCS processes defined on top of CONTEXT.",
"The only difference between congruence and precongruence is that the former must be an equivalence (reflexive, symmetric, transitive), while the latter can be just a preorder (reflexive, transitive): congruence R equivalence R x y ctx.",
"CONTEXT ctx R x y R (ctx x) (ctx y)[congruence_def] [precongruence_def] precongruence R PreOrder R x y ctx.",
"CONTEXT ctx R x y R (ctx x) (ctx y) For example, we can prove that, strong bisimilarity ($\\sim $ ) and rooted bisimilarity ($\\approx ^c$ ) are both congruence by above definition: (the transitivity proof of rooted bisimilarity is actually not easy.)",
"congruence STRONG_EQUIV[STRONG_EQUIV_congruence] congruence OBS_CONGR[OBS_CONGR_congruence] Although weak bisimilarity ($\\approx $ ) is not congruence with respect to CONTEXT, it is indeed “congruence” with respect to GCONTEXT (or if the CCS syntax were defined with only guarded sum operator [32]) as weak bisimilarity ($\\approx $ ) is indeed preserved by weakly-guarded sums."
],
[
"Coarsest (pre)congruence contained in $\\approx $ ({{formula:beecd12d-0426-4bd1-985f-a2b8c8b8eb29}} )",
"As bisimilarity ($\\approx $ ) is not congruence, for this reason rooted bisimilarity has been introduced (Def.",
"REF ).",
"In this subsection we discuss two proofs of an important result stating that rooted bisimilarity is the coarsest congruence contained in bisimilarity [10], [13], [22] (thus it is the best one): $\\forall p\\ \\ q.\\ p\\ \\mathrel {\\approx ^{\\rm {c}}}\\ \\!",
"q\\ \\Longleftrightarrow \\ ( \\forall r.\\ p\\ +\\ r\\ \\approx \\ q\\ +\\ r )\\hspace{5.0pt}.$ Actually the coarsest congruence contained in (weak) bisimilarity, namely the bisimilarity congruence [10], can be constructed as the composition closure (CC) of (weak) bisimilarity: WEAK_CONGR = CC WEAK_EQUIV[WEAK_CONGR] CC R = (g h. c. CONTEXT c R (c g) (c h))[CC_def] Indeed, for any relation $R$ on CCS processes, the composition closure of $R$ is always finer (i.e.",
"smaller) than $R$ , no matter if $R$ is (pre)congruence or notBut if $R$ is equivalence (or preorder), the composition closure of $R$ must be congruence (or precongruence).",
"Also there is no need to put $R\\ g\\ h$ in the antecedent of CC_def, as this is anyhow obtained from the trivial context $(\\lambda x.\\,x)$ .",
": (here $\\subseteq _r$ stands for relational subset) R. CC R R[CC_is_finer] Furthermore, we prove that any (pre)congruence contained in $R$ (which itself may not be) is contained in the composition closure of $R$ (hence the closure is the coarsest one): R R. congruence R R R R CC R[CC_is_coarsest] R R. precongruence R R R R CC R[CC_is_coarsest'] Given the central role of the sum operator, we also consider the closure of bisimilarity under such operator, called equivalence compatible with sums (SUMEQUIV): SUM_EQUIV = (p q. r. p $+$ r $\\approx $ q $+$ r)[SUM_EQUIV] Rooted bisimilarity $\\mathrel {\\approx ^{\\rm {c}}}$ (a congruence contained in $\\approx $ ), is now contained in WEAKCONGR, which in turn is trivially contained in SUMEQUIV, as shown in Fig.",
"REF .",
"Thus, to prove (REF ), the crux is to prove that SUMEQUIV implies rooted bisimilarity ($\\mathrel {\\approx ^{\\rm {c}}}$ ), making all three relations ($\\mathrel {\\approx ^{\\rm {c}}}$ , WEAKCONGR and SUMEQUIV) equivalent: $\\forall p\\ \\ q.\\ ( \\forall r.\\ p\\ +\\ r \\;\\approx \\; q\\ +\\ r ) \\ \\Rightarrow \\ p\\ \\mathrel {\\approx ^{\\rm {c}}}\\ \\!",
"q\\hspace{5.0pt}.$ Figure: Relationship between the equivalences mentionedThe standard argument [22] requires that $p$ and $q$ do not use up all available labels (i.e.",
"visible actions).",
"Formalising such an argument requires however a detailed treatment on free and bound names of CCS processes (with the restriction operator being a binder), not done yet.",
"However, the proof of (REF ) can be carried out just assuming that all immediate weak derivatives of $p$ and $q$ do not use up all available labels.",
"We have formalised this property and called it the free action property: free_action p a. p. (p $=$ label a$\\Rightarrow $ p)[free_action_def] With this property, the actual formalisation of (REF ) says: [COARSEST_CONGR_RL] free_action p free_action q (r. p $+$ r $\\approx $ q $+$ r) p $\\approx ^c\\!$ q With an almost identical proof, rooted contraction ($\\mathrel {\\succeq ^{\\rm {c}}_{\\rm {bis}}}$ ) is also the coarsest precongruence contained in bisimilarity contraction ($\\mathrel {\\succeq _{\\rm {bis}}}$ ) (the other direction of (REF ) is trivial): [COARSEST_PRECONGR_RL] free_action p free_action q (r. p $+$ r $\\succeq _{bis}\\!$ q $+$ r) p $\\succeq ^c_{bis}\\!$ q The formal proofs of above two results precisely follow Milner [22].",
"If only $p$ (or $q$ ) has free actions while the other uses up all available labels, the classic assumption $\\mathrm {fn}(p) \\cup \\mathrm {fn}(q) \\ne {L}$ (here $\\mathrm {fn}$ stands for free names) does not hold, and the proof cannot be completed.",
"Our assumption is a bit weaker in the sense that, $p$ and $q$ do not really need to have the same free action (also, $a$ and $\\overline{a}$ are different actions).",
"There exists a different, more complex proof of (REF ), given by van Glabbeek [10], which does not require any additional assumption.",
"The core lemma says, for any two processes $p$ and $q$ , if there exists a stable (i.e.",
"$\\tau $ -free) process $k$ which is not bisimilar with any derivative of $p$ and $q$ , then SUMEQUIV indeed implies rooted bisimilarity ($\\mathrel {\\approx ^{\\rm {c}}}$ ): (k. STABLE k (p u. p $=$ u$\\Rightarrow $ p (p $\\approx $ k)) q u. q $=$ u$\\Rightarrow $ q (q $\\approx $ k)) (r. p $+$ r $\\approx $ q $+$ r) p $\\approx ^c\\!$ q[PROP3_COMMON] STABLE p u p. p $-$ u$\\rightarrow $ p u $\\tau $[STABLE] To actually get this process $k$ , the proof relies on arbitrary infinite sum of processes and uses transfinite induction to obtain an arbitrary large sequence of processes (firstly introduced by Jan Willem Klop [10]) that are all pairwise non-bisimilar.",
"We have partially formalised this proof, because the typed logic implemented in various HOL systems (including Isabelle/HOL) is not strong enough to define a type for all possible ordinal values [26], thus we have replaced transfinite induction with plain induction.",
"As a consequence, the final result is about a restricted class of processes (which we have taken to be the finite-state processes).",
"This proof uses extensively HOL's predset theory [19] and has an interesting mix of CCS and pure mathematics in it.",
"(c.f.",
"[37] for more details.)"
],
[
"Unique solution of contractions",
"A delicate point in the formalisation of the results about unique solution of contractions are the proof of Lemma REF and lemmas alike; in particular, there is an induction on the length of weak transitions.",
"For this, rather than introducing a refined form of weak transition relation enriched with its length, we found it more elegant to work with traces (a motivation for this is to set the ground for extensions of this formalisation work to trace equivalence in place of bisimilarity).",
"A trace is represented by the initial and final processes, plus a list of actions so performed.",
"For this, we first define the concept of label-accumulated reflexive transitive closure (LRTC).",
"Given a labeled transition relation R on CCS, LRTC R is a label-accumulated relation representing the trace of transitions: LRTC R a l b P. (x. P x [] x) (x h y t z. R x h y P y t z P x (h::t) z) P a l b[LRTC_DEF] The trace relation for CCS can be then obtained by calling LRTC on the (strong, or single-step) labeled transition relation TRANS ($\\overset{\\mu }{\\rightarrow }$ ) defined by SOS rules: TRACE = LRTC TRANS[TRACE_def] If the list of actions is empty, that means that there is no transition and hence, if there is at most one visible action (i.e., a label) in the list of actions, then the trace is also a weak transition.",
"Here we have to distinguish between two cases: no label and unique label (in the list of actions).",
"The definition of “no label” in an action list is easy (here MEM tests if a given element is a member of a list): NO_LABEL L l. MEM (label l) L[NO_LABEL_def] The definition of “unique label” can be done in many ways, the following definition (a suggestion from Robert Beers) avoids any counting or filtering in the list.",
"It says that a label is unique in a list of actions if and only if there is no label in the rest of list: UNIQUE_LABEL u L L1 L2.",
"(L1 [u] L2 = L) NO_LABEL L1 NO_LABEL L2[UNIQUE_LABEL_def] The final relationship between traces and weak transitions is stated and proved in the following theorem (where the variable $acts$ stands for a list of actions); it says, a weak transition $P\\overset{u}{\\Rightarrow }P^{\\prime }$ is also a trace $P\\overset{acts}{\\longrightarrow }P^{\\prime }$ with a non-empty action list $acts$ , in which either there is no label (for $u = \\tau $ ), or $u$ is the unique label (for $u \\ne \\tau $ ): P $=$ u$\\Rightarrow $ P acts.",
"TRACE P acts P NULL acts if u = $\\tau $ then NO_LABEL acts else UNIQUE_LABEL u acts[WEAK_TRANS_AND_TRACE] Now the formalised version of Lemma REF : [UNIQUESOLUTIONOFCONTRACTIONSLEMMA] (E. WGS E P $\\succeq _{bis}\\!$ E P Q $\\succeq _{bis}\\!$ E Q) C. GCONTEXT C (l R. C P $=$ label l$\\Rightarrow $ R C. GCONTEXT C R $\\succeq _{bis}\\!$ C P (WEAK_EQUIV (x y. x $=$ label l$\\Rightarrow $ y)) (C Q) (C Q)) R. C P $=$$\\tau $$\\Rightarrow $ R C. GCONTEXT C R $\\succeq _{bis}\\!$ C P (WEAK_EQUIV EPS) (C Q) (C Q) Traces are actually used in the proof of above lemma via the following “unfolding lemma”: GCONTEXT C WGS E TRACE ((C FUNPOW E n) P) xs P LENGTH xs n C. GCONTEXT C (P = C P) Q.",
"TRACE ((C FUNPOW E n) Q) xs (C Q)[unfolding_lemma4] It roughly says, for any context $C$ and weakly-guarded context $E$ , if $C [\\, E^n[P]\\,] \\overset{xs}{\\Longrightarrow } P^{\\prime }$ and the length of actions $xs \\leqslant n$ , then $P$ has the form of $C^{\\prime }[P]$ (meaning that $P$ is not touched during the transitions).",
"Traces are used for reasoning about the number of intermediate actions in weak transitions.",
"For instance, from Def.",
"REF , it is easy to see that, a weak transition either becomes shorter or remains the same when moving between $\\mathrel {\\succeq _{\\rm {bis}}}$ -related processes.",
"This property is essential in the proof of Lemma REF .",
"We show only one such lemma, for the case of non-$\\tau $ weak transitions passing into $\\mathrel {\\succeq _{\\rm {bis}}}$ (from left to right): P $\\succeq _{bis}\\!$ Q xs l P. TRACE P xs P UNIQUE_LABEL (label l) xs xs Q.",
"TRACE Q xs Q P $\\succeq _{bis}\\!$ Q LENGTH xs LENGTH xs UNIQUE_LABEL (label l) xs[contracts_AND_TRACE_label] With all above lemmas, we can thus finally prove Theorem REF : WGS E P Q. P $\\succeq _{bis}\\!$ E P Q $\\succeq _{bis}\\!$ E Q P $\\approx $ Q [UNIQUE_SOLUTION_OF_CONTRACTIONS]"
],
[
"Unique solution of rooted contractions",
"The formal proof of “unique solution of rooted contractions theorem” (Theorem REF ) has the same initial proof steps as Theorem REF ; it then requires a few more steps to handle rooted bisimilarity in the conclusion.",
"Overall the two proofs are very similar, mostly because the only property we need from (rooted) contraction is its precongruence.",
"Below is the formally verified version of Theorem REF (having proved the precongruence of rooted contraction, we can use weakly-guarded expressions, without constraints on sums; that is, WG in place of WGS): [UNIQUE_SOLUTION_OF_ROOTED_CONTRACTIONS] WG E P Q. P $\\succeq ^c_{bis}\\!$ E P Q $\\succeq ^c_{bis}\\!$ E Q P $\\approx ^c\\!$ Q Having removed the constraints on sums, the result is similar to Milner's original `unique solution of equations' theorem for strong bisimilarity ($\\sim $ ) — the same weakly guarded context (WG) is required: WG E P Q. P $\\sim $ E P Q $\\sim $ E Q P $\\sim $ Q[STRONG_UNIQUE_SOLUTION] In contrast, Milner's “unique solution of equations” theorem for rooted bisimilarity ($\\mathrel {\\approx ^{\\rm {c}}}$ ) has more severe constraints (must be both strongly guarded and sequential): [OBS_UNIQUE_SOLUTION] SG E SEQ E P Q. P $\\approx ^c\\!$ E P Q $\\approx ^c\\!$ E Q P $\\approx ^c\\!$ Q"
],
[
"Related work on formalisation",
"Monica Nesi did the first CCS formalisations for both pure and value-passing CCS [24], [25] using early versions of the HOL theorem prover.Part of this work can now be found at https://github.com/binghe/HOL-CCS/tree/master/CCS-Nesi.",
"Her main focus was on implementing decision procedures (as a ML program, e.g.",
"[7]) for automatically proving bisimilarities of CCS processes.",
"Her work is the working basis of ours.",
"However, the differences are substantial, especially in the way of defining bisimilarities.",
"We greatly benefited from features and standard libraries in recent versions of HOL4, and our formalisation has covered a much larger spectrum of the (pure) CCS theory.",
"Bengtson, Parrow and Weber did a substantial formalisation work on CCS, $\\pi $ -calculi and $\\psi $ -calculi using Isabelle/HOL and its nominal logic, with main focus on the handling of name binders [3], [4], [27].",
"Other formalisations in this area include the early work of T. F. Melham [20] and O.A.",
"Mohamed [23] in HOL, Compton [8] in Isabelle/HOL, Solangehttps://github.com/coq-contribs/ccs in Coq and Chaudhuri et al.",
"[5] in Abella, the latter focuses on `bisimulation up-to' techniques for CCS and $\\pi $ -calculus.",
"Damien Pous [28] also formalised up-to techniques and some CCS examples in Coq.",
"Formalisations less related to ours include Kahsai and Miculan [17] for the spi calculus in Isabelle/HOL, and Hirschkoff [14] for the $\\pi $ -calculus in Coq."
],
[
"Conclusions and future work",
"In this paper, we have highlighted a formalisation of the theory of CCS in the HOL4 theorem prover (for lack of space we have not discussed the formalisation of some basic algebraic theory, of the basic properties of the expansion preorder, and of a few versions of `bisimulation up to' techniques).",
"The formalisation has focused on the theory of unique solution of equations and contractions.",
"It has also allowed us to further develop the theory, notably the basic properties of rooted contraction, and the unique solution theorem for it with respect to rooted bisimilarity.",
"The formalisation brings up and exploits similarities between results and proofs for different equivalences and preorders.",
"We think that the statements in the formalisation are easy to read and understand, as they are very close to the original statements found in standard CCS textbooks [13], [22].",
"For the future work, it would be worth extending to multi-variable equations/contractions.",
"A key aspect could be using unguarded constants as free variables (FV) and defining guardedness directly on expressions of type CCS (instead of CCS $\\rightarrow $ CCS), then linking to contexts.",
"For instance, an expression is weakly-guarded when each of its free variables, replaced by a hole, results in a weakly-guarded context: weakly_guarded1 E X. X FV E e. CONTEXT e (e (var X) = E) WG e Formalising other equivalences and preorders could also be considered, notably the trace equivalences, as well as more refined process calculi such as value-passing CCS.",
"On another research line, one could examine the formalisation of a different approach [9] to unique solutions, in which the use of contraction is replaced by semantic conditions on process transitions such as divergence."
],
[
"Acknowledgements",
"We have benefitted from suggestions and comments from several people from the HOL community, including (in alphabet order) Robert Beers, Jeremy Dawson, Ramana Kumar, Michael Norrish, Konrad Slind, and Thomas Türk.",
"The second half of this paper was written in memory of Michael J. C. Gordon, the creator of HOL theorem prover."
]
] | 1808.08652 |
[
[
"Asymptotically good edge correspondence colouring"
],
[
"Abstract We prove that every simple graph with maximum degree $\\Delta$ has an edge correspondence colouring with $\\Delta+o(\\Delta)$ colours."
],
[
"Introduction",
"Graph colouring is one of the richest and most fundamental fields of graph theory.",
"In its most basic form, one must assign a colour from a given set to each vertex of a graph so that the endpoints of each edge get different colours.",
"Many variations have arisen, one of the most fruitful being list colouring: Vizing [29] and Erdős, Rubin Taylor [12] independently suggested that rather than assigning colours to all vertices from a single set, we can give each vertex $v$ its own list of permissable colours, $L(v)$ .",
"This very natural variation grew into a prominent subfield of graph colouring.",
"Recently, Dvor̆ák and Postle [10] introduced another natural variation, correspondence colouring.",
"Rather than using the same colouring rule for all edges, each edge can forbid a different set of pairs of colours on its endpoints.",
"The only requirement is that no colour can be forbidden to a vertex by two pairs on the same edge.",
"Specifically, each edge $uv$ is given a partial matching $M_{uv}$ between $L(u),L(v)$ .",
"The goal is to assign to each vertex $v$ a colour from $L(v)$ so that for every edge $uv$ , the colours assigned to $u$ and $v$ are not paired in $M_{uv}$ .",
"Note that if every edge $uv$ simply matches each colour in $L(u)\\cap L(v)$ to itself then we have the usual list colouring.",
"Several studies of correspondence colouring have already appeared, eg.",
"[4], [5], [6], [7], [8], [13], [21].",
"See [7] for a discussion of how correspondence colouring can be more challenging than list colouring, including how some common useful approaches to list colouring do not apply to correspondence colouring.",
"In an instance of correspondence colouring, if every list of colours has the same size, $k$ , then we can assume that each list is $\\lbrace 1,...,k\\rbrace $ .",
"To see this, consider an instance where the lists differ.",
"For every vertex $v$ take a bijection $\\sigma _v:L(v)\\rightarrow \\lbrace 1,...,k\\rbrace $ and for every edge $uv$ , replace each $(i,j)\\in M_{uv}$ with $(\\sigma _v(i),\\sigma _v(j))$ .",
"Similarly, when the lists have different sizes, we can assume that the list of each vertex $v$ is $\\lbrace 1,...,|L(v)|\\rbrace $ .",
"So there is no difference between e.g.",
"the correspondence number and the list correspondence number of a graph.",
"One of the most pursued open questions in list colouring is: When edge-colouring a simple graph (i.e.",
"assigning colours to the edges so that every two edges which share a vertex must get different colours), are the identical lists the most difficult lists?",
"In other words, is the list edge chromatic number of a simple graph equal to the edge chromatic number?",
"This has been answered in the affirmative for specific classes of graphs (eg.",
"[14], [26]), but is still open for general graphs.",
"In a seminal paper [18], Kahn proved that the two numbers are asymptotically equal: the list edge chromatic number of a simple graph with maximum degree $\\Delta $ is equal to $\\Delta +o(\\Delta )$ .",
"In a followup paper [19] he proved that the two numbers are asymptotically equal for multigraphs as well.",
"Molloy and Reed [23] showed that, for simple graphs, the $o(\\Delta )$ term is at most $\\sqrt{\\Delta }{\\rm poly}(\\log \\Delta )$ .",
"See [16] for a more thorough background to list colouring.",
"Correspondence colouring can be defined for edge colouring in a natural way: each pair of edges that share a vertex is given a list of forbidden pairs (this is defined more formally below).",
"Bernshteyn and Kostochka [7] showed that the edge correspondence number of a simple graph can exceed the edge chromatic number.",
"In fact, every $\\Delta $ -regular simple graph has edge correspondence number at least $\\Delta +1$ , whereas many such graphs have edge chromatic number $\\Delta $ .",
"However, we show here that Kahn's result holds in this context; i.e.",
"every simple graph with maximum degree $\\Delta $ has edge correspondence number $\\Delta +o(\\Delta )$ .",
"The previous best bound in this direction was $(2-\\varepsilon )\\Delta $ for a constant $\\varepsilon >0$ , which follows from work in [9] (in particular, the correspondence colouring version of their Theorem 1.6).",
"To set things up formally: We are given a simple graph $G$ and a set of colours ${\\cal Q}=\\lbrace 1,...,q\\rbrace $ .",
"For each pair of incident edges $e,f$ , we are given a partial matching $M_{e,f}$ on $({\\cal Q},{\\cal Q})$ ; i.e.",
"a collection of at most $q$ pairs $(\\alpha ,\\alpha ^{\\prime })\\in {\\cal Q}\\times {\\cal Q}$ such that each colour $\\alpha $ is the first element of at most one pair and the second element of at most one pair.",
"$M_{f,e}$ will consist of the reversal of all pairs in $M_{e,f}$ , so there is only one matching on each pair of incident edges.",
"This collection of partial matchings is called an edge correspondence.",
"An edge correspondence colouring is an assignment to each edge $e\\in E(G)$ of a colour $\\sigma (e)\\in {\\cal Q}$ , such that for every two incident edges $e,f$ , the pair $(\\sigma (e),\\sigma (f))$ is not in $M_{e,f}$ .",
"The edge correspondence number of a graph $G$ is the minimum $q$ such that an edge correspondence colouring exists for every edge correspondence.",
"We denote this by $\\chi ^{\\prime }_{DP}(G)$ , following the notation of Bernshteyn and Kostochka who refer to correspondence colouring as DP-colouring, using the intials of the founders.",
"Theorem 1 Let $G$ be any simple graph with maximum degree $\\Delta $ .",
"Then $\\chi ^{\\prime }_{DP}(G)=\\Delta +o(\\Delta )$ .",
"Remark: Throughout the paper, asymptotic notation is with respect to $\\Delta \\rightarrow \\infty $ .",
"We colour the edges using an iterative procedure, first introduced in Kahn's proof [18] and since then adapted to a very large number of results (see eg. [24]).",
"At each step, we colour a small proportion (roughly $\\frac{1}{ \\ln \\Delta }$ ) of the edges.",
"We do so by considering a random colour assignment to those edges.",
"If we name a particular vertex, then a probabilistic analysis shows that the colours of the edges near that vertex will likely satisfy certain properties, for example that each edge has many remaining colours that can still be legally assigned to it.",
"We apply the Lovasz Local Lemma to obtain a colouring in which the colours of the edges near every vertex satisfies those properties.",
"Eventually we will have coloured almost all the edges; the remaining edges will be such that they are easily dealt with.",
"One useful aspect to our procedure: we carry out the random colouring so that for any edge $e=uv$ , the effect of random choices involving edges incident to $u$ is independent of the effect of choices on edges incident to $v$ , and we track the cummulative affects of those choices seperately.",
"This is where we make critical use of the facts: (i) $G$ is simple, and (ii) edge colouring has the nice structural property that the neighbourhood of each edge consists of two disjoint cliques."
],
[
"Setup",
"Let $\\varepsilon $ be any sufficiently small constant.",
"We will prove that there exists $\\Delta (\\varepsilon )$ such that if $\\Delta (G)\\ge \\Delta (\\varepsilon )$ then $\\chi _{DP}^{\\prime }(G)\\le (1+\\varepsilon )\\Delta (G)$ .",
"This is enough to establish Theorem REF .",
"We do not name $\\Delta (\\varepsilon )$ explicitly; instead we just assume that $\\Delta (G)$ is large enough to satisfy various inequalities that depend on $\\varepsilon $ .",
"So we are given a graph $G$ with maximum degree $\\Delta $ , colours ${\\cal Q}=\\lbrace 1,...,(1+\\varepsilon )\\Delta \\rbrace $ and an edge correspondence.",
"Our goal is to prove that, so long as $\\Delta $ is sufficiently large in terms of $\\varepsilon $ , there must be an edge correspondence colouring."
],
[
"Probabilistic tools",
"We often use the following straightforward bound: ${a\\atopwithdelims ()b}\\le \\left(\\frac{ea}{b}\\right)^b.$ We also rely on the following standard tool of the probabilistic method.",
"The Lovász Local Lemma [11].",
"Let ${\\cal A}=\\lbrace A_1,...,A_n\\rbrace $ be a set of random events such that for each $1\\le i\\le n$ : $\\mbox{\\bf Pr}(A_i)\\le p$ ; and $A_i$ is mutually independent of all but at most $d$ other events.",
"If $pd\\le \\frac{1}{ 4}$ then $\\mbox{\\bf Pr}(\\mbox{$\\overline{ A_1}$}\\cap ...\\cap \\mbox{$\\overline{ A_n}$})>0$ .",
"$BIN(n,p)$ is the sum of $n$ independent random boolean variables where each is equal to 1 with probability $p$ .",
"The following is a simplified special case of Chernoff's bound.",
"It follows from, e.g.",
"Corollary A.1.10 and Theorem A.1.13 from Appendix A of [3].",
"The Chernoff Bound.",
"For any $0<t\\le np$ : $\\mbox{\\bf Pr}(|BIN(n,p)-np|>t)<2e^{-t^2/3np}.$ Theorem 2.3 from [22] generalizes the Chernoff Bound.",
"Parts (b,c) of that theorem imply: Lemma 2 Suppose that we have independent random variables $Z_1,...,Z_n$ , with $0\\le Z_i\\le 1$ for each $i$ .",
"Set $Z=\\sum _{i=1}^n Z_i$ .",
"For any $0<t\\le {\\bf E}(Z)$ : $\\mbox{\\bf Pr}(|Z-{\\bf E}(Z)|>t)<2e^{-t^2/3{\\bf E}(Z)}.$ Our final concentration tool is Talagrand's Inequality, which often provides a stronger bound when the expectation of a random variable is much smaller than the number of trials that determine it.",
"The following reworking of Talagrand's original statement from [28], was proved in the appendix of [25]A similar statement appears as Talagrand's Inequality II in [24].",
"Regretfully, there is an error in the proof of that statement and so we use this one instead.",
"This is also discussed in [20].",
"Talagrand's Inequality.",
"Let $X$ be a non-negative random variable determined by the independent trials $T_1,...,T_n$ .",
"Suppose that for every set of possible outcomes of the trials, we have: changing the outcome of any one trial can affect $X$ by at most $\\ell $ ; and for each $s>0$ , if $X\\ge s$ then there is a set of at most $rs$ trials whose outcomes certify that $X\\ge s$ .",
"Then for any $t\\ge 0$ we have $\\mbox{\\bf Pr}(|X-{\\bf E}(X)|>t + 20\\ell \\sqrt{r\\mbox{\\bf E}(X)} + 64\\ell ^2 r)\\le 4e^{-\\frac{t^2}{8\\ell ^2r({\\bf E}(X)+t)}}.$"
],
[
"Adapting previous work",
"We will find an edge correspondence colouring of the given graph using a common randomized procedure.",
"One feature of that procedure is that when a edge gets a colour, any conflicting colours are removed from the lists of available colours for all neighbouring edges.",
"We would like to have applied the argument from [23] to prove that $\\chi ^{\\prime }_{DP}(G)=\\Delta +\\sqrt{\\Delta }{\\rm poly}(\\log \\Delta )$ .",
"The hurdle we could not overcome is as follows: The procedure in [23] begins by reserving a set of colours at each vertex which cannot be assigned to any edges incident to that vertex.",
"In the context of list edge colouring, this ensures that at the end of the procedure, each uncoloured edge $uv$ can be assigned any of the colours that were reserved at both $u$ and $v$ ; however, this is not true for correspondence colouring.",
"So instead we followed what is, at heart, the argument from [18], although presented as in [23], [24].",
"One difference is as follows: in that argument, one kept track of a parameter $T(v,c)$ which was the set of edges incident to $v$ which could still receive the colour $c$ .",
"That parameter was important because $T(u,c)$ and $T(v,c)$ comprised the edges which could cause $c$ to be removed from the list of the edge $e=uv$ .",
"In the context of correspondence colouring, we need to redefine that parameter.",
"For each edge $e=uv$ we define $T(e,v,c)$ to be the set of edges incident to $v$ which can still receive a colour that will cause $c$ to be removed from the list of $e$ .",
"Once that parameter is defined, the remainder of the argument is a simple adaptation of those from [18], [23]."
],
[
"A quick result",
"The following result is a very simple variation on the main result of [27] (improved in [15]).",
"It will be used at the end of our proof, just as the result of [27] is used at the end of many similar proofs.",
"As mentioned above, one can assume that each edge has the same list of permissible colours.",
"Nevertheless, it will be convenient to extend the definition of an edge correspondence, and an edge correspondence colouring in the obvious way to the case where the lists may differ.",
"We use $f\\sim e$ to denote that edges $f,e$ are adjacent.",
"For a pair of incident edges $e,f$ , we say that $\\alpha \\in L(e)$ has a partner in $M_{e,f}$ if $\\alpha $ is the first element of one of the pairs in $M_{e,f}$ ; i.e.",
"if assigning $\\alpha $ to $e$ forbids a colour to be assigned to $f$ .",
"Lemma 3 We are given a simple graph $G$ ; a list $L(e)$ of size at least $L$ on each edge $e$ ; and an edge correspondence such that for each edge $e$ and colour $\\alpha \\in L(e)$ , there are at most $T$ edges $f\\sim e$ such that $\\alpha $ has a partner in $M_{e,f}$ .",
"If $L\\ge 8T$ then there is an edge correspondence colouring.",
"The proof is essentially identical to that from [27]; we include it here for completeness.",
"It also follows easily from Theorem 2 of [15], with the constant 8 improved to 2.",
"Proof Assign to each edge $e$ a uniformly random colour from $L(e)$ .",
"For each pair of incident edges $e,f$ and pair of colours $(\\alpha ,\\alpha ^{\\prime })\\in M_{e,f}$ we define $A_{e,f,\\alpha ,\\alpha ^{\\prime }}$ to be the event that $e$ is assigned $\\alpha $ and $f$ is assigned $\\alpha ^{\\prime }$ .",
"The probability of each such event is at most $1/L^2$ .",
"Each event $A_{e,f,\\alpha ,\\alpha ^{\\prime }}$ is easily seen to be mutually independent of all events which do not involve $e$ or $f$; i.e.",
"of all but at most $2LT$ other events.",
"Since $\\frac{1}{ L^2}\\times 2LT\\le \\frac{1}{ 4}$ , the Lovász Local Lemma implies that with positive probability none of these events hold; i.e.",
"we obtain an edge correspondence colouring.",
"$\\Box $"
],
[
"A random colouring procedure",
"We colour the graph randomly through a series of iterations, as described in the introduction.",
"Roughly speaking, at each iteration we colour a small proportion of the edges.",
"When an edge receives a colour then we remove any conflicting colours from the lists of incident edges.",
"If two incident edges receive conflicting colours then both are uncoloured.",
"A few technical clarifications: (a) When an edge $e$ receives a colour then conflicting colours are removed from all incident edges even if that colour is removed from $e$.",
"This is often refered to as wasteful since some colours are needlessly removed from lists.",
"We do this because it simplifies the analysis.",
"Furthermore, because such a small proportion of edges are coloured, a vanishing proportion of coloured edges have their colour removed.",
"As a result, the number of colours removed needlessly from each list is negligible.",
"(b) We allow each edge to receive multiple colours.",
"For each edge $e$ and each colour $c\\in L(e)$ , the current list for $e$ , we assign $c$ to $e$ with probability $1/(|L(e)|\\ln \\Delta )$ ; the choice of whether to assign $c$ to $e$ is independent of the choices for all other colours in $L(e)$ .",
"So the probability that $e$ gets at least one colour is roughly $1/\\ln \\Delta $ .",
"Making these assignments independently simplifies the analysis.",
"And the probability that $e$ gets at least two colours is $O(\\ln ^{-2}\\Delta )$ which is small enough to be negligible.",
"We believe that this technique was first used by Johansson in [17].",
"(c) It is very convenient if, at each iteration, all lists have the same size and the probability that a colour $c$ is removed from $L(e)$ is the same for every $c,e$ .",
"We enforce this by truncating some lists and by carrying out so-called equalizing coin flips which round up the probability of a colour being removed from a list.",
"Our procedure makes use of the parameters $L_i,T_i, {\\rm Eq}_i(e,c)$ .",
"They will be defined formally below, as their definitions will be more intuitive after reading the procedure.",
"For now, the main things to understand are: (i) our analysis will enforce that at the beginning of each iteration $i$ , every edge $e$ has $|L(e)|\\ge L_i$ ; (ii) ${\\rm Eq}_i(e,c)$ is the value required for the equalizing coin flips described above.",
"If, during a particular iteration, colour $c$ is assigned to $e$ (in Step 2(b.i)) and colour $c$ is not unassigned from $e$ (in Step 2(b.ii) or Step 2(c)) then we say that $e$ retains $c$ .",
"At any iteration, an edge is considered uncoloured if it did not retain any colour during the previous iterations.",
"Recall from Section REF that ${\\cal Q}=\\lbrace 1,...,(1+\\varepsilon )\\Delta \\rbrace $ .",
"Initialize for every edge $e=uv$ and colour $c$ : $L(e)={\\cal Q}$ , $T(e,v,c)$ is the set of all edges incident to $e$ at $v$ .",
"For each $i\\ge 1$ until $L_i<\\Delta ^{9/10},T_i<\\Delta ^{9/10}$ or $L_i>10 T_i$: For every uncoloured edge $e$ with $|L(e)|>L_i$ , remove $|L(e)|-L_{i}$ arbitrary colours from $L(e)$ .",
"For every uncoloured edge $e$ and every colour $c\\in L(e)$ : assign $c$ to $e$ with probability $1/(L_i\\ln \\Delta )$ .",
"If $c$ was assigned to $e$ then for every $f\\sim e$ , if there is a colour $c^{\\prime }\\in L(f)$ with $c^{\\prime }:f$ blocking $c:e$ then remove $c^{\\prime }$ from $L({\\color {black}{f}})$ ; and if $c^{\\prime }$ was assigned to $f$ then unassign $c^{\\prime }$ from $f$ For every colour $c$ still in $L(e)$ , with probability $1-{\\rm Eq}_i(e,c)$ : remove $c$ from $L(e)$ , and if $c$ was assigned to $e$ then unassign $c$ from $e$ .",
"When this procedure terminates, each edge that has is not uncoloured is given one of the colours that it retained.",
"We will argue that it will terminate because $L_i>10T_i$ which will imply that this partial edge correspondence colouring can be completed using Lemma REF .",
"Consider an edge $e=uv$ .",
"As mentioned in the introduction, we wish to seperate the random choices related to the effect on $e$ of edges around $v$ from the effect of the edges around $f$ .",
"So to carry out the choice in line 2(c) whether to keep $c$ in $L(e)$ , we will in fact make two independent random coin flips $F(e,u,c),F(e,v,c)$ , which return 1 with probabilities ${\\rm Eq}_i(e,u,c)$ and ${\\rm Eq}_i(e,v,c)$ , respectively.",
"If either returns 0 then $c$ is removed from $L(e)$ .",
"(The values of those probabilities are specified below.)",
"For each edge $e=uv$ and colour $c$ , we define the following sets at the beginning of step 2 of iteration $i$ , i.e.",
"after the lists have been truncated: $L_i(e)&=&\\mbox{ the set of colours remaining in the list on } e\\\\T_i(e,v,c)&=&\\mbox{ the set of {\\color {black}{uncoloured}} edges $f$ containing $v$ for which}\\\\&&\\qquad \\mbox{ there is a colour $c^{\\prime }\\in L_i(f)$ such that $c^{\\prime }: f$ blocks $c: e$.",
"}$ We will recursively define parameters $L_i,T_i$ and enforce that for each iteration $i$ : $|L_i(e)|= L_i \\mbox{ and } |T_i(e,v,c)|\\le T_i \\mbox{ for every edge $e$, {\\color {black}{ endpoint $v$ of $e$}} and colour $c\\in L_i({\\color {black}{e}})$.",
"}$ Note that the first condition means that $|L(e)|\\ge L_i$ at the beginning of iteration $i$ .",
"Recalling that we wish to focus seperately on colours removed from $L(e)$ because of edges around $u$ and those removed because of colours around $v$ , we introduce the following terminology: Definition 4 For an edge $e=uv$ with $c\\in L(e)$ .",
"We say that $L(e)$ loses $c$ at $v$ during iteration $i$ if either (a) some edge $f$ with endpoint $v$ is assigned a colour $c^{\\prime }$ where $c^{\\prime }:f$ blocks $c:e$ or (b) the equalizing coin flip $F(e,v,c)$ returns 0.",
"Note that if $e$ is assigned the colour $c$ in step 2(b) then $c$ is unassigned from $e$ iff $L(e)$ loses $c$ at $u$ or $L(e)$ loses $c$ at $v$ .",
"Suppose that (REF ) holds at the beginning of step 2 of iteration $i$ .",
"Thus the probability that no colour $c^{\\prime }$ is assigned to an edge $f=wv$ where $c^{\\prime }:f$ blocks $c:e$ , is $\\left(1-\\frac{1}{ L_i\\ln \\Delta }\\right)^{|T_i(e,v,c)|}\\ge \\left(1-\\frac{1}{ L_i\\ln \\Delta }\\right)^{2T_i}$ .",
"This inspires us to define ${\\rm Keep }_i&=&\\left(1-\\frac{1}{ L_i\\ln \\Delta }\\right)^{T_i}\\\\{\\rm Eq}_i(e,u,c)&=&{\\rm Keep }_i/\\left(1-\\frac{1}{ L_i\\ln \\Delta }\\right)^{|T(e,u,c)|}\\\\{\\rm Eq}_i(e,v,c)&=&{\\rm Keep }_i/\\left(1-\\frac{1}{ L_i\\ln \\Delta }\\right)^{|T(e,v,c)|}\\\\{\\rm Eq}_i(e,c)&=&{\\rm Eq}_i(e,u,c)\\times {\\rm Eq}_i(e,v,c)$ So the probability that $L(e)$ loses $c$ at $v$ during iteration $i$ is exactly $1-{\\rm Keep }_i$ , and the event that it loses $c$ at $v$ is independent of the event that it loses $c$ at $u$ (since the graph is simple).",
"Thus, the probability that $c$ remains in $L(e)$ at the end of iteration $i$ is exactly ${\\rm Keep }^2_i$ , and so the expected number of such colours remaining on $L(e)$ is $L_i\\times {\\rm Keep }^2_i$ .",
"We now turn our attention to $T_{i+1}(e,v,c)$ .",
"We cannot show that this parameter is concentrated because it is possible for the assignment of a single colour to some $f\\sim e$ to cause $T_{i+1}(e,v,c)$ to drop to $\\emptyset $ .",
"So instead, we focus on a related parameter which essentially removes the influence of edges incident to $v$ .",
"$T^{\\prime }_{i+1}(e,v,c)$ is defined to be the set of edges $f=vw\\in T_{i}(e,v,c)$ such that (a) $f$ does not retain a colour during iteration $i$ and (b) $L(f)$ does not lose $c^{\\prime }$ at $w$ during iteration $i$ , where $c^{\\prime }$ is the unique colour in $L(f)$ such that $c^{\\prime }:f$ blocks $c:e$ .",
"Note that $T_{i+1}(e,v,c)\\subseteq T^{\\prime }_{i+1}(e,v,c)$ .",
"So an upper bound on $|T^{\\prime }_{i+1}(e,v,c)|$ will provide an upper bound on $|T_{i+1}(e,v,c)|$ .",
"The fact that each colour in the list of an edge is assigned to that edge independently, makes it simple to bound the expectation of $|T^{\\prime }_{i+1}(e,v,c)|$ : For any edge $f=vw$ and any colour $\\alpha \\in L_i(f)$ , let $Z(\\alpha ,f)$ be the event that $\\alpha $ is assigned to $f$ and let $Y^v(\\alpha ,f), Y^w(\\alpha ,f)$ be the events that $L(f)$ loses $\\alpha $ at $v$ , and $L(f)$ loses $\\alpha $ at $w$ during iteration $i$ .",
"The following observation is very helpful: Observation 5 The events $\\lbrace Z(\\alpha ,f), Y^v(\\alpha ,f), Y^w(\\alpha ,f): \\alpha \\in L_i(f)\\rbrace $ are mutually independent.",
"Proof First, by the way we carry out Step 2(b), the events $Z(c,e)$ over all edges $e$ and $c\\in L_i(e)$ are determined by independent trials.",
"$Y^v(\\alpha ,f)$ is determined by the events $Z({\\color {black}{h}},\\alpha ^{\\prime })$ for all edges ${\\color {black}{h}}\\in T(f,v,\\alpha )$ and colours $\\alpha ^{\\prime }\\in L(g)$ such that $\\alpha ^{\\prime }:{\\color {black}{h}}$ blocks $\\alpha :f$ .",
"By the nature of correspondence colouring, $\\alpha ^{\\prime }:{\\color {black}{h}}$ can block $\\alpha :f$ for at most one colour $\\alpha $ .",
"Since the graph is simple, no edge ${\\color {black}{h}}$ is relevant to both a $Y^v(\\cdot ,f)$ event and a $Y^w(\\cdot ,f)$ event.",
"So these events are determined by disjoint sets of trials.",
"$\\Box $ Now consider any $f\\in T_i(e,v,c)$ , where $c^{\\prime }:f$ blocks $c:e$ .",
"Suppose that (REF ) holds at the beginning of iteration $i$ .",
"Then Observation REF implies (see explanation below): $\\mbox{\\bf Pr}\\left(f\\in T^{\\prime }_{i+1}(e,v,c)\\right)&=&{\\rm Keep }_i\\times \\left(1-\\frac{1}{ \\ln \\Delta L_i}{\\rm Keep }_i\\right)\\times \\prod _{\\alpha \\in L_i(f), \\alpha \\ne c^{\\prime }}\\left(1-\\frac{1}{ \\ln \\Delta L_i}{\\rm Keep }^2_i\\right)\\\\&<&{\\color {black}{\\left(1-\\frac{1}{ \\ln \\Delta L_i}{\\rm Keep }^2_i\\right)^{L_i}\\qquad \\qquad \\qquad \\qquad \\mbox{since ${\\rm Keep }_i<1$}}}\\\\&<&{\\rm Keep }_i\\times \\left(1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right).$ Explanation: The first term is the probability that $L(f)$ does not lose $c^{\\prime }$ at $w$ .",
"The second term is the probability that if $f$ is assigned $c^{\\prime }$ then $L(f)$ loses $c^{\\prime }$ at $v$ and so $c$ is removed from $f$ .",
"The third term is the probability that $f$ does not retain any other colour.",
"This yields that if (REF ) holds for iteration $i$ then: ${\\bf E}[ |T^{\\prime }_{i+1}(e,v,c)|]< |T_{i}(e,v,c)| \\times \\left(1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right)\\times {\\rm Keep }_i.$ We will prove in section that $|T^{\\prime }_{i+1}(e,v,c)|$ and the number of colours removed from $L(e)$ during step 2 are both concentrated.",
"This leads us to recursively define: $L_0=(1+\\varepsilon )\\Delta , T_0=\\Delta $ and $L_{i+1}&=& L_i\\times {\\rm Keep }^2_i - \\Delta ^{2/3}\\\\T_{i+1}&=& T_i\\times \\left(1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right)\\times {\\rm Keep }_i + \\Delta ^{2/3}.$ Remark: Recall that our procedure halts if $L_i$ or $T_i$ drops below $\\Delta ^{9/10}$ .",
"It is not hard to show that ${\\rm Keep }_i=1-o(1)$ (see (REF ) below).",
"So for all relevant values of $i$ , $L_i$ is positive and $\\Delta ^{2/3}$ is a second-order term in (REF ) and ().",
"We will prove: Lemma 6 For every $i\\ge 0$ , every edge $e$ that is uncoloured at the beginning of iteration $i$ , each endpoint $v$ of $e$, and every $c\\in L_i(e)$ : if (REF ) holds for iteration $i$ and $L_i,T_i>\\Delta ^{9/10}$ then with probability at least $1-\\Delta ^{-10}$ , at the beginning of iteration $i+1$ we will have $|L(e)|\\ge L_{i+1}$ ; and $|T(e,v,c)|\\le T_{i+1}$ .",
"The Lovász Local Lemma then implies that, with positive probability, the conditions of Lemma REF hold simultaneously for every such $e,c$ and so: Lemma 7 If (REF ) holds for iteration $i$ and $L_i,T_i>\\Delta ^{9/10}$ then with positive probability (REF ) holds for iteration $i+1$ .",
"Proof For each edge $e=uv$ and colour $c\\in L_i(e)$ , we define $A(e)$ to be the event that $|L(e)|<L_{i+1}$ at the beginning of iteration $i+1$ , and $B(e,v,c)$ to be the event that $|T(e,v,c)|>T_{i+1}$ at the beginning of iteration $i+1$ .",
"If none of these events hold, then (REF ) holds for iteration $i+1$ .",
"Lemma REF says that the probability of each such event is at most $p:=\\Delta ^{-10}$ .",
"$A(e)$ is determined by colour assignments and equalizing coin flips for edges incident with $e$ ; $B(e,v,c)$ is determined by colour assignments and equalizing coin flips for edges within distance two of $e$ .",
"So each event is mutually independent of all events involving edges at distance greater than four, and thus is mutually independent of all but at most $d:=2\\Delta ^4 L_i<\\Delta ^5$ other events (see e.g.",
"the Mutual Independence Principle in [24]).",
"Since $pd<\\frac{1}{ 4}$ for large $\\Delta $ , the Local Lemma completes the proof.",
"$\\Box $ A simple analysis of our recursive equations shows that $T_i$ decreases more quickly than $L_i$ , and so eventually their ratio will be large enough to allow us to apply Lemma REF .",
"We must show that this happens before $L_i<\\Delta ^{9/10}$ , as our procedure stops running if $L_i$ drops below this value.",
"Lemma 8 For every sufficiently small $\\varepsilon >0$ , there is an $X=X(\\varepsilon )$ such that for $I=X\\ln \\Delta $ we have $L_I>10 T_I$ and $L_I,T_I>\\Delta ^{9/10}$.",
"Proof Note that $L_1/T_1=1+\\varepsilon $ .",
"We will prove inductively that $L_i/T_i$ increases with $i$ .",
"Our first useful bound is: If $L_i/T_i\\ge 1+\\varepsilon $ then: $1\\ge {\\rm Keep }_i\\ge 1-\\frac{T_i}{L_i\\ln \\Delta }> 1-\\frac{1}{(1+\\varepsilon )\\ln \\Delta }.$ Therefore for any constant $X$ and $i\\le X\\ln \\Delta $ , if $L_i/T_i\\ge 1+\\varepsilon $ and $L_j,T_j\\ge \\Delta ^{9/10}$ for all $0\\le j\\le i$ then: $\\nonumber L_i>T_i &>& T_1\\prod _{j=1}^{i-1}\\left(1-\\frac{1-\\varepsilon /2}{\\ln \\Delta }{\\rm Keep }^2_j\\right){\\rm Keep }_j\\\\& >&T_1\\left( 1-\\frac{1}{\\ln \\Delta }\\right)^{I}\\left( 1-\\frac{1}{(1+\\varepsilon )\\ln \\Delta }\\right)^{I}>\\Delta e^{-2X}>\\Delta ^{9/10},$ for $\\Delta $ sufficiently large in terms of $X,\\varepsilon $ .",
"We will prove that for any constant $X$ , if for all $0\\le j\\le i\\le X\\ln \\Delta $ we have $L_j/T_j\\ge 1+\\varepsilon $ and $L_j,T_j\\ge \\Delta ^{9/10}$ then $\\frac{L_{i+1}}{T_{i+1}}\\ge \\frac{L_i}{T_i}\\times \\left(1+\\frac{\\varepsilon }{{\\color {black}{4}}\\ln \\Delta }\\right),$ for $\\Delta $ sufficiently large in terms of $X,\\varepsilon $ .",
"It follows inductively that for all $1\\le i\\le I=X\\ln \\Delta $ we have $L_i/T_i\\ge 1+\\varepsilon $ and, by (REF ), $L_i,T_i\\ge \\Delta ^{9/10}$ .",
"So the bound in (REF ) holds for all $1\\le i\\le I=X\\ln \\Delta $ .",
"To prove (REF ), we first establish bounds on our recursive equations for $L_i,T_i$ .",
"The assumptions that $L_i/T_i>1+\\varepsilon $ (and so (REF ) holds) and $L_i,T_i>\\Delta ^{9/10}$ imply: $L_{i+1}&=&L_i\\times {\\rm Keep }^2_i - \\Delta ^{2/3}>L_i\\times {\\rm Keep }^2_i (1- \\Delta ^{-1/5}) \\\\\\nonumber T_{i+1}&=& T_i\\times \\left(1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right)\\times {\\rm Keep }_i + \\Delta ^{2/3}\\\\&<&T_i\\times \\left(1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right)\\times {\\rm Keep }_i (1 + \\Delta ^{-1/5}).",
"$ Therefore $\\frac{L_{i+1}}{T_{i+1}}&\\ge &\\frac{L_i}{T_i}\\times \\frac{{\\rm Keep }_i}{1-\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i}\\times \\frac{1- \\Delta ^{-1/5}}{1+ \\Delta ^{-1/5}}\\\\&>&\\frac{L_i}{T_i}\\times \\left(1-\\frac{1}{(1+\\varepsilon )\\ln \\Delta }\\right)\\times \\left(1+\\frac{{\\color {black}{1-\\varepsilon /2}}}{\\ln \\Delta }{\\rm Keep }^2_i\\right)\\times \\left(1- 2\\Delta ^{-1/5}\\right)\\\\&>&\\frac{L_i}{T_i}\\times \\left(1-(1-\\varepsilon +\\varepsilon ^2)\\frac{1}{\\ln \\Delta }\\right)\\times \\left(1+\\frac{{\\color {black}{1-2\\varepsilon /3}}}{\\ln \\Delta }\\right)\\qquad \\mbox{{\\color {black}{by (\\ref {ekk})}}}\\\\&>&\\frac{L_i}{T_i}\\times \\left(1+\\frac{\\varepsilon }{{\\color {black}{4}}\\ln \\Delta }\\right),$ for $\\varepsilon <{\\color {black}{\\frac{1}{ 12}}}$ and $\\Delta $ sufficiently large.",
"This establishes (REF ).",
"Therefore, if $I=X\\ln \\Delta $ where $X$ is a constant that is sufficiently large in terms of $\\varepsilon $ , $\\frac{L_I}{T_I}>\\frac{L_0}{T_0}\\times \\left(1+\\frac{\\varepsilon }{{\\color {black}{4}}\\ln \\Delta }\\right)^I>(1+\\varepsilon )\\times \\left(1+\\frac{\\varepsilon }{{\\color {black}{4}}}\\right)^X>10.$ This and (REF ) prove the lemma.",
"$\\Box $ Our main theorem follows immediately: Proof of Theorem REF : Setting $I=X\\ln \\Delta $ as in Lemma REF , (REF ) says that $T_I>\\Delta ^{9/10}$ for $\\Delta $ sufficiently large in terms of $\\varepsilon $ , so our procedure runs for at least $I$ iterations.",
"Since $L_i\\ge T_i$ , and by the looping rule of our procedure, we have $L_i,T_i>\\Delta ^{9/10}$ at every iteration.",
"So Lemma REF shows inductively that with positive probability (REF ) holds at the beginning of every iteration.",
"Thus Lemma REF and the fact that $L_i/T_i$ is increasing (as shown in the proof of Lemma REF ) yields that with positive probability, when the algorithm terminates, we will have $|L(e)|\\ge L_i$ and $|T(e,v,c)|\\le T_i$ for every uncoloured edge $e$ , endpoint $v$ of $e$ and colour $c\\in L(e)$ where $L_i>10 T_i$ .",
"Now Lemma REF shows that we can complete the colouring.",
"This establishes that for every $\\varepsilon >0$ , there exists $\\Delta (\\varepsilon )$ such that every graph of maximum degree $\\Delta \\ge \\Delta (e)$ has edge correspondence number at most $(1+\\varepsilon )\\Delta $ .",
"This implies our main theorem.",
"$\\Box $"
],
[
"Concentration",
"In this section, we prove our concentration lemma: Proof of Lemma REF : Part (a): For any colour $c$ remaining in $L(e)$ after step 2(a) of iteration $i$ , the probability that $c$ is not removed from $L(e)$ during the remaining steps of iteration $i$ is exactly ${\\rm Keep }^2_i$, as explained in section .",
"Observation 1: The event that $c$ is not removed from $L(e)$ is mutually independent of the corresponding events for any other colours of $L(e)$ .",
"This follows immediately from: (i) for every edge $f\\sim e$ and $c^{\\prime }\\in L(f)$ there is at most one $c\\in L(e)$ such that $c^{\\prime }:f$ blocks $c:e$ , and (ii) whether $c^{\\prime }$ is assigned to $f$ is independent of the choice to assign any other colour to $f$ or to assign any colour to any other edge.",
"So the number of colours remaining after those steps is distributed like ${\\rm Bin}(L_i,{\\rm Keep }^2_i)$ .",
"Our hypothesis states $L_i>\\Delta ^{9/10}$ and we know ${\\rm Keep }_i=1-o(1)$ by (REF ).",
"So the Chernoff Bounds imply the probability that fewer than ${\\color {black}{L_{i+1}}}=L_i\\times {\\color {black}{{\\rm Keep }^2_i}} - \\Delta ^{2/3}$ colours remain is at most $2e^{-\\Delta ^{4/3}/3 L_i{\\color {black}{{\\rm Keep }_i^2}}}<\\Delta ^{-11},$ for large $\\Delta $ since $L_i<2\\Delta $ and ${\\rm Keep }_i<1$.",
"Part (b): Let $\\Omega _0$ be the set of edges $f\\in T_i(e,v,c)$ that are not assigned any colours during interation $i$ .",
"Let $\\Omega _1$ be the set of edges $f\\in T_i(e,v,c)$ that are assigned at least one colour and fewer than $\\Delta ^{1/10}$ colours during interation $i$ .",
"The expected number of edges in $T_i(e,v,c)\\backslash (\\Omega _0\\cup \\Omega _1)$ is at most $T_i{L_i\\atopwithdelims ()\\Delta ^{1/10}}\\left(\\frac{1}{ L_i\\ln \\Delta }\\right)^{\\Delta ^{1/10}}<T_i\\left(\\frac{e}{\\Delta ^{1/10}\\ln \\Delta }\\right)^{\\Delta ^{1/10}}<\\Delta ^{-11}.$ So by Markov's Inequality, $\\Pr [\\Omega _0\\cup \\Omega _1\\ne T_i(e,v,c)] <\\Delta ^{-11}.$ Recall that an edge $f=vw\\in T_i(e,v,c)$ is not in ${\\color {black}{T^{\\prime }}}_{i+1}(e,v,c)$ if (a) $f$ is assigned and keeps a colour, or (b) $L(f)$ loses $c^{\\prime }$ at $w$ , where $c^{\\prime }$ is the unique colour in $L(f)$ such that $c^{\\prime }:f$ blocks $c:e$ ; if (b) occurs then we say that $f$ loses the colour blocking $c:e$ .",
"We define: $X_0$ is the number of edges $f\\in \\Omega _0$ such that $f$ loses the colour blocking $c:e$ ; $X_1$ is the number of edges $f\\in \\Omega _1$ such that all colours assigned to $f$ are then removed from $f$ ; $X_2$ is the number of edges $f\\in \\Omega _1$ such that $f$ loses the colour blocking $c:e$ and all colours assigned to $f$ are then removed from $f$ .",
"Note that if $\\Omega _0\\cup \\Omega _1=T_i(e,v,c)$ then ${\\color {black}{|T_{i+1}^{\\prime }(e,v,c)|}}= |\\Omega _0|-X_0+X_1-X_2.$ We will prove that $\\Omega _0,X_0,X_1,X_2$ are all concentrated around their means.",
"Then (REF ) and (REF ) will imply part (b).",
"We start with $\\Omega _0$ .",
"Each edge $f\\in T_i(e,v,c)$ goes into $\\Omega _0$ with probability $p_0=(1-\\frac{1}{ L_i\\ln \\Delta })^{L_i}\\approx 1-\\frac{1}{ \\ln \\Delta }$ and independently of whether any other edges enter $\\Omega _0$ .",
"So the Chernoff bound yields: $\\mbox{\\bf Pr}\\left({\\Large | }|\\Omega _0|- |T_i(e,v,c)|p_0 {\\Large |} > \\frac{1}{ 8}\\Delta ^{2/3}\\right) < 2e^{-\\Delta ^{4/3}/48 T_ip_0}<\\Delta ^{-11},$ for sufficiently large $\\Delta $ .",
"To prove that $X_0,X_1,X_2$ are concentrated, we will expose the colour assignments and equalizing coin flips in two phases: Phase 1.",
"We expose the colour assignments to all edges incident to $v$ .",
"Phase 2.",
"We expose the colour assignments to all remaining edges, and carry out all equalizing coin flips.",
"We use the notation $\\mbox{\\bf E}_j,\\Pr _j$ to denote expectation and probability over the random choices made during Phase $j=1,2$ ; $\\mbox{\\bf E},\\Pr $ denotes the expectation and probability over the entire colour assignment.",
"Let $\\Psi $ denote the colour assignments made in Phase 1.",
"The analysis of Phase 2 will be conditional on $\\Psi $ and so we are interested in the following variables, defined for $t=0,1,2$ .",
"$u_t=\\mbox{\\bf E}_2(X_t).$ Each $u_t$ is determined by $\\Psi $ .",
"Simple properties of conditional expectations imply that $\\mbox{\\bf E}_1(u_t)=\\mbox{\\bf E}(X_t).$ For each edge $f\\in T_i(e,v,c)$ , let $\\rho (f)$ denote the number of colours assigned to $e$ by $\\Psi $ , and set $\\rho ^+(f)=\\rho (f)$ if $f$ is assigned the colour blocking $c:e$ at $f$ , $\\rho ^+(f)=\\rho (f)+1$ otherwise.",
"By Observation 1 from part (a), we have: $u_0&=&(1-{\\rm Keep }_i)|\\Omega _0|\\\\u_1&=&\\sum _{f\\in \\Omega _1}(1-{\\rm Keep }_i)^{\\rho (f)}\\\\u_2&=&\\sum _{f\\in \\Omega _1}(1-{\\rm Keep }_i)^{\\rho ^+(f)}$ We now show that $\\mbox{\\bf E}_1(u_i)$ is large for each $i$ .",
"Since we halt if $L_i>10 T_i$ , we have ${\\rm Keep }_i\\le \\left(1-\\frac{1}{ 10T_i\\ln \\Delta }\\right)^{T_1}<1-\\frac{1}{ 11\\ln \\Delta }$ , for $\\Delta $ sufficiently large.",
"It is easy to show that for each $f\\in \\Omega _1$ , with probability at least ${1\\over 2}$ we have $\\rho (f)=1,\\rho ^+(f)=2$ .",
"Using the calculations from (REF ), this yields: $\\mbox{\\bf E}_1(u_0)\\ge \\frac{1}{ 11\\ln \\Delta }\\mbox{\\bf E}_1(|\\Omega _0|)> \\frac{1}{ 20\\ln \\Delta }T_i,$ $\\mbox{\\bf E}_1(u_1)\\ge \\frac{1}{ 20\\ln \\Delta }\\mbox{\\bf E}_1(|\\Omega _1|)\\approx \\frac{1}{ 30\\ln ^2\\Delta }T_i,$ $\\mbox{\\bf E}_1(u_2)\\ge \\frac{1}{ 200\\ln ^2\\Delta }\\mbox{\\bf E}_1(|\\Omega _1|)\\approx \\frac{1}{ 300\\ln ^3\\Delta }T_i.$ In each case, the expectation is at least $\\Delta ^{4/5}$ since we have $T_i\\ge \\Delta ^{9/10}$ by (REF ).",
"We analyze the colour assignments of Phase 1 to prove that for each $t=0,1,2$ : $\\mbox{$\\Pr _1$}[|u_t-{\\bf E}_1(u_t)|>\\frac{1}{ 8}\\Delta ^{2/3}]<\\Delta ^{-11}.$ For $i=0$ , this bound follows immediately from (REF ).",
"For $i=1,2$ , note that $u_i$ is the sum of $|T_i(u,v,c)|$ independent variables, each bounded between 0 and 1; for example, when $i=1$ the random variable corresponding to $f$ is equal to 0 if $f\\notin \\Omega _1$ and $(1-{\\rm Keep }_i)^{\\rho (f)}$ otherwise.",
"So (REF ) follows from Lemma REF .",
"We next analyse the colour assignments and equalizing flips of Phase 2 to prove that for each $i=0,1,2$ : $\\mbox{$\\Pr _2$}[|X_i-{\\bf E}_2(X_i)|>\\frac{1}{ 8}\\Delta ^{2/3}]<\\Delta ^{-11}.$ We will use Talagrand's Inequality.",
"Note first that the assignment of a colour $\\alpha $ to any edge $w_1w_2$ can only affect whether at most two edges, $w_1v$ and $w_2v$ , are counted by $X_0,X_1,X_2$ .",
"So each colour assignment affects each of these variables by at most $\\ell =2$ .",
"Next, suppose that $X_0\\ge s$ ; i.e.",
"at least $s$ different edges in $\\Omega _0$ lose the colour blocking $c:e$ .",
"Then each of $s$ edges loses the colour blocking $c:e$ because of a colour assignment to at least one incident edge not incident with $v$ or because of an equalizing coin flip.",
"So there are $s$ colour assignments or flips which certify that $X_0\\ge s$ .",
"If $X_1\\ge s$ then each of a set of $s$ edges had every colour that was assigned to it removed; since each edge in $\\Omega _1$ was assigned fewer than $\\Delta ^{1/10}$ colours, there is a set of fewer than $\\Delta ^{1/10}s$ colour assignments or equalizing flips that certify $X_1\\ge s$ .",
"Finally, if $X_2\\ge s$ then there are at most $\\Delta ^{1/10}s$ colour assignments or flips that certify $X_2\\ge s$ - one showing that each of the $s$ edges loses the colour blocking $c:e$ and fewer than $\\Delta ^{1/10}$ showing that it lost all its assigned colours.",
"Note that each of these three variables has expectation at most $T_i\\le \\Delta $ and so we can apply Talagrand's Inequality with $\\ell =2, r=\\Delta ^{1/10}$ and $t=\\frac{1}{ 8}\\Delta ^{2/3}$ , noting that $t + 20\\ell \\sqrt{r\\mbox{\\bf E}({\\color {black}{X_i}})} + 64\\ell ^2 r<\\frac{1}{ 4}\\Delta ^{2/3}$ to establish (REF ) for each $i=0,1,2$ : $\\Pr [|X_i-{\\bf E}(X_i)|>\\frac{1}{ {\\color {black}{8}}}\\Delta ^{2/3}]<4e^{-\\Delta ^{4/3}/8^3\\cdot 4\\Delta ^{1/10}(\\Delta +\\frac{1}{ 4}\\Delta ^{2/3})}<\\Delta ^{-{\\color {black}{11}}}.$ Combining (REF ), (REF ), (REF ), (REF ) and (REF ) establishes that $|T_i^{\\prime }(e,v,c)|$ is concentrated around $\\mbox{\\bf E}(|T_i^{\\prime }(e,v,c)|)$ , which is bounded by (REF ).",
"$T_{i+1}(e,v,c)\\subseteq T^{\\prime }_{i+1}(e,v,c)$ and so $T_{i+1}(e,v,c)\\le T^{\\prime }_{i+1}(e,v,c)$ .",
"This yields the upper bound of part (b) for $\\Delta $ sufficiently large.",
"$\\Box $"
],
[
"Hypergraphs",
"We close by remarking that our main theorem also holds for linear $k$ -uniform hypergraphs for $k=O(1)$ .",
"I.e., for any constant $k$ , and any hypergraph $H$ where every hyperedge contains exactly $k$ vertices, every pair of vertices lies in at most one hyperedge, and every vertex lies in at most $\\Delta $ hyperedges, we have $\\chi ^{\\prime }_{DP}(G)=\\Delta +o(\\Delta )$ .",
"The proof is a very straightforward adaptation of the proof of Theorem REF .",
"We outline it here: We first remark that our definintion of $\\chi ^{\\prime }_{DP}$ extends naturally to linear hypergraphs.",
"In the statement of Lemma REF , the constant 8 changes to $4k$ .",
"So in the halting condition of our procedure, $L_i>10T_i$ becomes $L_i>5k T_i$ , and the constant 10 is replaced with $5k$ as appropriate throughout the proof.",
"For each hyperedge $e$ , we define $T_i(e,v,c),T^{\\prime }_i(e,v,c),{\\rm Eq}_i(e,v,c)$ for each of the $k$ vertices $v$ in $e$ .",
"Equalizing coinflips ensure that for every $v\\in e$ and $c \\in L(e)$ , the probability that $L(e)$ loses $c$ at $v$ is exactly $1-{\\rm Keep }_i$ .",
"So now the probability that $c$ remains in $L(e)$ is ${\\rm Keep }_i^k$ .",
"So our recursive equation for $L_i$ becomes: $L_{i+1}= L_i\\times {\\rm Keep }_i^{k}-\\Delta ^{2/3}.$ This time, $T^{\\prime }_{i+1}(e,v,c)$ is the number of edges $f\\subseteq T_{i}(e,v,c)$ such that during iteration $i$ , (a) $f$ does not retain a colour and (b) $L(f)$ does not lose $c^{\\prime }$ at any of its $k-1$ vertices other than $v$, where $c^{\\prime }$ is the unique colour in $L(f)$ such that $c^{\\prime }:f$ blocks $c:e$ .",
"So equation (REF ) becomes: ${\\bf E}[ |T^{\\prime }_{i+1}(e,v,c)|]< |T_{i}(e,v,c)| \\times \\left(1-\\frac{1-\\varepsilon /2}{\\ln \\Delta }{\\rm Keep }^k_i\\right)\\times {\\rm Keep }^{k-1}_i.$ So our recursive equation for $T_i$ becomes: $T_{i+1}= T_i\\times \\left(1-\\frac{1-\\varepsilon /2}{\\ln \\Delta }{\\rm Keep }^k_i\\right)\\times {\\rm Keep }^{k-1}_i+\\Delta ^{2/3}.$ The statement of Observation REF is modified to include $k$ $Y$ -events.",
"The fact that $H$ is linear ensures that this Observation still holds.",
"The proof of Lemma REF changes only in very straightforward places; eg.",
"in the definitions of $u_1,u_2$ we multiply the exponent by $k-1$ and in the application of Talagrand's Inequality, $\\ell =2$ becomes $\\ell =k$ .",
"(The fact that $H$ is linear is important in the place where we set $\\ell =k$ .)",
"The same is true for Lemma REF .",
"The change in our recursive equations results in very minor changes to the calculations in the proof of Lemma REF .",
"Those three lemma statements remain the same, except for changing 10 to $5k$ in Lemma REF .",
"This modified proof yields: Theorem 9 Let $H$ be any linear uniform graph with maximum degree $\\Delta $ .",
"Then $\\chi ^{\\prime }_{DP}(H)=\\Delta +o(\\Delta )$ ."
],
[
"Acknowledgement",
"I am grateful to Runrun Liu and two anonymous referees for providing many corrections and improvements to the first draft.",
"This research is supported by an NSERC Discovery Grant."
]
] | 1808.08594 |
[
[
"Analysis of adversarial attacks against CNN-based image forgery\n detectors"
],
[
"Abstract With the ubiquitous diffusion of social networks, images are becoming a dominant and powerful communication channel.",
"Not surprisingly, they are also increasingly subject to manipulations aimed at distorting information and spreading fake news.",
"In recent years, the scientific community has devoted major efforts to contrast this menace, and many image forgery detectors have been proposed.",
"Currently, due to the success of deep learning in many multimedia processing tasks, there is high interest towards CNN-based detectors, and early results are already very promising.",
"Recent studies in computer vision, however, have shown CNNs to be highly vulnerable to adversarial attacks, small perturbations of the input data which drive the network towards erroneous classification.",
"In this paper we analyze the vulnerability of CNN-based image forensics methods to adversarial attacks, considering several detectors and several types of attack, and testing performance on a wide range of common manipulations, both easily and hardly detectable."
],
[
"Introduction",
"In the era of social networks, images have become a dominant communication vehicle.",
"They convey information with higher immediacy and depth than text, and have the potential to elicit strong responses in the observers.",
"Unfortunately, with modern media editing tools, tampering with images has become very easy.",
"The manipulated images can be used to discredit people, direct public opinion, even change the course of political events, and pass easily unnoticed from ordinary people.",
"A number of multimedia forensic tools have been proposed in the last years to detect image manipulations [1].",
"In particular, methods based on high-order statistics of image residuals have drawn great attention since long time [2], [3].",
"Indeed, when a pristine image is modified, by inserting or removing objects, or modifying global characteristics, several low-level operations are usually involved, like linear or non-linear filtering, resizing, or compression.",
"All these operations leave subtle but distinctive traces in the image micro-patterns, which can be discovered by means of suitable image descriptors extracted from the high-pass image residual.",
"To this end, the SPAM (subtractive pixel adjacency matrix) features [4] and the SRM (spatial rich models) [5] have shown great potential for many image forensics tasks [6], [7], [8].",
"In particular, excellent results [6], [9], [8] can be obtained even by considering one specific single model from [5], the one computing 4-pixel co-occurrences on the residuals of 3rd order linear filter.",
"Given their similarity wirh SPAM features, here for the sake of brevity we will refer to them as S3SPAM or simply SPAM features.",
"Nonetheless, the current trend in forensics, and in multimedia processing in general, is to abandon handcrafted features in favor of deep learning.",
"Given a sufficiently large training set, deep nets, typically convolutional neural networks (CNN), learn from the data which features best address the given task, reaching usually impressive performance.",
"The first CNN-based detector of image manipulation was proposed in [10], inspired to previous work in steganalysis.",
"Its main peculiarity is an ad hoc first layer, comprising a bank of filters constrained to extract high-pass features.",
"Since the most relevant information for discrimination is hidden in the high-pass image content, such filters speed up convergence to a satisfactory solution.",
"In [11], instead, it was proven that S3SPAM features can be extracted by a simple shallow CNN.",
"Besides reproducing the very good results of the original detector, the resulting net can be further improved by fine tuning on a specific dataset, providing a very good performance even with a small training set.",
"Very recently, another deep learning solution has been proposed, aimed at detecting the processing history of JPEG images [12].",
"All the above networks, though very effective, are relatively shallow.",
"Very deep architectures can be expected to provide a further performance boost.",
"Tellingly, in a recent competition on camera model identification organized by the IEEE Signal Processing society on the Kaggle platformhttps://www.kaggle.com/c/sp-society-camera-model-identification, all top-ranking teams proposed solutions based on an ensemble of very deep networks.",
"Likewise, very deep networks have shown top performance and higher robustness [13] in detecting images manipulated by generative adversarial networks (GAN).",
"Figure: Our reference scenario.",
"The subtle traces left during image forgery can be detected by a forensic tool (red cross in the bottom box).However, an attacker can conceal such traces by injecting suitable adversarial noise, thus misleading the detector into authenticating the image as pristine (green checkmark).Although deep learning holds great potential for multimedia forensics, one should not rely on a safe environment, counting on the attacker's naivete.",
"On the contrary, the risks incurred by counter-forensic actions, aimed at neutralizing forensic tools (see Fig.1), must be taken into serious account and analyzed in depth.",
"Some recent papers [14], [15], for example, propose to attack SPAM-based detectors by means of iterative gradient descent algorithms, which prove very effective, although definitely slow.",
"Attacking CNNs, however, has proven to be much simpler [16].",
"By exploiting the intrinsic differentiability of the loss function, a suitable adversarial noise can be easily generated and added to the input image to modify the network decision, without visible image impairments.",
"Following this seminal paper, many more attacks based on adversarial noise have been devised.",
"In addition, deep learning can be used itself for counter-forensics.",
"In [17], a GAN-based architecture was proposed to conceal the traces of 3$\\times $ 3 median filtering.",
"Such a study, though limited to a very special case, opens the way to interesting developments.",
"Here, we investigate on the effectiveness of adversarial attacks to CNN-based detectors.",
"We consider a large set of manipulations, both easily detectable and more challenging, and several CNN-based detectors.",
"Specific adversarial noise is generated for each detector, and the effects are assessed both on the target detector and on non-target ones.",
"The performance of GAN-based restoration is also assessed, with reference to the especially challenging case of median filtering.",
"To the best of our knowledge, this is the first study on this topic.",
"In the rest of the paper, we describe the detectors (Section 2), the attacks (Section 3), and the experimental analysis (Section 4), before drawing conclusions (Section 5)."
],
[
"CNN-based detectors of image manipulation",
"In this Section we briefly recall some relevant CNN-based detectors, with their main features.",
"However we also consider a baseline conventional detector, using handcrafted features [5] and support vector machine (SVM) classification."
],
[
"SPAM+SVM",
"To extract the residual features proposed in [5] the original image is high-pass filtered and quantized with a small number of bins.",
"Then, co-occurrences are computed, encoded, and collected in a linear histogram feature, normalized to unit energy.",
"Depending on the specific parameters of this process, different features are obtained, collectively called rich models [5].",
"As said before, we consider one single model here, with third-order linear filter, 5-bin quantization, and 4-lag co-occurrences (s3_spam14hv) and will refer to it as SPAM features from now on."
],
[
"Bayar2016",
"In [10] a relatively small CNN is proposed for image manipulation detection, referred to as Bayar2016 from now on, comprising three convolutional layers, two max-pooling layers, and three fully-connected layers.",
"In order to immediately extract residual-based features, as suggested by the literature, filters of the first layer, with 5$\\times $ 5 receptive field, are constrained to respect the following rule $\\left\\lbrace \\begin{array}{l}w(0,0) = -1 \\\\\\sum _{l,m \\ne 0} w(l,m) = 1\\end{array} \\right.$ Therefore, the sum of all weights is 0, enforcing the high-pass nature of the filters.",
"In particular, the off-center pixels are combined to compute a prediction of the center pixel, so the output of the filter can be regarded as a prediction error."
],
[
"Cozzolino2017",
"The main result of [11] is that a large class of conventional features can be computed exactly by suitable convolutional networks.",
"Although the result is quite general, the work focuses on the SPAM feature described before.",
"Exact SPAM feature extraction requires only two convolutional layers, followed by hardmax and average pooling.",
"The extracted features could then be used to train an external SVM.",
"However, a full-fledged CNN-based detector is also built in [11], by complementing the feature extractor subnet with a fully connected layer which replaces the external SVM classifier.",
"Then, the hardmax is also replaced by softmax to ensure differentiability, allowing further training by backpropagation.",
"Besides the theoretical result, the CNN proposed in [11] can faithfully replicate the SPAM-SVM suite, and improve upon it by means of quick fine tuning over a very small training set.",
"This latter version, referred to as Cozzolino2017, is considered here."
],
[
"Very deep nets: Xception",
"In recent years, a large experimental evidence has accumulated showing that network depth plays a fundamental role for generalization ability.",
"State-of-the-art architectures in computer vision and related fields, such as ResNet, DenseNet, InceptionNet, XceptionNet, all comprise from several dozens to hundreds of layers.",
"Our own experience in forensic applications [13] confirms the superior robustness of deep nets to challenging and off-training conditions.",
"On the down side, deep nets require very large datasets for correct training, a condition not always met in practice.",
"To include a deeper net in our comparative assessment we selected Xception [18], comprising a total of 42 layers, 36 convolutional, 5 pooling, and one fully connected.",
"Its main architectural innovation is the use of separable filters.",
"That is, 3D convolutions are obtained by the cascade of 2D spatial and 1D cross-map convolutions.",
"Thanks to this constraint, the number of free parameters drops significantly w.r.t.",
"competing nets or, under a different point of view, a deeper architecture can be adopted for the same level of complexity, allowing the use of such a deep net even with a relatively small training set."
],
[
"Attacking forensic detectors",
"In this Section, we describe some possible strategies to attack image forensic detectors, in particular gradient descent algorithms for SPAM+SVM; generation of adversarial noise for CNN-based detectors; GAN-based restoration of manipulated images.",
"Although we focus on targeted attacks, designed against a specific detector, universal counter-forensic methods are also studied, e.g., [19]"
],
[
"Attacking a SPAM-based detector by gradient descent",
"Let $X^{prist}$ and $X^0$ be the pristine and manipulated images, with $f^{prist}$ and $f^0$ the corresponding feature vectors, SPAM in our case.",
"Lacking perfect knowledge on the classifier, the attacker wants to modify $X^0$ into a new image, $\\hat{X}$ , similar to $X^0$ , to limit distortion, but whose feature, $\\hat{f}$ , is so close to $f^{prist}$ to fool the detector, see Fig.REF (left).",
"In formulas, the problem can be cast as $\\hat{X} = \\arg \\min _X \\psi (f(X),f^{prist}), \\hspace{11.38109pt} {\\rm s.t.",
"}\\;\\; \\phi (X,X^0)<T$ where $\\phi (\\cdot ,\\cdot )$ and $\\psi (\\cdot ,\\cdot )$ are image and feature space distances, and $T$ a suitable threshold on distortion.",
"In [14] an iterative algorithm is proposed, where the objective function is minimized through local changes on the image, like in the iterated conditional modes method.",
"This approach is effective but quite slow, because the feature must be recomputed at each new step, due to its complex nonlinear relationship with the image.",
"Note that, if the classifier is perfectly known, one can target $f^{close}$ , the feature closest to $f^0$ across the decision boundary, rather than $f^{prist}$ , as shown in Fig.REF (right).",
"This speeds up convergence considerably, but reduces robustness with respect to off-target detectors, as also depicted in Fig.REF (right)."
],
[
"Attacking CNNs by adversarial noise",
"Experiments in computer vision [16], [20] have clearly established the vulnerability of CNN-based detectors to adversarial attacks.",
"A suitable adversarial noise pattern can be added to the input image to mislead the classifier, see Fig.1, without impairing its visual quality.",
"With CNNs, some simple algorithms for the generation of adversarial noise are available.",
"The Fast Gradient Sign Method (FGSM), proposed in [16], exploits the differentiability of the loss function.",
"The gradient of the loss with respect to each pixel of the input image is first computed by backpropagation.",
"Then, each pixel is modified by a small quantity, $\\pm \\epsilon $ , taking the sign of the local gradient.",
"Neglecting higher order effects, all perturbations increase the loss, and hence a large change in output can be obtained with very low-variance adversarial noise.",
"Following this early, and simple, method, more sophisticated solutions have been proposed.",
"DeepFool [21] is based on a local linearization of the classifier under attack, which allows one to project the input image on the approximate decision boundary, and to introduce the minimum perturbation necessary to cross it.",
"The Jacobian-based Saliency Map Attack (JSMA) [20] relies on a greedy iterative procedure.",
"Unlike FGSM, it attacks only the pixels that contribute most to the correct classification, identified by a suitable saliency map.",
"In [22], adversarial noise generation is formulated as a min-max optimization, with the double aim of generating effective adversarial examples and training robust classifiers.",
"The resulting algorithm, projected gradient descent (PGD), provides the optimum adversarial examples when the network is perfectly known.",
"Noteworthy, FGSM can be regarded as a single-step scheme to solve the maximization step of PGD.",
"In the experiments, we will consider only the FGSM algorithm, because of its low complexity (JSMA and PGD are orders of magnitude slower) and easy interpretation.",
"Note that, in a realistic setting, images must be rounded to integer values to be stored or transmitted, so, unlike in theoretical analyses, we consider only integer values for the $\\epsilon $ parameter.",
"Table: TPR with adversarial noise (FGSM, ϵ=1\\epsilon =1).",
"Target: Bayar2016.Table: TPR with adversarial noise (FGSM, ϵ=1\\epsilon =1).",
"Target: Cozz.2017.Table: TPR with adversarial noise (FGSM, ϵ=1\\epsilon =1).",
"Target: Xception."
],
[
"Attacks based on GANs",
"Starting from the 2015 seminal work of Goodfellow et al.",
"[16] generative adversarial networks have gained a major role in deep learning, providing remarkable results in a large number of tasks involving image synthesis and/or manipulation.",
"The basic idea is to train in parallel two competing nets, a generator, which tries to synthesize images with a natural appearance, and a discriminator, which tries to tell apart natural from synthetic images.",
"This competition gradually improves the performance of both nets.",
"Ideally, at convergence, the generator should be able to produce images that are indistinguishable from natural ones.",
"Recently, a GAN-based method has been proposed [17] for the restoration of median filtered images.",
"For this application, the generator does not start from a random noise vector to synthesize the output, as usual with GANs, but takes in input the manipulated image and restores its natural features.",
"Accordingly, the generator loss includes not only an adversarial term, which measures its ability to fool the detector, but also two image quality terms.",
"These measure objective quality (distance from the original) and perceptual quality of the generated image.",
"We refer the reader to the original paper for all details of the method, underlining only that the generator relies heavily on residual connections to improve stability and speed up convergence."
],
[
"Experimental analysis",
"To carry out our experimental analysis we generate a dataset taking 200 images from each of 9 different devices, and 192 partially overlapping $128\\times 128$ patches from each image.",
"We consider 4 types of image manipulation: Gaussian blurring, JPEG compression, median filtering, and resizing, with two different settings for each case corresponding to “easy” and “challenging” tasks.",
"For example, a Gaussian filter with $\\sigma =1.10$ causes easily detectable blurring, unlike with $\\sigma =0.50$ .",
"In each binary classification task, patches from 6 devices chosen at random are used for training, the others for testing.",
"Overall, each training set comprises more than 200k+200k patches, still relatively small for deep learning applications.",
"In Tab.REF we report, for the considered detectors, false positive rate (FPR), true positive rate (TPR), and overall accuracy (ACC), in the absence of counter-forensic attacks.",
"For easy cases (top), accuracies are always close to 100%, only Xception shows a somewhat worse performance, very likely due to the limited training set.",
"For more challenging manipulations (bottom), larger differences are observed, with some poorer results on JPEG compression (Bayar2016, SPAM) and Gaussian blurring (SPAM, Xception).",
"Nonetheless, a very good detection performance is still observed, in general.",
"In Tables REF through REF we study the case in which adversarial noise is added to the manipulated images, using FGSM with $\\epsilon =1$ , namely, the weakest adversarial noise which survives the image rounding.",
"Since neither the pristine images nor the detectors change, we report only the TPR for the attacked manipulated images.",
"Note also that the PSNR is always 48.13 dB (MSE=1), hence no visual impairment can be appreciated.",
"The attack is very effective when the same net is used to generate the adversarial noise and to detect the manipulation (boldface entries).",
"Only Cozzolino2017, and only for the 7$\\times $ 7 median filtering, keeps providing a good TPR.",
"In the absence of alignment, however, the attack is much less effective, especially for median filtering, both 3$\\times $ 3 and 7$\\times $ 7, for which both SPAM and Cozzolino2017 provide a TPR close to 100%.",
"These results suggest that, at least in such cases, the adversarial noise is not restoring the features of pristine images disrupted by the manipulation, but only exploiting some detector weaknesses.",
"This latter consideration further motivates us to explore the GAN-based attack, which has the very goal of restoring manipulated images.",
"In Tab.REF we report results only for the critical median filtering cases.",
"They seem to confirm a better ability of the GAN-based method to attack uniformly all detectors.",
"Actually, the original architecture proposed in [17] works well only in the 3$\\times $ 3 case, and never fools Xception.",
"However, if we replace the original discriminator with a VGG net [23], the attack becomes more effective for all detectors, and none of them reaches a 50% TPR.",
"Table: TPR for median filtering after GAN-based restoration.",
"Top: Kim2018 discriminator.",
"Bottom VGG discriminator."
],
[
"Conclusions",
"We have presented an investigation on adversarial attacks to CNN-based image manipulation detectors.",
"Even a rather simple attack can completely mislead the target detector and largely reduce the detection performance of off-target detectors.",
"As only exception, the adversarial noise attack was not able to conceal 7$\\times $ 7 median filtering, which deeply modifies the image fine structures.",
"However, a suitable GAN-based attack proves to work well even in this challenging case.",
"Obviously, these early results represent only a proof of concept, and more thorough analyses are necessary to gather a solid understanding of the relevant issues.",
"More sophisticated attacks must be considered, and more detectors tested, on a wider range of manipulations.",
"In particular, realistic applications over social networks, involving resizing and compression, should be considered."
]
] | 1808.08426 |
[
[
"Resonating quantum three-coloring wavefunctions for the kagome quantum\n antiferromagnet"
],
[
"Abstract Motivated by the recent discovery of a macroscopically degenerate exactly solvable point of the spin-$1/2$ $XXZ$ model for $J_z/J=-1/2$ on the kagome lattice [H. J. Changlani et al.",
"Phys.",
"Rev.",
"Lett 120, 117202 (2018)] -- a result that holds for arbitrary magnetization -- we develop an exact mapping between its exact quantum three-coloring wavefunctions and the characteristic localized and topological magnons.",
"This map, involving resonating two-color loops, is developed to represent exact many-body ground state wavefunctions for special high magnetizations.",
"Using this map we show that these exact ground state solutions are valid for any $J_z/J \\geq -1/2$.",
"This demonstrates the equivalence of the ground-state wavefunction of the Ising, Heisenberg and $XY$ regimes all the way to the $J_z/J=-1/2$ point for these high magnetization sectors.",
"In the hardcore bosonic language, this means that a certain class of exact many-body solutions, previously argued to hold for purely repulsive interactions ($J_z \\geq 0$), actually hold for attractive interactions as well, up to a critical interaction strength.",
"For the case of zero magnetization, where the ground state is not exactly known, we perform density matrix renormalization group calculations.",
"Based on the calculation of the ground state energy and measurement of order parameters, we provide evidence for a lack of any qualitative change in the ground state on finite clusters in the Ising ($J_z \\gg J$), Heisenberg ($J_z=J$) and $XY$ ($J_z=0$) regimes, continuing adiabatically to the vicinity of the macroscopically degenerate $J_z/J=-1/2$ point.",
"These findings offer a framework for recent results in the literature, and also suggest that the $J_z/J=-1/2$ point is an unconventional quantum critical point whose vicinity may contain the key to resolving the spin-$1/2$ kagome problem."
],
[
"Introduction",
"Quantum frustrated magnetism presents one of the most intriguing and intricate examples of the interplay between spatial geometry and quantum mechanics.",
"This results in a rich multitude of competing exotic phases such as valence bond solids, topological phases including several spin liquids, and magnetically ordered phases.",
"Slight changes in the material composition or geometry can lead to a dramatic change in its phase, making frustrated magnets ideal playgrounds to study quantum phase transitions.",
"The building blocks of many of these systems are lattices of magnetic ions made from motifs of connected triangles.",
"Prominent amongst these is the kagome lattice, a lattice of corner sharing triangles which has been intensely studied owing to its relevance to materials such as Herbertsmithite (a kagome lattice of Cu$^{2+}$ ions) .",
"Experiments on Herbertsmithite , – of which the idealized kagome Heisenberg antiferromagnet is known to be a good model – find that spins do not order even at the lowest investigated temperatures (50 mK, a small fraction of the exchange energy of 200 K), tantalizingly suggesting the picture of a two-dimensional spin-liquid ground state.",
"However, in spite of several theoretical efforts devoted to the idealized model, there is no universal consensus on the precise nature of the spin liquid ground state , , , , , , , , , , and recent work even suggests that larger lattices should stabilize an ordered state .",
"To reconcile some of these observations, it has been suggested that the kagome Heisenberg model lies at or close to a critical point in the phase diagram in a suitably chosen parameter space of model Hamiltonians , .",
"Previous work (by two of us, HJC and BKC in collaboration with others) contributed to the understanding of the kagome phase diagram through the discovery of an extensively quantum degenerate exactly solvable point .",
"While the classical extensive degeneracy for the kagome and hyper-kagome lattice has a long history, the connection to the quantum case in the spin-1/2 $XXZ$ Hamiltonian, $H_{XXZ}[J_z] = J \\sum _{\\langle i,j \\rangle } S^{x}_{i} S^{x}_{j} + S^{y}_{i} S^{y}_{j} +J_{z} \\sum _{\\langle i,j \\rangle } S^{z}_{i} S^{z}_{j}$ at $H_\\textrm {XXZ}[J_z=-1/2,J=1]$ (notated as $H_{XXZ0}$ ), has not been entirely explored.",
"$S_i$ are spin-1/2 operators on site $i$ , $\\langle i,j \\rangle $ refer to nearest neighbor pairs and $J$ (set to 1 throughout the paper) and $J_z$ are the $XY$ and Ising couplings respectively.",
"Ref.",
"showed that the degeneracy exists in all $S_z$ sectors and all finite (or infinite) system sizes.",
"Numerical investigations on the highly symmetric $36d$ cluster showed how the $XXZ0$ point on the kagome lattice is embedded in the wider phase diagram.",
"Figure: Acknowledgements"
]
] | 1808.08633 |
[
[
"Contextual Parameter Generation for Universal Neural Machine Translation"
],
[
"Abstract We propose a simple modification to existing neural machine translation (NMT) models that enables using a single universal model to translate between multiple languages while allowing for language specific parameterization, and that can also be used for domain adaptation.",
"Our approach requires no changes to the model architecture of a standard NMT system, but instead introduces a new component, the contextual parameter generator (CPG), that generates the parameters of the system (e.g., weights in a neural network).",
"This parameter generator accepts source and target language embeddings as input, and generates the parameters for the encoder and the decoder, respectively.",
"The rest of the model remains unchanged and is shared across all languages.",
"We show how this simple modification enables the system to use monolingual data for training and also perform zero-shot translation.",
"We further show it is able to surpass state-of-the-art performance for both the IWSLT-15 and IWSLT-17 datasets and that the learned language embeddings are able to uncover interesting relationships between languages."
],
[
"Introduction",
"Neural Machine Translation (NMT) directly models the mapping of a source language to a target language without any need for training or tuning any component of the system separately.",
"This has led to a rapid progress in NMT and its successful adoption in many large-scale settings [35], [9].",
"The encoder-decoder abstraction makes it conceptually feasible to build a system that maps any source sentence in any language to a vector representation, and then decodes this representation into any target language.",
"Thus, various approaches have been proposed to extend this abstraction for multilingual MT [23], [10], [18], [15], [11].",
"Prior work in multilingual NMT can be broadly categorized into two paradigms.",
"The first, universal NMT [18], [15], uses a single model for all languages.",
"Universal NMT lacks any language-specific parameterization, which is an oversimplification and detrimental when we have very different languages and limited training data.",
"As verified by our experiments, the method of [18] suffers from high sample complexity and thus underperforms in limited data settings.",
"The universal model proposed by [15] requires a new coding scheme for the input sentences, which results in large vocabulary sizes that are difficult to scale.",
"The second paradigm, per-language encoder-decoder [23], [11], uses separate encoders and decoders for each language.",
"This does not allow for sharing of information across languages, which can result in overparameterization and can be detrimental when the languages are similar.",
"In this paper, we strike a balance between these two approaches, proposing a model that has the ability to learn parameters separately for each language, but also share information between similar languages.",
"We propose using a new contextual parameter generator (CPG) which (a) generalizes all of these methods, and (b) mitigates the aforementioned issues of universal and per-language encoder-decoder systems.",
"It learns language embeddings as a context for translation and uses them to generate the parameters of a shared translation model for all language pairs.",
"Thus, it provides these models the ability to learn parameters separately for each language, but also share information between similar languages.",
"The parameter generator is general and allows any existing NMT model to be enhanced in this way.In fact, it could likely be applied in other scenarios, such as domain adaptation, as well.",
"In addition, it has the following desirable features: Simple: Similar to [18] and [15], and in contrast with [23] and [11], it can be applied to most existing NMT systems with some minor modification, and it is able to accommodate attention layers seamlessly.",
"Multilingual: Enables multilingual translation using the same single model as before.",
"Semi-supervised: Can use monolingual data.",
"Scalable: Reduces the number of parameters by employing extensive, yet controllable, sharing across languages, thus mitigating the need for large amounts of data, as in [18].",
"It also allows for the decoupling of languages, avoiding the need for a large shared vocabulary, as in [15].",
"Adaptable: Can adapt to support new languages, without requiring complete retraining.",
"State-of-the-art: Achieves better performance than pairwise NMT models and [18].",
"In fact, our approach can surpass state-of-the-art performance.",
"We first introduce a modular framework that can be used to define and describe most existing NMT systems.",
"Then, in Section , we introduce our main contribution, the contextual parameter generator (CPG), in terms of that framework.",
"We also argue that the proposed approach takes us a step closer to a common universal interlingua."
],
[
"Background",
"We first define the multi-lingual NMT setting and then introduce a modular framework that can be used to define and describe most existing NMT systems.",
"This will help us distill previous contributions and introduce ours."
],
[
"Setting.",
"We assume that we have a set of source languages $S$ and a set of target languages $T$ .",
"The total number of languages is $L = |S \\cup T|$ .",
"We also assume we have a set of $C \\le |S| \\times |T|$ pairwise parallel corpora, $\\lbrace P_1, \\dots , P_C\\rbrace $ , each of which contains a set of sentence pairs for a single source-target language combination.",
"The goal of multilingual NMT is to build a model that, when trained using the provided parallel corpora, can learn to translate well between any pair of languages in $S \\times T$ .",
"The majority of related work only considers pairwise NMT, where $|S|=|T|=1$ ."
],
[
"NMT Modules",
"Most NMT systems can be decomposed to the following modules (also visualized in Figure REF ).",
"Figure: Overview of an NMT system, under our modular framework.",
"Our main contribution lies in the parameter generator module (i.e., coupled or decoupled — each of the boxes with blue titles is a separate option).",
"Note that gg denotes a parameter generator network.",
"In our experiments, we consider linear forms for this network.",
"However, our contribution does not depend on the choices made regarding the rest of the modules; we could still use our parameter generator with different architectures for the encoder and the decoder, as well as using different kinds of vocabularies."
],
[
"Preprocessing Pipeline.",
"The data preprocessing pipeline handles tokenization, cleaning, normalizing the text data and building a vocabulary, i.e.",
"a two-way mapping from preprocessed sentences to sequences of word indices that will be used for the translation.",
"A commonly used proposal for defining the vocabulary is the byte-pair encoding (BPE) algorithm which generates subword unit vocabularies [32].",
"This eliminates the notion of out-of-vocabulary words, often resulting in increased translation quality."
],
[
"Encoder/Decoder.",
"The encoder takes in indexed source language sentences, and produces an intermediate representation that can later be used by a decoder to generate sentences in a target language.",
"Generally, we can think of the encoder as a function, ${(enc)}$ , parameterized by ${\\bf \\theta }^{(enc)}$ .",
"Similarly, we can think of the decoder as another function, ${(dec)}$ , parameterized by ${\\bf \\theta }^{(dec)}$ .",
"The goal of learning to translate can then be defined as finding the values for ${\\bf \\theta }^{(enc)}$ and ${\\bf \\theta }^{(dec)}$ that result in the best translations.",
"A large amount of previous work proposes novel designs for the encoder/decoder module.",
"For example, using attention over the input sequence while decoding [3], [24] provides significant gains in translation performance.Note that depending on the vocabulary that is used and on whether it is one shared vocabulary across all languages, or one vocabulary per language, the output projection layer of the decoder (which produces probabilities over words) may be language dependent, or common across all languages.",
"In our experiments, we used separate vocabularies and thus this layer was language-dependent."
],
[
"Parameter Generator.",
"All modules defined so far have previously been used when describing NMT systems and are thus easy to conceptualize.",
"However, in previous work, most models are trained for a given language pair, and it is not trivial to extend them to work for multiple pairs of languages.",
"We introduce here the concept of the parameter generator, which makes it easy to define and describe multilingual NMT systems.",
"This module is responsible for generating ${\\bf \\theta }^{(enc)}$ and ${\\bf \\theta }^{(dec)}$ for any given source and target language.",
"Different parameter generators result in different numbers of learnable parameters and can thus be used to share information across different languages.",
"Next, we describe related work, in terms of the parameter generator for NMT: [noitemsep,leftmargin=*] Pairwise: In the simple and commonly used pairwise NMT setting [35], [9], the parameter generator would generate separate parameters, ${\\bf \\theta }^{(enc)}$ and ${\\bf \\theta }^{(dec)}$ , for each pair of source-target languages.",
"This results in no parameter sharing across languages, and thus $\\mathcal {O}(ST)$ parameters.",
"Per-Language: In the case of [10], [23] and [11], the parameter generator would generate separate encoder parameters, ${\\bf \\theta }^{(enc)}$ , for each source language, and separate decoder parameters, ${\\bf \\theta }^{(dec)}$ , for each target language.",
"This leads to a reduction in the number of learnable parameters for multilingual NMT, from $\\mathcal {O}(ST)$ to $\\mathcal {O}(S+T)$ .",
"On one hand, [10] train multiple models as a one-to-many multilingual NMT system that translates from one source language to multiple target languages.",
"On the other hand, [23] and [11] perform many-to-many translation.",
"[23], however, only report results for a single language pair and do not attempt multilingual translation.",
"[11] propose an attention mechanism that is shared across all language pairs.",
"We generalize the idea of multi-way multilingual NMT with the parameter generator network, described later.",
"Universal: In the case of [15] and [18], the authors propose using a single common set of encoder-decoder parameters for all language pairs.",
"While [15] embed words in a common semantic space across languages, [18] learn language embeddings that are in the same space as the word embeddings.",
"Here, the parameter generator would provide the same parameters ${\\bf \\theta }^{(enc)}$ and ${\\bf \\theta }^{(dec)}$ for all language pairs.",
"It would also create and keep track of learnable variables representing language embeddings that are prepended to the encoder input sequence.",
"As we observed in our experiments, this system fails to perform well when the training data is limited.",
"Finally, we believe that embedding languages in the same space as words is not intuitive; in our approach, languages are embedded in a separate space.",
"In contrast to all these related systems, we provide a simple, efficient, yet effective alternative – a parameter generator for multilingual NMT, that enables semi-supervised and zero-shot learning.",
"We also learn language embeddings, similar to [18], but in our case they are separate from the word embeddings and are treated as a context for the translation, in a sense that will become clear in the next section.",
"This notion of context is used to define parameter sharing across various encoders and decoders, and, as we discuss in our conclusion, is even applicable beyond NMT."
],
[
"Proposed Method",
"We propose a new way to share information across different languages and to control the amount of sharing, through the parameter generator module.",
"More specifically, we propose contextual parameter generators."
],
[
"Contextual Parameter Generator.",
"Let us denote the source language for a given sentence pair by $\\ell _s$ and the target language by $\\ell _t$ .",
"Then, when using the contextual parameter generator, the parameters of the encoder are defined as $\\theta ^{(enc)}\\triangleq g^{(enc)}(\\bf {l}_s)$ , for some function $g^{(enc)}$ , where $\\bf {l}_s$ denotes a language embedding for the source language $\\ell _s$ .",
"Similarly, the parameters of the decoder are defined as $\\theta ^{(dec)}\\triangleq g^{(dec)}(\\bf {l}_t)$ for some function $g^{(dec)}$ , where $\\bf {l}_t$ denotes a language embedding for the target language $\\ell _t$ .",
"Our generic formulation does not impose any constraints on the functional form of $g^{(enc)}$ and $g^{(dec)}$ .",
"In this case, we can think of the source language, $\\ell _s$ , as a context for the encoder.",
"The parameters of the encoder depend on its context, but its architecture is common across all contexts.",
"We can make a similar argument for the decoder, and that is where the name of this parameter generator comes from.",
"We can even go a step further and have a parameter generator that defines $\\theta ^{(enc)}\\triangleq g^{(enc)}(\\bf {l}_s, \\bf {l}_t)$ and $\\theta ^{(dec)}\\triangleq g^{(dec)}(\\bf {l}_s, \\bf {l}_t)$ , thus coupling the encoding and decoding stages for a given language pair.",
"In our experiments we stick to the previous, decoupled, form, because unlike [18], it has the potential to lead to an interlingua.",
"Concretely, because the encoding and decoding stages are decoupled, the encoder is not aware of the target language while generating it.",
"Thus, we can take an encoded intermediate representation of a sentence and translate it to any target language.",
"This is because, in this case, the intermediate representation is independent of any target language.",
"This makes for a stronger argument that the intermediate representation produced by our encoder could be approaching a universal interlingua, more so than methods that are aware of the target language when they perform encoding."
],
[
"Parameter Generator Network",
"We refer to the functions $g^{(enc)}$ and $g^{(dec)}$ as parameter generator networks.",
"Even though our proposed NMT framework does not rely on a specific choice for $g^{(enc)}$ and $g^{(dec)}$ , here we describe the functional form we used for our experiments.",
"Our goal is to provide a simple form that works, and for which we can reason about.",
"For this reason, we decided to define the parameter generator networks as simple linear transforms, similar to the factored adaptation model of [26], which was only applied to the bias terms of the output softmax: $g^{(enc)}(\\bf {l}_s) & \\triangleq \\bf {W^{(enc)}} \\bf {l}_s, \\\\g^{(dec)}(\\bf {l}_t) & \\triangleq \\bf {W^{(dec)}} \\bf {l}_t,$ where $\\bf {l}_s, \\bf {l}_t \\in \\mathbb {R}^M$ , $\\bf {W^{(enc)}} \\in \\mathbb {R}^{P^{(enc)} \\times M}$ , $\\bf {W^{(dec)}} \\in \\mathbb {R}^{P^{(dec)} \\times M}$ , $M$ is the language embedding size, $P^{(enc)}$ is the number of parameters of the encoder, and $P^{(dec)}$ is the number of parameters of the decoder.",
"Another way to interpret this model is that it imposes a low-rank constraint on the parameters.",
"As opposed to our approach, in the base case of using multiple pairwise models to perform multilingual translation, each model has $P = P^{(enc)} + P^{(dec)}$ learnable parameters for its encoder and decoder.",
"Given that the models are pairwise, for $L$ languages, we have a total of $L(L - 1)$ learnable parameter vectors of size $P$ .",
"On the other hand, using our contextual parameter generator we have a total of $L$ vectors of size $M$ (one for each language), and a single matrix of size $P \\times M$ .",
"Then, the parameters of the encoder and the decoder, for a single language pair, are defined as a linear combination of the $M$ columns of that matrix."
],
[
"Controlled Parameter Sharing.",
"We can further control parameter sharing by observing that the encoder/decoder parameters often have some “natural grouping”.",
"For example, in the case of recurrent neural networks we may have multiple weight matrices, one for each layer, as well as attention-related parameters.",
"Based on this observations, we now propose a way to control how much information is shared across languages.",
"The language embeddings need to represent all of the language-specific information and thus may need to be large in size.",
"However, when computing the parameters of each group, only a small part of that information is relevant.",
"Let $\\theta ^{(enc)} = \\smash{\\lbrace \\theta ^{(enc)}_j\\rbrace _{j=1}^G}$ and $\\theta ^{(enc)}_j \\in \\mathbb {R}^{P^{(enc)}_j}$ , where $G$ denotes the number of groups.",
"Then, we define: $\\theta ^{(enc)}_j & \\triangleq \\bf {W^{(enc)}_j} \\bf {P^{(enc)}_j} \\bf {l}_s,$ where $\\smash{\\bf {W^{(enc)}_j}} \\in \\mathbb {R}^{P^{(enc)}_j \\times M^{\\prime }}$ and $\\smash{\\bf {P^{(enc)}_j}} \\in \\mathbb {R}^{M^{\\prime } \\times M}$ , with $M^{\\prime }<M$ (and similarly for the decoder parameters).",
"We can see now that $\\smash{\\bf {P^{(enc)}_j}}$ is used to extract the relevant information (size $M^{\\prime }$ ) for parameter group $j$ , from the larger language embedding (size $M$ ).",
"This allows us to control the parameter sharing across languages in the following way: if we want to increase the number of per-language parameters (i.e., the language embedding size) we can increase $M$ while keeping $M^{\\prime }$ small enough so that the total number of parameters does not explode.",
"This would not have been possible without the proposed low-rank approximation for $\\bf {W^{(enc)}}$ , that uses the parameter grouping information."
],
[
"Alternative Options.",
"Given that our proposed approach does not depend on the specific choice of the parameter generator network, it might be interesting to design models that use side-information about the languages that are being used (such as linguistic information about language families and hierarchies).",
"This is outside the scope of this paper, but may be an interesting future direction."
],
[
"Semi-Supervised and Zero-Shot Learning",
"The proposed parameter generator also enables semi-supervised learning via back-translation.",
"Concretely, monolingual data can be used to train the shared encoder/decoder networks to translate a sentence from some language to itself (similar to the idea of auto-encoders by [34]).",
"This is possible and can help learning because of the fact that many of the learnable parameters are shared across languages.",
"Furthermore, zero-shot translation, where the model translates between language pairs for which it has seen no explicit training data, is also possible.",
"This is because the same per-language parameters are used to translate to and from a given language, irrespective of the language at the other end.",
"Therefore, as long as we train our model using some language pairs that involve a given language, it is possible to learn to translate in any direction involving that language."
],
[
"Potential for Adaptation",
"Let us assume that we have trained a model using data for some set of languages, $\\ell _1, \\ell _2, \\hdots , \\ell _m$ .",
"If we obtain data for some new language $\\ell _n$ , we do not have to retrain the whole model from scratch.",
"In fact, we can fix the parameters that are shared across all languages and only learn the embedding for the new language (along with the relevant word embeddings if not using a shared vocabulary).",
"Assuming that we had a sufficient number of languages in the beginning, this may allow us to obtain reasonable translation performance for the new language, with a minimal amount of training.This is due to the small number of parameters that need to be learned in this case.",
"To put this into perspective, in most of our experiments we used language embeddings of size 8."
],
[
"Number of Parameters",
"For the base case of using multiple pairwise models to perform multilingual translation, each model has $P + 2WV$ parameters, where $P = P^{(enc)} + P^{(dec)}$ , $W$ is the word embedding size, and $V$ is the vocabulary size per language (assumed to be the same across languages, without loss of generality).",
"Given that the models are pairwise, for $L$ languages, we have a total of $L(L - 1)(P + 2WV) = \\mathcal {O}(L^2P +2L^2WV)$ learnable parameters.",
"For our approach, using the linear parameter generator network presented in Section REF , we have a total of $\\mathcal {O}(PM + LWV)$ learnable parameters.",
"Note that the number of encoder/decoder parameters has no dependence on $L$ now, meaning that our model can easily scale to a large number of languages."
],
[
"Experiments",
"In this section, we describe our experimental setup along with our results and key observations."
],
[
"Setup.",
"For all our experiments we use as the base NMT model an encoder-decoder network which uses a bidirectional LSTM for the encoder, and a two-layer LSTM with the attention model of [3] for the decoder.",
"The word embedding size is set to 512.",
"This is a common baseline model that achieves reasonable performance and we decided to use it as-is, without tuning any of its parameters, as extensive hyperparameter search is outside the scope of this paper.",
"Table: Comparison of our proposed approach (shaded rows) with the base pairwise NMT model (PNMT) and the Google multilingual NMT model (GML) for the IWSLT-15 dataset.",
"The Percent Parallel row shows what portion of the parallel corpus is used while training; the rest is being used only as monolingual data.",
"Results are shown for the BLEU and Meteor metrics.",
"CPG* represents the same model as CPG, but trained without using auto-encoding training examples.",
"The best score in each case is shown in bold.During training, we use a label smoothing factor of 0.1 [35] and the AMSGrad optimizer [29] with its default parameters in TensorFlow, and a batch size of 128 (due to GPU memory constraints).",
"Optimization was stopped when the validation set BLEU score was maximized.",
"The order in which language pairs are used while training was as follows: we always first sample a language pair (uniformly at random), and then sample a batch for that pair (uniformly at random).We did not observe any “forgetting” effect, because we keep “re-visiting” all language pairs throughout training.",
"It would be interesting to explore other sampling schemes, but it is outside the scope of this paper.",
"During inference, we employ beam search with a beam size of 10 and the length normalization scheme of [35].",
"We want to emphasize that we did not run experiments with other architectures or configurations, and thus this architecture was not chosen because it was favorable to our method, but rather because it was a frequently mentioned baseline in existing literature.",
"All experiments were run on a machine with a single Nvidia V100 GPU, and 24 GBs of system memory.",
"Our most expensive experiment took about 10 hours to complete, which would cost about $\\$25$ on a cloud computing service such as Google Cloud or Amazon Web Services, thus making our results reproducible, even by independent researchers.",
"Table: Comparison of our proposed approach (shaded rows) with the base pairwise NMT model (PNMT) and the Google multilingual NMT model (GML) for the IWSLT-17 dataset.",
"Results are shown for the BLEU metric only because Meteor does not support It, Nl, and Ro.",
"CPG8 represents CPG using language embeddings of size 8.",
"The “C4” subscript represents the low-rank version of CPG for controlled parameter sharing (see Section ), using rank 4, etc.",
"The best score in each case is shown in bold."
],
[
"Experimental Settings.",
"The goal of our experiments is to show how, by using a simple modification of this model, (i) we can achieve significant improvements in performance, while at the same time (ii) being more data and computation efficient, and (iii) enabling support for zero-shot translation.",
"To that end, we perform three types of experiments: Supervised: In this experiment, we use full parallel corpora to train our models.",
"Plain pairwise NMT models (PNMT) are compared to the same models modified to use our proposed decoupled parameter generator.",
"We use two variants: (i) one which does not use auto-encoding of monolingual data while training (CPG*), and (ii) one which does (CPG).",
"Please refer to Section REF for more details.",
"Low-Resource: Similar to the supervised experiments except that we limit the size of the parallel corpora used in training.",
"However, for GML and CPG the full monolingual corpus is used for auto-encoding training.",
"Zero-Shot: In this experiment, our goal is to evaluate how well a model can learn to translate between language pairs that it has not seen while training.",
"For example, a model trained using parallel corpora between English and German, and English and French, will be evaluated in translating from German to French.",
"PNMT can perform zero-shot translation in this setting using pivoting.",
"This means that, in the previous example, we would first translate from German to English and then from English to French (using two pairwise models for a single translation).",
"However, pivoting is prone to error propagation incurred when chaining multiple imperfect translations.",
"The proposed CPG models inherently support zero-shot translation and require no pivoting.",
"For the experiments using the CPG model without controlled parameter sharing, we use language embeddings of size 8.",
"This is based merely on the fact that this is the largest model size we could fit on one GPU.",
"Whenever possible, we compare against PNMT, GML by [18],We use our own implementation of GML in order to obtain a fair comparison, in terms of the whole MT pipeline.",
"We have modified it to use the same per-language vocabularies that we use for our approaches, as the proposed shared BPE vocabulary fails to perform well for the considered datasets.",
"and other state-of-the-art results."
],
[
"Datasets.",
"We use the following datasets: [noitemsep,leftmargin=*] IWSLT-15: Used for supervised and low-resource experiments only (this dataset does not support zero-shot learning).",
"We report results for Czech (Ch), English (En), French (Fr), German (De), Thai (Th), and Vietnamese (Vi).",
"This dataset contains 90,000-220,000 training sentence pairs (depending on the language pair), 500-900 validation pairs, and 1,000-1,300 test pairs.",
"IWSLT-17: Used for supervised and zero-shot experiments.",
"We report results for Dutch (Nl), English (En), German (De), Italian (It), and Romanian (Ro).",
"This dataset contains 220,000 training sentence pairs (for all language pairs except for the zero-shot ones), 900 validation pairs, and 1,100 test pairs."
],
[
"Data Preprocessing.",
"We preprocess our data using a modified version of the Moses tokenizer [19] that correctly handles escaped HTML characters.",
"We also perform some Unicode character normalization and cleaning.",
"While training, we only consider sentences up to length 50.",
"For both datasets, we generate a per-language vocabulary consisting of the most frequently occurring words, while ignoring words that appear less than 5 times in the whole corpus, and capping the vocabulary size to 20,000 words."
],
[
"Results.",
"Our results for the IWSLT-15 experiments are shown in Table REF .",
"It is clear that our approach consistently outperforms both the corresponding pairwise model and GML.",
"Furthermore, its advantage grows larger in the low-resource setting (up to $5.06$ BLEU score difference, or a $2.4\\times $ increase), which is expected due to the extensive parameter sharing in our model.",
"For this dataset, there exist some additional published state-of-the-art results not shown in Tables REF and REF .",
"[17] report a BLEU score of 28.07 for the En$\\mathrel {\\hbox{\\rule []{3pt}{.4pt}}\\hspace{-2.22214pt}\\hbox{\\usefont {U}{lasy}{m}{n})}}$ Vi task, while our model is able to achieve a score of 29.03.",
"Furthermore, [15] report a BLEU score of 25.87 for the En$\\mathrel {\\hbox{\\rule []{3pt}{.4pt}}\\hspace{-2.22214pt}\\hbox{\\usefont {U}{lasy}{m}{n})}}$ De task, while our model is able to achieve a score of 26.77.We were unable to find reported state-of-the-art results for the rest of the language pairs.",
"Our results for the IWSLT-17 experiments are shown in Table REF .Note that, our results for IWSLT-17 are not comparable to those of the official challenge report [5], as we use less training data, a smaller baseline model, and our evaluation pipeline potentially differs.",
"However, the numbers presented for all methods in this paper are comparable, as they were all obtained using the same baseline model and evaluation pipeline.",
"Again, our method consistently outperforms both PNMT and GML, in both the supervised and the zero-shot settings.",
"Furthermore, the results indicate that our model performance is robust to different sizes of the language embeddings and the choice of $M^{\\prime }$ for controllable parameter sharing.",
"It only underperforms in the degenerate case where $M^{\\prime }=1$ .",
"It is also worth noting that, in the fully supervised setting, GML, the current state-of-the-art in the multilingual setting, underperforms the pairwise models.",
"The presented results provide evidence that our proposed approach is able to significantly improve performance, without requiring extensive tuning."
],
[
"Language Embeddings.",
"An important aspect of our model is that it learns language embeddings.",
"In Figure REF we show pairwise cosine distances between the learned language embeddings for our fully supervised experiments.",
"There are some interesting patterns that indicate that the learned language embeddings are reasonable.",
"For example, we observe that German (De) and Dutch (Nl) are most similar for the IWSLT-17 dataset, with Italian (It) and Romanian (Ro) coming second.",
"Furthermore, Romanian and German are the furthest apart for that dataset.",
"These relationships agree with linguistic knowledge about these languages and the families they belong to.",
"We see similar patterns in the IWSLT-15 results but we focus on IWSLT-17 here, because it is a larger, better quality, dataset with more supervised language pairs.",
"These results are encouraging for analyzing such embeddings to discover relationships between languages that were previously unknown.",
"For example, perhaps surprisingly, French (Fr) and Vietnamese (Vi) appear to be significantly related for the IWSLT-15 dataset results.",
"This is likely due to French influence in Vietnamese because to the occupation of Vietnam by France during the 19th and 20th centuries [25].",
"Figure: Pairwise cosine distance for all language pairs in the IWSLT-15 and IWSLT-17 datasets.",
"Darker colors represent more similar languages."
],
[
"Implementation and Reproducibility",
"Along with this paper we are releasing an implementation of our approach and experiments as part of a new Scala framework for machine translation.https://github.com/eaplatanios/symphony-mt It is built on top of TensorFlow Scala [28] and follows the modular NMT design (described in Section REF ) that supports various NMT models, including our baselines (e.g., [18]).",
"It also contains data loading and preprocessing pipelines that support multiple datasets and languages, and is more efficient than other packages (e.g., tf-nmthttps://github.com/tensorflow/nmt).",
"Furthermore, the framework supports various vocabularies, among which we provide a new implementation for the byte-pair encoding (BPE) algorithm [32] that is 2 to 3 orders of magnitude faster than the released one.https://github.com/rsennrich/subword-nmt All experiments presented in this paper were performed using version 0.1.0 of the framework."
],
[
"Related Work",
"Interlingual translation [30] has been the object of many research efforts.",
"For a long time, before the move to NMT, most practical machine translation systems only focused on individual language pairs.",
"Since the success of end-to-end NMT approaches such as the encoder-decoder framework [33], [3], [8], recent work has tried to extend the framework to multi-lingual translation.",
"An early approach was [10] who performed one-to-many translation with a separate attention mechanism for each decoder.",
"[23] extended this idea with a focus on multi-task learning and multiple encoders and decoders, operating in a single shared vector space.",
"The same architecture is used in [4] for translation across multiple modalities.",
"[36] flipped this idea with a many-to-one translation model, however requiring the presence of a multi-way parallel corpus between all the languages, which is difficult to obtain.",
"[22] used a single character-level encoder across multiple languages by training a model on a many-to-one translation task.",
"Closest to our work are more recent approaches, already described in Section [11], [18], [15], that attempt to enforce different kinds of parameter sharing across languages.",
"Parameter sharing in multilingual NMT naturally enables semi-supervised and zero-shot learning.",
"Unsupervised learning has been previously explored with key ideas such as back-translation [31], dual learning [16], common latent space learning [21], etc.",
"In the vein of multilingual NMT, [2] proposed a model that uses a shared encoder and multiple decoders with a focus on unsupervised translation.",
"The entire system uses cross-lingual embeddings and is trained to reconstruct its input using only monolingual data.",
"Zero-shot translation was first attempted in [12] who performed zero-zhot translation using their pre-trained multi-way multilingual model, fine-tuning it with pseudo-parallel data generated by the model itself.",
"This was recently extended using a teacher-student framework [6].",
"Later, zero-shot translation without any additional steps was attempted in [18] using their shared encoder-decoder network.",
"An iterative training procedure that leverages the duality of translations directly generated by the system for zero-shot learning was proposed by [20].",
"For extremely low resource languages, Gu:2018 proposed sharing lexical and sentence-level representations across multiple source languages with a single target language.",
"Closely related is the work of [7] who proposed the joint training of source-to-pivot and pivot-to-target NMT models.",
"[14] are probably the first to introduce a similar idea to that of having one network (called a hypernetwork) generate the parameters of another.",
"However, in that work, the input to the hypernetwork are structural features of the original network (e.g., layer size and index).",
"[1] also propose a related method where a neural network generates the parameters of a linear model.",
"Their focus is mostly on interpretability (i.e., knowing which features the network considers important).",
"However, to our knowledge, there is no previous work which proposes having a network generate the parameters of another deep neural network (e.g., a recurrent neural network), using some well-defined context based on the input data.",
"This context, in our case, is the language of the input sentences to the translation model, along with the target translation language."
],
[
"Conclusion and Future Directions",
"We have presented here a novel contextual parameter generation approach to neural machine translation.",
"Our resulting system, which outperforms other state-of-the-art systems, uses a standard pairwise encoder-decoder architecture.",
"However, it differs from earlier approaches by incorporating a component that generates the parameters to be used by the encoder and the decoder for the current sentence, based on the source and target languages, respectively.",
"We refer to this novel component as the contextual parameter generator.",
"The benefit of this approach is that it dramatically improves the ratio of the number of parameters to be learned, to the number of training examples available, by leveraging shared structure across different languages.",
"Thus, our approach does not require any extra machinery such as back-translation, dual learning, pivoting, or multilingual word embeddings.",
"It rather relies on the simple idea of treating language as a context within which to encode/decode.",
"We also showed that the proposed approach is able to achieve state-of-the-art performance without requiring any tuning.",
"Finally, we performed a basic analysis of the learned language embeddings, which showed that cosine distances between the learned language embeddings reflect well known similarities among language pairs such as German and Dutch.",
"In the future, we want to extend the concept of the contextual parameter generator to more general settings, such as translating between different modalities of data (e.g., image captioning).",
"Furthermore, based on the discussion of Section REF , we hope to develop an adaptable, never-ending learning [27] NMT system."
],
[
"Acknowledgments",
"We would like to thank Otilia Stretcu, Abulhair Saparov, and Maruan Al-Shedivat for the useful feedback they provided in early versions of this paper.",
"This research was supported in part by AFOSR under grant FA95501710218."
]
] | 1808.08493 |
[
[
"Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo"
],
[
"Abstract In this work, we establish $\\mathrm{L}^2$-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler.",
"The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics.",
"As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance."
],
[
"Introduction",
"Consider a probability distribution $\\pi $ defined on the Borel $\\sigma $ -field $\\mathcal {X}$ of some domain $\\mathsf {X}= \\mathbb {R}^d$ or $\\mathsf {X}= \\mathbb {T}^d$ where $\\mathbb {T}= \\mathbb {R}/ \\mathbb {Z}$ .",
"Assume that $\\pi $ has a density with respect to the Lebesgue measure also denoted $\\pi $ and of the form $ \\pi = \\mathrm {e}^{-U} / \\int _{\\mathbb {R}^d} \\mathrm {e}^{-U(y)} \\mathrm {d}y $ where $U\\colon \\mathsf {X}\\rightarrow \\mathbb {R}$ is a continuously differentiable function and is referred to as the potential associated with $\\pi $ .",
"Sampling from such distributions is of interest in computational statistical mechanics and in Bayesian statistics and allows one, for example, to compute efficiently expectations of functions $f: \\mathsf {X}\\rightarrow \\mathbb {R}$ with respect to $\\pi $ by invoking empirical process limit theorems, e.g.",
"the law of large numbers.",
"In practical set-ups, sampling exactly from $\\pi $ directly is either impossible or computationally prohibitive.",
"A standard and versatile approach to sampling from such distributions consists of using Markov Chain Monte Carlo (MCMC) techniques [30], [41], [54], where the ability of simulating realizations of ergodic Markov chains leaving $\\pi $ invariant is exploited.",
"Markov Process Monte Carlo (MPMC) methods are the continuous time counterparts of MCMC but their exact implementation is most often impossible on computers and requires additional approximation, such as time discretization of the process in the case of the Langevin diffusion.",
"A notable exception, which has recently attracted significant attention, is the class of MPMC relying on Piecewise Deterministic Markov Processes (PDMP) [17], which in addition to being simpler to simulate than earlier MPMC, are nonreversible, offering the promise of better performance.",
"We now briefly introduce a class of processes covering existing algorithms.",
"The generic mathematical notation we use in the introduction is fairly standard and fully defined at the end of the section.",
"Known PDMP Monte Carlo methods rely on the use of the auxiliary variable trick, that is the introduction of an instrumental variable and probability distribution $\\mu $ defined on an extended domain, of which $\\pi $ is a marginal distribution, which may facilitate simulation.",
"In the present set-up, one introduces the velocity variable $v \\in \\mathsf {V}\\subset \\mathbb {R}^d$ associated with a probability distribution $\\nu $ defined on the $\\sigma $ -field $\\mathcal {V}$ of $\\mathsf {V}$ , where the subset $\\mathsf {V}$ is assumed to be closed.",
"Standard choices for $\\nu $ include the centered normal distribution with covariance matrix $m_2 \\operatorname{I}_d$ , where $\\operatorname{I}_d$ is the $d$ -dimensional identity matrix, the uniform distribution on the unit sphere $\\mathbb {S}^{d-1}$ , or the uniform distribution on $\\mathsf {V}=\\lbrace -1, 1 \\rbrace ^d$ .",
"Let $\\mathsf {E}= \\mathsf {X}\\times \\mathsf {V}$ and define the probability measure $\\mu = \\pi \\otimes \\nu $ .",
"The aim is now to sample from the probability distribution $\\mu $ .",
"We denote by $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ the set of bounded functions of $\\mathrm {C}^2(\\mathsf {E})$ .",
"The PDMP Monte Carlo algorithms we are aware of fall in a class of processes associated with generators of the form, for $f\\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and $(x,v) \\in \\mathsf {E}$ , $\\mathcal {L}_1 f(x,v) = v^\\top \\nabla _xf(x,v) + \\sum _{k=1}^K \\lambda _k(x,v) \\left( \\mathcal {B}_k - \\operatorname{Id}\\right)f(x,v) +m_2^{{{1}{2}}} \\lambda _{\\rm ref}(x)\\mathcal {R}_v f(x,v)\\;,$ where $K\\in \\mathbb {N}$ , $\\lambda _k : \\mathsf {E}\\rightarrow \\mathbb {R}_+$ for $k \\in \\lbrace 1,\\ldots , K\\rbrace $ , $\\lambda _{\\rm ref}: \\mathsf {X}\\rightarrow \\mathbb {R}_+$ , $(\\mathcal {R}_v,\\mathrm {D}(\\mathcal {R}_v))$ and $(\\mathcal {B}_k,\\mathrm {D}(\\mathcal {B}_k))$ for $k \\in \\lbrace 1,\\ldots , K\\rbrace $ are operators we specify below, and $m_2 = \\int _{\\mathsf {V}} v_1^2 \\ \\mathrm {d}\\nu (v) \\;,$ which is assumed to be finite.",
"For any $k \\in \\lbrace 1,\\ldots , K\\rbrace $ , $\\lambda _k$ will be referred to as a jump rate and $\\lambda _{\\rm ref}$ as the refreshment rate.",
"In the case where $\\mathsf {V}= \\mathbb {R}^d$ and $\\nu $ is the zero-mean Gaussian distribution on $\\mathbb {R}^d$ with covariance matrix $m_2 \\operatorname{I}_d$ , we also consider generators of the form, for any $f\\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and $(x,v) \\in \\mathsf {E}$ , $\\mathcal {L}_2 f(x,v) = \\mathcal {L}_1 f(x,v) - m_2 F_0(x)^{\\top } \\nabla _v f(x,v) \\;,$ where $F_0 : \\mathsf {X}\\rightarrow \\mathbb {R}^d$ .",
"For any $k \\in \\lbrace 1,\\ldots , K\\rbrace $ , the jump operators $\\mathcal {B}_k$ we consider are associated with continuous vector fields $F_k : \\mathsf {X}\\rightarrow \\mathbb {R}^d$ of the form, for any $f: \\mathsf {E}\\rightarrow \\mathbb {R}$ and $(x,v) \\in \\mathsf {E}$ , $ \\mathcal {B}_k f(x, v) = f\\left( x,v - 2 \\big (v^\\top \\mathrm {n}_k(x)\\big ) \\, \\mathrm {n}_k(x) \\right) \\;, \\quad \\mathrm {n}_k(x) = {\\left\\lbrace \\begin{array}{ll}F_k(x)/\\left|F_k(x) \\right| & \\text{ if $F_k(x) \\ne 0$} \\;,\\\\0 & \\text{ otherwise \\;.}\\end{array}\\right.",
"}$ These operators correspond to reflections of the velocity through the hyperplanes orthogonal to $F_k(X)$ at the event position $X$ , i.e.",
"a flip of the component of the velocity in the direction given by $F_k$ inducing an elastic “bounce” of the position trajectory with the hyperplane.",
"As we shall see, the $K+1$ vector fields $F_k$ are tied to the potential $U$ by the relation $\\nabla _xU = \\sum _{k=0}^K F_k$ , required to ensure that $\\mu $ is left invariant by the associated semi-group.",
"Informally, assuming for the moment that $\\lambda _{\\rm ref}=0$ and $F_0 = \\nabla _xU_0$ for some $U_0 \\colon \\mathsf {X}\\rightarrow \\mathbb {R}$ , the corresponding process follows the solution of Hamilton's equations $(\\dot{x}_t,\\dot{v}_t) = \\big (v_t, -\\nabla _xU_0(x_t)\\big )$ for a random time of distribution governed by an inhomogeneous Poisson process with rate $(x,v) \\mapsto \\sum _{k=1}^K \\lambda _k(x,v)$ .",
"When an event occurs and the current state of the process is $(X,V)$ , one chooses between the $K$ possible updates of the state available, with probability proportional to $\\lambda _1(X,V),\\ldots ,\\lambda _K(X,V)$ , with the particularity here that the position $X$ is left unchanged.",
"The vector fields $\\lbrace F_k : \\mathsf {X}\\rightarrow \\mathbb {R}^d \\, ; \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace \\rbrace $ and jump rates $\\lbrace \\lambda _k : \\mathsf {E}\\rightarrow \\mathbb {R}_+ \\, ; \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace \\rbrace $ are linked by the relations $\\lambda _k(x,v) -\\lambda _k(x,-v)=v^\\top F_k(x)$ for $k \\in \\lbrace 1,\\ldots , K\\rbrace $ and $(x,v) \\in \\mathsf {E}$ , together with other conditions, required to ensure that $\\mu $ is an invariant distribution of the associated semi-group.",
"A standard choice, sometimes referred to as canonical, consists of choosing jump rates $\\lambda _k(x,v) =[v^{\\top }F_k(x)]_+$ for $k \\in \\lbrace 1,\\ldots , K\\rbrace $ and $(x,v) \\in \\mathsf {E}$ .",
"Denote by $\\mathrm {L}^2(\\mu )$ the set of measurable functions $g: \\mathsf {E}\\rightarrow \\mathbb {R}$ such that $\\int _{\\mathsf {E}} g^2 \\, \\mathrm {d}\\mu < +\\infty $ .",
"We let ${\\cdot }$ be the norm induced by the scalar product $ \\text{for all } f, g\\in {\\mathrm {L}^2(\\mu )}\\;, \\quad \\left\\langle f, g \\right\\rangle _2 = \\int _\\mathsf {E}f\\, g\\, \\mathrm {d}\\mu \\;,$ making $\\mathrm {L}^2(\\mu )$ a Hilbert space.",
"The operator $\\mathcal {R}_v$ will be referred to as the refreshment operator, a standard example of which is $\\mathcal {R}_v = \\Pi _v- \\operatorname{Id}$ where $\\Pi _v$ is the following orthogonal projector in ${\\mathrm {L}^2(\\mu )}$ : for any $f \\in {\\mathrm {L}^2(\\mu )}$ , $\\Pi _vf(x,v) = \\int _\\mathsf {V}f(x, w) \\, \\mathrm {d}\\nu (w) \\;,$ in which case the velocity is drawn afresh from the marginal invariant distribution, while the position is left unchanged.",
"In this scenario the informal description of the process given above carries on with $\\lambda _{\\rm ref}\\ne 0$ added to the rate $(x,v) \\mapsto \\sum _{k=1}^K \\lambda _k(x,v)$ , $\\Pi _v$ an additional possible update to the velocity chosen with probability proportional to $\\lambda _{\\rm ref}$ .",
"Another possible choice is the generator of an Ornstein-Uhlenbeck operator leaving $\\nu $ invariant.",
"In all the paper we assume the following condition to hold for either $\\mathcal {L}_1$ or $\\mathcal {L}_2$ , a condition satisfied by the examples covered in this manuscript.",
"A 1 The operator $\\mathcal {L}$ is closed in $\\mathrm {L}^2(\\mu )$ , generates a strongly continuous contraction semi-group $(P_t)_{t \\ge 0}$ on $\\mathrm {L}^2(\\mu )$ , i.e.",
"$P_0 = \\operatorname{Id}$ , for any $t,s \\in \\mathbb {R}_+$ , $P_{s+t} = P_sP_t$ , for any $f \\in {\\mathrm {L}^2(\\mu )}$ , ${P_tf} \\le {f}$ and $\\lim _{t \\rightarrow 0} {P_t f -f} = 0$ .",
"$\\mu $ is a a stationary measure for $(P_t)_{t \\ge 0}$ , i.e.",
"for any $t \\in \\mathbb {R}_+$ , $\\mu P_t = \\mu $ .",
"There exists a core $\\mathsf {C}$ for $\\mathcal {L}$ such that $\\mathsf {C}$ is dense in $\\mathrm {L}^2(\\mu )$ and $\\mathsf {C}\\subset \\mathrm {D}(\\mathcal {L}) \\cap \\mathrm {D}(\\mathcal {L}^{\\star })$ , where $(\\mathcal {L}^{\\star },\\mathrm {D}(\\mathcal {L}^{\\star }))$ is the adjoint of $\\mathcal {L}$ on $\\mathrm {L}^2(\\mu )$ .",
"Note that if $\\mathcal {L}$ generates a strongly continuous contraction semi-group then $\\mathrm {D}(\\mathcal {L})$ is dense by [27] and the adjoint of $\\mathcal {L}$ on $\\mathrm {L}^2(\\mu )$ is therefore well-defined and closed by [49], and $\\mathrm {D}(\\mathcal {L}^{\\star })$ is dense.",
"We now describe how various choices of $K$ and $F_k$ lead to known algorithms.",
"For simplicity of exposition, we assume for the moment that $\\mathsf {V}=\\mathbb {R}^d$ , $\\nu $ is the zero-mean Gaussian distribution with covariance matrix $m_2 \\operatorname{I}_d$ and $\\mathcal {R}_v=\\Pi _v-\\operatorname{Id}$ , but as we shall see later our results cover more general scenarios.",
"The particular choice $K=0$ and $F_0 = \\nabla _xU$ corresponds to the procedure described in [23] as a motivation for the popular hybrid Monte Carlo method.",
"This process is also known as the Linear Boltzman/kinetic equation in the statistical physics literature [5] or randomized Hamiltonian Monte Carlo [11].",
"In this scenario the process follows the isocontours of $\\mu $ for random times distributed according to an inhomogeneous Poisson law of parameter $\\lambda _{\\rm ref}>0$ , triggering events where the velocity is sampled afresh from $\\nu $ .",
"The scenario where $K=d$ , $F_0 = 0$ and for $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $x \\in \\mathsf {X}, \\ F_k(x) = \\partial _k U(x) \\mathbf {e}_k$ where $(\\mathbf {e}_k)_{k \\in \\lbrace 1,\\ldots ,d\\rbrace }$ is the canonical basis, corresponds to the Zig-Zag (ZZ) process [6], where the $x$ component of the process follows straight lines in the direction $v$ which remains constant between events.",
"In this scenario, the choice of $\\mathcal {B}_k$ to update the velocity, consists of negating its $k$ -th component; see also [29] for related ideas motivated by other applications.",
"The standard Bouncy Particle Sampler (BPS) of [51], extended by [12], correspond to the choice $K=1$ , $F_0 = 0$ and $F_1 = \\nabla _xU$ .",
"More elaborate versions of the ZZ and BPS processes, motivated by computational considerations, take advantage of the possibility to decompose the energy as $U=\\sum _{k=0}^K U_k$ and corresponds to the choice $F_k = \\nabla _xU_k$ [43], [12], where in the former the sign flip operation is replaced with a component swap.",
"It should be clear that one can consider more general deterministic dynamics with $F_0\\ne 0$ , effectively covering the Hamiltonian Bouncy Particle Sampler, suggested in [55].",
"We remark that the well-known Langevin algorithm corresponds to $K=0$ , $F_0 = \\nabla _xU$ and the situation where $\\mathcal {R}_v$ is the Ornstein-Uhlenbeck process.",
"More general bounces involving randomization (see [55], [58], [44]) can also be considered in our framework, at the cost of additional complexity and reduced tightness of our bounds.",
"The main aim of the present paper is the study of the long time behaviour for the class of processes described above using hypercoercivity methods popularized by [57].",
"More precisely, consider $(P_t)_{t \\ge 0}$ the semigroup associated to the PDMP with generator $\\mathcal {L}\\in \\lbrace \\mathcal {L}_1,\\mathcal {L}_2\\rbrace $ defined above, we aim to find simple and verifiable conditions on $U,F_k,\\mathcal {R}_v$ and $\\lambda _{\\rm ref}$ ensuring the existence of $A\\ge 1$ and $\\alpha > 0$ , and their explicit computation in terms of characteristics of the data of the problem, such that for any $f\\in \\mathrm {L}_0^2(\\mu ):=\\left\\lbrace g\\in {\\mathrm {L}^2(\\mu )}\\,:\\;\\ \\int _{\\mathsf {E}} g\\, \\mathrm {d}\\mu = 0 \\right\\rbrace $ and $t \\ge 0$ , ${ P_t f} \\le A \\mathrm {e}^{-\\alpha t} { f} \\;.$ Establishing such a result is of interest to practitioners for multiple reasons.",
"Explicit bounds may provide insights into expected performance properties of the algorithm in various situations or regimes.",
"In particular the above leads to an upper bound on the integrated autocorrelation, which is a performance measure of Monte Carlo estimators of $\\int _{\\mathsf {E}} f\\, \\mathrm {d}\\mu $ , $f \\in \\mathrm {L}^2_0(\\mu )$ , defined by $\\lim _{T\\rightarrow \\infty } T \\, {\\rm Var}_{\\mu }\\left(T^{-1} \\int _0^T f(X_t,V_t) \\, \\mathrm {d}t \\right) / { f}^2 \\le \\left.",
"2A / \\alpha \\right.",
"\\;,$ where $(X_t,V_t)_{t \\ge 0}$ is a trajectory of a PDMP process of generator $\\mathcal {L}$ with $(X_0,V_0)$ distributed according to $\\mu $ .",
"For a class of problems of, say, increasing dimension $d\\rightarrow \\infty $ , weak dependence of $A$ and $\\alpha $ on $d$ indicates scalability of the method.",
"It is worth pointing out that the result above is equivalent to the existence of $A\\ge 1$ and $\\alpha > 0$ such that for any measure $\\rho _0\\ll \\mu $ such that ${\\mathrm {d}\\rho _0/\\mathrm {d}\\mu }<\\infty $ $\\Vert \\rho _0 P_{t}-\\mu \\Vert _{\\mathrm {TV}} = \\int _{\\mathsf {E}} \\left|\\mathrm {d}(\\rho _0 P_{t})/\\mathrm {d}\\mu -\\operatorname{1} \\right| \\mathrm {d}\\mu \\le {\\mathrm {d}(\\rho _0 P_{t})/\\mathrm {d}\\mu -\\operatorname{1}}\\le A \\mathrm {e}^{-\\alpha t} {\\mathrm {d}\\rho _0/\\mathrm {d}\\mu -\\operatorname{1}} \\;,$ where the leftmost inequality is standard and a consequence of the Cauchy-Schwarz inequality.",
"Our hypocoercivity result therefore also allows characterization of convergence to equilibrium of PDMPs in various scenarios and regimes, leading in particular to the possibility to compare performance of algorithms started from the same initial distribution.",
"Establishing similar results for different metrics may be a useful complement to our characterization of algorithmic computational complexity and is left for future work.",
"In [46], [57], convergence of the type (REF ) is established using an appropriate $\\mathrm {H}^1$ -norm associated with $\\mu $ .",
"The method which was developed in these papers is closely related to hypoellipticity theory [39], [26], [37] for Partial Differential Equation and in particular the kinetic Fokker-Planck equation.",
"Convergence for linear Boltzman equations was first derived in [36], [46].",
"Since then, several works have extended and completed these results [21], [35], [1], [14], [28], [45]."
],
[
"Notation and conventions",
"The canonical basis of $\\mathbb {R}^d$ is denoted by $(\\mathbf {e}_i)_{i \\in \\lbrace 1,\\ldots ,d\\rbrace }$ and the $d$ -dimensional identity matrix $\\operatorname{I}_d$ .",
"The Euclidean norm on $\\mathbb {R}^d$ or $\\mathbb {R}^{d \\times d}$ is denoted by $\\vert \\cdot \\vert $ , and is associated with the usual Frobenius inner product ${\\rm Tr}(\\Phi ^\\top \\Gamma )$ for any $\\Phi ,\\Gamma $ in $\\mathbb {R}^d$ or $\\mathbb {R}^{d \\times d}$ .",
"Let $\\mathsf {M}$ be a smooth submanifold of $\\mathbb {R}^n$ , for $n \\in \\mathbb {N}$ .",
"For any $k \\in \\mathbb {N}$ , denote by $\\mathrm {C}^k(\\mathsf {M},\\mathbb {R}^m)$ the set of $k$ -times differentiable functions from $\\mathsf {M}$ to $\\mathbb {R}^m$ , $\\mathrm {C}_{\\operatorname{b}}^k(\\mathsf {M},\\mathbb {R}^m)$ stands for the subset of bounded functions in $\\mathrm {C}^k(\\mathsf {M},\\mathbb {R}^m)$ with bounded differentials up to order $k$ .",
"$\\mathrm {C}^k(\\mathsf {M})$ and $\\mathrm {C}_{\\operatorname{b}}^k(\\mathsf {M})$ stand for $\\mathrm {C}^k(\\mathsf {M},\\mathbb {R})$ and $\\mathrm {C}_{\\operatorname{b}}^k(\\mathsf {M},\\mathbb {R})$ respectively.",
"For $f :\\mathsf {X}\\rightarrow \\mathbb {R}$ and $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $x \\mapsto \\partial _{x_i}f(x)$ stands for the partial derivative of $f$ with respect to the $i^{\\text{th}}$ -coordinate, if it exists.",
"Similarly, for $f:\\mathsf {X}\\rightarrow \\mathbb {R}$ , $i,j \\in \\lbrace 1,\\ldots ,d\\rbrace $ , denote by $\\partial _{x_i,x_j} f = \\partial _{x_i} \\partial _{x_j} f$ when $\\partial _{x_i} \\partial _{x_j} f$ exists.",
"For $f =(f_1,\\ldots ,f_m) \\in \\mathrm {C}^1(\\mathsf {X},\\mathbb {R}^m)$ , $\\nabla _xf$ stands for the gradient of $f$ defined for any $x \\in \\mathsf {X}$ by $\\nabla _xf(x) = (\\partial _{x_j} f_i(x))_{i\\in \\lbrace 1,\\ldots ,m\\rbrace , \\, j \\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}^{d \\times m}$ .",
"For ease of notation, we also denote by $(\\nabla _x,\\mathrm {D}(\\nabla _x))$ the densely defined closed extension of $(\\nabla _x,\\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {X}))$ on $\\mathrm {L}^2(\\pi )$ , see [40].",
"For any $f \\in \\mathrm {C}^k(\\mathsf {X}, \\mathbb {R}^m)$ , $k \\in \\mathbb {N}$ and $p \\ge 0$ , define ${f}_{k,p} = \\sup _{x \\in \\mathsf {X}} \\, \\sup _{(i_1,\\ldots ,i_k) \\in \\lbrace 1,\\ldots ,d\\rbrace ^k} \\left\\lbrace {\\partial _{x_{i_1}, \\ldots ,x_{i_k}} f(x)}/(1+{x}^p) \\right\\rbrace \\;.$ We set for $k \\ge 0$ , $\\mathrm {C}_{\\mathrm {poly}}^k(\\mathsf {X}, \\mathbb {R}^m) = \\left\\lbrace f \\in \\mathrm {C}^k(\\mathsf {X},\\mathbb {R}^m)\\,:\\;\\inf _{p \\ge 0} {f}_{k,p} < +\\infty \\right\\rbrace \\;,$ and $ \\mathrm {C}_{\\mathrm {poly}}^k(\\mathsf {X})$ simply stands for $ \\mathrm {C}_{\\mathrm {poly}}^k(\\mathsf {X}, \\mathbb {R})$ .",
"For any $f \\in \\mathrm {C}^2(\\mathsf {X},\\mathbb {R})$ , we let $\\Delta _x f$ denote the Laplacian of $f$ .",
"$\\operatorname{Id}$ stands for the identity operator.",
"For two self-adjoint operators $(\\mathcal {A},\\mathrm {D}(\\mathcal {A}))$ and $(\\mathcal {B},\\mathrm {D}(\\mathcal {B}))$ on a Hilbert space $\\mathsf {H}$ equipped with the scalar product $\\left\\langle \\cdot ,\\cdot \\right\\rangle $ and norm ${\\cdot }$ , denote by $\\mathcal {A}\\succeq \\mathcal {B}$ if $\\left\\langle f, \\mathcal {A}f\\right\\rangle \\ge \\left\\langle f, \\mathcal {B}f\\right\\rangle $ for all $f\\in \\mathrm {D}(\\mathcal {A}) \\cap \\mathrm {D}(\\mathcal {B})$ .",
"Then, define $(\\mathcal {A}\\mathcal {B}, \\mathrm {D}(\\mathcal {A}\\mathcal {B}))$ with domain, if not specified, $\\mathrm {D}(\\mathcal {A}\\mathcal {B})= \\mathrm {D}(\\mathcal {B}) \\cap \\lbrace \\mathcal {B}^{-1}\\mathrm {D}(\\mathcal {A}) \\rbrace $ .",
"For a bounded operator $\\mathcal {A}$ on $\\mathsf {H}$ , we let ${ \\mathcal {A}}= \\sup _{f\\in \\mathsf {H}} {\\mathcal {A}f}/{f}$ .",
"$\\Pi $ is said to be an orthogonal projection if $\\Pi $ is a bounded symmetric operator $\\mathsf {H}$ and $\\Pi ^2=\\Pi $ .",
"An unbounded operator $(\\mathcal {A},\\mathrm {D}(\\mathcal {A}))$ is said to be symmetric (respectively anti-symmetric) is for any $f,g \\in \\mathrm {D}(\\mathcal {A})$ , $\\left\\langle \\mathcal {A}f ,g \\right\\rangle = \\left\\langle f,\\mathcal {A}g \\right\\rangle $ (respectively $\\left\\langle \\mathcal {A}f ,g \\right\\rangle = -\\left\\langle f,\\mathcal {A}g \\right\\rangle $ ).",
"If $\\mathcal {A}$ is densily defined, $\\mathcal {A}$ is said to be self-adjoint if $\\mathcal {A}= \\mathcal {A}^{\\star }$ .",
"If in addition $\\mathcal {A}$ is closed, $\\mathsf {C}\\subset \\mathrm {D}(\\mathcal {A})$ is said to be a core for $\\mathcal {A}$ if the closure of $\\mathchoice{{\\displaystyle \\mathcal {A}}_{\\scriptstyle \\mathsf {C}}}{\\mathcal {A}\\,\\smash{\\vrule height .8depth .85}}{_}{\\,\\mathsf {C}}$ A C A C A C$ is $ A$.",
"Denote by $ 1F$ the constant function equals to $ 1$ from a set $ F$ to $ R$.",
"For any unbounded operator $ (A,D(A))$, we denote by $ Ran(A) = {Af :f D(A)}$ and $ Ker(A) = {f D(A) :Af = 0}$.",
"For any probability measure $ m$ on a measurable space $ (M,F)$, we denote by $ L2(m)$ the Hilbert space of measurable functions $ f$ satisfying $ M f2 dm< +$, equipped with the inner product $ f , g m = M fg dm$, and $ L20(m) = { f L2(m) : M f dm= 0}$.",
"We will use the same notation for vector and matrix fields $ ,(Rd)M$ or $ (Rd d)M$, \\textit {i.e.",
"}~$ , m = MTr() dm$ and no confusion should be possible.",
"When $ m=$ we replace $ m$ with $ 2$ in this notation.For any $ x M$ denote by $ x$ the Dirac distribution at $ x$.We define the total variation distance between two probability measures $ m1,m2$ on $ (M,F)$ by $ m1-m2 TV = AF m1(A)-m2(A) $.",
"For a square matrix $ A$ we let $ diag(A)$ be its main diagonal and for a vector $ v Rd$ we let $ diag(v)$ be the square matrix of diagonal $ v$ and with zeros elsewhere.",
"For $ a,b R$ we let $ a b$ denote their minimum.",
"For $ a,b Rd$ ($ A,B Rd d$), we denote by $ a b Rd$ ($ A B Rd d$) the Hadamard product between $ a$ and $ b$ defined for any $ i {1,...,d}$ ($ i,j {1,...,d}$) by $ (ab)i = ai bi$ ($ (AB)i,j = Ai,jBi,j$).",
"For any $ i,j N$, $ i,j$ denotes the Kronecker symbol which is $ 1$ if $ i=j$ and $ 0$ otherwise.",
"For any $ n1,n2 N$, $ n1 < n2$, we let $ n2n1 = 0$.$"
],
[
"Main results and organization of the paper",
"We now state our main results.",
"In the following, for any densely defined operator $(\\mathcal {C},\\mathrm {D}(\\mathcal {C}))$ we let $(\\mathcal {C}^{\\star },\\mathrm {D}(\\mathcal {C}^{\\star }))$ denote its ${\\mathrm {L}^2(\\mu )}$ -adjoint.",
"First we specify conditions imposed on the potential $U$ .",
"H 1 The potential $U \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ and satisfies there exists $c_1 \\ge 0$ such that, for any $x \\in \\mathsf {X}$ , $ \\nabla _x^2 U(x) \\succeq - c_1 \\operatorname{I}_d$ ; $\\liminf _{|x| \\rightarrow \\infty } \\left\\lbrace | \\nabla _xU (x)|^2/2 - \\Delta _x U(x) \\right\\rbrace > 0 \\;.$ From [50], [3], as:U-REF is equivalent to assuming that $\\pi $ satisfies a Poincaré inequality on $\\mathsf {X}$ , that is the existence of $C_{\\operatorname{P}}>0$ such that, for any $f\\in \\mathrm {C}^2(\\mathsf {X})$ satisfying $\\int _{\\mathsf {X}} f\\mathrm {d}\\pi = 0$ , $ { \\nabla _xf}^2 \\ge C_{\\operatorname{P}}{ f}^2 \\;.",
"$ Further, as:U-REF also implies the existence of $c_2 > 0$ and $\\varpi \\ge 0$ such that for any $x \\in \\mathsf {X}$ , $\\Delta _x U(x) \\le c_2 d^{1+\\varpi } + | \\nabla _xU(x) |^2/2 \\;.$ as:U-REF indeed implies that the quantity considered is bounded from below, the scaling in $d$ in front of $c_2$ will appear natural in the sequel.",
"We have opted for this formulation of the assumption required of the potential to favour intuition and link it to the necessary and sufficient condition for geometric convergence of Langevin diffusions, but our quantitative bounds below will be given in terms of the Poincaré constant $C_{\\operatorname{P}}$ for simplicity (see [4] for quantitative estimates of $C_{\\operatorname{P}}$ depending on potentially further conditions on $U$ ).",
"as:U-REF is realistic in most applications, can be checked in practice and has the advantage of leading to simplified developments.",
"It is possible to replace this assumption with $\\sup _{x \\in \\mathsf {X}}\\lbrace \\vert \\nabla _x^2U(x) \\vert / (1+\\vert \\nabla _xU(x) \\vert )\\rbrace <\\infty $ and rephrase our results in terms of any finite upper bound of this quantity (see [21]).",
"Finally the Poincaré inequality (REF ) implies by [4] that there exists $s >0$ such that $\\int _{\\mathbb {R}^d}\\mathrm {e}^{s \\left|x \\right|} \\, \\mathrm {d}\\pi (x) < +\\infty \\;.$ H 2 The family of vector fields $\\lbrace F_k : \\mathsf {X}\\rightarrow \\mathbb {R}^d \\, ; \\, k \\in \\lbrace 0,\\ldots ,K\\rbrace \\rbrace $ satisfies for $k \\in \\lbrace 0,\\ldots ,K\\rbrace $ , $F_k \\in \\mathrm {C}^2(\\mathsf {X},\\mathbb {R}^d)$ ; for all $x \\in \\mathsf {X}$ , $ \\nabla _xU(x) = \\sum _{k=0}^K F_k(x)$ ; for all $k\\in \\lbrace 0,\\ldots ,K\\rbrace $ there exists $a_k \\ge 0$ such that for all $x \\in \\mathsf {X}$ , $|F_k|(x) \\le a_k \\left\\lbrace 1 + |\\nabla _xU |(x) \\right\\rbrace \\;.$ This assumption is in particular trivially true for the Zig-Zag and the Bouncy Particle Samplers.",
"In turn we assume the jump rates to be related to the family of vector fields $\\lbrace F_k : \\mathsf {X}\\rightarrow \\mathbb {R}^d \\, ; \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace \\rbrace $ through the following conditions.",
"H 3 There exist a continuous function $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ , $C_{\\varphi } \\ge 1$ and $c_{\\varphi } \\ge 0$ satisfying for any $s \\in \\mathbb {R}$ , $\\varphi (s) - \\varphi (-s) = s \\;, \\qquad \\text{and} \\qquad \\left|s \\right| \\le \\varphi (s) + \\varphi (-s) \\le c_{\\varphi } m_2^{{1}{2}}+ C_{\\varphi } \\left|s \\right| \\;,$ such that for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ and $(x,v) \\in \\mathsf {E}$ , $\\lambda _k(x,v)= \\varphi \\big (v^{\\top }F_k(x)\\big )$ .",
"We note that the canonical choice $\\varphi (s) = (s)_+$ satisfies these conditions and that the first condition of (REF ) is equivalent to $\\varphi (s) - (s)_+ = \\varphi (-s) -(-s)_+$ , implying that $\\varphi (s)\\ge (s)_+$ for all $s\\in \\mathbb {R}$ and therefore that the left hand side inequality in (REF ) is automatically satisfied.",
"If we further assume the existence of $C,c \\ge 0$ such that for all $s \\in \\mathbb {R}$ , $ \\varphi (s) \\le c m_2^{{1}{2}}+ C \\, (s)_+$ then the second inequality is satisfied with $C_{\\varphi } = C$ and $c_{\\varphi } = 2c$ .",
"As remarked in [2], the first condition of (REF ) holds for rates based on the choice $\\varphi (s):=-\\log \\left(\\phi \\big (\\exp (-s)\\big )\\right),$ such that $\\phi \\colon \\mathbb {R}_+ \\rightarrow [0,1]$ satisfies $r\\phi (r^{-1}) = \\phi (r)$ for all $r \\in \\mathbb {R}_+ \\setminus \\lbrace 0\\rbrace $ .",
"The canonical choice corresponds to $\\phi (r) = 1\\wedge r$ , but the (smooth) choice $\\phi (r) = r/(1+r)$ is also possible.",
"H 4 Assume that $\\mathsf {V}$ and $\\nu $ satisfy the following conditions.",
"$\\mathsf {V}$ is stable under bounces, i.e.",
"for all $(x,v) \\in \\mathsf {E}$ and $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ , $v - 2 (v^\\top \\mathrm {n}_k(x))\\, \\mathrm {n}_k(x) \\in \\mathsf {V}$ , where $\\mathrm {n}_k(x)$ is defined by (REF ).",
"For any $\\mathsf {A}\\in \\mathcal {V}$ , $x \\in \\mathsf {X}$ , we have $\\nu \\left(\\left\\lbrace \\operatorname{Id}-2\\mathrm {n}_k(x) \\mathrm {n}_k(x)^{\\top } \\right\\rbrace \\mathsf {A} \\right) =\\nu (\\mathsf {A})$ , for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ .",
"For any bounded and measurable function $g :\\mathbb {R}^2 \\rightarrow \\mathbb {R}$ , $i,j \\in \\lbrace 1,\\ldots ,d\\rbrace $ such that $i \\ne j$ , $\\int _{\\mathsf {V}} g(v_i,v_j) \\, \\mathrm {d}\\nu (v) = \\int _{\\mathsf {V}} g(-v_1,v_2) \\, \\mathrm {d}\\nu (v)$ ; $\\nu $ has finite fourth order marginal moment $m_4 = (1/3) { v_1^2}^2 = (1/3) \\int _{\\mathsf {V}} v_1^4 \\, \\mathrm {d}\\nu (v) < +\\infty \\;,$ and for any $i,j,k,l\\in \\lbrace 1,\\ldots ,d\\rbrace $ such that $\\operatorname{card}(\\lbrace i,j,k,l\\rbrace )>2$ , $\\int _{\\mathsf {V}}v_i v_j v_k v_l\\, \\mathrm {d}\\nu ( v) = 0$ .",
"Note that in the case where $\\mathsf {V}$ and $\\nu $ are rotation invariant, i.e.",
"for any rotation $O$ on $\\mathbb {R}^d$ , $O \\mathsf {V}= \\mathsf {V}$ and for any $\\mathsf {A}\\in \\mathcal {V}$ , $\\nu (O \\mathsf {A}) = \\nu (\\mathsf {A})$ , then as:radial-REF -REF -REF are automatically satisfied.",
"By as:radial-REF , we have $\\int _{\\mathsf {V}} v_1 v_2 \\mathrm {d}\\nu (x) = 0$ taking $g(v_1,v_2) = v_1v_2$ for any $(v_1,v_2) \\in \\mathbb {R}^2$ and therefore for any $i,j \\in \\lbrace 1,\\ldots ,d\\rbrace $ such that $i \\ne j$ , $\\int _{\\mathsf {V}} v_iv_j \\, \\mathrm {d}\\nu (v) = 0$ .",
"In addition, under as:radial-REF , from the Cauchy-Schwarz inequality, we obtain that $m_{2,2} = { v_1 v_2 }^2 = \\int _{\\mathsf {V}} v_1^2 v_2^2 \\, \\mathrm {d}\\nu (v) < \\infty \\;,$ and note that in the Gaussian case we have the relation $m_4 = m_{2,2} = m_2^2$ .",
"Finally, under as:radial, for any $f,g \\in \\mathrm {L}^2(\\mu )$ and $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ , $\\left\\langle \\mathcal {B}_k f,g \\right\\rangle _2 = \\left\\langle f,\\mathcal {B}_k g \\right\\rangle _2$ , that is $\\mathcal {B}_k$ is symmetric on $\\mathrm {L}^2(\\mu )$ .",
"In this paper we consider operators $(\\mathcal {R}_v,\\mathrm {D}(\\mathcal {R}_v))$ on $\\mathrm {L}^2(\\mu )$ satisfying the following conditions.",
"H 5 Functions depending only on the position belong to the kernel of $\\mathcal {R}_v$ : $\\mathrm {L}^2(\\pi ) \\subset \\mathrm {D}(\\mathcal {R}_v)$ and for any $f \\in \\mathrm {L}^2(\\pi )$ , $\\mathcal {R}_v f = 0$ ; $\\mathcal {R}_v$ satisfies the detailed balance condition: $ \\mathcal {R}_v = \\mathcal {R}_v^{\\star }$ and $\\mathrm {C}^2_{\\mathrm {poly}}(\\mathsf {E}) \\subset \\mathrm {D}(\\mathcal {R}_v)$ ; $\\mathcal {R}_v$ admits a spectral gap of size 1 on ${\\mathrm {L}_0^2(\\nu )}$ : for any $g \\in \\mathrm {L}^2_0(\\nu )\\cap \\mathrm {D}(\\mathcal {R}_v)$ , $\\left\\langle -\\mathcal {R}_v g,g \\right\\rangle _2 \\ge {g}^2$ ; in addition, for any $f \\in \\mathrm {L}^2(\\pi )$ , it holds for any $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $v_i f \\in \\mathrm {D}(\\mathcal {R}_v)$ and $ -\\mathcal {R}_v (v_i f) = v_i f$ .",
"Typically, $\\mathcal {R}_v$ is of the form $\\operatorname{Id}\\otimes \\tilde{\\mathcal {R}}_v$ where $(\\tilde{\\mathcal {R}}_v,\\mathrm {D}(\\tilde{\\mathcal {R}}_v))$ is a self-adjoint operator on $\\mathrm {L}^2(\\nu )$ with spectral gap equals 1.",
"Then, condition as:operator on velocities-REF is equivalent to $\\tilde{\\mathcal {R}}_v(\\operatorname{1}_{\\mathsf {V}}) = 0$ , which implies that for any $g \\in \\mathrm {D}(\\tilde{\\mathcal {R}}_v)$ , we have $\\int _{\\mathsf {V}} \\mathcal {R}_v g \\, \\mathrm {d}\\nu = \\left\\langle \\operatorname{1}_{\\mathsf {E}},\\mathcal {R}_v g \\right\\rangle _2 = \\left\\langle \\mathcal {R}_v^{\\star }(\\operatorname{1}_{\\mathsf {V}}),g \\right\\rangle _2 = \\left\\langle \\mathcal {R}_v(\\operatorname{1}_{\\mathsf {V}}),g \\right\\rangle _2 = 0 \\;,$ so that the process associated with $\\tilde{\\mathcal {R}}_v$ preserves the probability measure $\\nu $ .",
"Note that as:operator on velocities-REF implies that $\\mathcal {R}_v \\Pi _v= 0$ , whereas as:operator on velocities-REF implies that $-\\mathcal {R}_v (v_1 \\Pi _v) = v_1 \\Pi _v$ , where $\\Pi _v$ is defined by (REF ).",
"Assumption as:operator on velocities is satisfied when $\\mathcal {R}_v=\\Pi _v$ , or $\\mathcal {R}_v = \\operatorname{Id}\\otimes \\tilde{\\mathcal {R}}_v$ with $\\tilde{\\mathcal {R}}_v$ the generator of the Ornstein-Uhlenbeck process defined for any $g \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathbb {R}^d)$ by $\\tilde{\\mathcal {R}}_v g = -\\nabla _v g^{\\top }v + \\Delta _v g \\;.$ H 6 The refreshment rate $\\lambda _{\\rm ref}: \\mathsf {X}\\rightarrow \\mathbb {R}_+$ is bounded from below and from above as follows: there exist $\\underline{\\lambda }>0$ and $c_{\\lambda } \\ge 0$ such that for all $x \\in \\mathsf {X}$ , $0 < \\underline{\\lambda }\\le \\lambda _{\\rm ref}(x) \\le \\underline{\\lambda }(1+c_{\\lambda }|\\nabla _xU(x)|) \\;.$ Under the previous assumptions we can prove exponential convergence of the semigroup.",
"Theorem 1 Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ given by (REF ) or (REF ) satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and as:U, as:Fk, as:intensities, as:radial, as:operator on velocities and as:refreshment hold.",
"Then there exist $A > 0$ and $\\alpha > 0$ such that, for any $f\\in {\\mathrm {L}_0^2(\\mu )}$ , and $t \\in \\mathbb {R}_+$ , ${ P_t f} \\le A \\, \\mathrm {e}^{-\\alpha t} { f} \\;.$ The constants $A$ and $\\alpha $ are given in explicit form in (REF ) in thm:DMSmain (Section ), in terms of the constant appearing in as:U, as:Fk, as:radial, as:operator on velocities and as:refreshment, where $\\epsilon $ can be taken to be $\\epsilon _0$ given in (REF ), $ \\lambda _v = \\underline{\\lambda }$ , $\\lambda _x = C_{\\operatorname{P}}/ (1+C_{\\operatorname{P}}) $ and $R_0 = (4+2\\sqrt{3}) \\vee (\\underline{\\lambda }/2^{{{1}{2}}}) \\vee \\overline{R}_0 $ where $\\overline{R}_0 = \\frac{\\sqrt{2m_{2,2}+3(m_{4}-m_{2,2})_{+}}}{m_2}\\left\\lbrace \\frac{2^{1/2} (1+C_{\\varphi })\\kappa _1}{\\kappa _2} \\sum _{k=1}^K a_k + \\kappa _1 \\right\\rbrace + \\frac{\\underline{\\lambda }}{2^{{1}{2}}} \\left\\lbrace 1+ \\frac{2 c_{\\lambda }\\kappa _1}{ \\kappa _2} \\right\\rbrace + \\frac{c_\\varphi K}{2 ^{{1}{2}}}\\;,$ $\\kappa _1= (1 + c_1/2)^{{{1}{2}}}$ and $\\kappa _2^{-1} = C_{\\operatorname{P}}^{-1}(1+4 c_2 d^{1+\\varpi } + 16 C_{\\operatorname{P}}^2)^{{{1}{2}}}$ .",
"The proof is postponed to ss:proof-main-result.",
"The following details the expected scaling behaviour with $d$ of $A$ and $\\alpha $ .",
"The proof can be found in sec:proof-theor-refthm:s. Corollary Consider the assumptions and notation of thm:hypocoercivity.",
"Further suppose that there exists $m_b>0$ satisfying $ m_2^{-1}\\sqrt{2m_{2,2}+3(m_{4}-m_{2,2})_{+}} \\le m_b \\;,$ which together with $C_{\\operatorname{P}},c_1,c_2$ and ${a}_{\\infty }=\\sup _{k \\in \\lbrace 1,\\ldots ,K\\rbrace } a_k$ are independent of $d$ .",
"Then $A \\le 3^{{{1}{2}}}$ and there exists $C^{\\alpha }(C_{\\operatorname{P}},c_1,c_2,{a}_{\\infty },m_b) >0$ , independent of $d,\\underline{\\lambda },c_\\lambda $ and $C_\\varphi ,c_\\varphi $ , such that for $d$ large enough, $\\alpha > C^{\\alpha }(C_{\\operatorname{P}},c_1,c_2,{a}_{\\infty },m_b) \\, \\underline{\\lambda }\\, m_2^{{{1}{2}}} \\big [\\lbrace c_{\\varphi }K \\rbrace \\vee \\lbrace (1+C_{\\varphi })d^{(1+\\varpi )/2} K + 1\\rbrace \\vee \\lbrace \\underline{\\lambda }(1+c_{\\lambda } d^{(1+\\varpi )/2})\\rbrace \\big ]^{-2} \\;.$ Thus, if $\\underline{\\lambda }$ , $c_{\\lambda }$ , $C_{\\varphi }$ and $c_{\\varphi }$ are fixed, we get that $\\alpha ^{-1}$ is in general at most of order $\\mathcal {O}(m_2^{-{{1}{2}}} d^{1+\\varpi } K^2)$ if $K \\ge 1$ .",
"We now discuss the assumptions of the theorem, and application of its conclusion to various instances of PDMP-MC and two examples of potentials.",
"Assumption as:U is problem dependent and verifiable in practice, while as:Fk, as:radial, as:operator on velocities and as:refreshment are user controllable and we have already discussed standard choices satisfying these conditions.",
"More delicate may be establishing that as:generator holds and that $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ is indeed a core for the generator $\\mathcal {L}$ .",
"As shown in [25], BPS and ZZ are well defined Markov process whose generators admit $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ as a core and similar arguments can be used to establish that it is also a core for the RHMC.",
"Further, it is not difficult to show that for the class of processes described earlier, for any $f\\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\left\\langle \\mathcal {L}f,\\mathbf {1} \\right\\rangle _2=0$ , therefore implying that $\\mu $ is an invariant distribution and that as:generator holds.",
"First we note that the spectral gap is indeed expected to be proportional to $m_2^{{1}{2}}$ , since if $(X_t,V_t)_{t \\ge 0}$ is a PDMP with generator of the form (REF ) or (REF ) for $m_2=1$ , then $(X_{m^{{{1}{2}}} t},m^{{{1}{2}}} V_{m^{{{1}{2}}} t})_{t \\ge 0}$ is a PDMP with generator of the same form with $m_2=m$ .",
"We therefore set $m_2=1$ below, a condition satisfied when $\\nu $ is the uniform distribution on the sphere $\\sqrt{d} \\, \\mathbb {S}^{d-1}$ or $\\lbrace -1,1\\rbrace ^d$ , or the $d$ -dimensional zero-mean Gaussian distribution with covariance matrix $\\operatorname{I}_d$ , all of which also satisfy (REF ).",
"More generally, by lem:moments-spherically-symmetric in app:radial, property (REF ) is satisfied if $\\nu $ is a spherically symmetric distribution on $\\mathbb {R}^d$ corresponding to random variables $V=B^{{{1}{2}}} W$ for $W$ uniformly distributed on the hypersphere $ \\sqrt{d} \\, \\mathbb {S}^{d-1}$ and $B$ a non-negative random variable independent of $W$ and of first and second order moments $\\gamma _1$ and $\\gamma _2$ respectively such that $\\gamma _2^{1/2}/\\gamma _1$ is upper bounded by a constant independent of the dimension.",
"By [4], independence of $C_{\\operatorname{P}}$ on $d$ is satisfied for strongly convex potentials $U$ : i.e.",
"whenever there exists $m>0$ such that $\\nabla _x^2U(x) \\succeq m \\operatorname{I}_d$ for any $x \\in \\mathbb {R}^d$ which implies that one can take $C_{\\operatorname{P}}=m$ .",
"This is the case for $U(x)=\\sum _{i=1}^{d}\\big (1+x_{i}^{2}\\big )^{\\beta }/2$ or $U(x)=(1+\\vert x\\vert ^{2})^{\\beta }$ with $\\beta \\ge 1$ , for which (REF ) is also satisfied with $\\varpi =0$ and $\\varpi = 1-1/\\beta $ respectively (see lemma:potential:independent and lemma:potential:polynomial in sec:examples-potentials).",
"We note that from the Holley-Stroock perturbation principle [38], uniformly bounded perturbations of a strongly convex potential lead to independence of $C_{\\operatorname{P}}$ on $d$ .",
"For $\\beta \\in [1/2,1)$ $C_{\\operatorname{P}}>0$ , but is dependent on $d$ , see [4].",
"However recent progress in the precise quantitative estimation of spectral gaps of certain probability measures [9], [10] allows for the strong convexity property to be relaxed to simple convexity and beyond, but leads to a dependence of $C_{\\operatorname{P}}$ on $d$ which can be characterised.",
"Now further assume that $C_\\varphi ,c_\\varphi $ and that the refreshment rate are uniformly bounded in the position $x$ , implying $c_{\\lambda }=0$ .",
"Then by thm:scaling-with-d-(REF ), there exists $C^{\\alpha }(C_{\\operatorname{P}},c_1,c_2,{a}_{\\infty },m_b,c_\\varphi , C_\\varphi )>0$ such that for $d$ sufficiently large $\\alpha \\ge C^{\\alpha }(C_{\\operatorname{P}},c_1,c_2,{a}_{\\infty },m_b,c_\\varphi , C_\\varphi ) \\,\\left\\lbrace \\big [\\underline{\\lambda }\\big (1+K^2 d^{1+\\varpi }\\big )^{-{{1}{2}}}\\big ] \\wedge \\underline{\\lambda }^{-1} \\right\\rbrace \\;,$ from which we deduce the optimal scaling of the refreshment rate, namely $C_1^{\\underline{\\lambda }} \\, \\big (1+K^2 d^{1+\\varpi }\\big )^{{1}{2}}\\le \\underline{\\lambda }\\le C_2^{\\underline{\\lambda }} \\, \\big (1+K^2 d^{1+\\varpi }\\big )^{{1}{2}}$ for $C_1^{\\underline{\\lambda }}, C_2^{\\underline{\\lambda }}>0$ (which we denote $\\Theta \\big ((1+K^2 d^{1+\\varpi })^{{1}{2}}\\big )$ hereafter to alleviate notation).",
"Using the description of RHMC, ZZ and BPS provided in the introduction we deduce the first three lines of Table REF , where $\\alpha =\\omega (s)$ is used as a short hand notation for $\\alpha \\ge C^{\\alpha }(C_{\\operatorname{P}},c_1,c_2,{a}_{\\infty },m_b,c_\\varphi , C_\\varphi ) s$ for $s \\rightarrow 0$ .",
"The fourth line uses our specialised results of Section , showing that the conclusion of Theorem is not optimal for ZZ.",
"In [7] scaling limits of particular functionals of the ZZ and BPS processes are studied, leading to quantitative estimates of the time required to achieve near independence at equilibrium.",
"More specifically they consider the scenario where the target distribution is a centred normal distribution of covariance matrix $\\operatorname{I}_d$ and focus on the angular momentum, the negative log-target density and the first coordinate of the process.",
"Our more general results, obtained using a different argument, are in agreement after noticing that [7] considered the scenario $m_2=d^{-1}$ and using our earlier remark on the dependence of our estimate of the absolute spectral gap on $m_2^{{1}{2}}$ .",
"In [19] it is shown, again using an approach different from ours, that the RHMC has dimension free convergence rate in a scenario similar to ours.",
"Table: Left hand side: summary of the dependence of α\\alpha on dd for C P ,c 1 ,c 2 ,∥a∥ ∞ C_{P},c_{1},c_{2},\\Vert a\\Vert _{\\infty }constant, m 2 =1m_{2}=1 and optimal choice of λ ̲\\underline{\\lambda }.Right hand side: summary of application to two examples of potentials.While nonreversibily of the processes considered here may be practically beneficial, it is only recently that the tools allowing our work have been developed [56], [57].",
"Our method of proof relies on the framework proposed recently in [20], [21], [13] to study the solutions of the forward Kolmogorov equation associated with the linear kinetic process, but we study the dual backward Kolmogorov equation for a broader class of processes as is the case in [31], [32], [33] who provide the first rigorous derivation of the results of [20], [21], [13].",
"This, combined with the flexibility of the framework of [21], [13] explains the differing inner product used throughout, which we have found to lead to simpler computations while yielding identical conclusions.",
"The estimate (REF ) (with constant $A=1$ ) would follow straightforwardly from a Grönwall argument if the generator $\\mathcal {L}$ of the semigroup was coercive, that is it satisfied $\\left\\langle \\mathcal {L}f,f \\right\\rangle _2 \\le -a {f}^2$ for some $a>0$ and any $f$ in a core of $\\mathcal {L}$ .",
"Unfortunately, the symmetric part of the generator corresponding to a PDMP is degenerate in general, in the sense that it has a nontrivial null space.",
"Hence, the aforementioned coercivity clearly fails to hold.",
"However, it is possible to equip $\\mathrm {L}^2(\\mu )$ with an equivalent scalar product derived from $\\left\\langle \\cdot ,\\cdot \\right\\rangle _2$ with respect to which $\\mathcal {L}$ is coercive.",
"The constant $\\alpha $ is then given by the coercivity bound, while the constant $A$ can be obtained from estimates relating the two equivalent scalar products.",
"The paper is organised as follows.",
"In sec:DMSabstract we develop our framework for hypocoercivity suited to PDMP-MC processes, based on the ideas of [21].",
"In addition to providing a rigorous framework we further optimize the constants involved, ultimately leading to thm:hypocoercivity.",
"The proofs of thm:hypocoercivity and its corollary are given in sec:postponed-proofs.",
"In ss:scaling-ZZ, we specialize our results to the case of the Zig-Zag process for which better estimates are possible, leading to attractive scaling properties with the dimension $d$ .",
"Various intermediate technical results have been moved to Appendices where, for completeness, we have also included classical facts from functional analysis."
],
[
"The DMS framework for hypocoercivity",
"As stated above our results rely on the ideas proposed by [20], [21], [13] for which a rigorous framework was subsequently given in [31], [32], [33], [34].",
"We derive here a novel proof, which borrows elements of [31], [32], [33], [34] but leads to a different set of conditions motivated by our application to PDMP-Monte Carlo methods.",
"We further provide explicit and optimized estimates of the constants involved in terms of accessible characteristics of the process.",
"We first present abstract results which form the core of all of our proofs and then establish more specific ones common to all the processes considered in this paper, implying some of the abstract conditions.",
"More specific results relating to the Zig-Zag process are treated in ss:scaling-ZZ."
],
[
"Abstract DMS results",
"We let $\\mathcal {S}$ and $\\mathcal {T}$ be the ${\\mathrm {L}^2(\\mu )}$ -symmetric and ${\\mathrm {L}^2(\\mu )}$ -anti-symmetric parts of a generator $\\mathcal {L}$ satisfying as:generator, that is $\\mathcal {S}= (\\mathcal {L}+\\mathcal {L}^{\\star })/2 \\quad \\text{and} \\quad \\mathcal {T}= (\\mathcal {L}-\\mathcal {L}^{\\star })/2 \\;, \\;\\text{ defined on } \\mathrm {D}(\\mathcal {S}) = \\mathrm {D}(\\mathcal {T}) = \\mathsf {C}\\;.$ Consider the following additional assumption to as:generator.",
"A 2 $\\Pi _v\\mathsf {C}\\subset \\mathsf {C}$ and $(\\mathcal {T}\\Pi _v,\\mathsf {C})$ is a closable operator, where $\\Pi _v$ is defined by (REF ) and $\\mathsf {C}$ is given in as:generator.",
"Note that since $\\Pi _v\\mathsf {C}\\subset \\mathsf {C}$ , we have $\\mathsf {C}\\subset \\mathrm {D}(\\mathcal {T}\\Pi _v)$ and the restriction of $\\mathcal {T}\\Pi _v$ to $\\mathsf {C}$ exists.",
"Under as:generator and ass:stabilitycpiv, lem:definversedms in ss:spctral bounds justifies the definition of the operator $\\mathcal {A}$ , $\\mathcal {A}= \\left( m_2 \\operatorname{Id}+ (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) \\right)^{-1} (-\\mathcal {T}\\Pi _v)^{\\star }\\;, \\quad \\mathrm {D}(\\mathcal {A}) = \\mathrm {D}((\\mathcal {T}\\Pi _v)^{\\star }) \\;,$ where $m_2$ is given by (REF ) and $(\\overline{\\mathcal {T}\\Pi _v}, \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v}))$ and $((\\mathcal {T}\\Pi _v)^{\\star },\\mathrm {D}((\\mathcal {T}\\Pi _v)^{\\star }))$ are the closure and the adjoint of $(\\mathcal {T}\\Pi _v,\\mathsf {C})$ respectively.",
"Key properties are that $\\operatorname{Ran}(\\mathcal {A}) \\subset \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})$ , $\\mathcal {A}$ is closable with $\\overline{\\mathcal {A}}$ bounded, and $\\overline{\\mathcal {T}\\Pi _v} \\mathcal {A}$ is also closable of bounded closure.",
"To show this result we adapt [31] since their lemma assumes that $(\\mathcal {T},\\mathrm {D}(\\mathcal {T}))$ is closed whereas, motivated by our applications, we assume $(\\mathcal {T}\\Pi _v, \\mathsf {C})$ to be a densely defined and closable operator instead.",
"Lemma Let $(\\mathcal {T},\\mathrm {D}(\\mathcal {T}))$ be a anti-symmetric densely defined operator on $\\mathrm {L}^2(\\mu )$ .",
"Assume that there exists $\\mathsf {D}\\subset \\mathrm {D}(\\mathcal {T}\\Pi _v) \\cap \\mathrm {D}(\\mathcal {T})$ , such that $(\\mathcal {T}\\Pi _v,\\mathsf {D})$ is a densely defined closable operator.",
"The closure of $(\\mathcal {T}\\Pi _v,\\mathsf {D})$ , $(\\overline{\\mathcal {T}\\Pi _v}, \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v}))$ satisfies $\\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v}) \\subset \\mathrm {D}((\\Pi _v\\mathcal {T})^{\\star })$ and for any $f \\in \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})$ , $(\\Pi _v\\mathcal {T})^{\\star } f= -\\overline{\\mathcal {T}\\Pi _v}f$ , where $((\\mathcal {T}\\Pi _v)^{\\star },\\mathrm {D}((\\mathcal {T}\\Pi _v)^{\\star } ))$ is the adjoint of $(\\mathcal {T}\\Pi _v,\\mathsf {D})$ .",
"The operator $\\mathcal {A}$ defined by (REF ) satisfies $\\operatorname{Ran}(\\mathcal {A}) \\subset \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})$ , is closable and its closure $\\overline{\\mathcal {A}}$ is a bounded operator on $\\mathrm {L}^2(\\mu )$ with ${ \\overline{\\mathcal {A}}} \\le 1/ (2 m_2)^{{{1}{2}}}$ and $\\Pi _v\\overline{\\mathcal {A}}= \\overline{\\mathcal {A}}$ on ${\\mathrm {L}^2(\\mu )}$ .",
"Assume in addition that for any $f \\in \\mathsf {D}$ , $\\Pi _v\\mathcal {T}\\Pi _vf=0$ .",
"Then, $(\\overline{\\mathcal {T}\\Pi _v} \\mathcal {A},\\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v} \\mathcal {A}))$ is also closable and its closure $\\mathcal {E}$ is bounded and satisfies for any $f \\in \\mathrm {L}^2(\\mu )$ , ${ \\mathcal {E}f} \\le {(\\operatorname{Id}-\\Pi _v)f}$ .",
"To establish this result, we make use of classical results on unbounded operators in Hilbert spaces which for completeness, are given in ss:spctral bounds.",
"Since $\\mathcal {T}$ is assumed to be anti-symmetric, we have for any $f \\in \\mathrm {D}(\\mathcal {T}\\Pi _v)$ , $ g \\in \\mathsf {D}$ , $\\left\\langle \\Pi _v\\mathcal {T}f ,g \\right\\rangle _2 = - \\left\\langle f,\\mathcal {T}\\Pi _vg \\right\\rangle _2$ since $\\Pi _vg \\in \\mathrm {D}(\\mathcal {T})$ as $\\mathsf {D}\\subset \\mathrm {D}(\\mathcal {T}\\Pi _v)$ .",
"By definition of $(\\mathcal {T}\\Pi _v)^{\\star }$ , we obtain that $\\mathsf {D}\\subset \\mathrm {D}((\\Pi _v\\mathcal {T})^{\\star })$ , and for any $f \\in \\mathsf {D}$ , $ \\mathcal {T}\\Pi _vf = -(\\Pi _v\\mathcal {T})^{\\star } f$ .",
"Therefore $\\lbrace (f, \\mathcal {T}\\Pi _vf) \\, : \\, f \\in \\mathsf {D}\\rbrace \\subset \\lbrace (f,-(\\Pi _v\\mathcal {T})^{\\star } f) \\, : \\, f \\in \\mathrm {D}((\\Pi _v\\mathcal {T})^{\\star })\\rbrace $ , and we obtain the desired result by definition of $(\\overline{\\mathcal {T}\\Pi _v}, \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v}))$ since $-(\\Pi _v\\mathcal {T})^{\\star }$ is closed by [49].",
"The fact that $\\operatorname{Ran}(\\mathcal {A}) \\subset \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})$ , $\\mathcal {A}$ is closable and the bound follow directly from lem:definversedms and prop:abstract bound-REF -REF .",
"We turn to the statement $\\Pi _v\\overline{\\mathcal {A}}= \\overline{\\mathcal {A}}$ .",
"By lem:definversedms, the operator $\\mathcal {C}= (m_2 \\operatorname{Id}+ (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}))^{-1}$ is well-defined, bounded and $\\operatorname{Ran}(\\mathcal {C}) = \\mathrm {D}( (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) )$ .",
"Therefore using lem:grothauslemme23-REF (since $\\mathcal {T}\\Pi _v$ is densely defined), we have for any $f \\in \\mathrm {D}(\\mathcal {T})$ , $\\mathcal {A}f = \\mathcal {C}\\Pi _v\\mathcal {T}f = m_2^{-1}\\left\\lbrace \\operatorname{Id}- (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) \\mathcal {C} \\right\\rbrace \\Pi _v\\mathcal {T}f \\;.$ Therefore, by applying $\\Pi _v$ to both sides and using lem:grothauslemme23-REF , we deduce that for any $f \\in \\mathrm {D}(\\mathcal {T})$ , $ \\Pi _v\\mathcal {A}f = \\mathcal {A}f$ .",
"The proof is then concluded upon noting that $\\mathrm {D}(\\mathcal {T})$ is dense and $\\Pi _v$ is continuous.",
"For any $f \\in \\mathsf {D}$ , since $\\Pi _v\\mathcal {T}\\Pi _vf=0$ , (REF ) becomes $\\mathcal {A}f = \\mathcal {C}\\Pi _v\\mathcal {T}(\\operatorname{Id}- \\Pi _v) f = m_2^{-1}\\left\\lbrace \\operatorname{Id}- (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) \\mathcal {C} \\right\\rbrace \\Pi _v\\mathcal {T}(\\operatorname{Id}- \\Pi _v) f \\;.$ Therefore, we get for any $f\\in \\mathsf {D}$ , ${\\mathcal {A}f}^2 &= m_2^{-1}\\left\\lbrace \\left\\langle \\Pi _v\\mathcal {T}(\\operatorname{Id}- \\Pi _v) f,\\mathcal {A}f \\right\\rangle _2 - \\left\\langle (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) \\mathcal {C}\\Pi _v\\mathcal {T}(\\operatorname{Id}- \\Pi _v) f ,\\mathcal {A}f \\right\\rangle _2 \\right\\rbrace \\\\& = m_2^{-1}\\left\\lbrace \\left\\langle -( \\mathcal {T}\\Pi _v)^{\\star }(\\operatorname{Id}- \\Pi _v) f,\\mathcal {A}f \\right\\rangle _2 - \\left\\langle (\\mathcal {T}\\Pi _v)^{\\star } (\\overline{\\mathcal {T}\\Pi _v}) \\mathcal {C}\\Pi _v\\mathcal {T}(\\operatorname{Id}- \\Pi _v) f,\\mathcal {A}f \\right\\rangle _2 \\right\\rbrace \\\\& = m_2^{-1}\\left\\lbrace -\\left\\langle (\\operatorname{Id}- \\Pi _v) f,(\\overline{ \\mathcal {T}\\Pi _v})\\mathcal {A}f \\right\\rangle _2 - {(\\overline{ \\mathcal {T}\\Pi _v}) \\mathcal {A}f}^2 \\right\\rbrace \\;,$ using successively that $(\\operatorname{Id}-\\Pi _v)f \\in \\mathrm {D}(\\mathcal {T}) $ since $f\\in \\mathsf {D}\\subset \\mathrm {D}(\\mathcal {T}\\Pi _v)$ , lem:grothauslemme23 and $\\mathcal {A}f \\in \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})$ .",
"Using the Cauchy-Schwarz inequality we obtain that for any $f \\in \\mathsf {D}$ , ${(\\overline{\\mathcal {T}\\Pi _v})\\mathcal {A}f } \\le {(\\operatorname{Id}-\\Pi _v)f}$ .",
"Using that $\\mathsf {D}$ is dense in ${\\mathrm {L}^2(\\mu )}$ together with the bounded linear transformation extension theorem [53] concludes the proof.",
"The main result of [21] can be formulated under the following abstract assumption, which we shall assume to hold from now on, and the proof of our main theorem relies on optimized estimates of the constants involved.",
"A 3 (DMS abstract conditions) Let $\\mathsf {C}$ be as in as:generator.",
"Assume further that it satisfies ass:stabilitycpiv and the following conditions there exists $\\lambda _v>0$ satisfying for any $f\\in \\mathsf {C}$ $-\\left\\langle \\mathcal {S}f, f \\right\\rangle _2 \\ge \\lambda _v m_2^{{{1}{2}}} { (\\operatorname{Id}-\\Pi _v) f}^2 \\;;$ there exists $\\lambda _x\\in \\left(0,1\\right)$ satisfying for any $f\\in \\mathsf {C}$ $-\\left\\langle \\overline{\\mathcal {A}}\\mathcal {T}\\Pi _vf,f \\right\\rangle _2 \\ge \\lambda _x { \\Pi _vf}^2 \\;;$ there exists $R_0 \\ge 0$ satisfying for any $f\\in \\mathsf {C}$ $\\left|\\left\\langle \\overline{\\mathcal {A}}\\mathcal {T}(\\operatorname{Id}-\\Pi _v)f,f \\right\\rangle _2+\\left\\langle \\overline{\\mathcal {A}}\\mathcal {S}f,f \\right\\rangle _2 \\right|\\le R_0{(\\operatorname{Id}-\\Pi _v)f} {\\Pi _vf} \\;;$ for any $f \\in \\mathsf {C}$ , $\\Pi _v\\mathcal {T}\\Pi _vf =0 $ ; finally, $\\operatorname{Ran}(\\Pi _v) \\subset \\operatorname{Ker}(\\mathcal {S}^{\\star })$ .",
"Theorem 2 Assume as:generator, ass:stabilitycpiv and as:DMSabstract.",
"Then, for any $f\\in \\mathrm {L}^2_0(\\mu )$ , $t \\in \\mathbb {R}_+$ and $\\epsilon \\in (0, (2^{{{1}{2}}} \\lambda _v)^{-1} \\wedge \\lbrace 4 \\lambda _x /(4\\lambda _x+R_0^2)\\rbrace )$ ${ P_t f} \\le A(\\epsilon ) \\mathrm {e}^{-\\alpha (\\epsilon ) t} { f} \\;,$ with $\\alpha (\\epsilon ) = \\lambda _v m_2^{{1}{2}}\\frac{\\Lambda (\\epsilon )}{1+2^{{{1}{2}}} \\lambda _v \\epsilon }>0 \\quad \\mbox{and} \\quad A(\\epsilon ) = \\sqrt{\\frac{1+2^{{1}{2}}\\lambda _v \\epsilon }{1-2^{{1}{2}}\\lambda _v \\epsilon }} \\;,$ where $\\Lambda (\\epsilon )=\\frac{1-\\epsilon (1-\\lambda _{x})-\\sqrt{[1-\\epsilon (1-\\lambda _{x})]^{2}-4\\epsilon \\lambda _{x}(1-\\epsilon )+\\epsilon ^{2}R_{0}^{2}}}{2} \\;.$ Further, if $2^{{1}{2}}R_0 \\ge \\lambda _v $ then $\\alpha \\colon \\big (0, 4 \\lambda _x /(4\\lambda _x+R_0^2) \\big ) \\rightarrow \\mathbb {R}_+$ has a unique maximum at $\\epsilon ^{\\star }$ such that $\\alpha (\\epsilon _0) < \\alpha (\\epsilon ^{\\star }) <3\\alpha (\\epsilon _0)$ , with $\\epsilon _0 = \\frac{1+\\lambda _x - (1-\\lambda _x) \\sqrt{\\frac{R_0{^2}}{R_0{^2}+4 \\lambda _x}}}{(1+\\lambda _x)^2 + R_0{^2}} \\in (0, (2^{{{1}{2}}} \\lambda _v)^{-1} \\wedge \\lbrace 4 \\lambda _x /(4\\lambda _x+R_0^2)\\rbrace ) \\;,$ so that $A(\\epsilon _0)<+\\infty $ is well defined.",
"In addition, if $R_0 \\ge 2$ then $\\epsilon _0 < 3 \\lambda _x/(4 \\lambda _x + R_0^2)$ .",
"The main idea of [21] behind the proof of thm:DMSmain is the introduction of an equivalent norm for $\\varepsilon \\in \\mathbb {R}_+$ (instead of the ${\\mathrm {L}^2(\\mu )}$ norm, which corresponds to $\\varepsilon =0$ ) $ {H}_{\\varepsilon }(f) = (1/2) { f}^2 + \\varepsilon \\left\\langle f, \\overline{\\mathcal {A}}f \\right\\rangle _2,$ for which $(P_t)_{t \\ge 0}$ is exponentially contracting.",
"More precisely, [21] shows that for some $\\varepsilon \\in (-(m_2/2)^{{{1}{2}}},(m_2/2)^{{{1}{2}}})$ there exists $\\alpha (\\varepsilon ) >0$ such that for any $f\\in \\mathrm {L}^2_0(\\mu )$ , ${H}_{\\varepsilon }(P_tf) \\le \\mathrm {e}^{-\\alpha (\\varepsilon )t} {H}_{\\varepsilon }(f)$ .",
"Then, the convergence in $\\mathrm {L}^2_0(\\mu )$ follows by lem:boundedA-REF which implies that ${H}_{\\varepsilon }(\\cdot )$ defines a norm which is equivalent to ${ \\cdot }$ : for $\\varepsilon \\in (-(m_2/2)^{{{1}{2}}},( m_2/2)^{{{1}{2}}})$ and for any $f\\in \\mathrm {L}^2(\\mu )$ , it holds $ ( 1-(m_2/2)^{-{{1}{2}}} \\varepsilon ) { f}^2 \\le 2{H}_{\\varepsilon }(f) \\le (1+(m_2/2)^{-{{1}{2}}} \\varepsilon ) { f}^2.$ Therefore, for a family $\\big \\lbrace f_t \\in {\\mathrm {L}^2(\\mu )}\\big \\rbrace _{t\\ge 0}$ , exponential decay of $t\\mapsto {H}_{\\varepsilon }(f_t)$ is equivalent to that of $t\\mapsto { f_t }^2$ , a property exploited in the following proof.",
"We first establish the following results which give estimates of the functional $\\lbrace {F}_i \\, : \\, i \\in \\lbrace 1,2,3\\rbrace \\rbrace $ defined for any $g \\in \\mathrm {D}(\\mathcal {L})$ by ${F}_1(g) = \\left\\langle \\mathcal {L}g,g \\right\\rangle _2 \\;, \\quad {F}_2(g) = \\left\\langle \\mathcal {L}g, \\overline{\\mathcal {A}}g \\right\\rangle _2 \\;, \\quad {F}_3(g) = \\left\\langle \\overline{\\mathcal {A}}\\mathcal {L}g, g \\right\\rangle _2\\;.$ Lemma Assume that $\\mathcal {L}$ satisfies as:generator, ass:stabilitycpiv, and as:DMSabstract.",
"Then, for any $g \\in \\mathrm {D}(\\mathcal {L})$ , we have $\\begin{aligned}{F}_1(g) \\le & -\\lambda _v m_2^{{{1}{2}}}{(\\operatorname{Id}-\\Pi _v)g}^2 \\;, \\quad {F}_2(g) \\le {(\\operatorname{Id}-\\Pi _v)g}^2 \\;, \\\\& {F}_3(g) \\le -\\lambda _x {\\Pi _vg}^2 +R_0{(\\operatorname{Id}-\\Pi _v)g}{\\Pi _vg} \\;.\\end{aligned}$ Note that since $\\mathsf {C}$ is a core for $\\mathcal {L}$ and $\\overline{\\mathcal {A}}$ and $\\Pi _v$ are bounded, we only need to show that (REF ) holds for all $g \\in \\mathsf {C}$ .",
"In addition, since $\\overline{\\mathcal {A}}$ is an extension of $\\mathcal {A}$ by lem:boundedA-REF , and for any $g \\in \\mathsf {C}\\subset \\mathrm {D}((\\mathcal {T}\\Pi _v)^{\\star }) = \\mathrm {D}(\\mathcal {A})$ from lem:grothauslemme23-REF as $\\Pi _v(\\mathsf {C}) \\subset \\mathsf {C}= \\mathrm {D}(\\mathcal {T})$ by ass:stabilitycpiv, we deduce $\\overline{\\mathcal {A}}g = \\mathcal {A}g \\;.$ Using that $\\mathcal {S}$ is symmetric, $\\mathcal {T}$ is anti-symmetric and $\\mathsf {C}\\subset \\mathrm {D}(\\mathcal {L})\\cap \\mathrm {D}(\\mathcal {L}^{\\star })$ , we get that for any $g \\in \\mathsf {C}$ , ${F}_1(g) = \\left\\langle \\mathcal {S}g,g \\right\\rangle _2 \\le -\\lambda _vm_2^{{{1}{2}}} {(\\operatorname{Id}-\\Pi _v)g}$ by as:DMSabstract-REF .",
"Second, using that $\\Pi _v\\overline{\\mathcal {A}}= \\overline{\\mathcal {A}}$ by lem:boundedA-REF and (REF ), we have for any $g \\in \\mathsf {C}$ , ${F}_2(g) = \\left\\langle \\Pi _v\\mathcal {A}g,\\mathcal {S}g \\right\\rangle _2 + \\left\\langle \\Pi _v\\mathcal {A}g,\\mathcal {T}g \\right\\rangle _2 = \\left\\langle \\Pi _v\\mathcal {A}g,\\mathcal {T}g \\right\\rangle _2 \\;,$ where the last equality follows from $\\operatorname{Ran}(\\Pi _v) \\subset \\operatorname{Ker}(\\mathcal {S}^{\\star })$ .",
"In addition, since $\\Pi _v$ is symmetric, $\\Pi _v\\mathcal {T}\\Pi _vg = 0$ , $\\operatorname{Ran}(\\mathcal {A}) \\subset \\mathrm {D}(\\overline{\\mathcal {T}\\Pi _v})\\subset \\mathrm {D}((\\Pi _v\\mathcal {T})^{\\star })$ by lem:boundedA-REF -REF , so $(\\Pi _v\\mathcal {T})^{\\star } \\mathcal {A}= -\\overline{\\mathcal {T}\\Pi _v} \\mathcal {A}$ by lem:boundedA-REF and ${\\overline{\\mathcal {T}\\Pi _v} \\mathcal {A}g} \\le {(\\operatorname{Id}-\\Pi _v)g}$ by lem:boundedA-REF , we obtain for any $g \\in \\mathsf {C}$ , ${F}_2(g) = \\left\\langle \\mathcal {A}g,\\Pi _v\\mathcal {T}(\\operatorname{Id}-\\Pi _v)g \\right\\rangle _2 = \\left\\langle (\\Pi _v\\mathcal {T})^{\\star } \\mathcal {A}g,(\\operatorname{Id}-\\Pi _v)g \\right\\rangle _2\\\\ = -\\left\\langle \\overline{\\mathcal {T}\\Pi _v} \\mathcal {A}g,(\\operatorname{Id}- \\Pi _v) g \\right\\rangle _2 \\le {(\\operatorname{Id}-\\Pi _v)g}^2 \\;.$ Finally, using as:DMSabstract-REF -REF we have that for any $g \\in \\mathsf {C}\\subset \\mathrm {D}(\\mathcal {L}) \\cap \\mathrm {D}(\\mathcal {L}^{\\star }) \\cap \\mathrm {D}(\\mathcal {T}\\Pi _v)$ , ${F}_3(g) = \\left\\langle \\overline{\\mathcal {A}}\\mathcal {T}\\Pi _vg,g \\right\\rangle _2 + \\left\\langle \\overline{\\mathcal {A}}\\mathcal {T}(\\operatorname{Id}-\\Pi _v) g,g \\right\\rangle _2 + \\left\\langle \\overline{\\mathcal {A}}\\mathcal {S}g,g \\right\\rangle _2 \\le - \\lambda _x{\\Pi _vg}^2 + R_0 {(\\operatorname{Id}-\\Pi _v)g}{\\Pi _vg} \\;.$ [Proof of Theorem REF ] The first part of the proof follows along the same lines as [31].",
"Let $f\\in \\mathrm {L}^2(\\mu )$ satisfying $\\int _{\\mathsf {E}} f\\mathrm {d}\\mu = 0$ and $\\varepsilon >0$ .",
"For ease of notation, set for any $t \\ge 0$ , $f_t = P_t f$ .",
"From the Dynkin formula [27], for any $t >0$ $f_t \\in \\mathrm {D}(\\mathcal {L})$ and ${\\mathrm {d}}f_{t}/{\\mathrm {d}}t=\\mathcal {L}f_{t}$ .",
"Therefore, for any $t >0$ , $-\\frac{\\mathrm {d}}{\\mathrm {d}t} {H}_{\\varepsilon }(f_t) = - [{F}_1(f_t) + \\varepsilon \\left\\lbrace {F}_2(f_t)+{F}_3(f_t) \\right\\rbrace ] \\;,$ where $\\lbrace {F}_i \\, : \\, i \\in \\lbrace 1,2,3\\rbrace \\rbrace $ are defined in (REF ).",
"Then by lem:preliproofdmsboundfscr, we obtain that for any $t >0$ , $\\nonumber & -\\frac{\\mathrm {d}}{\\mathrm {d}t} {H}_{\\varepsilon }(f_t) \\\\\\nonumber & \\qquad \\ge \\lambda _v m_2^{{{1}{2}}} { (\\operatorname{Id}-\\Pi _v) f_t }^2 + \\varepsilon \\left[ \\lambda _x { \\Pi _vf_t }^2 - { (\\operatorname{Id}-\\Pi _v) f_t }^2 - R_0 { (\\operatorname{Id}-\\Pi _v) f_t } { \\Pi _vf_t } \\right]\\\\\\nonumber & \\qquad =\\begin{pmatrix} { \\Pi _vf_t} \\\\ { (\\operatorname{Id}-\\Pi _v) f_t} \\end{pmatrix}^\\top \\begin{pmatrix} \\varepsilon \\lambda _x & - \\varepsilon R_0/2 \\\\ - \\varepsilon R_0/2\\qquad & \\lambda _v m_2^{{{1}{2}}} - \\varepsilon \\end{pmatrix}\\begin{pmatrix} { \\Pi _vf_t} \\\\ { (\\operatorname{Id}-\\Pi _v) f_t} \\end{pmatrix} \\ge \\Lambda _0(\\varepsilon ) {f_t}^2 \\;,$ where $\\Lambda _0(\\varepsilon ) = \\frac{\\lambda _v m_2^{{{1}{2}}} -\\varepsilon (1 - \\lambda _x) - \\sqrt{(\\lambda _v m_2^{{{1}{2}}} -\\varepsilon (1-\\lambda _x))^2 - [4\\varepsilon \\lambda _{x}(\\lambda _{v}m_{2}^{{{1}{2}}}-\\varepsilon )-\\varepsilon ^{2}R_{0}^{2}] }}{2} \\;,$ is the smallest eigenvalue of the symmetric matrix, positive for $0\\le \\varepsilon \\le 4 \\lambda _x \\lambda _v m_2^{{{1}{2}}} /(4\\lambda _x+R_0^2)$ from lem:derivativeLambda0 in sec:optim-varepsilon (as $\\lambda _x \\le 1$ by as:DMSabstract-REF ).",
"Using (REF ), we get $-\\frac{\\mathrm {d}}{\\mathrm {d}t} {H}_{\\varepsilon }(f_t) \\ge \\frac{2\\Lambda _0(\\varepsilon )}{1+(m_2/2)^{-{{1}{2}}} \\varepsilon } {H}_{\\varepsilon }(f_t)\\;.$ From Grönwall's lemma and (REF ), we obtain for $0\\le \\varepsilon \\le (m_2/2)^{{{1}{2}}} \\wedge \\lbrace 4 \\lambda _x \\lambda _v m_2^{{{1}{2}}} /(4\\lambda _x+R_0^2)\\rbrace $ , ${f_t} \\le C_0(\\varepsilon ) \\mathrm {e}^{-\\alpha _0(\\varepsilon )t} {f_0}, \\text{ where } \\alpha _0(\\varepsilon )= \\frac{\\Lambda _0(\\varepsilon )}{1+(m_2/2)^{-{{1}{2}}} \\varepsilon } \\; \\text{and} \\; C_0(\\varepsilon ) = \\sqrt{\\frac{1+( m_2/2)^{-{{1}{2}}}\\varepsilon }{1-(m_2/2)^{-{{1}{2}}}\\varepsilon }} \\;.$ For notational simplicity we let $\\epsilon = \\varepsilon /(\\lambda _v m_2^{{1}{2}})$ and note that with the definitions in (REF )-(REF ), for $\\epsilon < 4 \\lambda _x /(4\\lambda _x+R_0^2)$ , $\\alpha (\\epsilon )=\\alpha _0(\\varepsilon )>0$ and $\\lambda _v m_2^{{1}{2}}\\Lambda (\\epsilon )=\\Lambda _0(\\varepsilon )>0$ , and for $\\epsilon \\le (2^{{{1}{2}}} \\lambda _v)^{-1}$ the two norms are equivalent and $A(\\epsilon )=A_0(\\varepsilon )$ is well defined.",
"This concludes the proof of REF .",
"From prop:bound-on-Phi-epsilon-star and associated notation in sec:optim-varepsilon, $\\epsilon \\mapsto \\alpha (\\epsilon )$ has a unique, but intractable, maximum, $\\epsilon ^{\\star }\\in (0, 4 \\lambda _x /(4\\lambda _x+R_0^2) )$ .",
"However from lem:second-derivative-Lambda0-REF and prop:bound-on-Phi-epsilon-star the unique maximum $\\epsilon _0 \\in (\\epsilon ^{\\star }, 4 \\lambda _x /(4\\lambda _x+R_0^2))$ of $\\epsilon \\mapsto \\Lambda (\\epsilon )$ , defined by (REF ), provides us with a tractable proxy such that $\\alpha (\\epsilon _0) < \\alpha (\\epsilon ^{\\star }) <3\\alpha (\\epsilon _0)$ .",
"In addition, since $\\lambda _x \\le 1$ and for $2^{{1}{2}}R_0 \\ge \\lambda _v$ we get $\\epsilon _0 < \\frac{(1+\\lambda _x)}{(1+\\lambda _x)^2 + R_0^2} \\le (2R_0)^{-1} \\le (2^{{{1}{2}}}\\lambda _v)^{-1} \\;,$ which implies that $A(\\epsilon _0)$ is well defined (and the two norms equivalent).",
"The last statement follows from lem:second-derivative-Lambda0-REF in sec:optim-varepsilon.",
"The following lemma provides us with simple estimates of $\\alpha (\\epsilon _0)$ and $A(\\epsilon _0)$ defined in thm:DMSmain.",
"Lemma Let $\\epsilon \\mapsto \\alpha (\\epsilon ), A(\\epsilon )$ and $\\epsilon _0$ be as in thm:DMSmain and let $\\lambda _x \\in \\left(0,1\\right)$ .",
"Then for any $R_0 \\ge 4 + 12^{{{1}{2}}}$ , $\\lambda _x/(1 +R_0^{2}) \\le \\epsilon _0 \\le 2/(4 +R_0^2) \\le 1/(4 R_0)\\;,$ for any $R_0 \\ge (4+12^{{{1}{2}}}) \\vee (\\lambda _v/2^{{{1}{2}}})$ , $A(\\epsilon _0) \\le 3^{{1}{2}}\\quad \\text{and} \\quad \\lambda _v \\lambda _x m_2^{{1}{2}}\\epsilon _0/6 \\le \\alpha (\\epsilon _0) \\le 4 \\lambda _v \\lambda _x m_2^{{1}{2}}\\epsilon _0 \\;.$ The proof is postponed to sec:proof:behaviourR0."
],
[
"DMS for PDMP: generic results",
"Proposition Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ , defined by (REF ) or (REF ), with $\\mathcal {B}_k$ given in (REF ), satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ together with as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment.",
"Then the ${\\mathrm {L}^2(\\mu )}$ -adjoint of $\\mathcal {L}_i$ for $i \\in \\lbrace 1,2\\rbrace $ defined by (REF ) or (REF ) is given for any $f\\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ by $\\mathcal {L}^{\\star }_i f= - v^\\top \\nabla _xf+ _{i,2} m_2 F_0^\\top \\nabla _vf+ \\sum _{k=1}^K \\varphi \\big (-v^\\top F_k\\big )[ (\\mathcal {B}_k-\\operatorname{Id})f]+ m_2^{{{1}{2}}}\\lambda _{\\rm ref}\\mathcal {R}_v f\\;.$ We only consider the case $i=2$ since the proof for $i=1$ follows along the same lines.",
"In addition, since $\\mathcal {R}_v$ is self-adjoint by as:operator on velocities and $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E}) \\subset \\mathrm {D}(\\mathcal {R}_v)$ , we can consider the case $\\lambda _{\\rm ref}(x) = 0$ for any $x \\in \\mathsf {X}$ .",
"Based on (REF )-(REF ), using that for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ , $\\mathcal {B}_k$ is symmetric on $\\mathrm {L}^2(\\mu )$ , for any $(x,v) \\in \\mathsf {E}$ , $\\mathcal {B}_k \\lambda _k(x,v) = \\lambda _k(x,-v)$ and by integration by part, for any $f,g \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we obtain $& \\left\\langle g,\\mathcal {L}f \\right\\rangle _2 = \\left\\langle - v^\\top \\nabla _xg + (v^\\top \\nabla _xU)g + m_2 F_0^\\top \\nabla _vg - (v^\\top F_0)g + \\textstyle \\sum _{k=1}^K (\\mathcal {B}_k - \\operatorname{Id}) [\\lambda _k(x,v)g] ,f \\right\\rangle _2 \\\\& = \\left\\langle - v^\\top \\nabla _xg + [v^\\top (\\nabla _xU-F_0)] g + m_2 F_0^\\top \\nabla _vg + \\textstyle \\sum _{k=1}^K\\lbrace \\lambda _k(x, -v) \\mathcal {B}_k g - \\lambda _k(x,v) g\\rbrace ,f \\right\\rangle _2 \\\\& = \\left\\langle \\mathcal {L}_i^{\\star } g,f \\right\\rangle _2 + \\left\\langle [v^\\top (\\nabla _xU-F_0)] g + g \\textstyle \\sum _{k=1}^K\\lbrace \\lambda _k(x,-v) - \\lambda _k(x,v)\\rbrace ,f \\right\\rangle _2 \\;.$ Using that $\\sum _{k=0}^{K} F_k= \\nabla _xU$ by as:Fk-REF and that $\\lambda _k(x,v) - \\lambda _k(x,-v) = v^{\\top }F_k(x)$ for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ and $(x,v) \\in \\mathsf {E}$ by as:intensities, concludes the proof.",
"The following provides expressions for the ${\\mathrm {L}^2(\\mu )}$ -symmetric and ${\\mathrm {L}^2(\\mu )}$ -anti-symmetric parts of $\\mathcal {L}$ for all the PDMP processes considered in this paper.",
"Define $\\lambda _k^{\\mathrm {e}} : \\mathsf {E}\\rightarrow \\mathbb {R}_+$ for any $(x,v) \\in \\mathsf {E}$ and $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ by $\\lambda ^{\\mathrm {e}}_k(x,v) := \\lambda _k(x,v) + \\lambda _k(x,-v) \\;.$ Proposition Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ , defined by (REF ) or (REF ), with $\\mathcal {B}_k$ given in (REF ), satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ together with as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment.",
"Let $\\mathcal {S}$ and $\\mathcal {T}_i$ be the symmetric and anti-symmetric parts of $\\mathcal {L}_i$ respectively, defined by (REF ).",
"Then for any $f\\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\mathcal {T}_i f= \\tilde{\\mathcal {T}}_if$ and $\\mathcal {S}_i f= \\tilde{\\mathcal {S}}f$ where $\\tilde{\\mathcal {T}}_i$ and $\\tilde{\\mathcal {S}}$ are the operators defined for any $g \\in \\mathrm {C}^2_{\\mathrm {poly}}(\\mathsf {E})$ by $\\tilde{\\mathcal {T}}_i g &= v^\\top \\nabla _xg - \\delta _{i,2} m_2 F_0^\\top \\nabla _vg + \\frac{1}{2} \\sum _{k=1}^K (v^\\top F_k) \\, (\\mathcal {B}_k-\\operatorname{Id}) g \\;, \\\\\\tilde{\\mathcal {S}}g &= \\frac{1}{2} \\sum _{k=1}^K \\lambda _k^{\\mathrm {e}} \\, (\\mathcal {B}_k-\\operatorname{Id}) g + m_2^{{{1}{2}}} \\lambda _{\\rm ref}\\mathcal {R}_v g \\;.$ $\\mathcal {S}$ satisfies as:DMSabstract-REF .",
"$\\mathrm {C}^1_{\\mathrm {poly}}(\\mathsf {E}) \\subset \\mathrm {D}(\\mathcal {T}_i^\\star ) \\cap \\mathrm {D}(\\mathcal {S}^\\star )$ and for any $f \\in \\mathrm {C}^1_{\\mathrm {poly}}(\\mathsf {E})$ , $\\mathcal {T}_i^{\\star }f = -\\tilde{\\mathcal {T}}_i f$ and $\\mathcal {S}^{\\star } f = \\tilde{\\mathcal {S}}f$ .",
"Note that the symmetric parts of $\\mathcal {L}_i$ for $i \\in \\lbrace 1,2\\rbrace $ are the same and equal to $\\mathcal {S}$ .",
"REF follows from prop:adjoint-calL and the definitions of $\\mathcal {S}$ and $\\mathcal {T}$ in (REF ).",
"REF is a direct consequence of the first result and the definition of $(\\mathcal {S}^{\\star },\\mathrm {D}(\\mathcal {S}^{\\star }))$ .",
"Simple integration by parts and definitions of $(\\mathcal {S}^{\\star },\\mathrm {D}(\\mathcal {S}^{\\star }))$ , $(\\mathcal {T}_i^{\\star },\\mathrm {D}(\\mathcal {T}_i^{\\star }))$ imply REF .",
"We define the directional derivative operator $\\text{ for any } f\\in \\mathrm {D}(\\mathcal {D}) = \\mathrm {C}_{\\operatorname{b}}^{1}(\\mathsf {E}) \\;, \\,\\mathcal {D}f(x,v) := v^{\\top } \\nabla _x f(x,v) \\;.$ The operators $(\\mathcal {D}, \\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {E}))$ and $(\\mathcal {D}\\Pi _v, \\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {E}))$ are densely defined on $\\mathrm {L}^2(\\mu )$ and closable.",
"The proof is similar to that for the operator $\\nabla _x$ and is omitted, see for example [40].",
"Note that by (REF ), a simple computation gives that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and $i \\in \\lbrace 1,2\\rbrace $ , since $\\Pi _v f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\mathcal {T}_i \\Pi _v f = \\mathcal {D}\\Pi _v f \\;.$ Lemma Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ , defined by (REF ) or (REF ), with $\\mathcal {B}_k$ given in (REF ), satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ together with as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment.",
"Then, with $\\mathcal {T}_i$ the anti-symmetric part of $\\mathcal {L}_i$ defined by (REF ) and the operator $\\mathcal {A}_i$ defined by (REF ) relative to $\\mathcal {T}_i$ , it holds: $\\mathcal {T}_i$ satisfies ass:stabilitycpiv and as:DMSabstract-REF with $\\mathsf {C}=\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and $((\\overline{\\mathcal {T}_i \\Pi _v}),\\mathrm {D}(\\overline{\\mathcal {T}_i \\Pi _v})) = ((\\overline{\\mathcal {D}\\Pi _v}), \\mathrm {D}(\\overline{\\mathcal {D}\\Pi _v}))$ ; $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E}) \\subset \\mathrm {D}((\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v})$ and for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $ (\\mathcal {T}_i \\Pi _v)^{\\star } \\mathcal {T}_i \\Pi _vf = m_2 \\nabla _x^{\\star } \\nabla _x \\Pi _vf$ ; $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v}\\rbrace ^{-1} \\Pi _v= m_2^{-1} \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _v$ on ${\\mathrm {L}^2(\\mu )}$ ; $\\mathcal {A}^{\\star }_i = m_2^{-1} (\\overline{\\mathcal {D}\\Pi _v}) \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _v$ and for any $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ , there exists a unique function $u \\in \\mathrm {C}_{\\mathrm {poly}}^3(\\mathsf {X})$ , such that $m_2^{-1}\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _vf = u$ and $\\mathcal {A}^{\\star }_i f = -v^{\\top }\\nabla _x u = -m_2^{-1} (\\overline{\\mathcal {D}\\Pi _v}) \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _v\\;.$ REF First note that $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ is a core for $(\\overline{\\mathcal {D}\\Pi _v},\\mathrm {D}(\\overline{\\mathcal {D}\\Pi _v}))$ since for any $f \\in \\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {E})$ , there exists a sequence of functions $(f_n)_{n \\in \\mathbb {N}}$ such that for any $n \\in \\mathbb {N}$ , $f_n \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\lim _{n \\rightarrow +\\infty } {f-f_n}= 0$ and $\\lim _{n\\rightarrow +\\infty }{\\nabla _xf - \\nabla _x f_n} = 0$ .",
"Then the proof is completed upon using (REF ) and (REF ).",
"REF By (REF ), we have for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , that $\\mathcal {T}_i \\Pi _vf = v^{\\top }\\nabla _x \\Pi _vf$ .",
"It suffices then to verify that with $g : (x,v) \\mapsto v^{\\top }\\nabla _x \\Pi _vf(x,v)$ , then $g \\in \\mathrm {D}((\\mathcal {T}_i \\Pi _v)^{\\star })$ and $(\\mathcal {T}_i \\Pi _v)^{\\star } g = m_2 \\nabla ^{\\star }_x g$ , i.e.",
"for any $h \\in \\mathrm {D}(\\mathcal {T}_i \\Pi _v)$ , $\\left\\langle \\mathcal {T}_i \\Pi _vh,g \\right\\rangle _2 = m_2 \\left\\langle h, \\nabla _x^{\\star }g \\right\\rangle _2$ .",
"But $\\mathrm {D}(\\mathcal {T}_i \\Pi _v) = \\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ by assumption and definition see (REF ), and then the result is just a straightforward consequence of (REF ), (REF ) and an integration by part.",
"REF Note that we only need to show that the two operators $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v}\\rbrace ^{-1}$ and $ m_2^{-1} \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1}$ coincide on a dense subset of $\\mathrm {L}^2(\\pi )$ since they are bounded.",
"We prove that this statement is true choosing the subset $ m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace (\\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X}))$ .",
"First, for any $h \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we have using REF , REF and the definition (REF ) that $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v} \\rbrace h= \\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\mathcal {T}_i \\Pi _v\\rbrace h= m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\Pi _v\\rbrace h \\;.$ Second, for any $g \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ , there exists a sequence $(g_n)_{n \\in \\mathbb {N}}$ such that for any $n \\in \\mathbb {N}$ , $g_n \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {X})$ , $(g_n)_{n\\in \\mathbb {N}}$ , $(\\nabla _x g_n)_{n \\in \\mathbb {N}}$ and $(\\nabla _x^2 g_n)_{n \\in \\mathbb {N}}$ converge in $\\mathrm {L}^2(\\pi )$ to $g$ , $\\nabla _x g$ and $\\nabla _x^2g$ respectively, which implies that $\\lbrace [m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v} ]g_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace m_2 [\\operatorname{Id}+ (\\nabla _x)^{\\star } \\nabla _x ] g_n\\rbrace _{n \\in \\mathbb {N}}$ converge in $\\mathrm {L}^2(\\pi )$ .",
"Therefore, since $ \\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v} \\rbrace $ and $m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace $ are closed, we get that $\\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ is included in the domain of these two operators and (REF ) holds for any $h \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ .",
"[48] or [15]Note that the result is stated for functions $f \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathbb {R}^d)$ but the proof can be easily extended to $f \\in \\mathrm {C}^2_{\\mathrm {poly}}(\\mathsf {X})$ show that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {X})$ , there exists $u \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ such that $m_2\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace u = f$ .",
"Therefore, it holds that $ \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {X}) \\subset m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace (\\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})) $ so $ m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace (\\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})) $ is dense in $\\mathrm {L}^2(\\pi )$ .",
"In addition, since we have shown that the operators $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star }\\overline{\\mathcal {T}_i\\Pi _v} \\rbrace $ and $m_2\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace $ coincide on $\\mathrm {C}_{\\mathrm {poly}}^3(\\mathsf {X})$ , $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star }\\overline{\\mathcal {T}_i\\Pi _v} \\rbrace ^{-1}$ and $m_2^{-1}\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace ^{-1}$ coincide on $m_2 \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace (\\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})) $ .",
"REF As $\\mathcal {A}_i$ is bounded, it is sufficient to show that $\\mathcal {A}_i^{\\star }$ and $m_2^{-1} (\\overline{\\mathcal {D}\\Pi _v}) \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star }\\nabla _x \\rbrace ^{-1} \\Pi _v$ coincide on a dense subset of $\\mathrm {L}^2(\\mu )$ .",
"First, for all $f,g \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we get that $\\left\\langle \\mathcal {A}_i g,f \\right\\rangle _2= \\left\\langle \\Pi _v\\mathcal {A}_i g,f \\right\\rangle _2$ by lem:boundedA-REF .",
"Now using the definition of $\\mathcal {A}_i$ (REF ), that $\\Pi _v$ and $\\lbrace m_2 \\operatorname{Id}+ (\\mathcal {T}_i \\Pi _v)^{\\star } \\overline{\\mathcal {T}_i \\Pi _v}\\rbrace ^{-1}$ are bounded and self-adjoint, since $\\Pi _v$ is an orthogonal projection and by prop:abstract bound-REF -REF , we get for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\left\\langle \\mathcal {A}_i g,f \\right\\rangle _2 = m_2^{-1} \\left\\langle (-\\Pi _v\\mathcal {T}_i)^{\\star }g,\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _v f \\right\\rangle _2 = m_2^{-1} \\left\\langle \\mathcal {T}_i \\Pi _vg,\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _v f \\right\\rangle _2\\;,$ where we have used lem:grothauslemme23-REF for the last equality and $\\mathrm {D}(\\mathcal {T}) = \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"[48] or [15] show that there exists $u \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ satisfying $m_2\\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x\\rbrace u = \\Pi _v f$ and therefore, we get that $\\left\\langle \\mathcal {A}_i g ,f \\right\\rangle _2 = \\left\\langle \\mathcal {T}_i \\Pi _vg,u \\right\\rangle _2 = -\\left\\langle g, v^{\\top }\\nabla _x u \\right\\rangle _2 \\;,$ using an integration by part for the last identity.",
"This result shows that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we have that $\\mathcal {A}_i^{\\star } f= - v^{\\top }\\nabla _x u$ .",
"In addition, for any $g \\in \\mathrm {C}_{\\mathrm {poly}}^1(\\mathsf {E})$ , there exists a sequence $(f_n)_{n \\in \\mathbb {N}}$ such that $f_n \\in \\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {E})$ and $\\lim _{n\\rightarrow +\\infty } {g-f_n}=0$ , $\\lim _{n \\rightarrow +\\infty } {\\nabla _xg - \\nabla _x f_n} = 0$ .",
"Therefore we get that $\\mathrm {C}_{\\mathrm {poly}}^1(\\mathsf {E}) \\subset \\mathrm {D}(\\overline{\\mathcal {D}\\Pi _v})$ and for any $g \\in \\mathrm {C}^1_{\\mathrm {poly}}(\\mathsf {E})$ , $\\overline{\\mathcal {D}\\Pi _v} g(x,v)= v^{\\top }\\nabla _xg(x,v)$ for any $(x,v) \\in \\mathsf {E}$ .",
"Therefore, we get the desired conclusion that $ \\mathcal {A}_i^{\\star } f= - v^{\\top }\\nabla _x u= - m_2^{-1} (\\overline{\\mathcal {D}\\Pi _v}) \\lbrace \\operatorname{Id}+ \\nabla _x^{\\star } \\nabla _x \\rbrace ^{-1} \\Pi _vf $ , which completes the proof.",
"Establishing as:DMSabstract-REF (referred to as microscopic coercivity in [21]) for the processes considered is fairly straightforward in the present framework.",
"Proposition Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ given by (REF ) or (REF ), where $\\mathcal {B}_k$ is defined in (REF ) satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"Assume in addition that as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment hold.",
"Let $\\mathcal {S}$ be the symmetric part of $\\mathcal {L}_i$ defined by (REF ).",
"Then as:DMSabstract-REF is satisfied with $\\lambda _v = \\underline{\\lambda }$ and $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"From as:operator on velocities-REF and as:refreshment, it holds that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we have $-\\left\\langle \\lambda _{\\rm ref}m_2^{{1}{2}}\\mathcal {R}_vf,f \\right\\rangle _2 \\ge \\underline{\\lambda }m_2^{{{1}{2}}} \\left\\langle (\\operatorname{Id}- \\Pi _v)f,f \\right\\rangle _2 \\;.$ In addition, any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ satisfies $\\max _{k \\in \\lbrace 1,\\ldots ,K\\rbrace } {v^{\\top } F_k f} < +\\infty $ by as:U,(REF ) and (REF ), then by as:intensities for any $k \\in \\lbrace 1, \\ldots ,K\\rbrace $ , $\\sup _{k \\in \\lbrace 1,\\ldots ,K\\rbrace }{ \\lambda ^{\\mathrm {e}}_k f} < +\\infty $ .",
"Therefore, using the Cauchy-Schwarz inequality, that $\\mathcal {B}_k$ is a symmetric involution on $\\mathrm {L}^2(\\mu )$ by as:radial, and $\\mathcal {B}_k \\lambda ^{\\mathrm {e}}_k = \\lambda ^{\\mathrm {e}}_k$ by definition (REF ), we obtain for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ and $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\left\\langle \\lambda _k^{\\mathrm {e}} \\, \\mathcal {B}_k f,f \\right\\rangle _2 &\\le {(\\lambda _k^{\\mathrm {e}})^{{1}{2}}f}{(\\lambda _k^{\\mathrm {e}})^{{1}{2}}\\mathcal {B}_k f} = {(\\lambda _k^{\\mathrm {e}})^{{{1}{2}}} f}^2 \\;.$ As a result, we deduce $\\left\\langle \\lambda _k^{\\mathrm {e}} \\, (\\operatorname{Id}- \\mathcal {B}_k)f,f \\right\\rangle _2 \\ge 0$ .",
"Combining this result and (REF ) in the expression for $\\mathcal {S}$ given in () in prop:dmspdmp1 completes the proof.",
"The following lemma establishes equivalence between as:DMSabstract-REF and the Poincaré inequality as:U , which allows one to refer to the expansive body of literature on the topic and implies dependence on the properties of the potential $U$ only.",
"Proposition Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ given by (REF ) or (REF ), where $\\mathcal {B}_k$ as in (REF ) satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"Assume in addition that as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment hold.",
"Let $\\mathcal {T}_i$ be the anti-symmetric part of $\\mathcal {L}_i$ defined by (REF ) and $\\mathcal {A}_i$ be defined by (REF ) relative to $\\mathcal {T}_i$ .",
"Then, as:DMSabstract-REF , i.e.",
"(REF ), holds with $\\lambda _x = C_{\\operatorname{P}}/(1+C_{\\operatorname{P}}) \\;.$ From the assumed Poincaré inequality (REF ) we have for any $f\\in \\mathrm {C}_{\\operatorname{b}}^1(\\mathsf {E})$ ${m_{2}^{-{{1}{2}}}\\mathcal {D}\\Pi _vf}^{2}={\\nabla _x\\Pi _vf}^{2}\\ge C_{\\operatorname{P}}{\\Pi _vf}^{2} \\;.$ Then, by definition of $\\overline{\\mathcal {D}\\Pi _v}$ this inequality holds also for any $f \\in \\mathrm {D}(\\overline{\\mathcal {D}\\Pi _v})$ replacing $\\mathcal {D}\\Pi _vf$ by $\\overline{\\mathcal {D}\\Pi _v} f$ .",
"Therefore, we obtain since $(\\mathcal {D}\\Pi _v)^{\\star \\star } = \\overline{\\mathcal {D}\\Pi _v}$ that for any $f\\in \\mathrm {D}((\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v})$ , $\\left\\langle f,m_{2}^{-1}(\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v} f \\right\\rangle _2\\ge C_{\\operatorname{P}}{\\Pi _vf}^{2} \\;.$ In addition by [49], $({\\mathcal {D}\\Pi _v})^{\\star } \\overline{\\mathcal {D}\\Pi _v}$ is a self-adjoint operator.",
"These results and (REF ) imply that $\\operatorname{Spec}(m_{2}^{-1}({\\mathcal {D}\\Pi _v})^{\\star } \\overline{\\mathcal {D}\\Pi _v})\\subseteq \\left[C_{\\operatorname{P}},\\infty \\right)$ by [16].",
"On the other hand, since by lem:relationmctmcd-REF , $\\overline{\\mathcal {D}\\Pi _v} = \\overline{{\\mathcal {T}_i \\Pi _v}}$ , we have $(\\mathcal {D}\\Pi _v)^{\\star } = ({\\mathcal {T}_i \\Pi _v})^{\\star }$ and $\\mathcal {A}_i = -\\big (m_{2}\\operatorname{Id}+(\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v}\\big )^{-1}(\\mathcal {D}\\Pi _v)^{\\star } \\;.$ Therefore, for any $f\\in \\mathrm {D}((\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v})$ , $- \\overline{\\mathcal {A}}_i \\, \\overline{\\mathcal {D}\\Pi _v} f= - \\mathcal {A}_i \\overline{ \\mathcal {D}\\Pi _v} f=\\big (m_{2}\\operatorname{Id}+(\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v}\\big )^{-1}(\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v} f= \\Phi \\big (m_{2}^{-1}(\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v}\\big ) f\\;,$ where $\\Phi (z)= z / (1+z)$ .",
"Since $\\mathrm {D}((\\mathcal {D}\\Pi _v)^{\\star }\\overline{\\mathcal {D}\\Pi _v})$ is a core for $\\overline{\\mathcal {D}\\Pi _v}$ by [49], from the spectral mapping theorem [16], and the fact that $\\Phi \\colon \\left[0,\\infty \\right)\\rightarrow \\left[0,1\\right]$ is non-decreasing, we get that $- \\overline{\\mathcal {A}}_i \\, \\overline{ \\mathcal {D}\\Pi _v}$ can be extended on $\\mathrm {L}^2(\\mu )$ as a self-adjoint bounded operator $\\mathcal {E}$ and ${\\operatorname{Spec}}(\\mathcal {E})\\subseteq \\left[\\Phi (C_{\\operatorname{P}}),1\\right)$ .",
"Finally, from the fact that $\\Pi _v$ is a projector, we deduce from lem:boundedA-REF that $ - \\overline{\\mathcal {A}}_i \\mathcal {T}_i \\Pi _vf= -\\Pi _v\\overline{\\mathcal {A}}_i \\, \\overline{\\mathcal {D}\\Pi _v} \\Pi _vf= \\Pi _v\\mathcal {E}\\Pi _vf$ for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E}) \\subset \\mathrm {D}(\\overline{\\mathcal {D}\\Pi _v})$ and therefore, we get that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ $- \\left\\langle \\Pi _vf,\\overline{\\mathcal {A}}\\mathcal {T}_i\\Pi _vf \\right\\rangle _2 =\\left\\langle \\Pi _vf,\\mathcal {E}\\Pi _vf \\right\\rangle _2 \\ge \\frac{C_{\\operatorname{P}}}{1+C_{\\operatorname{P}}}{\\Pi _vf}^{2}=\\lambda _{x}{\\Pi _vf}^{2} \\;,$ which concludes the proof.",
"as:DMSabstract-REF is usually a more involved condition to check.",
"For $f\\in {\\mathrm {L}^2(\\mu )}$ denote by $u_f = m_2^{-1} ( \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf \\;.$ In the scenarios considered here, condition as:DMSabstract-REF relies on estimates of ${u_f}$ , ${\\nabla _x u_f}$ and ${\\nabla _x^2 u_f}$ which are obtained by noticing that by definition $u_f$ is solution of the following partial differential equation $m_2 ( \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)u_f=\\Pi _vf\\;.$ In the next section, we show how general, but potentially rough, estimates can be obtained, while in ss:scaling-ZZ we show how tighter bounds can be obtained in specific scenarios where we can take advantage of the structure at hand, in particular when interested in the scaling properties of the algorithm with $d$ ."
],
[
"Computation of $R_{0}$ in the general setting",
"In all this section, we consider $u_f$ defined for any $f \\in \\mathrm {L}^2(\\mu )$ by (REF ).",
"Recall that from lem:relationmctmcd-REF , if $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ then $u_f \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathbb {R}^d)$ and satisfies (REF ).",
"Lemma Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ given by (REF ) or (REF ), where $\\mathcal {B}_k$ is given in (REF ), satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"Assume in addition that as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment hold.",
"Let $\\mathcal {S}$ be the symmetric part of $\\mathcal {L}_{i}$ defined by (REF ) and the operator $\\mathcal {A}_i$ defined by (REF ) relative to $\\mathcal {T}_i$ .",
"For any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $\\vert \\left\\langle \\overline{\\mathcal {A}}_i\\mathcal {S}(\\operatorname{Id}-\\Pi _v)f, f \\right\\rangle _2\\vert \\le {(\\operatorname{Id}-\\Pi _v)f}{(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {S}}\\mathcal {A}_i^{\\star } f} \\;,$ where $\\tilde{\\mathcal {S}}$ is given by ().",
"For any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , ${(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {S}}\\mathcal {A}_i^{\\star } f}={\\mathbf {G}^{\\top }\\nabla _xu_f} \\;,$ with $\\mathbf {G}$ given for any $(x,v) \\in \\mathsf {E}$ by $\\mathbf {G}(x,v)=\\sum _{k=1}^{K}\\lambda _k^{\\mathrm {e}}(x,v) \\big (\\mathrm {n}_{k}^{\\top }(x)v\\big ) \\mathrm {n}_{k}+m_2^{{{1}{2}}}\\lambda _{\\rm ref}(x) v \\;,$ and $u_f,\\lbrace \\lambda _k^{\\mathrm {e}} : \\mathsf {E}\\rightarrow \\mathbb {R}_+ \\, : \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace \\rbrace $ are defined by (REF ) and (REF ) respectively.",
"In addition ${\\mathbf {G}^{\\top }\\nabla _{x}u_f}\\le m_2 \\big ( { \\lambda _{\\rm ref}\\nabla _xu_f } + c_\\varphi K {\\nabla _xu_f}\\big )\\\\+ C_\\varphi \\sqrt{2 m_{2,2} + 3(m_4-m_{2,2})_+} \\sum _{k=1}^K {F_k^\\top \\nabla _xu_f } \\;.$ We only consider the case $i=2$ since the case $i=1$ is obtained by taking $F_0=0$ .",
"REF By lem:boundedA-REF , $\\overline{\\mathcal {A}}_i$ is a bounded operator.",
"Therefore, we have for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ that $\\langle \\overline{\\mathcal {A}}_i \\mathcal {S}(\\operatorname{Id}-\\Pi _v) f,f \\rangle _2 = \\langle \\mathcal {S}(\\operatorname{Id}-\\Pi _v) f,\\mathcal {A}_i^\\star f \\rangle _2$ .",
"Then, by lem:relationmctmcd-REF , we have that $\\mathcal {A}_i^\\star f = -v^\\top \\nabla _x u_f$ , with $u_f \\in \\mathrm {C}_{\\mathrm {poly}}^3(\\mathsf {E})$ .",
"This result, prop:dmspdmp1-REF , and the fact that $\\operatorname{Id}-\\Pi _v$ is an orthogonal projector imply that $\\left\\langle \\overline{\\mathcal {A}}_i\\mathcal {S}(\\operatorname{Id}-\\Pi _v)f, f \\right\\rangle _2 =\\left\\langle (\\operatorname{Id}-\\Pi _v)f,(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {S}}\\mathcal {A}_i^{\\star } f \\right\\rangle _2 \\;,$ with $\\nonumber \\tilde{\\mathcal {S}}\\mathcal {A}_2^{\\star } f &= - \\left( \\frac{1}{2} \\sum _{k=1}^K \\lambda _k^{\\mathrm {e}} (\\mathcal {B}_k - \\operatorname{Id}) + m_2^{{{1}{2}}} \\lambda _{\\rm ref}\\mathcal {R}_v \\right) v^\\top \\nabla _xu_f\\\\& = \\sum _{k=1}^K \\lambda _k^{\\mathrm {e}} \\, (v^\\top \\mathrm {n}_{k}) (\\mathrm {n}_{k}^\\top \\nabla _xu_f) + m_2^{{{1}{2}}} \\lambda _{\\rm ref}v^\\top \\nabla _xu_f= \\mathbf {G}^{\\top }\\nabla _xu_f\\;,$ where we have used as:operator on velocities-REF for the last equality.",
"The proof is completed upon using the Cauchy-Schwarz inequality.",
"REF Combining (REF ) and the fact that $\\Pi _v\\tilde{\\mathcal {S}}\\mathcal {A}_2^{\\star } f = 0$ completes the proof of (REF ).",
"We now show (REF ) for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"But it is a direct consequence of the triangle inequality, the definition of $\\lbrace \\lambda ^{\\mathrm {e}}_k : \\mathsf {E}\\rightarrow \\mathbb {R}_+ \\, ; \\, k \\in \\lbrace 1,\\ldots ,K\\rbrace \\rbrace $ given in (REF ), as:intensities, the Cauchy-Schwarz inequality, lemma:velocities norm bound and the identity $F_k = \\mathrm {n}_k \\left|F_k \\right|$ for any $k \\in \\lbrace 1,\\ldots ,K\\rbrace $ : $\\begin{aligned}{ \\mathcal {S}\\mathcal {A}_2^{\\star } f } &\\le m_2^{{{1}{2}}}{ \\lambda _{\\rm ref}v^\\top \\nabla _xu_f }+ \\sum _{k=1}^K \\left\\lbrace C_\\varphi { (v^\\top \\mathrm {n}_{k})^2 \\, F_k^\\top \\nabla _xu_f } +c_\\varphi m_2^{{1}{2}}{ (v^\\top \\mathrm {n}_{k}) \\, \\mathrm {n}_k^\\top \\nabla _xu_f } \\right\\rbrace \\\\&= m_2{ \\lambda _{\\rm ref}\\nabla _xu_f } +m_2 c_\\varphi K {\\nabla _xu_f}+ C_\\varphi \\sqrt{2 m_{2,2} + 3(m_4-m_{2,2})_+} \\sum _{k=1}^K {F_k^\\top \\nabla _xu_f } .\\\\\\end{aligned}$ Lemma Assume that $\\mathcal {L}_i$ , $i \\in \\lbrace 1,2\\rbrace $ given by (REF ) or (REF ), where $\\mathcal {B}_k$ is given in (REF ), satisfies as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"Assume in addition that as:U, as:Fk, as:intensities, as:radial as:operator on velocities and as:refreshment hold.",
"Let $\\mathcal {T}_i$ be the anti-symmetric part of $\\mathcal {L}_{i}$ defined by (REF ) and the operator $\\mathcal {A}_i$ defined by (REF ) relative to $\\mathcal {T}_i$ .",
"Then, For any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , we get $\\vert \\left\\langle \\overline{\\mathcal {A}}_i\\mathcal {T}_i(\\operatorname{Id}-\\Pi _v)f, f \\right\\rangle _2\\vert \\le {(\\operatorname{Id}-\\Pi _v)f}{(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {T}}_i \\mathcal {A}_i^{\\star } f} \\;,$ where $\\tilde{\\mathcal {T}}_i$ is given in (REF ).",
"For any $f\\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ ${(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {T}}\\mathcal {A}_i^{\\star } f}= 2m_{2,2}{\\mathbf {M}}^{2}+3(m_{4}-m_{2,2}){{\\rm diag}(\\mathbf {M})}^{2} \\;,$ with $\\mathbf {M}=\\nabla _{x}^{2}u_f +\\sum _{k=1}^{K} (F_{k}^{\\top } \\nabla _{x}u) \\mathrm {n}_{k}\\mathrm {n}_{k}^{\\top } \\;,$ and $u_f$ defined by (REF ).",
"Remark A general, but potentially rough, bound on the right hand side of (REF ) can be obtained as follows.",
"From the fact that ${{\\rm diag}(\\mathbf {M})}\\le {\\mathbf {M}}$ , it holds that ${(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {T}}_i \\mathcal {A}_i^{\\star } f}\\le \\sqrt{2m_{2,2}+3(m_{4}-m_{2,2})_{+}}{\\mathbf {M}}$ where from the triangle inequality and the property $\\vert \\mathrm {n}_{k}(x)\\mathrm {n}_{k}(x)^{\\top } \\vert =1$ ${\\mathbf {M}}\\le {\\nabla _{x}^{2}u_f}+\\sum _{k=1}^{K}{F_{k}^{\\mathsf {\\top }}\\nabla _{x}u_f} \\;.$ Remark Specific scenarios lead to simplifications of these bounds and the bounds in Lemma REF : from Lemma for radial distributions $m_4=m_{2,2}$ leading to a simplification of this bound, further if $\\nu $ is the centred normal distribution of covariance $m_2 \\operatorname{I}_d$ , then $m_{2,2}=m_2^2$ , leading to further simplifications, if $K=0$ , and hence $F_0 = \\nabla _xU$ , the scenario considered by [21], then one finds that the bound depends on ${\\nabla _x^2 u_f}$ only.",
"We proceed as in the proof of lemma:RS bounded.",
"We only consider the case $i=2$ since the case $i=1$ is obtained by taking $F_0=0$ .",
"REF By lem:boundedA-REF , $\\overline{\\mathcal {A}}_2$ is a bounded operator.",
"Therefore, we have for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ that $\\langle \\overline{\\mathcal {A}}_2 \\mathcal {T}_2(\\operatorname{Id}-\\Pi _v) f,f \\rangle _2 = \\langle \\mathcal {T}_2(\\operatorname{Id}-\\Pi _v) f,\\mathcal {A}_2^\\star f \\rangle _2$ .",
"Then, by lem:relationmctmcd-REF , we have that $\\mathcal {A}_2^\\star f = -v^\\top \\nabla _x u_f$ , with $u_f \\in \\mathrm {C}_{\\mathrm {poly}}^3(\\mathsf {E})$ .",
"This result, prop:dmspdmp1-REF , the fact that $\\operatorname{Id}-\\Pi _v$ is an orthogonal projector and $F_{k}=\\mathrm {n}_{k}\\vert F_{k}\\vert $ , imply that for any $\\left\\langle \\overline{\\mathcal {A}}_2\\mathcal {T}_2(\\operatorname{Id}-\\Pi _v)f, f \\right\\rangle _2 =\\left\\langle (\\operatorname{Id}-\\Pi _v)f,(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}_2 \\mathcal {A}_2^{\\star } f \\right\\rangle _2 \\;,$ with for any $(x,v) \\in \\mathsf {E}$ , $- \\tilde{\\mathcal {T}}_2\\mathcal {A}_2^{\\star } f(x,v) & =v^{\\top }\\nabla _{x}^{2}u_f(x)v-m_{2}F_{0}^{\\top }(x)\\nabla _{x}u_f(x)-\\sum _{k=1}^{K}(v^{\\top }F_{k}(x))\\big (\\mathrm {n}_{k}(x)\\mathrm {n}_{k}(x)^{\\top }v\\big )^{\\top }\\nabla _{x}u_f(x)\\\\& =v^{\\top }\\mathbf {M}(x) v-m_{2}F_{0}^{\\top }(x)\\nabla _{x}u_f(x) \\;.$ The proof is completed upon using the Cauchy-Schwarz inequality.",
"REF By (REF ), we obtain that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , $(x,v) \\in \\mathsf {E}$ , $- (\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}_2\\mathcal {A}_i^{\\star } f(x,v) = v^{\\top }\\mathbf {M}(x) v - m_2 \\operatorname{Tr}(\\mathbf {M}(x)) \\;.$ Combining this result and lemma:velocities norm bound, we deduce ${ (\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}_2 \\mathcal {A}_i^{\\star } f}^{2} & =2m_{2,2}{\\mathbf {M}}^{2}+3(m_{4}-m_{2,2}){{\\rm diag}(\\mathbf {M})}^{2}\\\\& \\le \\big [2m_{2,2}+3(m_{4}-m_{2,2})_+\\big ]{\\mathbf {M}}^{2} \\;,$ which completes the proof.",
"Remark Combining coro:bound regularization and coro:bound Fk in sec:ellipt-regul-estim, by definition of $u_f$ in (REF ) and using as:refreshment, we obtain that $m_2 {\\nabla _xu_f} \\le 2^{-{{1}{2}}}{ \\Pi _vf}$ , $\\sum _{k=1}^K { F_k^\\top \\nabla _xu_f } &\\le \\frac{2^{{{1}{2}}}\\kappa _1}{m_2 \\kappa _2} \\sum _{k=1}^K a_k { \\Pi _vf } \\;, \\\\m_2 {\\lambda _{\\rm ref}\\nabla _xu_f } & \\le \\underline{\\lambda }\\left\\lbrace 2^{-{{1}{2}}}+ \\frac{2^{{{1}{2}}}c_{\\lambda }\\kappa _1}{ \\kappa _2} \\right\\rbrace { \\Pi _vf } \\;.$"
],
[
"Proof of Theorem ",
"In this section we prove that ass:stabilitycpiv and as:DMSabstract holds for the dynamics described in s:main results in order to obtain thm:hypocoercivity as a consequence of the abstract thm:DMSmain.",
"Under the assumptions of the theorem, we can set $\\mathsf {C}$ to be $\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ .",
"ass:stabilitycpiv and as:DMSabstract-REF hold by lem:relationmctmcd-REF .",
"as:DMSabstract-REF follows from prop:microscopic coercivity with $\\lambda _v = \\underline{\\lambda }$ .",
"as:DMSabstract-REF follows from lem:linkPoincare-DMSabstract with $\\lambda _x =C_{\\operatorname{P}}/(1+C_{\\operatorname{P}})$ .",
"as:DMSabstract-REF follows from prop:dmspdmp1-REF .",
"We are left with checking as:DMSabstract-REF .",
"By lemma:RS bounded-REF , lemma:RT bounded-REF , rem:RT bounded, we get setting $m = \\sqrt{2m_{2,2}+3(m_{4}-m_{2,2})_{+}}$ , for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ that ${\\tilde{\\mathcal {S}}\\mathcal {A}_i^{\\star } f} &+ {(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}_i \\mathcal {A}_i^{\\star } f } \\\\& \\le m \\left\\lbrace {\\nabla _{x}^{2}u_f}+(1+C_{\\varphi }) \\sum _{k=1}^{K}{F_{k}^{\\mathsf {\\top }}\\nabla _{x}u_f} \\right\\rbrace + m_2 {\\lambda _{\\rm ref}\\nabla _xu_f } + m_2 c_\\varphi K {\\nabla _xu_f} \\\\& \\le \\left[ \\frac{m}{m_2}\\left\\lbrace \\frac{2^{1/2}(1+C_{\\varphi })\\kappa _1}{\\kappa _2} \\sum _{k=1}^K a_k + \\kappa _1 \\right\\rbrace + \\frac{\\underline{\\lambda }}{2^{{1}{2}}} \\left\\lbrace 1 + \\frac{2 c_\\lambda \\kappa _1}{\\kappa _2} \\right\\rbrace + \\frac{c_\\varphi K}{2^{{1}{2}}} \\right] {\\Pi _vf} \\;,$ where we have used that ${\\nabla _x^2 u_f} \\le m_2^{-1} \\kappa _1{ \\Pi _vf } $ by prop:bound hessian in sec:ellipt-regul-estim and rem:RS bounded, with $\\kappa _1$ and $\\kappa _2$ given in (REF ) and (REF ) respectively.",
"The proof of as:DMSabstract-REF is then completed using lemma:RT bounded-REF and lemma:RS bounded-REF ."
],
[
"Proof of lemma:behaviourrateR0",
"[Proof of lemma:behaviourrateR0] Fix $\\lambda _x \\in \\left(0,1\\right)$ .",
"Using that $t \\mapsto (1+t)/\\big [(1+t)^2+R_0^2\\big ]$ is nondecreasing on $\\mathbb {R}_+$ , we obtain that for any $R_0 \\ge 4+2 \\sqrt{3}$ , (REF ) is satisfied.",
"Since for any $a >0$ , $s \\mapsto (s+a)/(s-a)$ for $s>a$ is nonincreasing, we deduce from above that for $R_0 \\ge (4+2 \\sqrt{3}) \\vee (\\lambda _v/2^{{{1}{2}}})$ , $A(\\epsilon _0)^2 \\le \\frac{4 R_0 +2^{{1}{2}}\\lambda _v}{4 R_0 - 2^{{1}{2}}\\lambda _v} \\le \\frac{2^{3/2} \\lambda _v +2^{{1}{2}}\\lambda _v}{2^{3/2} \\lambda _v - 2^{{1}{2}}\\lambda _v} < 3^{{{1}{2}}}\\;.$ For the second part of the statement, first note that $\\Lambda (\\epsilon ) = 2^{-1}[1-\\epsilon (1-\\lambda _x)]\\big [1-\\big (1-\\epsilon b_{\\Lambda }(\\epsilon )\\big )^{{{1}{2}}}\\big ] \\;,$ where $b_{\\Lambda }(\\epsilon )=\\big [ 4 \\lambda _x (1-\\epsilon )- \\epsilon R_0^2 \\big ]/[1-\\epsilon (1-\\lambda _x)]^2 \\in \\left[0,\\epsilon ^{-1}\\right]$ for $\\epsilon \\le (2^{{1}{2}}\\lambda _v)^{-1} \\wedge \\lbrace 4 \\lambda _x /(4\\lambda _x+R_0^2)\\rbrace $ .",
"Using that for any $a \\in \\left[0,1\\right]$ , $a/2 \\le 1-(1-a)^{1/2} \\le a$ we deduce that for $\\epsilon \\le (2^{{1}{2}}\\lambda _v)^{-1} \\wedge \\lbrace 4 \\lambda _x /(4\\lambda _x+R_0^2)\\rbrace $ , $4^{-1}[1-\\epsilon (1-\\lambda _x)] \\epsilon b_{\\Lambda }(\\epsilon ) \\le \\Lambda (\\epsilon ) \\le 2^{-1}[1-\\epsilon (1-\\lambda _x)] \\epsilon b_{\\Lambda }(\\epsilon ) \\;.$ Further for $R_0 \\ge (4+2 \\sqrt{3}) \\vee (\\lambda _v/2^{{{1}{2}}})$ we have $\\epsilon _0 \\le (2^{{1}{2}}\\lambda _v)^{-1} \\wedge \\lbrace 3 \\lambda _x /(4\\lambda _x+R_0^2)\\rbrace $ from thm:DMSmain-REF , leading to $\\lambda _x /[1-\\epsilon _0(1-\\lambda _x)]^2 \\le b_{\\Lambda }(\\epsilon _0) \\le 4\\lambda _x /[1-\\epsilon _0(1-\\lambda _x)]^2 \\;,$ and consequently, using (REF ), $\\epsilon _0 \\lambda _x/4 \\le \\Lambda (\\epsilon _0) \\le 2\\lambda _x \\epsilon _0/[1-2(1-\\lambda _x)/(4+R_0^2)] \\le 4 \\lambda _x \\epsilon _0 \\;,$ where we have used that $\\lambda _x \\le 1$ for the last inequality.",
"Finally we note that from (REF ) $\\frac{2}{3} \\le \\frac{1}{1 + 2^{3/2} \\lambda _v/(4+R_0^2)} \\le \\frac{1}{1+2^{{1}{2}}\\lambda _v \\epsilon _0} \\le 1 \\;,$ where the leftmost inequality follows from the fact that for $2^{{1}{2}}R_0 \\ge \\lambda _v$ $\\frac{2^{3/2} \\lambda _v}{4+R_0^2} \\le \\frac{2^{3/2} \\lambda _v}{4+2^{-1} \\lambda _v^2} \\le 1/2 \\;.$"
],
[
"Proof of thm:scaling-with-d",
"[Proof of Theorem ] Since $\\lambda _v = \\underline{\\lambda }$ and $R_0 \\ge (4+2 \\sqrt{3}) \\vee (\\underline{\\lambda }/2^{{{1}{2}}})$ by thm:hypocoercivity, from thm:DMSmain and lemma:behaviourrateR0, $A < 3^{{{1}{2}}}$ while with $\\lambda _x=C_{\\operatorname{P}}/(1+C_{\\operatorname{P}})$ $\\underline{\\lambda }\\lambda _x m_2^{{1}{2}}\\epsilon _0/6 \\le \\alpha (\\epsilon _0) \\quad \\text{with} \\quad \\lambda _x/(1 +R_0^{2}) \\le \\epsilon _0 \\le 2/(4 +R_0^2) \\;.$ By (REF ), if $c_1,c_2, {a}_{\\infty },m_b$ are fixed, there exist $C^R_1(C_{\\operatorname{P}},c_1,c_2, {a}_{\\infty },m_b) >0$ , independent of $d,\\underline{\\lambda }$ , $c_\\lambda $ , $C_{\\varphi }$ and $c_{\\varphi }$ such that $\\overline{R}_0 \\le C^R_1(C_{\\operatorname{P}},c_1,c_2, {a}_{\\infty },m_b) \\overline{R}_1 \\;,$ where $\\overline{R}_1= c_\\varphi K + (1+C_{\\varphi }) d^{(1+\\varpi )/2} K+\\underline{\\lambda }(1+c_{\\lambda } d^{(1+\\varpi )/2})$ .",
"Combining this bound with (REF ) concludes the proof."
],
[
"The Zig-Zag sampler–optimization",
"In this section, we specify our results in the case of the Zig-Zag sampler for which better estimates can be obtained, leading to better scaling properties with respect to $d$ .",
"The Zig-Zag process corresponds to the instantiation of (REF ) for which $F_0=0$ , $K=d$ , $F_i(x) = \\partial _{x_i} U(x) \\mathbf {e}_i$ , $\\mathrm {n}_{i}(x)=\\mathbf {e}_{i}$ , $\\lambda _{\\rm ref}(x) = \\underline{\\lambda }>0$which corresponds to $c_{\\lambda } = 0$ in as:refreshment, for $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ and $x \\in \\mathsf {X}$ , and $\\mathcal {R}_v = \\Pi _v-\\operatorname{Id}$ .",
"The corresponding generator takes the simplified form, for $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ and any $(x,v) \\in \\mathsf {E}$ $\\mathcal {L}f(x,v) = v^{\\top } \\nabla _xf(x) + \\sum _{i=1}^d \\varphi \\big (v_i \\partial _{x_i} U (x)\\big )\\big [f\\big (x,(\\operatorname{Id}- 2 \\mathbf {e}_i \\mathbf {e}_i^{\\top })v\\big ) - f(x,v)\\big ]+ \\lambda _{\\rm ref}(x) m_2^{{1}{2}}\\mathcal {R}_v f(x,v) \\;,$ where $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ is a continuous function and satisfies (REF ) in as:intensities.",
"In the next two subsections we first consider general velocity distributions and then show how our results can be specialized to the scenario where $\\mathsf {V}= \\lbrace -m_2^{{1}{2}},+m_2^{{1}{2}}\\rbrace ^d$ for $m_2>0$ and $\\nu $ is the uniform distribution on $\\mathsf {V}$ ."
],
[
"General velocity distribution",
"Theorem 3 Consider the Zig-Zag process with generator defined by (REF ) with $\\lambda _{\\rm ref}=\\underline{\\lambda }$ , $\\mathcal {R}_v = \\Pi _v-\\operatorname{Id}$ and $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ is a continuous function satisfying (REF ) in as:intensities.",
"Assume as:generator with $\\mathsf {C}=\\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , as:U, as:Fk, as:radial, as:operator on velocities, as:refreshment hold and that there exists $c_3\\ge 0$ such that for any $g \\in {\\mathrm {L}^2(\\pi )}^d$ $ \\left\\langle g,\\big [\\nabla _x^{2}U-{\\rm diag}(\\nabla _x^{2}U)\\big ]g \\right\\rangle _2\\ge -c_3 {g}^{2} \\;.$ Then, thm:DMSmain holds with $\\lambda _x$ as in (REF ), $\\lambda _v=\\underline{\\lambda }$ and $R_0 = \\frac{(6 m_4)^{{1}{2}}(2+C_\\varphi )}{m_2} \\left(\\left(1+c_1/2\\right)^{{1}{2}}+1+ (c_3/2)^{{1}{2}} \\right) + \\frac{\\underline{\\lambda }+ c_\\varphi }{2^{{1}{2}}} \\;.$ Remark From as:U we have for any $g \\in {\\mathrm {L}^2(\\pi )}^ d$ $\\left\\langle g,\\nabla _x^{2}Ug \\right\\rangle _2\\ge -c_{1}{g}^{2}$ and therefore (REF ) holds if there exist $\\overline{c}_1>0$ such that for any $g\\in {\\mathrm {L}^2(\\pi )}^ d$ , $\\left\\langle g,{\\rm diag}(\\nabla _x^{2}U)g \\right\\rangle _2\\le \\overline{c}_1{g}^{2} \\;,$ which is itself implied by $\\overline{c}_1 \\operatorname{Id}\\succeq {\\rm diag}(\\nabla _x^{2}U(x))$ for all $x \\in \\mathsf {X}$ , since ${\\rm diag}(\\nabla _x^{2}U(x))$ is symmetric.",
"Note that this is the case when $\\vert {\\rm diag}(\\nabla _x^{2}U(x)) \\vert \\le \\overline{c}_1$ or $\\vert \\nabla _x^{2}U(x) \\vert \\le \\overline{c}_1$ for all $x \\in \\mathsf {X}$ , for example.",
"The proof is very similar to the proof thm:hypocoercivity and follows from applying thm:DMSmain.",
"Checking ass:stabilitycpiv and as:DMSabstract-REF -REF -REF -REF is identical to the work done in the proof of thm:hypocoercivity with the constants $\\lambda _v = \\underline{\\lambda }$ and $\\lambda _x$ given by (REF ).",
"We are left with checking as:DMSabstract-REF .",
"By the improved bounds from lemma:ZZ RS bounded and lemma:ZZ RT bounded, we have for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , ${\\tilde{\\mathcal {S}}\\mathcal {A}^{\\star } f} +{(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}\\mathcal {A}^{\\star } f } \\\\ \\le (6 m_4)^{{1}{2}}(2 + C_\\varphi )\\left({\\nabla _x^{2}u_f}+{\\nabla _x^{*}\\nabla _xu_f}+c_3^{{1}{2}}{\\nabla _xu_f}\\right) + \\big (\\underline{\\lambda }+ c_\\varphi \\big ) m_2 {\\nabla _xu_f} \\;.$ Using prop:bound hessian and coro:bound Fk, we obtain that for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , ${(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {S}}\\mathcal {A}^{\\star } f} +{(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}\\mathcal {A}^{\\star } f } \\\\\\le \\left\\lbrace \\frac{(6 m_4)^{{1}{2}}(2+C_\\varphi )}{m_2} \\left(\\left(1+c_1/2\\right)^{{1}{2}}+1+ (c_3/2)^{{1}{2}} \\right) + \\frac{\\underline{\\lambda }+ c_\\varphi }{2^{{1}{2}}} \\right\\rbrace {\\Pi _vf} \\;,$ The proof is then completed by lemma:RS bounded-REF and lemma:RT bounded-REF .",
"We discuss in the following the dependence on the dimension of the convergence rate $\\alpha (\\epsilon _0)$ and the constant $A(\\epsilon _0)$ given by thm:DMSmain based on the constant provided by theo:ZZ spectral gap.",
"Similarly to the general case, we need to impose some conditions on $m_2,m_4$ .",
"Here, we assume that $m_4^{1/2}/m_2$ does not depend on $d$ , which holds in the case where $\\nu $ is the uniform distribution on $\\mathsf {V}= \\lbrace -1,1\\rbrace ^d$ or the $d$ -dimensional zero-mean Gaussian distribution with covariance matrix $\\operatorname{I}_d$ .",
"In the case where $\\pi $ is the i.i.d.",
"product of one-dimensional distributions $\\pi _i$ on $(\\mathbb {R},\\mathcal {B}(\\mathbb {R}))$ associated with potentials $U_i : \\mathbb {R}\\rightarrow \\mathbb {R}$ satisfying as:U, i.e.",
"for any $x \\in \\mathsf {X}$ , $U(x) = \\sum _{i=1}^d U_i(x_i)$ , $\\nabla _x^2 U(x) = \\operatorname{diag}(\\nabla _x^2U(x))$ for any $x \\in \\mathsf {X}$ and therefore (REF ) holds with $c_3=0$ .",
"Then, the convergence rate $\\alpha (\\varepsilon _0)$ and the constant $A(\\varepsilon _0)$ in thm:DMSmain do not depend on the dimension but only on the constants $c_1$ , $c_2$ , $\\underline{\\lambda }$ , $c_\\lambda $ and $C_{\\operatorname{P}}$ associated to each $U_i$ .",
"Consider now the case where the potential $U$ is strongly convex and gradient Lipschitz, i.e.",
"there exist $m,L >0$ such that $m \\operatorname{I}_d\\preceq \\nabla _x^2 U(x) \\preceq L \\operatorname{I}_d$ for any $x \\in \\mathsf {X}$ .",
"Then, since for any $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ and $x \\in \\mathsf {X}$ , $\\partial _{x_i,x_i} U(x) = \\mathbf {e}_i^{\\top }\\nabla _x^2 U(x) \\mathbf {e}_i \\le L$ by assumption, rem:ZZ spectral gap implies that (REF ) holds for $c_3 = L-m$ .",
"In addition, as:U holds with $c_1=0$ and $c_2= L$ and by [4], $U$ satisfies (REF ) with $C_{\\operatorname{P}}=m$ .",
"Then, the convergence rate $\\alpha (\\varepsilon _0)$ and the constant $A(\\varepsilon _0)$ in thm:DMSmain do not depend on the dimension but only on $L$ , $m$ , $\\underline{\\lambda }$ and $\\overline{\\lambda }$ .",
"In addition, we observe that the larger $L-m$ is, the larger $R_0$ given in (REF ) is, which in turn make the convergence rate $\\alpha (\\varepsilon _0)$ worse since it is of order $\\mathcal {O}(1/R_0^2)$ as $R_0 \\rightarrow +\\infty $ by lemma:behaviourrateR0.",
"This result is expected in the Gaussian case $U(x) = x^{\\top } \\Sigma x$ for any $x \\in \\mathsf {X}$ , since $L-m$ is the diameter of the set of eigenvalues of $\\Sigma $ which is a characterization of the conditioning of the problem.",
"Lemma Consider the Zig-Zag process with generator $\\mathcal {L}$ defined by (REF ) with $\\lambda _{\\rm ref}=\\underline{\\lambda }$ , $\\mathcal {R}_v = \\Pi _v-\\operatorname{Id}$ and $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ is a continuous function satisfying (REF ) in as:intensities.",
"Assume as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , as:U, as:Fk, as:radial, as:operator on velocities, as:refreshment and (REF ) hold.",
"Let $\\mathcal {S}$ and $\\mathcal {T}$ be the symmetric and anti-symmetric parts of $\\mathcal {L}$ respectively and $\\mathcal {A}$ the operator defined by (REF ) relative to $\\mathcal {T}$ .",
"Then for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , ${(\\operatorname{Id}-\\Pi _v)\\tilde{\\mathcal {S}}\\mathcal {A}^{\\star } f} \\\\\\le (6 m_4)^{{1}{2}}C_\\varphi \\left( {\\nabla _x^2 u_f} + {\\nabla _x^{\\star } \\nabla _xu_f }+ c_3^{{1}{2}}{\\nabla _xu_f} \\right) + \\big (\\underline{\\lambda }+ c_\\varphi \\big ) m_2 { \\nabla _xu_f } \\;,$ where $u_f$ is given by (REF ).",
"We use lemma:RS bounded and its notation, where $K=d$ , for $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $F_k = \\partial _{x_k} U$ and $\\mathrm {n}_k = \\operatorname{sgn}(\\partial _{x_k} U) \\mathbf {e}_k$ .",
"In this setting and by (REF ), it follows that for any $(x,v)\\in \\mathsf {E}$ , $\\mathbf {G}(x,v)= \\sum _{k=1}^d \\lambda ^\\mathrm {e}_k(x,v) v_k \\mathbf {e}_k + \\underline{\\lambda }m_2^{1/2} v \\;.$ By the triangle inequality and since for $i,j \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $i \\ne j$ , $\\int _{\\mathsf {V}} g(v_i) g(v_j) v_iv_j {\\rm d}\\nu (v)= 0$ by as:radial-REF for any even measurable bounded function $g: \\mathbb {R}\\rightarrow \\mathbb {R}$ , we get ${\\mathbf {G}^{\\top }\\nabla _xu_f} & \\le {\\textstyle {\\sum }_{k=1}^d \\lbrace \\varphi (v_k \\partial _{x_k} U) + \\varphi (-v_k \\partial _{x_k} U)\\rbrace v_k \\partial _{x_k} u_f} + \\underline{\\lambda }m_2 { \\nabla _xu_f } \\\\& = \\left[ \\sum _{k=1}^d { \\lbrace \\varphi (v_k \\partial _{x_k} U) + \\varphi (-v_k \\partial _{x_k} U)\\rbrace v_k \\partial _{x_k} u_f}^2 \\right]^{{{1}{2}}} + \\underline{\\lambda }m_2 { \\nabla _xu_f } \\;.$ Then by as:intensities, as:radial-REF , the triangle inequality (on $\\mathrm {L}^2(\\mu )^d$ ) and since for any $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $\\int _{\\mathsf {V}}v_i^4 \\mathrm {d}\\nu (v) = 3 m_4$ by as:radial-REF we obtain $\\nonumber {\\mathbf {G}^{\\top }\\nabla _xu_f} & \\le \\left[ \\sum _{k=1}^d { (c_{\\varphi } m_2^{{1}{2}}+ C_{\\varphi } \\left|v_k \\partial _{x_k} U \\right|) v_k \\partial _{x_k} u_f}^2 \\right]^{{{1}{2}}} + \\underline{\\lambda }m_2 { \\nabla _xu_f } \\\\\\nonumber & \\le c_{\\varphi } m_2^{{1}{2}}\\left[ \\sum _{k=1}^d { v_k \\partial _{x_k} u_f}^2 \\right]^{{{1}{2}}} + C_{\\varphi } \\left[ \\sum _{k=1}^d {\\left|v_k \\partial _{x_k} U \\right| v_k \\partial _{x_k} u_f}^2 \\right]^{{{1}{2}}} + \\underline{\\lambda }m_2 { \\nabla _xu_f } \\\\& \\le ( c_{\\varphi } + \\underline{\\lambda })m_2 {\\nabla _x u_f} + C_{\\varphi }(3m_4)^{{{1}{2}}} \\left[ \\sum _{k=1}^d { \\partial _{x_k} U \\partial _{x_k} u_f}^2 \\right]^{{{1}{2}}} \\;.$ To bound the sum we note that for $k\\in \\lbrace 1,\\ldots ,d\\rbrace $ $\\partial _{x_{k}}U\\partial _{x_{k}}u_f =\\partial _{x_{k}}^{2}u_f+\\partial _{x_{k}}^{*}\\partial _{x_{k}}u_f$ by lem:adjoint-nax-REF , which together with the fact $(a+b)^2 \\le 2(a^2+b^2)$ leads to ${\\partial _{x_{i}}U\\partial _{x_{i}}u_f}^2 \\le 2\\big ( {\\partial _{x_{i}}^{2}u_f}^2+{\\partial _{x_{i}}^{*}\\partial _{x_{i}}u_f}^2 \\big ) \\;.$ Then, using that for $a,b\\ge 0$ $\\sqrt{a+b} \\le \\sqrt{a} +\\sqrt{b}$ twice and (REF ), we deduce $\\nonumber \\left(\\sum _{k=1}^d{\\partial _{x_{k}}U\\partial _{x_{k}}u_f}^2\\right)^{{1}{2}}& \\le 2^{{{1}{2}}} \\left\\lbrace \\sum _{k=1}^d \\left( {\\partial _{x_{k}}^{2}u_f}^2+{\\partial _{x_{k}}^{*}\\partial _{x_{k}}u_f}^2 \\right) \\right\\rbrace ^{{1}{2}}\\\\\\nonumber & \\le 2^{{{1}{2}}} \\left\\lbrace \\left( \\sum _{k=1}^d {\\partial _{x_{k}}^{2}u_f}^2 \\right)^{{1}{2}}+ \\left( \\sum _{k=1}^d {\\partial _{x_{k}}^{*}\\partial _{x_{k}}u_f}^2 \\right)^{{1}{2}} \\right\\rbrace \\\\& \\le 2^{{{1}{2}}} \\left({\\nabla _x^2 u_f}+{\\nabla _x^{*}\\nabla _xu_f}+c_3^{{1}{2}}{\\nabla _xu_f} \\right) \\;.$ Then combining (REF ) and (REF ) completes the proof by lemma:RS bounded-REF .",
"Lemma Consider the Zig-Zag process with generator $\\mathcal {L}$ defined by (REF ) with $\\lambda _{\\rm ref}=\\underline{\\lambda }$ , $\\mathcal {R}_v = \\Pi _v-\\operatorname{Id}$ and $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ a continuous function satisfying (REF ) in as:intensities.",
"Assume as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , as:U, as:Fk, as:radial, as:operator on velocities, as:refreshment and (REF ) hold.",
"Let $\\mathcal {T}$ be the anti-symmetric part of $\\mathcal {L}$ and $\\mathcal {A}$ the operator defined by (REF ) relative to $\\mathcal {T}$ .",
"Then for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ ${(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}\\mathcal {A}^{\\star } f} \\le [6(4m_4-m_{2,2})]^{{1}{2}}\\left({\\nabla _x^{2}u_f}+{\\nabla _x^{*}\\nabla _xu_f}+c_3^{{1}{2}}{\\nabla _xu_f}\\right) \\;,$ where $u_f$ is defined by (REF ).",
"We use lemma:RT bounded and its notations, where $K=d$ , for $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $F_k = \\partial _{x_k} U \\mathbf {e}_k$ and $\\mathrm {n}_k = \\operatorname{sgn}(\\partial _{x_k} U) \\mathbf {e}_k$ .",
"In this setting and by (REF ), it follows that $\\mathbf {M}(x)=\\nabla _x^{2}u_f(x)+{\\rm diag}\\big (\\nabla _xu_f\\odot \\nabla _xU\\big ),$ Since ${\\mathbf {M}}^{2}={{\\rm diag}(\\mathbf {M})}^{2}+{\\mathbf {M}-{\\rm diag}(\\mathbf {M})}^{2}$ , we obtain $\\nonumber 2m_{2,2}{\\mathbf {M}}^{2}+3(m_{4}-m_{2,2}){{\\rm diag}(\\mathbf {M})}^{2} & =2m_{2,2}{\\mathbf {M}-{\\rm diag}(\\mathbf {M})}^{2}+(3m_{4}-m_{2,2}){{\\rm diag}(\\mathbf {M})}^{2} \\\\& \\le 2m_{2,2}{\\nabla _x^{2}u_f}^{2}+(3m_{4}-m_{2,2}){{\\rm diag}(\\mathbf {M})}^{2}\\;.$ We now bound ${{\\rm diag}(\\mathbf {M})}^{2}$ .",
"First, we apply the triangle inequality and use lem:adjoint-nax-REF , to deduce that $\\nonumber & {\\operatorname{diag}(\\mathbf {M})}^2 = \\sum _{k=1}^d {2\\partial _{x_{k}}^{2}u_f-\\partial _{x_{k}}^{2}u_f+\\partial _{x_{k}}U\\partial _{x_{k}}u_f}^{2} \\\\& \\le \\sum _{k=1}^d \\left(2{\\partial _{x_{k}}^{2}u_f}+{-\\partial _{x_{k}}^{2}u_f+\\partial _{x_{k}}U\\partial _{x_k}u_f}\\right)^{2}\\le \\sum _{k=1}^d \\left(8{\\partial _{x_{k}}^{2}u_f}^{2}+2{\\partial _{x_{k}}^{*}\\partial _{x_{k}}u_f}^{2} \\right)\\;,$ where we have used for the last inequality that $(a+b)^2 \\le 2a^2 +2b^2$ for any $a,b \\in \\mathbb {R}$ .",
"By lem:adjoint-nax-REF , (REF ), (REF ) and the fact that $U \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ using as:U, using that same reasoning as to establish (REF ), it holds for any $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ , ${\\partial _{x_k}^{\\star } \\partial _{x_k} u_f}^2 &= {\\partial ^2_{x_k}u_f}^2 + \\left\\langle \\partial _{x_k} u_f, \\partial _{x_k,x_k} U \\, \\partial _{x_k} u_f \\right\\rangle _2 \\;, \\\\{\\nabla _x^{*}\\nabla _xu_f}^{2} & ={\\nabla _x^{2}u_f}^{2}+\\left\\langle \\nabla _xu_f,\\nabla _x^{2}U\\nabla _xu_f \\right\\rangle _2 \\;.$ These identities and the condition (REF ) imply $\\nonumber &\\sum _{i=1}^{d}{\\partial _{x_{i}}^{*}\\partial _{x_{i}}u_f}^{2} ={{\\rm diag}\\big (\\nabla _x^{2}u_f\\big )}^{2}+\\left\\langle \\nabla _xu_f,{\\rm diag}\\big (\\nabla _x^{2}U\\big )\\nabla _xu_f \\right\\rangle _2\\\\&\\le {\\nabla _x^{2}u_f}^{2}+\\left\\langle \\nabla _xu_f,{\\rm diag}\\big (\\nabla _x^{2}U\\big )\\nabla _xu_f \\right\\rangle _2 \\le {\\nabla _x^{*}\\nabla _xu_f}^{2}-\\left\\langle \\nabla _xu_f,\\big (\\nabla _x^{2}U-{\\rm diag}(\\nabla _x^{2}U)\\big )\\nabla _xu_f \\right\\rangle _2\\\\&\\le {\\nabla _x^{*}\\nabla _xu_f}^{2} + c_3 {\\nabla _xu_f}^2 \\;.",
"$ Combining (REF ) and (REF ), we obtain ${\\operatorname{diag}(\\mathbf {M})}^2 \\le 8\\sum _{k=1}^d {\\partial _{x_{k}}^{2}u_f}^{2} + 2 ({\\nabla _x^{*}\\nabla _xu_f}^{2} + c_3 {\\nabla _xu_f}^2) \\;.$ From this inequality, (REF ) and lemma:RT bounded-REF , we deduce ${(\\operatorname{Id}-\\Pi _v) \\tilde{\\mathcal {T}}\\mathcal {A}^{\\star } f}^{2}&\\le 6(4m_4-m_{2,2}){\\nabla _x^{2}u_f}^{2}+2(3m_{4}-m_{2,2})\\left({\\nabla _x^{*}\\nabla _xu_f}^{2}+c_3{\\nabla _xu_f}^{2}\\right)\\\\&\\le 6(4m_4-m_{2,2}) \\left({\\nabla _x^{2}u_f}+{\\nabla _x^{*}\\nabla _xu_f}+c_3^{{1}{2}}{\\nabla _xu_f}\\right)^2 \\;,$ since for $a,b, c\\ge 0$ , $a^2+ b^2+c^2\\le (a+b+c)^2$ ."
],
[
"$d$ -dimensional Radmacher distribution",
"We now consider the case $\\mathsf {V}= \\lbrace -m_2^{{{1}{2}}},+m_2^{{{1}{2}}}\\rbrace ^d$ and $\\nu $ is the uniform distribution on $\\mathsf {V}$ which corresponds to the original setting of the Zig-Zag process.",
"This process has been proved to be ergodic [8] even in the absence of refreshment, that is $\\lambda _{\\rm ref}=0$ .",
"We note that in this scenario $m_4 = m_2^2/ 3$ and $m_{2,2} = m_2^2$ which leads to simplified expressions for the bounds in lemma:ZZ RS bounded and lemma:ZZ RT bounded upon revisiting their proofs.",
"However this has no qualitative impact.",
"In this section we show that hypocoercivity holds with our techniques for $\\lambda _{\\rm ref}(x)=0$ for “most of $\\mathsf {X}$ ” for a particular type of partial refreshment update.",
"Consider the scenario where $\\mathcal {R}_v$ is a mixture of the bounces $\\lbrace \\mathcal {B}_k, k=1,\\ldots ,d \\rbrace $ , for any $f \\in \\mathrm {L}^2(\\mu )$ , $(x,v) \\in \\mathsf {E}$ , $ \\lambda _{\\rm ref}\\mathcal {R}_v f(x,v) = \\sum _{k=1}^d \\lambda _{{\\rm ref}, k}(x) \\big [f\\big (x, v - 2 v_k \\mathbf {e}_k \\big ) - f(x,v)\\big ] \\;,$ with $\\lambda _{{\\rm ref}, k}\\colon \\mathsf {X}\\rightarrow \\mathbb {R}_+$ for $k \\in \\lbrace 1, \\ldots , d\\rbrace $ satisfying as:refreshment, and $\\lambda _{\\rm ref}= \\sum _{k=1}^d \\lambda _{{\\rm ref}, k}$ , that is when the process refreshes, $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ is chosen at random with probability proportional to $(\\lambda _{{\\rm ref}, 1},\\ldots ,\\lambda _{{\\rm ref}, d})$ and the component $v_k$ of $v$ is updated to $-v_k$ .",
"Proposition Consider the Zig-Zag process with generator $\\mathcal {L}$ and refreshment operator as in (REF ) and (REF ) respectively, with $\\varphi : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ is a continuous function satisfying (REF ) in as:intensities.",
"Assume as:generator with $\\mathsf {C}= \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , as:U, as:Fk, as:radial, as:operator on velocities, as:refreshment and (REF ) hold.",
"Let $\\mathcal {S}$ be the symmetric part of $\\mathcal {L}$ defined by (REF ).",
"the symmetric part of the generator is given for any $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ , $(x,v) \\in \\mathsf {E}$ by $\\mathcal {S}f(x,v) = \\sum _{k=1}^d \\left\\lbrace \\frac{\\varphi \\big (v_k\\partial _{x_k} U(x)\\big ) + \\varphi \\big (-v_k\\partial _{x_k} U(x)\\big ) }{2} + m_2^{{1}{2}}\\lambda _{{\\rm ref}, k}(x) \\right\\rbrace \\big [f\\big (x,v - 2 v_k \\mathbf {e}_k\\big ) - f(x,v)\\big ] \\;;$ the microscopic coercivity condition as:DMSabstract-REF is satisfied, i.e.",
"for any $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ , $(x,v) \\in \\mathsf {E}$ $-\\left\\langle \\mathcal {S}f, f \\right\\rangle _2\\ge \\lambda _v m_2^{{{1}{2}}} { (\\operatorname{Id}-\\Pi _v) f}^2 \\quad \\text{with} \\quad \\lambda _v = \\min _{k \\in \\lbrace 1,\\ldots ,d \\rbrace , x \\in \\mathsf {X}} \\left\\lbrace \\frac{ |\\partial _{x_k} U(x)|}{2} + \\lambda _{{\\rm ref}, k}(x) \\right\\rbrace \\;.$ Remark In other words as:DMSabstract-REF holds if for any $\\varepsilon >0$ , for all $k \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $\\lambda _{{\\rm ref}, k}$ vanishes everywhere, except on $\\lbrace x \\in \\mathsf {X}\\, :\\, \\exists k \\in \\lbrace 1,\\ldots ,d\\rbrace \\mid \\vert \\partial _{x_k} U \\vert (x) < \\varepsilon \\rbrace $ .",
"We also note that a similar result holds for the case where $\\mathcal {R}_v = \\Pi _v-\\operatorname{Id}$ , that is as:DMSabstract-REF holds whenever $\\lambda _{\\rm ref}$ vanishes everywhere, except on $\\lbrace x \\in \\mathsf {X}\\, :\\, \\exists k \\in \\lbrace 1,\\ldots ,d\\rbrace , \\, \\mid \\vert \\partial _{x_k} U \\vert (x) < \\varepsilon \\rbrace $ for $\\varepsilon >0$ .",
"The first statement is a direct application of prop:dmspdmp1-REF .",
"For the second statement, using that $\\nu $ is the uniform distribution on $\\mathsf {V}= \\lbrace -m_2^{{1}{2}}, m_2^{{1}{2}}\\rbrace ^d$ , from the polarization identity and since $\\varphi $ satisfies as:intensities, we get for any $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ , setting $\\varphi ^{\\mathrm {e}}(s) := \\varphi (s)+\\varphi (-s)$ , $-\\left\\langle \\mathcal {S}f, f \\right\\rangle _2= \\frac{1}{2} \\int _{\\mathsf {E}} \\sum _{k=1}^d \\left\\lbrace \\frac{\\varphi ^{\\mathrm {e}}(v_k\\partial _{x_k} U(x)) }{2} + m_2^{{1}{2}}\\lambda _{{\\rm ref}, k}(x) \\right\\rbrace \\big [f(x,v) - f\\big (x,(\\operatorname{Id}- 2 \\mathbf {e}_k \\mathbf {e}_k^{\\top })v\\big ) \\big ]^2 \\, \\mathrm {d}\\mu (x,v)\\\\\\ge ( \\lambda _v m_2^{{{1}{2}}}/2) \\int _{\\mathsf {E}} \\sum _{k=1}^d \\big [ f(x,v) - f\\big (x,(\\operatorname{Id}- 2 \\mathbf {e}_k \\mathbf {e}_k^{\\top })v\\big )\\big ]^2 \\, \\mathrm {d}\\mu (x,v) \\;,$ where $\\lambda _v$ is defined in (REF ).",
"Now by the Poincaré inequality for any $g \\in \\mathrm {L}^2_0(\\nu )$ , see e.g.",
"[47], it holds that $(1/2) \\int _{\\mathsf {V}}\\sum _{k=1}^d \\big [ g(v) - g\\big ((\\operatorname{Id}- 2 \\mathbf {e}_i \\mathbf {e}_i^{\\top })v\\big ) \\big ]^2 \\, \\mathrm {d}\\nu (v) \\ge \\int _{\\mathsf {V}}\\sum _{k=1}^d g^2(v) \\, \\mathrm {d}\\nu (v) \\;.$ Now since for any $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ , $ \\left\\langle \\mathcal {S}f, f \\right\\rangle _2 = \\left\\langle \\mathcal {S}(\\operatorname{Id}-\\Pi _v)f, (\\operatorname{Id}-\\Pi _v)f \\right\\rangle _2$ and for any $x \\in \\mathsf {X}$ , $v \\mapsto (\\operatorname{Id}-\\Pi _v)f(x,v) \\in \\mathrm {L}^2_0(\\nu )$ , then combining (REF ) and (REF ) and using Fubini's theorem concludes the proof of (REF )."
],
[
"Discussion and link to earlier work",
"As pointed out earlier the scenario $K=0$ where $F_0=\\nabla _xU$ is considered in [21] where the authors establish hypercoercivity but also in [11] where the authors establish geometric convergence, that is the existence of constants $A, \\alpha > 0$ and a measurable function $V\\colon \\mathsf {E}\\rightarrow \\mathbb {R}_+$ satisfying $\\mu \\big (\\lbrace V=\\infty \\rbrace \\big )=0$ , such that for any $(x,v)\\in \\mathsf {E}$ and $t \\ge 0$ , $ \\Vert P_{t}\\big ((x,v),\\cdot \\big )-\\mu (\\cdot ) \\Vert _{\\mathrm {TV}} \\le A V(x,v) \\mathrm {e}^{-\\alpha t} \\;.$ Similar results have been obtained in [18] and [24] for the Bouncy particle sampler and in [8] for the Zig-Zag process.",
"All these methods rely on guessing such a suitable Lyapounov function $V$ and establishing a so-called drift condition for this function, in conjunction with a minorization condition [42].",
"Here we have established ${\\mathrm {L}^2(\\mu )}$ -exponential convergence, or equivalently that there exists an absolute ${\\mathrm {L}^2(\\mu )}$ -absolute spectral gap [22] (by considering the skeleton of the process) and is therefore $\\mu $ -a.e.",
"uniformly convergent by [22], that is (REF ) holds with $V=\\mathbf {1}$ and $\\mu $ -a.e.. An advantage of our approach is that it provides explicit and relatively simple bounds in terms of interpretable quantities which, we show, are informative, and is in contrast with those on minorization and drift conditions in most scenarios.",
"One exception is the study of BPS on the torus carried out in [24] for $U=0$ , using an appropriate coupling argument, which leads to a rate of convergence for the total variation distance with a favourable $\\Theta (d^{1/2})$ scaling.",
"Although we have shown that for the Zig-Zag sampler with Rademacher distribution $\\lambda _{\\rm ref}$ is not required to be bounded away from zero on $\\mathsf {X}$ , the results of [8] hold with $\\lambda _{\\rm ref}=0$ .",
"It would be interesting to further investigate whether our results can be specialized to consider the scenario $\\lambda _{\\rm ref}=0$ .",
"Although we have shown that the theory developed in this paper covers numerous scenarios in a unified set-up, various possible extensions are possible.",
"For example we have restricted this first investigation to deterministic bounces of the type given in (REF ), but there does not seem to be any obstacle to the extension of our results to the more general set-ups such as considered in [55], [58], [44].",
"In the same vein, great parts of our calculations could be used to consider distributions of the velocity $\\nu $ that are neither Gaussian, nor the uniform distribution on the hypersphere.",
"For $\\nu $ of density proportional to $\\exp (-\\mathrm {K}(v))$ with $\\mathrm {K}: \\mathbb {R}^d \\rightarrow \\mathbb {R}$ the Liouville operator involved in the definition of (REF ) would take the form $\\nabla _v \\mathrm {K}(v)^\\top \\nabla _xf(x,v) - m_2 F_0^\\top \\nabla _v f(x,v)$ , leading to a different expression for $\\mathcal {T}$ .",
"Such modified kinetic energies have been proposed to speed up the computation, introducing the Modified Langevin Dynamics for which convergence to equilibrium has been studied in [52]."
],
[
"Optimization and estimates of the rate of convergence $\\alpha (\\epsilon )$",
"Consider the functions $R,\\tilde{\\alpha } : \\mathbb {R}_+^* \\rightarrow \\mathbb {R}_+^*$ given for any $\\epsilon \\ge 0$ by $R(\\epsilon ) &=[1-\\epsilon (1-\\lambda _{x})]^{2}-4\\epsilon \\lambda _{x}(1-\\epsilon )+\\epsilon ^{2}R_{0}^{2} =R_{1}^{2}\\left(\\epsilon -\\frac{1+\\lambda _{x}}{R_{1}^{2}}\\right)^{2}+1-\\frac{(1+\\lambda _{x})^{2}}{R_{1}^{2}}>0 \\;, \\\\\\tilde{\\alpha }(\\epsilon ) &= \\frac{ \\Lambda (\\epsilon )}{1+2^{{1}{2}}\\lambda _{v}\\epsilon } = \\frac{ 1 - \\epsilon (1-\\lambda _x) - R^{{{1}{2}}}(\\epsilon ) }{2(1+2^{{1}{2}}\\lambda _{v}\\epsilon )} \\;,$ where $R_{1}^{2}=(1+\\lambda _{x})^{2}+R_{0}^{2} \\;,$ and $\\Lambda $ is given in (REF ).",
"We show that optimizing $\\epsilon \\mapsto \\Lambda (\\epsilon )$ is a good enough proxy for optimizing $\\epsilon \\mapsto \\tilde{\\alpha }(\\epsilon )$ , whose maximum is unique, but intractable.",
"Since $\\epsilon \\mapsto \\alpha (\\epsilon )$ defined by (REF ) is proportional to $\\epsilon \\mapsto \\tilde{\\alpha }(\\epsilon )$ , the same conclusion holds for this function.",
"Lemma Let $\\Lambda \\colon \\mathbb {R}_+\\rightarrow \\mathbb {R}$ be defined by (REF ).",
"Then with $\\lambda _{x}\\in (0,1)$ and $R_0 >0$ , $\\Lambda (\\epsilon )\\ge 0$ for $\\epsilon \\in \\left[0, 4\\lambda _{x}/(4\\lambda _{x}+R_{0}^{2})\\right]$ and $\\Lambda (0)=0$ .",
"$\\Lambda $ has first order derivative $\\Lambda ^{\\prime }(\\epsilon )=-(1/2)\\big [(1-\\lambda _{x})R^{{{1}{2}}}(\\epsilon )+\\epsilon R_{1}^{2}-(1+\\lambda _{x})\\big ]R^{-{{1}{2}}}(\\epsilon ) \\;,$ and $\\Lambda ^{\\prime }(0)=\\lambda _{x}>0$ .",
"$\\Lambda \\colon \\mathbb {R}_{+}\\rightarrow \\mathbb {R}$ has a unique stationary point ($\\Lambda ^{\\prime }(\\epsilon _{0})=0$ ) $\\epsilon _{0}=\\frac{(1+\\lambda _{x})-(1-\\lambda _{x})\\left[ R_{0}^{2}/(R_{0}^{2}+4\\lambda _{x}) \\right]^{{{1}{2}}}}{(1+\\lambda _{x})^{2}+R_{0}^{2}}>0 \\;,$ such that $\\Lambda (\\epsilon _{0})>0$ .",
"From (REF ) we see that $\\Lambda (\\epsilon )\\ge 0$ requires $0 \\le \\epsilon \\le \\frac{1}{1-\\lambda _{x}}\\wedge \\frac{4\\lambda _{x}}{4\\lambda _{x}+R_{0}^{2}}=\\frac{4\\lambda _{x}}{4\\lambda _{x}+R_{0}^{2}} \\;,$ where the equality follows from $\\lambda _{x}>0$ , which completes the proof of REF .",
"The proof of REF is a simple calculation and is omitted.",
"We now show REF .",
"If we set $\\Lambda ^{\\prime }(\\epsilon ) = 0$ , it implies that $\\epsilon >0$ satisfies $(1+\\lambda _{x})-\\epsilon R_{1}^{2} =R^{{{1}{2}}}(\\epsilon )(1-\\lambda _{x}) \\;,$ and imposes the condition $(1+\\lambda _{x})-\\epsilon R_{1}^{2} \\ge 0$ so $\\epsilon \\in \\left[0,\\frac{1+\\lambda _{x}}{(1+\\lambda _{x})^{2}+R_{0}^{2}}\\right] \\;.$ Squaring both sides of (REF ) implies the following sequence of equalities using (REF ) $(1-\\lambda _{x})^{2}R(\\epsilon ) & =\\left[\\epsilon R_{1}^{2}-(1+\\lambda _{x})\\right]^{2},\\\\(1-\\lambda _{x})^{2}\\left[R_{1}^{2}\\epsilon ^{2}-2(1+\\lambda _{x})\\epsilon +1\\right] & =R_{1}^{4}\\epsilon ^{2}-2R_{1}^{2}(1+\\lambda _{x})\\epsilon +(1+\\lambda _{x})^{2} \\;,$ which is equivalent by (REF ) to $R_{1}^{2} \\epsilon ^{2}\\left[(1-\\lambda _{x})^{2}-R_{1}^{2}\\right]-2\\epsilon (1+\\lambda _{x})\\left[(1-\\lambda _{x})^{2}-R_{1}^{2}\\right]-4\\lambda _{x} & =0\\\\(1+\\lambda _{x})^{2}+R_{0}^{2} \\epsilon ^{2}\\left[-4\\lambda _{x}-R_{0}^{2}\\right]-2\\epsilon (1+\\lambda _{x})\\left[-4\\lambda _{x}-R_{0}^{2}\\right]-4\\lambda _{x} & =0\\\\((1+\\lambda _{x})^{2}+R_{0}^{2})R_{1}^{2}\\epsilon ^{2}-2(1+\\lambda _{x})\\epsilon +4\\lambda _{x}/(R_{0}^{2}+4\\lambda _{x}) & =0 \\;.$ The two strictly positive roots are $\\epsilon _{\\pm } & =\\frac{(1+\\lambda _{x})\\pm \\left[ (1+\\lambda _{x})^{2}-4\\lambda _{x}\\lbrace (1+\\lambda _{x})^{2}+R_{0}^{2}\\rbrace /(R_{0}^{2}+4\\lambda _{x}) \\right]^{{{1}{2}}}}{(1+\\lambda _{x})^{2}+R_{0}^{2}}>0,$ where the inequality follows from $\\lambda _{x}>0$ and $R_0>0$ .",
"Further $(1+\\lambda _{x})^{2}\\big (R_{0}^{2}+4\\lambda _{x}\\big )-4\\lambda _{x}\\big [(1+\\lambda _{x})^{2}+R_{0}^{2}\\big ]=R_{0}^{2}\\big [(1+\\lambda _{x})^{2}-4\\lambda _{x}\\big ] = R_0^2 [1-\\lambda _x]^2 \\;,$ and since $\\lambda _{x}\\le 1$ , this yields the simplified expression for the two roots $\\epsilon _{\\pm }=\\frac{(1+\\lambda _{x})\\pm (1-\\lambda _{x})\\left[ R_{0}^{2}/(R_{0}^{2}+4\\lambda _{x}) \\right]^{{{1}{2}}}}{(1+\\lambda _{x})^{2}+R_{0}^{2}} \\;\\;.$ From the conditions on $\\epsilon $ given by REF and (REF ), and the fact that $\\lambda _{x}\\le 1$ , we retain $\\epsilon _{0}=\\epsilon _{-}$ only.",
"The last statement follows from the second statement and the fact that $\\Lambda ^{\\prime }$ is continuous.",
"The following lemma establishes in particular that $\\epsilon _{0}$ is a global maximum.",
"Lemma Let $\\Lambda \\colon \\mathbb {R}_+^*\\rightarrow \\mathbb {R}$ be defined by (REF ).",
"Then with $\\lambda _{x}\\in (0,1)$ and $R_0 >0$ , or any $\\epsilon >0$ , $\\Lambda ^{\\prime \\prime }(\\epsilon )<0$ (implying concavity), $\\Lambda $ is maximized at $\\epsilon _{0}$ defined by (REF ) and $0<\\epsilon _{0}\\le (4\\lambda _{x})/(4\\lambda _{x}+R_{0}^{2})$ .",
"If in addition $R_0 \\ge 2$ , $\\epsilon _0 \\le 3 \\lambda _x/(4\\lambda _x+R_0^2)$ .",
"We differentiate $\\epsilon \\mapsto -2\\Lambda (\\epsilon )=-[1-\\epsilon (1-\\lambda _{x})]+R^{{{1}{2}}}(\\epsilon )$ twice, yielding the first order derivative $\\epsilon \\mapsto (1-\\lambda _{x})+(1/2)R^{\\prime }(\\epsilon )R^{-{{1}{2}}}(\\epsilon )$ and the second order derivative follows $\\epsilon \\mapsto (1/2)\\left(R^{\\prime \\prime }(\\epsilon )R^{-{{1}{2}}}(\\epsilon )-(1/2)[R^{\\prime }(\\epsilon )]^{2}R^{-3/2}(\\epsilon )\\right)=(1/4)R^{-3/2}(\\epsilon )\\left(2R^{\\prime \\prime }(\\epsilon )R(\\epsilon )-[R^{\\prime }(\\epsilon )]^{2}\\right) \\;\\;.$ Now from (REF ), $R(\\epsilon )=a\\psi (\\epsilon )$ with $\\psi (\\epsilon )=(\\epsilon -b)^{2}+c$ with all constants $b,c$ non-negative.",
"Further $\\psi ^{\\prime }(\\epsilon )=2(\\epsilon -b)$ and $\\psi ^{\\prime \\prime }(\\epsilon )=2$ and therefore $2\\psi ^{\\prime \\prime }(\\epsilon )\\psi (\\epsilon )-\\psi ^{\\prime }(\\epsilon )^{2} & =4[(\\epsilon -b)^{2}+c-(\\epsilon -b)^{2}]=4c > 0 \\;,$ which implies that $\\Lambda ^{\\prime \\prime }(\\epsilon ) \\le 0$ for any $\\epsilon \\ge 0$ .",
"From the concavity we deduce that $\\epsilon _{0}$ is a maximum, and the inequality on $\\epsilon _{0}$ follows from the fact that this is required for $\\Lambda (\\epsilon _{0})\\ge 0$ .",
"Using that for any $s\\ge 0$ , $(1+s)^{{{1}{2}}} \\le 1+s/2$ , and $4 \\lambda _x \\le (1+\\lambda _x)^2$ , we get that $\\epsilon _0 = R_0 \\frac{(1+\\lambda _x)(4\\lambda _x/R_0^2+1)^{{{1}{2}}} - (1-\\lambda _x)}{\\big [(1+\\lambda _x)^2+R_0^2\\big ](R_0^2 + 4 \\lambda _x)^{{{1}{2}}}} \\le \\frac{2 \\lambda _x R_0 +2 \\lambda _x (1+\\lambda _x)/R_0}{\\big [(1+\\lambda _x)^2+R_0^2\\big ]^{{1}{2}}(R_0^2 + 4 \\lambda _x)} \\\\\\le \\frac{2 \\lambda _x +2 \\lambda _x (1+\\lambda _x)/R_0^2}{R_0^2 + 4 \\lambda _x} \\;.$ The assumption $R_0 \\ge 2$ completes the proof.",
"Proposition The function $\\tilde{\\alpha }\\colon \\mathbb {R}_{+}\\rightarrow \\mathbb {R}_{+}$ , defined by (), has a unique maximizer $\\epsilon ^{\\star }\\in \\left(0,\\epsilon _{0}\\right)$ , where $\\epsilon _0$ is given in (REF ).",
"In addition, if $2^{{1}{2}}R_{0}\\ge \\lambda _{v}$ then $\\tilde{\\alpha }(\\epsilon _{0})\\le \\tilde{\\alpha }(\\epsilon ^{\\star })\\le 3\\tilde{\\alpha }(\\epsilon _{0})\\;.$ First note that for any $\\epsilon \\ge 0$ , $\\tilde{\\alpha }^{\\prime }(\\epsilon )=\\frac{\\Psi (\\epsilon )}{(1+2^{{1}{2}}\\lambda _{v}\\epsilon )^{2}}\\;,$ with $\\Psi (\\epsilon )=\\Lambda ^{\\prime }(\\epsilon )(1+2^{{1}{2}}\\lambda _{v}\\epsilon )-2^{{1}{2}}\\lambda _{v}\\Lambda (\\epsilon )\\;.$ Then from lem:second-derivative-Lambda0, $\\Psi (\\epsilon _0) = 2^{{1}{2}}\\lambda _{v}\\Lambda (\\epsilon _0) < 0\\;, \\qquad \\text{ and for any $\\epsilon \\ge 0$, } \\Psi ^{\\prime }(\\epsilon )=(1+2^{{1}{2}}\\lambda _{v}\\epsilon )\\Lambda ^{\\prime \\prime }(\\epsilon )<0\\;.$ Together with $\\Psi (0)=\\Lambda ^{\\prime }(0)=\\lambda _{x}>0$ , and the fact that $\\epsilon \\rightarrow \\Psi (\\epsilon )$ is continuous, we deduce the existence and uniqueness of $\\epsilon ^{\\star }\\in (0,\\epsilon _{0})$ satisfying $\\tilde{\\alpha }^{\\prime }(\\epsilon ^{\\star })=0$ , and maximizing $\\tilde{\\alpha }$ on $\\mathbb {R}_{+}$ .",
"Further since $\\tilde{\\alpha }^{\\prime }(\\epsilon ^{\\star })=0$ and $\\epsilon \\mapsto \\Psi (\\epsilon )$ is non-increasing, using the first equality of (REF ) and the definition of $\\tilde{\\alpha }$ given in (), we deduce $\\sup _{\\epsilon \\in [\\epsilon ^{\\star },\\epsilon _{0}]}\\vert \\tilde{\\alpha }^{\\prime }(\\epsilon )\\vert \\le \\frac{\\vert \\Psi (\\epsilon _{0})\\vert }{(1+2^{{1}{2}}\\lambda _{v}\\epsilon ^{\\star })^{2}}=2^{{1}{2}}\\lambda _{v}\\frac{1+2^{{1}{2}}\\lambda _{v}\\epsilon _{0}}{(1+2^{{1}{2}}\\lambda _{v}\\epsilon ^{\\star })^{2}}\\tilde{\\alpha }(\\epsilon _{0})\\;,$ From a Taylor's theorem, we obtain $\\tilde{\\alpha }(\\epsilon ^{\\star })-\\tilde{\\alpha }(\\epsilon _{0})\\le (\\epsilon _{0}-\\epsilon ^{\\star })2^{{1}{2}}\\lambda _{v}\\frac{1+2^{{1}{2}}\\lambda _{v}\\epsilon _{0}}{(1+2^{{1}{2}}\\lambda _{v}\\epsilon ^{\\star })^{2}}\\tilde{\\alpha }(\\epsilon _{0})\\;,$ from which we conclude that $\\tilde{\\alpha }(\\epsilon _{0})\\le \\tilde{\\alpha }(\\epsilon ^{\\star })\\le \\left[1+(\\epsilon _{0}-\\epsilon ^{\\star })2^{{1}{2}}\\lambda _{v}\\frac{1+2^{{1}{2}}\\lambda _{v}\\epsilon _{0}}{(1+2^{{1}{2}}\\lambda _{v}\\epsilon ^{\\star })^2}\\right]\\tilde{\\alpha }(\\epsilon _{0})\\;.$ Now if we use $2^{{1}{2}}R_{0}\\ge \\lambda _{v}$ we have by (REF ) that $\\lambda _{v}\\epsilon _{0}<\\frac{(1+\\lambda _{x})\\lambda _{v}}{(1+\\lambda _{x})^{2}+R_0^2}\\le \\lambda _v (2R_0)^{-1} \\le 2^{-{{1}{2}}} \\;,$ implying $(\\epsilon _{0}-\\epsilon ^{\\star })2^{{1}{2}}\\lambda _{v}\\frac{1+2^{{1}{2}}\\lambda _{v}\\epsilon _{0}}{(1+2^{{1}{2}}\\lambda _{v}\\epsilon ^{\\star })^{2}}\\le 2^{{1}{2}}\\lambda _{v}\\epsilon _{0}(1+2^{{1}{2}}\\lambda _{v}\\epsilon _{0})\\le 2 \\;,$ which completes the proof of (REF )."
],
[
"Some results on closed operators on Hilbert spaces",
"In this section we gather classical results concerning densely defined closed operators on a Hilbert space to which we repeatedly refer throughout the manuscript.",
"Proposition Let $\\mathcal {B}$ be a closed and densely defined operator on a Hilbert space $\\mathsf {H}$ of inner product $\\left\\langle \\cdot , \\cdot \\right\\rangle $ , induced norm ${\\cdot }$ and operator norm $ {\\cdot }$ .",
"$\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B}$ is a positive self-adjoint operator on $\\mathsf {H}$ bijective from $\\mathrm {D}(\\mathcal {B}^{\\star } \\mathcal {B})$ to $\\mathsf {H}$ .",
"In addition, $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ is a positive self-adjoint bounded operator on $\\mathsf {H}$ and $\\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ is a bounded operator.",
"For any $h \\in \\mathsf {H}$ , $\\Vert (\\operatorname{Id}+\\mathcal {B}^{\\star } \\mathcal {B})^{-1} h \\Vert ^2 + 2 \\, \\Vert \\mathcal {B}(\\operatorname{Id}+\\mathcal {B}^{\\star } \\mathcal {B})^{-1} h\\Vert ^2 \\le \\Vert h \\Vert ^2 \\;.$ $\\mathcal {B}^{\\star } \\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ is a bounded operator on $\\mathsf {H}$ which satisfies ${\\mathcal {B}^{\\star } \\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}} \\le 1$ .",
"The operator $((\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star },\\mathrm {D}(\\mathcal {B}^{\\star }))$ is closable, its closure is a bounded operator and ${\\overline{(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }}} \\le 1$ .",
"Remark Note that under the condition of prop:abstract bound, we get that $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }$ can be extended to a bounded operator and ${(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} } \\le 1 \\;, \\quad { \\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} } \\le 1/2^{{{1}{2}}} \\;.$ REF and REF follow from [49] and inspection of the proof.",
"We now show REF .",
"First note that $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B}-\\operatorname{Id}) (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} = \\operatorname{Id}- (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ , from which we deduce that it is a self-adjoint and bounded operator by the triangle inequality with norm less or equal than 2.",
"To prove the tighter upper bound we use [49] (twice), the identity for any $h \\in \\mathsf {H}$ $\\left|\\left\\langle \\mathcal {B}^{\\star } \\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} h , h \\right\\rangle \\right| = \\max \\left\\lbrace \\Vert h\\Vert ^2 -\\left\\langle (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}h, h \\right\\rangle , \\left\\langle (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}h, h \\right\\rangle - {h}^2 \\right\\rbrace \\;,$ that $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ is positive and $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\le 1$ from the first statement.",
"It remains to prove REF .",
"Since $\\mathcal {B}$ is closed and densily defined, $\\mathrm {D}(\\mathcal {B}^{\\star })$ is dense and therefore $\\lbrace (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }\\rbrace ^{\\star }$ is closed and densely defined by [49].",
"By REF , we have for any $h_1 \\in \\mathrm {D}(\\mathcal {B}^{\\star })$ and $h_2 \\in \\mathsf {H}$ , we have $\\left\\langle (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }h_1,h_2 \\right\\rangle _2 = \\left\\langle h_1,\\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} h_2 \\right\\rangle _2 \\;,$ which implies that $\\lbrace (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }\\rbrace ^{\\star } = \\mathcal {B}(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1}$ .",
"Therefore, $\\lbrace (\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }\\rbrace ^{**}$ is a bounded operator on $\\mathsf {H}$ .",
"The proof then follows by [49] which implies that $(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }$ is closable and $\\overline{(\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star }} = ((\\operatorname{Id}+ \\mathcal {B}^{\\star } \\mathcal {B})^{-1} \\mathcal {B}^{\\star })^{**}$ .",
"A similar result can be obtained by using that $\\mathcal {B}$ is closable only, as a consequence of the following lemma.",
"Lemma Assume that $(\\mathcal {B},\\mathrm {D}(\\mathcal {B}))$ is a densely defined closable operator.",
"Let $(\\overline{\\mathcal {B}}, \\mathrm {D}(\\overline{\\mathcal {B}}))$ be the closure of $(\\mathcal {B},\\mathrm {D}(\\mathcal {B}))$ and $m>0$ .",
"Then, the conclusions of prop:abstract bound hold changing $\\mathcal {B}$ to $\\overline{\\mathcal {B}}$ .",
"This result is a just a consequence of [49] which implies that $\\mathcal {B}^{\\star }$ is densely defined, $\\overline{\\mathcal {B}}=(\\mathcal {B}^{\\star })^{\\star }$ and $\\mathcal {B}^{\\star } = \\overline{\\mathcal {B}}^{\\, \\star }$ .",
"The densely defined and closed operator $\\nabla _x$ on $\\mathrm {L}^2(\\pi )$ can be extended as an operator on $\\mathrm {L}^2(\\pi )^d$ as follows: for any $(f_1,\\ldots ,f_d) \\in \\mathrm {L}^2(\\pi )^d$ , $f_1 \\in \\mathrm {D}(\\nabla _x)$ , $\\nabla _x f = \\nabla _x f_1$ .",
"Therefore a direct consequence of prop:abstract bound applied to the operator $m^{-{{1}{2}}} \\nabla _x$ for $m >0$ , on $\\mathrm {L}^2(\\pi )^d$ is the following.",
"Corollary Let $m >0$ .",
"The operators $\\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1}$ and $\\nabla _x^{\\star } \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1}$ are bounded on $\\mathrm {L}^2(\\pi )^d$ with ${ \\nabla _x( m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} } \\le 1/(2m)^{{{1}{2}}} \\;, \\, \\, {\\nabla _x^{\\star } \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1}} \\le 1 \\;.$ In addition, for any $f\\in \\mathrm {L}^2(\\pi )$ , ${(m\\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} f}^2 + (2/ m) \\, { \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} f}^2 \\le \\lbrace { f}/m\\rbrace ^2 \\;,$ and ${\\nabla _x^{\\star } \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1}f} \\le {f} \\;.$ We conclude this section by the following results which can be found in [31].",
"Lemma ([31]) Let $(\\mathcal {T},\\mathrm {D}(\\mathcal {T}))$ be a anti-symmetric operator on $\\mathrm {L}^2(\\mu )$ and $\\Pi $ be an orthogonal projection on $\\mathrm {L}^2(\\mu )$ .",
"Assume that there exists $\\mathsf {D}\\subset \\mathrm {D}(\\mathcal {T})$ such that $\\Pi (\\mathsf {D}) \\subset \\mathrm {D}(\\mathcal {T})$ and $\\mathsf {D}$ is dense in $\\mathrm {L}^2(\\mu )$ .",
"Then the following statements hold.",
"$\\mathrm {D}(\\mathcal {T}) \\subset \\mathrm {D}((\\mathcal {T}\\Pi )^{\\star })$ and for any $f \\in \\mathrm {D}(\\mathcal {T})$ , $(\\mathcal {T}\\Pi )^{\\star }f= -\\Pi \\mathcal {T}f$ .",
"For any $f \\in \\mathrm {D}((\\mathcal {T}\\Pi )^{\\star })$ , $\\Pi (\\mathcal {T}\\Pi )^{\\star } f = (\\mathcal {T}\\Pi )^{\\star }f$ ."
],
[
"Elliptic regularity estimates",
"We preface this section with some complements on the adjoint of $\\nabla _x$ seen as an operator on $\\mathrm {L}^2(\\pi )^d$ .",
"Lemma Assume as:U.",
"Consider the operator $(\\nabla _x,\\mathrm {D}(\\nabla _x))$ from the Hilbert space $\\mathrm {L}^2(\\pi )$ to $\\mathrm {L}^2(\\pi )^d$ endowed with the inner product defined by (REF ).",
"Then it holds for any $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ , the $\\mathrm {L}^2(\\pi )$ -adjoint of $\\partial _{x_i}$ is given for any $g \\in \\mathrm {C}^{1}_{\\mathrm {poly}}(\\mathsf {X})$ by $\\partial _{x_i}^{\\star } g = - \\partial _{x_i} g + g \\partial _{x_i} U \\;;$ the $\\mathrm {L}^2(\\pi )$ -adjoint of $\\nabla _x$ is given for any $G \\in \\mathrm {C}^1_{\\mathrm {poly}}(\\mathsf {X},\\mathbb {R}^d)$ by $\\nabla _x^{\\star }G = -\\operatorname{div}_xG + \\nabla _x U^\\top G \\;.$ Remark Note that lem:adjoint-nax implies that for any $g \\in \\mathrm {C}^{2}_{\\mathrm {poly}}(\\mathsf {X})$ and $G \\in \\mathrm {C}^{2}_{\\mathrm {poly}}(\\mathsf {X},\\mathbb {R}^d)$ , we have $\\nabla _x^{\\star } \\nabla _xg =-\\Delta _x g +\\nabla _xU^\\top \\nabla _xg \\text{ and }\\nabla _x\\nabla _x^{\\star } G =\\nabla _x^{\\star }\\nabla _xG + \\nabla _x^{2}U G \\;,$ where we have defined $\\nabla _x^{\\star }\\nabla _xG \\in \\mathrm {C}_{\\mathrm {poly}}(\\mathsf {E},\\mathbb {R}^d)$ for any $(x,v) \\in \\mathsf {E}$ and $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ by $\\lbrace \\nabla _x^{\\star }\\nabla _xG(x,v)\\rbrace _i = \\nabla _x^{\\star } \\partial _{x_i} G(x,v) = \\sum _{j=1}^d - \\partial _{x_j,x_i} G_j(x,v) + \\partial _{x_j}U(x) \\partial _{x_i}G(x,v) \\;.$ The proof just follows by integration by parts.",
"Proposition Let $m >0$ and assume as:U.",
"Then for any $f \\in \\mathrm {C}_{\\operatorname{b}}^2(\\mathsf {E})$ , ${ \\nabla _x^2 (m\\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf } \\le \\kappa _1{\\Pi _vf} \\quad {\\rm where} \\quad \\kappa _1= (1 + c_1/(2m))^{{{1}{2}}} \\;.$ Let $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ and consider $u =(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf$ .",
"By [48], $u \\in \\mathrm {C}^{3}_{\\mathrm {poly}}(\\mathsf {X})$ .",
"Therefore we obtain by (REF ), (REF ) and the fact that $U \\in \\mathrm {C}^3_{\\mathrm {poly}}(\\mathsf {X})$ using as:U, ${ \\nabla _x^2 u }^2 = \\langle \\nabla _x^2 u, \\nabla _x^2 u \\rangle _2 = \\langle \\nabla _xu, ( \\nabla _x^{\\star } \\nabla _x)[ \\nabla _xu] \\rangle _2= \\langle \\nabla _xu, (\\nabla _x\\nabla _x^{\\star })[ \\nabla _xu] - \\nabla _x^2 U \\nabla _xu \\rangle _2\\\\= {\\nabla _x^{\\star } \\nabla _xu}^2 -\\langle \\nabla _xu, \\nabla _x^2 U \\nabla _xu \\rangle _2 \\;.$ From the definition of $u$ , using coro:bound regularization and as:U-REF we conclude that ${\\nabla _x^2 u}^2 \\le { \\Pi _vf }^2 + c_1 {\\nabla _xu}^2\\le {f }^2 + c_1 {\\Pi _vf }^2 /(2m) \\;.$ In order to bound terms of the form $\\Vert F_k^\\top \\nabla _xu \\Vert $ in app:bound RT RS we need the following Lemma which is a quantitative version of [21].",
"Consider the function $W : \\mathbb {R}^d \\rightarrow \\mathbb {R}_+$ defined for any $x \\in \\mathbb {R}^d$ by $W(x) = \\left\\lbrace 1+\\left|\\nabla _xU(x) \\right|^2 \\right\\rbrace ^{{{1}{2}}} \\;.$ Lemma ([21]) Assume as:U.",
"Then for any $\\varphi \\in \\mathrm {D}(\\nabla _x)$ , ${ \\nabla _x\\varphi } \\ge \\left[ 4\\left(1 + c_2 d^{1+\\varpi } /(4C_{\\operatorname{P}}^2) \\right)^{{{1}{2}}} \\right]^{-1} { \\varphi \\nabla _xU } \\;,$ where $c_2$ and $C_{\\operatorname{P}}$ are defined in (REF ) and (REF ) respectively.",
"As a corollary, it holds for any $\\varphi \\in \\mathrm {D}(\\nabla _x)$ , ${ \\nabla _x\\varphi } \\ge \\kappa _2{ \\varphi W } \\;, {\\rm where \\, } \\kappa _2^{-1} = \\left( C_{\\operatorname{P}}^{-2} + 16(1 + c_2 d^{1+\\varpi } /(4C_{\\operatorname{P}}^2)) \\right)^{{{1}{2}}} \\\\ = C_{\\operatorname{P}}^{-1} \\left(1+4 c_2 d^{1+\\varpi } + 16 C_{\\operatorname{P}}^2 \\right)^{{{1}{2}}} \\ge C_{\\operatorname{P}}^{-1} \\;.$ Note that we only need to consider $\\varphi \\in \\mathrm {C}^{\\infty }_{\\mathrm {c}}(\\mathsf {X})$ since $\\mathrm {C}^{\\infty }_{\\mathrm {c}}(\\mathsf {X})$ is a core for $(\\nabla _x,\\mathrm {D}(\\nabla _x))$ .",
"First since $\\nabla _xU \\in \\mathrm {L}^2(\\mu )$ , for any $\\varepsilon > 0$ , we get $2 \\left\\langle \\varphi \\nabla _xU,\\nabla _x\\varphi \\right\\rangle _2 \\le \\varepsilon ^{-1}{\\nabla _x\\varphi }^2 + \\varepsilon {\\varphi \\nabla _xU}^2 \\;.$ We then bound from below the left-hand side.",
"Using the carré du champ identity, i.e.",
"for any $f,g \\in \\mathrm {C}_{\\mathrm {poly}}^{2}(\\mathsf {X})$ , $\\left\\langle \\nabla _xf,\\nabla _xg \\right\\rangle _2 = \\left\\langle \\nabla _xU^{\\top }\\nabla _xf - \\Delta _x f,g \\right\\rangle _2$ , we get using that $\\nabla _x[\\varphi ^2] = 2 \\varphi \\nabla _x\\varphi $ , $2 \\left\\langle \\varphi \\nabla _xU, \\nabla _x\\varphi \\right\\rangle _2 = \\left\\langle \\nabla _x[\\varphi ^2], \\nabla _xU \\right\\rangle _2 = { \\varphi \\nabla _xU }^2 - \\left\\langle \\varphi ^2, \\Delta _x U \\right\\rangle _2 \\;.$ By (REF ) and (REF ), we obtain $2 \\left\\langle \\varphi \\nabla _xU, \\nabla _x\\varphi \\right\\rangle _2 \\ge { \\varphi \\nabla _xU }^2/2 - c_2 d^{1+\\varpi } {\\varphi }^2 \\ge { \\varphi \\nabla _xU }^2/2 - (c_2 d^{1+\\varpi } /C_{\\operatorname{P}}^2) {\\nabla _x\\varphi }^2 \\;.$ From this result and (REF ), it follows that ${ \\varphi \\nabla _xU }^2/2 - (c_2 d^{1+\\varpi } /C_{\\operatorname{P}}^2) {\\nabla _x\\varphi }^2 \\le \\varepsilon ^{-1}{\\nabla _x\\varphi }^2 + \\varepsilon {\\varphi \\nabla _xU}^2 \\;.$ Rearranging terms and setting $\\varepsilon = 1/4$ completes the proof.",
"The last statement is a direct consequence of the first one using the definition of $W$ in (REF ).",
"Putting this with Proposition , this implies the following.",
"Corollary Let $m >0$ and assume as:U and as:Fk.",
"For any $f \\in {\\mathrm {L}^2(\\mu )}$ and $k \\in \\lbrace 1, \\ldots , K \\rbrace $ , we have ${ F_k^\\top \\lbrace \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf \\rbrace } \\le 2^{{{1}{2}}} a_k { W \\lbrace \\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf \\rbrace } \\le \\frac{2^{{{1}{2}}} a_k \\kappa _1}{\\kappa _2} { \\Pi _vf} \\;,$ where $a_k$ , $W$ , $\\kappa _1$ and $\\kappa _2$ are defined by (REF ), (REF ), (REF ) and (REF ) respectively.",
"Note first that since $\\nabla _x(m \\operatorname{Id}+\\nabla _x^{\\star } \\nabla _x)^{-1}$ is a bounded operator by coro:bound regularization, it is sufficient by density to show this result for $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ .",
"Let $f \\in \\mathrm {C}_{\\operatorname{b}}^{2}(\\mathsf {E})$ and $u = (m +\\nabla _x^{\\star } \\nabla _x)^{-1} \\Pi _vf$ .",
"By [48], $u \\in \\mathrm {C}_{\\mathrm {poly}}^3(\\mathsf {X})$ .",
"Second since for any $t,s \\ge 0, \\, s + t \\le 2^{{{1}{2}}} \\sqrt{s^2 + t^2}$ , as:Fk-REF implies for any $x \\in \\mathsf {X}$ , $|F_k| (x) \\le a_k (1 + |\\nabla _xU|(x)) \\le 2^{{{1}{2}}}a_k W(x) \\;.$ Therefore using lemma:dms 6 and prop:bound hessian successively, we obtain ${ F_k^\\top \\nabla _xu } \\le { \\, |F_k| \\, \\nabla _xu \\, } \\le 2^{{{1}{2}}} a_k { W \\nabla _xu } = 2^{{{1}{2}}} a_k \\left( \\sum _{i=1}^d { W \\partial _{x_i} u }^2 \\right)^{1/2} \\\\ \\le (2^{{{1}{2}}} a_k / \\kappa _2) \\left( \\sum _{i=1}^d { \\nabla _x\\left[ \\partial _{x_i} u \\right] }^2 \\right)^{1/2} = (2^{{{1}{2}}} a_k / \\kappa _2) { \\nabla _x^2 u } \\le (2^{{{1}{2}}} a_k \\kappa _1/ \\kappa _2) { \\Pi _vf } \\;.$"
],
[
"Radial distributions",
"The following gathers standard results on spherically symmetric distributions on $\\mathbb {R}^d$ for which we could not find a single reference.",
"In particular we establish that as:radial-REF and conditions required in Lemma are satisfied in this scenario.",
"Lemma Let $d\\ge 2$ .",
"Assume $\\nu $ is the uniform distribution on the unit hypersphere $\\mathbb {S}^{d-1}$ , then for $i,j,k,l\\in \\lbrace 1,\\ldots , d\\rbrace $ such that $\\operatorname{card}(\\lbrace i,j,k,l\\rbrace )>2$ , we have $\\int _{\\mathbb {S}^{d-1}} v_{i}v_{j}v_{k}v_{l}\\, \\mathrm {d}\\nu (v)=0$ , otherwise, $m_{2}=\\frac{1}{d} \\;, \\qquad m_{2,2}=\\int _{\\mathbb {S}^{d-1}} v_{1}^2v_{2}^2\\, \\mathrm {d}\\nu (v)=\\frac{1}{d(d+2)}\\text{ and } \\qquad m_{4}=\\frac{1}{3}\\int _{\\mathbb {S}^{d-1}} v_{1}^{4} \\, \\mathrm {d}\\nu (v)=\\frac{1}{d(d+2)} \\;.$ For any spherically symmetric distribution $\\nu $ i.e.",
"corresponding to random variables $V=B^{{{1}{2}}} W$ for $W$ uniformly distributed on the unit hypersphere $\\mathbb {S}^{d-1}$ and $B$ a non-negative random variable independent of $w$ and of first and second order moments $\\gamma _1$ and $\\gamma _2$ respectively, for $i,j,k,l\\in \\lbrace 1,\\ldots , d\\rbrace $ such that $\\operatorname{card}(\\lbrace i,j,k,l\\rbrace )>2$ , we have $\\int _{\\mathbb {R}^d} v_{i}v_{j}v_{k}v_{l}\\, \\mathrm {d}\\nu (v)=0$ , otherwise, $m_{2}=\\frac{\\gamma _1}{d}\\;, \\qquad m_{2,2}=\\frac{\\gamma _2}{d(d+2)}\\text{ and } \\qquad m_{4}=\\frac{\\gamma _2}{d(d+2)} \\;.$ Remark Naturally the zero-mean $d$ -dimensional Gaussian distribution on $\\mathbb {R}^d$ with covariance matrix $\\operatorname{I}_d$ .",
"corresponds to $B$ distributed according to $\\chi ^{2}(d)$ , in which case $m_4=m_{2,2}=m_2^2$ .",
"We use the polar parametrization of the multivariate normal distribution.",
"Let $v(\\phi )=\\big (\\cos \\phi _{1},\\sin \\phi _{1}\\cos \\phi _{2},\\ldots ,\\cos (\\phi _{k})\\prod _{i=1}^{k-1}\\sin (\\phi _{i}),\\ldots ,\\prod _{i=1}^{d-1}\\sin (\\phi _{i})\\big ),$ $\\phi \\in [0,]^{d-2}\\times [0,2]$ .",
"The probability distribution for $\\phi $ ensuring uniformity of $v(\\phi )$ on the surface of the $d$ -sphere has density $f_{\\mathbb {S}}(\\phi )\\propto \\prod _{i=1}^{d-2}\\sin ^{d-i-1}(\\phi _{i}) \\mathbb {1}_{ [0,]^{d-2}\\times [0,2]} (\\phi ) \\;,$ with respect to the Lebesgue measure on $\\mathbb {R}^{d-1}$ .",
"Let $\\Phi $ be random variable with distribution $f_{\\mathbb {S}}$ .",
"Further let $B\\sim \\chi ^{2}(d)$ be independent of $\\Phi $ then it is standard knowledge that $W=B^{{{1}{2}}}v(\\Phi )$ follows the zero-mean $d$ -dimensional Gaussian distribution on $\\mathbb {R}^d$ with covariance matrix $\\operatorname{I}_d$ .",
"Therefore, by construction, $\\mathbb {E}\\big [W_{i}W_{j}W_{k}W_{l}\\big ]=\\mathbb {E}\\big [B^{2}v_{i}(\\Phi )v_{j}(\\Phi )v_{k}(\\Phi )v_{l}(\\Phi )\\big ]=\\mathbb {E}\\big [B^{2}\\big ]\\mathbb {E}\\big [v_{i}(\\Phi ) v_{j}(\\Phi )v_{k}(\\Phi ) v_{l}(\\Phi )\\big ]\\\\=d(d+2)\\mathbb {E}\\big [v_{i}(\\Phi ) v_{j}(\\Phi ) v_{k}(\\Phi ) v_{l} (\\Phi )\\big ] \\;,$ and the latter term vanishes when the leftmost term does.",
"We also deduce that $\\mathbb {E}\\big [W_{1}^{2}\\big ]\\mathbb {E}\\big [W_{2}^{2}\\big ] =\\mathbb {E}\\big [W_{1}^{2}W_{2}^{2}\\big ]=d(d+2)\\mathbb {E}\\big [v_{1}^{2} (\\Phi ) v_{2}^{2} (\\Phi )\\big ] \\;,$ from which we obtain $\\mathbb {E}\\big [v_{1}^{2}(\\Phi ) v_{2}^{2} (\\Phi )\\big ]$ .",
"Similarly using properties of the moments of the normal distribution, $3\\mathbb {E}\\big [W_{1}^{2}\\big ]^{2} =\\mathbb {E}\\big [W_{1}^{4}\\big ]=d(d+2)\\mathbb {E}\\big [v_{1}^{4}(\\Phi )\\big ] \\;,$ leading to the expression for $\\mathbb {E}\\big [v_{1}^{4} (\\Phi )\\big ]$ .",
"The last statement is straightforward."
],
[
"Expectation of quadratic forms of the velocity",
"This section provides expressions for second order moments of quadratic forms of $v$ for a large class of distributions for which we could not find adequate references.",
"Lemma Let $M \\in \\mathbb {R}^{d \\times d}$ be a symmetric matrix, $c \\in \\mathbb {R}$ and assume the distribution $\\nu $ of $v$ is such that for any bounded and measurable function $f :\\mathbb {R}^2 \\rightarrow \\mathbb {R}$ , $i,j \\in \\lbrace 1,\\ldots , d \\rbrace $ such that $i \\ne j$ , $\\int f(v_i,v_j) \\, \\mathrm {d}\\nu (v) = \\int f(v_1,v_2) \\, \\mathrm {d}\\nu (v)$ for $i,j,k,l\\in \\lbrace 1,\\ldots ,d\\rbrace $ , we have $\\int v_{i}v_{j}v_{k}v_{l}\\, \\mathrm {d}\\nu (v)=0$ whenever $\\operatorname{card}(\\lbrace i,j,k,l\\rbrace )>2$ .",
"Then $\\left\\Vert v^\\top M v - c \\right\\Vert _\\nu ^2 = 3(m_4 - m_{2,2}) \\mathrm {Tr}(M \\odot M) + \\left(m_2 \\mathrm {Tr}(M) - c \\right)^2 + 2 m_{2,2} \\mathrm {Tr}(M^2),$ where $\\odot $ denotes the Hadamard product.",
"Using that $M$ is symmetric, and the expectation symbol for expectations with respect to $\\nu $ , $\\begin{aligned}\\mathbb {E}\\left[ \\left(\\sum _{i,j=1}^d M_{ij} v_i v_j - c \\right)^2 \\right] &= \\sum _{i,j,k,\\ell =1}^d M_{ij} M_{k\\ell } \\mathbb {E}[v_i v_j v_k v_\\ell ] - 2 c \\sum _{i,j=1}^d M_{ij} \\mathbb {E}[v_i v_j] + c^2\\end{aligned}$ where $\\begin{aligned}\\sum _{i,j,k,\\ell =1}^d M_{ij} M_{k\\ell } \\mathbb {E}[v_i v_j v_k v_\\ell ] &= 3 m_4 \\sum _{i=1}^d M_{ii}^2 + m_{2,2} \\sum _{i\\ne j} M_{ii} M_{jj} + 2 m_{2,2} \\sum _{i\\ne j} M_{ij}^2 \\\\&= (3 m_4 - 3 m_{2,2}) \\sum _{i=1}^d M_{ii}^2 + m_{2,2} \\sum _{i,j=1}^d \\left( M_{ii} M_{jj} + 2 M_{ij}^2 \\right) \\\\&= (3 m_4 - 3 m_{2,2}) \\mathrm {Tr}(M \\odot M) + m_{2,2} \\left( \\mathrm {Tr}(M)^2 + 2 \\mathrm {Tr}(M^2) \\right).\\end{aligned}$ Therefore $\\begin{aligned}\\mathbb {E}\\left[ \\left(\\sum _{i,j=1}^d M_{ij} v_i v_j - c \\right)^2 \\right] &= (3 m_4 - 3 m_{2,2}) \\mathrm {Tr}(M \\odot M) + m_{2,2} \\mathrm {Tr}(M)^2 + 2 m_{2,2} \\mathrm {Tr}(M^2) \\\\&\\quad - 2 c m_2 \\mathrm {Tr}(M) + c^2,\\end{aligned}$ which implies the desired result.",
"Corollary Given a symmetric matrix $M \\in \\mathbb {R}^{d \\times d}$ and a constant $c \\in \\mathbb {R}$ , $\\left\\Vert v^\\top M v - m_2 \\mathrm {Tr}(M) \\right\\Vert _\\nu \\le \\sqrt{2m_{2,2} + 3 (m_4 - m_{2,2})_+} | M |.$"
],
[
"Examples of potentials",
"Lemma Assume that the potential $U$ is defined for any $x \\in \\mathsf {X}$ by $U(x)=\\sum _{i=1}^{d}\\big (1+x_{i}^{2}\\big )^{\\beta }/2$ , for $\\beta \\ge 1$ .",
"Then $U$ is strongly convex and there exists $c_{2}>0$ , dependent on $\\beta $ only, such that (REF ) is satisfied with $\\varpi = 0$ .",
"We have for $i,j\\in \\lbrace 1,\\ldots ,d\\rbrace $ and $x \\in \\mathsf {X}$ , $\\big [\\nabla _{x}U(x)\\big ]_{i}=\\beta x_{i}\\big (1+x_{i}^{2}\\big )^{\\beta -1}\\text{ and }\\big [\\nabla _{x}^{2}U(x)\\big ]_{i,j}=\\beta [1+(2\\beta -1)x_{i}^{2}]\\big (1+x_{i}^{2}\\big )^{\\beta -2} \\delta _{i,j} \\;,$ leading to $\\nabla _x^2 U(x) \\succeq \\beta \\operatorname{I}_d$ , and the strong convexity follows.",
"Using that $\\beta \\ge 1$ and for any $s \\ge 0$ and $c >0$ , $(1+s^2)^{\\beta -2} s^{2} \\le (1+c^2)^{\\beta -2}c^2\\mathbb {1}_{\\left[0,c\\right]}(s)+(1+s^2)^{2\\beta -2} s^{2}/(1+c^2)^{\\beta }\\mathbb {1}_{\\left(c,+\\infty \\right)}(s)$ and $ (1+s^2)^{\\beta -2} \\le \\lbrace 1\\vee (1+c^2)^{\\beta -2}\\rbrace \\mathbb {1}_{\\left[0,c\\right]}(s) + (1+s^2)^{2\\beta -2} (s/c)^{2} \\mathbb {1}_{\\left(c,+\\infty \\right)}(s)$ , we get for any $x \\in \\mathsf {X}$ , $\\Delta _x U(x) = \\operatorname{Tr}(\\nabla _x^2U(x)) = \\beta \\sum _{i=1}^d[1+(2\\beta -1)x_{i}^{2}]\\big (1+x_{i}^{2}\\big )^{\\beta -2} \\\\ \\le \\beta d \\left[ \\lbrace 1\\vee (1+c^2)^{\\beta -2}\\rbrace + (2\\beta -1)(1+c^2)^{\\beta -2}c^2 \\right]+ \\beta ^{-1}\\left|\\nabla _x U(x) \\right|^2\\left[ c^{-2} + (2\\beta -1)(1+c^2)^{-\\beta } \\right] \\;,$ which with $c \\ge (2 \\beta ^{-{{1}{2}}})\\vee 2^{1/\\beta }$ completes the proof.",
"Lemma Assume that the potential $U$ is defined for any $x \\in \\mathsf {X}$ by $U(x)=(1+\\vert x\\vert ^{2})^{\\beta }$ with $\\beta \\ge 1$ .",
"Then $U$ is strongly convex and there exists $c_{2}>0$ , dependent on $\\beta $ only, such that (REF ) is satisfied with $\\varpi = 1-1/\\beta $ .",
"First, we have that $\\nabla _{x}U(x) =2\\beta (1+\\vert x\\vert ^{2})^{\\beta -1}x=2\\beta U(x){}^{1-1/\\beta }x,$ and $\\nabla _{x}^{2}U(x)=2\\beta \\left[(1-1/\\beta )U^{-1/\\beta }(x) \\nabla _{x}U(x) x^{\\top } +U^{1-1/\\beta }(x) \\operatorname{I}_d\\right] \\;.$ As a result, and since $\\beta \\ge 1$ , $(1-\\beta ^{-1})U^{-1/\\beta }(x)\\nabla _{x}U(x) x^\\top =2\\beta U(x){}^{1-2/\\beta }x x^\\top \\succeq 0, \\qquad \\;, U^{1-1/\\beta }(x)I \\succeq \\operatorname{I}_d\\;,$ from which we conclude that for any $x\\in \\mathsf {X}$ , $\\nabla _{x}^{2}U(x)\\succeq 2\\beta \\operatorname{I}_d$ .",
"It remains to show that (REF ) holds.",
"First we have for any $x \\in \\mathsf {X}$ , $\\operatorname{Tr}\\left(\\nabla _{x}^{2}U(x)\\right) & =2(\\beta -1) U^{-1/\\beta } (x) x^\\top \\nabla _xU(x)+2 \\beta d\\,U^{1-1/\\beta }(x)\\\\& \\le 2(\\beta -1)\\vert \\nabla _{x}U(x)\\vert \\frac{\\vert x\\vert }{1+\\vert x\\vert ^{2}} +2 \\beta d\\,U^{1-1/\\beta }(x) \\;.$ Using that for any $s \\ge 0$ and $a >0$ , $2s \\le a^{-2}+(as)^2$ , $(1+s^2)^{\\beta -1} \\le (1+(2d / \\beta )^{1/\\beta })^{\\beta -1} \\mathbb {1}_{[0,(2d / \\beta )^{1/\\beta }]}(s^2) + (2d / \\beta )^{-1} s^{2 \\beta }(1+s^2)^{\\beta -1} \\mathbb {1}_{((2d / \\beta )^{1/\\beta },+\\infty )}(s^2) \\le (1+(2d / \\beta )^{1/\\beta })^{\\beta -1} \\mathbb {1}_{[0,(2d / \\beta )^{1/\\beta }]}(s^2) + (2d / \\beta )^{-1} s^{2}(1+s^2)^{2\\beta -2} \\mathbb {1}_{((2d / \\beta )^{1/\\beta },+\\infty )}(s^2)$ , (REF )-(REF ), we get for any $x \\in \\mathsf {X}$ , $\\operatorname{Tr}\\left(\\nabla _{x}^{2}U(x)\\right) & \\le 2(\\beta -1) \\vert \\nabla _{x}U(x)\\vert \\vert x\\vert (1+\\vert x\\vert ^{2})^{-1} +2 \\beta d\\,U^{1-1/\\beta }(x)\\\\& \\le (\\beta -1) (4\\beta +\\left|\\nabla _x U(x) \\right|^2/(4\\beta )) +2 \\beta d [(1+(2d/\\beta )^{1/\\beta })^{\\beta -1} + \\left|\\nabla _x U(x) \\right|^2/(8 d \\beta )] \\\\& \\le 4(\\beta -1) \\beta +2^{\\beta -1}\\beta d(1+(2d/\\beta )^{1-1/\\beta }) + \\left|\\nabla _x U(x) \\right|^2/2 \\;,$ where we used in the last step which completes the proof, that $(a +b)^{\\beta -1} \\le 2^{\\beta -2}(a^{\\beta -1}+ b^{\\beta -1})$ for any $a,b \\ge 0$ , applying Hölder inequality, since $\\beta \\ge 1$ ."
],
[
"Acknowledgments",
"JR would like to thank Pierre Monmarché for showing him how ZZ and BPS fall under a general framework.",
"CA acknowledges support from EPSRC “Intractable Likelihood: New Challenges from Modern Applications (ILike)” (EP/K014463/1).",
"All the authors acknowledge the support of the Institute for Statistical Science in Bristol.",
"AD acknowledges support from the Chaire BayeScale “P.",
"Laffitte”."
]
] | 1808.08592 |
[
[
"Directed flow in an extended multiphase transport model"
],
[
"Abstract We have studied the rapidity-odd directed flow in $^{197}$Au+$^{197}$Au collisions in the beam energy range from $\\sqrt{s_{NN}}$ = 7.7 to 39 GeV within the framework of an extended multiphase transport model with both partonic and hadronic mean-field potentials incorporated.",
"Effects of the partonic scatterings, mean-field potentials, hadronization, and hadronic evolution on the directed flow are investigated, and it is found that the final directed flow is mostly sensitive to the partonic scatterings and the hadronization mechanism.",
"Our study shows that a negative slope of the proton directed flow does not necessarily need the equation of state with a first-order phase transition."
],
[
"Introduction",
"Understanding the properties of the quark-gluon plasma as well as the hadron-quark phase transition is one of the main purposes of relativistic heavy-ion collision experiments.",
"The directed flow $v_1$ , especially its rapidity-odd component to be discussed in the present study, is an important probe characterizing the dynamics in these collisions.",
"Both hydrodynamic [1] and transport model [2] studies have shown that the baryon $v_1(y)$ in the midrapidity region ($y\\sim 0$ ) is sensitive to the equation of state (EoS) of the produced matter.",
"With the increasing collision energies, these calculations predicted that the slope of $v_1(y)$ near midrapidity region changes from positive to negative [3], [4], [5], and argued that this is the result of a soft EoS due to the first-order hadron-quark phase transition.",
"The recent directed flow results from RHIC beam energy scan (BES) program [6], [7] seem to support this argument, where the slope of the proton directed flow changes sign from positive to negative between $\\sqrt{s_{NN}}$ = 7.7 and 11.5 GeV, while the slope of the net proton directed flow changes sign twice between $\\sqrt{s_{NN}}$ = 11.5 and 39 GeV, and has a minimum between $\\sqrt{s_{NN}}$ = 11.5 and 19.6 GeV.",
"Considerable efforts have been devoted to this topic with various approaches, e.g., a hybrid approach with the fluid dynamics model for the partonic phase and the ultra-relativistic quantum molecular dynamics model for the hadronic phase [8], the parton-hadron-string-dynamics model and a 3-fluid hydrodynamics approach [9], and a pure hadronic transport approach but with the collision term modified to mimic the softness of the EoS [10].",
"However, none of them have described the experimental data of the directed flow at various collision energies very satisfactorily.",
"Obviously, so far the studies have shown that the relation between the $v_1(y)$ slope and the EoS is not as simple as expected, and the former is also sensitive to other factors, which needs to be further investigated in detail before a definite conclusion can be drawn.",
"For this purpose, we investigate the directed flow in relativistic heavy-ion collisions based on the framework of an extended multiphase transport (AMPT) model [11], which was developed from the original AMPT model [12] by incorporating the partonic mean-field potential based on the 3-flavor Nambu-Jona-Lasinio (NJL) model [13], [14] and the hadronic mean-field potential based the relativistic mean-field model and the chiral effective field theory [15].",
"This model has been used to study the elliptic flow splitting between particles and their antiparticles due to their different mean-field potentials [14], [11], and thus it is of great interest to see how the mean-field potentials affect their directed flows.",
"Furthermore, this model provides the possibility to investigate in detail the effects from the partonic scatterings, hadronization, and the hadronic interaction on the directed flow.",
"We found that the mean-field potential, which is related to the EoS through the energy density functional, has only moderate effect on the directed flow, while the partonic scatterings and the hadronization mechanism dominate the final directed flow.",
"The rest of the paper is organized as follows.",
"Section provides a brief description of the structure of the extended AMPT model.",
"The detailed analysis and discussions of the directed flow results are given in Sec. .",
"A summary and final remark is given in Sec.",
"."
],
[
" An extended AMPT model",
"The initial momentum distribution of partons in the extended AMPT model is generated by melting hadrons from the heavy-ion jet interaction generator (HIJING) model [16], while the spatial distribution of these partons are modified since they are expected to be more expanded in the beam direction at the RHIC-BES energies, compared to the treatment in the original AMPT model for ultra-relativistic heavy-ion collisions.",
"Similar to Ref.",
"[17], we sample the longitudinal coordinate of initial partons uniformly within $(-lm_N/\\sqrt{s_{NN}}, lm_N/\\sqrt{s_{NN}})$ , where $l=14$ fm is approximately the diameter of the Au nucleus, and $m_N=0.938$ GeV is the nucleon mass.",
"A more realistic treatment of the finite thickness effect can be found in Ref. [18].",
"The evolution of the partonic phase is described by a 3-flavor NJL transport model, with a Lagrangian given by [13], [14] $\\mathcal {L_{NJL}} &=& \\bar{q}(i\\lnot {\\partial }-M)q+\\frac{G_S}{2}\\sum _{a=0}^{8}\\bigg [(\\bar{q}\\lambda ^aq)^2+(\\bar{q}i\\gamma _5\\lambda ^aq)^2\\bigg ]\\nonumber \\\\&-&\\frac{G_V}{2}\\sum _{a=0}^{8}\\bigg [(\\bar{q}\\gamma _\\mu \\lambda ^aq)^2+(\\bar{q}\\gamma _\\mu \\gamma _5\\lambda ^aq)^2\\bigg ]\\nonumber \\\\&-&K\\bigg [{\\rm det}_f\\bigg (\\bar{q}(1+\\gamma _5)q\\bigg )+{\\rm det}_f\\bigg (\\bar{q}(1-\\gamma _5)q\\bigg )\\bigg ],$ where $q=(u, d, s)^T$ is the quark fields, $M={\\rm diag}(m_u, m_d, m_s)$ is the current quark mass matrix in flavor space, and $\\lambda ^{a}$ is the Gell-Mann matrices in $SU(3)$ flavor space with $\\lambda ^0=\\sqrt{2/3}I$ .",
"$G_S$ and $G_V$ are the strength of the scalar and vector coupling, respectively.",
"The last term in Eq.",
"(REF ), with det$_f$ denoting the determinant in the flavor space, is the Kobayashi-Maskawa-t'Hooft (KMT) interaction [19].",
"In the present study, we employ the parameters of the current quark mass $m_u=m_d=3.6$ MeV and $m_s=87$ MeV, and the values of coupling constants $G_S$ and $K$ from $G_S\\Lambda ^2=3.6$ and $K\\Lambda ^{5}=8.9$ , with the cutoff value in the momentum integral $\\Lambda =750$ MeV given in Refs.",
"[20], [21].",
"As in the previous studies, we define $R_V=G_V/G_S$ as the relative strength of the vector coupling.",
"From the mean-field approximation and some algebras based on the finite-temperature field theory, we can extract the single-quark Hamiltonian and the thermodynamic properties of quark matter (see more details in ).",
"The scalar mean-field potential enlarges the in-medium quark mass through the quark condensate, and it is the same for quarks and antiquarks and attractive in the non-relativistic reduction.",
"The time component of the vector potential for positive $G_V$ value is repulsive for quarks and attractive for antiquarks, while its space component has the opposite effect.",
"A constant and isotropic cross section of 3 mb is used for the parton elastic scattering process, and it is the same for all kinds of partons.",
"The test-particle method [22], [23] is used to calculate the average phase-space distribution function and thus the mean-field potential, from averaging over parallel events at the same impact parameter.",
"A mix-event spatial coalescence algorithm is used to describe the hadronization, which allows quarks and antiquarks in one event to coalesce with nearest quarks or antiquarks from all parallel events at the end of the partonic phase, when the central energy density becomes lower than about $0.8$ GeV/fm$^{3}$ .",
"Combinations from three nearest quarks (antiquarks) are chosen to form baryons (antibaryons), before nearest quark and antiquark pairs from the rest partons are chosen to form mesons, and the species of formed hadrons are determined by both the flavors and the invariant mass of their valence quarks and/or antiquarks.",
"The hadronization scheme produces not only ground-state hadrons but also excited-state resonances with a finite mass width.",
"For more details about the hadron production in the AMPT model, we refer the reader to Ref. [12].",
"An additional formation time of about 0.7 fm/c is used for all hadrons.",
"After the formation of initial hadrons, a relativistic transport model (ART) [24] with various elastic, inelastic, and decay channels is used to describe the evolution of the hadronic phase, where the mean-field potentials for hadrons are further implemented [15].",
"The mean-field potentials for nucleons and antinucleons are from the relativistic mean-field theory based on G-parity invariance [25].",
"The mean-field potentials for kaons and antikaons are from the chiral effective Lagrangian [26].",
"The $s$ -wave mean-field potentials for pions are from the chiral perturbation theory up to the two-loop order [27].",
"For more details about this model, we refer the reader to Ref.",
"[11]."
],
[
" Results and analysis",
"In the present study, we employ the extended multiphase transport model to investigate the directed flow in midcentral ($c = 10 - 40\\%$ ) $^{197}$ Au+$^{197}$ Au collisions at the typical RHIC-BES energies, i.e., $\\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27, and 39 GeV.",
"With the total inelastic cross section $\\sigma _{in}$ of about 705 fm$^2$ for Au+Au collisions, the corresponding impact parameters are about $\\text{b}=4.7 - 9.5$ fm, from the empirical relation $c=\\pi b^2/\\sigma _{in}$ [28].",
"From the analysis in , the partonic phase doesn't pass through the spinodal region, which corresponds to the softening of the EOS, at the above collision energies from the parameterization of the NJL model.",
"From the description of the hadronization process in Sec., there is no hadron-quark mixed phase, and thus no softness of the EoS during the phase transition, either.",
"In the following analysis, the directed flow is calculated by averaging the azimuthal angle $\\phi $ of particle momenta with respect to the theoretical reaction plane, i.e., $v_1=\\langle \\cos (\\phi ) \\rangle $ ."
],
[
"Directed flow in the partonic phase",
"To begin with the discussion on the directed flow in the partonic phase, we first show the directed flow of quarks and antiquarks in midcentral Au+Au collisions at $\\sqrt{s_{NN}}$ = 7.7 and 39 GeV at different time steps in Fig.",
"REF , with the parton scattering cross section 3 mb together the scalar potential but without the vector potential.",
"In the early stage of $t=0.5$ fm/c, the $v_1$ slopes of quarks and antiquarks at midrapidities at both collision energies are small positive values.",
"As the system evolves, the slope of the directed flow at $\\sqrt{s_{NN}}$ = 39 GeV changes sign at around 1.5 fm/c and then becomes saturated, while the slope at $\\sqrt{s_{NN}}$ = 7.7 GeV keeps growing to a maximum positive value.",
"The behavior of the directed flow at 39 GeV is similar to that observed in Ref.",
"[29], where the $v_1$ slope at midrapidities keeps going negative and then saturates in the later stage at 39 GeV due to repulsive partonic scatterings in the absence of the mean-field potential.",
"This behavior is due to the transfer of partons among different rapidity regions [29], i.e., partons contributing to the positive flow are scattered to large rapidities and only those contributing to the negative flow stay at midrapidities.",
"At 7.7 GeV, the scatterings are not strong enough for the parton transfer among different rapidity regions due to the lower parton density/pressure, and the slope of the directed flow keeps going positive.",
"Here we have already observed that the directed flow of partons changes sign with the increasing collision energy without a first-order phase transition.",
"The final parton directed flow is not expected to change by much if we vary the criterion for the end of the partonic evolution, since it is mostly saturated around $t=2.5$ fm/c.",
"Figure: (Color online) Directed flow v 1 v_1 of freeze-out quarks [(a), (c)] and antiquarks [(b), (d)] versus rapidity yy from the parton scattering cross sections of σ\\sigma = 1, 3, and 10 mb in midcentral Au+Au collisions at s NN \\sqrt{s_{NN}} = 7.7 [(a), (b)] and 39 GeV [(c), (d)].",
"Note the different scales for v 1 v_1 at 7.7 and 39 GeV.From our argument above, the partonic scatterings are important in determining the $v_1$ slope of freeze-out partons.",
"Figure REF displays the directed flow of final quarks and antiquarks from the parton scattering cross sections of 1, 3, and 10 mb together with the scalar potential but in the absence of the vector potential.",
"A larger parton scattering cross section is expected to enhance the interaction in both the early stage and the later stage, depending on the parton number density.",
"It is seen that the slope of the directed flow becomes more positive at 7.7 GeV but more negative at 39 GeV from a larger parton scattering cross section.",
"At 39 GeV, it is interesting to see that a small cross section of 1 mb leads to a positive slope of $v_1$ , while a larger cross section leads to a negative slope.",
"The partonic scatterings thus dominate the directed flow of final partons, or even change the slope sign especially at higher collision energies.",
"In addition, the stronger interaction from partonic scatterings reveals more explicitly the difference in the initial phase-space distributions between quarks and antiquarks, and thus their final directed flows.",
"Figure: (Color online) Directed flow v 1 v_1 of freeze-out quarks [(a), (c)] and antiquarks [(b), (d)] versus rapidity yy from only partonic scatterings (cascade), partonic scatterings together with the scalar potential (scalar), and partonic scatterings together with both scalar and vector potentials (scalar+vector) in midcentral Au+Au collisions at s NN \\sqrt{s_{NN}} = 7.7 [(a), (c)] and 39 GeV [(b), (d)].",
"Note the different scales for v 1 v_1 at 7.7 and 39 GeV.In order to understand the effects of the mean-field potential on the parton directed flow, we display in Fig.",
"REF the directed flow of final quarks and antiquarks with only partonic scatterings, partonic scatterings together with the scalar potential, and partonic scatterings together with both scalar and vector potentials with $R_V = 1.1$ at $\\sqrt{s_{NN}}$ = 7.7 GeV and 39 GeV.",
"It is found that neither the scalar potential nor the vector potential changes the slope sign of parton $v_1$ , and this shows that the mean-field potential has only moderate effects on the parton directed flow.",
"It is also seen that the scalar and the vector potential have different effects on the directed flows of quarks and antiquarks.",
"The attractive scalar potential leads to less negative slope of the directed flow for both quarks and antiquarks at 39 GeV.",
"At 7.7 GeV, it has small effects on the directed flow of quarks, but somehow enhances the directed flow of antiquarks, likely due to the enhancement of the scatterings.",
"It is seen that the vector potential has a larger effect on the directed flow at 39 GeV than at 7.7 GeV, and the effect can be seen at large rapidities for quarks and in the whole rapidity region for antiquarks.",
"Note that the scalar potential and the time component of the vector potential related to the quark number density are stronger in the early stage but weaker in the later stage, while the space component of the vector potential related to the quark flux density is expected to be weaker in the early stage but stronger in the later stage.",
"The results displayed here show that there are interplays between effects from the mean-field potentials and partonic scatterings, and the effect of the mean-field potential on the directed flow is different from that on the elliptic flow [14], [11] due to their different dynamic mechanisms."
],
[
"Directed flow in the hadronic phase",
"The directed flow of final partons discussed in previous subsections will be further modified in the hadronization process and the hadronic evolution, and this has been investigated in our previous study [29] in the absence of the mean-field potential.",
"Based on the framework of the extended AMPT model, we compared the directed flows of initial protons and antiprotons from the mix-event spatial coalescence approach as in the extended AMPT model and a dynamical coalescence approach [30], [31] in Fig.",
"REF .",
"In the dynamical coalescence approach, the probability to form a hadron is proportional to the quark Wigner function of that hadron, and partons that are close in phase space, i.e., both coordinate and momentum space, have a larger probability to form hadrons.",
"The dynamical coalescence approach has been successfully used to explain reasonable well spectra and elliptic flows of hadrons [30].",
"In the mix-event spatial coalescence approach as employed in the extended AMPT model, hadrons are formed by nearest combinations of partons in coordinate space in parallel events, and all the partons are force to be used up after hadronization.",
"The difference of the directed flows for protons and antiprotons from the two hadronization approaches is observed, especially for protons at 39 GeV where different slope signs are obtained.",
"As discussed in detail in Ref.",
"[29], a pure coalescence in momentum space always keeps the slope sign of the directed flow from final partons to initial hadrons, while there are competition effect from the coalescence in coordinate space, and this depends on the phase-space distribution in the freeze-out stage of the partonic evolution.",
"With the Gaussian width of the Wigner function fitted by the root-mean-square radius of the hadron, the dynamical coalescence approach is expected to describe more properly the competition effect of the coalescence in both the coordinate and momentum space.",
"Our finding that the hadron directed flow is sensitive to the hadronization scenario is consistent with that in Ref. [8].",
"So far the dynamical coalescence approach is limited to the study of the directed flow of initial hadrons without further hadronic evolution.",
"Incorporations of a more realistic hadronization criterion together with a proper hadronization scenario in the transport model need further investigations to give unambiguous results.",
"Figure: (Color online) Directed flow v 1 v_1 of final partons as well as initial and final protons and antiprotons versus rapidity yy in midcentral Au+Au collisions at s NN \\sqrt{s_{NN}} = 7.7, 11.5, 19.6, 27, and 39 GeV.The directed flows of final partons as well as the initial and final protons and antiprotons from the mix-event spatial coalescence and the hadronic evolution in midcentral Au+Au collisions at $\\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27, and 39 GeV are compared in Fig.",
"REF , where the solid lines are from a cubic fit of the rapidity dependence of the final directed flow, i.e., $v_1(y) = F_1 y + F_3y^3$ .",
"From the spatial coalescence approach, the $v_1$ slopes of initial protons are positive at all energies, although the directed flows of final quarks have negative slopes at higher collision energies.",
"On the other hand, the hadronic evolution with the hadronic mean-field potentials properly incorporated only slightly enhances the magnitude the directed flow, no matter whether its slope is positive or negative.",
"Figure: (Color online) Directed flow slope dv 1 /dy| y=0 dv_1/dy|_{y=0} of protons, antiprotons, and net protons versus collisions energy in midcentral Au+Au collisions with R V =0R_V = 0 (a) and R V =1.1R_V = 1.1 (b).",
"Results of star symbols are experimental data measured by the STAR Collaboration , and the lines are the directed flow slope from the dynamical coalescence approach.We summarize the energy dependence of the directed flow slopes at midrapidities for protons and antiprotons in Fig.",
"REF , and the experimental results from the STAR Collaboration [6] are also plotted in the same figure.",
"From the spatial coalescence at higher collision energies, the proton directed flows always have a positive slope, although the slope of freeze-out quark $v_1$ is negative at higher collision energies, as can be seen from Figs.",
"REF and REF .",
"The slope of the antiproton directed flow is positive at lower collision energies but rather negative at higher collision energies.",
"From a dynamical coalescence approach but in the absence of the hadronic evolution, the directed flows of both protons and antiprotons have a positive slope at lower collision energies but a negative slope at higher collision energies, reflecting the $v_1$ slope of quarks and antiquarks in the final stage of the partonic phase.",
"The hadronic evolution after the hadronization described by the dynamical coalescence approach, if properly incorporated, is expected to modify slightly the magnitude but not change the slope sign of the directed flow.",
"Results with the vector potential ($R_V=1.1$ ) show a similar behavior from the spatial coalescence approach, while it changes the relative slope of proton and antiproton $v_1$ from the dynamical coalescence approach.",
"In order to obtain a negative $v_1$ slope of protons or net protons, the coalescence in momentum space, such as that in the dynamical coalescence approach, should be taken into account, while the EoS with a first-order phase transition is not necessarily needed."
],
[
" Summary",
"We have studied in detail the effects of partonic scatterings as well as the mean-filed potential, hadronization, and hadronic evolution on the directed flow in relativistic heavy-ion collisions within the framework of the extended multiphase transport model.",
"As the system evolves, a non-monotonic behavior of the directed flow is observed at higher collision energies, as a result the later dynamics which changes the slope sign of the directed flow from positive to negative at midrapidity region.",
"The final directed flow of partons, particularly the sign of its slope at midrapidities, is dominated by the partonic scatterings especially at higher collision energies.",
"The mean-field potentials in the partonic phase, which can be attractive or repulsive, have only moderate effects on the parton directed flow.",
"The hadron directed flow is significantly affected by the hadronization mechanism, while it is only slightly affected by the hadronic evolution with the mean-field potentials for hadrons incorporated.",
"Based on this study, we found that a large parton scattering cross section and a coalescence approach in momentum space may result in a negative slope of the final directed flow.",
"Although the EoS with a first-order phase transition may lead to a negative slope of the directed flow, the present study shows that the negative slope of the directed flow does not necessarily need the softness of the EoS.",
"Before drawing a conclusion on the EoS and the order of the hadron-quark phase transition from the experimental data, deeper understandings on various stages in relativistic heavy-ion collisions are needed.",
"We thank Chen Zhong for maintaining the high-quality performance of the computer facility.",
"This work was supported by the Major State Basic Research Development Program (973 Program) of China under Contract Nos.",
"2015CB856904 and 2014CB845401, the National Natural Science Foundation of China under Grant Nos.",
"11475243 and 11421505, and the Shanghai Key Laboratory of Particle Physics and Cosmology under Grant No.",
"15DZ2272100."
],
[
"EFormulism of the NJL model The single-particle Hamiltonian for quarks (antiquarks) with flavor $i$ ($i=u, d, s$ ) from the NJL Lagrangian [Eq.",
"(REF )] is written as Here only the flavor-singlet state of the vector interaction is considered.",
"In this way, quarks and antiquarks are affected by the vector mean-field potential generated by partons of different flavors.",
"$H_i = \\sqrt{M_i^2+{p^*}^2} \\pm \\frac{2}{3}G_V \\rho ^0,$ where $M_i$ is the quark constituent mass, $\\vec{p^*} = \\vec{p} \\mp \\frac{2}{3}G_V \\vec{\\rho }$ is the real momentum of the particle with $\\vec{\\rho }$ being the space component of the vector density, and $\\rho ^0$ is the time component of the vector density.",
"The upper (lower) sign in the above equations are for quarks (antiquarks).",
"The quark constituent mass $M_i$ is given by the gap equation as $M_i &=&m_i-2G_S\\sigma _i+2K\\sigma _j\\sigma _k,$ where $\\sigma _i$ is the quark condensate expressed as $\\sigma _i&=&-2N_c\\int _0^\\Lambda \\frac{d^3p}{(2\\pi )^3}\\frac{M_i}{E_i}(1-f_i-\\bar{f}_i),$ where the factor $2N_c$ represents the spin and color degeneracy, $E_i=\\sqrt{{p^*}^2 +M_i^2}$ is the quark energy, and $f_i$ and $\\bar{f}_i$ are respectively the phase-space distributions of quarks and antiquarks.",
"They can be obtained by counting parton numbers in the local phase-space cell through the test-particle method, and in the thermodynamic limit they can be expressed as the Fermi-Dirac distributions, i.e., $f_i &=& \\frac{1}{1+e^{\\beta (E_i-\\tilde{\\mu }_i)}},\\\\\\bar{f_i} &=& \\frac{1}{1+e^{\\beta (E_i+\\tilde{\\mu }_i)}},$ where $\\beta =1/T$ represents the temperature, and the effective chemical potential $\\tilde{\\mu }_i$ is defined as $\\tilde{\\mu }_i&=&\\mu _i-\\frac{2}{3}G_V \\rho ^0.$ The 4-component net quark number density of the flavor $i$ can be calculated from $f_i$ and $\\bar{f}_i$ via $\\rho ^\\nu _i=2N_c\\int ^\\Lambda _0(f_i-\\bar{f_i})\\frac{p_i^\\nu }{E_i}\\frac{d^3p}{(2\\pi )^3},$ with $\\rho ^\\nu = \\rho ^\\nu _u + \\rho ^\\nu _d + \\rho ^\\nu _s$ being the total number density.",
"The time component ($\\nu =0$ ) of the above density is the net quark number density, while the space component ($\\nu =1,2,3$ ) is the net quark flux density.",
"As seen from Eq.",
"(REF ), the scalar mean-field potential is generated from the quark condensate.",
"The space and time components of the vector mean-field potential are $\\mp \\frac{2}{3} G_V \\vec{\\rho }$ and $\\pm \\frac{2}{3} G_V \\rho ^0$ terms in Eq.",
"(REF ), respectively.",
"From the mean-field approximation and some algebras based on the finite-temperature field theory, the thermodynamic potential $\\Omega _{NJL}$ of static quark matter at finite temperature and quark chemical potential can be expressed as $\\Omega _{NJL} &=& -2N_c\\sum _{i=u,d,s}\\int _0^\\Lambda \\frac{d^3p}{(2\\pi )^3}[E_i+T\\ln (1+e^{-\\beta (E_i-\\tilde{\\mu }_i)})\\\\&+&T\\ln (1+e^{-\\beta (E_i+\\tilde{\\mu }_i)})]+G_S(\\sigma _u^2+\\sigma _d^2+\\sigma _s^2)\\\\&-&4K\\sigma _u\\sigma _d\\sigma _s-\\frac{1}{3}G_V{\\rho ^0}^2.$ The energy density $\\varepsilon _{NJL}$ from the NJL model can be written as $\\varepsilon _{NJL} &=& -2N_c\\sum _{i=u,d,s}\\int _0^\\Lambda \\frac{d^3p}{(2\\pi )^3}E_i(1-f_i-\\bar{f}_i)\\\\&-& \\sum _{i=u,d,s}(\\tilde{\\mu }_i-\\mu _i)\\rho ^0+G_S(\\sigma _u^2+\\sigma _d^2+\\sigma _s^2)\\\\&-& 4K\\sigma _u\\sigma _d\\sigma _s-\\frac{1}{3}G_V{\\rho ^0}^2-\\varepsilon _0,$ where $\\varepsilon _0$ is introduced to ensure $\\varepsilon _{NJL} = 0$ in vacuum.",
"For a given net quark number density $\\rho ^0$ and temperature $T$ , the energy density $\\epsilon _{NJL}$ of quark matter can be obtained from Eq.",
"(REF ) by solving self-consistently Eqs.",
"(REF )$-$ (REF ).",
"In the partonic phase described by the NJL transport model, by assuming that the central region is static and in thermal equilibrium, we can calculate the temperature $T$ inversely from the energy density $\\epsilon _{NJL}$ and the net quark number density $\\rho ^0$ .",
"In this way, we can obtain a trajectory in the $(\\rho ^0, T)$ plane for the central region of the partonic phase in relativistic heavy-ion collisions, and this is shown in Fig.",
"REF for midcentral Au+Au collisions at $\\sqrt{s_{NN}}$ = 7.7 and 39 GeV, where the pressure $P = -\\Omega _{NJL}$ in the $(\\rho ^0, T)$ plane is also displayed.",
"Due to the different NJL parameterizations and collision energies in the present study compared to that in Ref.",
"[17], neither of the trajectories enters the spinodal region, i.e., $(\\partial P/\\partial \\rho ^0)_T<0$ , as marked approximately in Fig.",
"REF .",
"This means that the central region in the partonic phase doesn't pass through a first-order liquid-gas phase transition in this study.",
"The EoS from $R_V=1.1$ is even stiffer, and the spinodal region disappears [17].",
"Figure: (Color online) Equation of state in the [number density (ρ 0 \\rho ^0), temperature (TT)] plane of the partonic phase from the NJL model with R V =0R_V = 0, and the corresponding trajectories of the central region of the partonic phase in midcentral Au+Au collisions at s NN \\sqrt{s_{NN}} = 7.7 and 39 GeV based on the extended AMPT model.",
"The spinodal region with (∂P/∂ρ 0 ) T <0(\\partial P/\\partial \\rho ^0)_T<0 is also approximately marked."
]
] | 1808.08334 |
[
[
"Larger mutual inclinations for the shortest-period planets"
],
[
"Abstract The {\\it Kepler} mission revealed a population of compact multiple-planet systems with orbital periods shorter than a year, and occasionally even shorter than a day.",
"By analyzing a sample of 102 {\\it Kepler} and {\\it K2} multi-planet systems, we measure the minimum difference $\\Delta I$ between the orbital inclinations, as a function of the orbital distance of the innermost planet.",
"This is accomplished by fitting all the planetary signals simultaneously, constrained by an external estimate of the stellar mean density.",
"We find $\\Delta I$ to be larger when the inner orbit is smaller, a trend that does not appear to be a selection effect.",
"We find that planets with $a/R_\\star$<5 have a dispersion in $\\Delta I$ of $6.7\\pm 0.6$~degrees, while planets with $5 < a/R_\\star < 12$ have a dispersion of $2.0\\pm 0.1$~degrees.",
"The planetary pairs with higher mutual inclinations also tend to have larger period ratios.",
"These trends suggest that the shortest-period planets have experienced both inclination excitation and orbital shrinkage."
],
[
"Introduction",
"One of the revelations of the Kepler mission was that Sun-like stars often host planets with sizes between those of Earth and Neptune and orbital periods shorter than a year [4].",
"The formation of these short-period planets and their relationship to wider-orbiting planets are not understood.",
"An interesting clue is that the population of planets with the shortest periods ($\\lesssim $ 10 days) is different, in some respects, from the population with longer periods.",
"One difference is in the planet occurrence rate.",
"The function $d\\log N/d\\log P$ , where $N$ is the mean number of planets per star and $P$ is the orbital period, increases with period from 0.2–10 days before leveling off to a constant value out to at least 100 days [34].",
"Another difference is that stars hosting sub-Neptune planets with periods shorter than about 10 days tend to have higher metallicities than those hosting planets with longer periods [31], [34], [50].",
"A third difference is in the period ratios between adjacent planets.",
"When the inner planet's period is shorter than a few days, the period ratio tends to be larger than when both planets have longer periods [44].",
"This Letter describes another clue, which we found in the distribution of mutual inclinations.",
"Several scenarios have been proposed to explain the shortest-period planets.",
"In almost all these scenarios, the planet's orbit is initially wider, because of the theoretical difficulty of building a rocky core in close proximity to the star.",
"Some of the proposed mechanisms to shrink the orbits also involve raising the inclination [21], [35], while others predict low inclinations [25].",
"Previous studies of Kepler systems concluded that the mutual inclinations are typically $\\lesssim $$5^\\circ $ , based on population statistics [46], [12], [13].",
"Here, we focus on systems with the closest-orbiting planets, and attempt to measure the mutual inclination of each system directly by fitting the transit light curves."
],
[
"Sample Selection",
"Any sample of planets detected with the transit method is strongly biased against systems with large mutual inclinations.",
"However, as the planet's orbit becomes smaller, the range of transiting inclinations becomes larger.",
"The requirement is $\\cos I < R_\\star /a$ , where $I$ is the inclination, $R_\\star $ is the stellar radius, and $a$ is the orbital distance.",
"For $a/R_\\star =4$ (corresponding to $P\\approx 1$ day for a Sun-like star), transits can occur for inclinations between 75–90$^\\circ $ .",
"Therefore, by measuring the inclinations of innermost pair of transiting planets around the same star, we can place a lower bound on the mutual inclination ranging up to $15^\\circ $ for the most favorable cases.",
"For this study, we selected the Kepler multiple-planet systems with apparent Kepler magnitude ($Kp$ ) brighter than 14, for which the innermost planet has a radius smaller than $4\\,R_\\oplus $ , a transit signal-to-noise ratio (SNR) greater than 20, and $a/R_\\star < 12$ .",
"The limit on $Kp$ and SNR ensures that the stars have been well characterized and the transit signals can be modeled precisely.",
"The limit on planet size excludes giant planets, which may have a completely different history of formation and evolution than smaller planets.",
"The limit on $a/R_\\star $ corresponds to an allowed range of inclinations of 85–90$^\\circ $ .",
"We augmented our sample with planets with periods $<$ 1 day discovered with K2 data.",
"We expect the Kepler and K2 systems to be drawn from similar populations, since the observations were made with the same telescope and achieved nearly equivalent photometric precision after correcting for K2 systematics.",
"Table 1 reports the most important characteristics of the sample."
],
[
"Constraints on mean stellar density",
"When fitting a transit light curve, there is often a strong covariance between $I$ and $a/R_\\star $ .",
"To reduce this covariance, we enforced an external constraint on the stellar mean density $\\rho _\\star $ , which is related to $a/R_\\star $ through Kepler's third law.",
"The external constraint came from fitting stellar-evolutionary models to the observed spectroscopic parameters, $K$ -band apparent magnitude, and parallax.",
"We used spectroscopic parameters from the California-Kepler survey [33] whenever available, and from other sources as needed (see Table 1).",
"We imposed a minimum uncertainty of 100 K in the effective temperature to account for possible systematic errors.",
"The $K$ magnitudes were from the Two Micron All Sky Survey [40], after correcting for extinction using $A_K=0.443\\,E(B-V)$ with the value of $E(B-V)$ from [6] (and adopting a 30% uncertainty).",
"The parallaxes were from the Gaia Data Release 2 [19], although in practice we used the distance estimates provided by [1].",
"The stellar-evolutionary models were from the Dartmouth Stellar Evolution Database [8] and were compared to the data using the isochrones package by [30].",
"The results are given in Column 2 of Table 1.",
"The typical uncertainty is 8%.",
"As a consistency check, we compared the isochrone-based mean densities with those that were derived from asteroseismology by [47].",
"On average, the isochrone densities are 5% smaller than the asteroseismic densities, with a dispersion of 5%.",
"This suggests that one or both methods are subject to systematic errors of a few percent.",
"This level of error does not have an appreciable effect on the subsequent results, as we verified directly, by repeating the analysis using the asteroseismic densities in place of the isochrone-based densities whenever both were available."
],
[
"Light-curve analysis",
"For the Kepler systems we used the Pre-search Data Conditioning light curves, and for the K2 systems we used the target pixel files, both were obtained from the Mikulski Archive for Space Telescopeshttps://archive.stsci.edu.",
"To construct the K2 time series and mitigate the systematics from the rolling motion of the spacecraft, we used the code described by [7].",
"Prior to transit modeling, we removed any long-term photometric trends due to stellar variability or instrumental effects.",
"We masked out the known transits, and fitted the transit-free light curve with a cubic spline of width 0.75 days.",
"In cases for which a nearby stellar companion was reported by [18], we corrected the light curve for the “diluting” effect of the companion.",
"For each transit, we isolated the segment of data spanning three times the transit duration, centered on the midpoint.",
"We visually inspected each transit and removed those few that were obviously damaged by systematic effects.",
"We used the Batman code for transit modeling [24].",
"The free parameters were the orbital period $P$ , the midtransit time ($T_{\\rm c}$ ), the planet-to-star radius ratio ($R_{\\rm p}/R_\\star $ ), the scaled orbital distance ($a/R_\\star $ ), and $\\cos I$ .",
"We adopted a quadratic limb darkening profile, with coefficients subject to Gaussian priors with widths of 0.3 and mean values determined with EXOFASTastroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml.",
"[10].",
"For the long-cadence data, we computed the model light curve at 1-minute intervals and averaged it to 30 minutes before comparing it to the data.",
"To account for possible transit-timing variations, we used an iterative process.",
"First, a constant-period transit model was optimized based on all the data.",
"Next, this model was used as a template to derive individual transit times, by refitting each transit with the free parameters limited to the midtransit time and a quadratic function of time to account for stellar variability.",
"Then, we fitted a linear function of epoch to the transit times, to see if this fit was satisfactory, or if a sinusoidal model provided a better description.",
"We combined all the data to create a phase-folded light curve, where the folding was based on either a constant period, or the individually measured transit times if TTVs had been detected.",
"This phase-folded light curve was then fitted, leading to an improved template for measuring individual transit times.",
"This process converged after 2-3 iterations.",
"We did not end up identifying any TTVs that had not already been flagged by [32].",
"Finally, we fitted the phase-folded light curves for all the planets in each system simultaneously, while imposing an external constraint on the stellar mean density (Section ).",
"This was done with the Markov Chain Monte Carlo method as implemented by [15].",
"We imposed a Jeffreys prior on $R_{\\rm p}/R_\\star $ and a uniform prior on $\\cos I$ .",
"We restricted $\\cos I>0$ because transit data alone cannot be used to determine the sign of $\\cos I$ .",
"We assumed the orbit of the innermost planet to be circular, since tidal circularization is expected to be rapid for $a/R_\\star <12$ .",
"The outer planets were allowed to have eccentric orbits.",
"Uniform priors were adopted for $\\sqrt{e}\\cos \\omega $ and $\\sqrt{e}\\sin \\omega $ , where $e$ is the eccentricity and $\\omega $ is the argument of pericenter.",
"Table 1 reports the results for $a/R_\\star $ and $I$ .",
"For a visual appreciation of the task of measuring inclinations, Figure REF shows three representative light curves.",
"Also shown are the residuals between the data and two different models: one in which the inclination is allowed to vary freely, and one in which it is held fixed at $90^\\circ $ .",
"As a test of robustness, we repeated the entire analysis, allowing eccentric orbits for all planets.",
"Naturally, this broadened the allowed range of inclinations, but led to no qualitative changes in the results.",
"For further validation, we performed inject-and-recovery simulations.",
"We focused on the planets with periods shorter than one day, for which the coarse time sampling is of greatest concern.",
"Beginning with the original time series (prior to any detrending), we injected a transit signal into the data with the parameters of the best-fitting model but a different midtransit time and orbital inclination.",
"The fake signals were subjected to the same procedures as the real signals.",
"This was repeated for 5 choices of $\\cos I$ over the range for which transits occur.",
"Figure REF compares the injected and recovered inclinations, with error bars based on the 68% credible intervals.",
"The agreement is almost always within 1-$\\sigma $ , lending confidence to the results.",
"Toward $90^\\circ $ , the recovered inclination always falls below the identity line, but this is only because we chose to plot the median of the posterior, which is necessarily lower than $90^\\circ $ .",
"Figure: Results of the inject-and-recover test for orbital inclination.Filled circles are for systems observed in short-cadence mode (1-minute sampling)and open circles are for long-cadence mode (30-minute).Figure: Signatures of inclination in the light curves.",
"Shown arethree representative phase-folded light curves, along with thebest-fitting model (red curve), the best fitting model withI=90 ∘ I=90^\\circ (blue curve), and the corresponding residuals.",
"In allthese cases, there are patterned residuals in lowest panel,indicating that I=90 ∘ I=90^\\circ is disfavored."
],
[
"Minimum Mutual Inclinations",
"For each system, we computed $\\Delta I =|I_1-I_2|$ , where $I_1$ and $I_2$ are the fitted orbital inclinations.",
"In general, $\\Delta I$ is a lower bound on the mutual inclination.",
"It is equal to the mutual inclination only if the two planets' trajectories across the stellar disk are parallel and on the same hemisphere of the star.",
"Figure REF shows $\\Delta I$ as a function of $a/R_\\star $ for the innermost planet.",
"Among the systems with the closest-orbiting planets, there are about 10 systems for which $\\Delta I > 5^\\circ $ , larger than the typical mutual inclinations that have been inferred for wider-orbiting planets.The relatively large mutual inclination of EPIC 248435473b ($\\Delta I = 12.67^{+0.68}_{-0.75}$ deg) was also noted by [36].",
"The planets with $a/R_\\star >5$ almost all have $\\Delta I$ < 5$^{\\circ }$ , even though values of 5–10$^\\circ $ could have been detected in many cases.",
"This is consistent with previous studies of the Kepler multi-planet systems, which concluded that the mutual inclinations are likely to be a few degrees or smaller [46], [12], [13].",
"The planets with smaller values of $a/R_\\star $ have a broader distribution of $\\Delta I$ , nearly filling the full range of inclinations compatible with transits.",
"Figure REF also shows $\\Delta I$ as a function of the period ratio between the innermost two planets.",
"Higher values of $\\Delta I$ are associated with larger period ratios.",
"This reflects a pattern noted by [44]: the period ratios tend to be higher when the inner planet's period is shorter than about one day.",
"In that sense, the innermost planets are dynamically more separated.",
"Our results show that these planets are also associated with minimum mutual inclinations ranging up to 10$^\\circ $ , higher than that of the broader population of multi-planet systems.",
"We quantified these impressions with a hierarchical Bayesian analysis.",
"We compared the Bayesian evidence for the following models for the $\\Delta I$ distribution: A Rayleigh distribution, with width $\\sigma $ .",
"A Rayleigh distribution in which the width varies with orbital distance: $\\sigma = \\sigma _0~(a/R_\\star )^m$ .",
"A Rayleigh distribution in which the width changes abruptly from $\\sigma _{\\rm in}$ to $\\sigma _{\\rm out}$ at a critical value of $a/R_\\star $ .",
"A Rayleigh distribution in which the width changes abruptly from $\\sigma _{\\rm in}$ to $\\sigma _{\\rm out}$ at a critical value of the period ratio, $P_2/P_1$ .",
"To compute the likelihood as a function of the model parameters, we followed the procedure of [16], using the approximation $p (\\mathrm {obs} | \\theta ) \\propto \\prod _{k=1}^{K} \\frac{1}{N} \\sum _{n=1}^{N} \\frac{p(\\Delta I_k^{n} | \\theta )}{p_0(\\Delta I_k^n)},$ where “obs” represents the data, $k$ specifies the planetary system, $n$ specifies random samples from the posterior of $\\Delta I$ , and $\\theta $ is the set of hyperparameters.",
"The function $p_0$ is the prior probability in our transit modeling (Section ).",
"Since we adopted uniform and independent priors for $\\cos I_1$ and $\\cos I_2$ , it can be shown that $p_0(\\Delta I) = \\frac{1}{16}(\\pi - 2\\Delta I)\\cos (\\Delta I).$ All the hyperparameters were subject to log-uniform priors, except for the exponent $m$ in Model 2 for which we used a uniform prior.",
"For each model, we determined the credible intervals for the hyperparameters and the Bayesian evidence $Z$ using the nested sampling code MultiNest [14].",
"Table 2 gives the results.",
"Models 2 and 3 are both favored over Model 1 by $\\Delta \\log Z > 26$ , corroborating the visual impression that $\\Delta I$ is larger for smaller orbits.",
"Model 4 is not as successful as Models 2 and 3 but still favored over Model 1 by $\\Delta \\log Z > 14$ .",
"Figure: Upper Panel: inclination difference ΔI\\Delta I versus a/R ☆ a/R_\\star of the innermostplanet.",
"Filled points are for systems observed in short-cadence mode and hollow points are for long-cadence.",
"The black line is the boundary above which the innerplanet would not transit, assuming i 2 =90 ∘ i_2=90^\\circ .",
"The orange linerepresents Model 2.",
"The green zone represents the 68% credibleregion for the critical value of a/R ☆ a/R_\\star , in Model 3.",
"Highlightedin red are those data points that are more than 3 standard deviations away fromzero: Kepler-10, EPIC 248435473, K2-223, Kepler-312, WASP-47,KOI-2393 and Kepler-653.",
"Lower Panel: ΔI\\Delta I versus orbital periodratio P 2 /P 1 P_2/P_1.",
"The green zone is the68% credible interval for the critical value of P 2 /P 1 P_2/P_1, inModel 3."
],
[
"Discussion",
"We found that when the innermost planet has $a/R_\\star \\lesssim 5$ (or $P= 1.3$ days for a Sun-like star), the minimum mutual inclination is often 5–10$^\\circ $ .",
"This is somewhat higher than the typical value of a few degrees that has been previously estimated for the more general population of Kepler systems.",
"We also found $\\Delta I$ to decrease with the orbital separation of the innermost planet.",
"This observation does not appear to be purely a selection effect because for planets with $a/R_\\star $ between 5 and 10, we could have detected mutual inclinations larger than 5$^\\circ $ , and we did not.",
"These results may be related to some previously noted trends.",
"[43] found that the Kepler sample of “hot Earths” without additional transiting companions is larger than what one would obtain by drawing planets randomly from the multiple-transiting systems.",
"A related observation by [49] is that the fraction of Kepler systems with multiple transiting planets is lower when the innermost planet has a period shorter than a few days.",
"Our results offer a natural explanation: the shortest-period planets tend to have larger mutual inclinations, and thus, are more likely to be observed to transit even when the wider-orbiting companions do not transit.",
"[52] also found evidence for relatively high mutual inclinations in some Kepler systems, based on the observed frequencies of multiple transiting planets and TTVs.",
"In their model, the inclination dispersion depends on the total number of planets in the system, ranging from $0.8^\\circ $ in five-planet systems to $\\sim $ 10$^\\circ $ for two-planet systems.",
"It would be interesting to try and extend the model of [52] to allow the mutual inclination to depend on orbital separation in addition to, or instead of, the total number of planets.",
"[5] proposed that the inner protoplanetary disk may have a flat (rather than flared) geometry in which case the innermost planets tend to form with larger mutual inclinations.",
"We emphasize that $\\Delta I$ only represents a lower bound on the mutual inclination.",
"Moreover, if there exist systems with much larger mutual inclinations, they would be unlikely to appear in our sample, because the joint transit probability is low.",
"Despite these limitations, our results indicate that the shortest-period planets have a different orbital architecture, with higher mutual inclinations and larger period ratios.",
"This suggests that whatever processes led to the extremely tight orbits of these planets were also responsible for tilting the orbit to higher inclination.",
"Several theories have been offered for the formation of very short-period sup-Neptune planets, which differ in their predictions for mutual inclinations.",
"[25] proposed that the magnetospheric truncation radius determines the innermost orbit where planets can form.",
"In this scenario, planets begin with nearly-circular, well-aligned orbits, and the innermost planet undergoes tidal orbital decay.There is no obvious agent for exciting inclinations, and therefore this scenario does not provide an explanation for the larger mutual inclinations of the shortest-period planets.",
"Likewise, the formation scenarios proposed by [45] and [39] do not provide an obvious way to excite inclinations.",
"[42] proposed a scenario that does involve inclination excitation.",
"If the host star were initially rotating rapidly, with a non-zero obliquity, the planets' orbits would undergo nodal precession at different rates and become misaligned, with the innermost planet being most strongly affected.",
"The star would only need to be tilted by a few degrees to explain the observed values of $\\Delta I$ .",
"It is not clear whether this scenario would result in an association between higher mutual inclination and larger period ratios, as we have observed.",
"In the “secular chaos” or “high-$e$ migration” scenario proposed by [35], the innermost planet of a multi-planet system is launched into a high-eccentricity orbit via chaotic secular interactions.",
"If the period is short and the eccentricity becomes high enough ($\\approx $ 0.8), tidal interactions with the host star shrink the orbit.",
"Since eccentricity and inclination are excited together, this theory predicts that the shortest-period planets should have larger mutual inclinations, in qualitative agreement with our results.",
"A potential problem with this picture is that for systems in mean-motion resonance (MMR), the dynamics may be dominated by the resonance instead of secular interactions.",
"The sample of USP systems has a decent fraction of systems that are in or near MMR (5 out of 13 systems with at least 2 exterior planets).",
"Another possibility is forced-eccentricity migration, in which the interaction with outer companions (secular-forcing, or MMR) continually excites the eccentricity of the innermost planet.",
"This allows eccentricity tides to dissipate energy and shrink the orbit [21].",
"Since the planet's eccentricity never exceeds a few percent, the inclinations are only excited to a few degrees, perhaps not enough to be compatible with our results.",
"We thank Vincent Van Eylen, Daniel Fabrycky, Bonan Pu, Songhu Wang, and Cristobal Petrovich for helpful discussions.",
"We are also grateful to the referee for a prompt report.",
"This work made use of the gaia-kepler.fun crossmatch database created by Megan Bedell.",
"Work by F.D.",
"and J.N.W.",
"was supported by the Heising-Simons Foundation.",
"Work by K.M.",
"was performed in part under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute.",
"ccccccccccccccc Transit Modeling Results Planet $\\rho _\\star $ (g cm$^{-3}$ ) Source $P_1$ (days) $P_2/P_1$ $a/R_\\star $ $i (^{\\circ })$ $\\Delta I (^{\\circ })$ Cadence TTV KOI-1843.03 $ 5.115 ^{+ 0.155 }_{- 0.117 }$ [27]+Gaia $ 0.177 $$ 23.72 $$ 2.03 ^{+ 0.02 }_{- 0.02 } $$ 87.17 ^{+ 2.10 }_{- 2.86 } $$ 1.71 ^{+ 2.91 }_{- 1.22 } $ SC EPIC-211305568b $ 6.868 ^{+ 0.214 }_{- 0.186 }$ [9]+Gaia $ 0.198 $$ 58.39 $$ 2.42 ^{+ 0.02 }_{- 0.02 } $$ 77.71 ^{+ 7.35 }_{- 5.05 } $$ 10.19 ^{+ 5.16 }_{- 7.18 } $ LC K2-141b $ 3.089 ^{+ 0.130 }_{- 0.110 }$ [2]+Gaia $ 0.280 $$ 27.63 $$ 2.33 ^{+ 0.03 }_{- 0.03 } $$ 86.81 ^{+ 2.34 }_{- 3.41 } $$ 2.08 ^{+ 2.17 }_{- 1.42 } $ LC Kepler-42c $ 56.740 ^{+ 15.410 }_{- 15.410 }$ [27]+Gaia $ 0.453 $$ 2.68 $$ 6.71 ^{+ 0.05 }_{- 0.11 } $$ 89.14 ^{+ 0.63 }_{- 0.91 } $$ 2.95 ^{+ 0.60 }_{- 0.77 } $ SC K2-183b $ 1.452 ^{+ 0.100 }_{- 0.087 }$ [29] $ 0.469 $$ 23.00 $$ 2.55 ^{+ 0.05 }_{- 0.05 } $$ 85.11 ^{+ 3.25 }_{- 4.03 } $$ 4.19 ^{+ 3.93 }_{- 3.27 } $ LC K2-223b $ 1.611 ^{+ 0.061 }_{- 0.072 }$ [29] $ 0.506 $$ 9.03 $$ 2.79 ^{+ 0.04 }_{- 0.04 } $$ 73.80 ^{+ 1.93 }_{- 1.15 } $$ 14.72 ^{+ 1.50 }_{- 2.11 } $ LC Kepler-990c $ 1.460 ^{+ 0.409 }_{- 0.343 }$ CKS+Gaia $ 0.538 $$ 18.42 $$ 2.79 ^{+ 0.12 }_{- 0.13 } $$ 84.86 ^{+ 3.67 }_{- 3.74 } $$ 3.82 ^{+ 3.89 }_{- 2.96 } $ LC K2-106b $ 1.426 ^{+ 0.094 }_{- 0.086 }$ [20]+Gaia $ 0.571 $$ 23.38 $$ 2.91 ^{+ 0.06 }_{- 0.06 } $$ 82.65 ^{+ 5.03 }_{- 3.94 } $$ 6.31 ^{+ 4.02 }_{- 4.82 } $ LC K2-229b $ 2.451 ^{+ 0.149 }_{- 0.147 }$ [38]+Gaia $ 0.584 $$ 14.25 $$ 3.53 ^{+ 0.06 }_{- 0.07 } $$ 87.01 ^{+ 2.14 }_{- 2.54 } $$ 1.84 ^{+ 2.26 }_{- 1.28 } $ LC KOI-787.03 $ 1.104 ^{+ 0.154 }_{- 0.140 }$ [11]+Gaia $ 0.589 $$ 7.52 $$ 2.75 ^{+ 0.10 }_{- 0.11 } $$ 80.86 ^{+ 5.82 }_{- 3.99 } $$ 7.55 ^{+ 4.07 }_{- 5.61 } $ LC KOI-2250.02 $ 2.506 ^{+ 0.155 }_{- 0.178 }$ CKS+Gaia $ 0.626 $$ 4.69 $$ 3.74 ^{+ 0.08 }_{- 0.08 } $$ 84.51 ^{+ 3.14 }_{- 2.30 } $$ 2.27 ^{+ 1.46 }_{- 1.49 } $ LC Kepler-607b $ 2.047 ^{+ 0.152 }_{- 0.108 }$ CKS+Gaia $ 0.638 $$ 62.18 $$ 3.53 ^{+ 0.07 }_{- 0.07 } $$ 83.68 ^{+ 3.98 }_{- 3.09 } $$ 5.89 ^{+ 3.06 }_{- 3.96 } $ LC EPIC-248435473b $ 3.166 ^{+ 0.143 }_{- 0.105 }$ [36]+Gaia $ 0.658 $$ 9.27 $$ 4.17 ^{+ 0.05 }_{- 0.05 } $$ 76.40 ^{+ 0.26 }_{- 0.27 } $$ 12.67 ^{+ 0.68 }_{- 0.75 } $ LC d, e Kepler-1340b $ 1.421 ^{+ 0.116 }_{- 0.134 }$ CKS+Gaia $ 0.665 $$ 7.62 $$ 3.14 ^{+ 0.08 }_{- 0.07 } $$ 87.62 ^{+ 1.73 }_{- 2.26 } $$ 1.47 ^{+ 1.60 }_{- 1.07 } $ LC KOI-191.03 $ 1.535 ^{+ 0.179 }_{- 0.142 }$ CKS+Gaia $ 0.709 $$ 3.41 $$ 3.49 ^{+ 0.07 }_{- 0.07 } $$ 85.73 ^{+ 2.09 }_{- 1.41 } $$ 1.39 ^{+ 1.13 }_{- 0.90 } $ SC Kepler-32f $ 4.577 ^{+ 0.164 }_{- 0.130 }$ [27]+Gaia $ 0.743 $$ 3.90 $$ 5.11 ^{+ 0.05 }_{- 0.05 } $$ 87.68 ^{+ 1.38 }_{- 1.46 } $$ 1.64 ^{+ 1.33 }_{- 1.16 } $ SC b, c, e KOI-2248.03 $ 2.343 ^{+ 0.365 }_{- 0.352 }$ [11]+Gaia $ 0.762 $$ 3.47 $$ 4.15 ^{+ 0.16 }_{- 0.19 } $$ 85.58 ^{+ 2.85 }_{- 2.42 } $$ 3.39 ^{+ 4.45 }_{- 2.32 } $ LC Kepler-1067b $ 1.792 ^{+ 0.098 }_{- 0.127 }$ CKS+Gaia $ 0.762 $$ 7.12 $$ 3.80 ^{+ 0.09 }_{- 0.07 } $$ 81.58 ^{+ 2.69 }_{- 1.73 } $$ 6.75 ^{+ 1.67 }_{- 2.60 } $ LC KOI-1360.03 $ 2.879 ^{+ 0.131 }_{- 0.183 }$ CKS+Gaia $ 0.764 $$ 19.09 $$ 4.46 ^{+ 0.08 }_{- 0.07 } $$ 83.83 ^{+ 3.35 }_{- 2.34 } $$ 5.23 ^{+ 2.37 }_{- 3.28 } $ LC KOI-2393.02 $ 2.809 ^{+ 0.148 }_{- 0.169 }$ CKS+Gaia $ 0.767 $$ 6.00 $$ 4.44 ^{+ 0.08 }_{- 0.08 } $$ 83.34 ^{+ 1.76 }_{- 1.11 } $$ 5.59 ^{+ 1.30 }_{- 1.86 } $ LC K2-187b $ 1.889 ^{+ 0.113 }_{- 0.123 }$ [29] $ 0.774 $$ 3.71 $$ 3.91 ^{+ 0.08 }_{- 0.08 } $$ 82.61 ^{+ 2.48 }_{- 1.55 } $$ 5.42 ^{+ 1.78 }_{- 2.60 } $ LC KOI-1239.01 $ 1.628 ^{+ 0.183 }_{- 0.196 }$ CKS+Gaia $ 0.783 $$ 4.05 $$ 3.70 ^{+ 0.09 }_{- 0.11 } $$ 87.68 ^{+ 1.69 }_{- 2.18 } $$ 1.42 ^{+ 1.36 }_{- 0.98 } $ LC WASP-47e $ 0.980 ^{+ 0.083 }_{- 0.079 }$ [3]+Gaia $ 0.790 $$ 5.27 $$ 3.20 ^{+ 0.04 }_{- 0.03 } $$ 84.76 ^{+ 0.98 }_{- 0.78 } $$ 4.00 ^{+ 0.64 }_{- 0.84 } $ SC b, c Kepler-10b $ 1.068 ^{+ 0.001 }_{- 0.007 }$ Astreoseismology $ 0.837 $$ 54.08 $$ 3.406 ^{+ 0.005 }_{- 0.005 } $$ 84.02 ^{+ 0.12 }_{- 0.11 } $$ 5.82 ^{+ 0.16 }_{- 0.17 } $ SC c Kepler-10b $ 1.010 ^{+ 0.077 }_{- 0.063 }$ CKS+Gaia $ 0.838 $$ 54.08 $$ 3.32 ^{+ 0.07 }_{- 0.06 } $$ 82.78 ^{+ 1.03 }_{- 0.85 } $$ 7.03 ^{+ 0.84 }_{- 1.02 } $ SC c KOI-1499.02 $ 2.060 ^{+ 0.179 }_{- 0.185 }$ KIC[28]+Gaia $ 0.841 $$ 7.39 $$ 4.25 ^{+ 0.12 }_{- 0.13 } $$ 84.70 ^{+ 2.95 }_{- 2.25 } $$ 3.91 ^{+ 2.33 }_{- 2.69 } $ LC Kepler-732c $ 5.421 ^{+ 0.150 }_{- 0.122 }$ [27]+Gaia $ 0.893 $$ 10.60 $$ 6.12 ^{+ 0.05 }_{- 0.05 } $$ 87.92 ^{+ 0.64 }_{- 0.48 } $$ 1.71 ^{+ 0.55 }_{- 0.67 } $ SC Kepler-653c $ 0.807 ^{+ 0.062 }_{- 0.059 }$ CKS+Gaia $ 0.900 $$ 16.34 $$ 3.26 ^{+ 0.08 }_{- 0.07 } $$ 75.99 ^{+ 1.68 }_{- 1.16 } $$ 12.38 ^{+ 1.52 }_{- 1.72 } $ LC EPIC-206024342b $ 1.994 ^{+ 0.131 }_{- 0.167 }$ [51]+Gaia $ 0.912 $$ 16.06 $$ 4.42 ^{+ 0.11 }_{- 0.10 } $$ 84.60 ^{+ 3.62 }_{- 3.16 } $$ 3.95 ^{+ 3.05 }_{- 3.06 } $ LC HD-3167b $ 1.916 ^{+ 0.114 }_{- 0.107 }$ [48]+Gaia $ 0.960 $$ 31.10 $$ 4.53 ^{+ 0.08 }_{- 0.08 } $$ 87.38 ^{+ 1.69 }_{- 1.50 } $$ 2.04 ^{+ 1.46 }_{- 1.54 } $ LC Kepler-1322b $ 1.437 ^{+ 0.228 }_{- 0.240 }$ KIC+Gaia $ 0.963 $$ 6.71 $$ 4.15 ^{+ 0.19 }_{- 0.19 } $$ 85.59 ^{+ 2.60 }_{- 2.35 } $$ 2.10 ^{+ 1.89 }_{- 1.50 } $ LC KOI-3145.02 $ 3.129 ^{+ 0.189 }_{- 0.183 }$ KIC+Gaia $ 0.977 $$ 4.64 $$ 5.41 ^{+ 0.12 }_{- 0.10 } $$ 86.64 ^{+ 2.23 }_{- 2.24 } $$ 1.80 ^{+ 2.12 }_{- 1.21 } $ LC Kepler-80f $ 3.026 ^{+ 0.107 }_{- 0.136 }$ CKS+Gaia $ 0.987 $$ 3.11 $$ 5.35 ^{+ 0.05 }_{- 0.06 } $$ 88.85 ^{+ 0.89 }_{- 1.00 } $$ 0.89 ^{+ 0.90 }_{- 0.65 } $ SC b, c Kepler-755b $ 2.031 ^{+ 0.149 }_{- 0.096 }$ CKS+Gaia $ 1.269 $$ 2.25 $$ 5.57 ^{+ 0.11 }_{- 0.11 } $$ 84.12 ^{+ 0.41 }_{- 0.36 } $$ 1.58 ^{+ 2.67 }_{- 1.07 } $ SC Kepler-198d $ 1.695 ^{+ 0.146 }_{- 0.145 }$ KIC+Gaia $ 1.312 $$ 13.56 $$ 5.34 ^{+ 0.16 }_{- 0.12 } $$ 87.15 ^{+ 2.04 }_{- 1.45 } $$ 2.43 ^{+ 1.41 }_{- 1.91 } $ SC Kepler-207b $ 0.334 ^{+ 0.027 }_{- 0.025 }$ CKS+Gaia $ 1.612 $$ 1.91 $$ 3.60 ^{+ 0.10 }_{- 0.10 } $$ 81.77 ^{+ 1.55 }_{- 1.23 } $$ 5.93 ^{+ 1.91 }_{- 2.33 } $ LC Kepler-342e $ 0.661 ^{+ 0.056 }_{- 0.054 }$ CKS+Gaia $ 1.644 $$ 9.23 $$ 4.46 ^{+ 0.09 }_{- 0.11 } $$ 88.67 ^{+ 1.00 }_{- 1.51 } $$ 0.85 ^{+ 1.19 }_{- 0.58 } $ LC c, d Kepler-322b $ 2.222 ^{+ 0.119 }_{- 0.151 }$ CKS+Gaia $ 1.654 $$ 2.62 $$ 6.86 ^{+ 0.13 }_{- 0.14 } $$ 86.85 ^{+ 1.32 }_{- 0.84 } $$ 2.12 ^{+ 0.92 }_{- 1.21 } $ LC Kepler-323b $ 1.135 ^{+ 0.109 }_{- 0.102 }$ KIC+Gaia $ 1.678 $$ 2.12 $$ 5.49 ^{+ 0.08 }_{- 0.12 } $$ 88.51 ^{+ 1.01 }_{- 1.37 } $$ 0.82 ^{+ 1.01 }_{- 0.55 } $ LC Kepler-969c $ 2.375 ^{+ 0.113 }_{- 0.146 }$ CKS+Gaia $ 1.683 $$ 20.31 $$ 7.08 ^{+ 0.11 }_{- 0.11 } $$ 88.68 ^{+ 0.88 }_{- 0.96 } $$ 0.82 ^{+ 0.96 }_{- 0.60 } $ LC Kepler-1047c $ 0.371 ^{+ 0.037 }_{- 0.030 }$ CKS+Gaia $ 1.721 $$ 1.85 $$ 3.87 ^{+ 0.12 }_{- 0.10 } $$ 78.25 ^{+ 1.78 }_{- 1.20 } $$ 6.30 ^{+ 3.55 }_{- 2.84 } $ LC Kepler-312b $ 0.371 ^{+ 0.031 }_{- 0.030 }$ CKS+Gaia $ 1.772 $$ 11.14 $$ 3.95 ^{+ 0.11 }_{- 0.10 } $$ 79.32 ^{+ 0.62 }_{- 0.64 } $$ 9.12 ^{+ 1.19 }_{- 1.38 } $ LC c Kepler-524c $ 0.410 ^{+ 0.037 }_{- 0.031 }$ CKS+Gaia $ 1.889 $$ 4.22 $$ 4.26 ^{+ 0.10 }_{- 0.10 } $$ 87.24 ^{+ 1.91 }_{- 1.97 } $$ 1.65 ^{+ 1.82 }_{- 1.16 } $ LC b Kepler-1371c $ 1.440 ^{+ 0.086 }_{- 0.066 }$ CKS+Gaia $ 2.005 $$ 1.45 $$ 6.73 ^{+ 0.11 }_{- 0.12 } $$ 85.91 ^{+ 2.20 }_{- 1.08 } $$ 1.59 ^{+ 1.65 }_{- 1.08 } $ LC Kepler-142b $ 1.012 ^{+ 0.087 }_{- 0.074 }$ CKS+Gaia $ 2.024 $$ 2.35 $$ 6.06 ^{+ 0.10 }_{- 0.15 } $$ 88.02 ^{+ 1.21 }_{- 1.02 } $$ 0.89 ^{+ 0.81 }_{- 0.62 } $ SC Kepler-326b $ 1.068 ^{+ 0.073 }_{- 0.078 }$ CKS+Gaia $ 2.248 $$ 2.04 $$ 6.62 ^{+ 0.16 }_{- 0.15 } $$ 84.59 ^{+ 0.90 }_{- 0.83 } $$ 3.59 ^{+ 1.36 }_{- 1.51 } $ LC Kepler-406b $ 1.133 ^{+ 0.105 }_{- 0.090 }$ CKS+Gaia $ 2.426 $$ 1.91 $$ 6.97 ^{+ 0.13 }_{- 0.15 } $$ 89.07 ^{+ 0.62 }_{- 0.92 } $$ 1.02 ^{+ 0.97 }_{- 0.77 } $ LC Kepler-314b $ 1.845 ^{+ 0.137 }_{- 0.134 }$ CKS+Gaia $ 2.461 $$ 2.42 $$ 8.35 ^{+ 0.19 }_{- 0.18 } $$ 88.65 ^{+ 0.88 }_{- 1.05 } $$ 0.79 ^{+ 0.73 }_{- 0.52 } $ LC Kepler-213b $ 1.110 ^{+ 0.098 }_{- 0.096 }$ CKS+Gaia $ 2.462 $$ 1.96 $$ 7.10 ^{+ 0.22 }_{- 0.18 } $$ 83.99 ^{+ 0.37 }_{- 0.32 } $$ 0.94 ^{+ 0.37 }_{- 0.37 } $ SC Kepler-1311c $ 0.298 ^{+ 0.022 }_{- 0.018 }$ CKS+Gaia $ 2.536 $$ 4.41 $$ 4.72 ^{+ 0.08 }_{- 0.08 } $$ 87.22 ^{+ 1.65 }_{- 1.47 } $$ 1.53 ^{+ 1.60 }_{- 1.03 } $ LC Kepler-1398b $ 0.611 ^{+ 0.048 }_{- 0.044 }$ CKS+Gaia $ 2.788 $$ 1.48 $$ 6.27 ^{+ 0.13 }_{- 0.15 } $$ 88.45 ^{+ 1.03 }_{- 1.22 } $$ 1.45 ^{+ 1.29 }_{- 0.98 } $ LC Kepler-221b $ 2.417 ^{+ 0.132 }_{- 0.159 }$ CKS+Gaia $ 2.796 $$ 2.04 $$ 10.00 ^{+ 0.22 }_{- 0.19 } $$ 88.18 ^{+ 0.51 }_{- 0.32 } $$ 0.98 ^{+ 0.51 }_{- 0.51 } $ SC e Kepler-1542c $ 0.837 ^{+ 0.067 }_{- 0.051 }$ CKS+Gaia $ 2.892 $$ 1.37 $$ 7.18 ^{+ 0.15 }_{- 0.15 } $$ 85.49 ^{+ 0.72 }_{- 0.52 } $$ 1.77 ^{+ 1.88 }_{- 1.34 } $ LC Kepler-411b $ 2.630 ^{+ 0.132 }_{- 0.139 }$ CKS+Gaia $ 3.005 $$ 2.61 $$ 10.78 ^{+ 0.17 }_{- 0.13 } $$ 87.71 ^{+ 0.19 }_{- 0.16 } $$ 1.23 ^{+ 0.14 }_{- 0.15 } $ SC c Kepler-1271b $ 0.951 ^{+ 0.105 }_{- 0.090 }$ KIC+Gaia $ 3.026 $$ 1.79 $$ 7.74 ^{+ 0.24 }_{- 0.23 } $$ 88.10 ^{+ 1.22 }_{- 1.03 } $$ 1.04 ^{+ 1.10 }_{- 0.72 } $ LC Kepler-141b $ 2.311 ^{+ 0.153 }_{- 0.106 }$ CKS+Gaia $ 3.108 $$ 2.26 $$ 10.49 ^{+ 0.17 }_{- 0.18 } $$ 89.24 ^{+ 0.50 }_{- 0.70 } $$ 0.52 ^{+ 0.53 }_{- 0.36 } $ LC Kepler-203b $ 1.167 ^{+ 0.120 }_{- 0.107 }$ CKS+Gaia $ 3.163 $$ 1.70 $$ 8.53 ^{+ 0.24 }_{- 0.25 } $$ 84.92 ^{+ 0.28 }_{- 0.28 } $$ 3.39 ^{+ 1.18 }_{- 1.30 } $ LC Kepler-107b $ 0.581 ^{+ 0.049 }_{- 0.049 }$ Astreoseismology $ 3.180 $$ 1.54 $$ 6.71 ^{+ 0.06 }_{- 0.11 } $$ 88.97 ^{+ 0.65 }_{- 0.86 } $$ 1.62 ^{+ 0.80 }_{- 0.94 } $ SC Kepler-107b $ 0.519 ^{+ 0.038 }_{- 0.045 }$ CKS+Gaia $ 3.180 $$ 1.54 $$ 6.54 ^{+ 0.15 }_{- 0.14 } $$ 87.74 ^{+ 1.09 }_{- 0.69 } $$ 0.99 ^{+ 0.89 }_{- 0.66 } $ SC Kepler-140b $ 0.904 ^{+ 0.080 }_{- 0.067 }$ CKS+Gaia $ 3.254 $$ 28.07 $$ 8.01 ^{+ 0.23 }_{- 0.20 } $$ 87.45 ^{+ 1.08 }_{- 0.55 } $$ 2.31 ^{+ 0.57 }_{- 1.06 } $ LC Kepler-337b $ 0.278 ^{+ 0.023 }_{- 0.019 }$ CKS+Gaia $ 3.293 $$ 2.94 $$ 5.45 ^{+ 0.13 }_{- 0.12 } $$ 86.51 ^{+ 1.57 }_{- 0.87 } $$ 2.16 ^{+ 1.27 }_{- 1.31 } $ SC Kepler-111b $ 1.002 ^{+ 0.085 }_{- 0.078 }$ CKS+Gaia $ 3.342 $$ 67.26 $$ 8.48 ^{+ 0.18 }_{- 0.22 } $$ 88.40 ^{+ 0.77 }_{- 0.67 } $$ 1.32 ^{+ 0.65 }_{- 0.75 } $ SC Kepler-101b $ 0.311 ^{+ 0.027 }_{- 0.022 }$ CKS+Gaia $ 3.488 $$ 1.73 $$ 5.87 ^{+ 0.17 }_{- 0.15 } $$ 84.22 ^{+ 0.73 }_{- 0.46 } $$ 2.19 ^{+ 2.23 }_{- 1.31 } $ SC b Kepler-18b $ 1.612 ^{+ 0.143 }_{- 0.130 }$ CKS+Gaia $ 3.505 $$ 2.18 $$ 10.14 ^{+ 0.24 }_{- 0.20 } $$ 86.32 ^{+ 0.21 }_{- 0.19 } $$ 1.89 ^{+ 0.12 }_{- 0.14 } $ SC c, d Kepler-363b $ 0.396 ^{+ 0.034 }_{- 0.033 }$ CKS+Gaia $ 3.615 $$ 2.09 $$ 6.49 ^{+ 0.17 }_{- 0.18 } $$ 84.02 ^{+ 0.64 }_{- 0.54 } $$ 4.09 ^{+ 1.36 }_{- 1.06 } $ LC Kepler-218b $ 1.116 ^{+ 0.095 }_{- 0.072 }$ CKS+Gaia $ 3.619 $$ 12.35 $$ 9.16 ^{+ 0.19 }_{- 0.19 } $$ 88.77 ^{+ 0.86 }_{- 0.81 } $$ 0.80 ^{+ 0.81 }_{- 0.58 } $ LC Kepler-20b $ 1.756 ^{+ 0.155 }_{- 0.132 }$ CKS+Gaia $ 3.696 $$ 1.65 $$ 11.00 ^{+ 0.15 }_{- 0.17 } $$ 88.19 ^{+ 0.23 }_{- 0.22 } $$ 0.20 ^{+ 0.35 }_{- 0.16 } $ SC b, c Kepler-466c $ 1.505 ^{+ 0.241 }_{- 0.259 }$ CKS+Gaia $ 3.709 $$ 13.77 $$ 10.50 ^{+ 0.32 }_{- 0.37 } $$ 88.76 ^{+ 0.79 }_{- 0.78 } $$ 1.02 ^{+ 0.81 }_{- 0.71 } $ SC Kepler-89b $ 0.596 ^{+ 0.055 }_{- 0.047 }$ CKS+Gaia $ 3.743 $$ 2.78 $$ 7.70 ^{+ 0.13 }_{- 0.10 } $$ 88.09 ^{+ 0.62 }_{- 0.39 } $$ 0.28 ^{+ 0.38 }_{- 0.18 } $ SC c, d, e Kepler-217d $ 0.289 ^{+ 0.023 }_{- 0.023 }$ CKS+Gaia $ 3.887 $$ 1.38 $$ 6.13 ^{+ 0.16 }_{- 0.16 } $$ 84.18 ^{+ 0.64 }_{- 0.52 } $$ 3.53 ^{+ 1.52 }_{- 1.86 } $ LC Kepler-380b $ 0.757 ^{+ 0.074 }_{- 0.060 }$ CKS+Gaia $ 3.931 $$ 1.94 $$ 8.54 ^{+ 0.24 }_{- 0.24 } $$ 86.31 ^{+ 0.80 }_{- 0.53 } $$ 1.10 ^{+ 1.54 }_{- 0.78 } $ LC Kepler-402b $ 1.068 ^{+ 0.099 }_{- 0.092 }$ CKS+Gaia $ 4.029 $$ 1.52 $$ 9.74 ^{+ 0.29 }_{- 0.26 } $$ 88.06 ^{+ 1.03 }_{- 0.67 } $$ 0.77 ^{+ 0.74 }_{- 0.50 } $ LC Kepler-202b $ 3.014 ^{+ 0.115 }_{- 0.146 }$ CKS+Gaia $ 4.069 $$ 4.00 $$ 13.82 ^{+ 0.20 }_{- 0.20 } $$ 88.06 ^{+ 0.16 }_{- 0.14 } $$ 0.91 ^{+ 0.16 }_{- 0.14 } $ SC Kepler-625c $ 0.457 ^{+ 0.047 }_{- 0.035 }$ KIC+Gaia $ 4.165 $$ 1.86 $$ 7.49 ^{+ 0.23 }_{- 0.21 } $$ 85.06 ^{+ 0.36 }_{- 0.37 } $$ 3.34 ^{+ 1.08 }_{- 1.25 } $ LC Kepler-208b $ 0.745 ^{+ 0.068 }_{- 0.060 }$ CKS+Gaia $ 4.229 $$ 1.77 $$ 8.90 ^{+ 0.26 }_{- 0.22 } $$ 87.30 ^{+ 0.87 }_{- 0.50 } $$ 0.68 ^{+ 0.71 }_{- 0.43 } $ LC Kepler-783b $ 1.939 ^{+ 0.165 }_{- 0.142 }$ CKS+Gaia $ 4.293 $$ 1.64 $$ 12.36 ^{+ 0.34 }_{- 0.30 } $$ 88.47 ^{+ 0.98 }_{- 0.81 } $$ 1.26 ^{+ 1.39 }_{- 0.88 } $ LC Kepler-219b $ 0.840 ^{+ 0.075 }_{- 0.069 }$ CKS+Gaia $ 4.585 $$ 4.95 $$ 9.84 ^{+ 0.28 }_{- 0.27 } $$ 87.46 ^{+ 0.48 }_{- 0.37 } $$ 1.85 ^{+ 0.36 }_{- 0.43 } $ SC b Kepler-356b $ 0.562 ^{+ 0.049 }_{- 0.046 }$ KIC+Gaia $ 4.613 $$ 2.84 $$ 8.59 ^{+ 0.25 }_{- 0.22 } $$ 85.50 ^{+ 0.34 }_{- 0.29 } $$ 1.80 ^{+ 0.30 }_{- 0.32 } $ LC Kepler-1365c $ 0.360 ^{+ 0.032 }_{- 0.027 }$ CKS+Gaia $ 4.775 $$ 1.61 $$ 7.57 ^{+ 0.21 }_{- 0.21 } $$ 85.32 ^{+ 0.70 }_{- 0.53 } $$ 2.09 ^{+ 1.54 }_{- 0.95 } $ LC Kepler-321b $ 1.441 ^{+ 0.120 }_{- 0.120 }$ CKS+Gaia $ 4.915 $$ 2.66 $$ 12.30 ^{+ 0.36 }_{- 0.30 } $$ 87.67 ^{+ 0.28 }_{- 0.24 } $$ 0.25 ^{+ 0.21 }_{- 0.16 } $ LC b Kepler-376b $ 0.500 ^{+ 0.047 }_{- 0.040 }$ KIC+Gaia $ 4.920 $$ 2.88 $$ 8.70 ^{+ 0.23 }_{- 0.23 } $$ 87.79 ^{+ 0.99 }_{- 0.70 } $$ 0.80 ^{+ 0.97 }_{- 0.60 } $ LC Kepler-634b $ 0.472 ^{+ 0.045 }_{- 0.044 }$ KIC+Gaia $ 5.169 $$ 1.57 $$ 8.69 ^{+ 0.17 }_{- 0.20 } $$ 88.93 ^{+ 0.72 }_{- 0.82 } $$ 1.06 ^{+ 0.96 }_{- 0.69 } $ LC Kepler-392b $ 0.881 ^{+ 0.069 }_{- 0.066 }$ CKS+Gaia $ 5.342 $$ 1.47 $$ 10.97 ^{+ 0.23 }_{- 0.25 } $$ 88.97 ^{+ 0.72 }_{- 0.76 } $$ 0.78 ^{+ 0.77 }_{- 0.52 } $ LC Kepler-526b $ 0.478 ^{+ 0.038 }_{- 0.038 }$ CKS+Gaia $ 5.459 $$ 1.26 $$ 9.11 ^{+ 0.24 }_{- 0.24 } $$ 86.89 ^{+ 0.49 }_{- 0.36 } $$ 1.16 ^{+ 1.26 }_{- 0.84 } $ LC Kepler-197b $ 0.907 ^{+ 0.052 }_{- 0.052 }$ Astreoseismology $ 5.599 $$ 1.85 $$ 11.46 ^{+ 0.17 }_{- 0.20 } $$ 89.06 ^{+ 0.59 }_{- 0.52 } $$ 0.42 ^{+ 0.53 }_{- 0.30 } $ SC Kepler-197b $ 0.815 ^{+ 0.067 }_{- 0.059 }$ CKS+Gaia $ 5.599 $$ 1.85 $$ 11.13 ^{+ 0.29 }_{- 0.24 } $$ 88.57 ^{+ 0.73 }_{- 0.52 } $$ 0.35 ^{+ 0.45 }_{- 0.24 } $ SC Kepler-381b $ 0.482 ^{+ 0.045 }_{- 0.045 }$ CKS+Gaia $ 5.629 $$ 1.47 $$ 9.31 ^{+ 0.31 }_{- 0.27 } $$ 86.62 ^{+ 0.52 }_{- 0.41 } $$ 2.77 ^{+ 1.44 }_{- 1.81 } $ LC Kepler-33b $ 0.353 ^{+ 0.031 }_{- 0.027 }$ CKS+Gaia $ 5.668 $$ 2.32 $$ 8.38 ^{+ 0.16 }_{- 0.20 } $$ 88.35 ^{+ 1.24 }_{- 0.98 } $$ 0.76 ^{+ 1.02 }_{- 0.53 } $ SC d, e, f Kepler-116b $ 0.575 ^{+ 0.054 }_{- 0.048 }$ CKS+Gaia $ 5.969 $$ 2.19 $$ 10.32 ^{+ 0.29 }_{- 0.29 } $$ 86.63 ^{+ 0.27 }_{- 0.26 } $$ 0.71 ^{+ 1.71 }_{- 0.59 } $ SC Kepler-135b $ 0.630 ^{+ 0.056 }_{- 0.054 }$ KIC+Gaia $ 6.003 $$ 1.91 $$ 10.68 ^{+ 0.30 }_{- 0.29 } $$ 87.69 ^{+ 0.71 }_{- 0.46 } $$ 1.01 ^{+ 0.93 }_{- 0.71 } $ SC Kepler-132b $ 1.237 ^{+ 0.253 }_{- 0.212 }$ CKS+Gaia $ 6.178 $$ 2.92 $$ 13.59 ^{+ 0.72 }_{- 0.62 } $$ 88.49 ^{+ 0.88 }_{- 0.59 } $$ 0.69 ^{+ 0.69 }_{- 0.51 } $ LC d Kepler-1581b $ 0.572 ^{+ 0.047 }_{- 0.045 }$ KIC+Gaia $ 6.284 $$ 1.45 $$ 10.63 ^{+ 0.28 }_{- 0.25 } $$ 87.99 ^{+ 1.00 }_{- 0.68 } $$ 0.88 ^{+ 0.96 }_{- 0.60 } $ LC Kepler-335b $ 0.281 ^{+ 0.024 }_{- 0.022 }$ CKS+Gaia $ 6.562 $$ 10.34 $$ 8.61 ^{+ 0.21 }_{- 0.21 } $$ 84.97 ^{+ 0.28 }_{- 0.23 } $$ 4.54 ^{+ 0.34 }_{- 0.31 } $ LC c Kepler-431b $ 0.410 ^{+ 0.032 }_{- 0.028 }$ CKS+Gaia $ 6.802 $$ 1.28 $$ 10.02 ^{+ 0.24 }_{- 0.21 } $$ 87.41 ^{+ 0.84 }_{- 0.59 } $$ 0.90 ^{+ 1.13 }_{- 0.65 } $ LC Kepler-100b $ 0.454 ^{+ 0.004 }_{- 0.006 }$ Astreoseismology $ 6.887 $$ 1.86 $$ 10.43 ^{+ 0.04 }_{- 0.04 } $$ 87.32 ^{+ 0.06 }_{- 0.06 } $$ 1.39 ^{+ 0.54 }_{- 0.38 } $ SC c Kepler-100b $ 0.428 ^{+ 0.033 }_{-0.027 }$ CKS+Gaia $ 6.887 $$ 1.86 $$ 10.24 ^{+ 0.23 }_{- 0.19 } $$ 87.12 ^{+ 0.26 }_{- 0.21 } $$ 1.66 ^{+ 0.46 }_{- 0.36 } $ SC c Kepler-403b $ 0.317 ^{+ 0.025 }_{- 0.022 }$ CKS+Gaia $ 7.031 $$ 1.94 $$ 9.25 ^{+ 0.17 }_{- 0.17 } $$ 89.22 ^{+ 0.58 }_{- 0.70 } $$ 0.56 ^{+ 0.63 }_{- 0.40 } $ LC Kepler-60b $ 0.491 ^{+ 0.044 }_{- 0.040 }$ CKS+Gaia $ 7.133 $$ 1.25 $$ 11.00 ^{+ 0.30 }_{- 0.29 } $$ 88.15 ^{+ 1.06 }_{- 0.66 } $$ 0.73 ^{+ 1.08 }_{- 0.53 } $ LC b, c, d Kepler-853b $ 0.540 ^{+ 0.045 }_{- 0.043 }$ CKS+Gaia $ 7.169 $$ 7.58 $$ 11.38 ^{+ 0.29 }_{- 0.30 } $$ 88.43 ^{+ 0.82 }_{- 0.58 } $$ 1.08 ^{+ 0.66 }_{- 0.72 } $ LC Kepler-450d $ 0.478 ^{+ 0.064 }_{- 0.064 }$ Astreoseismology $ 7.514 $$ 2.05 $$ 11.29 ^{+ 0.32 }_{- 0.30 } $$ 88.55 ^{+ 0.73 }_{- 0.50 } $$ 0.39 ^{+ 0.40 }_{- 0.26 } $ SC b, c Kepler-450d $ 0.438 ^{+ 0.029 }_{- 0.028 }$ CKS+Gaia $ 7.514 $$ 2.05 $$ 11.06 ^{+ 0.20 }_{- 0.18 } $$ 88.16 ^{+ 0.37 }_{- 0.29 } $$ 0.99 ^{+ 0.38 }_{- 0.40 } $ SC b, c Kepler-216b $ 0.425 ^{+ 0.038 }_{- 0.034 }$ CKS+Gaia $ 7.694 $$ 2.26 $$ 11.09 ^{+ 0.28 }_{- 0.30 } $$ 88.68 ^{+ 0.71 }_{- 0.54 } $$ 0.65 ^{+ 0.59 }_{- 0.47 } $ SC Kepler-200b $ 1.300 ^{+ 0.124 }_{- 0.109 }$ CKS+Gaia $ 8.595 $$ 1.19 $$ 17.19 ^{+ 0.53 }_{- 0.52 } $$ 88.34 ^{+ 0.39 }_{- 0.30 } $$ 0.84 ^{+ 0.56 }_{- 0.41 } $ LC Kepler-338e $ 0.309 ^{+ 0.034 }_{- 0.034 }$ Astreoseismology $ 9.342 $$ 1.47 $$ 11.28 ^{+ 0.31 }_{- 0.36 } $$ 88.50 ^{+ 0.69 }_{- 0.52 } $$ 0.74 ^{+ 0.68 }_{- 0.49 } $ SC Kepler-338e $ 0.293 ^{+ 0.024 }_{- 0.022 }$ CKS+Gaia $ 9.342 $$ 1.47 $$ 11.07 ^{+ 0.29 }_{- 0.23 } $$ 88.20 ^{+ 0.53 }_{- 0.37 } $$ 1.00 ^{+ 0.58 }_{- 0.66 } $ SC Kepler-804c $ 1.168 ^{+ 0.109 }_{- 0.098 }$ CKS+Gaia $ 9.652 $$ 1.49 $$ 17.67 ^{+ 0.42 }_{- 0.41 } $$ 89.56 ^{+ 0.32 }_{- 0.39 } $$ 0.36 ^{+ 0.40 }_{- 0.26 } $ LC Kepler-36b $ 0.361 ^{+ 0.024 }_{- 0.022 }$ CKS+Gaia $ 13.850 $$ 1.17 $$ 15.41 ^{+ 0.13 }_{- 0.19 } $$ 89.48 ^{+ 0.32 }_{- 0.30 } $$ 0.29 ^{+ 0.28 }_{- 0.21 } $ SC b, c Kepler-277b $ 0.316 ^{+ 0.027 }_{- 0.025 }$ CKS+Gaia $ 17.324 $$ 1.91 $$ 15.99 ^{+ 0.13 }_{- 0.18 } $$ 89.75 ^{+ 0.17 }_{- 0.25 } $$ 1.06 ^{+ 0.64 }_{- 0.80 } $ SC b lcll Model comparison with Hierarchical Bayesian Modeling Model for $\\Delta I$ Bayesian Evidence log$(Z)$ Parameters Prior 1: $\\Delta I \\sim P(\\sigma _0)$PDF of a Rayleigh distribution $-41.9$ $\\sigma _0 =$ 0.0504$^{+0.0024}_{-0.0025}$ (RMS $\\Delta I = 4.05^{+0.19}_{-0.20}~^{\\circ })$ $\\sigma : $ log-uniform [-5,5] 2: $\\Delta I \\sim P(\\sigma _0 (\\frac{a}{R_\\star })^m)$ $-8.3$$\\sigma _0=0.382^{+0.080}_{-0.063}$ $\\sigma _0: $ log-uniform [-5,5] $m=-1.28\\pm 0.10$ $m$ : uniform [-5,5] 3: $\\Delta I \\sim P(\\sigma _1)$ if $\\frac{a}{R_\\star }<\\frac{a}{R_\\star }^\\prime $ $-15.1$ $\\sigma _1=$ 0.0830$^{+0.0079}_{-0.0069}$ (RMS $\\Delta I = 6.68^{+0.64}_{-0.55}~^{\\circ })$ $\\sigma _1: $ log-uniform [-5,5] $\\Delta I \\sim P(\\sigma _2)$ if $\\frac{a}{R_\\star }>\\frac{a}{R_\\star }^\\prime $ $\\sigma _2=$ 0.0250$^{+ 0.0016}_{-0.0014}$ (RMS $\\Delta I =2.01^{+0.13}_{-0.11}~^{\\circ })$ $\\sigma _2: $ log-uniform [-5,5] $\\frac{a}{R_\\star }^\\prime =$ 4.65$^{+ 0.27}_{-0.10}$ $\\frac{a}{R_\\star }^\\prime $ : log-uniform [0,2] 4: $\\Delta I \\sim P(\\sigma _1)$ if $\\frac{P_2}{P_1}<\\frac{P_2}{P_1}^\\prime $ $-27.1$ $\\sigma _1=$ 0.0311$^{+ 0.0021}_{-0.0022}$ (RMS $\\Delta I = 2.50^{+0.17}_{-0.18}~^{\\circ })$ $\\sigma _1: $ log-uniform [-5,5] $\\Delta I \\sim P(\\sigma _2)$ if $\\frac{P_2}{P_1}>\\frac{P_2}{P_1}^\\prime $ $\\sigma _2=$ 0.0765$^{+0.0081}_{-0.0066}$ (RMS $\\Delta I = 6.15^{+0.65}_{-0.53}~^{\\circ })$ $\\sigma _2: $ log-uniform [-5,5] $\\frac{P_2}{P_1}^\\prime =$ 5.38$^{+0.56}_{-0.86}$ $\\frac{P_2}{P_1}^\\prime $ : log-uniform [0,2]"
]
] | 1808.08475 |
[
[
"Symmetry deduction from spectral fluctuations in complex quantum systems"
],
[
"Abstract The spectral fluctuations of complex quantum systems, in appropriate limit, are known to be consistent with that obtained from random matrices.",
"However, this relation between the spectral fluctuations of physical systems and random matrices is valid only if the spectra are desymmetrized.",
"This implies that the fluctuation properties of the spectra are affected by the discrete symmetries of the system.",
"In this work, it is shown that in the chaotic limit the fluctuation characteristics and symmetry structure for any arbitrary sequence of measured or computed levels can be inferred from its higher-order spectral statistics without desymmetrization.",
"In particular, we consider a spectrum composed of $k>0$ independent level sequences with each sequence having the same level density.",
"The $k$-th order spacing ratio distribution of such a composite spectrum is identical to its nearest neighbor counterpart with modified Dyson index $k$.",
"This is demonstrated for the spectra obtained from random matrices, quantum billiards, spin chains and experimentally measured nuclear resonances with disparate symmetry features."
],
[
"Analytical expression",
"For a given sequence of uncorrelated eigenvalues, $E_1\\le E_2\\le \\cdots E_N$ , the spacings between nearest neighbours is defined as $s_i=E_{i+1}-E_i, i=1,2, \\cdots N$ .",
"The distribution of these spacings is of the form $P(s)=e^{-s}$ , and hence distributions of spacings and spacing ratios for integrable quantum systems are termed Poissonian.",
"The ratios of nearest neighbour spacings for these systems are defined as $r_i=s_{i+1}/s_i, i=1,2, \\cdots N$ , and the distribution of these ratios is of the form $P(r)=1/(1+r)^2$[25].",
"Ratios of higher order spacings may be defined as $r_i^{(k)} = \\frac{s_{i+k}^{(k)}}{s_{i}^{(k)}} = \\frac{E_{i+2k}-E_{i+k}}{E_{i+k}-E_i}, ~~~~~~~i,k=1,2,3,\\dots .$ To obtain a form for the distribution of $r^{(k)}$ , the higher order spacings may be expressed in terms of nearest neighbour spacings as $s_{i}^{(k)}&=E_{i+k}-E_i \\\\ \\nonumber &=E_{i+k}-E_{i+k-1}+E_{i+k-1}-E_{i+k-2}+ \\cdots +E_i \\\\ \\nonumber &= s_{k}+ \\cdots +s_{i+1}+s_i.$ Then the distribution of $s_i^{(k)}$ may be calculated as the distribution of a sum of $k$ random variables $s_i$ , each of which is distributed as $P(s)=e^{-s}$ .",
"For simplicity, $s_i^{(k)}$ is denoted as $z$ below.",
"The distribution of $z$ is given by $P(z)= \\frac{e^{-z}z^{k-1}}{(k-1)!",
"}$ Then the distribution of higher order spacing ratios is simply the distribution of the quotient of two random variables, each of which is distributed as Eq.",
"REF .",
"This distribution may be calculated as $P_P^{(k)}(r)=\\int |z|P(rz)P(z)dz$ Substituting for $P(z)$ and $P(rz)$ from Eq.",
"REF , $P_P^{(k)}(r)&=\\int _0^\\infty |z| \\frac{e^{-rz}(rz)^{k-1}}{(k-1)!}",
"\\frac{e^{-z}z^{k-1}}{(k-1)!}",
"dz \\nonumber \\\\&= \\frac{r^{k-1}}{(k-1)!^2}\\int _0^\\infty z^{2k-1} e^{-z(r+1)} dz.$ This can be evaluated in terms of the incomplete gamma function $\\Gamma (x)$ as $P_P^{(k)}(r)&=\\frac{\\Gamma (2k)}{(k-1)!^2}\\frac{r^{k-1}}{(1+r)^{2k}} \\nonumber \\\\&= \\frac{(2k-1)!",
"}{\\big ((k-1)!\\big )^2}\\frac{r^{k-1}}{(1+r)^{2k}}.$ For $k=1$ , it reduces to the familiar form $\\frac{1}{(1+r)^2}.$ For $k=2$ , $P_P^{(2)}(r)=\\frac{6r}{(1+r)^4},$ for $k=3$ , $P_P^{(3)}(r)=\\frac{30r^2}{(1+r)^6},$ and for $k=4$ , $P_P^{(4)}(r)=\\frac{140r^3}{(1+r)^8}.$"
],
[
"Comparison of analytical form of $P_P^{(k)}(r)$ with results from physical systems",
" Figure: Higher order spacing ratio distributions for k=2k=2 to 4, for uncorrelated eigenvalues (upper panel, indigo), circular billiards (lower panel, red)and integrable spin chain obtained by setting η=0\\eta =0 in Eq.",
"6 of the main paper (lower panel, black).",
"The corresponding analytical result (Eq.",
"4 in themain paper) is also shown in all cases (upper and lower panels, broken blue curve).The effect of missing levels in a given sequence of superposed spectra is studied by randomly deleting a fixed percentage of levels, and then calculating higher order spacing ratios, from a superpostion of GOE spectra of dimension $N=40000$ .",
"In each case, $D(\\beta ^{\\prime })$ is calculated, and the value of $\\beta ^{\\prime }$ corresponding to the minima of $D(\\beta ^{\\prime })$ corresponds to the best fit.",
"Figure: β ' \\beta ^{\\prime } (for which D(β ' )D(\\beta ^{\\prime }) is minimum) as a function of percentage of missing levels obtained by evaluating the second (fourth)order spacing ratio distribution for a superposition of two(four) GOE spectra, plotted in red(blue).",
"The dashed lines correspond tothe value of β ' \\beta ^{\\prime } as predicted by Eq.",
"4 of the main paper.Fig.",
"REF shows the value of $\\beta ^{\\prime }$ (evaluated in steps of 0.1) plotted against the percentage of missing levels, when $P^k(r,1,m)$ is evaluated for a superposition of $m$ GOE spectra, where $m=2$ (blue) and $m=4$ (red).",
"According to Eq.",
"4 of the main paper, namely, $P^k(r,1,m)=P(r,\\beta ^{\\prime }), ~\\mbox{where}~ \\beta ^{\\prime }=m=k$ , the expected value of $\\beta ^{\\prime }$ is 2 (for $k=2$ ) and 4 (for $k=4$ ) respectively, given by the blue and red dashed lines in Fig REF .",
"It may be observed that assuming even a $10\\%$ fluctuation in the numerical evaluation of $\\beta ^{\\prime }$ , a significant deviation from the predicted $\\beta ^{\\prime }$ occurs only when about $20\\%$ of the levels are missing.",
"A similar behavior was seen for spin chains with 2 and 4 irreps as well (not shown)."
]
] | 1808.08541 |
[
[
"Rolling and no-slip bouncing in cylinders"
],
[
"Abstract The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations.",
"A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down.",
"The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart.",
"Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact.",
"When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration.",
"Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded.",
"This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions.",
"While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics."
],
[
"Introduction",
"A classical example of a non-holonomic mechanical system consists of a ball that rolls against the inner side of a vertical cylinder with enough speed so as not to lose contact with the surface.",
"We imagine that the surface of the ball is ideally rough, or rubbery, so that a kind of conservative static friction causes it to roll without slipping.",
"Contrary to common intuition, the ball does not fall to the ground, but oscillates harmonically up and down.",
"This ideal behavior is approximately reproduced in lab experiments; see, for example, [9].",
"(For a study of this system in the context of the theory of nonholonomic systems see also [2] and [13].",
"We give below a fairly complete description of it in a form that will serve our present needs.)",
"Rather than rolling, we imagine that the idealized ball bounces off the cylinder's inner wall in a kind of grazing motion, but still under the same kind of conservative static friction constraint that couples the linear and rotational components of the motion.",
"Would the ball still defy gravity, so to speak, as in the rolling process?",
"This question introduces a number of issues.",
"First, how should we define such a discrete system in a way that is reasonably well-motivated, starting from general physical principles such as energy conservation and time reversibility?",
"A good candidate has been studied before under the name of no-slip billiard systems.",
"For some early papers, see [8], [3], [16].",
"See also [10], [12], as an example of how no-slip collision models are used in situations where exchange of linear and angular momentum of particles is desired.",
"Recently, the authors have begun to pursue a more systematic investigation of the dynamics of no-slip billiards; see [5], [6], [7].",
"Except for [5]—a differential geometric study of rigid collisions in $\\mathbb {R}^n$ in which no-slip billiards arise in a natural way—we are only aware of studies that are restricted to dimension 2.",
"Clearly, we need here to consider such systems in dimension 3 (or greater, if one is also interested, as we are, in related problems of a more differential geometric flavor.)",
"Another issue to consider is whether any particular choice of dynamical system defined by sequences of impacts can be said to serve as a good model for a theory of discrete nonholonomic dynamics.",
"Concretely, does the discrete model exhibit similar properties as the continuous time system, such as having bounded (e.g., not falling to the ground) trajectories; and do grazing trajectories approximate the well-studied rolling motion?",
"The purpose of this paper is twofold.",
"First, we wish to pursue the problem of existence of bounded trajectories of no-slip billiards in generalized cylinders in dimensions 2 and greater.",
"By a cylinder we mean a domain in $\\mathbb {R}^n$ that has translation symmetry along an axis.",
"(The 3-dimensional circular cylinder will have some prominence here, but we also consider other domains.)",
"In dimension 2, [3] showed that the no-slip billiard motion in an infinite strip is bounded, and in [6], [7] we extended and refined this observation in a way that provides some insight into the dynamics of general polygonal no-slip billiards.",
"As might be expected, the higher dimensional story is more subtle; we describe in this paper some of the new phenomena that arise beyond dimension 2.",
"Another goal is to make a direct comparison between the rolling and no-slip (bouncing) dynamics.",
"One of our main results here, which is specific to circular cylinders in dimension 3, is that for a set of initial conditions that we refer to as transversal rolling impact, the billiard motion is indeed bounded.",
"We also show numerically that the discrete system very closely approximates rolling under the just mentioned class of initial conditions.",
"On the other hand, if these conditions are not satisfied, it is observed numerically that the ball will acquire an overall acceleration under an external force.",
"There are many natural questions that we do not yet pursue in this study.",
"For example, when can the discrete system be said to be integrable?",
"(See [2] for a proof of boundedness of orbits for rolling in 3-dimensional cylinders of arbitrary cross-section using the existence of integrals of motion; our proof that no-slip billiard motion between parallel hyperplanes in $\\mathbb {R}^n$ under constant force is bounded also makes used of conserved quantities in a suggestive way.)",
"What can be said about the preservation of the canonical Liouville measure?",
"(We give sufficient conditions for the invariance of this measure in [5], and show that the measure is preserved for no-slip billiards in dimension 2, but the question is as yet open in higher dimensions.",
"The non-preservation of the Liouville measure is a feature of many non-holonomic mechanical systems.)",
"Moreover, proving that the differential equations for rolling can be obtained as a limit of the discrete equations of the no-slip billiard with short intercollision flights is a natural question that has eluded us so far.",
"Our numerical experiments suggest that a certain transversal rolling impact defect parameter introduced below should play a central role.",
"The focus of the present paper is not on such general issues but is aimed mainly at pointing out interesting phenomena that can be observed when comparing rolling and bouncing motion in such velocity constrained systems, as well as proving results on boundedness of orbits for generalized cylinders that do not have a rolling counterpart.",
"The rest of the paper is outlined as follows.",
"The next section summarizes the paper's main observations.",
"Section sets up the geometric notation and background for defining the no-slip collision map.",
"Section gives a self-contained overview of rolling on cylinders with general cross section.",
"Sections , , and give proofs of the main results.",
"We conclude in Section with some numerical observations and a proposal for future work on no-slip billiards in cylinders whose transverse dynamics exhibit chaotic behavior."
],
[
"Main definitions and results",
"We begin with a few definitions and, in particular, recall the notion of no-slip billiards.",
"Let $D=D_0(r)$ denote the ball of radius $r$ centered at the origin in $\\mathbb {R}^n$ .",
"It is given a mass distribution measure $\\mu $ having total mass $m$ .",
"We assume that the first moment $\\overline{x}=\\int _Dx\\, d\\mu (x)$ is zero and let $L=(l_{ij})$ be the matrix of second moments per unit mass: $l_{ij}:=\\frac{1}{m} \\int _{D} x_ix_j\\, d\\mu (x).",
"$ Only mass distributions for which $L$ is scalar, $L=\\lambda I$ , where $I$ is the identity matrix in dimension $n$ , will be considered here.",
"When $\\mu $ is rotationally symmetric and has a continuous density relative to the volume measure, then, expressed in terms of the density as a function of the radial coordinate, $ \\lambda =\\frac{V_n}{m}\\int _0^r s^{n+1} \\rho (s)\\, ds$ where $V_n$ is the volume of an $n$ -dimensional ball of radius 1.",
"It is convenient to introduce the parameter $\\gamma =\\sqrt{2\\lambda }/r$ .",
"For example, $\\lambda = \\frac{r^2}{n+2}$ and $\\gamma =\\sqrt{\\frac{2}{n+2}}$ if $\\rho =m/V_n r^n$ is constant.",
"It is not difficult to see that, for a general rotationally symmetric mass distribution, $0\\le \\lambda \\le r^2/n$ and $0\\le \\gamma \\le \\sqrt{\\frac{2}{n}}$ .",
"Let $\\mathcal {B}_0$ be an open connected region in $\\mathbb {R}^n$ .",
"Let $\\mathcal {B}$ be the closure of the set of $x\\in \\mathbb {R}^n$ for which the ball of center $x$ and radius $r$ is contained in $\\mathcal {B}_0$ .",
"We assume that $\\mathcal {B}$ is a manifold with corners as defined in [11].",
"We refer to $\\mathcal {B}$ as the billiard domain and, occasionally, to $\\mathcal {B}_0$ as the enlarged billiard domain.",
"For regular points $a$ of the boundary $\\partial \\mathcal {B}$ (that is, a point at which a tangent space is defined) define $\\nu (a) := \\nu _a$ to be the unit normal vector field pointing towards the interior of $\\mathcal {B}$ .",
"Denote by $SE(n)$ the Euclidean group of positive isometries of $\\mathbb {R}^n$ .",
"Its elements will be written as pairs $(A,a)\\in SO(n)\\times \\mathbb {R}^n$ acting on $\\mathbb {R}^n$ by affine transformations $x\\mapsto (A,a)x:=Ax+a$ on $\\mathbb {R}^n$ .",
"The configuration manifold of a spherical particle of radius $r$ with center in $\\mathcal {B}$ is the manifold with corners $M\\subset SE(n)$ consisting of all $(A,a)$ such that $a\\in \\mathcal {B}$ .",
"Naturally, a boundary point of $M$ is a pair $(A,a)$ such that $a\\in \\partial \\mathcal {B}$ .",
"In this paper we are interested in rolling and bouncing motion in cylinders, by which we will mean the following, unless further geometric assumptions are made: Definition 1 (Cylinders in $\\mathbb {R}^n$ ) A solid cylinder in $\\mathbb {R}^n$ with axis vector $e$ is a domain $\\mathcal {B}$ such that $a+se\\in \\mathcal {B}$ for all $a\\in \\mathcal {B}$ and $s\\in \\mathbb {R}$ .",
"Here $e$ is a unit vector in $\\mathbb {R}^n$ that defines which direction is up.",
"Writing $\\overline{\\mathcal {B}}=\\mathcal {B}\\cap e^\\perp $ , we have $\\mathcal {B} = \\overline{\\mathcal {B}}\\times \\mathbb {R}e$ .",
"The boundary cylinder will be written $S=\\overline{S}\\times \\mathbb {R}e$ , where $S=\\partial \\mathcal {B}$ .",
"As far as the rolling process is concerned, all we will need is the boundary cylinder $S$ ; the solid cylinder will be needed for the no-slip billiard systems.",
"For rolling we also assume that $S$ is a smooth hypersurface in $\\mathbb {R}^n$ , whereas for the billiard motion we may allow singular points so long as we consider orbits that avoid them.",
"From the spherically symmetric mass distribution $\\mu $ with second moments matrix $L=\\lambda I$ , $\\lambda = \\frac{1}{2}(r\\gamma )^2$ , we define the kinetic energy Riemannian metric on $M$ as follows: Let $\\xi =(U_\\xi A, u_\\xi )$ and $\\eta =(U_\\eta A, u_\\eta )$ be vectors tangent to $M$ at $(A,a)$ .",
"Then $\\left\\langle \\xi ,\\eta \\right\\rangle :=m\\left\\lbrace \\frac{(r\\gamma )^2}{2}\\text{Tr}\\left(U_\\xi U_\\eta ^\\dagger \\right) + u_\\xi \\cdot u_\\eta \\right\\rbrace .",
"$ It is easily checked that this bilinear form is (the restriction to $TM$ of) a left-invariant Riemannian metric on $SE(n)$ .",
"Observe that if $(A(t),a(t))$ is a differentiable curve in $M$ , then its derivative at $t=0$ is given by $(\\dot{A},\\dot{a})$ where $\\dot{a}=u\\in \\mathbb {R}^n$ and $\\dot{A}=UA$ , where $U\\in \\mathfrak {so}(n)$ is an element of the Lie algebra of the rotation group.",
"The kinetic energy of the moving ball with state $\\xi $ at configuration $(A,a)$ is then written in the norm associated to the Riemannian metric as $\\frac{1}{2}\\Vert \\xi \\Vert ^2$ .",
"Notice that the metric and the kinetic energy function do not depend on $A$ .",
"The boundary of the configuration manifold $M$ has a special structure that will be important for our concerns, which we call the no-slip bundle.",
"It is a vector subbundle of the tangent bundle to $\\partial M$ , denoted $\\mathfrak {S}$ , and defined as follows.",
"Definition 2 (The no-slip bundle) At each regular point $q=(A,a)\\in SE(n)$ , $a\\in S=\\partial \\mathcal {B}$ , describing a boundary configuration of the moving particle system, define $\\mathfrak {S}_q=\\left\\lbrace (UA,u)\\in T_qM: u=rU \\nu _a \\right\\rbrace $ where $r$ is the radius of the particle.",
"We call $\\mathfrak {S}_q$ the no-slip space at $q$ .",
"This definition has a clear motivation if we note that the point of contact $x= a-r\\nu _a$ of the moving particle with the boundary of the extended domain $\\mathcal {B}_0$ has velocity $ v_x = U(x-a)+u = -rU\\nu _a+ rU\\nu _a = 0.$ Thus a state of the system (that is, a tangent vector at some point of $M$ ) at a boundary configuration $q$ is in the no-slip bundle if the point of contact of the particle with the boundary of the domain $\\mathcal {B}_0$ has 0 velocity.",
"(The reader should keep in mind the distinction between the billiard domain $\\mathcal {B}$ and the enlarged domain $\\mathcal {B}_0$ ; the former contains the center of masses of the particle, and the latter contains the entire ball of radius $r$ around those centers in $\\mathcal {B}$ .)",
"Definition 3 (No-slip rolling) A particle whose motion is described by a smooth curve $q(t)\\in \\partial M$ is said to undergo (no-slip) rolling if $\\dot{q}(t)\\in \\mathfrak {S}_{q(t)}$ for each $t$ .",
"We also say in this case that the particle rolls with no slip on the boundary surface of the (enlarged) billiard domain.",
"We consider next the dynamical equations describing the motion of a particle of radius $r$ , mass $m$ , and rotationally symmetric mass distribution with parameter $\\gamma $ , that rolls with no-slip on the boundary of the enlarged domain $\\mathcal {B}_0$ .",
"With some innocuous abuse of language we speak of rolling on $S=\\partial \\mathcal {B}$ , the locus of the centers of mass of the moving particle, rather than the hypersurface of contact points.",
"Keeping this in mind, we will avoid when possible referring to $\\mathcal {B}_0$ .",
"For the details on non-holonomically constrained systems we suggest [1].",
"Let $f$ be a vector field on $M$ which we interpret as a force field.",
"The motion of the unconstrained system is governed by Newton's equation $m\\frac{\\nabla \\dot{q}}{dt}=f$ , where $\\nabla $ is the Levi-Civita connection for the kinetic energy Riemannian metric (REF ).",
"The constraint (rolling on the boundary hypersurface $S$ ) can be imposed by adding a force field $N$ taking values in $\\mathfrak {S}^\\perp $ .",
"This is the force needed to keep $\\dot{q}$ in $\\mathfrak {S}_q$ .",
"As $\\left\\langle N, \\dot{q}\\right\\rangle _q=0$ , the constraint force does no work.",
"Definition 4 (Constrained Newton's equation) A smooth path $q(t)\\in SO(n)\\times S$ satisfies the constrained Newton's equation with force field $f$ and non-holonomic constraint defined by the no-slip bundle $\\mathfrak {S}$ if $\\dot{q}(t)\\in \\mathfrak {S}_{q(t)}$ for all $t$ and $\\frac{\\nabla \\dot{q}}{dt} = m^{-1}f +N $ where $N=N(q,\\dot{q})$ lies in $\\mathfrak {S}^\\perp $ .",
"So far the interior of $\\mathcal {B}$ has not been relevant since in the rolling process the center of the moving particle must remain on the boundary hypersurface $S$ .",
"This will change, of course, as we consider next the notion of no-slip bouncing.",
"Before stating the definition, we recall the set-up of no-slip billiard systems.",
"(The reader is referred to [5] for a detailed account of what is briefly skimmed over below, and to [7] for some dynamical results for 2-dimensional systems.)",
"In a billiard system in $\\mathcal {B}$ , the motion of the particle (of radius $r$ and spherically symmetric mass distribution of total mass $m$ and distribution parameter $\\gamma $ ) consists of a sequence of flight segments in the interior of $M\\subset SE(n)$ separated by instantaneous collisions with the boundary $\\partial M$ .",
"The inter-collision flights are described by the unconstrained Newton's equation $m\\frac{\\nabla \\dot{q}}{dt}=f$ , or by geodesic motion $\\frac{\\nabla \\dot{q}}{dt}=0$ if no forces are present.",
"Collisions at a given point $q\\in \\partial M$ are defined by a linear map $C_q:T_qM\\rightarrow T_qM$ that sends pre-collision vectors, that is, vectors in $\\lbrace v\\in T_qM: \\langle v,\\mathbb {n}_q\\rangle \\le 0\\rbrace $ to post-collision vectors in $\\lbrace v\\in T_qM: \\langle v,\\mathbb {n}_q\\rangle \\ge 0\\rbrace $ , where $\\mathbb {n}_q$ is the inward pointing unit vector at a boundary point $q\\in M$ .",
"The map $C_q$ will be selected based on the following physical assumptions: (1) Collision is energy preserving; this means that $C_q$ is an orthogonal map relative to the Riemannian norm on $M$ .",
"(2) It is time reversible, which forces $C_q$ to be a linear involution.",
"(3) The orthogonal component of the pre-collision velocity in the no-slip subspace $\\mathfrak {S}_q$ is not affected by $C_q$ .",
"This third requirement is natural since an (instantaneously) rolling collision, for which the point of contact is stationary, should not cause a change in linear or angular velocities due to an exchange of momentum at impact, except in the direction $\\mathbb {n}$ .",
"In fact, a more careful analysis of the impact event (as in [5]) identifies the orthogonal complement of $\\mathfrak {S}_q$ as the space containing impulse vectors.",
"Ordinary billiard systems are those for which $C_q$ is the identity not only on $\\mathfrak {S}_q$ but also on the subspace of $\\mathfrak {S}^\\perp _q$ perpendicular to $\\mathbb {n}_q$ , in which case rotation and translation components of the motion are de-coupled and the rotation part may be ignored.",
"In other words, for standard billiards, the collision maps $C_q$ are specular reflection (in the kinetic energy inner product).",
"This corresponds to a perfectly slippery contact between the moving particle and the boundary of the billiard domain.",
"If instead we wish to model the behavior of a perfectly elastic ball with a perfectly rough (or rubbery) surface, for which angular and linear velocities may be partly exchanged at collision, then we are forced under the other assumptions to require $C_q$ to be the negative of the identity map on the full orthogonal complement of the no-slip subspace (which here includes $\\mathbb {n}_q$ .)",
"With this in mind we state the following definition.",
"Definition 5 (No-slip bounce) A no-slip bounce at $q\\in \\partial M$ is the correspondence $v^-\\mapsto v^+$ sending a pre- to a post-collision velocity at $q$ defined by the no-slip collision map $C_q$ .",
"The latter is the orthogonal (relative to the kinetic energy Riemannian metric (REF )) linear involution of $T_qM$ equal to the identity on $\\mathfrak {S}_q$ and minus the identity on this space's full orthogonal complement.",
"Due to the spherical symmetry of the moving particle's mass distribution and the discrete nature of the billiard process, it is less relevant for no-slip billiard systems that we keep track of the actual rotation matrix $A$ .",
"In fact, prior to each collision, we can imagine that the particle is rotated back to a fixed reference orientation in space, keeping linear and angular velocities unaffected.",
"This leads to the notion of reduced phase space $\\mathcal {N}$ at boundary configurations.",
"With $S=\\partial \\mathcal {B}$ , we define $\\mathcal {N}:= \\left.T\\mathcal {B}\\right|_{S} \\times \\mathfrak {so}(n)$ whose elements are triples $(a, u, U)$ , $u\\in T_a\\mathcal {B}$ .",
"In other words, we may assume that, at each collision, the rotation matrix is the identity and only keep track of the matrix $U\\in \\mathfrak {so}(n)$ of infinitesimal angular velocities, in addition to the point $(a,u)$ representing the velocity of the center of mass at $a\\in S$ .",
"Since $A$ will not be involved, we write $C_a$ rather than $C_q$ when viewing it as a map on $\\mathcal {N}_a:=T_a\\mathcal {B}\\times \\mathfrak {so}(n).$ We now turn to no-slip billiards on solid cylinders $\\mathcal {B}$ with axis vector $e$ .",
"If $a$ is a point in $\\mathcal {B}$ , we occasionally write $\\bar{a}$ for its projection to the cross-section $\\overline{\\mathcal {B}}$ ; similarly, if $u$ is a tangent vector at $a$ , its projection to a vector at $\\bar{a}$ on the cross-section may be written $\\bar{u}$ .",
"The cross-section domain $\\overline{\\mathcal {B}}$ has its own no-slip billiard system, with a ball of dimension $n-1$ as the moving particle.",
"The latter will also have a spherically symmetric mass distribution (if this is the case for the $n$ -dimensional particle) given by the marginal mass distribution after integrating along $e$ .",
"In this case the parameter $\\gamma $ for the $(n-1)$ -dimensional particle is the same as for the $n$ -dimensional one.",
"For standard billiard systems on a cylinder (for which reflection is specular), it is both clear and unremarkable that trajectories of the $n$ -dimensional system should project to trajectories of the billiard system on $\\overline{\\mathcal {B}}$ .",
"This is due to conservation of momentum resulting from the translation symmetry along $e$ .",
"For the no-slip billiard, this component of momentum is no longer conserved.",
"Nevertheless this projection property still holds.",
"Theorem 6 Let $\\mathcal {N}$ be the reduced phase space of the no-slip billiard system on the solid cylinder domain $\\mathcal {B}\\subset \\mathbb {R}^n$ , and let $\\overline{\\mathcal {N}}$ be the reduced phase space for the associated transverse billiard system.",
"Then trajectories of the no-slip billiard on $\\mathcal {N}$ , possibly with a constant force in the longitudinal direction, project to trajectories of the no-slip billiard map on $\\overline{\\mathcal {N}}$ , where the latter system is given the same mass distribution parameter $\\gamma $ as the billiard in dimension $n$ .",
"Figure: No-slip billiard system between two parallel plates under gravity have bounded orbits in all dimensions.",
"Far left: a simple periodic orbit with gravity turned off.",
"Orbits shown from left to right are under the influence of increasing force.",
"The above theorem only expresses part of a very useful decoupling of the full set of linear and angular velocity components into two that will greatly facilitate the study of the displacement of the particle's center of mass along the longitudinal direction.",
"(See Proposition REF .)",
"To anticipate what is to come, we note that the Lie algebra $\\mathfrak {se}(n)$ of the Euclidean group $SE(n)$ splits into the Lie algebra $\\mathfrak {se}(n-1)$ of $SE(n-1)$ , the group associated with the transversal system, and the orthogonal complement $\\mathfrak {m}$ of the latter Lie algebra with respect to the kinetic energy metric.",
"The no-slip collision map will factorize according to this orthogonal decomposition, and our main interest will be on the factor $\\mathfrak {m}$ since this is the one that contains the $e$ -component of the center of mass velocity.",
"The subspace $\\mathfrak {m}$ has dimension $n$ ; this means that, so long as we are concerned with the longitudinal motion, we only need to focus on $n$ out of $\\frac{n(n+1)}{2}$ velocity components (the latter number being the dimension of $SE(n)$ ).",
"Therefore, an investigation of the motion of the moving particle naturally splits into a study of the transverse billiard system on $\\overline{\\mathcal {B}}$ and a reduced system containing the component of the motion along $e$ .",
"For 3-dimensional no-slip billiards in cylinders, this theorem says that the transversal part of the motion reduces to understanding no-slip billiard dynamics in dimension 2.",
"In the course of this paper we refer to a few results in dimension 2 established in our [7].",
"In general however the focus of the rest of the paper is on the motion along the axis of translation symmetry set by the axis vector $e$ , rather than on the transverse dynamic, and in particular on the question of boundedness of orbits.",
"Note that we will refer to the direction along $e$ as the longitudinal direction or the vertical direction, using the terms interchangeably.",
"We now summarize our main results concerning the question of whether or not trajectories are bounded in the longitudinal direction of the cylinder.",
"As will be seen, the general answer is: sometimes yes, and sometimes no.",
"The following theorem extends the main result of [3] from free motion in 2-dimensional strips to possibly forced motion in $n$ -dimensional regions bounded by parallel hyperplanes.",
"(This domain is a cylinder, according to our general definition.)",
"Theorem 7 Consider a domain whose boundary consists of two parallel hyperplanes in $\\mathbb {R}^n$ , $n\\ge 2$ .",
"Then a trajectory of the no-slip billiard system whose initial center of mass velocity is not parallel to the hyperplanes is bounded.",
"Trajectories remain bounded if a constant force is applied to the particle's center of mass along any direction parallel to those hyperplanes.",
"Next we ask wether the property of having bounded orbit holds for general cylinders.",
"We show that even in the absence of forces the motion may not be bounded.",
"Typically the height of particle's center of mass will possess a drift in one direction along the cylinder axis $e$ superimposed to an oscillatory part.",
"Figure: Conditions for a transversal period 2 orbit in dimension 3.",
"The projection to ℝ 2 \\mathbb {R}^2 of the velocity uu of the center of mass and the angular velocityθ ˙\\dot{\\theta } are related by |θ ˙|=(mr/𝒥)|usinφ||\\dot{\\theta }|=(m r/\\mathcal {J})|u\\sin \\phi |, where mm is the projected disc's mass, rr is itsradius, and ℐ\\mathcal {I} is its moment of inertia for the projected (or marginal) mass distribution.Before validating this claim, we observe that, due to Theorem REF , it makes sense to talk about transversely periodic trajectories of the cylindrical billiard.",
"In dimension 3, transversal period 2 orbits are very ubiquitous and easy to obtain.",
"(See Section 3 of [7].)",
"Figure REF shows the conditions under which they arise.",
"Observe that, for $n=3$ , the constraint equation on initial conditions for period 2 orbits involves only the projection of the linear velocity to the cross-section plane and of the $e$ -component of the angular velocity vector $\\omega $ (which is $\\dot{\\theta }$ as indicated in Figure REF ).",
"In particular, no conditions are imposed on the velocity components that appear in the description of the longitudinal motion.",
"What follows is a corollary of Theorem REF .",
"(The theorem holds for general cylinders in arbitrary finite dimension.)",
"Referring to Figure REF , we regard $\\tau $ and $\\nu $ as the tangent and normal vectors at the first collision point and $\\omega $ as the pre-collision angular velocity at that point; $\\phi $ is as indicated in that figure.",
"Corollary 8 A transversal period 2 orbit of a 3-dimensional general cylinder billiard, under no forces, with initial linear vertical velocity $\\sigma _0$ and initial angular velocity vector $\\omega $ has a vertical drift $ \\lim _{\\ell \\rightarrow \\infty }\\frac{h_\\ell }{\\ell }=\\frac{ \\sigma _0\\tan \\phi + \\gamma ^2 r\\left(\\omega _\\nu +\\omega _\\tau \\tan \\phi \\right)}{\\tan \\phi +2\\gamma ^2 },$ where $\\omega _\\nu $ and $\\omega _\\tau $ are the $\\nu $ and $\\tau $ components of $\\omega $ , and $h_\\ell $ is the height (ie.",
"signed vertical displacement) after $\\ell $ collisions.",
"The condition on the orbit for having transversal period 2 does not restrict the values $\\sigma _0, \\omega _\\nu , \\omega _\\tau $ .",
"Thus the motion is generically unbounded in the vertical direction.",
"If the vertical drift is 0, the motion is bounded.",
"We wish next to compare the no-slip billiard system in circular cylinders with the rolling process.",
"Let us first review the classical fact about bounded orbits for the rolling motion.",
"Proposition 9 Suppose that the cross-section of the 3-dimensional vertical cylinder is a differentiable simple closed curve and that $\\omega _e$ —the vertical component of the angular velocity vector, a constant of motion—is non-zero.",
"Then trajectories of the rolling motion under a constant force parallel to the axis of the cylinder are bounded.",
"Implicit in this statement is the assumption that the particle is constrained to remain in contact with the surface.",
"Figure: Cross-section of the stadium cylinder and the rolling sphere.",
"As an illustration, consider the stadium cylinder whose cross-section, depicted in Figure REF , consists of two circular caps connected by parallel line segments.",
"A glance at the second order equation for the height function $h$ in Theorem REF shows that the ball is essentially in free fall (with acceleration $g/(1+\\gamma ^2)$ ) while it rolls on the flat parts of the surface.",
"In order to remain bounded, it must rebound upward when it passes over the curved caps.",
"Figure REF shows a typical height function.",
"In the final section of the paper we will briefly explore numerically the no-slip billiard version of this example.",
"It will be apparent that the question whether (and under which initial conditions) orbits remain bounded is much more challenging when the transverse no-slip billiard system is chaotic.",
"Let us return to the circular cylinder.",
"Compared to rolling motion, the behavior of a no-slip billiard system inside a cylindrical billiard domain, in the presence of a constant force pulling the particle downward, seems to be more subtle.",
"On the one hand, it is possible, and typical for general cylinders, for the particle to accelerate downward (as one might expect).",
"See Figure REF .",
"Figure: Height of center of mass of rolling particle in a cylinder with stadium cross-section.",
"But in dimension 3 and for ordinary circular cylinders, we show that for a class of initial conditions satisfying what we call transversal rolling impact, the particle does not fall: its position along the axis of the cylinder remains bounded.",
"Prior to stating the definition of rolling impact, observe that if $(a,u,U)\\in \\mathcal {N}$ is the pre-collision state at a boundary point $a$ , then the velocity of the point of contact $x= a-r\\nu _a$ of the spherical billiard particle of radius $r$ and the boundary of $\\mathcal {B}_0$ is $v=u+U(x-a)=u-rU\\nu _a$ .",
"We say that the collision satisfies the rolling impact condition if the component of $v$ tangent to the boundary of $\\mathcal {B}$ at $a$ is zero.",
"Definition 10 (Transversal rolling impact) The pre-collision state $(a,u,U)\\in \\mathcal {N}$ will be said to satisfy the rolling impact condition if the orthogonal projection of $v=u - rU\\nu _a$ to $T_a\\partial \\mathcal {B}$ is zero.",
"If the billiard domain is a cylinder (not necessarily circular) whose axis is parallel to the unit vector $e\\in \\mathbb {R}^n$ , we say that $(a, u, U)$ satisfies the transversal rolling impact condition if the orthogonal projection of $v$ to $e^\\perp \\cap T_a\\partial \\mathcal {B}$ is zero.",
"Figure: Comparison of the height functions for the rolling motion in a circular cylinder under a constant downward force (solid line) and the corresponding no-slip billiard motion satisfying the transverse rolling impact condition (small circles).",
"Initial conditions are chosen so that the two processes rotate around the cylinder at the same rate.The above definition of rolling impact is equivalent to the following: at each boundary configuration $q\\in \\partial M$ of the no-slip billiard, an initial state $v\\in T_qM$ projects to a vector in the no-slip subspace $\\mathfrak {S}_q$ .",
"Here we are referring to the orthogonal projection relative to the kinetic energy inner product on $T_qM$ , whereas in Definition REF it is the ordinary Euclidean inner product that is being invoked.",
"Also notice that transversal rolling impact means that the rolling impact condition holds for the transversal billiard system, which is well-defined due to Theorem REF .",
"Figure: When the transversal rolling impact condition does not hold (see Definition ), the particle acquires an overall acceleration downward.",
"The apparent increase in thickness of the height function graph and of the particle's path is due to a small scale vertical zig-zagmotion of increasing amplitude.",
"See also Figure .For cylinders in dimension 3, let $\\tau _a$ denote the unit tangent vector to $\\partial \\mathcal {B}$ at $a$ , oriented so that $\\tau _a, \\nu _a, e$ form a positive basis.",
"Let $\\omega \\in \\mathbb {R}^3$ be the angular velocity vector.",
"(We recall that $\\omega $ is defined by $\\omega \\times x:= Ux$ for all $x\\in \\mathbb {R}^3$ .)",
"Then the rolling impact condition is in this case expressed by the equation $u-u\\cdot \\nu _a\\nu _a = r\\omega \\times \\nu _a $ and the transversal rolling impact condition by $u\\cdot \\tau _a +r\\omega \\cdot e=0$ as a simple algebraic manipulation involving the cross-product shows.",
"We take as a measure of the failure in satisfying this condition the quantity $-r\\omega \\cdot e/u\\cdot \\tau _a$ and call it the transversal rolling defect (evaluated on the initial velocities).",
"The following simple observation is essential.",
"Proposition 11 Consider a two-dimensional no-slip billiard system in a disc.",
"If the first collision satisfies the rolling impact condition, then all subsequent collisions also do, and the times between consecutive collisions are all equal.",
"Furthermore, the center of mass of the moving particle undergoes specular reflection at each collision.",
"The main theorem for no-slip billiards in circular cylinders is now the following.",
"Theorem 12 Consider a no-slip billiard system in a circular cylinder in $\\mathbb {R}^3$ whose moving particle is subject to a constant force directed along the axis of the cylinder.",
"If the first collision satisfies the transversal rolling impact condition and the first flight segment does not go through the axis of the cylinder, then the particle's trajectory is bounded.",
"Figure REF shows that the rolling motion and the billiard motion consisting of a sequence of short no-slip bounces, under the assumption that the initial velocities satisfy the transversal rolling impact condition are, in fact, very close.",
"One may be inclined to think that the equations for the billiard motion are a simple-minded discretization of the differential equation for the rolling motion, or that the latter is a straightforward limit of the former, but this seems not to be the case.",
"It is essential here to bear in mind the phenomenon illustrated in Figure REF , which shows that if the transversal rolling defect is not 1, then in the continuous limit one obtains a kind of motion that appears to be smooth but is very different from that of solutions of the rolling differential equation.",
"The small scale zig-zag motion shown more clearly in Figure REF , which has a close-up of segments of trajectories highlighting the effect of introducing a small transversal rolling defect, suggests a potential difficulty to overcome.",
"It would be most interesting to find a limit differential equation containing this rolling defect as an equation parameter, and the rolling equations as a special case when the parameter is 1.",
"Such an equation would, hopefully, suggest a possible physical interpretation of this key parameter.",
"We leave this as a problem to be addressed in a future work.",
"Figure: These graphs correspond to a fixed initial transversal rolling defect -rω·e/u·τ a =1.15.-r\\omega \\cdot e/u\\cdot \\tau _a=1.15.",
"(When the transversal rolling impact initial condition holds, this value is 1.",
")Here ω·e\\omega \\cdot e is the longitudinal component of the initial angular velocity vector and u·τ a u\\cdot \\tau _a is the tangential (to the boundary of the billiard domain) component of the center of mass velocity.As the intercollision flight becomes shorter and motion grazes the cylinder more and more closely,the height function becomes smooth but the falling rate remains essentially unchanged.The proofs of the above theorems and propositions, and of some of the more general facts to be stated shortly that are not restricted to dimension 3, will be given in the subsequent sections.",
"Although our more complete observations pertain to 3 dimensional domains, we have chosen to state and prove our results, whenever we can, in arbitrary dimensions.",
"We believe that this subject touches on a number of questions of geometric/dynamical interest, for example, concerning a possible theory of discrete non-holonomic systems, for which it would be too restrictive to remain in dimension 3.",
"Partly for this reason, but also for the sake of completeness, we have also included reasonably detailed proofs of properties of rolling motion, such as the fact that general cylinders in dimension 3 have bounded orbits, even though this is a classical fact.",
"We believe that our more geometric approach, which avoids relying too much on background material from mechanics (like the use of such concepts as Coriolis torque, for example; see [9]) and highlights the central role of the Euclidean group, is worthwhile recording.",
"Figure: Near grazing paths of no-slip billiard particle.",
"On the left-hand side, the transversal rolling impact condition holds for the initial bounce, but on the right-hand side a small deviation of this condition is introduced.",
"Notice the characteristic zig-zag nature of the curve and that it does not return all the way to the initial height.",
"The starting center of mass position is indicated by the small black dot."
],
[
"More definitions and basic facts",
"There are two Riemannian metrics we need to consider: the one on $M$ defined above by (REF ), and the metric on the hypersurface $S=\\partial \\mathcal {B}$ induced (by restriction) from the Euclidean metric in $\\mathbb {R}^n$ .",
"Although the context should make it clear which is being referred to at any given moment, for example, when stating that vectors have unit length or are mutually orthogonal, we will always refer to the former as the kinetic energy metric.",
"We define the cross-product in $\\mathbb {R}^n$ as the bilinear map $(a,b)\\in \\mathbb {R}^n\\times \\mathbb {R}^n\\mapsto a\\wedge b\\in \\mathfrak {so}(n)$ taking on values in the Lie algebra $\\mathfrak {so}(n)$ of the rotation group, given by $u\\mapsto (a\\wedge b)u:= (a\\cdot u)b-(b\\cdot u)a$ for all $u\\in \\mathbb {R}^n.$ If $a$ and $b$ are unit orthogonal vectors and $V$ is the plane linearly spanned by $a$ and $b$ , then $\\exp (\\theta (a\\wedge b))=\\Pi ^{\\perp }+ (\\cos \\theta )\\Pi +(\\sin \\theta ) a\\wedge b\\in SO(n)$ is a rotation matrix that restricts to the identity transformation on $V^\\perp $ and rotates vectors in $V$ by angle $\\theta $ , where $\\Pi $ and $\\Pi ^\\perp $ are the orthogonal projections to $V$ and $V^\\perp $ , respectively.",
"(We also use $\\Pi $ , possibly with additional sub- or superscripts, for other orthogonal projections to appear in the course of this paper.",
"It will be clear in each case to which projection we are referring.)",
"It is useful to note that $\\frac{1}{2} \\text{Tr}\\left((a\\wedge b)(c\\wedge d)^\\dagger \\right) = (a\\cdot c) (b\\cdot d) - (a\\cdot d)( b\\cdot c) $ where $\\dagger $ denotes transpose.",
"If moreover $U$ is an $n\\times n$ matrix then $\\frac{1}{2} \\text{Tr}\\left(U(a\\wedge b)^\\dagger \\right) = (Ua)\\cdot b.",
"$ Notice that, if $n=3$ , $(a\\wedge b) u=(a\\times b)\\times u,$ where $\\times $ is the ordinary cross-product of vector algebra.",
"The kinetic energy Riemannian metric (REF ) is a product metric on $M=SO(n)\\times \\mathcal {B}$ .",
"The trace part is a bi-invariant metric on the rotation group.",
"It is a standard and easy to prove fact that the Levi-Civita connection on $SO(n)$ associated to this metric satisfies $\\nabla _XY=\\frac{1}{2}[X,Y]$ for any pair $X,Y$ of left-invariant vector fields.",
"The following proposition is also a standard fact about bi-invariant metrics.",
"Proposition 13 Let $A(t)$ be a smooth path in $SO(n)$ where the rotation group is given the above bi-invariant trace metric.",
"Define $U(t)=A(t)^{-1}\\dot{A}(t)\\in \\mathfrak {so}(n)$ , where $\\dot{A}$ indicates derivative in $t$ of the matrix-valued function.",
"Then $ \\frac{\\nabla \\dot{A}}{dt}= A\\dot{U}.$ In particular, $A(t)$ is a geodesic iff $U(t)$ is constant and $A(t)=A(0)e^{tU}.$ We often find it convenient to express tangent vectors to the Euclidean group $SE(n)$ at a point $(A,a)$ in the form $(UA, u)$ , where $U\\in {so}(n)$ and $u\\in \\mathbb {R}^n$ .",
"In this form, the vector is obtained by a right-translation from the identity $(I,0)$ to $(A,a)$ of an element of the Lie algebra $\\mathfrak {se}(n)$ of $SE(n)$ .",
"The reader should be attentive to the distinction between $(UA,u)$ and $(AU,Au)$ .",
"The latter arises when we wish to think of $(U,u)\\in \\mathfrak {so}(n)$ as an infinitesimal rotation expressed in the so-called body frame; on the other hand, when writing a tangent vector as $(UA, u)$ , one has the following interpretation: a material point $b\\in D$ which in configuration $(A,a)$ is at $x=Ab+a\\in \\mathbb {R}^n$ , will have velocity at $x$ equal to $U(x-a)+u.$ In particular, $a$ and $u$ are, respectively, the position and the velocity of the center of mass of the ball $D_0(r)$ in the state $(UA,u)\\in T_{(A,a)}M$ .",
"If $A(t)$ is a smooth curve in $SO(n)$ and $U(t)$ is now defined by $U(t)= \\dot{A}(t)A(t)^{-1}$ , then by Proposition REF $A^{-1}\\frac{\\nabla \\dot{A}}{dt}= \\frac{d}{dt}\\left(A^{-1}\\dot{A}\\right)= \\frac{d}{dt}\\left(A^{-1}U A\\right)= -\\left(-A^{-1}\\dot{A}A^{-1}\\right)UA + A^{-1}\\dot{U}A+ A^{-1}U\\dot{A},$ and the last term simplifies to $-A^{-1}U^2A+A^{-1}\\dot{U}A+ A^{-1}U^2A=A^{-1}\\dot{U}A.$ Thus we immediately have the following remark: Proposition 14 For a smooth curve $q(t)=(A(t),a(t))$ in $M$ , write $\\dot{q}=(U(t)A(t), u(t))$ , where $u=\\dot{a}$ and $U=\\dot{A}A^{-1}$ .",
"Then the acceleration of $q(t)$ relative to the Levi-Civita connection $\\nabla $ of the left-invariant (kinetic energy) metric (REF ) is $ \\frac{\\nabla \\dot{q}}{dt}=\\left(\\dot{U}A, \\dot{u}\\right).$ When the focus is on the geometry of the boundary of $M$ with the induced metric, and $q(t)\\in S:= \\partial M$ , then $\\dot{u}$ is replaced with $\\frac{D u}{dt}=\\dot{u}-(\\dot{u}\\cdot \\nu _a) \\nu _a $ where $D$ is the Levi-Civita connection of the hypersurface $S\\subset \\mathbb {R}^n$ and $\\nu _a$ is a unit normal vector to $S$ at $a$ .",
"Recall from Definition REF the no-slip bundle $\\mathfrak {S}$ .",
"A simple computation gives the explicit form of its orthogonal complement relative to the kinetic energy metric.",
"Notice that the unit normal vector to $\\partial M$ pointing into $M$ is $\\mathbb {n}(A,a)= \\frac{1}{\\sqrt{m}} (0, \\nu _a)$ , where $m$ is the mass of the moving particle.",
"Clearly, $\\mathbb {n}(q)$ lies in that orthogonal complement.",
"However, in what follows, it will be convenient to reserve the notation $\\mathfrak {S}^\\perp _q$ for the orthogonal vectors to $\\mathfrak {S}_q$ contained in $T_q(\\partial M)$ .",
"Thus defined, we have $\\mathfrak {S}^\\perp _{(A,a)}=\\left\\lbrace \\left(\\frac{1}{r\\gamma ^2} (w\\wedge \\nu _a) A,w\\right): w\\in T_aS\\right\\rbrace .$ Properties (REF ) and (REF ) of the product $\\wedge $ are useful for verifications of this kind.",
"We give now a more explicit description of the no-slip collision map on the reduced phase space $\\mathcal {N}$ introduced in Definition REF .",
"For details we refer the reader to [5].",
"The abbreviations $c_\\beta $ and $s_\\beta $ will be used throughout the rest of the paper for the quantities: $c_\\beta :=\\cos \\beta :=\\frac{1-\\gamma ^2}{1+\\gamma ^2}, \\ \\ s_\\beta :=\\sin \\beta := \\frac{2\\gamma }{1+\\gamma ^2}.$ The angle $\\beta $ is defined by these relations.",
"When there is no chance of confusion we may simply write $c, s$ .",
"Recall that $\\nu _a$ is the unit inward pointing normal vector to $S$ .",
"Proposition 15 (No-slip collision map, [5]) For each $a\\in S$ , the no-slip collision map at $a$ is the linear map $C_a:\\mathcal {N}_a\\rightarrow \\mathcal {N}_a$ such that $C_a(u,U)= \\left(c_\\beta u -\\frac{s_\\beta }{\\gamma } (u\\cdot \\nu _a) \\nu _a + s_\\beta \\gamma r U\\nu _a ,\\frac{s_\\beta }{\\gamma r}\\nu _a\\wedge u + U - \\frac{s_\\beta }{\\gamma }\\nu _a \\wedge U \\nu _a\\right).",
"$ Finally, let us also recall the following definition.",
"Definition 16 (Shape operator) The shape operator $\\mathbb {S}_a$ of the hypersurface $S\\subset \\mathbb {R}^n$ at $a$ is the symmetric linear transformation of $T_aS$ mapping $v$ to $-D_v\\nu $ , where $\\nu $ is the vector field along $S$ of normal vectors and $D_v$ is being used here for the ordinary directional differentiation of $\\mathbb {R}^n$ -valued functions.",
"(Elsewhere in the paper we also use $D$ for the Levi-Civita connection on $S$ relative to t he metric induced by the ordinary dot product.)",
"The unit eigenvectors of $\\mathbb {S}_a$ are called the principal vectors at $a$ , and the eigenvalues are the principal curvatures."
],
[
"Rolling motion on cylinders",
"In this section we review general facts about rolling and give a self-contained discussion of the fact that orbits of the rolling motion on 3-dimensional cylinders with general cross-section.",
"This is a classical result in non-holonomic dynamics, but we wish briefly to re-derive it here (in the present more geometric setting) so as to see more clearly the parallels with similar bounded motion of the no-slip billiard counterpart.",
"Throughout the section $e$ will denote the axis vector of the cylinder, as in Definition REF .",
"Moreover, $\\nu (a) = \\nu _a$ continues to denote the inward pointing unit normal vector.",
"In order to simplify notation, we will write $\\nu $ except when emphasizing the dependence on boundary point.",
"The rolling particle is subject to a force $f$ (a vector field on $M$ ), assumed to have zero $\\mathfrak {so}(n)$ component and $\\mathbb {R}^n$ component $\\varphi e$ , where $\\varphi =-m g$ is constant and $g$ is interpreted as the acceleration due to gravity.",
"Such an $f$ does not directly affect the angular velocities and can be thought to act on the center of mass of the moving particle.",
"Returning to the constrained Newton's equation of Definition REF , notice that due to the description of $\\mathfrak {S}^\\perp $ in Equation (REF ), the constraint force $N\\in \\mathfrak {S}^\\perp $ at $q=(A,a)$ has the form $N=\\left(\\frac{1}{r\\gamma ^2}\\zeta \\wedge \\nu A, \\zeta \\right) $ where $\\zeta =\\zeta (q,\\dot{q})\\in T_aS$ must be determined from the constraint $\\dot{q}\\in \\mathfrak {S}_q$ .",
"Using Proposition REF , and writing $\\dot{q}=(UA, u)$ , where $U\\in \\mathfrak {so}(n)$ and $u\\in T_aS$ , we can split Newton's equation into separate equations for the angular velocity matrix $U$ and center of mass velocity $u$ : $\\dot{U}=\\frac{1}{r\\gamma ^2} \\zeta \\wedge \\nu , \\ \\ \\frac{Du}{dt}= -g e +\\zeta .$ As before, $Du/dt$ refers to the Levi-Civita connection on $S$ relative to the metric induced by the standard Euclidean metric (dot product) on $\\mathbb {R}^n$ .",
"The first equation and the definition of $\\wedge $ imply $\\zeta = -r\\gamma ^2 \\dot{U} \\nu $ while the constraint on $\\dot{q}$ (recall the expression for $\\mathfrak {S}$ given in (REF )) implies $u = rU\\nu .$ It is not difficult to solve for $\\zeta $ and obtain an explicit differential equation for $U(t)$ .",
"This is done in the next proposition.",
"Proposition 17 The path $q(t)=(A(t), a(t))$ that solves the constrained Newton's equation for the rolling process is the solution to an initial value problem for the system of the differential equations $\\dot{a}=r U \\nu _a, \\ \\ \\dot{A}=UA, \\ \\ \\dot{U}=-\\frac{1}{(1+\\gamma ^2)r} \\left( {r^2}U\\mathbb {S}_a U\\nu _a -{g}e \\right)\\wedge \\nu _a.$ Let us define for each $a\\in S$ the linear map $\\Gamma _a: U\\in \\mathfrak {so}(n)\\mapsto U-{\\gamma ^{-2}}\\left(U\\nu _a\\right)\\wedge \\nu _a\\in \\mathfrak {so}(n).$ This is an invertible map.",
"To check this claim, let $\\tau _1, \\dots , \\tau _n$ be an orthonormal basis of $\\mathbb {R}^n$ with $\\tau _n=\\nu _a$ .",
"Then a straightforward computation using the basic properties of $\\wedge $ leads, for $i<j\\le n$ , to $\\tau _i\\cdot \\Gamma _a(U)\\tau _j =\\left(1+\\frac{\\delta _{jn}}{\\gamma ^2}\\right)\\tau _i\\cdot U\\tau _j.$ Clearly, then, $\\Gamma _a$ is injective, hence invertible.",
"It may be helpful noting here that $\\tau _i\\cdot U\\tau _j$ is a constant multiple of the trace inner product $\\left\\langle U, \\tau _i\\wedge \\tau _j\\right\\rangle $ .",
"If in particular $U=w\\wedge \\nu $ , then $\\Gamma (w\\wedge \\nu )= (1+\\gamma ^2)/\\gamma ^2 w\\wedge \\nu .$ The first equation, for $\\dot{a}$ , comes from (REF ) and the second, for $\\dot{A}$ , is immediate from the definition $U=\\dot{A}A^{-1}$ .",
"Thus we only need to justify the third equation.",
"The second equation in (REF ) gives $\\zeta = \\frac{Du}{dt}+ge$ , and the first gives $\\dot{U}=\\frac{1}{r\\gamma ^2}\\left(\\frac{Du}{dt}+ge\\right)\\wedge \\nu .$ In the above we may replace the covariant derivative $\\frac{Du}{dt}$ with the ordinary derivative $\\dot{u}$ since the difference is a multiple of $\\nu $ , which vanishes after taking the cross product $\\wedge $ with $ \\nu $ .",
"Now notice that $ \\dot{\\nu }_a = D_u\\nu = - \\mathbb {S}_au$ , where $\\mathbb {S}_a$ is the shape operator of $S$ at $a$ .",
"(See Definition REF .)",
"Therefore, $ \\dot{u}=r\\dot{U}\\nu + rU\\dot{\\nu } = r\\dot{U}\\nu - rU\\mathbb {S}u = r\\dot{U}\\nu - r^2U\\mathbb {S}U\\nu ,$ from which we obtain $\\Gamma (\\dot{U})= \\dot{U}-\\frac{1}{\\gamma ^2}\\left(\\dot{U}\\nu \\right)\\wedge \\nu = -\\frac{1}{r\\gamma ^2}\\left(r^2U\\mathbb {S}U\\nu -ge\\right)\\wedge \\nu .$ The desired third equation follows from the definition of $\\Gamma $ .",
"Since we are particularly concerned with the issue of boundedness of trajectories, we need to obtain information about the function $h(a)=a\\cdot e$ , whose derivative in time is $u\\cdot e$ .",
"The next proposition shows how to get a handle on this quantity.",
"Proposition 18 Let $e$ be the axis vector of the cylinder $S$ , $\\nu $ the inward pointing unit normal vector field of $S$ , $q(t)=(A(t), a(t))$ , and $\\dot{q}(t)=(U(t)A(t), u(t))$ .",
"Then $\\frac{d}{dt}(e\\cdot u)= \\frac{\\gamma ^2}{1+\\gamma ^2}r^2 \\left(Ue\\right)\\cdot \\left(\\mathbb {S}_a U\\nu \\right) -\\frac{g}{1+\\gamma ^2}$ holds under the assumption that $q(t)$ satisfies the constrained Newton's equation for the rolling motion of the particle of radius $r$ , mass $m$ , mass distribution parameter $\\gamma $ , subject to a constant force $-mge$ .",
"Furthermore, $\\left(Ue\\right)\\cdot \\left(\\mathbb {S}_a U\\nu \\right)=\\sum ^{n-2}_{i=1}\\lambda _i(a)(\\tau _i\\cdot Ue)(\\tau _i\\cdot U\\nu ) $ where $\\tau _i=\\tau _i(a)$ , $i=1,\\dots , n-1$ , is an orthonormal basis of $T_aS$ consisting of principal vectors, $\\lambda _i(a)$ are the respective principal curvatures of $S$ , and $\\tau _{n-1}=e$ , $\\lambda _{n-1}=0$ .",
"In dimension 3 we have $\\tau _a:=\\tau _1(a)=\\nu _a\\times e$ .",
"Letting in this case $\\lambda (a)$ be the principal curvature in direction $\\tau _a$ , the above equation reduces to $\\frac{d}{dt}(e\\cdot u)= \\frac{\\gamma ^2}{1+\\gamma ^2}r^2 \\lambda (a)\\left(\\tau _a\\cdot Ue\\right) \\left(\\tau _a\\cdot U\\nu _a\\right) -\\frac{g}{1+\\gamma ^2}.",
"$ By Proposition REF $r e\\cdot \\dot{U}\\nu =\\frac{1}{1+\\gamma ^2}\\left[-r^2\\left(Ue\\right)\\cdot \\left(\\mathbb {S}U\\nu \\right)-g\\right].",
"$ Now observe that $ \\frac{d (e\\cdot u)}{dt} = r\\frac{d(U\\nu )}{dt}= re\\cdot \\dot{U}\\nu + r e\\cdot U \\dot{\\nu }= re\\cdot \\dot{U}\\nu - r e\\cdot U \\mathbb {S} u= re\\cdot \\dot{U}\\nu - r^2 e\\cdot U \\mathbb {S} U\\nu $ hence $re\\cdot \\dot{U} \\nu = \\frac{d (e\\cdot u)}{dt} -r^2 (Ue)\\cdot (\\mathbb {S} U\\nu )$ .",
"The first equation of the proposition is now an immediate consequence.",
"The other claims follow easily from definitions.",
"In dimension 3 there is an orthonormal moving frame consisting of $\\tau , \\nu , e$ where $\\tau =\\nu \\times e$ .",
"The field $\\tau $ is parallel ($\\frac{D\\tau }{dt}=0$ ).",
"Furthermore $\\mathbb {S}_a\\tau _a=\\lambda (a)\\tau _a$ where $\\lambda (a)$ is a principal curvature, and $\\mathbb {S}_ae=0$ .",
"When $S$ is a circular cylinder of radius $R-r$ (and the extended billiard domain is a circular solid cylinder of radius $R$ ), $\\lambda (a)=1/(R-r)$ .",
"Theorem 19 Let $n=3$ and introduce the angular velocity vector $\\omega $ according to the definition $w\\mapsto U=\\omega \\times u.$ Under the conditions and notations of Proposition REF , the following system of equations hold: $ \\frac{d}{dt}\\left(u\\cdot e\\right) +\\frac{\\gamma ^2}{1+\\gamma ^2} r^2 \\lambda (a) (\\omega \\cdot e)(\\omega \\cdot \\nu ) +\\frac{g}{1+\\gamma ^2}=0$ and $ \\frac{d}{dt}\\left( \\omega \\cdot \\nu \\right) = \\lambda (a)(\\omega \\cdot e)(u\\cdot e),$ where $\\omega \\cdot e$ is constant in $t$ .",
"Notice that $u\\cdot e=\\dot{h}$ , where $h(a)=a\\cdot e$ is the height of the center of mass of the moving particle.",
"In terms of $h$ , the above becomes $ \\ddot{h}+\\frac{\\gamma ^2}{1+\\gamma ^2} r^2 \\lambda (a)\\omega _e(0)\\left[\\omega _\\nu (0)+\\omega _e(0)\\int _0^t\\lambda (a(s))\\dot{h}(s)\\, ds\\right]+\\frac{g}{1+\\gamma ^2}=0,$ where $\\omega _e=\\omega \\cdot e$ and $\\omega _\\nu =\\omega \\cdot \\nu $ .",
"Observe that $\\nu \\cdot \\mathbb {S} U\\nu = 0, \\ e\\cdot \\mathbb {S}U\\nu =(\\mathbb {S} e)\\cdot (U\\nu )=0, \\ \\tau \\cdot \\mathbb {S}U\\nu =(\\mathbb {S}\\tau )\\cdot (U\\nu )=\\lambda \\tau \\cdot U\\nu ,$ so that $(Ue)\\cdot (\\mathbb {S}_a U\\nu )= (\\tau \\cdot Ue)(\\tau \\cdot \\mathbb {S}U\\nu )+ (\\nu \\cdot Ue)(\\nu \\cdot \\mathbb {S}U\\nu )+(e\\cdot Ue)(e\\cdot \\mathbb {S}U\\nu )=\\lambda (\\tau \\cdot Ue)(\\tau \\cdot U\\nu ).$ The quantity $\\tau \\cdot U\\nu $ is constant in $t$ .",
"To verify this claim, first observe that, as $\\dot{\\tau }$ is collinear to $\\nu $ and $\\dot{\\nu }$ is collinear to $\\tau $ , $\\frac{d}{dt}\\left(\\tau \\cdot U\\nu \\right)= \\dot{\\tau }\\cdot U\\nu +\\tau \\cdot \\dot{U}\\nu +\\tau \\cdot U \\dot{\\nu } =\\tau \\cdot \\dot{U}\\nu $ From the third equation in Proposition REF and Equation (REF ) we obtain $\\left(1+\\frac{1}{\\gamma ^2}\\right)\\tau \\cdot \\dot{U}\\nu = \\tau \\cdot \\Gamma (\\dot{U})\\nu =-\\frac{1}{r\\gamma ^2}\\tau \\cdot \\left[\\left(r^2U\\mathbb {S}U\\nu -ge\\right)\\wedge \\nu \\right]\\nu =\\frac{r}{\\gamma ^2}\\tau \\cdot U\\mathbb {S}U\\nu .$ But $\\tau \\cdot U\\mathbb {S}U\\nu =-(U\\tau )\\cdot (\\mathbb {S} U\\nu )=-\\lambda (\\tau \\cdot U\\tau )(\\tau \\cdot U\\nu )=0.$ Therefore $\\tau \\cdot U\\nu $ is indeed constant.",
"It remains to understand the term $\\tau \\cdot Ue$ in $(Ue)\\cdot (\\mathbb {S}_a U\\nu )=\\lambda (\\tau \\cdot Ue)(\\tau \\cdot U\\nu ).$ Observe that $\\tau \\cdot \\dot{U} e=\\tau \\cdot \\left((r\\gamma ^2)^{-1}\\zeta \\wedge \\nu \\right)e=0$ and $\\dot{\\tau }=\\lambda u\\cdot \\tau \\nu $ , hence $\\frac{d}{dt}\\left(\\tau \\cdot U e\\right)= \\dot{\\tau }\\cdot U e + \\tau \\cdot \\dot{U}e=\\lambda (\\tau \\cdot u)( \\nu \\cdot Ue)=-\\lambda r (\\tau \\cdot U\\nu )(e\\cdot U\\nu )=-\\lambda (\\tau \\cdot U \\nu )(e\\cdot u),$ where the third and fourth equalities made use of Equation (REF ).",
"Finally, notice that $\\tau \\cdot U\\nu =-\\omega \\cdot e$ and $\\tau \\cdot U e=\\omega \\cdot \\nu $ .",
"We conclude the proof by applying these observations to the last equation in the statement of Proposition REF .",
"As a simple example, we see that the height of the rolling particle in a 3-dimensional vertical circular cylinder undergoes simple harmonic oscillations, so long as the constant of motion $\\omega _e$ is non-zero.",
"In particular, the motion is bounded.",
"In fact, suppose the cross-section of $S$ is a circle of radius $R-r$ .",
"In this case $\\lambda =1/(R-r)$ is constant, so $\\ddot{h} +\\frac{\\gamma ^2}{1+\\gamma ^2}\\left(\\frac{r}{R-r}\\right)^2 \\omega _e(0)^2 h + \\frac{\\gamma ^2}{1+\\gamma ^2}\\frac{r^2}{R-r}\\omega _e(0)\\left(\\omega _\\nu (0)-\\frac{1}{R-r}\\omega _e(0) h(0)\\right)+\\frac{g}{1+\\gamma ^2}=0.$ This has the form $\\ddot{h}+c_0h +c_1=0$ where $c_0$ is a positive constant (assuming $\\omega _e(0)\\ne 0$ ).",
"In terms of the variable $z:=h+c_1/c_0$ , the equation takes the form $\\ddot{z}+c_0z=0$ , whose solutions are the bounded functions $z(t)=C_1\\cos (\\sqrt{c_0}t)+C_2\\sin (\\sqrt{c_0}t).$ The following interesting observation was made in [9].",
"Let $T_h$ and $T_v$ denote, respectively, the periods of horizontal and vertical oscillation of the rolling ball in the circular vertical cylinder.",
"One easily finds that $T_h = 2\\pi (R-r)/r\\omega _e$ and $T_v=\\sqrt{\\frac{1+\\gamma ^2}{\\gamma ^2}}2\\pi (R-r)/r\\omega _e$ .",
"Therefore the ratio of these two periods only depends on the mass distribution parameter $\\gamma $ : $T_v/T_h= \\sqrt{\\frac{1+\\gamma ^2}{\\gamma ^2}}$ .",
"For example, $\\gamma ^2=2/5$ for the uniform distribution in dimension 3, so the period ratio in this case is $\\sqrt{7/2}$ .",
"We now restate and prove Proposition REF .",
"Proposition REF Suppose that the cross-section of the 3-dimensional vertical cylinder is a differentiable simple closed curve and that the constant of motion $\\omega _e$ —the vertical component of the angular velocity vector—is non-zero.",
"Then trajectories of the rolling motion under a constant force parallel to the axis of the cylinder are bounded.",
"Let $h=a\\cdot e$ denote, as before, the height of the center of mass of the rolling particle, and introduce $\\sigma =\\dot{h}$ and $w$ the $\\nu $ component of the angular velocity vector $\\omega $ .",
"According to Theorem REF , the function $h(t)$ can be obtained by solving an initial value problem for the system $\\dot{h} = \\sigma , \\ \\ \\dot{\\sigma }= -c_1\\lambda (t) w + c_3, \\ \\ \\dot{w}= c_2\\lambda (t)\\sigma , $ where $c_1, c_2, c_3$ are constants involving the parameters $\\gamma $ , $\\omega _e$ , $r$ and $g$ , and $c_1, c_2$ are positive.",
"The principal curvature $\\lambda (t)=\\lambda (a(t))$ is a periodic function of $t$ which is known in advance since it only depends on the point of contact at time $t$ along the cross-sectional boundary curve, and we know which point that is from the initial condition and the constant value of $\\omega _e$ .",
"(That boundary point moves at a constant rate $r\\omega _e$ .)",
"A simple rescaling of the variables gives the system $\\dot{x}_1= x_2, \\ \\ \\dot{x}_2= -\\eta (t) x_3 + 1, \\ \\ \\dot{x}_3=\\eta (t) x_2$ where $x_1$ is a constant multiple of $h$ and $\\eta (t)$ is a periodic function of $t$ whose period we may assume without loss of generality to be 1.",
"Introducing the complex variable $z=x_2+ix_3$ , we previous system reduces to $\\dot{x}_1= \\text{Re}(z), \\ \\ \\dot{z}= i\\eta z+1.$ For simplicity, let us assume 0 initial conditions.",
"Then the differential equation for $z$ has solution $z(t) = e^{i f(t)}\\int _0^t e^{-i f(s)}\\, ds,$ where $f(t)=\\int _0^t\\eta (s)\\, ds$ satisfies $f(t+1)=f(t) + 1$ .",
"Standard integral manipulations give $ z(n+t)=\\frac{e^{in}-1}{1-e^{-i}} e^{i f(t)} \\int _0^1 e^{-if(s)}\\, ds + e^{i f(t)}\\int _0^t e^{-i f(s)}\\, ds$ for all integers $n$ and $0\\le t<1$ .",
"The goal is to establish that the real part of $\\int _0^t z(s)\\, ds$ , which equals $x_1(t)$ , is a bounded function.",
"Let us verify this fact for $t=n$ , an integer.",
"Another straightforward manipulation of integrals leads to $\\int _0^n z(s)\\, ds = \\text{(bounded term)} -\\frac{n}{\\left| 1- e^{-i} \\right|^2}\\left\\lbrace (1-e^i)\\mathcal {I}_1- 2(1-\\cos 1)\\mathcal {I}_2 \\right\\rbrace .",
"$ where $\\mathcal {I}_1=\\int _0^1\\int _0^1 e^{i[f(t)-f(s)]}\\, ds\\, dt $ and $\\mathcal {I}_2=\\int _0^1\\int _0^t e^{i[f(t)-f(s)]}\\, ds\\, dt$ .",
"But $\\mathcal {I}_1=2\\int _0^1\\int _0^t \\cos (f(t)-f(s))\\, ds\\, dt.",
"$ One then notices that the real part of the term in braces in equation (REF ) must be zero."
],
[
"No-slip billiards in general cylinders",
"We now prove Theorem REF , reproduced below.",
"Theorem REF Let $\\mathcal {N}$ be the reduced phase space of the no-slip billiard system on the solid cylinder domain $\\mathcal {B}\\subset \\mathbb {R}^n$ , and let $\\overline{\\mathcal {N}}$ be the reduced phase space for the associated transverse billiard system.",
"Then trajectories of the no-slip billiard on $\\mathcal {N}$ , possibly with a constant force in the longitudinal direction, project to trajectories of the no-slip billiard map on $\\overline{\\mathcal {N}}$ , where the latter system is given the same mass distribution parameter $\\gamma $ as the billiard in dimension $n$ .",
"Given a vector space $W$ , it makes sense to write the Lie algebra of the special Euclidean group on $W$ , as a vector space, in the form $\\mathfrak {se}(W)=(W\\wedge W) \\oplus W$ where $W$ corresponds to infinitesimal translations and $W\\wedge W$ is the space spanned by elements $u\\wedge v$ , for all $u, v\\in W$ .",
"In this notation we have, for $W=\\mathbb {R}^{n-1} = e^\\perp $ , $\\mathfrak {se}(n)= \\mathfrak {se}(n-1)\\oplus \\left(\\mathbb {R}^{n-1}\\wedge e\\right) \\oplus \\mathbb {R} e$ and this direct sum decomposition is orthogonal with respect to the inner product on $\\mathfrak {se}(n)$ given above in Equation REF .",
"Also observe that the map $C_a$ (Proposition REF ), at each $a$ on the boundary of $\\mathcal {B}$ , respects the decomposition $\\mathfrak {se}(n)=\\mathfrak {se}(n-1)\\oplus \\mathfrak {se}(n-1)^\\perp $ since for all $w\\in W$ for which $w\\cdot \\nu _a=0$ we have $ C_a(0, w\\wedge e)= (0,w\\wedge e)$ and $C_a\\left(0, {\\nu _a\\wedge e}\\right)=\\left({r\\gamma }s_\\beta e, -c_\\beta {\\nu _a\\wedge e}\\right), \\ \\ C_a(e,0) = \\left(c_\\beta e, ({r\\gamma })^{-1}s_\\beta {\\nu _a\\wedge e}\\right).$ Here we are writing $C_a(u,U)$ , for $u\\in \\mathbb {R}^n$ and $U\\in \\mathfrak {so}(n)$ , on the reduced phase space $\\mathcal {N}$ .",
"Letting $C_{\\bar{a}}$ be the no-slip reflection map at $\\bar{a}$ of the system on $\\overline{\\mathcal {B}}$ , and writing $\\Pi $ for the orthogonal projection from $\\mathfrak {se}(n)$ to $\\mathfrak {se}(n-1)$ , it follows that $\\Pi \\circ C_a= C_{\\bar{a}}\\circ \\Pi .",
"$ (As already noted, we use the symbol $\\Pi $ to denote the orthogonal projection on various subspaces of $\\mathbb {R}^n$ ; the context will make it clear which subspace one is referring to at any given moment.)",
"From these observations we conclude that the natural projection from $\\mathcal {N}$ to the reduced phase space $\\overline{\\mathcal {N}}$ of the transverse billiard system commutes with the respective no-slip billiard maps.",
"The following notation will be used in our study of the longitudinal motion of no-slip billiards in general cylinders with axis vector $e$ .",
"Let $(a_j, u^-_j, U^-_j), (a_j, u_j, U_j)\\in \\mathcal {N}$ denote, respectively, the pre- and post-collision states at the $j$ th collision, $j=0, 1, \\dots $ , for a given trajectory of the no-slip system in $\\mathcal {B}$ ; we write $\\nu _j=\\nu (a_j)$ for the inward pointing normal vector to the boundary of $\\mathcal {B}$ at $a_j$ ; the time interval between consecutive collisions, from the $j$ th to the $j+1$ st collision, will be denoted $t_j$ ; we further introduce the velocity components $\\sigma _j:= u_j\\cdot e$ and $w_j:= \\gamma rU_je\\in e^\\perp =:W$ , and the longitudinal projection $h_j:=a_j\\cdot e$ .",
"Define the following elements of $\\mathbb {R}^n=\\mathbb {R}e\\oplus W$ : $\\Lambda _i=\\left(\\begin{array}{c}\\sigma _i \\\\w_i\\end{array}\\right), \\ \\ \\Lambda _i^-=\\left(\\begin{array}{c}\\sigma ^-_i \\\\w^-_i\\end{array}\\right), \\ \\ \\mathbb {1}=\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right),\\ \\ \\Phi =-g\\mathbb {1}.$ The special notation $\\mathbb {1}$ is used here for the first standard basis vector of $\\mathbb {R}^n$ in order to emphasize that we are dealing with a velocity space mixing linear and angular components, and not the ambient $\\mathbb {R}^n$ of the billiard domain.",
"Set $W_a:=\\lbrace w\\in W: w\\cdot \\nu _a=0\\rbrace $ , $a\\in \\partial \\mathcal {B}$ , let $\\Pi _a$ be the orthogonal projection to $W_a$ , and define $\\mathcal {A}(a):=\\left(\\begin{array}{cc}c_\\beta & -s_\\beta \\nu _a^\\dagger \\\\-s_\\beta \\nu _a & -c_\\beta \\nu _a \\nu _a^\\dagger + \\Pi _a\\end{array}\\right) $ Simple algebraic manipulation using the basic properties of $\\wedge $ and Proposition REF gives that $\\mathcal {A}(a)$ maps pre- to post-collision velocity components in the mixed velocity space $\\mathbb {R}^n$ .",
"Thus $\\Lambda _i=\\mathcal {A}_i\\Lambda _i^-$ , where $\\mathcal {A}_i:=\\mathcal {A}(a_i)$ .",
"Over the intercollision flight, the change in these $n$ mixed velocity components is: $\\Lambda ^-_i=\\Lambda _{i-1}+t_{i-1}\\Phi $ since $\\sigma _i^-=\\sigma _{i-1}- t_{i-1}g$ and $w_i^-=w_{i-1}$ (recall that $U$ does not change between collisions).",
"Therefore, Proposition 20 With the notation just introduced, the sequence of displacements $h_i$ along the cylinder's axis satisfies $h_i&=h_{i-1}+\\mathbb {1}^\\dagger \\left(t_{i-1}\\Lambda _{i-1}+\\frac{t^2_{i-1}}{2}\\Phi \\right) \\\\\\Lambda _i&=\\mathcal {A}_i\\left(\\Lambda _{i-1}+t_{i-1}\\Phi \\right).$ with initial conditions $\\Lambda _0$ and $h_0$ .",
"This is a simple consequence of the general form of $C_a$ given in Proposition REF , the above definitions, and the elementary properties of $\\wedge $ .",
"Observe that $\\mathcal {A}(a)$ , like $C_a$ , is an orthogonal involution.",
"It has eigenvalues $-1$ with multiplicity 1 and 1 with multiplicity $n-1$ .",
"In fact we have for all $w\\in W_a$ $\\mathcal {A}(a) \\left(\\begin{array}{c}0 \\\\ w\\end{array}\\right)= \\left(\\begin{array}{c}0 \\\\ w\\end{array}\\right),\\ \\ \\mathcal {A}(a) \\left(\\begin{array}{c}-1 \\\\ \\gamma \\nu _a\\end{array}\\right)= \\left(\\begin{array}{c}-1 \\\\ \\gamma \\nu _a\\end{array}\\right), \\ \\ \\mathcal {A}(a) \\left(\\begin{array}{c}\\gamma \\\\ \\nu _a\\end{array}\\right)= - \\left(\\begin{array}{c}\\gamma \\\\ \\nu _a\\end{array}\\right).$ We turn our attention now to the longitudinal motion when the no-slip billiard orbit is transversely periodic of period 2.",
"We wish to find an expression for the longitudinal drift in the absence of forces.",
"This is provided by the following theorem.",
"Theorem 21 For an orbit with transversal period 2, define $Q=\\mathcal {A}_1\\mathcal {A}_2$ , $\\mathcal {A}=\\mathcal {A}_1$ , and the row vector $\\xi = \\left(1+c_\\beta , -s_\\beta \\nu _1^\\dagger \\right)$ .",
"Then $Q\\in SO(n)$ .",
"Denote by $P$ the orthogonal projection onto the eigenspace of $Q$ for the eigenvalue 1.",
"Then $h_{\\ell }=\\hat{h}_{\\ell }+ \\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor \\xi P\\Lambda _0$ where the $\\hat{h}_{\\ell }$ are bounded terms of an oscillatory character that can be obtained explicitly if desired.",
"Consequently, $ \\lim _{\\ell \\rightarrow \\infty }\\frac{h_\\ell }{\\ell }=\\frac{1}{2}\\xi P\\Lambda _0$ In particular, if 1 is not in the spectrum of $Q$ (which may be the case in even dimensions), the system has bounded orbits.",
"On the other hand, if $\\xi $ is not orthogonal to the eigenspace for $Q$ associated to eigenvalue 1, then generically in the initial conditions orbits are not bounded.",
"In the above formulas, the constant intercollision time has been set to 1.",
"For transversal period 2 orbits, one has only to consider two values for $\\nu _j$ , and consequently only two values for $\\mathcal {A}(a)$ and a single value $t_j=t$ .",
"From $a_\\ell =a_{\\ell -1}+ tu_{\\ell -1}$ we obtain $h_\\ell = h_{\\ell -1}+t\\sigma _{\\ell -1}$ .",
"Setting $h_0=0$ and $t_0=1$ without loss of generality, we have $h_\\ell = \\sum _{j=0}^{\\ell -1}\\sigma _j= \\mathbb {1}^\\dagger \\sum _{j=0}^{\\ell -1} \\Lambda _j=\\mathbb {1}^\\dagger \\sum _{j=0}^{\\ell -1} \\mathcal {A}_j\\cdots \\mathcal {A}_0\\Lambda _0.",
"$ Here we are setting by convention $\\mathcal {A}_0$ to be the identity transformation.",
"For concreteness, let us assume that $\\ell $ is odd: $\\ell =2m+1$ .",
"Then, letting $\\mathcal {A}:=\\mathcal {A}_1$ and $Q=\\mathcal {A}_2\\mathcal {A}_1$ gives $h_{2m+1}=\\mathbb {1}^\\dagger \\left\\lbrace \\sum _{j=0}^{m} Q^j + \\mathcal {A} \\sum _{j=0}^{m-1} Q^j\\right\\rbrace \\Lambda _0=\\mathbb {1}^\\dagger Q \\Lambda _0+ \\mathbb {1}^\\dagger (I+\\mathcal {A}) \\sum _{j=0}^{m-1} Q^j\\Lambda _0.",
"$ Notice that $ \\sum _{j=0}^{m-1} Q^j\\Lambda _0= m P\\Lambda _0+ \\sum _{j=0}^{m-1} Q^jP^\\perp \\Lambda _0.$ The summation on the right-hand side of the above equation must be bounded.",
"In fact, further decomposing $P^\\perp $ into $2\\times 2$ or $1\\times 1$ blocks (the latter associated to eigenvalue $-1$ if it is present in the spectrum of $Q$ ), we end up with sums of a sequence of vectors generated by iterating a non-trivial rotation in dimension 2 or 1.",
"In particular, it follows that if 1 is not an eigenvalue of $Q$ (in even dimension) then trajectories having transversal period 2 are necessarily bounded.",
"To conclude, we note that $\\xi = \\mathbb {1}^\\dagger (I +\\mathcal {A})$ .",
"Figure: Notation for the proof of Corollary .From the above theorem we can now derive the claim made earlier that unbounded orbits actually exist, say in dimension 3, and obtain the explicit formula for the longitudinal drift shown in Corollary REF .",
"This requires that we obtain the explicit form of the rotation matrix $Q$ and find its spectral decomposition.",
"This is an entirely straightforward but somewhat tedious computation, whose details we omit.",
"Let us introduce the angle $\\alpha =2\\phi $ (see Figure REF ) and write $c_\\alpha =\\cos \\alpha , s_\\alpha =\\sin \\alpha $ .",
"Recall that $c$ and $s$ are reserved for the cosine and sine of the special angle $\\beta $ determined by the mass distribution parameter $\\gamma $ .",
"Also consider the normal and tangent vectors $\\nu _i$ and $\\tau _i$ at the two contact points, as indicated in Figure REF .",
"Then $Q$ assumes the following form $Q=\\left(\\begin{array}{cc}c^2-s^2 c_\\alpha & \\left[-sc(1+c_\\alpha )\\nu _2-ss_\\alpha \\tau _2\\right]^\\dagger \\\\[0.1cm]-sc(1+c_\\alpha )\\nu _1 + ss_\\alpha \\tau _1& (s^2-c^2c_\\alpha )\\nu _1\\nu _2^\\dagger +cs_\\alpha ( \\tau _1\\nu _2^\\dagger -\\nu _1\\tau _2^\\dagger )-c_\\alpha \\tau _1\\tau _2^\\dagger \\end{array}\\right).$ Now observe that $\\eta =\\frac{1}{ \\sqrt{s_\\alpha ^2+2\\gamma ^2 (1+c_\\alpha )} } \\left(\\begin{array}{c}s_\\alpha \\\\\\gamma (\\tau _1-\\tau _2)\\end{array}\\right)$ is a unit length eigenvector for the eigenvalue 1 of $Q$ .",
"Then an application of the limit formula for the vertical drift from Theorem REF gives the formula of Corollary REF .",
"Notice that $P$ in Theorem REF is in this case the rank-1 projection on the subspace spanned by $\\eta $ ."
],
[
"Forced billiard motion in a circular cylinder",
"In this section we restrict attention to circular cylinders in dimension $n=3$ .",
"The main goal is the prove Theorem REF , restated below after a couple of propositions.",
"Proposition 22 If the pre-collision state $(a, u, U)$ of a general (not necessarily a cylinder) no-slip billiard system satisfies the rolling impact condition, then the post-collision state is given by $C_a(u,U)=\\left(u-2 u\\cdot \\nu _a \\nu _a, U\\right).$ In words, the center of mass velocity of the moving particle is reflected specularly and the angular velocity matrix $U$ remains the same.",
"From the definition of the no-slip collision map $(u^+, U^+)=C_a(u^-,U^-)$ , the rolling impact condition $ rU^- = u-u\\cdot \\nu _a\\nu _a$ , and the relation $c_\\beta +\\gamma s_\\beta = 1$ we obtain $u^+ = c_\\beta u^--\\gamma ^{-1}{s_\\beta } u^-\\cdot \\nu _a\\nu _a + \\gamma s_\\beta r U^- \\nu _a=u^--2 u^-\\cdot \\nu _a \\nu _a$ and $U^+=\\frac{s_\\beta }{\\gamma r} \\nu _a \\wedge u^- + U^- -\\frac{s_\\beta }{\\gamma r} \\nu _a\\wedge r U^- \\nu _a= U^-,$ as claimed.",
"Next we restate and prove Proposition REF , which gives a broader context to a property observed in [6].",
"Proposition REF Consider a two-dimensional no-slip billiard system in a disc.",
"If the first collision satisfies the rolling impact condition, then all subsequent collisions also do, and the times between consecutive collisions are all equal.",
"Furthermore, the center of mass of the moving particle undergoes specular reflection at each collision.",
"Let $a$ and $a^{\\prime }$ be consecutive collision points on the boundary of $\\mathcal {B}$ .",
"Let $(u^-,\\omega ^-)$ denote pre-collision linear and angular velocities at $a$ and $(u^+, \\omega ^+)$ the post-collision velocities at $a$ .",
"Notice that the latter are also the pre-collision velocities at $a^{\\prime }$ .",
"Suppose that the rolling impact condition holds at $a$ .",
"Then as $\\omega ^-=\\omega ^+$ , we have $-r\\omega ^+ = -r\\omega ^-= u^-\\cdot \\tau _a = u^+\\cdot \\tau (a^{\\prime }) $ where the last equality is due to the post-collision velocity $u^+$ at $a$ being the specular reflection of $u^-$ .",
"Therefore the rolling impact condition also holds at $a^{\\prime }$ .",
"That intercollision times are all equation is a consequence of Proposition REF .",
"Theorem REF Consider a no-slip billiard system in a circular cylinder in $\\mathbb {R}^3$ whose moving particle is subject to a constant force directed along the axis of the cylinder.",
"If the first collision satisfies the transversal rolling impact condition and the first flight segment does not go through the axis of the cylinder, then the particle's trajectory is bounded.",
"Reviewing some notation, $\\mathcal {B}_0$ is here the cylinder of radius $R$ along $e=(0,0,1)^\\dagger $ so that $\\mathcal {B}$ is the cylinder of radius $R-r$ along $e$ .",
"A trajectory of the billiard system gives a sequence of post-collision states $(a_i, u_i, U_i)\\in \\mathcal {N}$ , $i=0,1, \\dots $ , and for this trajectory we have the unit normal vectors $\\nu _i=\\nu (a_i)= -\\bar{a}_i/|\\bar{a}_i|$ to $\\partial \\mathcal {B}$ where $\\bar{a}=a-a\\cdot e e$ , the tangent vectors $\\tau _i=\\tau (a_i)= \\nu _i\\times e$ to $\\partial \\mathcal {B}$ , the intercollision times $t_i$ between the $i$ th and $i+1$ st collisions, the longitudinal component of the center of mass velocities $\\sigma _i= u_i\\cdot e$ , the transversal angular velocity vectors $w_i = \\gamma r \\omega _i\\times e = \\gamma r U_ie$ , the position $h_i = a_i\\cdot e$ of the center of the moving particle along the cylinder's axis, and $\\bar{u}_i=u_i-u_i\\cdot e e$ .",
"The stating point of the proof are the equations (and notations) recorded in Proposition REF .",
"We specialize them to this situation by noting that the projection $\\Pi _a$ appearing in the lower-right block of the matrix $\\mathcal {A}(a)$ may be written here as $\\tau _a\\tau _a^\\dagger $ .",
"Thus $\\Lambda _i =\\left(\\begin{array}{c}\\sigma _i \\\\w_i\\end{array}\\right), \\ \\ \\mathcal {A}_i= \\left(\\begin{array}{cc}c & -s \\nu _i^\\dagger \\\\-s \\nu _i & -c \\nu _i \\nu _i^\\dagger + \\tau _i \\tau _i^\\dagger \\end{array}\\right), \\ \\ \\mathbb {1}=\\left(\\begin{array}{c}1 \\\\0 \\\\0\\end{array}\\right).$ The $\\Lambda _i$ are vectors in $\\mathbb {R}^3$ and the $\\mathcal {A}_i$ are $3\\times 3$ matrices.",
"Further, the $\\mathcal {A}_i$ are orthogonal matrices of determinant $-1$ , as is easily checked.",
"With $\\Phi =-g\\mathbb {1}$ , then by Proposition REF , $\\Lambda _i = \\mathcal {A}_i \\left(\\Lambda _{i-1} + t_{i-1}\\Phi \\right).$ Let $a_0=(R-r,0,0)$ be the initial position of the particle's center of mass, $u_0$ its initial velocity, and $\\bar{u}_0=u_0-u_0\\cdot e e$ the cross-sectional projection.",
"We define $\\theta $ as the angle between $\\bar{u}_0$ and $\\tau _0$ , as in Figure REF , and assume that $0<\\theta <\\pi /2$ .",
"Figure: Cross-sectional projection of initial velocity u ¯ 0 \\bar{u}_0 and definition of θ\\theta and δ\\delta .We also assume without further notice that the transversal rolling impact condition holds.",
"The free-flight times are all equal by Proposition REF ; the common value is $t_i = t = \\frac{2(R-r)\\tan \\theta }{|\\bar{u}_0|} $ and $\\nu _i= \\mathcal {R}(\\delta )\\nu _{i-1}, \\ \\ \\tau _i= \\mathcal {R}(\\delta )\\tau _{i-1}$ for $i\\ge 1$ , where $\\mathcal {R}(\\delta )=\\left(\\begin{array}{cr}\\cos \\delta & -\\sin \\delta \\\\\\sin \\delta & \\cos \\delta \\end{array}\\right)$ .",
"Observe that small values of $\\theta $ correspond to near grazing trajectories.",
"Define the $3\\times 3$ block diagonal matrix $\\mathcal {R}=\\text{diag}(1, \\mathcal {R}(\\delta ))\\in SO(3)$ .",
"A simple matrix multiplication shows that $\\mathcal {A}_i = \\mathcal {R}\\mathcal {A}_{i-1}\\mathcal {R}^{-1}$ and $\\mathcal {A}_0= \\left(\\begin{array}{rrc}c & -s & 0 \\\\-s & c & 0 \\\\0 & 0 & 1\\end{array}\\right)$ where, we recall, $c$ and $s$ are here the cosine and sine of the angle $\\beta $ .",
"In terms of mass distribution parameter $\\gamma $ , $c=(1-\\gamma ^2)/(1+\\gamma ^2)$ and $s=2\\gamma /(1+\\gamma ^2)$ .",
"All this notation in place, we now have $h_i= h_{i-1}+ t \\mathbb {1}^\\dagger \\left\\lbrace \\Lambda _{i-1}+\\frac{t}{2}\\Phi \\right\\rbrace , \\ \\ \\mathcal {A}_i=\\mathcal {R}^i\\mathcal {A}_0\\mathcal {R}^{-i}, \\ \\ \\Lambda _i=\\mathcal {A}_i\\left\\lbrace \\Lambda _{i-1}+t\\Phi \\right\\rbrace .$ We wish to show that the sequence of $h_i$ obtained by iterating these relations is bounded.",
"From Equation (REF ) we obtain $h_\\ell = h_0 - \\frac{\\ell t^2}{2}g + t\\mathbb {1}^\\dagger \\left\\lbrace \\Lambda _0+ \\cdots + \\Lambda _{\\ell -1}\\right\\rbrace .$ and $\\Lambda _i= \\mathcal {A}_i \\cdots \\mathcal {A}_1\\Lambda _0 +t\\left\\lbrace \\mathcal {A}_i\\cdots \\mathcal {A}_1+\\mathcal {A}_i\\cdots \\mathcal {A}_2+\\cdots +\\mathcal {A}_i \\mathcal {A}_{i-1}+ \\mathcal {A}_i \\right\\rbrace \\Phi .$ Define $\\mathcal {M}=\\mathcal {A}_0\\mathcal {R}^{-1}$ .",
"We also obtain from Equation (REF ), for $j>i$ , $\\mathcal {A}_j\\cdots \\mathcal {A}_{i}=\\mathcal {R}^j\\mathcal {M}^{j-i+1}\\mathcal {R}^{-i+1}.$ Equation (REF ) yields $\\begin{split}\\Lambda _0+\\cdots +\\Lambda _{i-1}&= \\left\\lbrace I + \\mathcal {A}_1 + \\mathcal {A}_2\\mathcal {A}_1+ \\cdots + \\mathcal {A}_{i-1}\\cdots \\mathcal {A}_{1}\\right\\rbrace \\Lambda _0\\\\&\\ \\ \\ +t\\left\\lbrace I\\right.\\\\&\\ \\ \\ +\\mathcal {A}_1\\\\&\\ \\ \\ +\\mathcal {A}_2+ \\mathcal {A}_2\\mathcal {A}_1\\\\&\\ \\ \\ + \\mathcal {A}_3+ \\mathcal {A}_3\\mathcal {A}_2+ \\mathcal {A}_3\\mathcal {A}_2\\mathcal {A}_1\\\\&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\cdots \\\\&\\ \\ \\ + \\left.\\mathcal {A}_{i-1}+ \\mathcal {A}_{i-1}\\mathcal {A}_{i-2}+\\cdots + \\mathcal {A}_{i-1}\\cdots \\mathcal {A}_{i}\\right\\rbrace \\Phi .\\end{split}$ Since $\\mathcal {R}\\mathbb {1}=\\mathbb {1}$ , it follows from Equations (REF ) and (REF ) that $\\begin{split}\\mathbb {1}^\\dagger \\left\\lbrace \\Lambda _0+\\cdots +\\Lambda _{\\ell -1}\\right\\rbrace &= \\mathbb {1}^\\dagger \\left\\lbrace I + \\mathcal {M}+\\mathcal {M}^2+\\cdots +\\mathcal {M}^{\\ell -1}\\right\\rbrace \\Lambda _0\\\\&\\ \\ \\ \\ \\ \\ + t\\mathbb {1}^\\dagger \\left\\lbrace (\\ell -1)\\mathcal {M}+(\\ell -2)\\mathcal {M}^2+\\cdots + 2\\mathcal {M}^{\\ell -2}+\\mathcal {M}^{\\ell -1}\\right\\rbrace \\Phi .\\end{split}$ It is now necessary to better understand $\\mathcal {M}$ .",
"This matrix is the product of two orthogonal matrices, hence orthogonal, with determinant $\\det \\mathcal {M}=(\\det \\mathcal {A}_0)(\\det \\mathcal {R}^{-1})=-1$ .",
"It has the explicit form $ \\mathcal {M}=\\left(\\begin{array}{cc}c & -s\\nu _1^\\dagger \\\\-s \\nu _0 & -c\\nu _0\\nu _1^\\dagger + \\tau _0\\tau _1^\\dagger \\end{array}\\right).$ Under the assumption $0<\\theta <\\pi /2$ , $\\nu _1\\ne \\pm \\nu _0$ .",
"Consider the orthonormal basis of $\\mathbb {R}^3$ defined by the vectors: $e_0&=\\frac{1}{\\sqrt{\\gamma ^2(1+\\nu _0\\cdot \\nu _1)^2+ 2(1+\\nu _0\\cdot \\nu _1)}} \\left(\\begin{array}{c}\\gamma (1+\\nu _0\\cdot \\nu _1) \\\\\\nu _0+\\nu _1\\end{array}\\right) \\\\e_1&=\\frac{1}{\\sqrt{2(1-\\nu _0\\cdot \\nu _1)}}\\left(\\begin{array}{c} 0 \\\\ \\nu _0-\\nu _1\\end{array}\\right)\\\\e_2&=\\frac{1}{\\sqrt{4 + 2\\gamma ^2\\left(1+\\nu _0\\cdot \\nu _1\\right)}}\\left(\\begin{array}{c}-2 \\\\\\gamma (\\nu _0+\\nu _1)\\end{array}\\right)$ Then $e_0$ is an eigenvector of $\\mathcal {M}$ associated to the eigenvalue $-1$ and the restriction of $\\mathcal {M}$ to $e_0^\\perp $ is a planar rotation.",
"Relative to the basis $\\lbrace e_1, e_2\\rbrace $ , this restriction has matrix form $\\left(\\begin{array}{cr}a & -b \\\\b & a\\end{array}\\right)$ where $a^2+b^2=1$ and $a= e_1\\cdot \\left(\\mathcal {M}e_1\\right), b= e_2\\cdot \\left(\\mathcal {M}e_1\\right).$ Explicitly, $a&= 1-\\frac{\\gamma ^2}{1+\\gamma ^2}\\left(1-\\nu _0\\cdot \\nu _1\\right)\\\\b&=-\\frac{\\gamma }{1+\\gamma ^2}\\sqrt{(1-\\nu _0\\cdot \\nu _1)(2+\\gamma ^2(1+\\nu _0\\cdot \\nu _1))}.$ Let $\\Pi _-$ and $\\Pi _\\perp $ denote, respectively, the orthogonal projections from $\\mathbb {R}^3$ to the line $\\mathbb {R}e_0$ and the plane $e_0^\\perp $ , and write $\\mathcal {M}_\\perp $ for the restriction of $\\mathcal {M}$ to $e_0^\\perp $ .",
"Notice that $\\mathcal {M}_\\perp $ cannot be the identity (the equation $\\mathcal {M}e_1=e_1$ implies $1-\\nu _0\\cdot \\nu _1=0$ , which is not the case).",
"As $\\mathcal {M}_\\perp $ is a planar rotation, $I-\\mathcal {M}_\\perp $ is nonsingular and $\\begin{split}\\lbrace \\cdots \\rbrace _1:= I + \\mathcal {M}+\\cdots +\\mathcal {M}^{\\ell -1}&= \\Pi _\\perp \\left\\lbrace I + \\mathcal {M}+\\cdots +\\mathcal {M}^{\\ell -1}\\right\\rbrace +\\Pi _-\\left\\lbrace I + \\mathcal {M}+\\cdots +\\mathcal {M}^{\\ell -1}\\right\\rbrace \\\\&= \\left\\lbrace I+\\mathcal {M}_\\perp +\\cdots +\\mathcal {M}_\\perp ^{\\ell -1}\\right\\rbrace \\Pi _\\perp + \\lbrace 1-1+\\cdots +(-1)^{\\ell -1}\\rbrace \\Pi _-\\\\&=(I-\\mathcal {M}_\\perp )^{-1}\\left(I-\\mathcal {M}_\\perp ^\\ell \\right) \\Pi _\\perp - {\\left\\lbrace \\begin{array}{ll} \\, 0 &\\text{if } \\ell =\\text{odd}\\\\ \\Pi _-&\\text{if } \\ell = \\text{even.}",
"\\end{array}\\right.",
"}\\end{split}$ Notice that $\\lbrace \\cdots \\rbrace _1$ is bounded.",
"Next, consider the expression $\\lbrace \\cdots \\rbrace _2:= (\\ell -1)\\mathcal {M}+(\\ell -2)\\mathcal {M}^2+\\cdots +2\\mathcal {M}^{\\ell -2}+\\mathcal {M}^{\\ell -1}.",
"$ Then $\\Pi _-\\lbrace \\cdots \\rbrace _2= \\lbrace -(\\ell -1)+(\\ell -2)-(\\ell -3)+ \\cdots + (-1)^{\\ell -1}\\rbrace \\Pi _-\\\\= -\\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor \\Pi _-$ where $\\lfloor \\cdot \\rfloor $ denotes the floor function.",
"We claim that $\\Pi _\\perp \\lbrace \\cdots \\rbrace _2 = \\left\\lbrace (\\ell -1)(I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp + (I-\\mathcal {M}_\\perp )^{-2}\\mathcal {M}_\\perp ^2 \\left(I-\\mathcal {M}_\\perp ^{\\ell -1}\\right)\\right\\rbrace \\Pi _\\perp $ This can be shown to hold as follows.",
"Let $z$ denote a complex variable.",
"Then $(\\ell -1)z+(\\ell -2)z^2+\\cdots + 2z^{\\ell -2}+ z^{\\ell -1}&=-z^{\\ell +1}\\frac{d}{dz}\\frac{1}{z}\\left\\lbrace 1+\\frac{1}{z}+\\cdots +\\frac{1}{z^{\\ell -2}}\\right\\rbrace \\\\&=(\\ell -1)\\frac{z}{1-z} - \\frac{z^2(1-z^{\\ell -1})}{(1-z)^2}.$ Identifying $\\mathbb {R}^2$ with $\\mathbb {C}$ and $\\mathcal {M}_\\perp $ with multiplication by some $z=e^{i\\lambda }$ gives the claimed identity.",
"Consequently, $\\begin{split}\\mathbb {1}^\\dagger \\lbrace \\cdots \\rbrace _2\\Phi &=\\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor g\\mathbb {1}^\\dagger \\Pi _-\\mathbb {1} +-(\\ell -1) g \\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp \\Pi _\\perp \\mathbb {1}\\\\&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -g\\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-2}\\mathcal {M}_\\perp ^2\\left(I-\\mathcal {M}_\\perp ^{\\ell -1}\\right)\\Pi _\\perp \\mathbb {1}.\\end{split}$ The third term on the right-hand side of Equation (REF ) is bounded.",
"The first term can be evaluated by noting that $\\mathbb {1}^\\dagger \\Pi _-\\mathbb {1} = \\left(e_0\\cdot \\mathbb {1}\\right)^2 = \\frac{\\gamma ^2(1+\\nu _0\\cdot \\nu _1)}{2 + \\gamma ^2(1+\\nu _0\\cdot \\nu _1)}.$ Concerning the second term, first observe that $\\Pi _\\perp \\mathbb {1}= \\mathbb {1}\\cdot e_1 e_1 + \\mathbb {1}\\cdot e_2 e_2=\\mathbb {1}\\cdot e_2 e_2=-\\frac{2}{\\sqrt{4+2\\gamma ^2(1+\\nu _0\\cdot \\nu _1)}}e_2$ so that $ \\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp \\Pi _\\perp \\mathbb {1}=\\frac{2}{2+\\gamma ^2(1+\\nu _0\\cdot \\nu _1)} e_2^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp \\Pi _\\perp e_2.",
"$ Since the rotation group in dimension 2 is commutative, the number $w^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp \\Pi _\\perp w$ does not depend on the unit vector $w$ .",
"Therefore $ e_2^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\mathcal {M}_\\perp \\Pi _\\perp e_2&=\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)^\\dagger \\left[\\left(\\begin{array}{cr}a & -b \\\\b & a\\end{array}\\right)\\left(I - \\left(\\begin{array}{cr}a & -b \\\\b & a\\end{array}\\right)\\right)^{-1}\\right]\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)\\\\&= \\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)^\\dagger \\left[\\frac{1}{2(1-a)}\\left(\\begin{array}{cc}a-1 & -b \\\\b & a-1\\end{array}\\right)\\right]\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)\\\\&=-\\frac{1}{2}.$ For concreteness, let us assume $\\ell =$ odd; the case when $\\ell $ is even will differ only by a bounded term.",
"For $\\ell $ odd we obtain $ \\mathbb {1}^\\dagger \\lbrace \\cdots \\rbrace _2\\Phi =+\\frac{\\ell -1}{2} g-g\\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-2}\\mathcal {M}_\\perp ^2\\left(I-\\mathcal {M}_\\perp ^{\\ell -1}\\right)\\Pi _\\perp \\mathbb {1}$ Returning now to Equation (REF ), and using the results so far contained in Equations (REF ), (REF ) and (REF ) we notice that the unbounded terms $-(\\ell t^2 g/2)$ cancel out and we are left with $h_\\ell = h_0 + t\\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-1}\\left(I-\\mathcal {M}_\\perp ^\\ell \\right)\\Pi _\\perp \\Lambda _0 - {(t^2g/2)}\\left\\lbrace 1 + 2\\mathbb {1}^\\dagger (I-\\mathcal {M}_\\perp )^{-2} \\left(\\mathcal {M}_\\perp ^2-\\mathcal {M}_\\perp ^{\\ell +1}\\right)\\Pi _\\perp \\mathbb {1}\\right\\rbrace , $ which is bonded.",
"This concludes the proof."
],
[
"Forced motion between parallel planes",
"Here we consider the billiard domain bounded by two infinite parallel affine codimension-1 subspaces of $\\mathbb {R}^n$ .",
"Let $\\nu $ denote the inward-pointing unit normal vector to one of the planes, so that $-\\nu $ is the (inward pointing) normal vector for the other plane.",
"Let $e$ be a unit vector perpendicular to $\\nu $ .",
"We suppose that the billiard particle is subject to a constant force $-gm e$ and wish to study the motion of the particle's center of mass along a direction $e$ .",
"The next theorem, which is a restatement of Theorem REF , asserts that this motion is bounded.",
"Theorem REF Consider a domain whose boundary consists of two parallel hyperplanes in $\\mathbb {R}^n$ , $n\\ge 2$ .",
"Then a trajectory of the no-slip billiard system whose initial center of mass velocity is not parallel to the hyperplanes is bounded.",
"Trajectories remain bounded if a constant force is applied to the particle's center of mass along any direction parallel to those hyperplanes.",
"This theorem admits a proof very similar to that of Theorem REF , but we give instead a more conceptual proof that makes use of a certain invariant quantity that we can identify for the two planes system, but whose possible counterpart for the circular cylinder is not yet apparent to us.",
"For any given set of initial conditions, the time between two consecutive collisions is constant throughout the orbit; we denote it by $t$ .",
"As before, we let $a_j$ denote the position of the center of mass of the moving particle at the $j$ th collision with the boundary of the billiard domain.",
"Due to Theorem REF , the proof may be approached by induction: we can focus on the motion in the plane spanned by $e$ and $\\nu $ and then argue by induction that trajectories are bounded for the transverse billiard system on $e^\\perp $ .",
"Set $\\omega _j:= w_j\\cdot \\nu $ , where $w_j:= \\gamma rU_je$ and $U_j$ is the post-collision angular velocity matrix at step $j$ .",
"Let the constant force be $-gm e$ , where $m$ is the particle's mass.",
"The component of $a_j$ in the direction $e$ is $h_j=a_j\\cdot e$ and the component of the post-collision velocity $u_j$ in the direction $e$ is $\\sigma _j:= u_j\\cdot e$ .",
"Then $ h_\\ell = h_{\\ell -1} + t \\sigma _{\\ell -1} - \\frac{t^2g}{2 }=h_0- \\frac{t^2 g}{2}\\ell + t \\sum _{j=1}^{\\ell -1} \\sigma _j.$ It is also useful to introduce the angular displacement $k_\\ell = k_0+t \\sum _{j =1}^{\\ell -1} \\omega _j.$ The following observation is key: For the billiard domain between two parallel planes, possibly with a transverse force, the ratio of angular to linear displacements remains constant after an even number of collisions.",
"In particular, $\\frac{\\Delta k}{\\Delta h}:=\\frac{k_{j+2}-k_j}{h_{j+2}-h_j}=(-1)^j\\gamma .",
"$ The proof of this claim is a calculation.",
"For any even $j$ we may reindex to $j=0$ , and $\\frac{\\Delta k}{\\Delta h}=\\frac{t(\\omega _0+\\omega _1)}{t(\\sigma _0+\\sigma _1)-t^2g}=\\frac{\\omega _0+\\omega _1}{\\sigma _0+\\sigma _1-tg}.$ As before, we write $c_\\beta =\\cos \\beta =\\frac{1-\\gamma ^2}{1+\\gamma ^2}$ and $s_\\beta =\\sin \\beta =\\frac{2\\gamma }{1+\\gamma ^2}.$ Due to Proposition REF , $\\sigma _0+\\sigma _1 = \\sigma _0+c_\\beta (\\sigma _0-tg)+s_\\beta \\omega _0, \\ \\ \\omega _0+\\omega _1 = \\omega _0 +s_\\beta (\\sigma _0-tg) - c_\\beta \\omega _0.$ Notice that $1-c_\\beta = \\gamma s_\\beta $ and $1+c_\\beta =\\gamma ^{-1}s_\\beta $ .",
"Thus we arrive at $ \\frac{\\Delta k}{\\Delta h}=\\frac{(1-c_\\beta )\\omega _0+s_\\beta (\\sigma _0-tg)}{(1+c_\\beta )(\\sigma _0-tg)+s_\\beta \\omega _0}= \\gamma .$ If $j$ is odd, a similar calculation yields $\\frac{\\Delta k}{\\Delta h}=-\\gamma $ .",
"It follows from this observation that $k_{2n}-k_0=\\gamma (h_{2n}-h_0), \\ \\ k_{2n+1}-k_1=-\\gamma (h_{2n+1}-h_1).$ We will refer to these $(h,k)$ lines as the lines of contact.",
"The constraint obtained from the existence of the lines of contact, combined with conservation of energy, bounds the orbits, as we now show.",
"Notice that the kinetic energy, expressed in terms of $\\sigma $ and the rescaled angular velocity $\\omega $ is $\\mathcal {K}=\\frac{1}{2} m\\left(\\sigma ^2+\\omega ^2\\right)$ .",
"Up to a common additive constant, the total energy at step $j$ is $ \\mathcal {E}_j=\\frac{1}{2}m\\left(\\sigma _j^2+\\omega _j^2\\right)+mgh_j=E$ where $E$ is the constant value of the total energy.",
"Setting $\\lambda = t^2g/2$ , we have $ \\left(h_{2n+1}-h_{2n}\\right)^2+ \\left(k_{2n+1}-k_{2n}\\right)^2=\\left( t\\sigma _{2n}-\\lambda \\right)^2 + \\left(t\\omega _{2n}\\right)^2.$ The linear relations given by Equations (REF ) yield $k_{2n+1}-k_{2n}= -\\gamma \\left(h_{2n+1}+ h_{2n}+ c\\right) $ where the constant $c$ only depends on initial values.",
"The above energy equation gives $t^2\\left(\\sigma ^2_{2n}+\\omega _{2n}^2\\right)= \\frac{2t^2E}{m}- 4\\lambda h_{2n}.",
"$ Inserting the previous two equations into (REF ), $ \\left(h_{2n+1}-h_{2n}\\right)^2+\\gamma ^2\\left(h_{2n+1}+h_{2n} +c\\right)^2+2\\lambda \\left(h_{2n+1}+h_{2n}\\right)=\\frac{2t^2E}{m} +\\lambda ^2.",
"$ This is the equation of an ellipse in the $(h_{\\text{\\tiny odd}}, h_{\\text{\\tiny even}})$ -plane.",
"A similar ellipse is the locus of $(h_{\\text{\\tiny even}}, h_{\\text{\\tiny odd}})$ .",
"Therefore we can conclude that the sequence $h_0, h_1, \\dots $ is bounded."
],
[
"Final comments: chaotic billiards",
"The examples of no-slip billiards considered so far in this paper (transversal period 2, parallel hyperplanes, circular cylinder) all share the property that the associated transversal systems have simple and well-understood behavior.",
"If we were to look for a notion of completely integrable no-slip billiard systems, these would be models to have in mind.",
"We wish now to consider a numerical example whose transversal dynamics can exhibit chaotic behavior, for which the problem of bounded orbits is likely to be much more challenging.",
"Let us revisit the stadium cylinder, whose cross section was shown in Figure REF .",
"The rolling motion, as already noted, is bounded, and has a typical quasi-periodic character (see Figure REF ), but the corresponding no-slip billiard behaves much differently.",
"Here we focus on a transition from simple bounded motion to a more chaotic regime at a natural bifurcation point (see Figure REF ) as an illustration of how different these two types of dynamics (namely, rolling motion versus no-slip billiards) can be.",
"To better appreciate the changes to orbits due to changes in initial conditions, it is useful to resort to a visualization device that we have called in [7] a velocity phase portrait.",
"We give here a brief review of this simple, but helpful, tool.",
"Figure: Transition from regular to chaotic motion.",
"The moving particle beginsfrom the middle of the lower flat side with linear velocity pointing up and a small angular velocity that causes it to move right after the first collision with the upper flat side.",
"For small values of the angular velocitytrajectories never touch the curved sides of the boundary, and the motion along the axis of the cylinder is bounded.If the initial angular velocity is large enough, trajectories move beyond the ends of the flat sidesand eventually becomes unstable.",
"For no-slip billiards in dimension 3, the associated transverse billiard system has a 3-dimensional reduced phase space $\\mathcal {N}$ .",
"(See the definition above in (REF ).)",
"This space is the product of a 1-dimensional manifold—the boundary of the planar billiard domain—and a hemisphere in $\\mathbb {R}^3$ representing the components of linear and angular velocities relevant to the transversal dynamics.",
"To make sense of this latter part, notice that the two components of the center of mass velocity and the single angular velocity of the planar billiard contribute two degrees of freedom due to conservation of kinetic energy.",
"(The constant force in the vertical direction does not affect the transversal motion due to Theorem REF .)",
"Vectors in this hemisphere are most conveniently expressed in the moving frame defined by the unit tangent vector $\\tau _a$ to the boundary of the planar billiard domain at a given point $a$ , the unit inward pointing normal vector $\\nu _a$ at the same point, and a third unit vector perpendicular to the first two representing a unit of angular velocity of the rotating disc (rescaled by a factor that turns the kinetic energy into a multiple of the ordinary Euclidean square norm in $\\mathbb {R}^3$ ).",
"Using this moving frame, $\\nu _a$ at each $a$ is identified with $(0,0,1)$ , and each hemisphere with the points of the unit sphere $S^2$ having positive last coordinate.",
"We further project this upper-hemisphere to the unit disc in $\\mathbb {R}^2$ .",
"In this way we have a bijection between points in the unit disc and (linear-angular) 3-velocities at each boundary point of the planar billiard domain.",
"Figure: Transition from regular to chaotic motion for the transverse dynamics of the stadium cylinder no-slip billiard system, as viewed in the velocity phase portrait.",
"The full velocity space is a disc of radius 1 as shown in the far right.",
"Initial conditions for the depicted orbits roughly compare to those of Figure .Three transversal orbit segments for the no-slip stadium-cylinder billiard are shown in Figure REF .",
"In each, the particle begins at the bottom flat side with linear velocity pointing up, and a small angular velocity that causes it to reflect rightward upon first collision.",
"All the other velocity components are set to 0.",
"When the angular velocity is sufficiently small, orbits are confined to the flat parts of the boundary and thus exhibit the bounded motion established in Theorem REF .",
"As the angular velocity increases, orbits eventually reach the curved parts, and soon transition to a chaotic regime in which a much larger region of the billiard phase space is explored, as suggested by the rightmost diagram of Figure REF .",
"Figure REF shows what happens during that transition using the velocity phase portrait.",
"The regular motion restricted to the flat sides of the boundary has the property that the linear-angular 3-velocity vector rotates in a simple fashion, forming a small circle around the north pole of $S^2$ .",
"As the trajectory barely crosses into the curved parts of the boundary, the possible linear-angular 3-velocity still remains in a small neighborhood of the north pole, but begins to behave in more interesting ways that are very sensitive to the initial velocities.",
"(See the middle diagram in Figure REF .)",
"As the angular velocity increases further, the linear-angular 3-velocity spreads throughout the velocity phase portrait as shown by the rightmost diagram in Figure REF .",
"The height function accordingly changes from simple bounded behavior (when the motion is limited to the flat boundary parts) to the rather more complicated motion over much wider distances shown in Figure REF .",
"This height function is likely not bounded; in fact, the graph in Figure REF suggests a type of “null-recurrent” behavior as in one-dimensional random walks.",
"Notice the short periods of fast falling and bouncing back up, separated by rough plateaux distributed in a seemingly random fashion.",
"We believe that trying to establish limit theorems for the longitudinal motion of chaotic transverse no-slip billiards in cylinders is a potentially fruitful direction to pursue.",
"Figure: Height function for the stadium cylinder for a trajectory in the chaotic regime, corresponding to the short orbit segment on the right of Figure and of Figure .",
"To give a sense of the scales involved, the diameter of the stadium is 8 and velocities are of order 1.",
"The number of time steps is 4×10 4 4\\times 10^4."
]
] | 1808.08448 |
[
[
"Automatic 3D bi-ventricular segmentation of cardiac images by a\n shape-refined multi-task deep learning approach"
],
[
"Abstract Deep learning approaches have achieved state-of-the-art performance in cardiac magnetic resonance (CMR) image segmentation.",
"However, most approaches have focused on learning image intensity features for segmentation, whereas the incorporation of anatomical shape priors has received less attention.",
"In this paper, we combine a multi-task deep learning approach with atlas propagation to develop a shape-constrained bi-ventricular segmentation pipeline for short-axis CMR volumetric images.",
"The pipeline first employs a fully convolutional network (FCN) that learns segmentation and landmark localisation tasks simultaneously.",
"The architecture of the proposed FCN uses a 2.5D representation, thus combining the computational advantage of 2D FCNs networks and the capability of addressing 3D spatial consistency without compromising segmentation accuracy.",
"Moreover, the refinement step is designed to explicitly enforce a shape constraint and improve segmentation quality.",
"This step is effective for overcoming image artefacts (e.g.",
"due to different breath-hold positions and large slice thickness), which preclude the creation of anatomically meaningful 3D cardiac shapes.",
"The proposed pipeline is fully automated, due to network's ability to infer landmarks, which are then used downstream in the pipeline to initialise atlas propagation.",
"We validate the pipeline on 1831 healthy subjects and 649 subjects with pulmonary hypertension.",
"Extensive numerical experiments on the two datasets demonstrate that our proposed method is robust and capable of producing accurate, high-resolution and anatomically smooth bi-ventricular 3D models, despite the artefacts in input CMR volumes."
],
[
"Introduction",
"Cardiac magnetic resonance (CMR) imaging is the gold standard for assessing cardiac chamber volume and mass for a wide range of cardiovascular diseases [1].",
"For decades, clinicians have been relying on manual segmentation approaches to derive quantitative measures such as left ventricle (LV) volume, mass and ejection fraction.",
"However, manual expert segmentation of CMR images is tedious, time-consuming and prone to subjective errors.",
"It becomes impractical when dealing with large-scale datasets.",
"As such, there is a demand for automatic techniques for CMR image analysis that can handle the scale and variability associated with large imaging studies [2], [3].",
"Recently, automatic segmentation based on deep neural networks has achieved state-of-the-art performance in the CMR domain [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].",
"For example, in the Automatic Cardiac Diagnosis Challenge (ACDC) [18] the 8 highest-ranked segmentation methods were all neural network-based methods.",
"Theoretically, 3D neural network-based segmentation methods may be designed with arbitrarily deep architectures.",
"In practice however, the size of cardiac images, especially that of high-resolution volumetric images [11], often presents a computational bottleneck at the training stage.",
"To deal with this, shallow 3D network architectures [11] or fewer feature/activation maps [5] are typically considered.",
"Also, to reduce the computational burden, most methods extract the region of interest (ROI) containing the whole heart as a first step to reduce the volume size [8], [9], [10], [11], [14], [15], [17], or train a 2D network to separately segment each short-axis slice in the volume [12], [13], [14], [15], [16].",
"However, there are fundamental problems associated with each of these workarounds.",
"For example, the use of shallow 3D network architectures or fewer feature maps is known to compromise segmentation accuracy.",
"The ROI extraction approach is carried out using ROI detection algorithms, whose robustness remains questionable [8].",
"In addition, as no 3D context is taken into account, 2D network-based methods suffer from lack of 3D spatial consistency between the segmented slices (leading to lack of smoothness in the long-axis direction), and may result in a false positive prediction at an image slice containing non-ventricular tissues that are similar to target ventricles [8].",
"Due to the limitations of standard clinical acquisition protocols, raw volumetric CMR images acquired from standard scans often contain several artefacts [19], including inter-slice shift (i.e.",
"respiratory motion), large slice thickness, and lack of slice coverage.",
"Most deep learning methods do not routinely account for imaging artefacts [4], [5], [6], [7], [8], [9], [10], [12], [13], [14], [16].",
"As such, these artefacts are inevitably propagated onto the resulting segmentations.",
"An example is given in Fig REF $e$ .",
"The figure shows the segmentation of a 3D volume (whose short- and long-axis views are shown in Fig REF $a$ and $b$ ) using a state-of-the-art CNN approach [13].",
"As can be seen, the segmentation Fig REF $e$ inherits the misalignment and staircase artefacts present in the original volumetric image due to cardiac motion and large slice thickness.",
"Further, holes exist at the apical region of the 3D model due to incomplete slice coverage of the whole heart.",
"Different approaches have been proposed to tackle each artefact accordingly before building a smooth model.",
"For example, misalignment was corrected using quadratic polynomials [15] or rigid registration [20]; Large slice thickness can be addressed by super-resolution techniques [21].",
"However, few studies have addressed different artefacts directly from an image segmentation perspective.",
"To date, we are aware of only one deep learning segmentation method [11] that takes into account different cardiac artefacts, but the method was tested on only simulated images of the LV, whose anatomy is less complex than the bi-ventricular anatomy.",
"It is thereby still an open problem as to how to build an artefact-free and smooth bi-ventricular segmentation model from real artefact-corrupted CMR volumes with novel image segmentation methods.",
"For clinical applications, segmentation algorithms need to maintain accuracy across diverse patient populations with varying disease phenotypes.",
"In the existing literature, however, most methods [5], [6], [9], [11], [12], [13], [14], [15] have been developed and validated over normal (healthy) hearts or mildly abnormal hearts.",
"Few studies have focused on hearts with very significant pathology with altered geometry and motion compared to healthy hearts.",
"In addition, most methods [4], [5], [6], [7], [9], [10], [12], [14], [15] tend to use small image datasets.",
"For example, four representative MICCAI challenges, namely the 2009 automatic LV segmentation challengehttp://www.cardiacatlas.org/challenges/lv-segmentation-challenge/ (also known as Sunnybrook cardiac data), the 2011 LV segmentation challengehttp://www.cardiacatlas.org/studies/sunnybrook-cardiac-data/ (organized as part of the STACOM workshop), the 2015 RV segmentation challenge [22] and the 2017 ACDC, were tested on only 30, 100, 48 and 100 CMR datasets respectively.",
"Given the small size of the datasets used for training and testing, whether the reported results can be generalised to larger cohorts remains questionable.",
"In this paper, we propose a segmentation pipeline to address the aforementioned limitations of current approaches.",
"Specifically, we make the following contributions: We propose a multi-task deep learning network that simultaneously predicts segmentation labels and anatomical landmarks in CMR volumes.",
"The network takes input volumetric images as multi-channel vector images (2.5D representation), requires no ROI extraction, and contains up to 15 convolutional layers.",
"As such, the network has the computational advantage of 2D networks and is able to address 3D issues without compromising accuracy and spatial consistency.",
"To our knowledge, this is the first work applying deep learning to CMR landmark localisation in a 3D context.",
"We introduce anatomical shape prior knowledge to the network segmentation, which is a refinement step that is carried out using atlas propagation with a cohort of high-resolution atlases.",
"As such, the pipeline is able to produce an accurate, smooth and clinically meaningful bi-ventricular segmentation model, despite the existing artefacts in the input volume.",
"Moreover, due to the use of landmarks detected by the network, the proposed pipeline is entirely automatic.",
"We demonstrate that the proposed pipeline can be readily generalised to segmenting volumetric CMR images from subjects with pulmonary hypertension (a cardiovascular disease).",
"We thoroughly assess the effectiveness and robustness of the proposed pipeline using a large-scale dataset, comprising 2480 short-axis CMR volumetric images for training and testing.",
"To our knowledge, this is one of the first CMR segmentation studies utilising a volumetric dataset of this size, and the technique introduced herein is the first automatic approach capable of producing a full high-resolution bi-ventricular model in 3D.",
"Figure: Illustrating the differences between a low-resolution CMR volume (top row) and a high-resolution CMR volume (bottom row).",
"The images in the short-axis view are shown in aa and cc, while those in the long-axis view are in bb and dd.",
"The corresponding segmentations are given in ee and ff.The proposed automatic segmentation pipeline handles two types of CMR volumetric inputs: low-resolution (LR) and high-resolution (HR) volumes.",
"Fig REF illustrates the differences between them.",
"The LR volume has a large slice thickness (10 mm), giving rise to a staircase effect in the long-axisIn a standard CMR acquisition, short-axis and long-axis images are acquired separately, both of which have high in-plane resolution.",
"However, in this paper, only CMR-acquired short-axis images are used, and a long-axis image denotes a vertical slice/cross-section of a stack of these short-axis images.",
"Large thickness between short-axis images would result in a poor resolution in the long-axis image.",
"An example is given in Fig REF .",
"view (Fig REF $b$ ).",
"Moreover, since the slices in Fig REF $b$ were acquired from multiple breath-holds, inconsistency of each breath-hold results in an inter-slice shift artefact.",
"In contrast, the cross plane resolution of the HR volume is 2 mm, making its long-axis image Fig REF $d$ relatively smooth.",
"In addition, HR imaging requires only one single 20-25 second breath-hold and therefore it introduces no inter-slice shift artefact.",
"However, HR imaging may not be feasible for pathological subjects who are unable to hold their breath for 20-25s during each scan.",
"Since HR imaging acquisition generates artefact-free cardiac volumes [23], it enables an accurate delineation of ventricular morphology, as shown in Fig REF $f$ .",
"In comparison, Fig REF $e$ shows that the segmentation of an LR volume contains different cardiac artefacts [19] (e.g.",
"inter-slice shift, large slice thickness, and lack of slice coverage).",
"Note that the in-plane resolution of both HR and LR volumes is about $1.3 \\times 1.3 $ mm, so their corresponding short-axis views Fig REF $a$ and $c$ are of relatively high quality.",
"The proposed pipeline has three main components: segmentation, landmark localisation and atlas propagation.",
"We term the proposed network used in the pipeline as the Simultaneous Segmentation and Landmark Localisation Network (SSLLN).",
"Further, the related terms SSLLN-HR and SSLLN-LR will be used to refer to versions of SSLLN trained with HR and LR volumetric data, respectively.",
"In Fig REF , we illustrate the pipeline schematically.",
"For an HR volume input, the trained SSLLN-HR is deployed to predict its segmentation labels as well as landmark locations.",
"Since the HR volume input is artefact-free, the resulting segmentation is an accurate and smooth bi-ventricular 3D model.",
"Afterwards, the HR volume and its corresponding SSLLN-HR outputs (landmarks and segmentation) are used as part of an HR atlas.",
"For an LR volume input, the pipeline consists of two steps: First, the trained SSLLN-LR predicts an initial segmentation of the LR volume.",
"In order to guarantee an artefact-free smooth segmentation output, a further refinement is carried out (second step).",
"In this step, multiple selected HR atlases derived from SSLLN-HR are propagated onto the initial LR segmentation to form a smooth segmentation.",
"This step explicitly fits anatomical shapes and is fully automatic due to the use of landmarks predicted from SSLLN-HR and -LR.",
"We detail each of the two steps in the next two subsections.",
"Figure: Pipeline for automatic bi-ventricular segmentation of low- and high-resolution volumetric images.",
"The pipeline includes segmentation, landmark localisation and atlas propagation.",
"It is capable of producing accurate, high-resolution and anatomically smooth bi-ventricular models, despite existing artefacts in input CMR volumes."
],
[
"Learning segmentation labels and landmark locations ",
"We treat the problem of predicting segmentation labels and landmark locations as a multi-class classification problem.",
"First, let us formulate the learning problem as follows: we denote the input volumetric training dataset by $S=\\lbrace (U_i, R_i, L_i), i=1,...,N_t\\rbrace $ , where $U_i=\\lbrace u^i_j,j=1,...,|U_i|\\rbrace $ is the raw input CMR volume (Fig REF left), $R_i=\\lbrace r^i_j,j=1,...,|R_i|\\rbrace $ , $r^i_j \\in \\lbrace 1,...,N_r\\rbrace $ denotes the ground-truth segmentation labels for volume $U_i$ ($N_r=5$ representing 4 tissue types and a background as shown in Fig REF right), $L_i=\\lbrace l^i_j,j=1,...,|L_i|\\rbrace $ , $l^i_j \\in \\lbrace 1,...,N_l\\rbrace $ stands for the ground-truth landmark labels for $U_i$ ($N_l=7$ representing 6 landmarks and a background as shown in Fig REF middle), and $N_t$ is the number of samples in the training data.",
"Figure: An exemplar raw volumetric CMR image, its ground-truth landmarks and segmentation labels, which are utilised as inputs to train the network in Fig .",
"On the left, three short-axis slices in the volume are highlighted, corresponding to basal, mid-ventricular, and apical locations (from top to bottom) of the heart.",
"In the middle, six landmarks are shown, coloured according to the following cardiac regions: the left ventricular lateral wall mid-point (yellow), two right ventricular insert points (red and blue), right ventricular lateral wall turning point (green), apex (pink) and centre of the mitral valve (cyan).",
"Together, they reflect the size, pose and shape of the heart.",
"On the right, a full anatomical bi-ventricular heart model is shown, coloured according to the left ventricular cavity (red), left ventricular wall (green), right ventricular cavity (yellow) and right ventricular wall (blue).Note that $|U_i|=|R_i|=|L_i|$ is the total number of voxels in a CMR volume.",
"We then define all network layer parameters as $\\textbf {W}$ .",
"In a supervised setting, we propose to solve the following minimisation problem via the standard (back-propagation) stochastic gradient descent $ {{\\bf {W}}^*} = \\mathop {{\\rm {argmin}}}\\limits _{\\bf {W}} ({L_D}({\\bf {W}}) + \\alpha {L_L}({\\bf {W}}) + \\beta \\Vert {\\bf {W}}\\Vert _F^2),$ where $\\alpha $ and $\\beta $ are weight coefficients balancing the three terms.",
"$L_D(\\textbf {W})$ is the segmentation loss that evaluates spatial overlap with ground-truth labels.",
"$L_L(\\textbf {W})$ is the landmark associated loss for predicting landmark locations.",
"$\\Vert \\textbf {W}\\Vert _F^2$ , known as the weight decay term, represents the Frobenius norm on the weights $\\textbf {W}$ .",
"This term is used to prevent over-fitting in the network.",
"The training problem is to estimate the parameters $\\textbf {W}$ associated with all the convolutional layers and by minimising (REF ) the network is able to simultaneously predict segmentation labels and landmark locations.",
"The definition of $L_D(\\textbf {W})$ above is first given as follows $ {L_D}({\\bf {W}}) = - \\sum \\limits _i {\\frac{{2\\sum \\limits _k {\\sum \\limits _j {{\\mathbb {1}_{\\left\\lbrace {r_j^i = k} \\right\\rbrace }} \\cdot P(r_j^i = k|{U_i},{\\bf {W}})} } }}{{\\sum \\limits _k {\\sum \\limits _j {\\left( {\\mathbb {1}_{\\left\\lbrace {r_j^i = k} \\right\\rbrace }^2 + {P^2}(r_j^i = k|{U_i},{\\bf {W}})} + \\epsilon \\right) } } }}},$ where $\\mathbb {1}_{\\lbrace \\cdot \\rbrace }$ is an indicator function.",
"$\\epsilon $ is a small positive value used to avoid dividing by zero.",
"$i$ , $k$ and $j$ respectively denote the training sample index, the segmentation label index and the voxel index.",
"$P(r^i_j=k|U_i,\\textbf {W})$ corresponds to the softmax probability estimated by the network for a specific voxel $j$ (subject to the restriction $r_j^i=k$ ), given the training volume $U_i$ and network weights $\\textbf {W}$ .",
"Note that (REF ) is known as the differentiable Dice loss [24], in which the summations are carried out over all voxels, labels and training samples.",
"For landmark localisation in a CMR volume, the primary challenge is the extreme imbalance between the proportion of voxels belonging to landmark regions and the proportion belonging to non-landmark regions (the 6 landmarks are represented by 6 voxels, while all the remaining voxels (numbering in the millions) represent background).",
"To solve this highly imbalanced classification problem, we propose the class-balanced weighted categorical cross-entropy loss $ {L_L}({\\bf {W}}) = - \\sum \\limits _{i} {\\sum \\limits _{k} {\\left( {w_k^i\\sum \\limits _{j \\in {Y_k^i}} {{\\rm {log}}P(l_j^i = k|{U_i},{\\bf {W}})} } \\right)}}.$ Here $k$ denotes the landmark label index, ranging from 1 to 7.",
"$Y_k^i$ represents the voxels in training sample $i$ that belong to the region for which the value of landmark label index is $k$ .",
"To automatically balance landmark and non-landmark classes, we use a weight $w_k^i$ for (REF ), where $w_k^i = 1 - {{\\left| {Y_k^i} \\right|} \\mathord {\\left\\bad.",
"{\\vphantom{{\\left| {Y_k^i} \\right|} {\\left| {{Y_i}} \\right|}}} \\right.",
"\\hspace{0.0pt}} {\\left| {{Y_i}} \\right|}}$ , $k=1,..,7$ .",
"Here $|Y_k^i|$ denotes the number of voxels in $Y_k^i$ , while $|Y_i|$ represents the total number of voxels in training sample $i$ .",
"Let us explain how the weighting process works intuitively.",
"For the voxel falling in any one of the 6 landmark locations, $\\left| {Y_k^i} \\right|$ is 1 and ${{\\left| {Y_k^i} \\right|} \\mathord {\\left\\bad.",
"{\\vphantom{{\\left| {Y_k^i} \\right|} {\\left| {{Y_i}} \\right|}}} \\right.",
"\\hspace{0.0pt}} {\\left| {{Y_i}} \\right|}}$ is close to zero.",
"Therefore, $1 - {{\\left| {Y_k^i} \\right|} \\mathord {\\left\\bad.",
"{\\vphantom{{\\left| {Y_k^i} \\right|} {\\left| {{Y_i}} \\right|}}} \\right.",
"\\hspace{0.0pt}} {\\left| {{Y_i}} \\right|}}$ is close to 1.",
"On the other hand, ${\\sum \\limits {{\\rm {log}}P(l_j^i = k|{U_i},{\\bf {W}})} }$ in (REF ) is very small as only one voxel contributes to this term.",
"Therefore, the product ${w_k^i\\sum \\limits {{\\rm {log}}P(l_j^i = k|{U_i},{\\bf {W}})} }$ ends up being a small value.",
"In contrast, for a voxel falling in background area, $1 - {{\\left| {Y_k^i} \\right|} \\mathord {\\left\\bad.",
"{\\vphantom{{\\left| {Y_k^i} \\right|} {\\left| {{Y_i}} \\right|}}} \\right.",
"\\hspace{0.0pt}} {\\left| {{Y_i}} \\right|}}$ is a very small value close to zero.",
"${\\sum \\limits {{\\rm {log}}P(l_j^i = k|{U_i},{\\bf {W}})} }$ is however very large as almost all voxels (excluding the 6 landmark voxels) contribute to this term.",
"Therefore, the product ${w_k^i\\sum \\limits {{\\rm {log}}P(l_j^i = k|{U_i},{\\bf {W}})} }$ becomes a small value.",
"As such.",
"the losses resulting from the landmark and non-landmark voxels are well balanced, which is crucial for successfully detecting merely 6 landmarks from a volume containing millions of voxels.",
"Figure: The architecture of the proposed SSLLN with 15 convolutional layers.",
"The network takes different CMR volumes as input, applies a branch of convolutions, learns image features from fine to coarse levels, concatenates multi-scale features and finally predicts the probability maps of segmentation and landmarks simultaneously.",
"These probability maps, together with the ground-truth segmentation labels and landmark locations, are then utilised in the loss function in () which is minimised via the stochastic gradient descent.",
"Here #\\#S, #\\#A, #\\#C, #\\#LK and GT represent the number of volume slices, the number of activation maps, the number of anatomies, the number of landmarks, and ground truth, respectively.In Fig REF , we show the architecture of SSLLN.",
"There are two major differences between our network architecture and existing 2D or 3D ones, which we highlight as novel contributions of this work.",
"First, 2D networks [4], [5], [6], [7], [9], [10], [12], [13], [14], [15], [16] are often trained using 2D short-axis slices separately.",
"Therefore, there is no 3D spatial consistency between the resulting segmented slices.",
"3D networks [5], [6], [7], [11], [24] often rely on 3D convolutions, which in practice leads to 5D tensors (e.g.",
"batch size $\\times $ [3D volume size] $\\times $ classification categories) during forward and backward propagations and requires far more GPU memory than their 2D counterparts.",
"Workarounds such as subsampling [25] or use of small batch size and fewer convolutional layers [5], [7], [11] are often considered when training 3D networks, but these either complicate the training process or cause loss of information and accuracy.",
"Unlike 2D networks, our network treats each input CMR volume as a multi-channel vector image, known as `2.5D' representation.",
"In this sense, 3D volumes rather than 2D short-axis slices are used to train our network.",
"As such, our network accounts for the spatial consistency between slices.",
"Retaining the 3D spatial relationship is crucial for landmark localisation as landmarks encode spatial information.",
"Unlike 3D networks, our network only involves 4D tensors (excluding the last layer).",
"After the input volume passes through the first convolutional layer, the subsequent convolutional operations (excluding the last layer) in our network function exactly the same as those in 2D methods.",
"Hence, the proposed network has the computational advantage of 2D networks, and also handles the input explicitly as a 3D volume (rather than a series of 2D slices), thus retaining accuracy and spatial consistency.",
"This will be demonstrated later in Section REF .",
"We also note that other network architecture, such as the multi-view CNN [26] that parses 3D data into different 2D components, may also suit our applications.",
"Second, our network predicts segmentation labels and landmark locations simultaneously as we integrate the two problems into a unified image classification problem for which we tailored a novel loss function (REF ).",
"We are not aware of any previous approach that detects cardiac landmarks using a deep learning-based classification method.",
"This is also the first work that focuses on segmentation and landmark localisation simultaneously.",
"After the network is trained, given an unseen CMR volume $f : \\Omega \\rightarrow {\\mathbb {R}^{\\#S}}$ ($\\#S$ is the number of short-axis slices in the volume) defined on the domain $\\Omega \\subset \\mathbb {R}^2$ , we deploy the network on it and obtain the probability maps of segmentation ($P_S$ ) and the probability maps of landmarks ($P_L$ ) from the last convolutional layer.",
"The binary segmentation and landmark labels are the indices of the maximum values of their probability maps along the channel direction, i.e.",
"$S = \\arg {\\max _{k = 1,...,{N_r}}}{P_S}$ and ${\\cal L} = \\arg {\\max _{k = 1,...,{N_l}}}{P_L}$ ."
],
[
"Introducing anatomical shape prior knowledge",
"Due to limitations of cardiac MR imaging, low-resolution (LR) volumetric training datasets often contain artefacts, such as inter-slice shift, large slice thickness, lack of slice coverage, etc.",
"Inevitably, the deployment of SSLLN-LR trained from such a dataset causes the propagation of these artefacts to the resulting segmentation.",
"An example can be found in Fig REF $d$ and $f$ .",
"In this section, we introduce shape prior knowledge through atlas propagation to overcome such artefacts in SSLLN-LR segmentation.",
"In Fig REF , we outline the shape refinement framework, including initial affine alignment, atlas selection, deformable registration and label fusion.",
"The framework involves using a cohort of high-resolution (HR) atlases produced from SSLLN-HR, each of which consists of an HR CMR volume ($1.25 \\times 1.25 \\times 2.0\\;\\rm {mm}$ ), and its corresponding landmarks and segmentation labels.",
"Next, we detail the framework.",
"Figure: A block diagram illustrating how to explicitly introduce an anatomical shape refinement to the SSLLN-LR segmentation.",
"As is evident in jj, such a shape refinement enables an accurate, smooth and clinically meaningful bi-ventricular segmentation model, despite the artefacts in the LR input volume dd.",
"The framework is fully automated due to the use of the landmarks detected from SSLLN-HR and -LR.Due to individual differences, the scanned heart often shows marked variations in size, pose and shape (as shown in Fig REF $a$ and $d$ ).",
"This poses difficulty for existing image registration algorithms due to their non-convex nature.",
"For this, the landmarks detected from SSLLN-HR and -LR were used to initialise the subsequent non-rigid algorithm between target and each atlas, which is similar to [27], [28].",
"An affine transformation with 12 degrees of freedom was first computed between the target landmarks (predicted by SSLLN-LR) and the atlas landmarks (predicted by SSLLN-HR).",
"In addition to initialising the non-rigid image registration, the resulting affine transformations were used to warp segmentations in all atlases to the target space for atlas selection.",
"According to the normalised mutual information (NMI) scores between the target segmentation and each of affinely warped atlas segmentations, $L$ most similar atlases can be selected to save registration time and to remove dissimilar atlases for label fusion.",
"Since the correspondences of structures across both target and atlas volumes are explicitly encoded in their segmentations, we only use segmentations for the following non-rigid registration.",
"Let $S$ and $l_n$ ($n=1,...,L$ ) be the SSLLN-LR segmentation and the $n$ th atlas segmentation, respectively.",
"Let $P_{S,{l_n}}(i,j)$ be the joint probability of labels $i$ and $j$ in $S$ and $l_n$ , respectively.",
"It is estimated as the number of voxels with label $i$ in $S$ and label $j$ in $l_n$ divided by the total number of voxels in the overlap region of both segmentations.",
"We then maximise the overlap of structures denoted by the same label in both $S$ and ${l_n}$ by minimising the following objective function $ \\Phi _n^* = \\arg \\min {\\cal C}\\left( {{S},{{{l_n}}}({\\Phi _n})} \\right)$ where $\\Phi _n$ is the transformation between $S$ and ${l_n}$ , which is modelled by a free-form deformation (FFD) based on B-splines [29].",
"${\\cal C}( {{S},{{{l_n}}}})=\\sum _{i=1}^{N_r} P_{S,{l_n}}(i,i)$ , representing the label consistency [30].",
"${\\cal C}$ in (REF ) is a similarity measure of how many labels, of all the labels in the atlas segmentation, are correctly mapped into the target segmentation.",
"With the affine transformation as initialisation, a multi-scale gradient descent was then used to minimise the objective function (REF ).",
"After the optimal $\\Phi ^*_n$ is found, the segmentations and volumes in the $n$ th atlas are warped to the target space.",
"The process is repeated until $n=L$ .",
"Lastly, we perform non-local label fusion to generate an accurate and smooth bi-ventricular model $\\tilde{S}$ for the imperfect SSLLN-LR segmentation $S$ .",
"Let us first denote the warped atlas volumes and segmentations as $\\lbrace (f_{n},l^{\\prime }_{n})|n=1,...,L\\rbrace $ , respectively.",
"Here, $n$ denotes the warped atlas index and $L$ is the number of selected atlases.",
"For each voxel $x$ in the target LR volume $f$ , a patch $f_x$ centred at $x$ can be constructed.",
"The aim of the label fusion task is to determine the label at $x$ in $f$ using $\\lbrace (f_{n},l^{\\prime }_{n})|n=1,...,L\\rbrace $ .",
"For each voxel $x$ in $f_{n}$ , we define $\\lbrace (f_{n,y},l_{n,y})|n=1,...,L, y \\in {\\cal N}(x)\\rbrace $ , where $y$ denotes a voxel in the search window ${\\cal N}(x)$ , $f_{n,y}$ denotes the patch centred at voxel $y$ in the $n$ th warped atlas, and $l_{n,y}$ denotes the corresponding label for voxel $y$ .",
"The resulting label at voxel $x$ in the target volume $f$ can be calculated as $ {S_x} = \\mathop {\\arg \\max }\\limits _{k = 1,..,{N_r}} {\\sum \\limits _{n}} {\\sum \\limits _{\\;\\;y \\in {\\cal N}(x)}} { {{e^{ - \\frac{{\\Vert {{f_x} - {f_{n,y}}}\\Vert _F^2}}{h}}} \\cdot {\\delta _{{l_{n,y}},k}}} }$ where $h$ denotes the bandwidth for the Gaussian kernel function and ${\\delta _{{l_{n,y}},k}}$ denotes the Kronecker delta, which is equal to one when $l_{n,y}=k$ and equal to zero otherwise.",
"The equation (REF ) can be understood as a form of weighted voting, where each of the patches from each of the atlases contributes a vote for the label.",
"It is a non-local method because it uses patch similarity formulation (i.e.",
"Gaussian kernel function), which is inspired by the non-local methods [31], [32], [33].",
"It has been shown in [34] that, in a Bayesian framework, (REF ) is essentially a weighted $K$ nearest neighbours (KNN) classifier, which determines the label by maximum likelihood estimation.",
"By aggregating high-resolution atlas shapes in this way, an explicit anatomical shape prior can be inferred.",
"The artefacts in the SSLLN-LR segmentation can thus be resolved, as shown in Fig.",
"REF $j$ .",
"In this section, we cover extensive experiments to evaluate (both qualitatively and quantitatively) the performance of the proposed pipeline on short-axis CMR volumetric images.",
"Dice index and Hausdorff distance [13] were employed for evaluating segmentation accuracy.",
"Dice varies from 0-1, with high values corresponding to a better results.",
"The Hausdorff distance is computed on an open-ended scale, with smaller values implying a better match.",
"We also validate the performance using clinical measures (ventricular volume and mass) derived from the segmentations.",
"In the following experiments, each component in the pipeline is studied separately."
],
[
"Clinical datasets",
"UK Digital Heart Project Dataset: This datasethttps://digital-heart.org/ (henceforth referred to as Dataset 1) is composed of 1831 cine HR CMR volumetric images from healthy volunteers, with corresponding dense segmentation annotations at the end-diastolic (ED) and end-systolic (ES) frames.",
"The ground-truth segmentation labels were manually annotated by a pair of clinical experts working together, and each volume was only annotated by one expert at a time.",
"For each volume at ED, 6 landmarks, as shown in Fig REF middle, were manually annotated by a clinician (inter-user 1).",
"The raw volumes were derived from healthy subjects, scanned at Hammersmith Hospital, Imperial College London using a 3D cine balanced steady-state free precession (b-SSFP) sequence [23] and has a resolution of $1.25 \\times 1.25 \\times 2 \\;\\rm {mm}$ .",
"As introduced in Section REF , HR imaging technique does not produce cardiac artefacts which are often seen in LR imaging acquisition [19].",
"Pulmonary Hypertension Dataset: This dataset (henceforth referred to as Dataset 2) was acquired at Hammersmith Hospital National Pulmonary Hypertension Centre, and composed of 649 subjects with pulmonary hypertension (PH) - a cardiovascular disease characterised by changes in bi-ventricular volume and geometry.",
"PH subjects often have breathing difficulties, therefore HR imaging was impractical for the majority of patients in this cohort due to the relatively long breath-hold time required.",
"Within the cohort, 629 of the 649 patients were scanned using conventional LR image acquisition, and this manner of image acquisition (over multiple short breath-holds) often leads to lower-resolution volumes and inter-slice shift artefacts.",
"In contrast, the remaining 20 subjects managed to perform a single breath-hold, and therefore HR volumes could be acquired for these subjects.",
"Coupled with these HR volumes, LR volumes were also acquired during scanning, forming 20 pairs of LR and HR cine CMR volumes.",
"The resolutions for LR and HR volumes are $1.38 \\times 1.38 \\times 10 \\;\\rm {mm}$ and $1.25 \\times 1.25 \\times 2 \\;\\rm {mm}$ , respectively.",
"For all 649 subjects, the manual ground-truth segmentation labels at ED and ES were generated, and 6 landmarks at ED were also annotated."
],
[
"Preprocessing and augmentation",
"Preprocessing: Image preprocessing was carried out to ensure: 1) the size of each volumetric image fits the network architecture; 2) the intensity distribution of each volume was in a comparable range so that each input could be treated equally importantly.",
"As such, each of the HR volumes in Dataset 1 was reshaped to common dimensions of $192 \\times 192 \\times 80$ with zero-padding if necessary, while each of LR volumes in Dataset 2 was interpolated to $1.25 \\times 1.25 \\times 2$ mm and then reshaped to $192 \\times 192 \\times 80$ .",
"For the best visual effect, the figures shown in experiments may be cropped manually.",
"However, no ROI detection algorithm (for localisation of the heart) was used in image preprocessing.",
"The intensity redistribution processes for both HR and LR volumes are the same.",
"After reshaping, we first clipped the extreme voxel values (i.e.",
"outliers) in each HR/LR volume.",
"We defined outliers as voxel values lying outside of the 1st to 99th percentile range of original intensity values.",
"Finally, the resulting voxel intensities of each volume were scaled to the $[0, 1]$ range.",
"Parameter selection: The following parameters were utilised for the experiments in this study: For training the network, each run was carried out for 50 epochs, with batch size of 8 volumes, learning rate of 0.001 and Adam stochastic gradient descent for optimisation.",
"The weight coefficients $\\alpha $ , $\\beta $ and $\\gamma $ in (REF ) are empirically set to 0.8, 0.2 and $5 \\times 10 ^ {-5}$ , respectively.",
"The small positive value in the Dice loss (REF ) is set to $1 \\times 10^{-8}$ .",
"According to [6] the exact network architecture only plays a minor role in improving segmentation accuracy.",
"Therefore, the network architecture, as shown in Fig REF , was used without significant modification.",
"For the non-local label fusion (REF ), we used a value of 10 for the bandwidth parameter $h$ , voxel dimensions $7 \\times 7 \\times 1$ for the patch window size and $7 \\times 7 \\times 3$ for the search window size.",
"For more details on parameter tuning in (REF ), we refer the reader to [34].",
"Finally, $L=5$ atlases were used for label fusion.",
"Using the parameter settings outlined above, we found the pipeline performed very well for our experiments, indicating its robustness to parameters tuning.",
"Augmentation: Since our network takes volumetric images as inputs, we performed 3D data augmentation on-the-fly during training.",
"At each iteration, augmentation included rescaling of voxel intensities in the input volume, and a 3D random affine transformation of the volume and corresponding labels and landmarks.",
"For simplicity, the affine transformation only involved in-plane translation, isotropic scaling and rotation along one random direction ($x$ -, $y$ - or $z$ -axis) at the central voxel of the volume.",
"Neither shearing nor volume flipping was used.",
"Data augmentation enables the network to see a large and diverse array of inputs by the end of training, and was implemented using the SimpleITK library in Python.",
"With an Nvidia Titan XP GPU, training (50 epochs) took approximately 20 and 10 hours for Datasets 1 and 2, respectively.",
"For inference, segmentation (without shape refinement) of an HR/LR volume for a single subject at ED took $<1s$ ."
],
[
"Segmentation of high-resolution volumes",
"First, we conducted experiments using Dataset 1, which includes 1831 HR CMR volumes.",
"We randomly split the dataset into two disjoint subsets of 1000/831.",
"The first subset was used to train SSLLN-HR, and the second subset was used for testing the accuracy of segmentation and landmark localisation, respectively.",
"During training, we only used ED instances (volumes, landmarks and segmentation labels).",
"Note that the proposed SSLLN-HR is a multi-task network that simultaneously outputs labels and landmarks.",
"Next we segmented a cardiac volume into 5 regions: the left ventricular cavity (LVC), right ventricular cavity (RVC), left ventricular wall (LVW), right ventricular wall (RVW) and background.",
"Our method is the first one capable of producing a full HR bi-ventricular segmentation (LVC+LVW+RVC+RVW) in 3D.",
"Figure: Visual comparison of segmentation results by 2D slice-by-slice FCN, 3D FCN and SSLLN-HR.",
"aa and bb: two views of a high-resolution volume; cc, dd and ee: results by 2D FCN; ff, gg and hh: results by 3D FCN; ii, jj and kk: SSLLN-HR.",
"SAX and LAX denote short-axis and long-axis, respectively.In Fig REF , we compare SSLLN-HR with two baseline methods for segmentation.",
"The first one is the 2D FCN proposed in [13], where the networkCode is publicly available at https://github.com/baiwenjia/ukbb$\\_$ cardiac was trained using each short-axis slice in the volume separately.",
"The second one is the 3D FCN, whose architecture is similar as in Fig REF .",
"To make the 3D FCN fit GPU memory, we halved the number of activation maps in each layer (excluding last one) and cropped the original image to a size of $112 \\times 112 \\times 64$ .",
"To focus exclusively on segmentation accuracy, we removed the landmark localisation activation maps in the last layer of the 3D FCN.",
"As Fig REF shows, 2D FCN produces a jagged appearance as shown in the long-axis view image Fig REF $d$ , and there are `cracks' in the corresponding 3D model as shown in Fig REF $e$ .",
"This problem is due to the fact that the 2D method does not consider 3D context of the volumetric image, leading to a lack of spatial consistency between segmented slices.",
"In contrast, both SSLLN-HR and 3D FCN account for the spatial consistency between slices, enabling smooth results.",
"Visually, SSLLN-HR is comparable to 3D FCN.",
"However, SSLLN-HR is less memory demanding and therefore can be directly implemented on non-cropped volumes with a faster training speed.",
"Table: Dice index and Hausdorff distance derived from 2D FCN, 3D FCN, and SSLLN-HR for segmenting 831 high-resolution short-axis volumetric images.",
"The mean ±\\pm standard deviation are reported.Table REF provides a summary of quantitative comparisons between 2D FCN, 3D FCN and SSLLN-HR, with statistics derived from 831 subjects.",
"Statistical significance of the observed differences in the evaluation metrics (Dice index and Hausdorff distance) between each pair of methods is assessed via the Wilcoxon signed-rank test.",
"The results in the table demonstrate the high consistency between automated and manual segmentations.",
"In terms of Dice and Hausdorff distance, SSLLN-HR and 3D FCN outperformed 2D FCN, and SSLLN-HR achieved comparable performance to 3D FCN.",
"Of note, all three methods achieved a relative low Dice score on the RVW anatomy.",
"This is due to the thinness of RVW and the fact that the Dice index is more sensitive to errors in this structure.",
"In Fig REF , boxplots visually depicting the results of Table are presented.",
"As these plots show, the 2D method produced large variation across different segmentations for the four anatomies, resulting in a inferior accuracy than the 2.5D and 3D methods.",
"SSLLN-SR achieved similar results to 3D FCN, with the segmentation accuracy of RVC and RVW slightly higher than that of 3D FCN.",
"Figure: Boxplot comparison of segmentation accuracy between 2D FCN, 3D FCN and SSLLN-HR on 831 high-resolution short-axis volumetric images.",
"The symbol `***' denotes pp ≪\\ll 0.001, and `*' denotes pp << 0.1.Figure: Testing 3D spatial consistency of the 2D FCN and SSLLN-HR methods.",
"1st column: target segmentation volumes with zero-filled gaps of different sizes; 2nd and 3rd columns: 2D FCN results; 4th and 5th columns: SSLLN-HR results.In Fig REF , we further compare the proposed SSLLN-HR with the 2D FCN.",
"We selected batches of $k$ consecutive short-axis slices in a volumetric image, with multiple settings of $k$ (=5,13, and 20).",
"In each case, we set intensities in the selected slices to zero, as shown in the 1st column.",
"The two methods under comparison were then applied to these partially zero-filled volumes, and the results are given in 2nd-5th columns.",
"As is evident, 2D FCN fails to segment these zero-filled slices, thus leaving gaps in the resulting 3D segmentations.",
"In contrast, SSLLN-HR demonstrates robustness to missing slices and has the capability of `inpainting' these gap regions.",
"However, as the gap (number of zero-filled slices) increases (from $k$ =5 to $k$ =20), the segmentation performance becomes worse.",
"These results further illustrate that the proposed network retains 3D spatial consistency, which the 2D FCN is unable to achieve.",
"Our method thus outperforms the 2D approach in this regard."
],
[
"Landmark localisation",
"To enable automatic alignment for subsequent non-rigid registration, we also predicted landmark locations (together with segmentation) for each input volume using SSLLN-HR.",
"Same as above, we used the split subsets 1000/831 for training and testing.",
"Note that SSLLN-HR was trained with manual landmarks carried out by inter-user 1 on each of the 1000 subjects.",
"For the 831 unseen test subjects, the automatically detected landmarks were compared with the manual ones from inter-user 1 using the point-to-point Euclidean distance.",
"Also, to study inter-user variability of landmarking, a second expert (inter-use 2) was recruited to manually annotate landmarks for each of 831 test subjects.",
"The annotations were then compared with those of inter-user 1.",
"Fig REF first shows a visual comparison of automated and manual (inter-user 1) landmarks.",
"Fig REF $b$ shows the landmark locations predicted by our SSLLN-HR.",
"As is evident, each landmark is represented by a few locally clustered voxels.",
"The central gravity (represented by a single voxel) of each landmark in Fig REF $b$ can be computed by averaging the positions of the voxels forming the true landmark.",
"The corresponding results are shown in Fig REF $c$ , where the two type of landmarks are superimposed.",
"The respective colour-coded single-voxel landmarks are shown in Fig REF $d$ , which were used for initial point-to-point affine registration, as shown in Fig REF .",
"In Fig REF $f$ , we superimposed the automated detected landmarks and manual landmarks (Fig REF $e$ ).",
"As can be seen, Fig REF $f$ demonstrates very good consistency between the automated and manual landmarks.",
"Figure: Landmark localisation using the proposed network.",
"aa: input volume; bb: landmarks detected by SSLLN-HR directly; cc: single-voxel landmark extraction from each clustered landmark in bb; dd: colour coded single-voxel landmarks; ee: ground-truth landmarks annotated by inter-user 1; ff: superimposed ground-truth (red) and automated (white) landmarks.In Table REF , we compare the landmark localisation errors between automated and manual methods, as well as between the two manual methods on 831 test volumes.",
"Using inter-user 1 as a baseline for comparison, we observe that the SSLLN-HR detections are more accurate than the annotations of inter-user 2.",
"The point-to-point distance errors between SSLLN-HR and inter-user 1 vary only from 3.67$\\pm $ 3.20 mm for Landmark-I to 8.18$\\pm $ 6.91 mm for Landmark-II.",
"In contrast, the errors between the two inter-users vary from 5.61$\\pm $ 2.62 mm for Landmark-V to 17.4$\\pm $ 9.27 mm for Landmark-II.",
"This confirms that computer-human difference can be smaller than human-human difference.",
"Table: Point-to-point (P2P) distance error statistics in landmark localisation over 831 volumes.",
"The second column shows the errors between automated (SSLLN-HR) and manual (inter-user 1) landmark localisations.",
"The third column shows the errors between two manual (inter-user 1 and 2) annotations.",
"The mean ±\\pm standard deviation in mm are reported.",
"The description of the 6 landmarks is given in Fig .",
"Figure: Cumulative error distribution curves of landmark localisation errors.",
"The left curves are derived from manual landmarks of inter-user 1 and inter-user 2, and the right curves are plotted based on automated (SSLLN-HR) and manual (inter-user 1) landmark localisations.Fig REF provides a simple visualisation of the relative error distribution in the test sample of 831 volumes.",
"The two plots show the cumulative distribution of point-to-point distance error for each landmark.",
"As can be seen, the curves in the right are more clustered and stacked vertically than those in the left, indicating superior accuracy of landmark localisations by SSLLN-HR.",
"For example, from the right plot we see that for all landmarks, about 92$\\%$ of test volumes had point-to-point distance error of $<$ 20 mm.",
"In contrast, only 60$\\%$ of test volumes reached point-to-point distance error of $\\sim $ 20 mm, as shown in the left plot.",
"Figure: Visualisation of landmarks in 3D.",
"aa: manual landmarks by inter-user 1; bb: manual landmarks by inter-user 2; cc: landmarks localised by the network, trained on the manual annotations of inter-user 1; dd: superimposed aa and bb; ee: superimposed aa and cc.",
"It is evident that aa and cc overlap to a greater degree than bb and cc.Figure: Impact of using landmarks in the proposed pipeline.",
"1st column: SSLLN-HR segmentation of an HR atlas volume; 2nd column: SSLLN-LR segmentation of an LR volume; 3rd column: superimposed SSLLN-HR and SSLLN-LR segmentation labels; 4th column: SSLLN-HR segmentation affinely warped to the SSLLN-LR segmentation based on their labels; 5th column: final SSLLN-LR segmentation refined by the non-rigid registration initialised with the label-based affine transform; 6th column: SSLLN-HR segmentation affinely warped to the SSLLN-LR segmentation based on their landmarks localised by the network in Fig ; 7th column: final SSLLN-LR segmentation refined by the non-rigid registration initialised with the landmark-based affine transform.",
"The 1st, 2nd and 3rd rows respectively show the short-axis view, long-axis view and corresponding 3D visualisation of segmentation.Fig REF provides a 3D visualisation of landmarks for all the 831 volumes.",
"These landmarks were acquired from inter-user 1, inter-user 2 and SSLLN-HR.",
"This figure further illustrates that SSLLN-HR, trained from manual annotations of a human, excellently matches the performance of that human on an unseen test set.",
"On the other hand, the discrepancy between human-human performance could be very large.",
"Fig REF , together with Fig REF , Table REF and Fig REF , provide an ample evidence that the proposed SSLLN-HR has the capability of detecting landmarks robustly and accurately, and that it tends to produce less variability in predictions relative to variability among human experts."
],
[
"Impact of landmarks",
"In this section, we show that landmarks localisation is a necessary step in our pipeline.",
"In Fig REF , we compare the SSLLN-LR segmentation results refined by the non-rigid deformation with different initialisations of affine transformation.",
"As shown in the 5th column of Fig REF , the non-rigid refinement failed completely if the affine transform is initialised from the tissue classes.",
"In contrast, initialising it directly on the landmarks resulted in an accurate refinement, as shown in the last column of Fig REF .",
"We propose two reasons for this observation: 1) the six anatomical landmarks defined in the study effectively reflect the underlying pose, size and shape of a heart.",
"As such, warping a heart with landmark-based affine transformation produces a very robust initialisation for the subsequent non-rigid registration; 2) Computing an affine transformation from a pair of landmarks is a convex least squares problem, a unique solution to which exits.",
"In contrast, initialising an affine transform directly on the tissue classes is a non-convex problem.",
"As such, the warped result is sometimes sub-optimal, which may negatively impact the non-rigid registration and increase uncertainty of the registration method.",
"Moreover, label-based affine registration is much more computational expensive than landmark-based affine registration as it needs to deal with millions of voxels in the 3D volumes.",
"In the IRTK implementationCode is publicly available at https://github.com/BioMedIA/IRTK, it took $\\sim $ 0.005s to compute an affine transformation on a pair of landmarks, whilst it took $\\sim $ 5s to perform an affine registration using the 3D segmentation labels with size $256 \\times 256 \\times 56$ .",
"We note that it may also be possible to detect landmarks automatically from segmentation labels.",
"In this case, the accuracy of landmarks will be conditioned on the accuracy of segmentation.",
"On the other hand, it may not be straightforward to determine which landmarks should be detected from segmentation labels for robust registration.",
"As such, directly detecting the six landmarks defined in the study using the proposed network is neater and better."
],
[
"Experiments on simulated low-resolution volumes",
"To quantitatively assess the performance of SSLLN-LR and shape refinement (SSLLN-LR+SR) in the pipeline (bottom path in Fig REF ), we developed a method to simulate different types of artefacts seen in LR cardiac volumes.",
"Specifically, in Fig REF an HR volume and its manual segmentation were first downsampled from $1.25 \\times 1.25 \\times 2 \\;\\rm {mm}$ to $1.25 \\times 1.25 \\times 10 \\;\\rm {mm}$ , as shown in the 1st and 2nd columns.",
"The downsampling produces a staircase artefact due to reduction in long-axis resolution.",
"Moreover, the segmentation (Fig REF $d$ ) around the apical region is now incomplete due to the lack of coverage of the whole heart.",
"We further simulated inter-slice shift artefact by randomly translating each 2D short-axis slice horizontally.",
"This step produced misalignment in the cardiac volume and its segmentation, as shown in the 3rd column.",
"Next, for training the SSLLN-LR, the LR volume Fig REF $e$ and its segmentation Fig REF $f$ were used as inputs.",
"Note that our method is capable of producing an HR smooth segmentation model even from misaligned inputs such as the example in Fig REF $f$ .",
"Since we have the smooth ground truth Fig REF $b$ for the simulated Fig REF $e$ , we can quantitatively assess the ability of our method to recover the original smooth shape.",
"For these simulation experiments, we split Dataset 1 into subsets (1000/600/231).",
"The first two subsets were corrupted with the simulated artefacts described above, which were used for training the SSLLN-LR and testing the proposed shape refinement (SC) component of the pipeline.",
"The HR atlas shapes (segmented by SSLLN-HR) in the last cohort ($n$ =231) were used to refine SSLLN-LR segmentations.",
"Here we highlight three reasons why we used SSLLN-HR network results as a reference atlas set for shape-refinement: 1) our SSLLN-HR is able to produce results that are very similar to the corresponding ground truth, as confirmed from Section REF and REF ; 2) Once the SSLLN-HR is trained, it can be readily deployed on an external dataset (where HR atlases are not available) to create new HR atlases so as to facilitate the running of our pipeline; 3) The atlas set can be enriched by adding more SSLLN-HR results, which will increase the possibility to select better atlases for the sequential registration-based refinement.",
"Figure: Simulating cardiac artefacts in real scenarios.",
"1st column: artefact-free high-resolution cardiac volume and ground-truth labels.",
"2nd column: downsampled versions of volumes in the 1st column.",
"3rd column: inter-slice shift is added to the downsampled volumes in the 2nd column.In Table REF , we compare the Dice index and Hausdorff distance between the SSLLN-HR and SSLLN-LR+SR results.",
"SSLLN-HR was directly evaluated on 600 artefact-free HR volumes at ED in Section REF , while SSLN-LR+SC was tested on the 600 corresponding simulated LR volumes where cardiac artefacts exist, as shown in Fig REF .",
"Although SSLLN-HR performs better than SSLLN-LR+SR, the performance gap between two approaches is minor.",
"For LVC, LVW and RVC, the Dice index of SSLLN-HR is only about 0.2 higher than that of SSLLN-LR+SR.",
"The Hausdorff distance of SSLLN-HR is about 0.5 mm smaller than that of SSLLN-LR+SR for all 4 regions.",
"Again due to the thin structure of RVW, the mean Dice values of the two methods are relatively low: 0.662 and 0.557, respectively.",
"This table shows that SSLLN-LR+SR achieves good segmentation results for imperfect LR input volumes, and the results are comparable to direct segmentation of artefact-free HR results.",
"Table: Comparison of Dice index and Hausdorff distance between SSLLN-HR and SSLLN-LR+SR (shape refinement).",
"SSLLN-HR was validated on 600 high-resolution short-axis volumetric images from Dataset 1, whilst SSLLN-LR+SR was validated on 600 low-resolution volumes, simulated from the corresponding high-resolution volumes.In Table REF , we report the mean and standard deviation of the measurements derived from the two automated methods and manual segmentation.",
"The table further demonstrates SSLLN-LR+SR results are comparable to SSLLN-HR results, proving that our proposed method can produce results comparable to direct segmentation of artefact-free HR volumes, even though target segmentation volumes are of low resolution and contain artefacts.",
"Moreover, the RVM measurement derived from the two methods is consistent with the manual RVM measurement, confirming adequate segmentation of RVW using the two methods despite relatively lower Dice scores, as shown in Table REF Table: Comparison of clinical measures between SSLLN-HR, SSLLN-LR+SR and manual measurements on 600 volumetric cardiac images.",
"SSLLN-HR was validated on high-resolution volumes from Dataset 1, whilst SSLLN-LR+SR was validated on 600 low-resolution volumes, simulated from the corresponding high-resolution volumes.",
"The 4th/5th columns show absolute difference between automated and manual measures.Table: Comparison of Dice index and Hausdorff distance between the proposed SSLLN-LR+SR and 5 state-of-the-art 3D approaches.",
"These methods were tested on 20 simulated LR volumes (∼\\sim 200 CMR images).",
"The ground-truth labels were obtained from high-resolution volumes acquired from same subjects, which do not contain cardiac artefacts.Next, we compare SSLLN-LR+SR with the 3D-seg model [11], 3D-UNet model [35], cascaded 3D-UNet and convolutional auto-encoder model (3D-AE) [36], 3D anatomically constrained neural network model (3D-ACNN) [11] as well as multi-atlas methodCode is publicly available at https://github.com/baiwenjia/CIMAS (MAM) [37].",
"To ensure a fair comparison, we used the same 20 CMR volumes as in [11] and the quantitative results are summarised in Table REF .",
"Since 3D-ACNN only segments the left ventricle (LV), the table only shows the results for the endocardium and myocardium of LV.",
"Among the methods compared, 3D-seg and 3D-UNet do not use shape information, while 3D-AE and 3D-ACNN infer shape constraints using an auto-encoder during network training.",
"As Dice shows, MAM is inferior to deep learning-based methods, shape-based models outperform those without shape priors, and our SSLLN-LR+SR achieved the best performance.",
"We propose three main reasons for this: 1): SSLLN-LR+SR uses atlas propagation to impose a shape refinement explicitly while 3D-AE and 3D-ACNN impose shape constraints in an implicit fashion.",
"When the initial segmentation by SSLLN-LR is of sufficiently adequate quality, such an explicit shape refinement is able to produce more accurate segmentation.",
"2): SSLLN-LR+SR is a 2.5D-based method which allows the use of deeper network architectures than the 3D-based methods (e.g.",
"ACNN-seg only uses 7 convolutional layers while SSLLN-LR+SR has 15), leading to improved segmentation accuracy.",
"3): SSLLN-LR+SR uses label-based non-rigid registration (REF ), which may be more accurate for segmentation purpose than the intensity-based non-rigid registration used in MAM."
],
[
"Experiments on pathological low-resolution volumes",
"In Section REF , we have quantitatively studied the performance of the proposed SSLLN-LR+SR using simulated LR cardiac volumes.",
"In this section, we will use real LR volumes.",
"In particular, we test SSLLN-LR+SR on volumetric data in patients with pulmonary hypertension (PH) in Dataset 2.",
"PH leads to a progressive deterioration in cardiac function and ultimately death, due to RV failure.",
"As such, it is critical to accurately segment different functional regions of the heart in PH so as to study PH patients quantitatively.",
"Fig REF shows the difference in two CMR volumes from a representative healthy subject and a PH subject.",
"In health, the RV is crescentic in short-axis views and triangular in long-axis views, wrapping around the thicker-walled LV.",
"In PH, the dilated RV pushes onto the LV causing deformation and loss of its circular shape.",
"The abnormal cardiac morphology of PH heart poses challenges for existing segmentation algorithms.",
"Figure: Illustrating the difference between a healthy subject (first two) and a PH subject (last two) from short- and long-axis views.",
"Both subjects were scanned using low-resolution acquisition.For training and testing, we use Dataset 2 introduced in Section REF .",
"This dataset includes 629 LR PH volumes and 20 pairs of LR and HR PH volumes.",
"We randomly split the 629 volumes into two disjoint subset of 429/200.",
"The first subset is used to train SSLLN-LR, while the second subset is used for visually testing the accuracy of SSLLN-LR+SR segmentations (due to lack of corresponding HR ground truths).",
"The 20 LR volumes are also used to quantitatively evaluate SSLLN-LR+SR using their HR volumes as ground-truth references.",
"231 HR atlases appearing in Section REF are used to refine SSLLN-LR segmentations.",
"200 greyscale PH volumes ($1.38 \\times 1.38 \\times 10$ mm) were segmented by SSLLN-LR+SR into HR smooth models ($1.25 \\times 1.25 \\times 2$ mm).",
"Results were visually assessed by one clinician with over five years' experience in CMR imaging and judged satisfactory in all cases.",
"We propose three reasons why the shape refinement works for PH cases: 1) the landmark-based affine and non-rigid registrations are collectively able to capture both global and local deformations between subjects; 2) for the non-rigid registration, we used label consistency as a loss function (REF ).",
"It is based on segmentation masks, which can provide stronger regional and edge information for an accurate registration; 3) multiple atlases (i.e.",
"the most similar to the subject) were selected for registration and fusion, and these selected atlases together vote for the final result, which further prevents diseased cases producing healthy results.",
"In Fig REF $a$ -$h$ and Fig REF , we show an exemplary bi-ventricular segmentation of a cardiac volume in PH.",
"We visually compare SSLLN-LR+SR with 2D FCN [13] and two approaches (nearest neighbour interpolation (NNI) and shape-based interpolation (SBI) [30], [38]) that interpolate the 2D FCN results.",
"Both 2D FCN and interpolation methods do not use anatomical shape information, so they performed worse than SSLLN-LR+SR in the long-axis view, as confirmed in Fig REF $f$ -$h$ .",
"Due to the high in-plane resolution, similar results in the short-axis view were achieved by different methods, as shown in Fig REF $b$ -$d$ .",
"Moreover, we observed from Fig REF that SSLLN-LR+SR gives a better 3D phenotype result which is smooth, accurate and artefact-free.",
"Figure: Bi-ventricular segmentation of volumetric images from two PH patients.",
"aa and ee: original low-resolution volume (two views) from patient I; bb and ff: 2D FCN+NNI results; cc and gg: 2D FCN+SBI results; dd and hh: SSLLR-LR+SC results.",
"ii and mm: original low-resolution volume from patient II; jj and nn: SSLLN-LR+SR results; kk and oo: original high-resolution volume from patient II; ll and pp: ground truth.",
"The proposed SSLLN-LR+SR is not only insensitive to cardiac artefacts (inter-slice shift, large slice thickness, and lack of slice coverage), but also robust against pathology-induced morphological changes.Next, we test SSLLN-LR+SR using 20 pairs of LR and HR cardiac volumetric images.",
"In Fig REF $i$ -$p$ , we first demonstrate a segmentation example on a pair of LR and HR volumes acquired from the same patient with PH.",
"The original low-resolution volume ($1.38 \\times 1.38 \\times 10$ mm) was segmented by SSLLN-LR+SR into a HR smooth model ($1.25 \\times 1.25 \\times 2$ mm).",
"The smooth segmentation is then visually compared with the ground truth, obtained directly from segmenting the corresponding HR volume of the patient.",
"As is evident, the paired segmentation results show a very good agreement in terms of their cardiac morphology.",
"Further, Table REF is provided, which shows a quantitative comparison between the SSLLN-LR+SR results and the ground-truth segmentations.",
"The automated measurements are quantitatively consistent with the manual measurements.",
"Comparing Table REF with Table REF , we observed that PH patients have a bigger RVC and a smaller LVC than healthy subjects, and that the RVW of PH patients is thicker than that of healthy subjects.",
"Note that the Dice scores computed from the paired LR and HR volumes are not applicable here due to the fact that they were acquired from subjects scanned at different positions with different breath-holds.",
"We also note that $p$ values in Table REF are relatively large.",
"This is likely due to the relatively low sample size of the dataset used in this experiment, in addition to the fact that automatic and manual measurements are not substantially different.",
"Figure: Visualisation of a 3D bi-ventricular model obtained through segmenting the volumetric image from a PH patient.",
"1st column: 2D FCN results; 2nd column: 2D FCN+NNI results; 3rd column: 2D FCN+SBI results; 4th colunm: SSLLR-LR+SC results.",
"The proposed approach is capable of producing accurate, high-resolution and anatomically smooth bi-ventricular models for pathological subjects.Table: Comparison of clinical measures derived from SSLLN-LR+SR and manual segmentations on 20 pairs of low-resolution and high-resolution volumetric images from Dataset 2.",
"SSLLN-LR+SR segmented 20 low-resolution volumes into high-resolution models, whilst manual segmentation was performed on 20 high-resolution cardiac volumes directly.",
"The 4th column shows absolute difference between SSLLN-LR+SR and manual measures."
],
[
"Discussion and Conclusion",
"In this paper, we developed a fully automatic pipeline for shape-refined bi-ventricular segmentation of short-axis CMR volumes.",
"In the pipeline, we proposed a network that learns segmentation and landmark localisation tasks simultaneously.",
"The proposed network combines the computational advantage of 2D networks and the capability of addressing 3D spatial consistency issues without loss of segmentation accuracy.",
"The pipeline also induces an explicit shape prior information, thus allowing accurate, smooth and anatomically meaningful bi-ventricular segmentations despite artefacts in the cardiac volumes.",
"Extensive experiments were conducted to validate the effectiveness of the proposed pipeline for both healthy and pathological cases.",
"However, there still exist limitations in the pipeline.",
"For example, the pipeline is a 2-stage approach, which is not end-to-end learning.",
"In such a case, the network parameters learned in stage 1 might not be optimal to generate high-resolution smooth segmentations in stage 2.",
"In addition, although the deployment of a trained network (SSLLN-HR or SSLLN-LR) in stage 1 took less than 1s, the shape refinement (SR) in stage 2 is relatively computationally expensive, which is a big disadvantage.",
"SR combines the computational costs from atlas selection, target-to-atlas non-rigid image registration, and non-local label fusion.",
"In our implementation, SR was performed in parallel for 5 selected atlases using multiple CPUs of a workstation and it took 15-20 mins per subject at ED.",
"In future work, we will investigate how to train a single network to compute smooth shapes from artefact-corrupted low-resolution cardiac volumes.",
"A simple solution would be training an end-to-end super-resolution network, as in [21], but with the segmentation labels acquired from our pipeline as the ground truth inputs.",
"We will also investigate how to improve the computational speed of Stage 2 in our pipeline.",
"For example, a GPU-based non-rigid image registration toolboxhttp://cmictig.cs.ucl.ac.uk/research/software/software-nifty/niftyreg could be utilised.",
"Besides the GPU-based implementation, deep hashing [39], [40] may be explored to select relevant atlas subjects instead of the brute force search technique (i.e.",
"nearest neighbour) currently used in our atlas selection process.",
"Another direction will be to investigate how to adapt the proposed network architecture for different tasks.",
"For example, a fully connected layer may be concatenated for classification of subjects into healthy versus pathological groups, which will be carried out simultaneously with segmentation and landmark localisation tasks.",
"Our pipeline treats landmarks as voxels and classifies them.",
"In future work, we will explore an alternative approach that treats landmarks as points and regresses their coordinates, which could be implemented with a fully connected layer."
],
[
"Acknowledgements",
"The research was supported by the British Heart Foundation (NH/17/1/32725, RE/13/4/30184); the EPSRC SmartHeart Programme (EP/P001009/1); the National Institute for Health Research (NIHR) Biomedical Research Centre based at Imperial College Healthcare NHS Trust and Imperial College London; and the Medical Research Council, UK.",
"We would like to thank Dr Simon Gibbs, Dr Luke Howard and Prof Martin Wilkins for providing the CMR image data.",
"The TITAN Xp GPU used for this research was kindly donated by the NVIDIA Corporation."
]
] | 1808.08578 |
[
[
"Secrecy Performance Analysis of UAV Transmissions Subject to\n Eavesdropping and Jamming"
],
[
"Abstract Unmanned aerial vehicles (UAVs) have been undergoing fast development for providing broader signal coverage and more extensive surveillance capabilities in military and civilian applications.",
"Due to the broadcast nature of the wireless signal and the openness of the space, UAV eavesdroppers (UEDs) pose a potential threat to ground communications.",
"In this paper, we consider the communications of a legitimate ground link in the presence of friendly jamming and UEDs within a finite area of space.",
"The spatial distribution of the UEDs obeying a uniform binomial point process (BPP) is used to characterize the randomness of the UEDs.",
"The ground link is assumed to experience log-distance path loss and Rayleigh fading, while free space path loss with/without the averaged excess path loss due to the environment is used for the air-to-ground/air-to-air links.",
"A piecewise function is proposed to approximate the line-of-sight (LoS) probability for the air-to-ground links, which provides a better approximation than using the existing sigmoid-based fitting.",
"The analytical expression for the secure connection probability (SCP) of the legitimate ground link in the presence of non-colluding UEDs is derived.",
"The analysis reveals some useful trends in the SCP as a function of the transmit signal to jamming power ratio, the location of the UAV jammer, and the height of UAVs."
],
[
"Introduction",
"UNMANNED aerial vehicles (UAVs), also known as drones, are a promising technology that offers reliable and cost-effective wireless communication solutions in a wide range of real-world scenarios [1].",
"Recently, the low-altitude UAVs with elevated height from hundreds of meters to several kilometers have drawn much research attention in surveillance, public safety and secure communications [2], [3], [4].",
"Compared to the existing terrestrial communication systems, UAV aided wireless networks have the potential to overcome the propagation constraints due to terrain characteristics, enhance the signal coverage and reduce operating cost [4].",
"Due to the broadcast nature of the wireless signal and the openness of the space, ground communications are susceptible to eavesdropping.",
"UAV eavesdroppers (UEDs) could pose a greater threat to the security of ground communications than ground eavesdroppers, since UEDs are less constrained by terrain characteristics and a higher chance of line-of-sight (LoS) link with stronger signal strength can be formed from the ground transmitter to the UEDs rather than to ground eavesdroppers.",
"Furthermore, many aspects of UAVs such as low production cost, high mobility, and ease of operation could incentivize the attackers to use UAVs as the major eavesdropping tools.",
"Consequently, it has become increasingly urgent and necessary to study the security of ground-based wireless communication in the presence of UEDs.",
"Among the many methods for securing wireless communication, physical layer security (PLS) has emerged as a promising technique for achieving a secure transmission by exploiting the channel characteristics through signal processing techniques and channel coding without the need of a shared secret key [5], [6].",
"By using an information theoretic formalism, PLS has been shown to support perfect secrecy under realistic condition [7].",
"Due to the fading effects of wireless channels and the unpredictable locations of eavesdroppers, probabilistic approaches have been used to characterize the likelihood of a link achieving a secrecy rate, namely, secure connection probability (SCP) [8].",
"Stochastic geometry has been exploited for the analysis of eavesdropping wireless networks by endowing the locations of the eavesdroppers with a probability distribution, such as the Poisson point process (PPP) or the binomial point process (BPP) [9], [10], [11].",
"For example, the spectrum sharing of super dense drone small cell networks modeled by a 3D PPP is studied in [12].",
"The authors of [13] consider the case that the number of UAVs is small and deployed to cover a given finite region with a more suitable homogeneous BPP for UAV networks under Nakagami-m fading.",
"Since it is common in deployment scenarios (especially in suburban and rural areas) to have a significantly stronger LoS component rather than reflected multipath components, the coverage probability in the absence of fading has been derived in [13].",
"Furthermore, to capture the performance of an air-to-ground (ATG) link between a ground device and a UAV, the channel propagation model incorporating blockages from buildings is required.",
"Based on the statistical model for building blockages [14], the LoS probability in the product of a sequence of terms and its approximation via sigmoid function is formulated, and the optimal altitude for deploying the UAV with maximum coverage is studied in [15].",
"There have been a lot of studies on either optimizing network resources or developing techniques for realistic system-level analysis of UAV networks on coverage, but it has been pointed out by [4] that very few studies have investigated the secrecy performance of UAV networks, and only ground-based eavesdroppers were considered in those scenarios.",
"In [16], the optimization of the secrecy rate of a UAV-enabled mobile relay system was formulated, and a performance gain over static relaying was achieved.",
"By jointly designing the trajectory and transmit power of the UAV, an optimization algorithm designed to achieve the average worst-case secrecy rate improvement in UAV-to-ground communications is proposed in [17].",
"Additionally, [4] investigated the secrecy performance of UAV networks working in the millimeter wave band, where the UAVs can be used either for information transmission or jamming, and it was revealed that the average achievable secrecy rate does not change monotonically with an increasing proportion of UAV jammers.",
"Although it is known that increasing the jamming power will pose a stronger interference to both the UEDs and the legitimate ground receiver, how the secrecy performance behaves with respect to this increase is still unknown.",
"Furthermore, the coverage and secrecy analysis in the previous literature has suffered from the tractability problem when the LoS probability is considered, since the analysis always results in multi-integral expressions."
],
[
"Contributions",
"This paper focuses on the secrecy performance of ground-based communications in the presence of a UAV jammer and UEDs.",
"The main contributions of the paper are summarized as follows: LoS model: This is the first work to propose an approximation to simplify the modeling of the probability of LoS channels, which allows us to get tractable formulations in the analysis of wireless networks concerning ATG channels.",
"The trend of the LoS probability with respect to the height of UAVs has been captured in a simplistic manner.",
"UAV jammer: We introduce a UAV jammer for improving the security of the ground communication, and we give the cumulative distribution function (CDF) of the signal-to-interference-plus-noise ratio (SIR) from a ground transmitter to a UED subject to interference by the UAV jammer.",
"SCP: We formulate the SCP of the ground link in the presence of randomly deployed UEDs for the non-colluding scenario.",
"The trends of the SCP in some environments with respect to different transmit signal to jamming power ratios, and the locations of the UAV jammer have been analysed.",
"The rest of the paper is structured as follows.",
"Section II begins with a description of the system model then addresses the LoS probability and its approximations.",
"Section III focuses on the derivation of the SCP.",
"Section IV focuses on the behaviour of the SCP for different parameters.",
"Section V gives the simulation results and discussion, and Section VI concludes the paper."
],
[
"Network layout",
"As shown in Fig.",
"REF , transmitter ${s}$ at $(0,0,0)\\in \\mathbb {R}^3$ connects to receiver ${d}$ at $(x_d,y_d,0)$ via the legitimate link, where $x_d$ and $y_d$ denotes the locations of $d$ on the x-axis and y-axis, respectively.",
"There are $n$ UEDs uniformly distributed in a disk, forming a uniform binomial point process (BPP) [13].",
"The disk is denoted as $O(t,R_{1})$ , where $O$ denotes the two-dimensional disk of radius $R_{1}$ centred at $t$ .",
"The coordinates of the origin of the disk are $(0,0, H)$ so that the center of the circular plane $t$ is $H$ meters right above the transmitter $s$ on the ground, and the disk is also parallel to the ground We assume that the ground between $s$ and $d$ is flat, and the effect due to the spherical surface of the earth is negligible..",
"The UEDs work independently to decode the received signal on their links.",
"Meanwhile, a UAV jammer $j$ is also located within the disk $O(t,R_{1})$ and continually sends a jamming signal.",
"The ground channel is assumed to experience Rayleigh fading and path loss, while the channel model for the ATG communications is based on the probabilistic LoS and non-line-of-sight (NLoS) links given by [15], [18].",
"The air-to-air communication channel is assumed to follow Friis free space transmission.",
"The transmit powers of $s$ and $j$ are given by $P_s$ and $P_j$ , respectively.",
"Figure: The system model for communication in the presence of UEDs.",
"The bold solid arrow denotes the legitimate link.",
"For UED e 1 e_1, the dashed arrow denotes the wire-tap link, and the dotted arrows denote the jamming links.Since no major obstacles are obstructing the communications in the sky, the air-to-air channel follows Friis free space transmission.",
"And the received jamming power from $j$ to $e$ is given by $P_{j,e} = P_{j} (\\lambda /4\\pi )^\\alpha l_{j,e}^{-2},$ where $\\alpha =2$ is the free space path loss (FSPL) exponent, and $l_{{x},{y}}$ is the distance between nodes $x$ and $y$ ."
],
[
"ATG channel models",
"The potential existence of buildings and other obstructions lying in the propagation path results in the presence of mixed LoS/NLoS channel conditions between an air terminal and a ground terminal.",
"According to [18], the ATG channel model will be mainly based on probabilistic LoS and NLoS links instead of following a classical fading channel.",
"Therefore, the corresponding received signal powers at $e$ from transmitter $s$ for LoS and NLoS links are written as $P_{s,e}={\\left\\lbrace \\begin{array}{ll}\\eta _{\\text{L}} P_s (\\lambda /4\\pi )^2 l_{s,e}^{-\\alpha }, & \\text{LoS link}\\\\\\eta _{\\text{N}} P_s (\\lambda /4\\pi )^2 l_{s,e}^{-\\alpha }, & \\text{NLoS link},\\end{array}\\right.",
"}$ and $\\eta _{\\text{L}}$ and $\\eta _{\\text{N}}$ refer to the mean values of excess path loss added to the FSPL, where $(\\eta _{\\text{L}},\\eta _{\\text{N}})$ can be measured at $f_c = 2$ GHz in dB to be $(-0.1,-21),(-1.0,-20),(-1.6,-23)$ , and $(-2.3,-34)$ for suburban, urban, dense urban, and highrise urban areas, respectively [15]."
],
[
"LoS probability and sigmoid fitting",
"The probabilities for a link $(s,e)$ to be either LoS or NLoS can be denoted as $\\mathbb {P}_{\\text{L}}^{s,e}$ and $\\mathbb {P}_{\\text{N}}^{s,e} =1 - \\mathbb {P}_{\\text{L}}^{s,e}$ , respectively.",
"According to [14], [15], $\\begin{split}\\mathbb {P}_{\\text{L}}^{s,e}=& \\prod _{l=0}^{f(r)} \\left[1- \\exp \\left(-\\frac{\\left(H-(l+1/2)H/(f(r)+1) \\right)^2}{2\\sigma ^2} \\right) \\right]\\\\\\approx &\\frac{1}{1+ C \\exp \\left(-B\\left(\\theta _{s,e} - C\\right)\\right)}\\end{split}$ where $f(r) = \\lfloor r \\sqrt{\\rho _1 \\rho _2 }-1\\rfloor $ , $r= l_{t,e}$ is the ground distance between $s$ and $e$ , $\\rho _1$ is the ratio of built-up land area to the total land area, $\\rho _2$ is the mean number of buildings per unit area (buildings/km$^2$ ), and $\\sigma $ is the scale parameter of the Rayleigh probability density function (PDF), which gives the height distribution of buildings in meters.",
"The environment parameters ($\\rho _1, \\rho _2, \\sigma $ ) for typical environments are suburban ($0.1, 750, 8$ ), urban ($0.3, 500, 15$ ), dense urban ($0.5, 300, 20$ ), and highrise urban ($0.5, 300, 50$ ), respectively.",
"Note that increasing $H$ will increase the smoothness of the plot of the product, and $\\mathbb {P}_{\\text{L}}^{s,e} $ can be considered as a continuous function of elevation angle and the environment parameters for a large $H$ [15].",
"The approximation to the LoS probability of an ATG link is given by a sigmoid function, where $C$ and $B$ are constant values depending on the aforementioned environment and $\\theta _{s,e}$ is the elevation angle in radians, which is given by $\\theta _{s,e} = {\\tan ^{-1}} \\left(\\frac{H}{l_{s,e}}\\right) $ .",
"The ATG channel model also applies to the received jamming power $P_{j,d}$ with LoS and NLoS probability given by $\\mathbb {P}_{\\text{L}}^{j,d}$ and $\\mathbb {P}_{\\text{N}}^{j,d}$ .",
"Figure: Plots of the LoS probabilities for different urban environments based on the exact product, the sigmoid fitting and the proposed piecewise fitting.",
"The goodness of fit measure by root-mean-square error is given inside the brackets."
],
[
"Proposed piecewise fitting",
"Although the sigmoid fitting provides a reasonably good approximation to the actual LoS probability, it provides a non-linear relationship to $l_{s,e}$ and has been shown to only give analytical forms which are non-trivial to evaluate.",
"Based on the value of the elevation angle, each curve of the LoS probability can be divided into three approximated regions: LoS region for the link in a pure LoS state with high elevation angle, NLoS region for the the link in a pure NLoS state with low elevation angle, and a transitional region where the link is in mixed LoS/NLoS states.",
"The transitional region can be further divided into two regions based on the slope of the curve as the elevation angle increases.",
"As a result, we propose a piecewise fitting model in (REF ), which not only gives a meaningful relationship between the LoS probability and $l_{s,e}$ but also provides a tractable way to calculate the SCP as we will show later.",
"The piecewise approximation is given by $ \\mathbb {P}_{\\text{L}}^{s,e}={\\left\\lbrace \\begin{array}{ll}1, &\\text{if } \\theta _{s,e} > \\tan ^{-1} \\frac{H}{{\\ell _1}} \\\\c_3\\cot (\\theta _{s,e}) +c_4, &\\text{if } \\tan ^{-1} \\frac{H}{{\\ell _2}}< \\theta _{s,e} \\le \\tan ^{-1} \\frac{H}{{\\ell _1}} \\\\0, &\\text{if } \\theta _{s,e} \\le \\tan ^{-1} \\frac{H}{{\\ell _3}} \\\\c_1\\tan (\\theta _{s,e}) +c_2, &\\text{otherwise}\\\\\\end{array}\\right.",
"}$ where ${\\ell _1} = H\\frac{1-c_4}{c_3}$ , ${\\ell _2} = \\frac{2Hc_1}{c_4-c_2-\\sqrt{(c_4-c_2)^2+4c_1c_3} } $ , ${\\ell _3} = -H\\frac{c_1}{c_2}$ , $c_1$ , $c_2$ , $c_3$ and $c_4$ are given in Table REF and obtained by Algorithm REF , and the corresponding results for different typical environments are plotted in Fig.",
"REF .",
"[t] KwDataGiven Environment parameters ($\\rho _1, \\rho _2, \\sigma $ ), thresholds: $c_l = 0.005$ , $c_m = 0.5$ , and $c_u = 0.995$ .",
"Generate $10^4$ evenly spaced samples from $[0, \\pi /2]$ , and obtain the corresponding samples of LoS probabilities with (REF ).",
"Create a new dataset $D_{t_1}$ from the samples of LoS probabilities whose values are bigger than $c_l$ and smaller than $c_m$ .",
"Find the coefficients $c_1$ and $c_2$ of the model function $c_1\\tan (\\theta _{s,e}) +c_2$ by fitting the function to $D_{t_1}$ with the trust region reflective least squares algorithm.",
"Let $c_1\\tan (\\theta _{s,e}) +c_2=0$ , the maximum elevation angle for having only the NLoS link is $\\tan ^{-1} \\left(-\\frac{c_2}{c_1} \\right)$ .",
"Create a new dataset $D_{t_2}$ from the samples of LoS probabilities whose values are bigger than $c_m$ and smaller than $c_u$ .",
"Find the coefficients $c_3$ and $c_4$ of the model function $c_3\\cot (\\theta _{s,e}) +c_4$ by fitting the function to $D_{t_2}$ with the trust region reflective least squares algorithm.",
"Let $c_3\\cot (\\theta _{s,e}) +c_4=1$ , the minimum elevation angle for having only the LoS link is $\\tan ^{-1} \\left(\\frac{c_3}{1-c_4} \\right)$ .",
"The elevation angle at the inflection point, where the LoS probability changes from being $c_1\\tan (\\theta _{s,e}) +c_2$ to $c_3\\cot (\\theta _{s,e}) +c_4$ is $\\tan ^{-1} \\left(\\frac{c_4-c_2-\\sqrt{(c_4-c_2)^2+4c_1c_3} }{2c_1} \\right)$ , is obtained by solving $c_1\\tan (\\theta _{s,e}) +c_2=c_3\\cot (\\theta _{s,e}) +c_4$ , and then selecting the root which provides a better fit to the samples of LoS probabilities.",
"Finding parameters $c_1, c_2, c_3 \\text{ and } c_4$ for link $(s,e)$ given environment parameters ($\\rho _1, \\rho _2, \\sigma $ ).",
"Table: The parameters for () and ().Since $\\tan (\\theta _{s,e}) = \\frac{H}{l_{t,e}}$ , (REF ) can be further written as $ \\mathbb {P}_{\\text{L}}^{s,e}={\\left\\lbrace \\begin{array}{ll}1, &\\text{if } l_{t,e} < {\\ell _1}\\\\c_3l_{t,e}/H +c_4, &\\text{if } {\\ell _1}\\le l_{t,e} < {\\ell _2}\\\\c_1 H/l_{t,e} +c_2, &\\text{if } {\\ell _2}\\le l_{t,e} < {\\ell _3}\\\\0, &\\text{otherwise}.\\end{array}\\right.",
"}$ In this figure, the proposed piecewise fitting outperforms the sigmoid fitting in the goodness of fit measured by root-mean-square error.",
"The former is at least four times better than the latter in the suburban environment and nine times better in the highrise urban environment.",
"The transition region is further approximated by a linear function of the variable $l_{t,e}$ and another linear function of the reciprocal of $l_{t,e}$ .",
"In the transition region, $\\mathbb {P}_{\\text{L}}^{s,e}$ increases as $l_{t,e}$ decreases."
],
[
"Ground-to-ground channel model",
"For ground communications, the log-distance path loss model is used to characterize the legitimate link from $s$ to ${d}$ .",
"The received power at $d$ from $s$ is written as $P_{{s,d}}=P_{s}(\\lambda /4\\pi )^2 l_{{s},{d}}^{-\\beta }|h_{{s,d}}|^2$ where $\\lambda $ is the carrier wavelength, $|h_{{s,d}}|^2$ is the channel gain associated with the Rayleigh fading, and $\\beta $ is the path loss exponent for ground communications."
],
[
"Secrecy capacity",
"Let $\\Phi =\\lbrace 1,2,...,n\\rbrace $ denote the collection of $n$ UEDs.",
"The secrecy capacity of a link is the difference between the capacity of the main link and the capacity achieved via a collection of wire-tap links.",
"The general form is given as [19] $\\begin{split}C_s& = \\left[\\log _2\\left({\\frac{1+\\frac{P_{s,d}}{P_{j,d}+N_0}}{ 1+\\max _{{e} \\in \\Phi } \\left(\\frac{P_{s,e}}{P_{j,e}+N_e} \\right)}}\\right) \\right]^+\\\\&\\ge \\left[\\log _2\\left({\\frac{\\Gamma _1}{ \\max _{{e} \\in \\Phi }\\Gamma _2}}\\right) \\right]^+\\\\\\end{split}$ where $N_0$ is the additive white Gaussian noise (AWGN) power at the ground receiver $d$ , $N_e$ is the AWGN power at a UED, $[x]^+$ denotes $\\max (0,x)$ and the approximation holds over interference-limited channels [20].",
"'$\\ge $ ' holds if the receivers on the UEDs have the low-noise figure by using the thermoelectric cooling and new materials [21], [22], [23].",
"$\\Gamma _1 = 1+\\frac{P_{s,d}}{P_{j,d}+N_0}$ represents one plus the signal-to-interference-plus-noise ratio ($1+\\text{SINR}$ ) from $s$ to $d$ , and $\\Gamma _2 = { 1+\\frac{P_{s,e}}{P_{j,e}} }$ represents $1 + \\text{SIR}$ from $s$ to $e$ , where the interference is due to UAV jammer $j$ ."
],
[
"SCP formulation",
"The SCP is defined as $\\begin{split}p_{c}&=\\mathbb {P} {\\left(C_s > \\mathcal {R}_t \\right)}\\ge \\mathbb {P} {\\left({\\frac{\\Gamma _1}{ \\max _{{e} \\in \\Phi }\\Gamma _2}} > 2^{\\mathcal {R}_t} \\right)}\\\\\\end{split}$ where $\\mathcal {R}_t \\ge 0$ is the target secrecy rate.",
"Besides, $p_{c}=\\mathbb {P} {\\left(C_s > \\mathcal {R}_t \\right)} \\approx \\mathbb {P} {\\left({\\frac{\\Gamma _1}{ \\max _{{e} \\in \\Phi }\\Gamma _2}} > 2^{\\mathcal {R}_t} \\right)}$ , when both $d$ and UEDs are working in an interference-limited environment as the result of a large jamming signal.",
"To evaluate the above inequality, we require the following calculations.",
"Since $\\Gamma _1$ is a random variable related to $|h|^2$ , the CDF of $\\Gamma _1$ can be written as $\\begin{split}F&_{\\Gamma _1}({\\gamma _1}) = \\mathbb {P} \\left( 1+\\frac{P_{s,d}}{P_{j,d}+N_0} <{\\gamma _1} \\right)\\\\=&\\mathbb {P} \\left( \\frac{|h|^2}{\\Lambda (\\eta _{\\text{L}}) } < {\\gamma _1}-1\\right)\\mathbb {P}_{\\text{L}}^{j,d} + \\mathbb {P} \\left( \\frac{|h|^2}{\\Lambda (\\eta _{\\text{N}})} < {\\gamma _1}-1 \\right)\\mathbb {P}_{\\text{N}}^{j,d}\\\\= &1- \\exp \\left( (1-{\\gamma _1})\\Lambda (\\eta _{\\text{L}})\\right)\\mathbb {P}_{\\text{L}}^{j,d} - \\exp \\left((1-{\\gamma _1}) \\Lambda (\\eta _{\\text{N}})\\right)\\mathbb {P}_{\\text{N}}^{j,d}\\\\\\end{split}$ where $\\Lambda (\\eta ) = \\frac{l_{s,d}^ \\beta }{P_s} \\left(\\eta l_{j,d}^{-\\alpha } {P_j} +N_0\\right)$ , $l_{j,d}^2 = l_{t,j}^2 + l_{s,d}^2 - 2l_{t,j} l_{s,d} \\cos \\left( \\varphi _{j^{\\prime }} - \\varphi _{d}\\right) + H^2$ , $j^{\\prime }$ denotes the projection of $j$ on the xy-plane.",
"$\\varphi _{j^{\\prime }}$ and $\\varphi _{d}$ denote the angles of $l_{s,j^{\\prime }}$ and $l_{s,d}$ measured counterclockwise from the x-axis.",
"Thus, the corresponding PDF is given by $f_{\\Gamma _1}({\\gamma _1}) = {\\frac{{\\rm {d}}}{{\\rm {d}}\\,{\\gamma _1}}} F_{\\Gamma _1}({\\gamma _1})$ such that $\\begin{split}f_{\\Gamma _1}({\\gamma _1})= &\\Lambda (\\eta _{\\text{L}})\\exp \\left((1-{\\gamma _1}) \\Lambda (\\eta _{\\text{L}}) \\right)\\mathbb {P}_{\\text{L}}^{j,d}+ \\Lambda (\\eta _{\\text{N}})\\exp \\left((1-{\\gamma _1}) \\Lambda (\\eta _{\\text{N}}) \\right)\\mathbb {P}_{\\text{N}}^{j,d}.\\\\\\end{split}$ Let $e^{\\prime }$ denote the projection of the node $e$ on the xy-plane, and $\\phi = \\varphi _{j^{\\prime }} - \\varphi _{e^{\\prime }}$ .",
"$\\Gamma _2$ is a random variable related to the position of one UED $e$ , which is uniformly distributed on the disk $O(t,R_{1})$ .",
"Thus, the CDF of $\\Gamma _2$ is given by $ \\begin{split}F_{\\Gamma _2}({\\gamma _2}) &= \\mathbb {E}_e\\left[F_1({\\gamma _2}, l_{t,e},\\phi ) \\right]\\\\&= \\frac{1}{\\pi R_{1}^2} \\int _0^{2\\pi } \\int _0^{R_{1}} F_1({\\gamma _2}, l_{t,e},\\phi ) l_{t,e} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\\\end{split}$ where $F_1({\\gamma _2},l_{t,e},\\phi ) ={g}({\\gamma _2}, \\eta _{\\text{L}},l_{t,e},\\phi )\\mathbb {P}_{\\text{L}}^{s,e} + {g}({\\gamma _2}, \\eta _{\\text{N}},l_{t,e},\\phi )\\mathbb {P}_{\\text{N}}^{s,e}$ and ${g} ({\\gamma },\\eta ,\\ell ,\\phi ) = \\mathbb {1} \\left(1+\\frac{(H^2+\\ell ^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+\\ell ^2-2 l_{t,j} \\ell \\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} \\le {\\gamma }\\right).$"
],
[
"Calculation of the SCP",
"When $\\mathbb {P}_{\\text{L}}^{s,e}$ is given by (REF ), it is hard to find a closed-form expression for (REF ).",
"To make progress, we use (REF ) instead, which can yield a closed-form expression for (REF ) with any given $R_{1}$ and $H$ .",
"With the help of (REF ), we can divide the LoS probability into multiple regions, and calculate their contributions to $F_{\\Gamma _2}({\\gamma _2})$ independently.",
"This decomposition technique to obtain (REF ) seems to be tedious, but necessary, because it leads to a tractable solution.",
"Throughout the following discussion, we refer to a number of propositions, which will be presented in Section REF .",
"For $R_{1} < {\\ell _1}$ , where $l_1$ is considered in (REF ), the ATG link between transmitter $s$ and UED $e$ is always in a pure LoS state since $l_{t,e}\\le R_{1}$ .",
"Following Proposition REF , we have $\\begin{split}F_{\\Gamma _2}({\\gamma _2})&=\\int _0^{2\\pi } \\int _0^{R_{1}} {g}({\\gamma _2}, \\eta _{\\text{L}},l_{t,e},\\phi )\\frac{l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi = F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{L}},R_{1}).\\end{split}$ For ${\\ell _1} \\le R_{1} < {\\ell _2}$ , the ATG link between transmitter $s$ and UED $e$ is in a pure LoS state if $l_{t,e}< {\\ell _1}$ , and the ATG link between transmitter $s$ and UED $e$ is in the mixed LoS/NLoS states with $\\mathbb {P}_{\\text{L}}^{s,e}=c_3 l_{t,e}/H +c_4$ if ${\\ell _1}\\le l_{t,e} \\le R_{1}$ .",
"Following Propositions REF and REF , we have $\\begin{split}F_{\\Gamma _2}({\\gamma _2})&=\\int _0^{2\\pi } \\int _0^{{{\\ell _1} }} {g}({\\gamma _2}, \\eta _{\\text{L}},l_{t,e},\\phi )\\frac{l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\, + \\int _0^{2\\pi } \\int _{{\\ell _1} }^{R_{1}} F_1({\\gamma _2}, l_{t,e},\\phi ) \\frac{l_{t,e}}{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\&= F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{L}},{{\\ell _1} }) +F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}},R_{1}) -F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, {\\ell _1}).\\end{split}$ For ${\\ell _2}\\le R_{1} < {\\ell _3}$ , the ATG link between transmitter $s$ and UED $e$ is in a pure LoS state if $l_{t,e}<{\\ell _1}$ , the ATG link between transmitter $s$ and UED $e$ is in the mixed LoS/NLoS states with $\\mathbb {P}_{\\text{L}}^{s,e}=c_3 l_{t,e}/H +c_4$ if ${\\ell _1}\\le l_{t,e} < {\\ell _2}$ , and the ATG link between transmitter $s$ and UED $e$ is in the mixed LoS/NLoS states with $\\mathbb {P}_{\\text{L}}^{s,e}=c_1 H/l_{t,e} +c_2$ if ${\\ell _2}\\le l_{t,e} \\le R_{1}$ .",
"Following Propositions REF , REF and REF , we have $\\begin{split}F&_{\\Gamma _2}({\\gamma _2})=\\int _0^{2\\pi } \\int _0^{{{\\ell _1} }} {g}({\\gamma _2}, \\eta _{\\text{L}},l_{t,e},\\phi )\\frac{l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\, + \\int _0^{2\\pi } \\int _{{\\ell _1} }^{{R_1}} F_1({\\gamma _2}, l_{t,e},\\phi ) \\frac{l_{t,e}}{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\,\\\\=& F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{L}},{{\\ell _1} }) +F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}},{\\ell _2}) - F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, {\\ell _1})+ F_{\\Gamma _4}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, R_{1}) -F_{\\Gamma _4}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, {\\ell _2}).\\end{split}$ For $R_{1} \\ge {\\ell _3}$ , the ATG link between transmitter $s$ and UED $e$ is in a pure LoS state if $l_{t,e}< {\\ell _1}$ , the ATG link between transmitter $s$ and UED $e$ is in the mixed LoS/NLoS states with $\\mathbb {P}_{\\text{L}}^{s,e}=c_3 l_{t,e}/H +c_4$ if ${\\ell _1}\\le l_{t,e} < {\\ell _2}$ , the ATG link between transmitter $s$ and UED $e$ is in the mixed LoS/NLoS states with $\\mathbb {P}_{\\text{L}}^{s,e}=c_1 H/l_{t,e} +c_2$ if ${\\ell _2}\\le l_{t,e} < {\\ell _3}$ , and the ATG link between transmitter $s$ and UED $e$ is in a pure NLoS state if $ {\\ell _3} \\le l_{t,e} \\le R_{1}$ .",
"Following Propositions REF , REF and REF , we have $\\begin{split}F_{\\Gamma _2}({\\gamma _2})=&\\int _0^{2\\pi } \\int _0^{{{\\ell _1} }} {g}({\\gamma _2}, \\eta _{\\text{L}},l_{t,e},\\phi )\\frac{l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\, + \\int _0^{2\\pi } \\int _{{\\ell _1} }^{{\\ell _3}} F_1({\\gamma _2}, l_{t,e},\\phi ) \\frac{l_{t,e}}{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\,+ \\\\& \\int _0^{2\\pi } \\int _{{\\ell _3} }^{R_{1}} {g}({\\gamma _2}, \\eta _{\\text{N}},l_{t,e},\\phi )\\frac{l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\= & F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{L}},{{\\ell _1} }) +F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}},{\\ell _2}) -F_{\\Gamma _5}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, {\\ell _1})+\\\\& F_{\\Gamma _4}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}},{\\ell _3}) - F_{\\Gamma _4}({\\gamma _2}, \\eta _{\\text{L}}, \\eta _{\\text{N}}, {\\ell _2})+ F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{N}}, R_{1}) - F_{\\Gamma _3}({\\gamma _2}, \\eta _{\\text{N}},{\\ell _3} ).\\end{split}$ As a result of the above decomposition, $p_c$ is lower bounded by $\\begin{split}\\mathbb {P} \\left(\\frac{\\Gamma _1}{\\max _{{e} \\in \\Phi }{\\Gamma _2}}>2^{\\mathcal {R}_t} \\right) = \\int _1^\\infty F_{\\Gamma _2}\\left(\\frac{\\gamma _1}{2^{\\mathcal {R}_t}}\\right)^{n} f_{\\Gamma _1} (\\gamma _1) \\, {\\rm {d}} \\gamma _1.\\\\\\end{split}$"
],
[
"Propositions",
"All three propositions given below rely on Lemma REF , which is given in the Appendix.",
"Table: Table of conditions.Proposition 1 Let ${r} \\le R_{1}$ , assuming that the link $(s,e)$ is always in the same channel state (i.e, pure LoS or pure NLoS) with gain $\\eta $ , the CDF of $\\Gamma _2$ conditioning on that $e$ is only active for $l_{t,e}\\le {r}$ is given by $ \\begin{split}F_{\\Gamma _3}(y, \\eta ,{r}) &= \\int _0^{2\\pi } \\int _0^{{r}} \\frac{{g}({\\gamma _2}, \\eta ,l_{t,e},\\phi )l_{t,e} }{\\pi R_{1}^2} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi = \\frac{S_1(y,\\eta , {r})}{\\pi R_{1}^2/2}\\end{split}$ where $S_1(y,\\eta , {r}) ={\\left\\lbrace \\begin{array}{ll}D_1 + D_2 -\\frac{1}{2}D_3, & \\text{ \\rm {cases 1, 9, 10} } \\\\D_2 +D_4, & \\text{ \\rm {cases 2, 7} } \\\\\\frac{1}{2}\\pi {r}^2 - D_3, & \\text{ \\rm {cases 3, 11} } \\\\D_1 +D_2 +\\frac{1}{2}D_3, & \\text{ \\rm {cases 4, 6, 14} } \\\\\\frac{1}{2}\\pi {r}^2, &\\text{ \\rm {cases 5, 12, 13, 19} }\\\\D_3, & \\text{ \\rm {cases 8, 15} } \\\\D_5, & \\text{ \\rm {cases 16, 17, 18} } \\\\0, & \\text{ \\rm {otherwise}}\\end{array}\\right.",
"}$ and all the cases with index numbers are given in Table REF , $D_1= \\frac{{A} B-l_{t,j}^2}{2 {A}^2} \\cot ^{-1}\\left(\\frac{{a_1}-2 B}{\\sqrt{4 l_{t,j}^2 {r}^2-{a_1}^2}}\\right)$ , ${A} = 1-\\left((y-1)\\frac{P_j}{\\eta P_s}\\right)^\\frac{2}{\\alpha }$ , $B= l_{t,j}^2- (1-{A}) H^2$ , ${a_1} ={A} {r}^2+B$ , $D_2= \\frac{\\pi {r}^2}{4} - \\frac{{A}}{4 {A}^2} \\sqrt{4 l_{t,j}^2 {r}^2-{a_1}^2} - \\frac{{a_1}{A}-l_{t,j}^2}{2 {A}^2} \\csc ^{-1}\\left(\\frac{2 l_{t,j} {r}}{{a_1}}\\right)$ , $D_3= \\frac{ l_{t,j}^2-{A} B}{2 {A}^2} \\pi $ , $D_4=\\frac{({a_1}-2 B) \\sqrt{4 j^2 {r}^2-{a_1}^2}}{4 {A}^2 {r}^2}$ , $D_5=\\frac{1}{2} {r}^2 \\sec ^{-1}\\left(\\frac{2 l_{t,j} {r}}{{a_1}}\\right)-\\frac{\\left(l_{t,j}^2-H^2\\right) \\sqrt{4 l_{t,j}^2 {r}^2 - {\\left(H^2-l_{t,j}^2\\right)^2} }}{8 l_{t,j}^2 }$ .",
"$\\begin{split}&\\frac{1}{\\pi R_{1}^2}\\int _0^{2\\pi } \\int _0^{{r}} {g}(y, \\eta _{\\text{L}},l_{t,e},\\phi ){l_{t,e} } \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\& =\\frac{1}{\\pi R_{1}^2} \\int _{o_\\phi (\\eta )} \\int _{o_{l_{t,e}}(\\eta )} {l_{t,e} } \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\& =\\frac{{r}^2}{R_{1}^2}- \\frac{1}{\\pi R_{1}^2} \\int _{{o^{\\prime }}_\\phi (\\eta )} \\int _{{o^{\\prime }}_{l_{t,e}}(\\eta )} {l_{t,e} } \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\end{split}$ where ${o_\\phi }(\\eta )$ and $o_{l_{t,e}}(\\eta )$ are the domains so that $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} \\le y$ , while ${{o^{\\prime }}_\\phi }(\\eta )$ and ${o^{\\prime }}_{l_{t,e}}(\\eta )$ are the domains so that $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} > y$ .",
"Given Lemma REF , (REF ) is achieved.",
"Thus, we conclude the proof.",
"Proposition 2 Let $r\\le R_{1}$ , assuming that the mixed LoS/NLoS states exist for link $(s,e)$ , and the (quasi-) LoS probability is $\\mathbb {P}_{\\text{L}}^{s,e}=c_1 H/l_{t,e} +c_2$ , the CDF of $\\Gamma _2$ conditioning on the UED $e$ being active within the disk $O(t,r)$ is given by $ \\begin{split}F_{\\Gamma _4}(y, \\eta _{\\text{L}}, \\eta _{\\text{N}},r) =& \\frac{2}{\\pi R_{1}^2}\\big ( c_2 S_1(y,\\eta _{\\text{L}}, r)+ c_1H S_2(y,\\eta _{\\text{L}}, r)+ (1-c_2) S_1(y,\\eta _{\\text{N}}, r)-c_1 H S_2(y,\\eta _{\\text{N}}, r)\\big )\\end{split}$ $S_2(y,\\eta , r) ={\\left\\lbrace \\begin{array}{ll}G_1 - G_2 +G_3 -G_4 , & \\text{ \\rm {cases 1, 2} } \\\\\\pi r + G_1- 2G_2, & \\text{ \\rm {cases 3} } \\\\G_3+ G_4 -G_2, & \\text{ \\rm {cases 4, 14} } \\\\\\pi r, &\\text{ \\rm {cases 5, 12, 13, 19} } \\\\G_3 +G_4 -G_5 , & \\text{ \\rm {cases 6, 7} } \\\\-G_5, & \\text{ \\rm {case 8} } \\\\G_3- G_4, & \\text{ \\rm {cases 9, 10} } \\\\\\pi r- G_2, & \\text{ \\rm {case 11} } \\\\-G_2, & \\text{ \\rm {case 15} } \\\\G_6, & \\text{ \\rm {cases 16, 17} } \\\\\\frac{1}{2}\\pi r, & \\text{ \\rm {case 18} }\\\\0, & \\text{ \\rm {otherwise}}\\end{array}\\right.",
"}$ and all the cases with index are given in Table REF , $G_0=\\frac{2 \\sqrt{-{A} B+l_{t,j}^2} }{-{A}},$ $G_1=G_0 \\mathsf {E}\\left(\\sec ^{-1}\\left(\\frac{-l_{t,j}}{\\sqrt{B{A}}}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right),$ $G_2=G_0 \\mathsf {E}\\left(\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right),$ $G_3=\\frac{\\sqrt{4 l_{t,j}^2 r^2-{a_1}^2}-2 {A} {r}^2 \\sec ^{-1}\\left(\\frac{2 l_{t,j} r}{{a_1}}\\right)}{-2{A}r},$ $G_4={G_0} \\mathsf {E}\\left(\\sec ^{-1}\\left(\\frac{2 l_{t,j} r}{{a_1}}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right)/{2},$ $G_5 =G_0 \\mathsf {E}\\left(\\sec ^{-1}\\left(\\frac{l_{t,j}}{\\sqrt{B{A}}}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right),$ $G_6= r \\cos ^{-1}\\left(\\frac{l_{t,j}^2-H^2}{2 l_{t,j} r}\\right) -\\frac{l_{t,j}^2-H^2}{2 l_{t,j}} { \\ln \\frac{ \\sqrt{4 l_{t,j}^2 r^2- {\\left(H^2-l_{t,j}^2\\right)^2}}+2l_{t,j}r }{\\left|l_{t,j}^2-H^2\\right|} }$ , $\\mathsf {E}(\\cdot )$ and $\\mathsf {E}(\\cdot \\mid \\cdot )$ denote complete and incomplete elliptic integral of the 2nd kind [24].",
"$\\begin{split}\\int _0^{2\\pi } &\\int _0^{{r}} ({g}(y, \\eta _{\\text{L}},l_{t,e},\\phi )\\mathbb {P}_{\\text{L}}^{s,e} + {g}(y, \\eta _{\\text{N}},l_{t,e},\\phi )\\mathbb {P}_{\\text{N}}^{s,e} ){l_{t,e} } \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\=&\\int _{o_{\\phi }(\\eta _\\text{L})} \\int _{o_{l_{t,e}}(\\eta _\\text{L})} \\left(c_1 H+ c_2{l_{t,e} } \\right)\\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + \\int _{o_{\\phi }(\\eta _\\text{N})} \\int _{o_{l_{t,e}}(\\eta _\\text{N})} \\left(-c_1 H+ (1-c_2){l_{t,e} } \\right) \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\= &{c_1 H }\\int _{o_{\\phi }(\\eta _\\text{L})} \\int _{o_{l_{t,e}}(\\eta _\\text{L})} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + 2 {c_2 S_1(y,\\eta _{\\text{L}}, {r})}- {c_1 H }\\int _{o_{\\phi }(\\eta _\\text{N})} \\int _{o_{l_{t,e}}(\\eta _\\text{N})} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + 2{(1-c_2)S_1(y,\\eta _{\\text{N}}, {r})} \\\\\\end{split}$ where $\\int _{o_{\\phi }(\\eta )} \\int _{o_{l_{t,e}}(\\eta )} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi = {2 \\pi {r}} - \\int _{{o^{\\prime }}_{\\phi }(\\eta )} \\int _{{o^{\\prime }}_{l_{t,e}}(\\eta )} \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi .$ Given the domains in Lemma REF and dividing (REF ) by $\\pi R_{1}^2$ , (REF ) is achieved.",
"Thus, we conclude the proof.",
"Proposition 3 Let ${r}\\le R_{1}$ , assuming that the mixed LoS/NLoS states exist for link $(s,e)$ , and the (quasi-) LoS probability is $\\mathbb {P}_{\\text{L}}^{s,e}=c_3 l_{t,e}/H +c_4$ , the CDF of $\\Gamma _2$ conditioning on the UED $e$ being active within the disk $O(t,r)$ is given by $ \\begin{split}F_{\\Gamma _5}(y, \\eta _{\\text{L}}, \\eta _{\\text{N}},{r}) =& \\frac{2}{\\pi R_{1}^2}\\bigg ( c_4 S_1(y,\\eta _{\\text{L}}, {r})+ \\frac{c_3}{3H} S_3(y,\\eta _{\\text{L}}, {r})+ (1-c_4) S_1(y,\\eta _{\\text{N}}, {r}) -\\frac{c_3}{3H} S_3(y,\\eta _{\\text{N}}, {r})\\bigg )\\end{split}$ $S_3(y,\\eta , {r}) ={\\left\\lbrace \\begin{array}{ll}M_3+M_4-M_1-M_2, & \\text{ \\rm {cases 1, 2} }\\\\\\pi {r}^3 -2 M_2 + M_3, & \\text{ \\rm {case 3} } \\\\M_1 - M_2 + M_4, & \\text{ \\rm {cases 4, 14} } \\\\\\pi {r}^3, &\\text{ \\rm {cases 5, 12, 13, 19} } \\\\M_1 - M_3 + M_4, & \\text{ \\rm {cases 6, 7} } \\\\-M_3, & \\text{ \\rm {case 8} } \\\\-M_1 + M_4, & \\text{ \\rm {cases 9, 10} }\\\\-M_2 + \\pi {r}^3, & \\text{ \\rm {case 11} } \\\\-M_2, & \\text{ \\rm {case 15} } \\\\\\pi {r}^3 -M_5 -M_6, & \\text{ \\rm {case 16} } \\\\M_5 +M_6, & \\text{ \\rm {case 17} } \\\\\\pi {r}^3/2, & \\text{ \\rm {case 18} } \\\\0, & \\text{ \\rm {otherwise}}\\end{array}\\right.",
"}$ and all the cases with index are given in Table REF , $M_{0}= \\frac{\\sqrt{-{A} B+l_{t,j}^2}}{-3 {A}^3 /2},$ $\\frac{2M_1}{M_{0}}= 4 B {A} \\mathsf {F}\\left(\\sec ^{-1}\\left(\\frac{l_{t,j} {r}}{{a_1}/2}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right)+ \\left(8 l_{t,j}^2 -7 B {A} \\right) \\mathsf {E}\\left(\\sec ^{-1}\\left(\\frac{l_{t,j} {r}}{{a_1}/2}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right) ,$ $M_2= M_{0}4 B {A} \\mathsf {K}\\left(\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right)+ M_{0}\\left(8 l_{t,j}^2 -7 B {A} \\right) \\mathsf {E}\\left(\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right),$ $M_3 = M_{0}\\left(8 l_{t,j}^2 -7 B {A} \\right) \\mathsf {E}\\left(\\sec ^{-1}\\left(\\frac{-l_{t,j}}{\\sqrt{B{A}}}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right) + M_{0}4 B {A} \\mathsf {F}\\left(\\sec ^{-1}\\left(\\frac{-l_{t,j}}{\\sqrt{B{A}}}\\right)|\\frac{l_{t,j}^2}{l_{t,j}^2-{A} B}\\right),$ $M_4 = {r}^3 \\cos ^{-1}\\frac{{A} {r}^2+B}{2 l_{t,j} {r}} +\\frac{M_{0}}{4r} \\left({A} (2 {a_1}-9 B)+8 l_{t,j}^2 \\right) \\sqrt{\\frac{4 l_{t,j}^2 {r}^2-{a_1}^2}{l_{t,j}^2-{A} B}},$ $M_5 = {r}^3 \\cos ^{-1}\\frac{\\left| H^2-l_{t,j}^2\\right| }{-2 l_{t,j} {r}},$ $M_6 = \\frac{ {r} | H^2-l_{t,j}^2| }{8 l_{t,j}^2} \\sqrt{4 l_{t,j}^2 {r}^2-\\left(H^2-l_{t,j}^2\\right)^2}+ \\frac{| H^2-l_{t,j}^2| ^3}{16 l_{t,j}^3} \\ln \\frac{\\sqrt{4 l_{t,j}^2 {r}^2-\\left(H^2-l_{t,j}^2\\right)^2}+2 l_{t,j} {r}}{\\left| H^2-l_{t,j}^2 \\right|}$ , $\\mathsf {K}(\\cdot )$ and $\\mathsf {F}(\\cdot \\mid \\cdot )$ denote complete and incomplete elliptic integral of the 1st kind [24].",
"$\\begin{split}\\int _0^{2\\pi }& \\int _0^{{r}} ({g}(y, \\eta _{\\text{L}},l_{t,e},\\phi )\\mathbb {P}_{\\text{L}}^{s,e} + {g}(y, \\eta _{\\text{N}},l_{t,e},\\phi )\\mathbb {P}_{\\text{N}}^{s,e} ){l_{t,e} } \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\=&\\int _{o_{\\phi }(\\eta _\\text{L})} \\int _{o_{l_{t,e}}(\\eta _\\text{L})}\\left( c_3 {l_{t,e} }^2/H+ c_4{l_{t,e} }\\right) {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + \\int _{o_{\\phi }(\\eta _\\text{N})} \\int _{o_{l_{t,e}}(\\eta _\\text{N})}\\left( -c_3 {l_{t,e} }^2/H+ (1-c_4){l_{t,e} } \\right) {\\rm {d}} l_{t,e} {\\rm {d}} \\phi \\\\= &\\frac{c_3 }{ H }\\int _{o_{\\phi }(\\eta _\\text{L})} \\int _{o_{l_{t,e}}(\\eta _\\text{L})} l_{t,e}^2 \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + 2 {c_4 S_1(y,\\eta _{\\text{L}}, {r})} - \\frac{c_3 }{ H }\\int _{o_{\\phi }(\\eta _\\text{N})} \\int _{o_{l_{t,e}}(\\eta _\\text{N})} l_{t,e}^2 \\; {\\rm {d}} l_{t,e} {\\rm {d}} \\phi + 2{(1-c_4)S_1(y,\\eta _{\\text{N}}, {r})}\\end{split}$ where $ \\int _{o_{\\phi }(\\eta )} \\int _{o_{l_{t,e}}(\\eta )} l_{t,e}^2 {\\rm {d}} l_{t,e} {\\rm {d}} \\phi +\\int _{{o^{\\prime }}_{\\phi }(\\eta )} \\int _{{o^{\\prime }}_{l_{t,e}}(\\eta )} l_{t,e}^2 {\\rm {d}} l_{t,e} {\\rm {d}} \\phi = \\frac{2}{ 3} \\pi {r}^3 .$ Given Lemma REF and dividing (REF ) by $\\pi R_{1}^2$ , (REF ) is achieved.",
"Thus, we conclude the proof."
],
[
"Trend of the SCP with respect to $\\frac{P_s}{P_j}$",
"This subsection investigates the trend of the SCP with respect to transmitting-to-jamming power ratio $\\frac{P_s}{P_j}$ for $d$ and UEDs working in an interference-limited regime.",
"Let variable $x = \\frac{P_{s}}{P_{j}}$ , and random variables ${d_1}= \\frac{ l_{{s},{d}}^{-\\beta }|h_{{s,d}}|^2}{l_{j,d}^{-\\alpha }\\eta _{j,d}}$ , ${d_2}= \\max _{{e} \\in \\Phi }\\left( { \\frac{l_{s,e}^{-\\alpha } \\eta _{s,e}}{l_{j,e}^{-\\alpha }}}\\right)$ , we have $\\begin{split}y=&\\frac{1+\\frac{P_{s,d}}{P_{j,d}}}{ 1+\\max _{{e} \\in \\Phi }\\left(\\frac{P_{s,e}}{P_{j,e}} \\right)} =\\frac{1 + {d_1}x }{1 + {d_2}x}\\\\\\end{split}$ for $x\\ge 0$ .",
"As shown in Fig.",
"REF , $y$ is a hyperbola for $d_1, d_2>0$ .",
"Its horizontal asymptote is $y = d_1/d_2$ .",
"A special point that the curve passes through the x-axis is $(-1/d_1,0)$ , and the vertical asymptote of the hyperbola is given by $x = -1/d_2$ .",
"Clearly, when $x=0$ , $y=1$ and the achievable secrecy rate is zero.",
"For $ {d_1}> {d_2}$ , the hyperbola $y$ is monotonically increasing in $x$ , and if $ {d_1} < {d_2}$ , $y$ is monotonically decreasing on $x$ .",
"Note that (REF ) defines the statistical comparison between (REF ) and $2^{\\mathcal {R}_t}$ where $\\mathcal {R}_t\\ge 0$ .",
"For ${d_1} < {d_2}$ , $y$ is always less than one.",
"Hence, it is smaller than $2^{\\mathcal {R}_t}$ .",
"Therefore, when ${d_1} < {d_2}$ , neither increasing nor decreasing $P_s/P_j$ will affect the results of the comparison between (REF ) and $2^{\\mathcal {R}_t}$ .",
"Consequently, the monotonicity of (REF ) is solely decided by ${d_1}> {d_2}$ .",
"As a result, the SCP is monotonically increasing as $P_s/P_j$ increases.",
"Meanwhile, when $\\mathcal {R}_t=0$ , neither increasing nor decreasing $P_s/P_j$ will affect the results of the comparison between (REF ) and $2^{\\mathcal {R}_t}$ because (REF ) is always larger than one for ${d_1} > {d_2}$ , while (REF ) is always smaller than one for ${d_1} < {d_2}$ .",
"To conclude, the analysis above shows that for a given $P_s$ , the less $P_j$ the better the SCP in the interference-limited environment.",
"As we will see in the following section, deploying the UAV jammer to a carefully chosen position would benefit the communication networks than the case without the UAV jammer.",
"Figure: The plot of hyperbola ().",
"The solid lines are the hyperbolas in the first quadrant and the dashed lines are horizontal asymptotes."
],
[
"Trend of the SCP with respect to jammer's position",
"This subsection investigates the trend of the SCP with respect to jammer's position when all ATG links are in a pure LoS state and $H^2 >> (R_{1}+ l_{s,d})^2$ .",
"An important aspect of UAV networks is the high likelihood of having significantly stronger LoS components than the reflected multipath components in some deployment scenarios (especially in suburban and rural morphologies) [13].",
"Hence, it is very important to study the performance of the SCP for the ATG links in a pure LoS state.",
"According to (REF ), for ATG links in a pure LoS state, as $H$ increases, they will remain in a pure LoS state.",
"Given $l_{j,d}^2 = l_{t,j}^2 + l_{s,d}^2 - 2l_{t,j} l_{s,d} \\cos \\left( \\varphi _{j^{\\prime }} - \\varphi _{d}\\right) + H^2$ and $l_{s,e}^2 =H^2+l_{t,e}^2$ , when $H^2 >> (R_{1}+ l_{s,d})^2$ and $R_{1}\\ge l_{t,j}$ , we have $l_{j,d}\\approx l_{s,e} \\approx H $ , and $ \\begin{split}&\\frac{1+\\frac{P_{s,d}}{P_{j,d}+N_0}}{ 1+\\max _{{e} \\in \\Phi }\\left(\\frac{P_{s,e}}{P_{j,e}+N_e} \\right)}\\approx \\frac{1+ \\frac{P_{s} l_{{s},{d}}^{-\\beta }|{h}_{{s,d}}|^2}{P_{j}H^{-{\\alpha }}\\eta _{L} +N_0}}{ 1+ \\frac{ P_{s}\\eta _{L}}{H^{\\alpha }} \\max _{{e} \\in \\Phi } \\left( \\frac{1}{P_{j}l_{j,e}^{-\\alpha }+N_e} \\right) }.\\\\\\end{split}$ According to (REF ), the performance of the SCP is related to $\\max _{{e} \\in \\Phi } {l_{j,e}}$ .",
"For a UED $e$ uniformly distributed within a disk, the PDF of $l_{e,j}$ conditoning on $l_{t,j}$ is proportional to the circular arc of the circle $O(j, l_{j,e})$ that is enclosed by the disk $O(t,R_{1})$ [25].",
"Assuming that for a given $l_{t,j}$ , the domain of $l_{j,e}$ with non-zero density is $[0, Z_1]$ , where $Z_1>0$ .",
"Then, as $l_{t,j}$ increases by $\\delta >0$ and $j$ still stays within the circle $O(t,R_{1})$ .",
"The new non-zero domain of $l_{j,e}$ is $[0, Z_1 + \\delta ]$ , and the density of $l_{j,e}$ is decreasing in $l_{t,j}$ for any $l_{j,e}\\in [0, Z_1]$ .",
"Thus, the mean of $l_{j,e}$ increases as $l_{t,j}$ increases.",
"As a result, when the number of the UEDs is one, placing UAV jammer $j$ to $t$ (i.e., $l_{t,j}=0$ ) will provide the optimum SCP for $H^2 >> (R_{1}+ l_{s,d})^2$ .",
"Meanwhile, for multiple UEDs, we have the following discussion.",
"Let $n$ i.i.d.",
"random variables $X_1,...,X_n$ denote the distances from $n$ UEDs to $j$ , the CDF of $Y=\\max (X_1,...,X_n)$ conditioning on $l_{t,j}$ is given by $P(Y\\le y \\mid l_{t,j}) &\\stackrel{i.i.d}{=} P(X_1\\le y \\mid l_{t,j})^n$ and the corresponding PDF is written as $f_Y(y\\mid l_{t,j}) = n(P(X_1\\le y \\mid l_{t,j}))^{n-1}f_{X_1}(y\\mid l_{t,j})$ where $f_{X_1}(y\\mid l_{t,j})$ is the conditional PDF of $X_1$ .",
"For an infinitesimal $ \\Delta \\delta $ , when the upper bound of the domain of $l_{t,j}$ with non-zero density increases from $Z_1$ to $Z_1 + \\Delta \\delta $ by moving $j$ away from $t$ , we have $f_Y(Z_1 + \\Delta \\delta \\mid l_{t,j} + \\Delta \\delta ) = nf_{X_1}(Z_1 + \\Delta \\delta \\mid l_{t,j}+ \\Delta \\delta )>0$ since $P(X_1 \\le Z_1+ \\Delta \\delta \\mid l_{t,j}+ \\Delta \\delta ) =1$ .",
"Meanwhile, as a result of the previous discussion for one UED, we have both $f_{X_1}(y\\mid l_{t,j}) \\ge f_{X_1}(y\\mid l_{t,j}+\\Delta \\delta )$ and $P(X_1\\le y \\mid l_{t,j})\\ge P(X_1\\le y \\mid l_{t,j}+ \\Delta \\delta )$ for $y\\in [0, Z_1]$ .",
"According to (REF ), $f_Y(y\\mid l_{t,j}+ \\Delta \\delta ) \\le f_Y(y\\mid l_{t,j})$ for $y\\in [0, Z_1]$ .",
"As a result of (REF ) and (REF ), the mean of $\\max _{e \\in \\Phi } l_{j,e}$ increases as $l_{t,j}$ increases.",
"Thus, placing UAV jammer $j$ at $t$ will obtain the optimum SCP for $H^2 >> (R_{1}+ l_{s,d})^2$ ."
],
[
"Simulation Results",
"To validate the derivation and the analysis of the SCP with respect to different variables, simulations based on LTE parameters were developed.",
"The parameters used in the simulation are given in Table REF unless otherwise specified.",
"The receiver $d$ and UEDs are assumed to have the same noise power.",
"The simulation results are obtained by averaging over $1 \\times 10^5$ independent Monte Carlo trials.",
"Table: System parametersFigure: SCP vs. ground location of UAV jammer for different P j P_j in watt as given by the figure legends.",
"For the same color, the solid line represents the numerical results with the UAV jammer, the markers represent the simulation results with the UAV jammer and the dashed line represents the simulation results without the UAV jammer.",
"j=(0,0,500)j=(0,0, 500).Figure: SCP vs. location of UAV jammer, where P j =0.01P_j = 0.01 W. The mesh grids represent simulation results, and the markers represent numerical results.Figure: SCP vs. location of UAV jammer, where P j =0.01P_j = 0.01 W. The mesh grids represent simulation results, and the markers represent numerical results.Fig.",
"REF shows the performance of the SCP in the simulation for UEDs at different heights.",
"The numerical results in this section are obtained by evaluating (REF ) for the same parameters as the simulations.",
"One can easily see that the numerical results act as the lower bound to the simulation results of the SCP when $P_j$ is small.",
"The simulation curves match well with the numerical results as $P_j$ reaches the interference-limited regime.",
"In this regime, the performance of the SCP is decreasing as $P_j$ increases.",
"This can be easily explained by the discussion on $P_s/P_j$ in (REF ).",
"Meanwhile, the same figure also shows that introducing a UAV jammer to the ground communications could change the performance of the SCP significantly.",
"On the one hand, for $H=500$ m and 1000 m, the SCP increases as $P_j$ increases until around $10^{-3}$ W and then starts to decrease until the SCP reaches zero.",
"Compared to the similar settings without the UAV jammer, introducing a UAV jammer with proper $P_j$ would allow the ground communications to achieve a higher SCP.",
"For $H=500$ m and 1000 m, the ground communications with a UAV jammer allow the maximum of 20% and 60% increase in SCP, respectively, over the case without a UAV jammer as given by the dashed lines.",
"On the other hand, for $H=100$ m, the SCP increases as $P_j$ increases.",
"By having a UAV jammer in this height, the SCP of the ground communications will be affected severely.",
"Meanwhile, notice the case without the UAV jammer, the achieved SCP in dashed lines show that the SCP at $H=100$ m is higher than the one at $H=500$ m, while this trend is reversed when comparing $H=500$ m to $H=1000$ m. It is because increasing $H$ will cause not only a higher LoS probability from the transmitter to a UED but also a longer receiving distance (i.e., a higher path loss).",
"In general, the ground communications prefer to have NLoS links towards the UEDs and have a maximum distance from it, however, these two objectives conflicts with each other.",
"If the LoS probability from the transmitter to a UED contributes more than the path loss does to the SCP as $H$ increases, the SCP will decrease.",
"Otherwise, the SCP will increase.",
"Fig.",
"REF depicts the simulation results on the SCP when all ATG links are in a pure LoS state.",
"When $H=2000$ , the performance of the SCP is optimum when placing the UAV jammer at $(0,0,H)$ .",
"This can be explained by the discussion on (REF ).",
"The performance curves from $H=300$ to $H=2000$ show that, for the UAV jammer with the same x and y coordinates, the SCP increases as $H$ increases.",
"This can also be explained by (REF ), where the right term decreases as $H$ increases for the ATG links in a pure LoS state.",
"Meanwhile, Fig.",
"REF gives the cases without the requirement on a pure LoS state for ATG links.",
"In general, the trend of the SCP with respect to the height of UAVs is hard to capture, because increasing $H$ will not only cause a higher LoS probability but also decreases the jamming and received signal powers.",
"A higher LoS probability from $s$ to UED will cause the SCP to drop while a smaller jamming power will cause the SCP to increase.",
"Meanwhile, the convex surface shows that the jammer should move away from the receiver $d$ in order to achieve a higher SCP.",
"Furthermore, by observing both Fig.",
"REF and Fig.",
"REF , the numerical results of the SCP matching with the simulation results suggest that both receiver $d$ and UEDs are working in the interference-limited regime for $P_j=0.01$ W. By increasing $H$ from 50 m to 200m, the SCP at the given x and y coordinates decreases first and then increases."
],
[
"Conclusion",
"In this paper, we investigated the secrecy performance of the ground link in the presence of randomly deployed non-colluding UEDs.",
"A piecewise fitting model has been proposed to characterize the LoS probability of an ATG link with higher accuracy than the existing sigmoid fitting.",
"The SCP has been formulated based on the piecewise fitting model so that the formulation become tractable.",
"The performance of the SCP with respect to transmitting-to-jamming power ratio, the height of the UAVs and the UAV jammer location has been investigated.",
"The effective jamming power and location of the UAV jammer have been pointed out.",
"The model and analysis frameworks presented in this paper will help to facilitate the analysis of the UAV-aided communications."
],
[
"Appendix",
"Lemma 1 For a UED $e$ which is uniformly distributed within the disk $O(t,{r})$ , denote $\\phi = \\varphi _{j^{\\prime }}- \\varphi _{e^{\\prime }}$ and $\\eta $ as the excess path loss, the domains of $\\phi $ and $l_{t,e}$ so that $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} \\le y$ is denoted as $\\lbrace {o}_{\\phi }(\\eta ), {o}_{l_{t,e}}(\\eta )\\rbrace $ , while the domains of $\\phi $ and $l_{t,e}$ so that $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} > y$ is denoted as $\\lbrace {o^{\\prime }}_{\\phi }(\\eta ), {o^{\\prime }}_{l_{t,e}}(\\eta ) \\rbrace $ .",
"Let $\\mathcal {F}_1 = {\\cos ^{-1}} \\frac{-\\sqrt{B{A}}}{l_{t,j}}$ , $\\mathcal {F}_2 = {\\cos ^{-1}} \\frac{{r}^2{A} + B}{2 {r} l_{t,j}}$ , ${A} = 1-\\left((y-1)\\frac{P_j}{\\eta P_s}\\right)^\\frac{2}{\\alpha }$ , $B= l_{t,j}^2- (1-{A}) H^2$ .",
"$x_1 = \\frac{ l_{t,j} \\cos \\phi + \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}}}{{A}}$ , and $x_2 = \\frac{ l_{t,j} \\cos \\phi - \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}}}{{A}}$ .",
"Based on different ${A}$ and $B$ , these domains are given as follows.",
"For ${A}<0, B \\le 0$ , we have ${o^{\\prime }}_{l_{t,e}}(\\eta ) = [x_1, x_2]$ and ${o^{\\prime }}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}\\left[\\mathcal {F}_1, \\mathcal {F}_2\\right], &\\text{if } {r}^2{A}\\le B, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j}\\\\\\left[\\mathcal {F}_1, \\pi \\right], &\\text{if } {r}^2{A} < B, \\sqrt{B{A}} \\le l_{t,j} < \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , &\\text{otherwise;}\\end{array}\\right.",
"}$ or $ {o^{\\prime }}_{l_{t,e}}(\\eta ) = [x_1, {r}]$ and ${o^{\\prime }}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}\\left(\\mathcal {F}_2, \\pi \\right], & \\text{if } {r}^2{A} \\le B, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j} \\\\\\left[\\mathcal {F}_2, \\pi \\right], &\\text{if } {r}^2{A} > B, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ For ${A}>0, B \\ge 0$ , we have ${o}_{l_{t,e}}(\\eta ) = [x_2, x_1]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}\\left[\\mathcal {F}_2, \\pi - \\mathcal {F}_1\\right],&\\text{if } {r}^2{A}\\ge B, l_{t,j} \\ge \\frac{{r}^2{A} + B}{2 {r}}\\\\\\left[0, \\pi - \\mathcal {F}_1\\right], &\\text{if } {r}^2{A} > B, \\sqrt{B{A}} \\le l_{t,j} < \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , &\\text{otherwise;}\\end{array}\\right.",
"}$ or ${o}_{l_{t,e}}(\\eta ) = [x_2, {r}]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}\\left[0,\\mathcal {F}_2 \\right], &\\text{if } {r}^2{A} \\le B, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} \\\\\\left[0, \\mathcal {F}_2\\right), &\\text{if } {r}^2{A} > B, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ For ${A}<0, B > 0$ , we have ${o^{\\prime }}_{l_{t,e}}(\\eta ) = [0, {r}]$ and ${o^{\\prime }}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} < B, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\\\left(\\mathcal {F}_2, \\pi \\right], & \\text{if } l_{t,j}\\ge \\frac{\\left|{r}^2{A} + B\\right|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise;}\\end{array}\\right.",
"}$ or ${o^{\\prime }}_{l_{t,e}}(\\eta ) =[0,x_2]$ and ${o^{\\prime }}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A}> B, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\left[0, \\mathcal {F}_2\\right], & \\text{if } l_{t,j}\\ge \\frac{\\left| {r}^2{A} + B\\right|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ For ${A}>0, B < 0$ , we have ${o}_{l_{t,e}}(\\eta ) =[0, {r}]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} > B, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\left[ 0, \\mathcal {F}_2 \\right], & \\text{if } l_{t,j}\\ge \\frac{|{r}^2{A} + B |}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise;}\\end{array}\\right.",
"}$ or ${o}_{l_{t,e}}(\\eta ) =[0, x_1]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} < B, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\\\left[\\mathcal {F}_2, \\pi \\right], & \\text{if } l_{t,j}\\ge \\frac{|{r}^2{A} + B|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ For ${A}=0$ , we have ${o}_{l_{t,e}}(\\eta ) = \\left[ \\frac{\\left| l_{t,j}^2 -H^2\\right|}{2 l_{t,j} } , {r}\\right]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}\\left[0, {\\cos ^{-1}} \\left(\\frac{ l_{t,j}^2 -H^2}{2 l_{t,j} l_{t,e}}\\right)\\right], & \\text{if } \\frac{\\left| l_{t,j}^2 -H^2\\right| }{2 l_{t,j} } \\le {r}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise;}\\end{array}\\right.",
"}$ or ${o}_{l_{t,e}}(\\eta ) = \\left[0, \\frac{ -l_{t,j}^2 +H^2}{2 l_{t,j} } \\right)$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } {r}\\ge \\frac{ -l_{t,j}^2 +H^2}{2 l_{t,j} } \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise;}\\end{array}\\right.",
"}$ or ${o}_{l_{t,e}}(\\eta ) = \\left[0, {r}\\right]$ and ${o}_{\\phi }(\\eta )={\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } {r}< \\frac{ -l_{t,j}^2 +H^2}{2 l_{t,j} } \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ For ${A}\\ne 0$ , the corresponding solutions of $l_{t,e}$ in $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} = y$ are denoted as $\\begin{split}x_1 = \\frac{ l_{t,j} \\cos \\phi + \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}}}{{A}}\\\\x_2 = \\frac{ l_{t,j} \\cos \\phi - \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}}}{{A}}\\end{split}$ where $x_1 \\le x_2$ for ${A}<0$ , while $x_2 \\le x_1$ for ${A}>0$ .",
"For $x_1$ and $x_2$ to exist, ${\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}}\\ge 0.$ As a result, $\\cos \\phi \\ge \\frac{\\sqrt{B{A}}}{l_{t,j}}$ or $\\cos \\phi \\le -\\frac{\\sqrt{B{A}}}{l_{t,j}}$ for $B{A}\\ge 0$ , and ${\\left( l_{t,j} \\cos \\phi \\right)^2 }\\ge 0 $ for $B{A}< 0$ .",
"Since $\\phi = \\varphi _{j^{\\prime }}- \\varphi _{e^{\\prime }}$ and the impact from $j$ to $e$ for $\\phi \\in [0,\\pi ]$ is the same as the case for $\\phi \\in [\\pi ,2\\pi ]$ .",
"Thus, we only consider $\\phi \\in [0,\\pi ]$ in the following analysis.",
"The domains of $\\phi $ in (REF ) are given by $\\phi \\le \\pi - \\mathcal {F}_1, \\text{ for } l_{t,j}\\ge \\sqrt{B{A}}, B{A}\\ge 0,\\\\$ $\\phi \\ge \\mathcal {F}_1, \\text{ for } l_{t,j}\\ge \\sqrt{B{A}} , B{A}\\ge 0, \\\\$ $\\phi \\in [0, \\pi ], \\text{ for } B{A}< 0.$ Meanwhile, $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos \\phi \\right)^\\frac{-\\alpha }{2}P_j} \\le y$ gives $\\begin{split}A l_{t,e}^2 - 2 l_{t,j} \\cos (\\phi ) l_{t,e} + l_{t,j}^2 - (1-A) H^2 \\le 0.\\end{split}$ Apparently, when $A<0$ , $l_{t,e} \\ge x_2$ or $l_{t,e} \\le x_1$ ; when $A>0$ , $x_2 \\le l_{t,e} \\le x_1$ .",
"To simplify analysis, we investigate $x_1 \\le l_{t,e} \\le x_2$ for $A<0$ and $x_2 \\le l_{t,e} \\le x_1$ for $A>0$ .",
"As a result, the former is the case for $A<0$ and $\\begin{split}A l_{t,e}^2 - 2 l_{t,j} \\cos (\\phi ) l_{t,e} + l_{t,j}^2 - (1-A) H^2 \\ge 0.\\end{split}$ For ${A}<0, B \\le 0$ : Assuming that (REF ) holds, comparing $x_1$ with 0 gives $\\begin{split}\\phi \\in \\left[\\frac{\\pi }{2}, \\pi \\right], \\text{ for } x_1 \\ge 0; \\text{ and }\\phi \\in \\left[0,\\frac{\\pi }{2} \\right],\\text{ for } x_1 <0.\\end{split}$ Assuming that (REF ) holds, comparing $x_2$ with 0 and considering that $x_1 \\le x_2$ for ${A}<0$ , we have $\\begin{split}\\phi \\in \\left[\\frac{\\pi }{2}, \\pi \\right], & \\,x_2 \\ge 0.\\end{split}$ Comparing $x_1$ with ${r}$ , we have $\\begin{split}x_1\\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\ge {r}{A}- l_{t,j} \\cos \\phi .\\\\\\end{split}$ If ${r}{A}- l_{t,j} \\cos \\phi \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\pi ], & \\text{if } l_{t,j} \\ge -{r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{if } l_{t,j} < -{r}{A} \\\\\\end{array}\\right.",
"}$ where $\\mathcal {F}_3 = {\\cos ^{-1}} \\frac{{r}{A}}{l_{t,j}}$ .",
"Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_2, \\pi ], & \\text{if } l_{t,j}\\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\pi ], & \\text{if } {r}^2{A} \\le B, \\, l_{t,j} \\ge {-{r}{A}} \\\\[\\mathcal {F}_2, \\pi ], & \\text{if } {r}^2{A} > B, \\, l_{t,j} \\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\pi ], & \\text{if } {r}^2{A} \\le B, \\, l_{t,j} \\ge {-{r}{A}} \\\\[\\mathcal {F}_2, \\pi ], & \\text{if } {r}^2{A} > B, \\, l_{t,j} \\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ If ${r}{A}- l_{t,j} \\cos \\phi < 0$ , then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\pi ], & \\text{if } l_{t,j} < -{r}{A}\\\\[0, \\mathcal {F}_3), & \\text{if } l_{t,j} \\ge -{r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_1, \\pi ], & \\text{if } {r}^2{A} < B, \\sqrt{B{A}} \\le l_{t,j} <-{r}{A}\\\\[\\mathcal {F}_1, \\mathcal {F}_3), & \\text{if } {r}^2{A} < B, l_{t,j} \\ge -{r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The unions of (REF ) and (REF ) give $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_2, \\pi ], & \\text{if } {r}^2{A} > B, \\, l_{t,j} \\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\[\\mathcal {F}_1, \\pi ], & \\text{if } {r}^2{A} \\le B, \\sqrt{B{A}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Comparing ${r}$ with $x_2$ , $\\begin{split}x_2> {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} > -{r}{A}+ l_{t,j} \\cos \\phi .\\\\\\end{split}$ If $-{r}{A}+ l_{t,j} \\cos \\phi \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\pi ], & \\text{if } l_{t,j} < -{r}{A}\\\\[0, \\mathcal {F}_3], & \\text{if } l_{t,j} \\ge -{r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_2, \\pi ], &\\text{if } l_{t,j}\\ge \\frac{-{r}^2{A} - B}{2 {r}}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_2, \\pi ], & \\text{if } {r}^2{A} < B, \\, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j}< -{r}{A} \\\\(\\mathcal {F}_2, \\mathcal {F}_3], & \\text{if } {r}^2{A} < B,\\, l_{t,j} \\ge -{r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Then, the intersections of (REF ) and (REF ) yield (REF ).",
"If $-{r}{A}+ l_{t,j} \\cos \\phi < 0$ , then $ \\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_3, \\pi ], &\\text{if } l_{t,j} \\ge -{r}{A}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_3, \\pi ] , & \\text{if } {r}^2{A} \\le B, l_{t,j} \\ge -{r}{A} \\\\[\\mathcal {F}_1, \\pi ], &\\text{if } {r}^2{A} > B, l_{t,j}\\ge \\sqrt{B{A}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Therefore, the regions for $x_2 > {r}$ are given by the unions of (REF ) and (REF ): $\\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_2, \\pi ], & \\text{if } {r}^2{A} \\le B, \\, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j}\\\\[\\mathcal {F}_1, \\pi ], &\\text{if } {r}^2{A} > B, l_{t,j}\\ge \\sqrt{B{A}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ Meanwhile, $\\begin{split}x_2\\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\le -{r}{A}+ l_{t,j} \\cos \\phi .\\\\\\end{split}$ If $-{r}{A}+ l_{t,j} \\cos \\phi \\ge 0$ , we have (REF ).",
"Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2], & \\text{if } l_{t,j} \\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\pi ] , & \\text{if } l_{t,j} < \\frac{-{r}^2{A} - B}{2 {r}}.",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2], & \\text{if } {r}^2{A} \\le B, l_{t,j} \\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\mathcal {F}_3], & \\text{if } {r}^2{A} > B,l_{t,j} \\ge -{r}{A} \\\\[0,\\pi ], & \\text{if } {r}^2{A} \\le B, l_{t,j} < \\frac{-{r}^2{A} - B}{2 {r}}\\\\[0,\\pi ], & \\text{if } {r}^2{A} > B, l_{t,j} < -{r}{A}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The intersections of (REF ) and (REF ) are given by (REF ).",
"As a result, when $0\\le x_1 \\le {r}$ and $x_2 \\le {r}$ , we have (REF ).",
"Meanwhile, when $0\\le x_1 \\le {r}$ and $x_2 > {r}$ , we have (REF ).",
"For ${A}>0, B \\ge 0$ : Assuming that (REF ) holds, comparing $x_2$ with 0 gives $\\begin{split}\\phi \\in \\left[\\frac{\\pi }{2}, \\pi \\right], \\text{ for } x_2 < 0; \\text{ and }\\phi \\in \\left[0,\\frac{\\pi }{2} \\right], \\text{ for } x_2 \\ge 0.\\end{split}$ Assuming that (REF ) holds, comparing $x_1$ with 0 gives $\\begin{split}\\phi \\in \\left[\\frac{\\pi }{2}, \\pi \\right], \\text{ for } x_1 < 0; \\text{ and }\\phi \\in \\left[0,\\frac{\\pi }{2} \\right], \\text{ for } x_1 \\ge 0.\\end{split}$ Since $x_1$ must be larger than 0 for a valid region to exist, $\\phi \\in \\left[0,\\frac{\\pi }{2} \\right]$ and (REF ) stands.",
"Comparing $x_2$ with ${r}$ , we have $\\begin{split}x_2\\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\ge l_{t,j} \\cos \\phi - {r}{A}.\\\\\\end{split}$ If $l_{t,j} \\cos \\phi - {r}{A}\\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3], & \\text{if } l_{t,j} \\ge {r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{if } l_{t,j} < {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2], & \\text{if } l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{if } l_{t,j}<\\frac{{r}^2{A} + B}{2 {r}}.",
"\\\\\\end{array}\\right.",
"}$ Then, combining (REF ), (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3], & \\text{if } {r}^2{A} > B, l_{t,j} \\ge {r}{A} \\\\[0, \\mathcal {F}_2], & \\text{if } {r}^2{A} \\le B, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise. }",
"\\\\\\end{array}\\right.",
"}$ If $l_{t,j} \\cos \\phi - {r}{A} < 0$ , then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\pi ], & \\text{if } l_{t,j} < {r}{A} \\\\(\\mathcal {F}_3, \\pi ], & \\text{if } l_{t,j} \\ge {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A}> B, \\sqrt{B{A}} \\le l_{t,j} < {r}{A}\\\\(\\mathcal {F}_3, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} > B, {{r}{A}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The unions of (REF ) and (REF ) yield $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2], & \\text{if } {r}^2{A} \\le B, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\[0, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} > B, \\sqrt{B{A}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Comparing ${r}$ with $x_1$ , $\\begin{split}x_1\\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\le {r}{A}- l_{t,j} \\cos \\phi .\\\\\\end{split}$ If ${r}{A}- l_{t,j} \\cos \\phi \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\pi ], & \\text{if } l_{t,j} <{r}{A} \\\\[\\mathcal {F}_3, \\pi ], & \\text{if } l_{t,j}\\ge {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\\\left[\\mathcal {F}_2,\\pi \\right], & \\text{if } l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}}.",
"\\\\\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], & \\text{if } {r}^2{A} < B, l_{t,j} < {{r}{A}}\\\\[0,\\pi ], & \\text{if } {r}^2{A} \\ge B, l_{t,j} < \\frac{{r}^2{A} + B}{2 {r}}\\\\[\\mathcal {F}_3, \\pi ], & \\text{if } {r}^2{A} < B, {{r}{A} \\le l_{t,j}}\\\\[\\mathcal {F}_2, \\pi ] , & \\text{if } {r}^2{A} \\ge B, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_2, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} \\ge B, l_{t,j} \\ge \\frac{{r}^2{A} + B}{2 {r}}\\\\[0, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} \\ge B, \\sqrt{B{A}} \\le l_{t,j} < \\frac{{r}^2{A} + B}{2 {r}}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ $\\begin{split}x_1> {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} > {r}{A}- l_{t,j} \\cos \\phi .\\\\\\end{split}$ If ${r}{A}- l_{t,j} \\cos \\phi \\ge 0,$ we have (REF ).",
"Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2), & \\text{if } l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\mathcal {F}_2), & \\text{if } {r}^2{A} > B, l_{t,j}\\ge {r}{A} \\\\[0, \\mathcal {F}_2), & \\text{if } {r}^2{A} > B, \\frac{{r}^2{A} + B}{2 {r} } \\le l_{t,j}< {r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ If ${r}{A}- l_{t,j} \\cos \\phi < 0$ , we have $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3) , &\\text{if } l_{t,j}\\ge {{r}{A}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The unions of (REF ) and (REF ) yield $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2), & \\text{if } {r}^2{A} > B, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r} } \\\\[0, \\mathcal {F}_3), &\\text{if } {r}^2{A} \\le B, l_{t,j}\\ge {{r}{A}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2), & \\text{if } {r}^2{A} > B, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r} } \\\\[0, \\pi - \\mathcal {F}_1] , &\\text{if } {r}^2{A} \\le B, l_{t,j}\\ge {\\sqrt{B{A}}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The feasible regions of $\\phi $ , where $x_2 \\le {r}$ and $x_1 \\le {r}$ , are the intersections of (REF ) and (REF ).",
"As a result, $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_2, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} \\ge B, l_{t,j} \\ge \\frac{{r}^2{A} + B}{2 {r}}\\\\[0, \\pi - \\mathcal {F}_1], & \\text{if } {r}^2{A} > B, \\sqrt{B{A}} \\le l_{t,j} < \\frac{{r}^2{A} + B}{2 {r}}\\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The feasible regions of $\\phi $ , where $x_2 \\le {r}$ and $x_1 > {r}$ , are the intersections of (REF ) and (REF ).",
"As a result, $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_2], &\\text{if } {r}^2{A} \\le B, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} \\\\[0, \\mathcal {F}_2), &\\text{if } {r}^2{A} > B, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ As a result, when $x_2 \\le {r}$ and $x_1 \\le {r}$ , we have (REF ).",
"Meanwhile, when $x_2 \\le {r}$ and $x_1 > {r}$ , we have (REF ).",
"For ${A}<0, B > 0$ : Comparing $x_1$ with 0 gives $\\begin{split}\\phi \\in \\left[0, \\pi \\right], \\text{ for } x_1 < 0; \\text{ and }\\phi = \\emptyset , \\text{ for }x_1 \\ge 0.\\end{split}$ Comparing $x_2$ with 0 gives $\\begin{split}\\phi = \\emptyset , \\text{ for } x_2 < 0; \\text{ and }\\phi \\in \\left[0, \\pi \\right], \\text{ for } x_2 \\ge 0.\\end{split}$ Comparing $x_2$ with ${r}$ , $\\begin{split}x_2 > {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} > l_{t,j} \\cos \\phi - {r}{A}.\\\\\\end{split}$ If $l_{t,j} \\cos \\phi - {r}{A} < 0$ , then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}(\\mathcal {F}_3, \\pi ] , & \\text{if } l_{t,j} \\ge -{r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{if } l_{t,j} < -{r}{A}.",
"\\\\\\end{array}\\right.",
"}$ If $l_{t,j} \\cos \\phi - {r}{A} \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3], & \\text{if } l_{t,j} \\ge -{r}{A} \\\\[0, \\pi ], & \\text{if } l_{t,j} < -{r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} <B, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\(\\mathcal {F}_2, \\pi ], & \\text{if } -{r}^2{A} \\ge B, l_{t,j}\\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\(\\mathcal {F}_2, \\pi ], & \\text{if } -{r}^2{A} < B, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $ \\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} \\ge \\frac{B}{3{r}^2}, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\[0,\\pi ], &\\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, l_{t,j}< -{r}{A} \\\\(\\mathcal {F}_2, \\pi ], & \\text{if } -{A} \\ge \\frac{B}{{r}^2}, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j} < -{r}{A} \\\\(\\mathcal {F}_2, \\pi ], & \\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} > \\frac{B}{3{r}^2}, \\frac{{r}^2{A} + B}{2 {r}} \\le l_{t,j} < -{r}{A} \\\\[0, \\mathcal {F}_3], & \\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, -{r}{A} \\le l_{t,j} < \\frac{{r}^2{A} + B}{2 {r}}\\\\(\\mathcal {F}_2, \\mathcal {F}_3], & \\text{if } -{A} \\ge \\frac{B}{{r}^2}, l_{t,j}\\ge -{r}{A} \\\\(\\mathcal {F}_2, \\mathcal {F}_3], & \\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} \\ge \\frac{B}{3{r}^2}, l_{t,j}\\ge -{r}{A} \\\\(\\mathcal {F}_2, \\mathcal {F}_3], & \\text{if } -{A} < \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, l_{t,j}\\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The unions of (REF ) and (REF ) yield $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} < B, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\(\\mathcal {F}_2, \\pi ], & \\text{if } l_{t,j}\\ge \\frac{\\left|{r}^2{A} + B\\right|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Meanwhile, $\\begin{split}x_2 \\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\le l_{t,j} \\cos \\phi - {r}{A}.\\\\\\end{split}$ If $l_{t,j} \\cos \\phi - {r}{A} \\ge 0$ , then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3], & \\text{if } l_{t,j} \\ge -{r}{A} \\\\[0, \\pi ], & \\text{if } l_{t,j} < -{r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} > B, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\mathcal {F}_2], & \\text{if } l_{t,j}\\ge \\frac{\\left| -{r}^2{A} - B\\right|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The intersections of (REF ) and (REF ) are given by (REF ).",
"Hence, the feasible regions for $x_2>{r}$ or $0\\le x_2\\le {r}$ are given by (REF ) and (REF ), respectively.",
"As a result, when $x_2>{r}$ , we have (REF ).",
"Meanwhile, when $0\\le x_2\\le {r}$ , we have (REF ).",
"For ${A}>0, B < 0$ : Comparing $x_1$ with 0 gives $\\begin{split}\\phi \\in \\left[0, \\pi \\right], \\text{ for } x_1 \\ge 0; \\text{ and } \\phi =\\emptyset , \\text{ for } x_1 < 0.\\end{split}$ Comparing $x_2$ with 0 gives $\\begin{split}\\phi =\\emptyset , \\text{ for } x_2 \\ge 0; \\text{ and }\\phi \\in \\left[0, \\pi \\right]\\text{ for } \\,x_2 < 0.\\end{split}$ Comparing $x_1$ with ${r}$ , $\\begin{split}x_1 > {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} > {r}{A} - l_{t,j} \\cos \\phi .\\\\\\end{split}$ If ${r}{A} - l_{t,j} \\cos \\phi < 0$ , then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0, \\mathcal {F}_3), & \\text{if } l_{t,j} \\ge {r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{if } l_{t,j} < {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ If ${r}{A} - l_{t,j} \\cos \\phi \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\pi ], & \\text{if } l_{t,j} \\ge {r}{A} \\\\[0, \\pi ], & \\text{if } l_{t,j} < {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} > B, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\mathcal {F}_2], & \\text{if } -{r}^2{A} \\le B, l_{t,j} \\ge \\frac{{r}^2{A} + B}{2 {r}} \\\\[0, \\mathcal {F}_2], & \\text{if } -{r}^2{A} > B, l_{t,j}\\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Then, combining (REF ) and (REF ) with intersections yields $ \\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} \\ge \\frac{B}{3{r}^2}, l_{t,j}< {r}{A} \\\\[0,\\pi ], &\\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\mathcal {F}_2], & \\text{if } -{A} \\le \\frac{B}{{r}^2}, \\frac{{r}^2{A} + B}{2 {r}}\\le l_{t,j} < {r}{A} \\\\[0, \\mathcal {F}_2], & \\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, \\frac{-{r}^2{A} - B}{2 {r}} \\le l_{t,j} < {r}{A}\\\\[\\mathcal {F}_3, \\pi ], & \\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} > \\frac{B}{3{r}^2}, {r}{A} \\le l_{t,j} < \\frac{-{r}^2{A} - B}{2 {r}}\\\\[\\mathcal {F}_3, \\mathcal {F}_2], & \\text{if } -{A} \\le \\frac{B}{{r}^2}, l_{t,j} \\ge {r}{A} \\\\[\\mathcal {F}_3, \\mathcal {F}_2], & \\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} \\ge \\frac{B}{3{r}^2}, l_{t,j}\\ge \\frac{-{r}^2{A} - B}{2 {r}} \\\\[\\mathcal {F}_3, \\mathcal {F}_2], & \\text{if } -{A} > \\frac{ B}{{r}^2}, -{A} < \\frac{B}{3{r}^2}, l_{t,j}\\ge {r}{A} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The unions of (REF ) and (REF ) yield $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} > B, l_{t,j}< \\frac{-{r}^2{A} - B}{2 {r}} \\\\[0, \\mathcal {F}_2], & \\text{if } l_{t,j}\\ge \\frac{|{r}^2{A} + B |}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Meanwhile, $\\begin{split}x_1 \\le {r} \\Rightarrow \\sqrt{\\left( l_{t,j} \\cos \\phi \\right)^2 - B{A}} \\le {r}{A} - l_{t,j} \\cos \\phi .\\\\\\end{split}$ If $ {r}{A} - l_{t,j} \\cos \\phi \\ge 0,$ then $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[\\mathcal {F}_3, \\pi ], & \\text{if } l_{t,j} \\ge {r}{A} \\\\[0, \\pi ], & \\text{if } l_{t,j} < {r}{A}.",
"\\\\\\end{array}\\right.",
"}$ Assuming that (REF ) and (REF ) hold, solving (REF ) gives $\\phi \\in {\\left\\lbrace \\begin{array}{ll}[0,\\pi ], &\\text{if } -{r}^2{A} < B, l_{t,j}< \\frac{{r}^2{A} + B}{2 {r}} \\\\[\\mathcal {F}_2, \\pi ], & \\text{if } l_{t,j}\\ge \\frac{|{r}^2{A} + B|}{2 {r}} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ The intersections of (REF ) and (REF ) is (REF ).",
"Hence, the feasible regions for $x_1>{r}$ or $0\\le x_1\\le {r}$ are given by (REF ) and (REF ), respectively.",
"As a result of the above discussions, when $x_1>{r}$ , we have (REF ).",
"Meanwhile, when $0\\le x_1\\le {r}$ , we have (REF ).",
"For ${A}= 0$ : The corresponding solution to $1+\\frac{(H^2+l_{t,e}^2)^\\frac{-\\alpha }{2}\\eta P_s}{\\left(l_{t,j}^2+l_{t,e}^2-2 l_{t,j} l_{t,e}\\cos (\\phi )\\right)^\\frac{-\\alpha }{2}P_j} \\le y$ is given by ${\\bf {{ \\cos \\phi }} }\\ge \\frac{ l_{t,j}^2 -H^2}{2 l_{t,j} l_{t,e}}.$ Thus, we have $\\phi \\in {\\left\\lbrace \\begin{array}{ll}\\left[0, {\\cos ^{-1}} \\left(\\frac{ l_{t,j}^2 -H^2}{2 l_{t,j} l_{t,e}}\\right) \\right], & \\text{if } \\frac{\\left| l_{t,j}^2 -H^2\\right|}{2 l_{t,j} } \\le l_{t,e} \\\\[0,\\pi ], &\\text{if } \\frac{ -l_{t,j}^2 +H^2}{2 l_{t,j} } >l_{t,e} \\\\\\lbrace \\emptyset \\rbrace , & \\text{otherwise.}\\end{array}\\right.",
"}$ Comparing $l_{t,e}$ from (REF ) with ${r}$ , we have (REF ) - (REF )."
]
] | 1808.08628 |
[
[
"New supercharacter theory for Sylow subgroups in orthogonal and\n symplectic groups"
],
[
"Abstract Applying the embedding of $A_{n-1}$ in $B_n$, $C_n$ and $D_n$ we construct a new supercharacter theory for the Sylow subgroups in orthogonal and symplectic groups over a finite field.",
"The constructed supercharacter appears to be a little bit more precise than the previously known one."
],
[
"The notion of a supercharacter theory was suggested by P.Diaconis and I.M.Isaaks in the paper [1].",
"A priori every group affords several supercharacter theories.",
"One of examples of a supercharacter theroy is the theory of irreducible characters.",
"Since for some groups (such as the unitriangular group, the Sylow subgroups in symplectic and orthogonal groups, the parabolic subgroups and others) the problem of classification of irreducible characters (representations) remains to be a very complicated \"wild\" problem, it appears reasonable to replace this problem by the problem of construction of a supercharacter theory, which provides the best approximation of theory of irreducible characters.",
"Let us formulate the definition of a supercharacter theory following the paper [1].",
"Let $G$ be a finite group, $1\\in G$ be a unit element.",
"Let ${\\mathfrak {Ch}}= \\lbrace \\chi _1, \\ldots , \\chi _N$ } be a system of complex characters (representations) of the group $G$ .",
"Definition 1.",
"The system of characters ${\\mathfrak {Ch}}$ determines a supercharacter theory of $G$ if there exists a partition ${\\mathcal {K}}= \\lbrace K_1,\\ldots , K_N\\rbrace $ of the group $G$ satisfying the following conditions: S1) the characters of ${\\mathfrak {Ch}}$ are pairwise disjoint (orthogonal); S2) each character $\\chi _i$ are constant on each subset $K_j$ ; S3) $\\lbrace 1\\rbrace \\in {\\mathcal {K}}$ .",
"Under this definition each character of ${\\mathfrak {Ch}}$ is referred to as a supercharacter, each subset of ${\\mathcal {K}}$ a superclass.",
"Observe that the number of supercharacters is equal to the number of superclasses.",
"The square table $\\lbrace \\chi _i(K_j)\\rbrace $ is called a supercharacter table.",
"For each supercharacter $\\chi _i$ , consider its support $X_i$ (the subset of all irreducible constituents of $\\chi _i$ ).",
"Observe that the condition S3) of Definition REF may be replaced by following condition: S3') The system of subsets ${\\mathcal {X}}=\\lbrace X_1,\\ldots , X_N\\rbrace $ is a partition of the system of irreducible characters ${\\mathrm {Irr}}(G)$ .",
"Moreover, here each character $\\chi _i$ differs from the character $\\sigma _i=\\sum _{\\psi \\in X_i} \\psi (1)\\psi $ by a constant factor (see [1], [9], [10]).",
"For the unitriangular group ${\\mathrm {UT}}(m,{\\mathbb {F}}_q)$ , the suitable supercharacter theory was constructed in the series of papers of C.A.M.",
"André [2], [3], [4].",
"This theory was generated for the algebra groups by P.Diaconis and I.M.Isaaks [1].",
"By definition, an algebra group is a group of the form $G=1+J$ , where $J$ is an associative finite dimensional nilpotent algebra.",
"Superclasses in the algebra group $G$ are the equivalence classes for the equivalence relation: $g\\sim g^{\\prime }$ , where $g=1+x$ and $g^{\\prime }=1+x^{\\prime }$ , if there exist $a,b\\in G$ such that $x^{\\prime }=axb$ .",
"The similar relation is defined for $J^*$ : by definition, ${\\lambda }\\sim {\\lambda }^{\\prime }$ if there exist $a,b\\in G$ such that ${\\lambda }^{\\prime }=a{\\lambda }b$ (here $a{\\lambda }b(x)={\\lambda }(bxa)$ ).",
"Fix a nontrivial character $t\\rightarrow {\\varepsilon }^t$ of the additive group of the field ${\\mathbb {F}}_q$ with values in the group of invertible elements of the field ${\\mathbb {C}}$ .",
"Supercharacters $\\chi _{\\lambda }$ of a given algebra group are the induced characters from linear characters $\\xi _{\\lambda }(1+x)={\\varepsilon }^{{\\lambda }(x)}$ of right stabilizers of ${\\lambda }\\in J^*$ .",
"The sets of characters $\\lbrace \\chi _{\\lambda }\\rbrace $ and classes $\\lbrace K(g)\\rbrace $ , where ${\\lambda }$ and $g$ run through the set of representatives of equivalence classes of $J^*$ and $G$ respectively, give rise to a supercharacter theory of the algebra group $G$ .",
"Supercharacters $\\chi _{\\lambda }$ afford the analog of A.A.Krillov formula (see [1], [10]): $\\chi _{\\lambda }(1+x) = \\frac{|G{\\lambda }|}{|G{\\lambda }G|}\\sum _{\\mu \\in G{\\lambda }G}{\\varepsilon }^{\\mu (x)}.$ An unipotent group is not an algebra group in general.",
"The outlined method is not valid for unipotent groups.",
"In this paper, we propose the new approach which can be applied for a large class of unipotent groups, hypothetically.",
"The application of this approach for the Sylow subgroups of orthogonal and symplectic groups enables to construct the supercharacter theory (see Theorem REF ), which is a bit better then the one suggested in the papers [5], [6], [7], [8].",
"Let us present the content of this approach.",
"Let $U$ be an unipotent group that is a semidirect product $U=U_1U_0$ with the normal subgroup $U_1$ .",
"Suppose that $U_0$ is an algebra group, i.e.",
"$U_0=1+{\\mathfrak {u}}_0$ , where ${\\mathfrak {u}}_0$ is an associated finite dimensional nilpotent algebra.",
"The Lie algebra ${\\mathfrak {u}}$ of the group $U$ is a direct sum of two subalgebras ${\\mathfrak {u}}={\\mathfrak {u}}_0 \\oplus {\\mathfrak {u}}_1$ , where ${\\mathfrak {u}}_0$ is an associated algebra, and ${\\mathfrak {u}}_1=\\mathrm {Lie}(U_1)$ is an ideal in ${\\mathfrak {u}}$ .",
"Since $U_0$ is an algebra group, for any $a\\in U_0$ and $x_0\\in {\\mathfrak {u}}_0$ , the elements $\\ell _a(x_0)=ax_0$ and $r_a(x_0)=x_0a$ also belongs to ${\\mathfrak {u}}_0$ .",
"The left and right actions of $U_0$ on ${\\mathfrak {u}}_0$ can be extended to the actions on ${\\mathfrak {u}}$ as follows $\\begin{array}{l} \\ell _a(x)= \\ell _a(x_0)+ {\\mathrm {Ad}}_a(x_1),\\\\r_a(x) = r_a(x_0) + x_1, \\end{array}$ where $ a\\in U_0$ and $x=x_0+x_1$ , $x_0\\in {\\mathfrak {u}}_0$ , $x_1\\in U_1$ .",
"Observe that the left and right actions of the subgroup $U_0$ commute, and $\\ell _ar_a^{-1}(x)={\\mathrm {Ad}}_a(x),$ where ${\\mathrm {Ad}}_a$ is the adjoint operator for $a\\in U_0$ .",
"Definition 2.",
"Let $x, x^{\\prime }\\in {\\mathfrak {u}}$ .",
"The element $x$ is equivalent to $x^{\\prime }$ if there exists a chain of transformations of forms 1) $x\\rightarrow \\ell _a(x)$ , where $a\\in U_0$ , 2) $x\\rightarrow {\\mathrm {Ad}}_u(x)$ , where $u\\in U$ , that maps $x$ to $x^{\\prime }$ .",
"Because of (REF ) we may substitute $r_a$ for $\\ell _a$ in 1).",
"Fix ${\\mathrm {Ad}}$ -invariant bijective map $f:U\\rightarrow {\\mathfrak {u}}$ , $f(1)=0$ .",
"As a map $f$ we can take the logarithm $\\ln $ (it requires strong restrictions of characteristic of the field; see below Definition REF for Sylow subgroups in orthogonal and symplectic groups).",
"Introduce the equivalence relation on $U$ as follows.",
"Definition 3.",
"Two elements $u_1$ and $u_2$ of the group $U$ are equivalent if the elements $f(u_1)$ and $f(u_2)$ from ${\\mathfrak {u}}$ are equivalent in the sense of Definition REF .",
"Consider the equivalence classes $\\lbrace K(u)\\rbrace $ ; hypothetically they are superclasses for some supercharacter theory.",
"Let us define the left and right actions of $U_0$ on the dual space ${\\mathfrak {u}}^*$ by the formulas $\\ell ^*_a{\\lambda }(x)={\\lambda }(r_a(x)),$ $r^*_a{\\lambda }(x)={\\lambda }(\\ell _a(x)).$ The equivalence relation on ${\\mathfrak {u}}^*$ is defined similarly to REF for ${\\mathfrak {u}}$ .",
"Definition 4.",
"Let ${\\lambda }, {\\lambda }^{\\prime }\\in {\\mathfrak {u}}^*$ .",
"The element ${\\lambda }$ is equivalent to ${\\lambda }^{\\prime }$ if there exists a chain of transformations of forms 1) ${\\lambda }\\rightarrow \\ell ^*_a({\\lambda })$ , where $a\\in U_0$ , 2) ${\\lambda }\\rightarrow {\\mathrm {Ad}}^*_u({\\lambda })$ , where $u\\in U$ , that maps $x$ to $x^{\\prime }$ .",
"As above $\\ell _a^* (r_a^*)^{-1} {\\lambda }= {\\mathrm {Ad}}_a^*{\\lambda }$ ; in definition, we may substitute $r^*_a$ for $\\ell ^*_a$ .",
"Denote by ${\\mathcal {O}}({\\lambda })$ the equivalence class of ${\\lambda }\\in {\\mathfrak {u}}^*$ .",
"In this paper, we present the classification of equivalence classes in ${\\mathfrak {u}}$ , $U$ and ${\\mathfrak {u}}^*$ for the Sylow subgroups in orthogonal and symplectic groups (see Theorems REF , REF , REF ).",
"Conjecture 5.",
"There exists a system of characters of a given finite unipotent group $U$ of the form $\\chi _{{\\lambda }}(u) = c({\\lambda }) \\sum _{\\mu \\in {\\mathcal {O}}({\\lambda })} {\\varepsilon }^{\\mu (f(u))}, ~~\\mbox{where}~~ c({\\lambda })\\in {\\mathbb {C}},~ c({\\lambda })\\ne 0,$ such that along with the partition of the group $U$ into the classes $\\lbrace K(u)\\rbrace $ , where ${\\lambda }$ and $u$ run through the sets of representatives of equivalence classes in ${\\mathfrak {u}}^*$ and $U$ respectively, give rise to a supercharacter theory of the group $U$ .",
"Remark 6 (see [8]).",
"Observe that the formula (REF ) defines the system of orthogonal functions on $U$ (since the characters $\\lbrace {\\varepsilon }^{{\\lambda }(x) }\\rbrace $ of the abelian group ${\\mathfrak {u}}$ are pairwise orthogonal).",
"Easy to see that the functions (REF ) are constant on the classes $K(u)$ .",
"From $f(1)=0$ it follows $K(1) =1$ .",
"So, the functions (REF ) always fulfil the conditions S1, S2, S3.",
"The main problem is to prove existence of constants $c({\\lambda })$ such that the formula (REF ) defines a character of some representation of the group $U$ .",
"The unitriangular group $G={\\mathrm {UT}}(m,{\\mathbb {F}}_q)$ consists of all upper triangular matrices of order $m$ with ones on the diagonal and entries from the finite field ${\\mathbb {F}}_q$ .",
"Assume that the characteristic of field $p>2$ .",
"The Lie algebra of the unitriangular group ${\\mathfrak {g}}={{\\mathfrak {u}}{\\mathfrak {t}}(m,{\\mathbb {F}}_q)}$ consists of upper triangular matrices with zeros on the diagonal.",
"Consider the matrices $I_n=\\left(\\begin{array}{ccc}0&\\dots &1\\\\ \\vdots &&\\vdots \\\\ 1&\\dots &0\\end{array}\\right)\\quad \\mbox{and }\\quad J_{2n}=\\left(\\begin{array}{cc} 0&I_n\\\\ -I_n&0\\end{array}\\right).$ Let $m$ denote the dimension of standard representation of Lie algebras of types $B_n$ , $C_n$ , and $D_n$ .",
"That is $m=2n+1$ for $B_n$ , and $m=2n$ for $C_n$ and $D_n$ .",
"The matrix algebra $\\mathrm {Mat}(m,{\\mathbb {F}}_q) $ affords the involutive antiautomorphism $X\\rightarrow X^\\dag $ , where $X^\\dag = I_{m}X^tI_{m}$ for $B_n$ and $D_n$ , and $X^\\dag = J_{2n}X^tJ_{2n}$ for $C_n$ .",
"The standard Sylow subgroup $U$ in orthogonal and symplectic group consists of all $g\\in G$ obeying $g^\\dag =g^{-1}$ .",
"Respectively, its Lie algebra ${\\mathfrak {u}}=\\lbrace x\\in {\\mathfrak {g}}:~ x^\\dag =-x\\rbrace $ .",
"We denote by $X^\\tau $ the matrix transposed to $X$ with respect to the second diagonal.",
"The Lie algebra ${\\mathfrak {u}}$ for $C_n$ and $D_n$ consists of matrices of the form ${\\mathfrak {u}}=\\left\\lbrace \\left( \\begin{array}{cc} X_0&X_1\\\\0&-X_0^\\tau \\end{array}\\right)\\right\\rbrace ,$ where $X_0\\in {{\\mathfrak {u}}{\\mathfrak {t}}(n,{\\mathbb {F}}_q)}$ , $X_1^\\tau =X_1$ for $C_n$ and $X_1^\\tau =-X_1$ for $D_n$ .",
"The Lie algebra ${\\mathfrak {u}}$ is a sum of two subalgebras ${\\mathfrak {u}}={\\mathfrak {u}}_0+{\\mathfrak {u}}_1$ , where ${\\mathfrak {u}}_0=\\left\\lbrace \\left( \\begin{array}{cc} X_0&0\\\\0&-X_0^\\tau \\end{array}\\right)\\right\\rbrace ,\\quad \\quad {\\mathfrak {u}}_1=\\left\\lbrace \\left( \\begin{array}{cc} 0&X_1\\\\0&0\\end{array}\\right)\\right\\rbrace .$ The subalgebra ${\\mathfrak {u}}_1$ is an ideal in ${\\mathfrak {u}}$ , and ${\\mathfrak {u}}_0$ is isomorphic to ${{\\mathfrak {u}}{t}(n,{\\mathbb {F}}_q)}$ , and, therefore, it has a natural structure of associative algebra.",
"The group $U$ is a semidirect product $U=U_1U_0$ , where $ U_1=\\left\\lbrace \\begin{pmatrix} E&B\\\\0&E\\end{pmatrix}\\right\\rbrace \\quad \\mbox{and}\\quad U_0=\\left\\lbrace \\begin{pmatrix} A&0\\\\0& (A^\\tau )^{-1}\\end{pmatrix}\\right\\rbrace ,$ $B^\\tau =B$ for $C_n$ and $B^\\tau =-B$ for $D_n$ , and $A\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ .",
"The subgroup $U_0$ is isomorphic to ${\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ , and, therefore, it is an algebra group.",
"In the case $B_n$ , the Lie algebra ${\\mathfrak {u}}$ consists of matrices of the form $\\left\\lbrace \\left( \\begin{array}{ccc}X_0&X_1&X_2\\\\0&0&-X_1^\\tau \\\\0&0&- X_0^\\tau \\end{array}\\right)\\right\\rbrace $ where $X_0\\in {{\\mathfrak {u}}{\\mathfrak {t}}(n,{\\mathbb {F}}_q)}$ , $X_1$ is a $n\\times 1$ column, $X_2$ is a $n\\times n$ matrix and $X_2^\\tau =-X_2$ .",
"As above the Lie algebra ${\\mathfrak {u}}$ is a sum of two subalgebras ${\\mathfrak {u}}={\\mathfrak {u}}_0+{\\mathfrak {u}}_1$ , where ${\\mathfrak {u}}_0 = \\left\\lbrace \\left( \\begin{array}{ccc}X_0&0&0\\\\0&0&0 \\\\0&0&- X_0^\\tau \\end{array}\\right)\\right\\rbrace ,\\quad {\\mathfrak {u}}_1 = \\left\\lbrace \\left( \\begin{array}{ccc}0&X_1&X_2\\\\0&0&-X_1^\\tau \\\\0&0&0\\end{array}\\right)\\right\\rbrace .$ The subalgebra ${\\mathfrak {u}}_1$ is an ideal in ${\\mathfrak {u}}$ , and ${\\mathfrak {u}}_0$ is isomorphic to ${{\\mathfrak {u}}{t}(n,{\\mathbb {F}}_q)}$ , and, therefore, it has a natural structure of associative algebra.",
"The group $U$ decomposes $U=U_1U_0$ , where $ U_1=\\left\\lbrace \\begin{pmatrix} E&v&-\\frac{1}{2}vv^\\tau +B\\\\0&1& -v^\\tau \\\\ 0&0&E \\end{pmatrix}\\right\\rbrace \\quad \\mbox{and}\\quad U_0=\\left\\lbrace \\begin{pmatrix} A&0&0\\\\0&1&0\\\\0&0& (A^\\tau )^{-1}\\end{pmatrix}\\right\\rbrace ,$ $B^\\tau =-B$ , and $v$ is a $n$ -column.",
"The subgroup $U_0$ is isomorphic to ${\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ , and it is an algebra group.",
"Let us define the left and right actions of the subgroup $U_0$ on ${\\mathfrak {u}}$ following the formula (REF ).",
"For $C_n$ and $D_n$ , $a=\\mathrm {diag}(A, (A^\\tau )^{-1})$ and $x\\in {\\mathfrak {u}}$ , we have $\\ell _a(x) = \\ell _a(x)= \\ell _a(x_0)+ {\\mathrm {Ad}}_a(x_1) =\\begin{pmatrix} AX_0&AX_1A^\\tau \\\\0&-X_0A^\\tau \\end{pmatrix}$ $r_a(x) = r_a(x_0) + x_1 = \\begin{pmatrix} X_0A&X_1\\\\0&-A^\\tau X^\\tau \\end{pmatrix}.$ Observe $ \\ell _a(x) = \\begin{pmatrix} A&0\\\\0&E\\end{pmatrix}\\begin{pmatrix} X_0&X_1\\\\0&-X^\\tau \\end{pmatrix}\\begin{pmatrix} E&0\\\\0&A^\\tau \\end{pmatrix} = a_1xa_1^\\dag , ~~\\mbox{where}~~ a_1=\\mathrm {diag}(A,E);$ $ r_a(x) = \\begin{pmatrix} E&0\\\\0&A^\\tau \\end{pmatrix}\\begin{pmatrix} X_0&X_1\\\\0&-X^\\tau \\end{pmatrix}\\begin{pmatrix} A&0\\\\0&E\\end{pmatrix} = a_2xa_2^\\dag , ~~\\mbox{where}~~ a_2=\\mathrm {diag}(E,A^\\tau ).$ For $B_n$ , $a=\\mathrm {diag}(A,1,(A^\\tau )^{-1})$ , and $x\\in {\\mathfrak {u}}$ , we have $\\ell _a(x) = \\ell _a(x_0)+ {\\mathrm {Ad}}_a(x_1) = \\begin{pmatrix} AX_0&AX_1&AX_2A^\\tau \\\\0&0&-X_1^\\tau A^\\tau \\\\0&0&- X_0^\\tau A^\\tau \\end{pmatrix},$ $r_a(x) = r_a(x) = r_a(x_0) + x_1 = \\begin{pmatrix} X_0A &X_1&X_2\\\\0&0&-X_1^\\tau \\\\0&0&- A^\\tau X_0^\\tau \\end{pmatrix}.$ Observe $ \\ell _a(x) =\\begin{pmatrix} A&0&0\\\\0&1&0\\\\ 0&0&E\\end{pmatrix}\\begin{pmatrix} X_0&X_1&X_2\\\\0&0&-X_1^\\tau \\\\0&0&- X_0^\\tau \\end{pmatrix} \\begin{pmatrix} E&0&0\\\\0&1&0\\\\ 0&0&A^\\tau \\end{pmatrix} = a_1xa_1^\\dag ,$ where $a_1=\\mathrm {diag}(A,1,E);$ $r_a(x) = \\begin{pmatrix} E&0&0\\\\0&1&0\\\\ 0&0&A^\\tau \\end{pmatrix}\\begin{pmatrix} X_0&X_1&X_2\\\\0&0&-X_1^\\tau \\\\0&0&- X_0^\\tau \\end{pmatrix} \\begin{pmatrix} A&0&0\\\\0&1&0\\\\ 0&0&E \\end{pmatrix}= a_2xa_2^\\dag ,$ where $a_2=\\mathrm {diag}(E,1,A^\\tau ).$ Denote by ${G^\\circ }$ the subgroup in $G={\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ generated by the subgroup $U$ and the matrices ${\\mathrm {diag}}(A_1,A_2)$ in the case $C_n$ and $D_n$ (respectively, ${\\mathrm {diag}}(A_1,1,A_2)$ in the case $B_n$ ).",
"Remark.",
"The elements $x,x^{\\prime }\\in {\\mathfrak {u}}$ are equivalent if and only if there exists $g\\in {G^\\circ }$ such that $x^{\\prime }=gxg^\\dag $ .",
"In referred to above papers [5], [6], [7], [8], the equivalence relation is a bit coarser: $x\\sim x^{\\prime }$ if there exists $g\\in G$ such that $x^{\\prime }=gxg^\\dag $ .",
"Denote by ${H^\\circ }$ the subgroup in $G={\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ generated by the subgroup $U_1$ and the matrices ${\\mathrm {diag}}(A_1,E)$ in the case $C_n$ and $D_n$ (respectively, ${\\mathrm {diag}}(A_1,1,E)$ in the case $B_n$ ).",
"Let us describe the subgroups ${G^\\circ }$ and ${H^\\circ }$ .",
"Proposition 7.",
"1) In the case $B_n$ , $ {G^\\circ }=\\left\\lbrace \\begin{pmatrix}A_0&A_1&A_2\\\\0&A_3&A_4\\\\0&0&A_5\\end{pmatrix}\\right\\rbrace \\quad \\mbox{and}\\quad {H^\\circ }=\\left\\lbrace \\begin{pmatrix}A_0&A_1&A_2\\\\0&A_3&A^{\\prime }_4\\\\0&0&E\\end{pmatrix}\\right\\rbrace ,$ where $A_0,A_5\\in {\\mathrm {UT}}(n-1,{\\mathbb {F}}_q)$ , $A_1,A_2,A_4$ are an arbitrary matrices of sizes $(n-1)\\times 3$ , $(n-1)\\times (n-1)$ , $3\\times (n-1)$ respectively, $A_4^{\\prime }$ is an arbitrary $3\\times n$ matrix with zero last row, $E$ is the unit $n\\times n$ matrix, $A_3$ is the $3\\times 3$ matrix of the form $\\begin{pmatrix}1&c&-\\frac{1}{2}c^2\\\\0&1&-c\\\\ 0&0&1\\end{pmatrix},\\quad c\\in {\\mathbb {F}}_q.$ 2) In the case $C_n$ , the subgroup ${G^\\circ }$ coincides with $G$ and $ {H^\\circ }=\\left\\lbrace \\begin{pmatrix}A_0&A_1\\\\0&E\\end{pmatrix}\\right\\rbrace .",
"$ 3) In the case $D_n$ , $ {G^\\circ }=\\left\\lbrace \\begin{pmatrix}A_0&A_1\\\\0&A_2\\end{pmatrix}\\right\\rbrace \\quad \\mbox{and}\\quad {H^\\circ }=\\left\\lbrace \\begin{pmatrix}A_0&A_1\\\\0&E\\end{pmatrix}\\right\\rbrace ,$ where $A_0,A_2\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ , and $A_1$ is an arbitrary $n\\times n$ matrix of the form $\\begin{pmatrix} *&*&\\cdots &*\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\c&*&\\cdots &*\\\\0&-c&\\cdots &*\\end{pmatrix},\\quad c\\in {\\mathbb {F}}_q.$ Proof.",
"We shall prove for the subgroup ${H^\\circ }$ (for ${G^\\circ }$ similarly).",
"Case $C_n$.",
"The subgroup ${H^\\circ }$ is generated by the matrices of the form $\\begin{pmatrix} A&0\\\\0&E\\end{pmatrix}$ , where $A\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ and $\\begin{pmatrix} E&B\\\\0&E\\end{pmatrix}$ with $B^\\tau = B$ .",
"Then the matrix $ \\begin{pmatrix} A&0\\\\0&E\\end{pmatrix}\\begin{pmatrix} E&B\\\\0&E\\end{pmatrix} \\begin{pmatrix} A^{-1}&0\\\\0&E\\end{pmatrix} = \\begin{pmatrix} E&AB\\\\0&E\\end{pmatrix}$ also belongs to ${H^\\circ }$ .",
"Easy to verify that the linear subspace, spanned by the matrices of the form $AB$ , where $A\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ and $B ^\\tau =B$ , coincides with ${\\mathrm {Mat}}(n,{\\mathbb {F}}_q)$ .",
"This proves statement 2).",
"Case $D_n$.",
"It is treated similarly.",
"Easy to verify that the linear subspace, spanned by the matrices of the form $AB$ , where $A\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ and $B^\\tau =-B$ , coincides with the subspace of matrices of the form (REF ).",
"This proves statement 3).",
"Case $B_n$.",
"For any two $n$ -columns $v_1$ and $v_2$ we consider the matrix $M(v_1,v_2) = \\begin{pmatrix} E&v_1&-\\frac{1}{2}v_1v_2^\\tau \\\\0&1& -v_2^\\tau \\\\ 0&0&E \\end{pmatrix}.$ The group ${H^\\circ }$ is generated by the matrices ${\\mathrm {diag}}(A,1,E)$ with $A\\in {\\mathrm {UT}}(n,{\\mathbb {F}}_q)$ , the matrices $M(v,v)$ , where $v$ is a $n$ -column, and $F(B) = \\begin{pmatrix}1&0&B\\\\0&1&0\\\\0&0&1\\end{pmatrix},$ where $B^\\tau =-B$ .",
"Analogically the case $D_n$ one can show that the matrices of the form $F(B)$ , where $B$ is a matrix (REF ), belong to ${H^\\circ }$ .",
"The equality $ \\begin{pmatrix}A&0&0\\\\ 0&1&0\\\\0&0&E\\end{pmatrix}\\cdot M(v,v)\\cdot \\begin{pmatrix}A^{-1}&0&0\\\\ 0&1&0\\\\0&0&E\\end{pmatrix}= M(Av, v)$ implies the subgroup ${H^\\circ }$ contains all matrices of the form $M(v_1,v_2)$ for all columns $v_1=\\begin{pmatrix}\\beta _1\\\\ \\vdots \\\\ \\beta _n\\end{pmatrix}, \\quad v_2=\\begin{pmatrix}\\beta _1^{\\prime }\\\\ \\vdots \\\\ \\beta _n^{\\prime }\\end{pmatrix}~~~ \\mbox{such~~ that}~~~ \\beta _n=\\beta _n^{\\prime }\\ne 0.",
"$ For any $n$ -columns $v_1,v_2,d_1,d_2$ we have an equality $M(v_1,v_2) M(d_1,d_2)= M(v_1+d_1,v_2+d_2) F(B_0),$ where $B_0=\\frac{1}{2}(-v_1d_2^\\tau + d_1v_2^\\tau )$ .",
"The equality (REF ) implies the subgroup ${H^\\circ }$ contains all matrices of the form $F(B)$ , where $B=(b_{ij})$ is an arbitrary $n\\times n$ matrix with $b_{n1}=0$ .",
"Finally, applying (REF ) one can verify ${H^\\circ }$ contains all matrices of the form $M(v_1,v_2)$ , where $\\beta _n=\\beta _n^{\\prime }$ .",
"This follows statement 1).",
"$\\Box $ Let us describe the equivalence classes in ${\\mathfrak {u}}$ and ${\\mathfrak {u}}^*$ (see Definitions REF and REF ).",
"Order the set of integers of the segment $[-n,n]$ as follows $1\\prec \\ldots \\prec n\\prec 0 \\prec -n\\prec \\ldots \\prec -1.$ Denote by $\\Delta ^+$ the set of following integer pairs from $[-n,n]$ : for $B_n$ $ {{\\Delta }^+}=\\lbrace (i,j):~~ 1\\leqslant i\\leqslant n,~~ i\\prec j\\prec -i\\rbrace ,$ for $C_n$ $ {{\\Delta }^+}=\\lbrace (i,j):~~ 1\\leqslant i\\leqslant n,~~ i\\prec j \\preccurlyeq -i,~~ j\\ne 0\\rbrace ,$ for $D_n$ $ {{\\Delta }^+}=\\lbrace (i,j):~~ 1\\leqslant i\\leqslant n,~~ i\\prec j\\prec -i,~~ j\\ne 0\\rbrace .$ We refer to elements from ${{\\Delta }^+}$ as positive roots, and to ${{\\Delta }^+}$ as the set of positive roots.",
"For any positive root ${\\alpha }=(i,j)\\in {{\\Delta }^+}$ , we call $i$ a row number (denote $i={\\mathrm {row}}({\\alpha })$ ) and $j$ a column number (denote $j={\\mathrm {col}}({\\alpha })$ ).",
"Definition 8.",
"The subset ${\\mathcal {D}}\\subset {{\\Delta }^+}$ is called basic if there is no more than one root from ${\\mathcal {D}}$ in each row and each column.",
"The other name is the set of rook placement type.",
"Definition 9.",
"We refer to a subset ${\\mathcal {D}}\\subset {{\\Delta }^+}$ as quasibasic if 1) there is no more than one root of ${\\mathcal {D}}$ in any column; 2) there is no more than one root of ${\\mathcal {D}}$ in any row except the cases of $B_n$ , $D_n$ and pairs of roots $(i,n)$ and $(i,-n)$ .",
"So, any quasibasic subset in $C_n$ is a basic subset.",
"For any positive root ${\\alpha }=(i,j)$ denote ${\\alpha }^{\\prime }=(-j,-i)$ (according to definition ${\\alpha }^{\\prime }$ is not a positive root).",
"For any matrix $x=(x_{\\alpha })\\in {\\mathfrak {u}}$ the entices $x_{\\alpha }$ and $x_{{\\alpha }^{\\prime }}$ differs by a sign $x_{\\alpha }=\\epsilon ({\\alpha })x_{{\\alpha }^{\\prime }}$ .",
"The Lie algebra ${\\mathfrak {g}}$ has the standard basis $\\lbrace E_{ij}:~~ 1\\leqslant i < j\\leqslant m\\rbrace $ .",
"The Lie algebra ${\\mathfrak {u}}$ also has the standard basis $\\lbrace {\\mathcal {E}}_{\\alpha }=E_{\\alpha }+\\epsilon ({\\alpha }) E_{{\\alpha }^{\\prime }}\\rbrace $ .",
"By a pair $({\\mathcal {D}},\\phi )$ , where ${\\mathcal {D}}$ is a quasibasic subset of ${{\\Delta }^+}$ and a map $\\phi : D\\rightarrow {\\mathbb {F}}_q^*$ , we define the element $ x_{{\\mathcal {D}},\\phi } = \\sum _{{\\alpha }\\in {\\mathcal {D}}} \\phi ({\\alpha }) {\\mathcal {E}}_{\\alpha }.$ Theorem 10.",
"1) Each element $x\\in {\\mathfrak {u}}$ is equivalent to some element $ x_{{\\mathcal {D}},\\phi }$ .",
"2) The pair $({\\mathcal {D}},\\phi )$ is uniquely determined by $x$ .",
"Proof.",
"1) Applying transformations $x\\rightarrow gxg^\\dag $ , $g\\in {G^\\circ }$ , we are able to obtain more zeros in the matrix $x$ , and finally get $ x_{{\\mathcal {D}},\\phi }$ .",
"2) Let us show that $({\\mathcal {D}},\\phi )$ is uniquely determined by $x$ .",
"Suppose that the elements $x_{{\\mathcal {D}},\\phi }$ and $x_{{\\mathcal {D}}^{\\prime },\\phi ^{\\prime }}$ are equivalent.",
"For any positive root $(i,j)$ , we consider the submatrix ${\\mathrm {Mat}}_{ij}(x)$ of $x$ with systems of rows and columns $\\lbrace k:~ i\\preccurlyeq k \\preccurlyeq j\\rbrace $ .",
"Easy to see that if $x\\sim x^{\\prime }$ , then the submatrices ${\\mathrm {Mat}}_{ij}(x)$ and ${\\mathrm {Mat}}_{ij}(x^{\\prime })$ have equal ranks.",
"Suppose that ${\\mathcal {D}}$ and ${\\mathcal {D}}^{\\prime }$ are basic subsets.",
"The equality of ranks implies that ${\\mathcal {D}}={\\mathcal {D}}^{\\prime }$ .",
"For each root ${\\alpha }=(i,j)\\in {\\mathcal {D}}$ , we consider the subset ${\\mathcal {D}}_{\\alpha }\\subset {\\mathcal {D}}$ that consists of $(k,m)\\in {\\mathcal {D}}$ , where $k\\succcurlyeq i$ and $m \\preccurlyeq j$ .",
"For ${\\alpha }\\in {\\mathcal {D}}$ we consider the minor $M_{{\\alpha }}$ of the matrix $x$ with systems of rows ${\\mathrm {row}}({\\mathcal {D}}_{\\alpha })$ and columns ${\\mathrm {col}}({\\mathcal {D}}_{\\alpha })$ .",
"It is not difficult to show that the equivalence of matrices $x_{{\\mathcal {D}},\\phi }$ and $x_{{\\mathcal {D}},\\phi ^{\\prime }}$ implies $M_{\\alpha }(x_{{\\mathcal {D}},\\phi }) = M_{\\alpha }(x_{{\\mathcal {D}},\\phi ^{\\prime }})$ .",
"Hence $\\phi =\\phi ^{\\prime }$ .",
"Assume that the quasibasic subset ${\\mathcal {D}}$ is not basic.",
"In this case, ${\\mathfrak {u}}$ is of the type $B_n$ or $D_n$ , and ${\\mathcal {D}}$ contains the pair of roots $\\beta _1=(i,n)$ and $\\beta _2=(i,-n)$ .",
"For $D_n$ the statement can be proves similarly the case of basic subset.",
"Consider the case of $B_n$ .",
"The equality of ranks of matrices ${\\mathrm {Mat}}_{ij}$ implies that ${\\mathcal {D}}\\setminus \\lbrace \\beta _2\\rbrace = {\\mathcal {D}}^{\\prime }\\setminus \\lbrace \\beta _2\\rbrace $ .",
"For ${\\alpha }\\in {\\mathcal {D}}\\setminus \\lbrace \\beta _2\\rbrace $ , we consider the system of roots ${\\mathcal {D}}_{\\alpha }\\in {\\mathcal {D}}$ that consists of $(k,m)\\in {\\mathcal {D}}\\setminus \\lbrace \\beta _2\\rbrace $ , where $k\\succcurlyeq i$ and $m \\preccurlyeq j$ .",
"By the subset ${\\mathcal {D}}_{\\alpha }$ we define the minor $M_{\\alpha }$ as above.",
"Since $M_{\\alpha }(x_{{\\mathcal {D}},\\phi }) = M_{\\alpha }(x_{{\\mathcal {D}},\\phi ^{\\prime }})$ , it follows $\\phi ({\\alpha })=\\phi ^{\\prime }({\\alpha })$ for each ${\\alpha }\\in {\\mathcal {D}}\\setminus \\lbrace \\beta _2\\rbrace $ .",
"It remains to show $\\beta _2\\in {\\mathcal {D}}^{\\prime }$ and $\\phi (\\beta _2)=\\phi ^{\\prime }(\\beta _2)$ .",
"For $\\beta _1$ , we construct the root system $\\widetilde{{\\mathcal {D}}}_{\\beta _1}$ that coincides with ${\\mathcal {D}}_{\\beta _1}$ if there is no root $(k,0)$ , $i<k\\leqslant n$ , in ${\\mathcal {D}}$ ; if such root exists, then $\\widetilde{{\\mathcal {D}}}_{\\beta _1} = {\\mathcal {D}}_{\\beta _1}\\cup {(k,0)}$ .",
"As above we define the minor $\\widetilde{M}_{\\beta _1}$ .",
"For $\\beta _0=(i,0)$ , we construct the system of roots $\\widetilde{{\\mathcal {D}}}_{\\beta _0}$ that coincides with $({\\mathcal {D}}_{\\beta _1}\\setminus \\lbrace \\beta _1\\rbrace )\\cup \\lbrace \\beta _0\\rbrace $ if there is no root $(k,0)$ , $i<k\\leqslant n$ , in ${\\mathcal {D}}$ ; if such root exists, then $\\widetilde{{\\mathcal {D}}}_{\\beta _0} = {\\mathcal {D}}_{\\beta _1}\\cup \\lbrace (k,-n)\\rbrace $ .",
"As above we define the minor $\\widetilde{M}_{\\beta _0}$ .",
"Take $\\widetilde{{\\mathcal {D}}}_{\\beta _2} = {\\mathcal {D}}_{\\beta _2}\\setminus \\lbrace \\beta _1\\rbrace $ and consider the corresponding minor $\\widetilde{M}_{\\beta _2}$ .",
"One can show that polynomial $I= \\widetilde{M}_{\\beta _1} \\widetilde{M}_{\\beta _2} + \\frac{1}{2}\\widetilde{M}^2_{\\beta _0}$ is constant on the equivalence class of element $x=x_{{\\mathcal {D}},\\phi }$ .",
"As $x\\sim x^{\\prime }$ , where $x^{\\prime }=x_{{\\mathcal {D}}^{\\prime },\\phi ^{\\prime }}$ , we have $I(x)=I(x^{\\prime })$ .",
"Observe that $\\widetilde{M}_{\\beta _0}(x)=\\widetilde{M}_{\\beta _0}(x^{\\prime })=0$ , and $\\widetilde{M}_{\\beta _1}(x) = \\widetilde{M}_{\\beta _1}(x^{\\prime }) \\ne 0$ .",
"Therefore, $\\widetilde{M}_{\\beta _2}(x) = \\widetilde{M}_{\\beta _2}(x^{\\prime }) \\ne 0$ .",
"Hence $\\beta _2\\in {\\mathcal {D}}^{\\prime }$ , the values of $\\phi $ and $\\phi ^{\\prime }$ on the root $\\beta _2$ coincide.",
"$\\Box $ The dual space ${\\mathfrak {g}}^*$ has the dual basis $\\lbrace E^*_{ij}:~~ 1\\leqslant i < j\\leqslant m\\rbrace $ .",
"The dual space ${\\mathfrak {u}}^*$ has the basis $\\lbrace E^*_{\\alpha }+\\epsilon ({\\alpha }) E^*_{{\\alpha }^{\\prime }}\\rbrace $ .",
"For $({\\mathcal {D}},\\phi )$ , we construct the element $ {\\lambda }_{{\\mathcal {D}},\\phi } = \\sum _{{\\alpha }\\in {\\mathcal {D}}} \\phi ({\\alpha }) \\left(E^*_{\\alpha }+\\epsilon ({\\alpha }) E^*_{{\\alpha }^{\\prime }}\\right).$ Theorem 11.",
"1) Each element ${\\lambda }\\in {\\mathfrak {u}}$ is equivalent to some element $ {\\lambda }_{{\\mathcal {D}},\\phi }$ .",
"2) The pair $({\\mathcal {D}},\\phi )$ is uniquely determined by ${\\lambda }$ .",
"Proof.",
"The proof is similar to the one of Theorem REF .",
"$\\Box $ Turn to the equivalence relation on the group $U$ .",
"We take the Springer map as a map $f$ in Definition REF .",
"Definition 12.",
"A map $f:G\\rightarrow {\\mathfrak {g}}$ is called Springer map if $f$ is a bijection and it obeys the following conditions: 1) $f(U)={\\mathfrak {u}}$ , 2) there exist $a_2,a_3,\\ldots $ from the field ${\\mathbb {F}}_q$ such that $f(1+x) = x+a_2x^2+a_3x^3+\\ldots $ for any $x\\in {\\mathfrak {g}}$ .",
"Examples of Springer map are as follows: 1) the logarithm map $\\ln (1+x) = \\sum _{i=1}^\\infty (-1)^{i+1}\\frac{x^i}{i}$ (it requires strong restrictions on the characteristic of field); 2) Cayley's map $f(1+x)=\\frac{2x}{x+2}$ (for $\\mathrm {char}\\,{\\mathbb {F}}_q\\ne 2)$ .",
"Denote $u_{{\\mathcal {D}},\\phi }=f^{-1}(x_{{\\mathcal {D}},\\phi })$ .",
"Classification of equivalence classes in group $U$ follows from Theorem REF .",
"Theorem 13.",
"1) Each element $u\\in U$ is equivalent to some $ u_{{\\mathcal {D}},\\phi }$ .",
"2) The pair $({\\mathcal {D}},\\phi )$ is uniquely determined by $u$ .",
"In the case $C_n$ and $D_n$ , the group ${G^\\circ }$ is an algebra group, and construction of supercharacters for the group $U$ doesn't differ from the approach of paper [8].",
"In the case $B_n$ , the group ${G^\\circ }$ is not an algebra group.",
"Consider the subgroup $S$ of all matrices $\\mathrm {diag}(E,M,E)$ , where $M$ is a matrix of the type (REF ).",
"The group ${G^\\circ }$ is a semidirect product ${G^\\circ }=S{G^\\diamond }$ , where ${G^\\diamond }$ is a subgroup of ${G^\\circ }$ that consists of matrices of the form (REF ) with $A_3=E$ .",
"The subgroup ${G^\\diamond }$ is an algebra group ${G^\\diamond }=1+{{\\mathfrak {g}}^\\diamond }$ , where ${{\\mathfrak {g}}^\\diamond }$ is the Lie subalgebra of the group ${G^\\diamond }$ .",
"The group $U$ is also a semidirect product $U=S{U^\\diamond }$ , where ${U^\\diamond }=U\\cap {G^\\diamond }$ .",
"The Lie algebra ${\\mathfrak {u}}$ decomposes ${\\mathfrak {u}}={\\mathfrak {s}}\\oplus {{\\mathfrak {u}}^\\diamond }$ , where ${\\mathfrak {s}}$ is the one dimensional Lie algebra spanned by the vector ${\\mathcal {E}}_{n,0}$ .",
"Take ${H^\\diamond }={G^\\diamond }\\cap {H^\\circ }$ .",
"The subgroup ${H^\\diamond }$ is an algebra group and ${H^\\circ }=S{H^\\diamond }$ .",
"Easy to see that in each case $B_n,~C_n,~D_n$ we have ${G^\\circ }=U{H^\\diamond }$ (see [8]).",
"For any linear form $\\eta \\in ({{\\mathfrak {g}}^\\diamond })^*$ , we define the following associative subalgebras in ${{\\mathfrak {g}}^\\diamond }$ : 1) $r_\\eta ^\\diamond =\\lbrace x\\in {{\\mathfrak {g}}^\\diamond }:~ \\eta (xy)=0~ ~\\mbox{for~~any}~~ y\\in {{\\mathfrak {h}}^\\diamond }\\rbrace $ ; 2) $\\ell _\\eta ^\\diamond =\\lbrace x\\in {{\\mathfrak {g}}^\\diamond }:~ \\eta (y^\\dag x)=0~ ~\\mbox{for~~any}~~ y\\in {{\\mathfrak {h}}^\\diamond }\\rbrace $ ; 3) ${\\mathfrak {g}}_\\eta ^\\diamond = r_\\eta ^\\diamond \\cap \\ell _\\eta ^\\diamond $ .",
"Since $(r_\\eta ^\\diamond )\\dag = \\ell _\\eta ^\\diamond $ , we have $({\\mathfrak {g}}_\\eta ^\\diamond )^\\dag = {\\mathfrak {g}}_\\eta ^\\diamond $ .",
"The subgroup ${G^\\circ }$ contains the algebra subgroups $R_\\eta ^\\diamond =1+r_\\eta ^\\diamond $ , $L_\\eta ^\\diamond =1+\\ell _\\eta ^\\diamond $ and $G_\\eta ^\\diamond =1+{\\mathfrak {g}}_\\eta ^\\diamond $ .",
"Lemma 14 [8].",
"$\\eta (xy)=0$ for any $x,~y\\in {\\mathfrak {g}}_\\eta ^\\diamond $ .",
"Proof.",
"Present $x$ in the form $x=x^{\\prime }+x^{\\prime \\prime }$ , where $x^{\\prime }_{ij}=0$ for $1\\preccurlyeq j\\preccurlyeq 0$ , and $x^{\\prime \\prime }_{ij}=0$ for $0\\prec j\\preccurlyeq -1$ .",
"Present $y$ in the form $y=y^{\\prime }+y^{\\prime \\prime }$ , where $y^{\\prime }_{ij}=0$ for $0\\prec i\\preccurlyeq -1$ , and $y^{\\prime \\prime }_{ij}=0$ for $1\\preccurlyeq i \\preccurlyeq 0$ .",
"Then $x^{\\prime }\\in ({{\\mathfrak {h}}^\\diamond })^\\dag $ , $y^{\\prime }\\in {{\\mathfrak {h}}^\\diamond }$ , $x^{\\prime }y^{\\prime }=0$ , $x^{\\prime \\prime }y^{\\prime \\prime }=0$ .",
"The equality $xy=x^{\\prime }y+xy^{\\prime }$ implies $\\eta (xy)=0$ .",
"$\\Box $ Let ${\\lambda }\\in ({{\\mathfrak {u}}^\\diamond })^*$ and $\\eta \\in {{\\mathfrak {g}}^\\diamond }$ be such that $\\eta ^\\dag =-\\eta $ and the restriction of $\\eta $ on ${{\\mathfrak {u}}^\\diamond }$ coincides with ${\\lambda }$ .",
"Define ${\\mathfrak {u}}_{\\lambda }^\\diamond = {\\mathfrak {u}}\\cap {\\mathfrak {g}}_\\eta ^\\diamond ~~\\mbox{and}~~U_{\\lambda }^\\diamond = U\\cap G_\\eta ^\\diamond .$ For $x\\in {{\\mathfrak {u}}^\\diamond }$ , we have $\\eta (y^\\dag x) =-\\eta ^\\dag (y^\\dag x) = -\\eta ((y^\\dag x)^\\dag ) = -\\eta (x^\\dag y)=\\eta (xy).$ It follows $ {\\mathfrak {u}}\\cap r_\\eta ^\\diamond = {\\mathfrak {u}}\\cap \\ell _\\eta ^\\diamond = {\\mathfrak {u}}_{\\lambda }^\\diamond $ .",
"By Lemma REF , the restriction of ${\\lambda }$ on the Lie algebra ${\\mathfrak {u}}_{\\lambda }^\\diamond $ is its character with values in the field ${\\mathbb {F}}_q$ .",
"Let $\\pi ^\\diamond $ be the natural projection $({{\\mathfrak {u}}^\\diamond })^*\\rightarrow ({\\mathfrak {u}}_{\\lambda }^\\diamond )^*$ .",
"The following statement follows from the paper [8].",
"For readers convenience we present it with complete proof.",
"Lemma 15.",
"For any ${\\lambda }\\in ({{\\mathfrak {u}}^\\diamond })^*$ the fiber $(\\pi ^\\diamond )^{-1} \\pi ^\\diamond ({\\lambda })$ equals to $ {H^\\diamond }\\centerdot {\\lambda }$ .",
"Proof.",
"Item 1.",
"Let $\\eta \\in ({{\\mathfrak {g}}^\\diamond })^*$ and $P$ is the natural projection $({{\\mathfrak {g}}^\\diamond })^*\\rightarrow (r_\\eta ^\\diamond )^*$ .",
"Let us show that $P^{-1}P(\\eta )={H^\\diamond }\\eta $ (here $h\\eta (x)=\\eta (xh)$ is the left action $h\\in {H^\\diamond }$ on $\\eta $ ).",
"Indeed, $P^{-1}P(\\eta )=\\eta + (r_\\eta ^\\diamond )^\\perp $ .",
"The definition of $r_\\eta ^\\diamond $ implies $r_\\eta ^\\diamond =\\lbrace x\\in {{\\mathfrak {g}}^\\diamond }:~~ y\\eta (x)=0 ~ \\mbox{for~any}~ y\\in {{\\mathfrak {h}}^\\diamond }\\rbrace = ({{\\mathfrak {h}}^\\diamond }\\eta )^\\perp .$ Hence $(r_\\eta ^\\diamond )^\\perp ={{\\mathfrak {h}}^\\diamond }\\eta $ and $P^{-1}P(\\eta )=\\eta + {{\\mathfrak {h}}^\\diamond }\\eta = (1+{{\\mathfrak {h}}^\\diamond })\\eta = {H^\\diamond }\\eta $ .",
"Item 2.",
"Let $\\Pi $ be the natural projection $({{\\mathfrak {g}}^\\diamond })^*\\rightarrow ({{\\mathfrak {u}}^\\diamond })^*$ .",
"Let $\\eta ^\\dag =-\\eta $ and ${\\lambda }= \\Pi (\\eta )$ .",
"Let us show that $\\Pi ({H^\\diamond }\\eta )={H^\\diamond }\\centerdot {\\lambda }$ .",
"Really, for $h=1+y\\in {H^\\diamond }$ and $x\\in {{\\mathfrak {u}}^\\diamond }$ , we obtain $h\\centerdot {\\lambda }(x) = (h{\\lambda }h^\\dag )(x) = {\\lambda }(h^\\dag xh) ={\\lambda }((1+y)^\\dag x(1+y)) = {\\lambda }(x) + {\\lambda }(y^\\dag x+xy) + {\\lambda }(y^\\dag x y).",
"$ Observe that $y^\\dag x y=0$ for any $x\\in {\\mathfrak {g}}$ and $y\\in {{\\mathfrak {h}}^\\diamond }$ .",
"Applying the equality (REF ) we obtain $h\\centerdot {\\lambda }(x) = \\eta (x)+2\\eta (xy) = (1+2y)\\eta (x).$ Hence ${H^\\diamond }\\centerdot {\\lambda }= \\Pi ({H^\\diamond }\\eta )$ ; this proves statement 2.",
"Item 3.",
"Since ${\\mathfrak {u}}_{\\lambda }^\\diamond = {{\\mathfrak {u}}^\\diamond }\\cap r_{\\lambda }^\\diamond $ , we have $ (\\pi ^\\diamond )^{-1} \\pi ^\\diamond ({\\lambda }) = \\Pi (P^{-1}P(\\eta )) = \\Pi ({H^\\diamond }\\eta ) = {H^\\diamond }\\centerdot {\\lambda }.",
"~~\\Box $ Let ${\\lambda }={\\lambda }_{{\\mathcal {D}},\\phi }$ and $\\eta =\\eta _{{\\mathcal {D}},\\phi }$ is the element $({\\mathfrak {g}})^*$ such that $\\Pi (\\eta )={\\lambda }$ and $\\eta ^\\dag =-\\eta $ .",
"Lemma 16.",
"Let ${\\lambda }$ and $\\eta $ are defined by ${\\mathcal {D}},\\phi $ as above.",
"Then ${\\mathfrak {s}}{\\mathfrak {g}}_{\\lambda }^\\diamond \\subset {\\mathfrak {g}}_{\\lambda }^\\diamond $ and ${\\mathfrak {g}}_{\\lambda }^\\diamond {\\mathfrak {s}}\\subset {\\mathfrak {g}}_{\\lambda }^\\diamond $ .",
"Proof.",
"Let $\\eta \\in ({{\\mathfrak {g}}^\\diamond })^*$ and $\\Pi (\\eta )={\\lambda }$ .",
"For each ${\\alpha }=(i,j)\\in {{\\Delta }^+}$ , where $1\\leqslant i<n$ and $i\\prec j$ , we define the subalgebra $r_{\\alpha }^\\diamond $ that consists of matrices $x\\in {{\\mathfrak {g}}^\\diamond }$ obeying the conditions: 1) if $1\\leqslant j<n$ , then $x_{ik}=0$ for all $i<k<j$ ; 2) if $j\\in \\lbrace n,0,-n\\rbrace $ , then $x_{ik}=0$ for all $i<k<n$ ; 3) if $ -n<j\\leqslant -1$ , then $x_{ik}=0$ for all $i\\prec k \\prec -n$ , and $x_{-j,k}=0$ for all $-j\\prec k \\prec -n$ .",
"Easy to see that the subalgebra $r_{\\alpha }^\\diamond $ is invariant with respect to the left and right multiplication by ${\\mathfrak {s}}$ .",
"The subalgebra $r_\\eta ^\\diamond $ coincides with intersection of subalgebras $r_{\\alpha }^\\diamond $ over all ${\\alpha }\\in {\\mathcal {D}}$ .",
"The subalgebra $r_\\eta ^\\diamond $ is also invariant with respect to the left and right multiplication by ${\\mathfrak {s}}$ .",
"$\\Box $ Define the subalgebra ${\\mathfrak {u}}_{\\lambda }$ in each of the following cases separately.",
"1) ${\\mathcal {D}}$ doesn't contain any roots of the form $(i,0)$ and $(i,n)$ .",
"In this case, we take ${\\mathfrak {g}}_{\\lambda }={\\mathfrak {s}}\\oplus {\\mathfrak {g}}_{\\lambda }^\\diamond $ and $G_{\\lambda }=SG_{\\lambda }^\\diamond $ .",
"Denote ${\\mathfrak {u}}_{\\lambda }= {\\mathfrak {g}}_{\\lambda }\\cap {\\mathfrak {u}}= {\\mathfrak {s}}\\oplus {\\mathfrak {u}}_{\\lambda }^\\diamond $ .",
"Take $U_{\\lambda }= G_{\\lambda }\\cap U =SU_{\\lambda }^\\diamond $ .",
"2) ${\\mathcal {D}}$ contains $(i,0)$ or $(i,n)$ .",
"In this case, we define ${\\mathfrak {g}}_{\\lambda }={\\mathfrak {g}}_{\\lambda }^\\diamond $ , ${\\mathfrak {u}}_{\\lambda }={\\mathfrak {u}}_{\\lambda }^\\diamond $ , and $G_{\\lambda }=U_{\\lambda }^\\diamond $ .",
"The associative subalgebra ${{\\mathfrak {h}}^\\diamond }$ is an ideal in the associative algebra ${\\mathfrak {g}}$ .",
"Therefore, ${\\mathfrak {h}}_{\\lambda }={\\mathfrak {g}}_{\\lambda }+ {\\mathfrak {h}}^\\diamond $ is its associative subalgebra in ${\\mathfrak {g}}$ .",
"Then the subgroup $H_{\\lambda }=G_{\\lambda }H = 1+{\\mathfrak {h}}_{\\lambda }$ is an algebra group in $G$ .",
"Let $\\pi $ be the natural projection ${\\mathfrak {u}}^*\\rightarrow {\\mathfrak {u}}_{\\lambda }^*$ .",
"Lemma 17.",
"1) For any ${\\lambda }={\\lambda }_{{\\mathcal {D}},\\phi }$ , the fiber $\\pi ^{-1}\\pi ({\\lambda })$ coincides with $ H_{\\lambda }\\centerdot {\\lambda }$ .",
"2) The formula $\\xi _{\\lambda }(u)={\\varepsilon }^{{\\lambda }(f(u))}$ defines a character of the subgroup $U_{\\lambda }$ .",
"Proof.",
"Item 1.",
"${\\mathcal {D}}$ doesn't contain any roots of the form $(i,0)$ and $(i,n)$ .",
"Then $\\eta ({\\mathfrak {s}}x)=\\eta (x{\\mathfrak {s}})=0$ for any $x\\in {\\mathfrak {g}}$ .",
"From Lemma REF it follows that $\\xi _{\\lambda }(g)={\\varepsilon }^{{\\lambda }(f(g))}$ is a character of $G_{\\lambda }$ ; this proves statement 2).",
"By direct calculations, we obtain $h\\eta h^\\dag (x)=\\eta (x)$ for any $x\\in {\\mathfrak {u}}_{\\lambda }$ and $h\\in H_{\\lambda }$ .",
"The Lemma REF implies statement 1).",
"Item 2.",
"${\\mathcal {D}}$ contains one of roots of type $(i,0)$ or $(i,n)$ .",
"Denote this root from ${\\mathcal {D}}$ by $\\gamma $ .",
"If $\\gamma = (i,0)$ , then we take $z=E_{-n,-i} $ ; if $\\gamma =(i,-n)$ , then we take $z=E_{0,-i}$ .",
"In each case $z\\in {\\mathfrak {g}}_{\\lambda }\\subset {\\mathfrak {h}}_{\\lambda }$ and $\\eta ({\\mathcal {E}}_{n0}z)=\\pm \\phi (\\gamma )\\ne 0$ .",
"The element $h_t=1+tz$ belongs to $H_{\\lambda }$ for any $t\\in {\\mathbb {F}}_q$ .",
"By direct calculations, we obtain $h_t\\eta h^\\dag _t(x)=\\eta (x)$ for any $x\\in {\\mathfrak {u}}^\\diamond $ , and $h_t\\eta h^\\dag _t ({\\mathcal {E}}_{n0}) = \\eta ({\\mathcal {E}}_{n0})\\pm 2t\\phi (\\gamma )$ .",
"It follows $\\Pi ^{-1}\\Pi (\\eta )$ is contained in $H_{\\lambda }\\centerdot {\\lambda }$ .",
"The Lemma REF implies statement 1).",
"Statement 2) follows from Lemma REF .",
"$\\Box $ By the linear form ${\\lambda }={\\lambda }_{D,\\phi }$ , we define a linear character of the subgroup $U_{\\lambda }$ as follows $\\xi _{\\lambda }(u)={\\varepsilon }^{{\\lambda }(f(u))}.",
"$ Consider the induced character $\\chi _{\\lambda }= {\\mathrm {Ind}}(\\xi _{\\lambda },U_{\\lambda }, U).$ Theorem 18.",
"If ${\\lambda }={\\lambda }_{D,\\phi }$ , then $\\chi _{\\lambda }(u) = \\frac{|H_{\\lambda }\\centerdot {\\lambda }|}{|{G^\\circ }\\centerdot {\\lambda }|}\\sum _{\\mu \\in {G^\\circ }\\centerdot {\\lambda }} {\\varepsilon }^{\\mu (f(u))}.$ Proof.",
"Let $\\dot{\\xi _{\\lambda }}$ define the function on the group $U$ equal to $\\xi _{\\lambda }$ on $U_{\\lambda }$ and zero outside of $U_{\\lambda }$ .",
"By definition $\\chi _{\\lambda }(u)=\\frac{1}{|U_{\\lambda }|}\\sum _{v\\in U} \\dot{\\xi _{\\lambda }}(vuv^{-1}).$ Applying Lemma REF for all $x\\in {\\mathfrak {u}}$ , we have $\\sum _{\\mu \\in \\pi ^{-1}\\pi ({\\lambda })} {\\varepsilon }^{\\mu (x)} = {\\varepsilon }^{{\\lambda }(x)}\\cdot \\sum _{\\nu \\in {\\mathfrak {u}}_{\\lambda }^\\perp } {\\varepsilon }^{\\nu (x)} =\\left\\lbrace \\begin{array}{l} \\frac{|U|}{|U_{\\lambda }|}\\cdot {\\varepsilon }^{{\\lambda }(x)}, ~~\\mbox{if}~~ x\\in {\\mathfrak {u}}_{\\lambda };\\\\0, ~~\\mbox{if}~~ x\\notin {\\mathfrak {u}}_{\\lambda }.\\end{array}\\right.$ Hence $\\dot{\\xi }_{\\lambda }(u)=\\frac{|U_{\\lambda }|}{|U|}\\sum _{\\mu \\in \\pi ^{-1}\\pi ({\\lambda })} {\\varepsilon }^{\\mu (f(u))} = \\frac{|U_{\\lambda }|}{|U|}\\sum _{\\mu \\in H_{\\lambda }\\centerdot {\\lambda }} {\\varepsilon }^{\\mu (f(u))}.$ Then $\\chi _{\\lambda }(u)=\\frac{1}{|U|}\\cdot \\sum _{\\mu \\in H_{\\lambda }\\centerdot {\\lambda },~v\\in U} {\\varepsilon }^{\\mu (vf(u)v^{-1})} = \\frac{|H_{\\lambda }\\centerdot {\\lambda }|}{|U|\\cdot |H_{\\lambda }|}\\sum _{h\\in H_{\\lambda },~ v\\in U}{\\varepsilon }^{h\\centerdot {\\lambda }(vf(u)v^{-1})} =\\\\\\frac{|H_{\\lambda }\\centerdot {\\lambda }|}{|U|\\cdot |H_{\\lambda }|}\\sum _{h\\in H_{\\lambda },~ v\\in U}{\\varepsilon }^{{\\lambda }(hvf(u)v^\\dag h^\\dag )}.$ Since ${G^\\circ }=UH_{\\lambda }$ , we get $\\chi _{\\lambda }(u) = \\frac{|H_{\\lambda }\\centerdot {\\lambda }|\\cdot |U\\cap H_{\\lambda }|}{|U|\\cdot |H_{\\lambda }|}\\sum _{g\\in {G^\\circ }}{\\varepsilon }^{{\\lambda }(gf(u)g^\\dag )} = \\frac{|H_{\\lambda }\\centerdot {\\lambda }|}{|{G^\\circ }|}\\sum _{g\\in {G^\\circ }}{\\varepsilon }^{{\\lambda }(gf(u)g^\\dag )} = \\\\ \\frac{|H_{\\lambda }\\centerdot {\\lambda }|}{|{G^\\circ }\\centerdot {\\lambda }|}\\sum _{\\mu \\in {G^\\circ }\\centerdot {\\lambda }} {\\varepsilon }^{\\mu (f(u))}.",
"~~\\Box $ Theorem 19.",
"Let $U$ be the Sylow subgroup in orthogonal or symplectic group.",
"The system of characters $\\lbrace \\chi _{\\lambda }\\rbrace $ , and the partition of the group $U$ into classes $\\lbrace K(u)\\rbrace $ , where ${\\lambda }$ and $u$ run through the sets of representatives of equivalence classes ${\\lambda }_{{\\mathcal {D}},\\phi }\\in {\\mathfrak {u}}^*$ and $u_{{\\mathcal {D}},\\phi }\\in U$ , give rise to a supercharacter theory of the group $U$ .",
"Proof.",
"The proof follows from Remark REF , Theorems REF and REF .",
"$\\Box $"
]
] | 1808.08539 |
[
[
"Fusion++: Volumetric Object-Level SLAM"
],
[
"Abstract We propose an online object-level SLAM system which builds a persistent and accurate 3D graph map of arbitrary reconstructed objects.",
"As an RGB-D camera browses a cluttered indoor scene, Mask-RCNN instance segmentations are used to initialise compact per-object Truncated Signed Distance Function (TSDF) reconstructions with object size-dependent resolutions and a novel 3D foreground mask.",
"Reconstructed objects are stored in an optimisable 6DoF pose graph which is our only persistent map representation.",
"Objects are incrementally refined via depth fusion, and are used for tracking, relocalisation and loop closure detection.",
"Loop closures cause adjustments in the relative pose estimates of object instances, but no intra-object warping.",
"Each object also carries semantic information which is refined over time and an existence probability to account for spurious instance predictions.",
"We demonstrate our approach on a hand-held RGB-D sequence from a cluttered office scene with a large number and variety of object instances, highlighting how the system closes loops and makes good use of existing objects on repeated loops.",
"We quantitatively evaluate the trajectory error of our system against a baseline approach on the RGB-D SLAM benchmark, and qualitatively compare reconstruction quality of discovered objects on the YCB video dataset.",
"Performance evaluation shows our approach is highly memory efficient and runs online at 4-8Hz (excluding relocalisation) despite not being optimised at the software level."
],
[
"Introduction",
"Indoor scene understanding and 3D mapping is a foundational technology that can enable autonomous real-world robotic task completion and also provide a common interface for more intelligent and intuitive human-map and human-robot interactions.",
"To enable this requires a careful choice of map representation.",
"One particularly useful representation is to build an object-oriented map.",
"We argue this is a natural and efficient way to represent the things that are most important for robotic scene understanding, planning and interaction; and it is also highly suitable as the basis for human-robot communication.",
"In an object level map, the geometric elements which make up an object are grouped together as instances and can be labelled and reasoned about as units, in contrast to approaches which independently label dense geometry such as surfels or points.",
"This approach also naturally paves the way towards interaction and dynamic object reasoning, although our system currently assumes a static environment and does not yet aim to track individual dynamic objects.",
"In this work we demonstrate an object-oriented online SLAM system with a focus on indoor scene understanding using RGB-D data.",
"We aim to produce semantically labelled TSDF reconstructions of object instances without strong a priori knowledge of the object types present in a scene.",
"We use Mask R-CNN [13], [40] to provide 2D instance mask predictions and fuse these masks online into the TSDF reconstruction (see Figure REF ) along with a 3D `voxel mask' to fuse the instance foreground (see Figure REF ).",
"Unlike many dense reconstruction systems [24], [39], [43], [38], [3], [8] we make no attempt to keep a dense representation of the entire scene.",
"Our persistent map consists of only reconstructed object instances.",
"This allows the use of rigid TSDF volumes for high-quality reconstructions to be combined with the flexibility of a pose-graph system without the complication of performing intra-TSDF deformations.",
"Each object is contained within a separate volume, allowing each one to have a different, suitable, resolution with larger objects integrated into lower fidelity TSDF volumes than their smaller counterparts.",
"It also enables tracking large scenes with relatively small memory usage and high-fidelity reconstructions by excluding large volumes of free-space.",
"A throw-away local TSDF of unidentified structure is used to assist tracking and model occlusions.",
"We capture a repeated loop of an indoor office scene to evaluate the system under conditions of occasional poorly constrained ICP tracking.",
"The scene also contains a large number and variety of objects which not only exhibit the generality of the approach but is useful for evaluating the memory and run-time scaling of the method with many objects.",
"While not optimised for real-time operation, we achieve $\\sim $ 4-8Hz operating performance (excluding relocalisation/graph optimisation) on our office sequence and are confident that with sufficient optimisation true real-time operation is possible.",
"We also quantitatively evaluate the trajectory error improvement of our system over a baseline approach on the RGB-D SLAM Benchmark [33].",
"In this work we make the following contributions: A generic object-oriented SLAM system which performs mapping as variable resolution 3D instance reconstruction.",
"Per-frame instance detections are robustly fused using voxel foreground masks and missing detections are accounted for with an “existence” probability.",
"We show high quality object reconstruction within globally consistent loop-closed object SLAM maps."
],
[
"Related work",
"For reconstruction, we follow the TSDF formulation of Curless and Levoy [6] and the KinectFusion approach of Newcombe et al.",
"[23] for local tracking.",
"Our approach to object-level reconstruction is related to the work of Zhou and Koltun [42], where “points of interest” were detected and the aim was to reconstruct the scene so as to preserve detail in these areas while distributing drift and registration errors throughout the rest of the environment.",
"In our work we analogously aim to optimise the quality of object reconstructions and allow residual error to be absorbed in the edges of the pose graph.",
"SLAM++ by Salas-Moreno et al.",
"[30] was an early RGB-D object-oriented mapping system.",
"They used point pair features for object detection and a pose graph for global optimisation.",
"The drawback was the requirement that the full set of object instances, with their very detailed geometric shapes, had to be known beforehand and pre-processed in an offline stage before running.",
"Stückler and Behnke [31] also previously tracked object models learned beforehand by registering them to a multi-resolution surfel map.",
"Tateno et al.",
"[35] used a pre-trained database of objects to generate descriptors, but they used a KinectFusion [23] TSDF to incrementally segment regions of a reconstructed TSDF volume and match 3D descriptors directly against those of other objects in the database.",
"A number of approaches to object discovery exist [5], [32], [4].",
"Most related to ours is the work of Choudhary et al.",
"[4] where they localised the camera in an online manner using discovered objects as landmarks in a pose-graph formulation similar to ours, although they used the point cloud centroid only whereas our pose-graph object landmark edges are full 6 DoF $SE(3)$ constraints provided from ICP on dense volumes.",
"They showed that the approach improves SLAM results by detecting loop closures.",
"However, unlike our work they use point-clouds rather than TSDFs and do not train an object detector but instead they use the unsupervised segmentation approach of Trevor et al. [36].",
"Another approach to object discovery is through dense change detection between successive mappings of the same scene [12], [19], [11].",
"Unlike these systems, our system is designed for online use and does not require changes to occur in a scene before objects are detected.",
"These approaches are complementary to our proposed approach, providing supervisory signals for CNN fine-tuning, and enabling additional object database filtering mechanisms.",
"In RGB-only SLAM for object detection, Pillai and Leonard [26] use ORB-SLAM [21] to assist object recognition.",
"They use a semi-dense map to produce object proposals and aggregate detection evidence across multiple views for object detection and classification.",
"MO-SLAM by Dharmasiri et al.",
"[9] focused on object discovery through duplicates.",
"They use ORB [28] descriptors to search for sets of landmarks which can be grouped by a single rigid body transformation.",
"This approach is similar to our relocalisation method, which uses BRISK features [18] but augmented with depth.",
"Very closely related to ours is work by Sünderhauf et al.",
"[34], who proposed an object-oriented mapping system composed of instances using bounding box detections from a CNN and an unsupervised geometric segmentation algorithm using RGB-D data.",
"Although the premise is closely related, there are a number of differences when compared to our system.",
"They use a separate SLAM system, ORB-SLAM2 [22], whereas in our system the discovered object instances are tightly integrated into the SLAM system itself.",
"We also fuse instances into separate TSDF volumes with a foreground mask from 2D instance mask detection rather than using point cloud segments.",
"A number of very recent related works have also been announced.",
"Pham et al.",
"[25] fuse a TSDF of the entire scene and semantically label voxels using a CNN followed by a progressive CRF.",
"To segment instances, instead of fusing native instance detections, they opt to cluster semantically labelled voxels in 3D.",
"This approach, although a natural next-step from dense 3D semantic mapping, is not suitable for object-level pose graph optimisation and reconstruction as the instances are embedded within a shared TSDF.",
"It also requires semantic recognition as a pre-requisite for object discovery which could prove problematic for similar or unrecognised objects in close proximity (Figure REF ).",
"Rünz and Agapito [29], as in our method, use Mask R-CNN predictions to detect object instances.",
"They aim to densely reconstruct and track moving instances using an ElasticFusion [38] surfel model for each object, as well as for the background static map.",
"Although using the same prediction model, the approach and goals of these two systems differ substantially.",
"Unlike the present work, they do not aim to reconstruct high-quality objects as pose-graph landmarks in room-scale SLAM.",
"We on the other hand do not currently tackle dynamic scenes and assume all objects to be static during an observation.",
"Clearly there is the long-term potential to combine these two approaches."
],
[
"Method",
"Our pipeline is visualised in Figure REF .",
"From RGB-D input, a coarse background TSDF is initialised for local tracking and occlusion handling (Section REF ).",
"If the pose changes sufficiently or the system appears lost, relocalisation (Section REF ) and graph optimisation (Section REF ) are performed to arrive at a new camera location, and the coarse TSDF is reset.",
"In a separate thread RGB frames are processed by Mask R-CNN and the detections are filtered and matched to the existing map (Section REF ).",
"When no match occurs, new TSDF object instances are created, sized, and added to the map for local tracking, global graph optimisation, and relocalisation.",
"On future frames, associated foreground detections are fused into the object's 3D `foreground' mask alongside semantic and existence probabilities (Section REF ).",
"Figure: Overview of the Fusion++ system."
],
[
"TSDF Object Instances",
"Our map is composed of object instances reconstructed within separate TSDFs, $\\mathcal {V}^o$ , each with a pose defined by a transformation, $\\mathbf {T}_{WO}\\in SE(3)$ , which maps coordinates of a point $_{O}\\mathbf {p}\\in \\mathbb {R}^3$ from object frame ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{O}}}$ to coordinates $_{W}\\mathbf {p}\\in \\mathbb {R}^3$ in World frame ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{W}}}$ .",
"For convenience of notation, homogeneous coordinates are assumed where appropriate (e.g.",
"in transformations), however when explicitly required they are denoted with italics, $_{O}{\\mathbfit {p}}= [_{O}\\mathbf {p}^\\intercal , 1]^\\intercal $ .",
"Object instance frames have an origin at the centre of the volume and are sized cubically with an edge-length, $s_o$ .",
"Initialisation and resizing: Detections not matched by the procedure described in REF are used to initialize an appropriately sized and positioned instance TSDF.",
"In the $k$th frame each detection $i$ produces a binary mask $M_i^k$ .",
"We project all the masked image coordinates $\\mathbf {u}=(u_1,u_2)$ into ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{W}}}$ using the depth map $D_k(\\mathbf {u})$ , $_{W}\\mathbf {p} = \\tilde{\\mathbf {T}}_{WC}^k \\mathbf {K}^{-1} D_k(\\mathbf {u}){\\mathbfit {u}}~,$ where $\\mathbf {K}$ denotes the $3\\times 3$ intrinsic camera matrix, $\\tilde{\\mathbf {T}}_{WC}^k\\in SE(3)$ the camera pose estimate.",
"To robustly size the TSDF in the presence of masks which can occasionally include far-away background surfaces, we do not directly accept the maximum and minimum of this point cloud.",
"Instead we use the 10th and 90th percentiles of this point cloud (separately for each axis) to define points $\\mathbf {p}_{10}$ and $\\mathbf {p}_{90}$ respectively, which are used to calculate the volume centre $\\mathbf {p}_o=\\frac{\\mathbf {p}_{90}+\\mathbf {p}_{10}}{2}$ and volume size $s_o=m\\Vert (\\mathbf {p}_{90}-\\mathbf {p}_{10})\\Vert _\\infty $ .",
"We use an $m$ of 1.5 to account for erosion and provide additional padding.",
"Each instance TSDF has an initial fixed resolution along a given axis of $r_o$ , which we choose to be 64, and $s_o$ is used to calculate the physical size of a voxel $v_o=\\frac{s_o}{r_o}$ .",
"Therefore, small objects will be reconstructed with fine details and large objects more coarsely, making the map as useful as possible for a given memory footprint.",
"During operation matched objects may need to be re-sized as new detections include additional areas.",
"To do this, the point cloud of the current mask described above is combined with a similarly eroded point cloud generated from the current TSDF reconstruction.",
"The 3D volume encompassing them both is used to calculate the new volume centre and size as before.",
"To avoid aliasing when re-sizing, we translate the volume centre by discrete multiples of $v_o$ , and maintain the same $v_o$ but increase $r_o$ , while maintaining an even parity.",
"We also limit the maximum voxel resolution to 128, by re-initialising the volume as though new if $r_o>128$ , and limit the maximum object size to be 3m.",
"Before initialising an instance we require the volume centre to be within 5m of the camera, and a 3D axis-aligned bounding box Intersection over Union (IoU) $< 0.5$ with any other volume already in the map.",
"When an object centre is moved, the pose-graph node and associated measurements are also updated as described in Section REF .",
"Integration: For integrating surface measurements from a depth map $D^k$ into $\\mathcal {V}^o$ we take an approach similar to Newcombe et al.",
"[23]Code based on https://github.com/GerhardR/kfusion.. $\\mathcal {V}^o$ stores at each discrete voxel location $\\mathbf {v}=(v_x,v_y,v_z)$ both the current normalised truncated signed distance value $S^o_{k-1}(\\mathbf {v})$ and its associated weight $W^o_{k-1}(\\mathbf {v})$ .",
"If $\\mathbf {v}$ projects into a camera frame pixel with a depth value less than the depth measurement plus the truncation distance, $\\mu $ (here chosen as $4v_o$ ), then that measurement is fused into the volume in a weighted average fashion.",
"Integration is performed on every frame where the TSDF volume is visible, when 50% of TSDF pixels are validly tracked and the ICP RMSE $< 0.03$ (these error metrics are described in more detail in Section REF ).",
"This is to maintain the reconstruction quality of instances when the camera frame may have drifted.",
"It is also important to note that the above surface integration is performed throughout the entire volume, regardless of whether it is a masked region or not.",
"To store which voxels correspond to this instance's `foreground' we also fuse instance mask detections.",
"We view each positive or negative detection as the result of a binomial trial sampled from a latent foreground probability, $p^o(\\mathbf {v} \\in \\text{foreground})$ .",
"We store foreground $F^o_{k-1}(\\mathbf {v})$ and not foreground $N^o_{k-1}(\\mathbf {v})$ detection counts as the $(\\alpha ,\\beta )$ shape parameters in a beta distribution conjugate prior which are initialised with $(1,1)$ .",
"When a new detection is matched and the depth measurement is within the truncation distance as above, then we also update the detection counts using the corresponding mask $i$ : $F^o_{k}(\\mathbf {v}) = F^o_{k-1}(\\mathbf {v}) + M^{i}_{k}(\\mathbf {K}{}\\pi (_{C}\\mathbf {p}(\\mathbf {v}))),$ $N^o_{k}(\\mathbf {v}) = N^o_{k-1}(\\mathbf {v}) + (1-M^{i}_{k}(\\mathbf {K}{}\\pi (_{C}\\mathbf {p}(\\mathbf {v})))),$ with $\\pi ([x,y,z]^\\intercal )=[x/z,y/z,1]^\\intercal $ denoting the projection.",
"Finally, to compute whether a voxel is part of the foreground we calculate the expectation, $E[p^o(\\mathbf {v})] = \\frac{F^o_{k-1}(\\mathbf {v})}{F^o_{k-1}(\\mathbf {v})+N^o_{k-1}(\\mathbf {v})},$ and use a decision threshold of $E[p^o(\\mathbf {v})] > 0.5$ .",
"A visualisation of this is shown in Figure REF .",
"Figure: Object volume foreground.",
"Note that if this value falls below 0.5 it is not rendered.Raycasting: For tracking, data association, and visualisation we render depth, normals, vertices, RGB, and object indices.",
"Within each object volume $\\mathcal {V}^o$ we step along the ray with a stepsize of $v^o_s$ (and $0.5v^o_s$ when $S_k^o(\\mathbf {v})<0.8$ , where $S_k^o(\\mathbf {v})$ is the SDF normalised by $\\mu $ ) and search for the zero-crossing point in $S_k^o(\\mathbf {v})$ where $E[p^o(\\mathbf {v})] > 0.5$ (both values are trilinearly interpolated from neighbouring voxels to smooth the representation).",
"We store the ray length of the nearest of these intersections to avoid searching past that point in another volume.",
"This alone results in occluding surfaces which are not part of the foreground failing to occlude the ray.",
"If a background TSDF is available, and either no intersection with a foreground object occurs or the intersection is farther than 5cm behind the background TSDF intersection, then the background TSDF ray intersection is used instead.",
"Existence Probability: To prevent spurious instances from building up over time, we also model the probability of each instance's existence as $p(o)$ using the Beta distribution, in a manner identical to the foreground mask.",
"For any frame where a predicted instance should be clearly visible (i.e.",
"our raycasted image has more than $50^2$ pixels of that instance), then if the instance has been associated to a detection its existence count $e_o$ is incremented, and if not its non-existence count, $d_o$ , is incremented.",
"If $E[p(o)]$ falls below $0.1$ , the instance is deleted and the object node with all associated edges are removed from the pose graph (described in Section REF ).",
"Semantic Labels: Each TSDF also stores a probability distribution over potential class labels $l_o$ .",
"Mask R-CNN provides a probability distribution $p(l_{o}|I_k)$ over the classes given the image, $I_k$ .",
"We found that the standard multiplicative Bayesian update scheme [15], [20]: $p(l^k_{o}|I_1, \\ldots , I_k)=Z^{-1}p(l_{o}|I_k)p(l_{o}|I_1, \\ldots , I_{k-1}),$ where $Z$ is a normalising constant, often leads to an overly confident class probability distribution, with scores unsuitable for ranking in object detection.",
"Instead here we fuse multiple associated detections by simple averaging: $p(l^k_{o}|I_1, \\ldots , I_k)=\\frac{1}{k}\\sum \\limits _{i=1}^{k} p(l_{o}|I_i),$ which produces a more even class probability distribution."
],
[
"Detection and Data Association",
"Detections from the Mask R-CNN model [13] for a given frame $k$ contain instances $i$ with a binary mask $M^{i}_{k}$ and class probability distribution $p(l_{i}|I_k)$ .",
"A forward pass takes $\\sim $ 250ms, and although our system is not real-time, this still represents a significant bottleneck and so can be performed in a parallel thread.",
"For GPU memory efficiency, we take only the top 100 detections (scored according to the region proposal network `object' score [27]) and filter for masks not near the image border (within 20 pixels) and where both $\\text{max}(p(l_{i}|I_k)) > 0.5$ and $\\sum M^{i}_{k} > 50^2$ .",
"After local tracking (Section REF ) we use the estimated camera pose and TSDFs already initialised in the map to raycast a binary mask $M^{o}_{k}$ for object instances $o$ in the current view.",
"We map each detection $i$ to a single instance $o$ by calculating the intersection of the two as a proportion of the detection's area, $a_\\mathrm {detect}(i,o)=\\frac{\\sum {M^{o}_{k} \\cap M^{i}_{k}}}{\\sum {M^{i}_{k}}}$ and assigning the detection to the largest intersection, $\\tilde{o}=\\operatornamewithlimits{argmax}_{o}a_\\mathrm {detect}(i,o)$ , where $a_\\mathrm {detect}(i,\\tilde{o}) > 0.2$ , otherwise the detection is unassigned.",
"For the integration step, each detection which has been mapped to the same instance is combined by taking the union of the detection masks, and the average of the class probabilities."
],
[
"Layered Local Tracking",
"We maintain an instance-agnostic coarse background TSDF, $a$ , to assist local frame-to-model tracking where/when there are no instances and to handle occlusions.",
"It has a resolution of $256^3$ with a voxel size of 2cm.",
"Its initialisation point ${}_{W}\\mathbf {p}_{a}=\\mathbf {T}^k_{WC}[0\\quad 0\\quad 2.56]^\\intercal $ , is 2.56m along the $z$ -axis in the camera frame ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{C}}}$ to prevent wasted volume as in [37].",
"The volume is reset when its new initialisation point exits a spherical threshold (1.28m) around the previous volume centre, i.e.",
"$\\Vert {}_{W}\\mathbf {p}_{a}-\\mathbf {T}^k_{WC}[0\\quad 0\\quad 2.56]^\\intercal \\Vert _2>1.28$ .",
"We combine the background TSDF with individual instances to raycast (Section REF ) a `layered' reference frame, denoted $r$ , with vertex map, $V_r$ , normal map, $N_r$ , and object index map, $X_r$ , from the previous camera pose, $\\mathbf {T}_{WC_{r}}$ , with vertices and normals defined in the world frame ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{W}}}$ .",
"The transform to the live frame, denoted $l$ , is estimated by aligning the live depth map, after bilateral filtering and projection to a vertex map $V_l$ and normal map $N_l$ with pixels $\\mathbf {u}_l$ , to the rendered maps with iterative closest point using projective data association and a point-to-plane error, $E_{\\mathrm {icp}}(\\tilde{\\mathbf {T}}_{WC_l})$ , as described in [23]: ${\\mathbfit {u}}_r = \\mathbf {K}\\pi (\\mathbf {T}_{WC_r}^{-1}\\tilde{\\mathbf {T}}_{WC_l} V_l(\\mathbf {u}_l)),$ $r_{\\mathrm {\\mathrm {icp}}}(\\tilde{\\mathbf {T}}_{WC_l},\\mathbf {u}_l) = N_r(\\mathbf {u}_r) \\cdot (V_r(\\mathbf {u}_r)-\\tilde{\\mathbf {T}}_{WC_l}V_l(\\mathbf {u}_l)),$ $E_{\\mathrm {\\mathrm {icp}}}(\\tilde{\\mathbf {T}}_{WC_l}) = \\sum \\limits _{\\mathbf {u}_l \\in V_{\\mathrm {valid}}} r_{\\mathrm {icp}}(\\tilde{\\mathbf {T}}_{WC_l},\\mathbf {u}_l)^2.$ Where $V_{\\mathrm {valid}}$ includes any $\\mathbf {u}_l$ with a corresponding vertex and normal, where there is a corresponding $\\mathbf {u}_r$ with a valid vertex and normal, and where $N_r(\\mathbf {u}_r)\\cdot N_l(\\mathbf {u}_l)<0.8$ and $\\Vert V_r(\\mathbf {u}_r)-\\tilde{\\mathbf {T}}_{WC_l}V_l(\\mathbf {u}_l) \\Vert _2 <0.1\\text{m}$ .",
"We minimize this non-linear least squares problem using the Gauss-Newton algorithm.",
"We linearise $\\tilde{\\mathbf {T}}_{WC_l}$ about the previous estimate with the perturbation, $\\zeta $ where $\\tilde{\\mathbf {T}}_{WC} = \\exp (\\zeta ) \\bar{\\mathbf {T}}_{WC}$ .",
"Each row of the $|V_{\\mathrm {valid}}| \\times 6$ Jacobian, $\\mathbf {J}_{\\mathrm {icp}}$ , corresponds to the residual of a given $\\mathbf {u}_l \\in V_{\\mathrm {valid}}$ : $\\frac{\\partial r_{\\mathrm {icp}}(\\zeta ,\\mathbf {u}_l)}{\\partial \\zeta }|_{\\zeta = 0} = -[N_r^\\intercal (\\mathbf {u}_r),(V_l(\\mathbf {u}_l) \\times N_r(\\mathbf {u}_r))^\\intercal ].$ The Gauss-Newton iteration can then be implemented as follows (with iteration index $t$ ): $\\zeta ^t = -(\\mathbf {J}_{\\mathrm {icp}}^\\intercal \\mathbf {J}_{\\mathrm {icp}})^{-1}\\mathbf {J}_{\\mathrm {icp}}^\\intercal \\mathbf {r}_{\\mathrm {icp}},$ $\\tilde{\\mathbf {T}}^{t+1}_{WC_l} = \\text{exp}(\\zeta ^t)\\bar{\\mathbf {T}}^{t}_{WC_l}.$ The $6\\times 6$ Hessian approximation, $\\mathbf {J}_{\\mathrm {icp}}^\\intercal \\mathbf {J}_{\\mathrm {icp}}$ , and $6\\times 1$ error Jacobian, $\\mathbf {J}_{\\mathrm {icp}}^\\intercal \\mathbf {r}_{\\mathrm {icp}}$ , are reduced in parallel on the GPU and solved on the CPU using SVD and back substitution.",
"We use a three-level coarse-to-fine pyramid scheme with 5 Gauss-Newton iterations per level.",
"We perform an additional reduction on the GPU to produce the same system of equations partitioned into pixels, $\\mathbf {u}_l$ , associated to each instance in $X_r(\\mathbf {u}_r)$ for pose-graph optimisation and to produce per-instance error metrics.",
"The error metrics are the ICP RMSE, $(|V_{\\mathrm {valid}}|^{-1}E_{\\mathrm {icp}}(\\tilde{\\mathbf {T}}_{WC_l}))^{\\frac{1}{2}}$ , and the proportion of validly tracked pixels $\\frac{|V_{\\mathrm {valid}}|}{|V_l|}$ .",
"These are used for instance integration and to check whether local tracking is lost.",
"We consider local tracking to be lost when the total ICP RMSE is greater than 0.05m or when at least 10% of the image consists of instance TSDFs and less than half of the pixels are validly tracked, in which case we enter relocalisation mode."
],
[
"Relocalisation",
"If the system is lost or we reset the coarse TSDF, we perform relocalisation to align the current frame to the current set of instances (if there are any).",
"We found direct dense ICP methods using only the volume reconstructions did not produce accurate results for wide baseline relocalisation as they are sensitive to the initial pose and small objects were often ambiguous without texture constraints.",
"Although alternative dense methods may also prove useful here, we took the approach of using snapshots of sparse BRISK featuresBRISK v.2 with homogeneous Harris scale space corner detection on only the highest image resolution.",
"(with a detection threshold of 10) projected to 3D using the depth map.",
"For a given detection of an object if there is no existing snapshot of the object within $15^\\circ $ view angle difference, we then add a new snapshot of the object from that pose (see Figure REF ).",
"To re-localise we perform 3D-3D RANSAC against each instance where the dot product with the predicted class distribution is greater than 0.6.",
"We use OpenGV [16] with a minimum of 5 inlier features (within 2cm) to match each object individually.",
"If we find one or more matching objects in the scene, we run a final 3D-3D RANSAC on every point in the scene (from all objects and the background jointly) with a minimum of 50 inlier features (within 5cm) to arrive at a final camera pose.",
"This pose is used to render a new reference image of the map to produce the constraints required for the pose graph optimisation described below.",
"Figure: Re-localisation snapshots around an instance."
],
[
"Object-Level Pose Graph",
"Our pose-graph formulation is similar to that of [30].",
"For every frame with a Mask R-CNN detection (including coarse TSDF resets), we add a new camera pose node to our graph.",
"When a new instance, index $o$ , is initialised, a corresponding landmark node is added to the graph, defined by the coordinate frame attached to the centre of the object's volume, $\\mathbf {p}_{o}$ .",
"The first camera pose node is fixed and defined to be the origin of the world frame, ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{W}}}$ .",
"Each node consists of a full $SE(3)$ transformation from object to World, $\\mathbf {T}_{WO}$ , or camera to world, $\\mathbf {T}_{WC}$ , and the measurements are $SE(3)$ relative pose constraints between nodes.",
"Each relative measurement is derived by employing only the ICP error terms which correspond to the pixels of the specific object $o$ (for object-camera constraints), or the instance-agnostic background $a$ (for camera-camera constraints).",
"To ensure that the measurement coincides with the minimum of the partitioned set's quadratically approximated error function, an additional Gauss-Newton step is performed using the partitioned $\\mathbf {J}^o_{\\mathrm {icp}}$ (see Section REF ) to produce `virtual' relative pose measurements $\\tilde{\\mathbf {T}}^{\\prime a}_{C_{k-1}C_k}$ , between camera nodes, and $\\tilde{\\mathbf {T}}^{\\prime o}_{OC_k}$ , between camera and landmark objects.",
"The resulting measurement errors for the graph factors are: $\\mathbf {e}_{\\mathrm {cc}}(\\mathbf {T}_{C_{k-1}W},\\mathbf {T}_{WC_{k}}) = \\text{log}((\\tilde{\\mathbf {T}}^{\\prime a}_{C_{k-1}C_k})^{-1}\\mathbf {T}_{C_{k-1}W}\\mathbf {T}_{WC_{k}}),$ $\\mathbf {e}_{\\mathrm {oc}}(\\mathbf {T}^{o}_{OW},\\mathbf {T}_{WC_{k}}) = \\text{log}((\\tilde{\\mathbf {T}}^{\\prime o}_{OC_k})^{-1}\\mathbf {T}^{o}_{OW}\\mathbf {T}_{WC_{k}}).$ For every relative measurement, we approximate the inverse measurement covariance by $\\Sigma ^{-1}= \\mathbf {J}_{\\mathrm {icp}}^{o\\intercal }\\mathbf {J}^{o}_{\\mathrm {icp}}$ .",
"However, since the way perturbations are modelled differs between the ICP algorithm and the employed pose graph optimiser we need to transform the covariance by considering the relation between the local perturbations.",
"The graph optimiser models perturbations $\\zeta _{\\mathrm {pg}}$ to relative pose measurements via $\\tilde{\\mathbf {T}}^{\\prime o}_{O^{\\prime }C_{k}}=\\tilde{\\mathbf {T}}^{\\prime o}_{OC_{k}}\\text{exp}(\\zeta _{\\mathrm {pg}})$ (equivalently for $\\tilde{\\mathbf {T}}^{\\prime a}_{C_{k-1}C_{k}}$ ).",
"To ensure our information matrix properly corresponds to perturbations $\\zeta _{\\mathrm {pg}}$ , it is necessary to convert $\\mathbf {J}_{\\mathrm {icp}}$ .",
"As can be seen in Eq.",
"REF , $\\mathbf {J}_{\\mathrm {icp}}$ is with respect to perturbations applied via $\\tilde{\\mathbf {T}}_{W^{\\prime }C_{k}}=\\text{exp}(\\zeta _{\\mathrm {icp}})\\tilde{\\mathbf {T}}_{WC_{k}}$ .",
"The relation between $\\zeta _\\mathrm {icp}$ and $\\zeta _{\\mathrm {pg}}$ is: $\\text{exp}(\\zeta _{\\mathrm {icp}})\\mathbf {T}_{WC_{k}}=\\mathbf {T}^{o}_{WO}\\tilde{\\mathbf {T}}^{\\prime o}_{OC_{k}}\\text{exp}(\\zeta _{\\mathrm {pg}}),$ $\\zeta _{\\mathrm {icp}}=\\text{log}(\\mathbf {T}_{WC_{k}}\\text{exp}(\\zeta _{\\mathrm {pg}})\\mathbf {T}^{-1}_{WC_{k}}) = \\text{Adj}_{\\mathbf {T}_{WC_{k}}}\\zeta _{\\mathrm {pg}},$ $\\mathbf {J}_{\\mathrm {pg}}=\\frac{\\partial \\zeta _{\\mathrm {icp}}}{\\partial \\zeta _{\\mathrm {pg}}}=\\text{Adj}_{\\mathbf {T}_{WC_{k}}},$ where $\\text{Adj}_{{\\mathbf {T}_{WC_k}}}$ is the Adjoint of ${\\mathbf {T}_{WC_k}}$ such that $\\text{exp}(\\text{Adj}_{{\\mathbf {T}_{WC_k}}}\\zeta _\\mathrm {pg}) = {\\mathbf {T}_{WC_k}}\\text{exp}(\\zeta _\\mathrm {pg}) {\\mathbf {T}^{-1}_{WC_k}}$ as described in [10].",
"The derivation for camera nodes results in the same transformation and the new information matrix therefore becomes, $\\mathbf {H}_\\mathrm {pg} = \\mathbf {J}^{\\intercal }_{\\mathrm {pg}} (\\mathbf {J}^{o \\intercal }_{\\mathrm {icp}} \\mathbf {J}^o_{\\mathrm {icp}}) \\mathbf {J}_{\\mathrm {pg}}.$ The final error to be minimised in the pose graph is the sum over all the edges from the camera to objects, $\\mathcal {O}$ , and camera to camera, $\\mathcal {C}$ , given their state, the measurement, and the information matrix, $E_{\\mathrm {pg}} = \\sum \\limits _{\\mathrm {cc}\\in \\mathcal {C}} L_\\sigma (\\mathbf {e}_{\\mathrm {cc}}^\\intercal \\mathbf {H}_{\\mathrm {pg}} \\mathbf {e}_{\\mathrm {cc}}) +\\sum \\limits _{\\mathrm {oc}\\in \\mathcal {O}} L_\\sigma (\\mathbf {e}_{\\mathrm {oc}}^\\intercal \\mathbf {H}_{\\mathrm {pg}}, \\mathbf {e}_{\\mathrm {oc}}),$ where $L_\\sigma $ denotes a robust Huber kernel.",
"We solve this graph in the g2o [17] framework using sparse Cholesky decomposition and Levenberg-Marquart.",
"After optimisation we update the pose of the instance TSDFs and the camera before initialising the new coarse TSDF to that pose and continuing with local tracking.",
"As described in Section REF , when a landmark is re-sized, its centre, $\\mathbf {p}_{o}$ , can also be adjusted from ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{O}}}$ to a new frame ${\\smash{\\protect \\underrightarrow{\\mathcal {F}}_{O^{\\prime }}}}$ via the transform $\\mathbf {T}_{O^{\\prime }O}$ .",
"In this case we also transform the corresponding node variable, $\\mathbf {T}^{o}_{WO^{\\prime }}=\\mathbf {T}^{o}_{WO}\\mathbf {T}_{OO}^{-1}$ , as well as the measurement for every edge connected to that node, $\\tilde{\\mathbf {T}}^{\\prime o}_{O^{\\prime }C} = \\mathbf {T}_{O^{\\prime }O}\\tilde{\\mathbf {T}}^{\\prime o}_{OC}$ .",
"Figure: Comparison of office sequence trajectory before loop-closure (left) and after loop-closure (right)."
],
[
"Experiments",
"We evaluate the performance and memory usage of our system on a Linux system with an Intel Core i7-5820K CPU at 3.30GHz, and an nVidia GeForce GTX1080 Ti GPU with 11.175GB of memory.",
"Our core pipeline is implemented in Python and uses Tensorflow for instance predictions, and Python wrappers around other core components which are developed in C++ and/or CUDA, such as KFusion, g2o, BRISK, and OpenGV.",
"Our input is standard $640\\times 480$ resolution RGB-D video.",
"To allow for reproducibility, instead of running an asynchronous CNN thread we here perform predictions synchronously every 30 frames.",
"Our Mask-RCNN uses the ResNet-101 base model [14] (up to the conv4_x block) and is finetuned from the publicly available tensorpack implementation and weights [40].http://models.tensorpack.com For finetuning on indoor scenes we use the NYUv2 dataset.",
"We lock the ResNet-101 weights from the COCO pre-training and fine-tune the remaining layers.",
"As the COCO dataset consists of 80 classes we re-size and reinitialise the class-specific upper layers of Mask R-CNN and Faster R-CNN.",
"We train using stochastic gradient descent with momentum of $0.9$ for 30 epochs with a learning rate of 0.001."
],
[
"Loop Closure and Map Consistency",
"To evaluate the performance of our system while repeatedly viewing a scene of instances we captured a 3,685 frame sequence of an indoor office scene.",
"We tailored this sequence to evaluate the consistency of our map in the presence of poorly constrained (planar floor) geometry and ICP drift, after which we loop over the same scene again.",
"The pose-graph and loop closure is shown in Figure REF , it can be seen that despite the accumulated drift, the system re-localises and corrects the pose graph, this allows the previously reconstructed objects to be correctly associated in future frames.",
"On the entirety of the trajectory our system reconstructed 105 landmark object instances, however, it must be noted that despite our filtering mechanisms, a build up of noisy partially reconstructed sub-objects still occurs."
],
[
"Reconstruction Quality",
"To evaluate the reconstruction quality we use objects from the YCB dataset which provides ground truth models [1] and reconstruct discovered objects from sequence 0001 of the public YCB video dataset [41].",
"Figure REF shows a qualitative comparison against the ground truth.",
"The missing portion of the cracker box was caused by an occlusion by another object, and a missed foreground detection on one of the few frames where the cracker box was unoccluded.",
"Figure: Reconstruction quality vs ground truth from sequence 0001 of the public YCB video dataset ."
],
[
"RGB-D SLAM Benchmark",
"We evaluate the trajectory error of our system against the baseline approach of simple coarse TSDF odometry, i.e.",
"using the same coarse resetting background without instances layered on top, and without loop-closure pose graph optimisation.",
"Table REF shows the results.",
"It can be seen that in all but one of the sequences evaluated our Fusion++ system improved upon the baseline approach (while providing an inventory of objects as Figure REF visualises for the fr2_desk sequence).",
"It is also worth noting that our system does not achieve state-of-the-art performance on these sequences such as [38], [22], and would require additional work, such as including joint depth and photometric tracking, to become competitive.",
"We focused on a usable object map here and leave accuracy of motion tracking for future work.",
"Table: RGB-D SLAM Benchmark ATE RMSE (mm)."
],
[
"Memory and Run-time Analysis",
"Memory usage: We use the office sequence to evaluate the run-time performance and memory usage of our system.",
"As memory usage scales cubically with the size of a TSDF, it is significantly more efficient to compose a map of many relatively small, highly detailed, volumes in dense areas of interest than to use one large one with a resolution equal to the smallest.",
"After loading the CNN and image buffers, our remaining $\\sim $ 7GB GPU memory budget (and 10 bytes per voxel) would allow a single $900^3$ volume or, as here, a $256^3$ background volume and up to 2.5K object volumes with dimension $64^3$ , 2MB.",
"Our object volumes dynamically vary up to $128^3$ and on our office sequence used 377MB for 105 objects ($\\sim $ 4MB/object), as shown in Figure REF .",
"Of course, more efficient alternatives such as an octree or voxel hashing can also be used to directly eliminate wasted free-space voxels, and are also directly applicable to our approach.",
"Figure: GPU memory usage and per-frame wall clock scaling by number of objects on the office sequence.Runtime performance: Our system, although not real-time, scales well with the number of objects.",
"Excluding re-localisation on the office sequence the average frame rate was 4-8Hz (shown in Figure REF ), with an average additional computational cost of 1ms per object.",
"A more detailed breakdown of the runtime performance of different components and their scaling factors is given in Table REF .",
"Table: Run-time analysis of system components (msms) with approximate scaling performance on office sequence."
],
[
"Conclusions",
"We have shown consistent instance mapping and classification of numerous objects of previously unknown shape in real, cluttered indoor scenes.",
"Our online and near real-time system, which is built from modules for image-based instance segmentation, TSDF fusion and tracking, and pose graph optimisation, makes a long-term map which focuses on the most important object elements of a scene with variable, object size-dependent resolution.",
"A number of shortcomings of the current approach remain to be addressed in future work.",
"There is a balance to be struck between filtering detections and providing good coverage of a scene, and even with the existence probability and deletion mechanism detailed here, over time spurious detections result in a growing clutter of partial object reconstructions.",
"More thorough object detection precision/recall evaluations as well as semantic accuracy metrics will assist in this.",
"A learned mechanism for filtering and reconstructing these objects, such as [7] may prove useful in this regard, or combining view-based segmentation and classification with 3D methods which take advantage of object databases such as ShapeNet [2].",
"There is also significant scope in future to better combine information from multiple duplicate objects seen from different views to reconstruct a single better model, rather than maintaining separate TSDFs for each.",
"Our object-oriented representation can also naturally be extended to model moving objects with individually changing poses.",
"This attribute would be particularly useful when reasoning about dynamic applications in robotics or augmented reality."
],
[
"Acknowledgements",
"This research was supported by Dyson Technology Ltd."
]
] | 1808.08378 |
[
[
"Protostellar classification using supervised machine learning algorithms"
],
[
"Abstract Classification of young stellar objects (YSOs) into different evolutionary stages helps us to understand the formation process of new stars and planetary systems.",
"Such classification has traditionally been based on spectral energy distributions (SEDs).",
"An alternative approach is provided by supervised machine learning algorithms.",
"We attempt to classify a sample of Orion YSOs into different classes, where each source has already been classified using multiwavelength SED analysis.",
"We used 8 different learning algorithms to classify the target YSOs, namely a decision tree, random forest, gradient boosting machine (GBM), logistic regression, na\\\"ive Bayes classifier, $k$-nearest neighbour classifier, support vector machine, and neural network.",
"The classifiers were trained and tested by using a 10-fold cross-validation procedure.",
"As the learning features, we employed ten continuum flux densities spanning from the near-IR to submm.",
"With a classification accuracy of 82% (with respect to the SED-based classes), a GBM algorithm was found to exhibit the best performance.",
"The lowest accuracy of 47% was obtained with a na\\\"ive Bayes classifier.",
"Our analysis suggests that the inclusion of the 3.6 $\\mu$m and 24 $\\mu$m flux densities is useful to maximise the YSO classification accuracy.",
"Although machine learning has the potential to provide a rapid and fairly reliable way to classify YSOs, an SED analysis is still needed to derive the physical properties of the sources, and to create the labelled training data.",
"The classification accuracies can be improved with respect to the present results by using larger data sets, more detailed missing value imputation, and advanced ensemble methods.",
"Overall, the application of machine learning is expected to be very useful in the era of big astronomical data, for example to quickly assemble interesting target source samples for follow-up studies."
],
[
"Introduction",
"An essential part of the star formation studies is to try to classify the young stellar objects (YSOs) into different evolutionary stages, and construct a coherent YSO evolutionary sequence.",
"Also, by determining the relative percentages of YSOs in different stages, the statistical time spent in each stage can be constrained, which in turn helps to quantify the overall timescale of the stellar birth process in different molecular cloud environments (e.g.",
"[29]; [28]).",
"Considering the formation of low-mass, solar-type stars, the YSOs have traditionally been classified into distinct stages on the basis of their infrared (IR) spectral slopes (e.g.",
"[55]) or bolometric temperatures ([70]).",
"In particular, the spectral energy distribution (SED) of a YSO, which is characterised by the bolometric temperature and luminosity, is commonly used to determine the evolutionary stage of the source, that is whether it is a so-called Class 0 or I protostar, or Class II or III pre-main sequence (PMS) star (e.g.",
"[54]; [2]; [7]; see also [8] for a review).",
"Indeed, an SED analysis is very useful, not just for the purpose of source classification, but to derive some of the key physical properties of the source, such as the dust temperature and dust mass.",
"However, modelling the source SEDs can be fairly time consuming, and hence, to quickly determine the evolutionary classes for a large sample of YSOs, an automated procedure that employs the observed source properties (i.e.",
"the flux densities) would be very useful.",
"In this regard, machine learning has the potential to yield a fast way to classify sources (as compared to an SED analysis) as long as the algorithm(s) in question can be trained with data sets composed of relevant flux densities and corresponding evolutionary classes of the target YSOs.",
"So far, machine learning based classification of astrophysical objects has mostly been applied in extragalactic research (e.g.",
"[53]; Aniyan & Thorat 2017; [88]; [12]; Pashchenko et al.",
"2018; [46]; Lukic et al.",
"2018; [5]; see also [58]), while Galactic machine learning studies have been relatively few in number (e.g.",
"[60]; [95]).",
"Hence, pilot studies about using machine learning in YSO classification, which the present work represents, are warranted.",
"In this paper, we report the results of our protostellar classification test using several different supervised machine learning algorithms.",
"The data set used in this study is described in Sect.",
"2, while the data analysis is presented in Sect. 3.",
"The results are presented and discussed in Sect.",
"4, and in Sect.",
"5 we summarise the key results and conclusions of this work."
],
[
"Data",
"The data analysed in this paper were taken from Furlan et al.",
"(2016, hereafter FFA16).",
"As part of the HerschelHerschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.",
"Orion Protostar Survey (HOPS; e.g.",
"[89]), FFA16 studied and modelled the SEDs of a large, homogeneous sample of 330 YSOs in the Orion molecular cloud complex (the authors assumed a uniform distance of 420 pc to the cloud complex).",
"At the time of writing, this is the largest available YSO sample investigated in a single star-forming cloud complex.",
"The photometric data employed by the authors included the $J$ , $H$ , and $K_{\\rm S}$ near-IR data from the Two Micron All Sky Survey (2MASS; [86]), Spitzer IR data obtained with the Infrared Array Camera (IRAC; 3.6–8.0 $\\mu $ m; [31]), the Multiband Imaging Photometer for Spitzer (MIPS; 24 $\\mu $ m; [78]), and the Infrared Spectrograph (IRS; 5.4–35 $\\mu $ m; [45]).",
"The Herschel satellite ([74]) was used to observe the 70, 100, and 160 $\\mu $ m far-IR bands with the Photodetector Array Camera and Spectrometer (PACS; [75]), while the 350 and 870 $\\mu $ m submillimetre data were obtained with the Atacama Pathfinder EXperiment (APEX; [37]) using its Submillimetre APEX BOlometer CAmera (SABOCA; Siringo et al.",
"2010) and Large APEX BOlometer CAmera (LABOCA; [84]).",
"The aforementioned bands cover the typical protostellar SED peak emission at $\\sim 100$ $\\mu $ m and its surrounding wavelengths, which is essential to determine reliable physical properties and evolutionary stage of the source (see below).",
"On the basis of their panchromatic SED analysis, FFA16 classified their YSO sample into 92 Class 0 protostars, 125 Class I protostars, 102 flat-spectrum sources (expected to be objects in transition between the Class I and II phases), and 11 Class II PMS stars.",
"The corresponding relative percentages are $27.9\\% \\pm 2.9\\%$ , $37.9\\% \\pm 3.4\\%$ , $30.9\\% \\pm 3.1\\%$ , and $3.3\\% \\pm 1.0\\%$ , respectively, where the quoted uncertainties represent the Poisson counting errors.",
"We note that one of the FFA16 sources, namely the Class 0 source HOPS 400, was first uncovered by Miettinen et al.",
"(2009; their source SMM 3) in their LABOCA imaging of the Orion B9 star-forming region (see also [65], and references therein).",
"In brief, the physical explanation of why the SED analysis presented by FFA16 can be used to classify YSOs into different evolutionary stages is as follows (see e.g.",
"[93]; [27] for reviews).",
"The youngest protostellar objects, or Class 0 objects, are characterised by a central protostar deeply embedded in its cold, dusty envelope ([7]; [6]).",
"Hence, the source is extremely faint in the optical ($\\lambda \\sim 0.4-0.7$ $\\mu $ m) and near-IR ($\\lambda \\gtrsim 0.7-5$ $\\mu $ m; traced by 2MASS and Spitzer/IRAC observations), but bright in the far-IR ($\\lambda \\sim 25-350$ $\\mu $ m; traced by Spitzer/MIPS and Herschel observations) and (sub-)millimetre ($\\lambda \\gtrsim 350-1\\,000$ $\\mu $ m; traced by APEX bolometer observations) dust emission.",
"The central protostar increases its mass by accreting gas from the surrounding envelope via a circumstellar disk.",
"When the envelope mass has dropped to that of the growing central protostar, the system is believed to transition from the Class 0 to the Class I stage.",
"Class I protostars are still surrounded by an accretion disk and a circumstellar envelope of gas and dust, and hence their SEDs peak in the far-IR.",
"However, another prominent SED bump can be seen in the mid-IR ($\\lambda \\sim 5-25$ $\\mu $ m; traced by Spitzer/IRAC and MIPS observations), which is an indication of hotter dust than in the previous Class 0 stage.",
"If the central protostar can be seen along the long axis of the protostellar outflow, the Class I object can be optically visible.",
"The intermediate stage between the Class I and Class II stages is characterised by the disappearing dust excess emission in the mid-IR, and hence the sources in this transition stage are known as flat-spectrum (SED) sources ([36]).",
"In the Class II stage, the envelope has dissipated, and an optically visible PMS star is surrounded by a tenuous disk.",
"The SEDs of Class II objects peak at visible or near-IR wavelengths, and the disk adds an IR excess to the SED."
],
[
"Source selection",
"One of the four YSO classes from FFA16, namely the Class II phase, is highly unbalanced with respect to the other three classes (Class 0, Class I, and flat sources).",
"Indeed, only $3.3\\%$ of the FFA16 sources were classified as Class II sources, which is about ten times less than the occurrence of other types of sources.",
"Although techniques to deal with imbalanced data sets exist (e.g.",
"[51] for a review; [40]), we do not consider the FFA16 Class II sources in the subsequent analysis because their relative rarity can lead to problems in the training of the supervised classification algorithms, and in the evaluation of the classifiers' performance.",
"After discarding the Class II sources, we are left with 319 sources, out of which $28.8\\% \\pm 3.0\\%$ are Class 0 sources, $39.1\\% \\pm 3.5\\%$ are Class I sources, and $32.0\\% \\pm 3.3\\%$ are flat-spectrum sources.",
"Hence, all these three classes are in good relative number balance with respect to each other."
],
[
"Missing value treatment",
"When developing machine learning models, it is important to handle the missing values in the analysed data set (e.g.",
"[94]; Saar-Tsechansky & Provost 2007).",
"A common approach is to replace the missing values by the mean of the non-missing values for the variable or feature in question (e.g.",
"[3]; [56]).",
"This imputation method does not change the sample mean of the variable.",
"We found that $69.3\\%$ , $54.9\\%$ , and $40.1\\%$ of the selected FFA16 sources missed the 2MASS $J$ , $H$ , and $K_{\\rm S}$ near-IR data.",
"Owing to these large percentages, we discarded the 2MASS data in our subsequent analysis.",
"Also, $31\\%$ of the selected FFA16 YSOs lacked the SABOCA 350 $\\mu $ m data, but we included this waveband in the analysis because it provides a useful data point in the Rayleigh-Jeans part of the source SED, and the band has also been covered by observations with other instruments, such as by Herschel submm surveys (e.g.",
"[9]).",
"All the remaining bands except the Herschel 70 $\\mu $ m band were also found to contain missing values, but in those cases only 0.3% to 16.6% (11% on average) of the sources had missing data.",
"To impute the missing values, we used the R package MICE (Multivariate Imputation via Chained Equations; [92]).",
"The usage of MICE is based on the assumption that the missing data are Missing at Random (MAR), and it imputes data on a variable by variable basis (i.e.",
"flux density by flux density basis in our case) by specifying an imputation model per variable.",
"To calculate the imputations, we used the predictive mean matching (PMM) method ([57]).",
"If $S_{\\rm miss}$ is the variable that contains missing data, and $S_{\\rm i}$ are the variables that do not suffer from missing data, the PMM algorithm works as follows: i) for cases with no missing data, a linear regression model of $S_{\\rm miss}$ is estimated on $S_{\\rm i}$ , which yields a set of coefficients $b$ ; ii) a random draw is taken from the distribution of $b$ values, which yields a new sample of coefficients $b^*$ ; this step is needed to generate some random variability in the imputed values; iii) the $b$ values are used to calculate the predicted values for the observed $S_{\\rm miss}$ values, while the $b^*$ values are used to calculate the predicted values for the missing $S_{\\rm miss}$ values; iv) for each case with a missing $S_{\\rm miss}$ value, a set of cases with $S_{\\rm miss}$ present is identified where the predicted values are closest to the predicted value for the case with a missing $S_{\\rm miss}$ value; v) from the latter cases, a random value is chosen, and it is then used to impute the missing value.",
"The PMM method is expected to lead to more reasonable estimates of the missing flux densities than simply using the sample averages, while still being a very fast imputation method.",
"Regarding the traditonal mean or median imputations, one might think that replacing the missing flux density values by the mean or median values for each protostellar class separately would be a better approach than using the full sample mean or median.",
"Indeed, this would be a more physical approach than using the full sample mean or median imputation because the flux density at a given wavelength can evolve as the source evolves (e.g.",
"the amount of dust in the protostellar envelope decreases as the source matures, which affects the far-IR and submm emission considered in the present study).",
"However, to do this, one would have to use information from the test set's labels or classes, and one should not leak such information (which is correlated with the labels) to the training procedure of a classifier.",
"More importantly, when analysing new, previously unseen data, the protostellar or YSO classes are not even known, but they are what one wants to determine or predict.",
"On the other hand, it is known that for example the far-IR and submm flux densities considered in the present work depend on each other via the frequency dependent dust opacity, $\\kappa _{\\nu } \\propto \\nu ^{\\beta }$ , where $\\beta $ is the dust emissivity index (e.g.",
"[83]).",
"Hence, the missing submm flux densities could also be estimated in a source-by-source fashion from the existing ones by assuming a value for $\\beta $ .",
"However, estimating the missing flux density values this way requires more feature value engineering, and hence is not as fast as for example the PMM.",
"Indeed, obtaining fast classifications (as compared to an SED analysis) is the key of applying machine learning in the first place."
],
[
"Training and test data sets",
"A supervised machine learning algorithm requires a so-called training set through which the algorithm tries to learn how the input values, or features (flux densities in our case), map to the response values, or labels (protostellar classes in the present work).",
"Another data set, the so-called test set, is then used to test the performance of the trained model by showing it only the input values, and see which corresponding classes the model predicts for the data it did not see during training.",
"A traditional way to create the training and test data sets is to fragment the original data into two parts.",
"For example, one common way is to use 80% of the data for training the algorithm, and the remaining 20% for testing it (the so-called 80/20 rule or the Pareto principle; e.g.",
"[13]).",
"However, the present data set is fairly small for a machine learning experiment (319 sources in total), and hence the aforementioned data splitting would not yield good training and test capabilities.",
"Instead, we used the technique of $k$ -fold cross-validation (CV), where the data set is randomly divided into $k$ roughly equal sized subsamples, or folds ([68]; see e.g.",
"[47], Sect.",
"5.1.3 therein).",
"The algorithm is trained using $k-1$ of the folds, and the resulting model is tested on the remaining part of the data.",
"This procedure is then repeated $k$ times.",
"We did the sampling with replacement, which means that the same source could be sampled more than once.",
"The final prediction performance was taken as the mean of the $k$ results.",
"A value of $k=10$ was used in the present work.",
"Another caveat of dealing with small data sets in machine learning is the problem of overfitting, which means that an algorithm starts to learn the details of the training data set, including the random noise features, such as outliers instead of (or besides) the underlying general rules, patterns, or relationships (e.g.",
"[39]).",
"This weakens the algorithm's ability to generalise well to previously unseen cases, because the aforementioned noise features are unlikely to occur in the test and new data sets.",
"However, the usage of $k$ -fold CV can reduce the degree of overfitting (but not fully prevent it) owing to the split of the data into multiple training and test sets (e.g.",
"[18])."
],
[
"Principal component analysis",
"Another data preprocessing step we did was principal component analysis (PCA; [72]; Hotelling 1933; [49]; Abdi & Williams 2010), which is a common technique in machine learning for extracting the most important variables among a large number of variables in a data set, and hence to overcome feature redundancy and reduce the dimensionality of the problem.",
"We note that if the analysed data set is split into separate labelled training set and test set, then the PCs need to be calculated on the training data set.",
"However, because in the present study we employed the method of $k$ -fold CV for training and testing the classifiers, we carried out the PCA using the full data set (i.e.",
"319 sources).",
"Because in the PCA the original data are projected onto directions that maximise the variance, the variables were scaled to have a variance equal to $\\sigma _{\\rm SD}^2=1$ , where $\\sigma _{\\rm SD}$ is the standard deviation, before the PCs were calculated.",
"In Fig.",
"REF , we plot the cumulative proportion of the variance in the data set explained by each PC.",
"In this analysis, all the other wavebands except those of 2MASS were taken into account (i.e.",
"four Spitzer/IRAC bands, one Spitzer/MIPS band, three Herschel/PACS bands, two APEX bands, and 16 Spitzer/IRS bands).",
"We found that the first ten PCs explain about 99.2% of the variance in the data, and hence we used only the first ten features in the subsequent analysis.",
"These ten features correspond to the continuum flux densities in the FFA16 photometric data, that is from the Spitzer/IRAC 3.6 $\\mu $ m to LABOCA 870 $\\mu $ m data, while the Spitzer/IRS data were discarded.",
"The distributions of the considered flux densities are presented in Fig.",
"REF , which shows a draftsman's plot, or a scatter matrix plot (see e.g.",
"NIST/SEMATECH e-Handbook of Statistical Methodshttps://www.itl.nist.gov/div898/handbook/eda/section3/ scatterb.htm ), and which enables to see the interrelations between variables in multivariate data (ten dimensional in our case).",
"The plot consists of an array of two-variable scatter diagrams.",
"For example, the plot demonstrates the strong correlation between the Herschel 100 $\\mu $ m and 160 $\\mu $ m flux densities.",
"Besides reducing the dimensionality of the problem, the feature selection enabled by PCA also helps to relieve the problem of overfitting discussed in Sect. 3.3.",
"The reason for this is that the trained model will be less complex the smaller the number of features is relative to the number of data cases (or rows).",
"Hence, although the present data set is fairly small and hence subject to overfitting, both the $k$ -fold CV and PCA employed in the present study can alleviate the influence of overfitting.",
"Figure: Cumulative proportion of the variance in the data explained by each PC.",
"The first ten PCs explain about 99.2% ofthe variance.Figure: Draftsman's plot showing the values of each of the considered continuum flux densities (in Jy) against each other."
],
[
"Machine learning classification",
"After data preparation, the following eight supervised classifiers were tested on the FFA16 data: a decision tree (e.g.",
"[76]; [69]), random forest ([41]; [15]), gradient boosting machine (GBM; [14]; [34]), logistic regression ([23]), naïve Bayes classifier (e.g.",
"[24]; [62]; [99]), $k$ -nearest neighbours ($k$ -NN; e.g.",
"Cover & Hart 1967; [4]), support vector machine (SVM; e.g.",
"[91]; Cortes & Vapnik 1995; Burges 1998; [20]; Scholkopf & Smola 2001), and artificial neural network (e.g.",
"McCulloch & Pitss 1943; [79]; Jeffrey & Rosner 1986; [98]).",
"We refer to Kotsiantis et al.",
"(2006b), Kotsiantis (2007), and Ball & Brunner (2010) for reviews of the aforementioned algorithms.",
"As mentioned in Sect.",
"3.3, the classifiers were trained and their performance was tested using the technique of 10-fold CV.",
"In what follows, is a brief description of each of the tested classifier."
],
[
"Decision tree",
"A decision tree classifier attempts to learn simple decision rules that can predict the label for an instance on the basis of its feature values.",
"The data are split according to the feature values in the so-called decision nodes, and the final leaf nodes contain the outputs (i.e.",
"the protostellar classes in our case).",
"One caveat of decision trees is that they are subject to overfitting if too complex trees are being built, and hence they might not generalise well.",
"To buid a decision tree classifier, we used the R package rpart (Recursive Partitioning and Regression Trees), which uses the Classification and Regression Trees (CART; Breinman et al.",
"1984) algorithm.",
"The CART algorithm employs binary splits on the input variables to grow the tree, and the splits were evaluted on the basis of the Gini index (a Gini score of zero means a perfect separation)."
],
[
"Random forest",
"Contrary to a single tree CART model, random forest takes random subsamples of both the observations and features from the training data (bagging), and trains decision trees on those cases.",
"A whole army of such decision trees are grown, and the most common outcome for each observation is used as the final result.",
"Such approach enable random forests to limit overfitting, which makes them very powerful classifiers.",
"For the purpose of random forest classification, we employed the R package randomForest, which implements Breinman's random forest algorithm.",
"As the number of trees, we used the randomForest's default value of $n_{\\rm tree}=500$ , and the subsamples were chosen randomly with replacement.",
"The number of variables that were randomly sampled was set at $\\sqrt{p}$ , where $p$ is the total number of variables in the original data set ($p=10$ in our case).",
"The winning class was defined as the one with the highest ratio of proportion of votes to the cutoff parameter, where the latter was set at $1/k$ , where $k$ is the number of classes ($k=3$ in our case).",
"The minimum terminal node size, which controls the depth of the tree (the larger the parameter is, the smaller the tree will be), was set at unity, while in terms of the number of terminal nodes, the trees were let to grow to the maximum possible size."
],
[
"Gradient boosting",
"While random forest is a bagging method with trees being run in parallel and without interaction, gradient boosting is an ensemble method where decision trees are added to learn the misclassification errors in existing models, and this sequentially boosts the training procedure.",
"Because gradient boosting is a greedy algorithm (i.e.",
"it makes the optimal choice at each step (locally optimal choice) as it tries to reach the overall optimal way to solve the classification problem), it can quickly overfit the training data set.",
"The boosting was performed using the R's gbm algorithm, whose implementation follows the Friedman's GBM ([34]).",
"Because our classification problem is composed of more than two classes, the analysis was carried out as a multinomial version.",
"As the metric of the information retrieval measure, we used the normalised discounted cumulative gain (metric = ndcg).",
"The total number of trees to fit, which corresponds to the number of iterations, was set at $n_{\\rm tree}=500$ as in the case of our random forest classification.",
"The value of $n_{\\rm tree}$ also corresponds to the number of basis functions that are being iteratively added in the boosting process (each additional basis function further reduces the loss function).",
"The interaction depth, which represents the maximum depth of variable interactions, was fixed at $k=3$ .",
"Hence, the number of terminal nodes or leaves, which is given by $J=k+1$ , was set at $J=4$ (for comparison, $J=2$ , or a decision stump, means that no interactions between variables is allowed).",
"The shrinkage parameter ($0<\\nu \\le 1$ ), which represent the learning rate, was set at $\\nu =0.005$ .",
"High learning rates of $\\nu \\simeq 1$ ($\\nu =1$ means no shrinking) are expected to result in overfit models, while small shrinkage parameter values ($\\nu \\le 0.1$ ) slow down the learning process, but are expected to lead to much lower generalisation error ([33]).",
"Finally, the minimum number of observations in the trees' terminal nodes was set to unity.",
"Such small value was chosen because our training samples were so small."
],
[
"Logistic regression",
"Despite its name, logistic regression is a classification algorithm rather than a regression technique.",
"Logistic regression uses a logistic function and the predictor feature values to model the probabilities for an instance to belong to different classes.",
"To estimate a multinomial logistic regression model, we used the R algorithm called multinom.",
"The algorithm predicted the probabilities for each source to belong to the three different classes (Class 0, Class I, and flat-spectrum sources), and the final assignment was done according to the highest predicted probability."
],
[
"Naïve Bayes",
"Similarly to logistic regression, naïve Bayes classifiers belong to a family of probabilistic classifiers.",
"Here, the classification relies on the Bayes' Theorem under the naïve assumption that the features are independent of each other.",
"To calculate the conditional posterior probabilities for our categorical protostellar class variable given the flux densities as predictor variables, we used the naiveBayes algorithm of R. No Laplace smoothing was applied, which would prevent the frequency-based probability estimate to be equal to zero as a result of a given class and feature value never occuring together in the training data.",
"The latter is not an issue for the present data set, because the missing feature values were imputed."
],
[
"$k$ -nearest neighbours",
"The $k$ -NN algorithm is a member of the so-called instance-based, lazy-learning algorithms (Mitchell 1997).",
"Here, a new, unseen instance is classified by comparing it with those $k$ training cases that are closest in feature space (i.e.",
"have similar properties with the new case), and the new case's class is then determined by a majority vote of its neighbours' classes.",
"To find these $k$ -nearest neighbours, the value of $k$ needs to be specified, and a distance metric is required.",
"To carry out a $k$ -NN classification, we used the R's knn3 algorithm.",
"We run our $k$ -NN classification by experimenting with different values of $k$ , ranging from $k=1$ to $k=15$ , and by adopting the Euclidean distance metric.",
"As shown in Fig.",
"REF , the best classification performance was reached when $k=1$ (considering only the closest neighbour).",
"The $y$ -axis of Fig.",
"REF shows the overall accuracy of the classification, which is defined as the ratio of cases that are correctly classified to the total number of cases.",
"However, a 1-NN classifier can lead to overfitting and does not generalise well enough to other YSO samples (a small $k$ means that noise has a higher effect on the classification).",
"Hence, we consider the next highest accuracy that was reached when $k=7$ in our subsequent comparison of different classifiers.",
"Figure: Accuracy of the kk-NN classifier as a function of the number of nearest neighbours.",
"The best peformance was reached when k=1k=1, but we selected the next best performace reached with k=7k=7 because the 1-NN classifier is subject tooverfitting (see text for details)."
],
[
"Support vector machine",
"The basic SVMs are binary classifiers, where the idea is to find the dividing hyperplane that both separates the two classes in the training data set and maximises the margin between the boundary members, that is between the SVs.",
"A new instance is classified by examining on which side of the hyperplane it falls.",
"In the case of non-linear classification, the goal is to find a hyperplane that is a non-linear function of the input variables.",
"This is done by the so-called kernel trick, where the input features are mapped into a higher dimensional feature space.",
"Besides binary classification, SVMs can also perform multiclass classification, which is the case in the present study ($k=3$ classes).",
"There are various options to do that, and we used a balanced one-against-one classification strategy, where three binary classifiers were trained (the number of classifiers is given by $k(k-1)/2$ ), and a simple voting strategy among them was applied to classify a new instance.",
"Our SVM classification was done using the ksvm algorithm of R, and to create a non-linear classifier we used a Gaussian radial basis function (RBF) kernel.",
"The algorithm was set to calculate the inverse kernel width for the RBF directly from the data (rather than specifying its value).",
"The value of the regularisation constant $C$ was set equal to unity, where the effect of $C$ is such that the larger it is, the narrower the margin between the SVs is, and hence the classifier is more prone to overfitting.",
"A smaller $C$ means a wider margin, and hence more misclassifications in the training set, which in turn can lead to underfitting issues."
],
[
"Neural network",
"A neural network classifier reads in the input features in the so-called input layer, and, in the case of a multi-layer perceptron, attempts to learn a non-linear function approximator to correctly classify the training cases' target variables appearing in the so-called output layer.",
"To build a neural network classifier, we used the R package nnet, which fits a single-hidden-layer neural network, that is there is only one hidden layer between the aforementioned input and output layers.",
"The feature data are being weighted, and transfered to the nodes or neurons in the hidden layer.",
"The hidden layer neurons process the sum of the weighted inputs by applying a so-called transfer function, and pass the results forward.",
"In the present work, we adopted a sigmoid shaped logistic transfer function, which is appropriate for discrete outputs such as protostellar classes.",
"The maximum number of iterations used was set to 100, and the weight decay in the weight update rule was set equal to zero, which means that the weights did not exponentially decay to zero in case of no other updates were being scheduled (during the training phase, the update steps modify the weights applied on the input features).",
"We experimented with different numbers of neurons in the hidden layer, ranging from two to nine, where the latter number is equal to the number of features (ten flux densities) minus one.",
"Using too many layers or neurons in the net can lead to overfitting, and hence we only employd a single-hidden-layer model.",
"As shown in the left panel in Fig.",
"REF , the best performance was found when there were eight neurons.",
"The corresponding neural net is shown in the right panel in Fig.",
"REF .",
"Figure: Left: Accuracy of the single-layer neural network classifier as a function of the number ofneurons or nodes in the hidden layer.",
"The red, filled circles indicates the highest accuracy, which was obtained with eight nodes.Right: Artifical neural net comprised of eight hidden layer nodes (labelled as H1 through H8), which wasfound to yield the best accuracy among the tested neural nets as shown in the left panel.",
"The input features (labelled as I1–I10)are the ten continuum flux densities from FFA16, and the outputs are the protostellar classes (Class 0, Class I, and flat-spectrum sources,labelled as O1, O2, and O3, respectively).",
"The black lines between the layers represent positive weights, while the grey lines indicatenegative weights.",
"The thickness of the lines is proportional to the magnitude of the weight with respect to all other weights."
],
[
"Performance metrics",
"To quantify and visualise the performance of each of the aforementioned classification algorithm, we derived their confusion matrices (see Fig.",
"REF ).",
"The columns of each matrix represent the instances in an actual, SED-based class (FFA16), while the rows represent the instances in a predicted class.",
"The diagonal elements of a confusion matrix show the cases where the predicted class is the same as the true (SED-based) class, while the off-diagonal elements represent the misclassified cases.",
"Hence, the larger the diagonal element numbers are, the better the classifier has performed.",
"Several different parameters that characterise the performance of a classifier can be calculated from the confusion matrix (e.g.",
"[30]), but here we focus only on four of them, namely the aforementioned overall accuracy, which tells how often the classifier is correct (the sum of the diagonal elements of the confusion matrix divided by the total number of cases), purity of a class (ratio between the correctly classified sources of a class and the number of sources classified in that class), contamination of a class (ratio between the misclassified sources in a class and the number of sources classified in that class), which is given by ${\\rm contamination}=1-{\\rm purity}$ (e.g.",
"[90]), and the Matthews correlation coefficient ([61]), or the phi coefficient, which is defined as ${\\rm MCC}=\\frac{{\\rm TP \\times TN - FP \\times FN}}{\\sqrt{{\\rm (TP+FP)\\times (TP+FN)\\times (TN+FP)\\times (TN+FN)}}}\\,,$ where TP, TN, FP, and FN are the numbers of true positives, true negatives, false positives, and false negatives, respectively.",
"The MCC can be considered a correlation coefficient between the true and predicted binary classifications, and its value lies in the range of $-1$ to $+1$ , where $-1$ means a full disagreement between the predicted and true classes, 0 is equivalent to random guess, and $+1$ indicates a perfect prediction performance.",
"Because we are dealing with a multiclass classification (rather than binary classification), we calculated the so-called micro-averaged MCCs, that is we summed all the TP, TN, FP, and FN values for each class to calculate the MCC.",
"More precisely, the TPs were derived by taking the sum of the confusion matrix diagonal elements, the TNs were calculated by removing the target class' row and column from the confusion matrix, and then taking the sum of all the remaining elements, the FPs were calculated as the sum of the respective column, minus the diagonal element for the class under consideration, and the FNs were computed by taking the sum of the respective row elements, minus the diagonal elements.",
"The performance metrics are tabulated in Table REF ."
],
[
"Performance of the tested classifiers",
"The evolutionary stages of our target protostellar objects were originally derived by FFA16 using an SED analysis.",
"Hence, by comparing our classifications with respect to these SED-based classes, we are assuming that the SED classes are correct.",
"Although panchromatic SEDs are expected to yield some of the most reliable source evolutionary stages (if not even the most reliable ones), it is still good to keep in mind the possibility that a machine learning classifier could predict a correct evolutionary stage for a source although it would differ from its SED class.",
"The SED-based source classification itself can depend on the exact method of how the analysis is performed (e.g.",
"modified blackbody fitting versus fitting based on radiative transfer models as done in FFA16).",
"Moreover, the assumptions about the dust grain properties affect the dust-based physical properties of the source, and hence the corresponding evolutionary stage.",
"From an observational point of view, the source inclination angle and variability might also affect the inferred evolutionary stage (e.g.",
"the central protostar might be visible if observed through the outflow cavity).",
"Related to the issues of the SED analysis, we remind the reader that all the FFA16 sources were assumed to lie at the same distance (Sect. 2).",
"Hence, we did not use the distance as a separate feature in our supervised source classification.",
"However, if the source sample is being drawn from different star-forming regions that lie at different distances from the Sun, the distance should be included as a feature because some of the fundamental source properties depend on it (e.g.",
"the mass scales as $d^2$ ).",
"In the following subsections, we briefly discuss the performance of each tested classifier.",
"The algorithms are discussed in the order of increasing classification accuracy, which were found to range from 47% to 82% (Table REF )."
],
[
"Naïve Bayes",
"The poorest job in the present classification analysis was done by the naïve Bayes classifier with an accuracy of only 47% and an MCC of 0.20.",
"For comparison, among three possible classes as in the present study (Class 0, Class I, and flat sources), the accuracy of random guess would be $\\sim 33\\%$ .",
"As described in Sect.",
"3.5.5, naïve Bayes classifier is based on the assumption that the predictors are independent of each other.",
"Considering the present set of features, which is composed of continuum flux densities, the assumption of their independence is certainly violated.",
"For example, as mentioned in Sect.",
"3.2, the far-IR and submm flux densities explored in the present work depend on each other via dust opacity."
],
[
"$k$ -nearest neighbours",
"With an accuracy of only 54% and an MCC of 0.31, the $k$ -NN classifier was found to be comparable to the naïve Bayes classifier.",
"As described in Sect.",
"3.5.6, the number of nearest neighbours we considered was set to $k=7$ .",
"The decision of how many neighbours to take into account controls the model's ability to generalise to future data instances.",
"Although the exact value of the optimal $k$ is dependent on the analysed data set, there are a few general things to keep in mind.",
"First, if only a single nearest neighbour is considered, the classifier is subject to noisy data features.",
"For this reason, we did not adopt the value $k=1$ although it yielded a better accuracy than using $k=7$ .",
"Secondly, while a large $k$ reduces the influence of noisy data, it is computationally more expensive and suffers from the possibility of ignoring important, small-scale patterns (and one might end up considering cases that are not even actual neighbours anymore).",
"Hence, the optimal $k$ is expected to lie somewhere between these two extreme cases (e.g.",
"[56]).",
"A common empirical rule of thumb is to set $k$ equal to $\\sqrt{n_{\\rm train}}$ , where $n_{\\rm train}$ is the size of the training data set (e.g.",
"[38], and references therein; [56]).",
"This will usually lead to large values of $k$ (i.e.",
"many neighbours), which reduces the effect of variance caused by noisy data.",
"In our 10-fold CV analysis, nine folds were used to train the $k$ -NN classifier, which means that $n_{\\rm train}$ was roughly $\\sim 270$ , which would suggest a value of $k\\simeq 16$ .",
"The optimal number of nearest neighbours we found (in terms of accuracy), $k=7$ , is over two times smaller than suggested by the aforementioned rule of thumb.",
"Hence, although possibly being time-consuming, the best value of $k$ should probably be searched using a cross-validation approach as in the present study (see Fig.",
"REF )."
],
[
"Support vector machine",
"The accuracy of our SVM classifier, 68%, is a factor of 1.26 better than that of our $k$ -NN classifier.",
"Moreover, the MCC of our SVM ($\\textrm {MCC}=0.52$ ) just exceeds a binary classification threshold of 0.50 between pure guessing ($\\textrm {MCC=0}$ ) and perfect prediction ($\\textrm {MCC=1}$ ).",
"By tuning the hyperparameters of the SVM, such as the $C$ parameter, the classification accuracy could potentially be improved, although the risk for overfitting might increase accordingly."
],
[
"Decision tree",
"A simple decision tree algorithm was found to perform fairly well as compared to the other algorithms tested in the present work.",
"The accuracy and MCC (0.71 and 0.57) of our decision tree classifier are comparable to those derived for the SVM."
],
[
"Logistic regression",
"Our multinomial logistic regression yielded a classifcation accuracy of 79% with fairly pure classes (purity is 0.81, 0.71, and 0.91 for the Class 0, Class I, and flat-spectrum sources, respectively).",
"The derived MCC of 0.68 is a factor of 1.19 larger than for our decision tree model."
],
[
"Neural network",
"Eighty percent of the test cases were correctly classified by our neural network classifier.",
"Also, an MCC of 0.70 derived for the classifier shows that its prediction performance is fairly good among the tested algorithms.",
"Overall, the performance metrics of our neural network are comparable to our logistic regression.",
"This is perhaps unsurprising, because the transfer function in our neural net was taken to be the logistic sigmoid function."
],
[
"Random forest",
"The second highest classification accuracy (81%) and an MCC (0.71) were derived for our random forest classifier.",
"As expected, a random forest classifier did a much better job than a simple decision tree, which is an indication that the former generalises better on unseen data than the latter."
],
[
"Gradient boosting",
"The best classification performance (82% accuracy) was obtained with a GBM.",
"Our gradient boosting classifier led to fairly pure classifications per class (0.84, 0.79, and 0.83 for the Class 0, Class I, and flat-spectrum sources), and hence low contaminations.",
"Its MCC of 0.73 also indicates a reasonable prediction performance.",
"As expected, GBM outperforms the simple decision tree (a factor of 1.15 better accuracy), while our GBM was found to be only marginally (factor of 1.01) more accurate than our random forest classifier with an accuracy of 81% and MCC of 0.71 (the second most accurate classifier in the present study).",
"Table: Performance metrics of the tested machine learning classifiers in order of increasing accuracy.Figure: Colour-coded confusion matrices showing the performance of the tested machine learning classifiers."
],
[
"Unveiling the most important wavelength bands for classifying young stellar objects via supervised learning algorithms",
"The PCA presented in Sect.",
"3.4 suggests that the 3.6 $\\mu $ m flux density is the most informative feature in the preset study (the feature explains 56.1% of the variance of the data; see Fig.",
"REF ).",
"For comparison, the next most important band was found to be the 4.5 $\\mu $ m band, which explains 18.1% of the data variance.",
"As an alternative approach to unveil the most important band for the present YSO classification, we employed the leave-one-out cross-validation (LOOCV) technique, that is the classifications were done by using nine out of the ten features (ranging from the Spitzer/IRAC 3.6 $\\mu $ m band to LABOCA 870 $\\mu $ m band), and this process was repeated ten times with a different wavelength band left out every time.",
"In Table REF , we tabulate the classification accuracies of each of the tested algorithm when one out of the ten most relevant features was being left out.",
"The results are presented visually in Fig.",
"REF .",
"We note that in the cases of the $k$ -NN and neural network classifiers, the hyperparameter settings (the $k$ value and number of neurons) were identical to those used for the full feature set.",
"The results suggest that if the Spitzer/IRAC 3.6 $\\mu $ m band is ignored, the classification accuracies drop with respect to the full feature set (though only by factors of 1–1.08), which conforms with the PCA result of 3.6 $\\mu $ m band being the most informative one.",
"The importance of the 3.6 $\\mu $ m band could, at least partly, be related to the shocked H$_2$ emission associated with protostellar outflows, although such shock emission is stronger at 4.5 $\\mu $ m (e.g.",
"[96]).",
"There is also a polycyclic aromatic hydrocarbon feature at 3.3 $\\mu $ m owing to C-H stretching that might contribute to the Spitzer/IRAC band 1 emission (e.g.",
"[26] for a review).",
"Also, the Spitzer/MIPS 24 $\\mu $ m band, which is sensitive to warm dust emission (e.g.",
"Rathborne et al.",
"2010), appears to be a fairly important feature for most of the classifiers; if the band is ignored, the classification accuracies drop by factors of 0.96–1.14.",
"For the decision tree and naïve Bayes classifiers, however, the classification accuracy was actually marginally higher when the 24 $\\mu $ m data were ignored (by a factor of 1.04 in both cases).",
"From a physical point of view, the most important wavelength bands are expected to be those probing the peak of the source SED (i.e.",
"around $\\sim 100$ $\\mu $ m; see Sect.",
"2), but this is not manifested in our LOOCV feature selection.",
"As mentioned above, in some cases it was found that the exclusion of a wavelength band actually increases the classification accuracy with respect to the case where all the ten features are being used.",
"Most notably, this happens when the Herschel/PACS 160 $\\mu $ m band is ignored from the $k$ -NN classification (the accuracy increases by a factor of 1.13; see the green curve in Fig.",
"REF ).",
"This suggests that the inclusion of the aforementioned band might have led to a slight overfitting effect in our $k$ -NN classification.",
"We also note that the random forest and GBM are generally found to yield the best classification accuracies when ignoring the different features, but in the case where the 3.6 $\\mu $ m band was ignored, our neural network appeared equally good as the GBM (76% accuracy for both).",
"In the latter case, the random forest was found to yield the highest classification accuracy of 79%, but the difference is only marginal with respect to the GBM.",
"Also, these small differences in the accuracies compared to the usage of all the ten features can not be considered significant owing to the random sampling nature of the both the random forest algorithm and the 10-fold CV technique.",
"Of course, there are many feature combinations among the considered flux densities that could be explored.",
"For example, there are 45 unique flux density pairs in the present set of ten different flux densities.",
"However, a thorough feature analysis is beyond the scope of the present study.",
"Considering the future YSO surveys where similar machine learning approaches could be used as in the present study, the observed wavelengths are likely to differ from those we have analysed (e.g.",
"the cryogenic phase of Spitzer operated from 2003 to 2009, while its warm mission (using the 3.6 $\\mu $ m and 4.5 $\\mu $ m IRAC bands only) is scheduled to end in March 2019 (e.g.",
"[43]; [97]), and the Herschel mission ended on 29 April 2013 when the satellite ran out of its helium coolant (e.g.",
"[81])).",
"Nevertheless, the aforementioned analysis suggests that bands near 3.6 $\\mu $ m and 24 $\\mu $ m would be informative if the other bands are comparable to those employed here.",
"Table: Classification accuracy when one of the ten continuum bands was left out of consideration.Figure: Behaviour of the classification accuracy when one of the ten continuum bands was left out of consideration.",
"For reference,the first data point from left shows the accuracy when all the ten features were used (see column (2) in Table )."
],
[
"Potential of using machine learning in classifying young stellar objects",
"Owing to the fact that we considered only ten out of the original 29 features (i.e.",
"the 2MASS and Spitzer/IRS flux densities were ignored), a protostellar classification accuracy of 82% with a GBM can be considered fairly good.",
"Also, although being the largest, homogeneous protostar sample drawn from a single molecular cloud system, the size of the employed data set is fairly small for a machine learning approach (only 319 sources in total), and undoubtedly a higher classification accuracy could be reached with a larger training sample.",
"On the other hand, as shown in Fig.",
"REF , the Orion Class 0, Class I, and flat-spectrum sources from FFA16 are not well separable in the two-dimensional projections of the feature space, which is an indication that learning their classification is demanding for at least some of the supervised classifiers.",
"A more detailed missing value imputation, such as estimating the missing far-IR to submm flux densities in a source-by-source fashion from the existing values by assuming a value of $\\beta $ (see Sect.",
"3.2), could lead to an improved performance.",
"However, this approach is also based on assumptions about the flux frequency dependence at different bands.",
"Finally, an application of more advanced ensemble methods, like the extreme gradient boosting (XGBoost; [19]), has the potential to lead to an improved accuracy.",
"As mentioned in Sect.",
"4.3, the future YSO surveys where machine learning assisted source classification could be used will undoubtedly be carried out at least partly at different wavelengths than considered here.",
"This also means that the models trained in the present work can not be employed as such, but they would need to be retrained (cf.",
"the deep learning knowledge transfer study by Domínguez Sánchez et al.",
"(2018)).",
"In fact, even in a hypothetical case where there would be a survey of YSOs carried out at exactly the same ten bands as we considered, the rms noise levels of the observations would probably still be different, which would again call for retraining the classifiers (for example, deeper surveys could detected weaker sources than those in the FFA16 Orion sample).",
"Regarding this issue, we note that the $10\\sigma $ limiting magnitudes in the Spitzer/IRAC 3.6, 4.5, 5.8, and 8 $\\mu $ m data, and the Spitzer/MIPS 24 $\\mu $ m data employed by FFA16 were 16.5, 16.0, 14.0, 13.0, and 8.5 mag, respectively ([64]; [87]), and Megeath et al.",
"(2012) derived the final magnitudes for all their Spitzer sources that were detected with uncertainties of $\\le 0.25$ mag in one of the four IRAC bands.",
"The properties of the Herschel and APEX data products employed by FFA16 will be described in more detail by B. Ali et al.",
"(in prep.)",
"and T. Stanke et al.",
"(in prep.",
"); see Fischer et al.",
"(2017).",
"Moreover, the source flux densities depend on the aperture sizes used to extract the photometry, and if the apertures differ from those used by FFA16 (who, for example, used the aperture radii of 96, 96, and 128 for the Herschel 70, 100, and 160 $\\mu $ m data, respectively), the present classifiers would again have to be retrained.",
"Although reaching high accuracies, a supervised machine learning classification cannot replace an SED-based YSO classification because an SED analysis also yields the important physical properties of the source, like the dust temperature and (envelope) dust mass (however, the SED fitting itself could also rely on machine learning regression).",
"Moreover, SED analyses are still expected to be needed to create the training data sets for machine learning applications.",
"Nevertheless, if appropriate training data sets are available, machine learning techniques can serve as a quick way to estimate the relative percentages of YSOs in different evolutionary stages in the era of big astronomical data (e.g.",
"[42]; [73]), and also to mine the YSO data to unveil interesting subsamples for more detailed follow-up observations (e.g.",
"[60])."
],
[
"Summary and conclusions",
"We used eight different supervised machine learning algorithms to classify the Orion protostellar objects from FFA16 into Class 0, Class I, and flat-spectrum sources.",
"On the basis of PCA, we employed only the IR and submm continuum photometric data from FFA16.",
"The training and testing of the classifiers were performed by using a 10-fold CV technique.",
"Using the SED-based classifications of FFA16 as the benchmark, we found that the highest classification accuracy is reached by a GBM algorithm (82% of the cases were correctly classified with $\\gtrsim 80\\%$ purity and an MCC of 0.73), while the poorest performance was that of naïve Bayes classification (47% accuracy).",
"Our analysis suggests that among the ten continuum emission bands used in the classification, the Spitzer 3.6 $\\mu $ m and 24 $\\mu $ m flux densities are the most informative features in terms of the source classification accuracy.",
"Hence, these two wavelength bands would be useful to include in a panchromatic YSO classification study, especially if the other bands available are comparable to those analysed in the present work (i.e.",
"4.5, 5.8, 8.0, 70, 100, 160, 350, and 870 $\\mu $ m).",
"Larger data sets, detailed missing value imputations, and more sophisticated learning algorithms have the potential to improve the classification accuracies.",
"Overall, machine learning algorithms can provide a fast (at least compared to an SED analysis) way to classify large samples of sources into different evolutionary stages, and hence estimate the statistical lifetimes of the sources, and also pick up subsamples of interesting sources for targeted follow-up studies.",
"However, an obvious challenge of supervised machine learning classification is the creation of training data sets, which requires classification of large numbers of YSOs into different evolutionary stages on the basis of their measured flux densities at the observed wavelengths.",
"Because the latter is based on SED fitting, which itself requires knowledge of the source distance and assumptions about the underlying dust grain model, an SED analysis might be a prerequisite to the usage of machine learning classifiers on new survey data sets.",
"I would like to thank the referee for providing constructive comments and suggestions that helped to improve the quality of this paper.",
"This research has made use of NASA's Astrophysics Data System and the NASA/IPAC Infrared Science Archive, which is operated by the JPL, California Institute of Technology, under contract with the NASA."
]
] | 1808.08371 |
[
[
"Whitney's Theorem, Triangular Sets and Probabilistic Descent on\n Manifolds"
],
[
"Abstract We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients.",
"The system of polynomials is assumed to be triangularized.",
"An application of Whitney's embedding theorem allows us to work in a reduced dimensional embedding space.",
"A numerical continuation method applied to the reduced-dimensional embedded manifold is used to drive the procedure."
],
[
"Introduction",
"In an optimization problem with equality constraints the feasible set often has nice geometric properties.",
"If we `stand' at a typical point the feasible set will locally look to us like a Euclidean space.",
"But this locally Euclidean space will be contained in a larger space.",
"For example, the feasible set could be a one-dimensional string contained in a hundred-dimensional ambient space.",
"It seems like a waste to work with a hundred variables when the geometric structure we're actually interested in is only one-dimensional.",
"Whiney's theorem tells us that in fact we could work in a three-dimensional ambient space.",
"The string can be projected from the original hundred dimensions into a random three-dimensional subspace.",
"Provided we can find a map from the projected string back to the original string we lose nothing by working in the three-dimensional space.",
"Sard's theorem is well known in optimization since it tells us that equality constraints often give a reasonable feasible set over which we work.",
"Or at least it will alert us when this may not be the case.",
"Whitney's theorem is perhaps less well known.",
"The theorem offers a potentially large decrease in the effective dimension one needs to optimize in, which is certainly an attractive proposal.",
"However this is not cost free.",
"Firstly, the decrease in dimensionality is achieved by a linear projection from the initial ambient space into a lower-dimensional working space.",
"This requires some sort of variable elimination from the set of equality constraints in order to do the projection.",
"The variable elimination is the nonlinear analogue of triangularizing a matrix.",
"Secondly, the reconstruction from the reduced dimension working space back into the original space is generally nonlinear.",
"A numerical implementation of the inverse (implicit) function theorem over a manifold is required.",
"This is a nonlinear version of basic and nonbasic variables.",
"The manifold in the reduced dimensional space serves as the basic variables and the embedding in the higher dimensional space represents the nonbasic variables.",
"A decade ago the author was interested in applying Whitney's theorem to optimization problems.",
"Unfortunately the required machinery was not quite sufficient enough to give any satisfactory results.",
"In the intervening years developments in polynomial decompositions over real numbers has made it possible to provide a sketch of the main ideas.",
"While the method is not quite complete it is complete enough to offer some potential applications and, more importantly, a clear map of what work still needs to be done.",
"In other words: it works but it may not yet work well enough.",
"The paper is organized as follows.",
"We give a simple example in Section that demonstrates the basic problem and principles we will be dealing with.",
"Section covers the necessary results from differential geometry.",
"The main result is Whitney's weak embedding theorem which gives an upper bound on the minimal embedding dimension of a manifold.",
"In Section we look at some needed results from real algebraic geometry.",
"In particular some real polynomials decompositions are introduced.",
"These put the constraint equations into a form convenient for applying Whitney's theorem, which we examine in Section .",
"Moving over manifolds in a controllable and intelligent way is necessary.",
"Section looks at some methods to accomplish this.",
"The full optimization procedure is laid out in Section with Section providing an example.",
"A discussion follows in Section ."
],
[
"Motivating Example",
"Here we give a simple example in order to motivate and demonstrate the methodology that will be developed.",
"Consider the optimization problem $\\min _{\\mathbf {z} \\in \\mathbb {R}^{4}} & f(\\mathbf {z})$ subject to the constraints $0 &= G_1(\\mathbf {z}) = z_4 + z_3 + z_2 + z_1 \\mbox{,} \\\\0 &= G_2(\\mathbf {z}) = z_4z_1 + z_4z_3 + z_3z_2 + z_2z_1 \\mbox{,} \\\\0 &= G_3(\\mathbf {z}) = z_4z_2z_1 + z_4z_3z_1 + z_4z_3z_2 + z_3z_2z_1 \\mbox{ and } \\\\0 &= G_4(\\mathbf {z}) = z_4z_3z_2z_1 - 1 \\mbox{.",
"}$ The constraint set can be rewritten in the form [1] $\\begin{split}0 &= g_1(x,u) = u^2 x^2 - 1 \\\\0 &= g_2(y_1,u) = y_1 + u \\\\0 &= g_3(y_2,x) = y_2 + x\\end{split}$ where we renamed the variables $u = z_1$ , $x = z_2$ , $y_1 = z_3$ and $y_2 = z_4$ .",
"Looking at the polynomials in (REF ) we see that they are in a `triangular form': $g_i$ only depends on $u$ , $x$ or $y_j$ where $j < i$ .",
"Looking at the new polynomials we see that it is not necessary to work in $\\mathbb {R}^{4}$ .",
"Only the $(x,u) \\in \\mathbb {R}^{2} $ need to be explicitly optimized over, treating $y_1(x,u) = - u$ and $y_2(x,u) = -x$ as functions of $u$ and $x$ .",
"The original optimization problem now becomes $\\min _{(x,u) \\in \\mathbb {R}^{2}} \\hat{f}(y_1(x,u),y_2(x,u),x,u)\\mbox{ subject to }g_1(x,u) = 0$ where $f(z_1,z_2,z_3,z_4) = \\hat{f}(z_3,z_4,z_2,z_1)$ .",
"[We will typically drop the `hat' notation on the objective function when the variables are simply permuted.",
"The context should make it clear when this occurs.]",
"This example highlights the key techniques we will employ.",
"First, we have an optimization problem initially formulated in a `large' Euclidean space.",
"The constraints are given by polynomials with rational coefficients.",
"A triangular decomposition can be performed on the original constraint statements to derive a new set of constraints.",
"This new constraint set is then used to reduce the effective dimension we need to optimize in.",
"The optimization can now proceed in a lower-dimensional Euclidean space versus the original one.",
"As part of the projection into the lower-dimensional space an inverse function is used to map from the feasible set in the lower dimension back to the original feasible set."
],
[
"Whitney's Theorem",
"A manifold $\\mathcal {M}$ is a subset of $\\mathbb {R}^{n}$ that looks locally like $\\mathbb {R}^{m}$ , $n > m$ .",
"Let $\\mathcal {I}$ be an index set and for each $i \\in \\mathcal {I}$ let $\\mathcal {V}_{i} \\subset \\mathcal {M} $ be an open subset where $\\bigcup _{i \\in \\mathcal {I}} \\mathcal {V}_{i} = \\mathcal {M} $ .",
"Additionally, let there be homeomorphisms $\\phi _{i} : \\mathcal {V}_{i} \\rightarrow \\mathbb {R}^{m} $ .",
"Each pair $(\\mathcal {V}_{i}, \\phi _{i})$ is called a chart and the set $\\lbrace (\\mathcal {V}_{i}, \\phi _{i}) \\rbrace _{i \\in \\mathcal {I}}$ is called an atlas.",
"When two open subsets $\\mathcal {V}_{i}$ and $\\mathcal {V}_{j}$ of $\\mathcal {M}$ overlap there will be a homeomorphic transition function $\\tau _{ij} : \\mathbb {R}^{m} \\rightarrow \\mathbb {R}^{m} $ on the overlap $\\mathcal {V}_{i} \\cap \\mathcal {V}_{j}$ defined by $\\tau _{ij} = \\phi _{j} \\circ \\phi _{i}^{-1}$ .",
"The transition functions allow us to stitch together the charts into a coherent whole.",
"This then defines a topological manifold.",
"The transition functions are often taken to be differentiable.",
"In this case we have a differentiable manifold.",
"If the $\\tau _{ij}$ are all $\\mathcal {C}^{k}$ , $k \\ge 1$ , then $\\mathcal {M}$ is said to be a $\\mathcal {C}^{k}$ -manifold.",
"If $\\mathcal {M}$ is a $\\mathcal {C}^{\\infty }$ -manifold then it is called smooth.",
"Every $\\mathcal {C}^{k}$ -manifold can be made smooth [2].",
"At every point $p \\in \\mathcal {M} \\subset \\mathbb {R}^{n} $ a tangent space $\\mathcal {T}_{p}\\mathcal {M}$ can be attached.",
"If an inner product $h: \\mathcal {T}_{p}\\mathcal {M} \\times \\mathcal {T}_{p}\\mathcal {M} \\rightarrow \\mathbb {R}$ is defined for every $p \\in \\mathcal {M} $ and varies smoothly over $\\mathcal {M}$ then we have a Riemannian manifold.",
"The function $h(\\cdot ,\\cdot )$ is called a metric and with it notions of lengths and angles in $\\mathcal {T}_{p}\\mathcal {M}$ are available over $\\mathcal {M}$ .",
"The manifolds we consider arise from equality constraints.",
"[Regular level sets] Let $\\mathbf {g}: \\mathbb {R}^{m+k} \\rightarrow \\mathbb {R}^{k} $ .",
"For the set $\\mathcal {M} = \\lbrace \\mathbf {z}:\\mathbf {g}(\\mathbf {z}) = \\mathbf {c} \\rbrace $ assume $\\nabla \\mathbf {g}(\\mathbf {z})$ is full rank.",
"Then $\\mathcal {M}$ is a regular level set of $\\mathbf {g}(\\mathbf {z})$ .",
"Additionally, the null space of $\\nabla \\mathbf {g}(\\mathbf {z})$ coincides with the tangent space to $\\mathcal {M}$ .",
"$\\square $ This paper will consider equality constraints given by members of $\\mathbb {Q}[\\mathbf {z}]$ , the polynomials in $\\mathbf {z} = [z_1,\\ldots ,z_{m+k}]$ with rational coefficients.",
"Then by Sard's theorem almost every level set is a Riemannian manifold.",
"[Regular level set manifolds and Sard's theorem [2]] Let $\\mathbf {g}: \\mathbb {R}^{m+k} \\rightarrow \\mathbb {R}^{k} $ be a $\\mathcal {C}^{\\infty }$ function.",
"Then almost every level set of $\\mathbf {g}(\\mathbf {z})$ is regular and a $m$ -dimensional manifold.",
"A regular level set acquires a metric from its embedding in $\\mathbb {R}^{m+k}$ by restricting the standard Euclidean metric to the tangent space $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} $ given by $\\mathrm {null}(\\nabla \\mathbf {g}(\\mathbf {p}))$ for all $\\mathbf {p} \\in \\mathcal {M} $ .",
"One issue is that the $k$ in Theorem may be much larger than $m$ .",
"So even though the manifold may be low-dimensional it could be embedded in a very high-dimensional space.",
"Whitney's theorem gives an upper bound on the minimal embedding dimensional for any manifold.",
"[Whitney's weak embedding theorem [2]] Let $\\mathcal {M}$ be an $m$ -dimensional manifold.",
"Then $\\mathcal {M}$ can be embedded in $\\mathbb {R}^{2m+1}$ .",
"Further, if $\\mathcal {M}$ is embedded in $\\mathbb {R}^{m+k}$ , $k > m + 1$ , a random projection into a $2m+1$ subspace gives an embedding of $\\mathcal {M}$ in $\\mathbb {R}^{2m+1}$ ."
],
[
"Triangular Sets",
"Here we cover the necessary background from real algebraic geometry.",
"Our main focus is putting a set of polynomials $P \\subset \\mathbb {Q}[\\mathbf {z}] $ into a convenient form for utilizing Whitney's theorem.",
"[Semi-algebraic sets, real algebraic varieties and Nash manifolds] A set $S \\subset \\mathbb {R}^{n} $ is a semi-algebraic set if it satisfies a set of polynomial equations $g_i(\\mathbf {z}) = 0$ , $i=1,\\ldots ,j$ , and polynomial strict inequalities $h_i(\\mathbf {z}) > 0$ , $i=1,\\ldots ,l$ , or a finite union of such sets.",
"A real algebraic variety is defined using only a set of polynomial equations $g_i(\\mathbf {z}) = 0$ , $i=1,\\ldots ,j$ , without finite unions.",
"Nash manifolds are semi-algebraic sets that are also manifolds.",
"$\\square $ This paper focuses on Nash manifolds that are also varieties.",
"In order to utilize Whitney's theorem the polynomials defining a Nash manifold will need to be decomposed into a friendlier form.",
"Let the Nash manifold $\\mathcal {N} \\subset \\mathbb {R}^{m+k} $ be defined by the set of polynomial equations $g_i(\\mathbf {z}) = 0$ , $i=1,\\ldots ,k$ .",
"Assign to the entries of $\\mathbf {z}$ the ordering $z_1 < z_2 < \\ldots < z_{m+k} \\mbox{.",
"}$ Define $Q_i = \\mathbb {Q}[z_1,\\ldots ,z_i]$ as the ring of polynomials in the variables $z_1,\\ldots ,z_i$ with rational coefficients.",
"$\\mathtt {deg}(g,z_i)$ is the degree of polynomial $g$ in the variable $z_i$ .",
"$\\mathtt {mvar}(g)$ is the greatest variable $z_i$ such that $\\mathtt {deg}(g,z_i) \\ne 0$ .",
"We can decompose a $g \\in Q_i$ into the form $g = \\iota z_i^d + \\tau $ where $d = \\mathtt {deg}(g,z_i) > 0$ , $\\iota \\in Q_{i-1}$ and $\\tau \\in Q_i$ with $\\mathtt {deg}(\\tau ,z_i) < d$ .",
"Then $\\mathtt {init}(g) = \\iota $ is the initial of $g$ , $\\mathtt {mdeg}(g) = d$ is the main degree of $g$ , $\\mathtt {rank}(g) = z_i^d$ is the rank of $g$ , $\\mathtt {tail}(g) = \\tau $ is the tail of $g$ and $\\mathtt {head}(g) = g - \\mathtt {tail}(g) = \\iota z_i^d$ is the head of $g$ .",
"Every polynomial is then decomposed as $p = \\mathtt {init}(p) \\mathtt {rank}(p) + \\mathtt {tail}(p) $ .",
"[Triangular sets] Let $T \\subset Q_n$ .",
"We call $T$ a triangular set if no member of $T$ is constant and for any pair $g,h \\in T$ , $g \\ne h$ , we have $\\mathtt {mvar}(g) \\ne \\mathtt {mvar}(h) $ .",
"If $z_i = \\mathtt {mvar}(g) $ for some $g \\in T$ then $z_i$ is called algebraic with respect to $T$ .",
"Otherwise $z_i$ is called free with respect to $T$ .",
"$\\square $ Triangular sets are a nonlinear generalization of triangulating a matrix.",
"They are very convenient for using Whitney's theorem.",
"Let $T = \\lbrace z_1^2 z_2^2 -1, z_3 + z_1, z_4 + z_2 \\rbrace $ .",
"This is a triangular set where $\\mathtt {mvar}(z_1^2 z_2^2 -1) &= z_2 \\mbox{,} \\\\\\mathtt {mvar}(z_3 + z_1) &= z_3 \\mbox{ and }\\\\\\mathtt {mvar}(z_4 + z_2) &= z_4 \\mbox{.",
"}$ Then $z_1$ is free and the other variables are algebraic.",
"$\\square $ One of the difficulties we face with using triangular sets is that most of the prior work on them has been with $\\mathbb {C}[\\mathbf {z}]$ , the polynomials with complex coefficients.",
"Our concern is only with the real solutions of a system of polynomials in $\\mathbb {Q}[\\mathbf {z}]$ which has seen recent development [3], [4].",
"We'll develop the necessary machinery for working with Nash manifolds but first we'll look at a standard solution method over $\\mathbb {C}$ .",
"The algorithm developed by Kalkbrener in [1] seems like a reasonable choice for calculating triangular sets [5], [6].",
"This method builds on a particular type of triangular set which requires some additional machinery.",
"[Saturated ideals] Given a triangular set $T = \\lbrace t_1,\\ldots ,t_k\\rbrace $ the ideal $\\langle T \\rangle $ generated by $T$ is given by $\\langle T \\rangle &=\\left\\lbrace q \\in \\mathbb {Q}[\\mathbf {z}] :q = \\textstyle \\sum _i p_i t_i \\mbox{, } p_i \\in \\mathbb {Q}[\\mathbf {z}]\\right\\rbrace \\mbox{.",
"}$ Let $h_T = \\textstyle \\prod _i \\mathtt {init}(t_i) $ .",
"The saturated ideal of $T$ is defined as $\\mathtt {sat}(T) &=\\left\\lbrace q \\in \\mathbb {Q}[\\mathbf {z}] :\\exists n \\in \\mathbb {N}_0 \\mbox{ such that } h_T^n q \\in \\langle T \\rangle \\right\\rbrace $ with $\\mathtt {sat}(\\varnothing ) = \\langle 0 \\rangle $ .",
"$\\square $ The ideal $\\langle T \\rangle $ gives all valid equality constraints that follow from those in $T$ .",
"We see that $\\langle T \\rangle \\subset \\mathtt {sat}(T) $ but $\\mathtt {sat}(T)$ can contain additional polynomials.",
"[Regular chains] Let $T = \\lbrace t_1,\\ldots ,t_k\\rbrace $ be a triangular set.",
"A $p \\in \\mathbb {Q}[\\mathbf {z}] $ is regular modulo $\\mathtt {sat}(T)$ if $p \\notin \\mathtt {sat}(T) $ and there does not exist a $q \\in \\mathbb {Q}[\\mathbf {z}] $ where $q \\notin \\mathtt {sat}(T) $ but $pq \\in \\mathtt {sat}(T) $ .",
"Define $T_{<j} = \\lbrace t_1,\\ldots ,t_{j-1}\\rbrace $ , $j < k$ .",
"$T$ is a regular chain if $T = \\varnothing $ , or $T_{<k}$ is a regular chain and $\\mathtt {init}(t_k)$ is regular modulo $\\mathtt {sat}(T_{<k})$ .",
"$\\square $ [Kalkbrener triangular decomposition[4]] Let $\\mathcal {T} = \\lbrace T_1,\\ldots ,T_l\\rbrace $ be a set of regular chains and $V(F)$ be an algebraic variety.",
"$\\mathcal {T}$ is a Kalkbrener triangular decomposition of $V(F)$ if $V(F) = \\bigcup _i V(\\mathtt {sat}(T_i))$ .",
"$\\square $ So $V(\\mathtt {sat}(T_i))$ is geometrically more relevant to Kalkbrener's method than $V(T_i)$ : each $V(\\mathtt {sat}(T_i))$ gives some piece of $V(F)$ .",
"We can relate $V(T)$ and $V(\\mathtt {sat}(T))$ .",
"[Regular zeros] Given a triangular set $T = \\lbrace t_1,\\ldots ,t_k\\rbrace $ let $h_T = \\textstyle \\prod _i \\mathtt {init}(t_i) $ .",
"The regular zeros $W(T)$ of the triangular set $T$ is defined as $W(T) = V(T) - V(h_T)$ .",
"$\\square $ For a triangular set $T$ it is the case that [7] $V(\\mathtt {sat}(T)) &= \\overline{W(T)}$ where closure is with respect to the Zariski topology.",
"If $V(T)$ is also a manifold then $\\overline{W(T)} = V(T) = V(\\mathtt {sat}(T))$ .",
"[Radical of $\\langle T \\rangle $ ] The radical of the ideal $\\langle T \\rangle $ is $\\sqrt{\\langle T \\rangle } &=\\left\\lbrace q \\in \\mathbb {Q}[\\mathbf {z}]:\\exists n \\in \\mathbb {N}_0 \\mbox{ such that } q^n \\in \\langle T \\rangle \\right\\rbrace \\mbox{.",
"}$ That is, $\\sqrt{\\langle T \\rangle }$ consists of all those polynomials that vanish on $V(T)$ .",
"$\\square $ Since $V(T) = V(\\mathtt {sat}(T))$ it follows from Hilbert's Nullstellensatz [8] that $\\sqrt{\\langle T \\rangle } = \\sqrt{\\mathtt {sat}(T)}$ .",
"From this we see that a Kalkbrener triangular decomposition $\\mathcal {T} = \\lbrace T_1,\\ldots ,T_l\\rbrace $ of $F \\subset \\mathbb {Q}[\\mathbf {z}] $ gives $\\sqrt{\\langle F \\rangle } = \\bigcap _i \\sqrt{\\mathtt {sat}(T_i)}$ .",
"A Kalkbrener triangular decomposition allows for complex solutions but Nash manifolds only work with the real solutions."
],
[
"Whitney's Theorem Applied to Triangular Sets",
"We first look at an easy to understand case.",
"Consider the linear system $A\\mathbf {z} = \\mathbf {b}$ .",
"This set of polynomials defines a special type of manifold that is globally like $\\mathbb {R}^{m}$ when $A \\in \\mathbb {R}^{k \\times (m+k)} $ .",
"Let $k > m+1$ for what follows.",
"It is convenient to have $A$ in a triangular form $A &=\\begin{blockarray}{cccc}k-m-1 & m+1 & m & \\\\\\begin{block}{[ccc]c}A_{11} & A_{12} & A_{13} & k-m-1 \\\\0 & A_{22} & A_{23} & m+1 \\\\\\end{block}\\end{blockarray}$ with $A_{11}$ and $A_{22}$ upper triangular.",
"Partition the $\\mathbf {z}$ as $\\mathbf {z} &=[\\mathbf {y}\\ \\mathbf {x}\\ \\mathbf {u}]$ with $\\mathbf {y} \\in \\mathbb {R}^{k-m-1} $ , $\\mathbf {x} \\in \\mathbb {R}^{m+1} $ and $\\mathbf {u} \\in \\mathbb {R}^{m} $ .",
"Whitney's theorem tells us that the space given by $\\mathbf {x}$ and $\\mathbf {u}$ is sufficient to embed the manifold implicitly defined by the set of polynomials $A\\mathbf {z} - \\mathbf {b}$ .",
"Let the original manifold defined in $\\mathbb {R}^{m+k}$ be denoted by $\\mathcal {M}$ and the projected image in $\\mathbb {R}^{2m+1}$ be denoted by $\\mathcal {M} ^{\\prime }$ .",
"Partition $\\mathbf {b}$ as $\\mathbf {b} = [\\mathbf {b}_1\\ \\mathbf {b}_2]$ .",
"We can solve the equation $\\left[A_{22}\\ A_{23}\\right]\\left[\\begin{array}{c}\\mathbf {x} \\\\\\mathbf {u}\\end{array}\\right]&=\\mathbf {b}_2$ to find the points on $\\mathcal {M} ^{\\prime }$ .",
"Call one of these $[\\mathbf {x}^*\\ \\mathbf {u}^*]^T \\in \\mathcal {M} ^{\\prime }$ .",
"This can then be mapped onto the corresponding point on $\\mathcal {M}$ by solving $\\left[A_{11}\\ A_{12}\\ A_{13}\\right]\\left[\\begin{array}{c}\\mathbf {y} \\\\\\mathbf {x}^* \\\\\\mathbf {u}^*\\end{array}\\right]&=\\mathbf {b}_1$ for $\\mathbf {y}$ .",
"While we may not want to do it in this case, the optimization can be carried out in $\\mathbb {R}^{2m+1}$ and we can then map from $\\mathcal {M} ^{\\prime }$ to $\\mathcal {M}$ by finding $\\mathbf {y} = \\mathbf {y}(\\mathbf {x},\\mathbf {u})$ .",
"Using the full vector $\\mathbf {z} = [\\mathbf {y}\\ \\mathbf {x}\\ \\mathbf {u}]$ we can pull back any function defined on $\\mathcal {M}$ to $\\mathcal {M} ^{\\prime }$ .",
"A function that is pulled back could be, e.g., the objective function or an inequality constraint.",
"Now we can repeat the above example for the general nonlinear case.",
"We will make the simplifying assumption that $\\mathcal {M}$ is path connected.",
"If there are multiple pieces of $\\mathcal {M}$ then the following argument still carries through for each individual piece.",
"After a possible reordering of the coordinates, partition the $\\mathbf {z} \\in \\mathbb {R}^{m+k} $ from Definition as $\\mathbf {z} &=[\\mathbf {y}\\ \\mathbf {x}\\ \\mathbf {u}]$ where $\\mathbf {y} \\in \\mathbb {R}^{k-m-1} $ , $\\mathbf {x} \\in \\mathbb {R}^{m+1} $ and $\\mathbf {u} \\in \\mathbb {R}^{m} $ .",
"We take the members of $\\mathbf {g}(\\mathbf {z})$ to be triangularized [4] in the $\\mathbf {x}$ and $\\mathbf {y}$ as $\\mathbf {g}^{\\star }(\\mathbf {x}, \\mathbf {u}) : \\mathbb {R}^{2m+1} &\\rightarrow \\mathbb {R}^{m+1}\\mbox{ where } \\\\g^{\\star }_i(\\mathbf {x}, \\mathbf {u}) &= g^{\\star }_i(x_1,\\ldots ,x_i,\\mathbf {u})$ for $i=1,\\ldots ,m+1$ and $\\mathbf {g}^{\\circ }(\\mathbf {y};\\mathbf {x}, \\mathbf {u}) : \\mathbb {R}^{k-m-1} &\\rightarrow \\mathbb {R}^{k-m-1}\\mbox{ where } \\\\g^{\\circ }_j(\\mathbf {y};\\mathbf {x}, \\mathbf {u}) &= g^{\\circ }_j(y_1,\\ldots ,y_j;\\mathbf {x}, \\mathbf {u})$ for $j=1,\\ldots ,k-m-1$ , with $\\mathbf {g} = [\\mathbf {g}^{\\star }\\ \\mathbf {g}^{\\circ }]^T$ .",
"In (REF ) the $\\mathbf {x}$ and $\\mathbf {u}$ are determined from (REF ) and are treated as parameters.",
"If there is at most one solution to $\\mathbf {g}^{\\circ }(\\mathbf {y}; \\mathbf {x}, \\mathbf {u}) = \\mathbf {0}$ for every $\\mathbf {x}$ and $\\mathbf {u}$ such that $\\mathbf {g}^{\\star }(\\mathbf {x}, \\mathbf {u}) = \\mathbf {0}$ , then $\\mathbf {g}^{\\star }(\\mathbf {x}, \\mathbf {u})$ gives a low-dimensional embedding of our manifold.",
"Theorem tells us that typically any choice for the $\\mathbf {x}$ and $\\mathbf {u}$ should accomplish this.",
"We see that the optimization can progress in $\\mathbb {R}^{2m+1}$ by using the $\\mathbf {x}$ and $\\mathbf {u}$ variables.",
"The manifold defined implicitly by (REF ) can then be mapped onto the original manifold by solving (REF ) for the unique $\\mathbf {y}(\\mathbf {x},\\mathbf {u})$ .",
"We then use $\\mathbf {z}(\\mathbf {u},\\mathbf {x}) &=[\\mathbf {y}(\\mathbf {x},\\mathbf {u})\\ \\mathbf {x}\\ \\mathbf {u}]$ to pull back the objective function and any inequality constraints from the manifold embedding in $\\mathbb {R}^{m+k}$ to the embedding in $\\mathbb {R}^{2m+1}$ .",
"The ability to triangularize the polynomials in $\\mathbf {g}(\\mathbf {z})$ is what allows Whitney's theorem to be easily implemented.",
"There are two things to notice about this triangularization.",
"First, all of this is restricted to working only with the real versus complex roots.",
"Second, as noted above, we only work on the highest dimensional subcomponent of the variety $V(\\mathbf {g})$ .",
"Finding the function $\\mathbf {y}(\\mathbf {x},\\mathbf {u})$ is the crucial step.",
"The most obvious way of doing this is the one we presented above.",
"However it's unimportant how this is done in practice.",
"Any numerical implementation of the implicit function theorem applied to $\\mathbf {g}^{\\circ }(\\mathbf {y}, \\mathbf {x}, \\mathbf {u})$ will suffice, where now $\\mathbf {x}$ and $\\mathbf {u}$ are treated as variables."
],
[
"Moving Over the Manifold",
"We will need a procedure to intelligently move over the manifold $\\mathcal {M} ^{\\prime } \\subset \\mathbb {R}^{2m+1} $ .",
"One advantage of working with a lower-dimensional embedding space is that methods that may be expensive when working in high-dimensions can become feasible.",
"This is counter-balanced by the need to map back into the original space, say with the $\\mathbf {y}(\\mathbf {x},\\mathbf {u})$ from Section .",
"But we're free to use different methods to accomplish these different goals.",
"We move along a path on the manifold $\\mathcal {M} ^{\\prime }$ with one technique and drag along a point on $\\mathcal {M} \\subset \\mathbb {R}^{m+k} $ with a different one.",
"Numerical continuation methods will likely play a key roll in any implementation employing Whitney's theorem.",
"We mention the references [9], [10], [11] in addition to the ones below.",
"On a Riemannian manifold $\\mathcal {M}$ a geodesic is the shortest path between two (nearby) points on $\\mathcal {M}$ and generalizes the notion of a straight line.",
"The coupled nonlinear equations for a geodesic on $\\mathcal {M}$ are [12] $\\ddot{z}^i + \\Gamma ^{i}_{jk} \\dot{z}^j \\dot{z}^k &= 0 \\\\\\Gamma ^{i}_{jk} &=\\left[ \\nabla \\mathbf {g}^+ \\right]^i\\frac{\\partial ^2 \\mathbf {g}}{\\partial z^j \\partial z^k}$ where $\\left[ \\nabla \\mathbf {g}^+ \\right]^i$ is the $i$ -th row of the pseudo-inverse of $\\nabla \\mathbf {g}$ .",
"The Einstein summation convention is used above where a repeated index in a term is summed over.",
"The $\\Gamma ^{i}_{jk}$ are the Christoffel symbols associated with $\\mathbf {g}$ .",
"The distance moved over the manifold in a unit time period is controlled by the length of the initial tangent vector $\\dot{\\mathbf {z}}_0$ .",
"For $\\mathcal {M} ^{\\prime } \\subset \\mathbb {R}^{2m+1} $ $\\mathbf {g}^\\star $ is substituted for $\\mathbf {g}$ and we only work with the $\\mathbf {x}$ and $\\mathbf {u}$ variables.",
"If $2m+1$ is `small' it's possible that finding geodesics on $\\mathcal {M} ^{\\prime }$ would be feasible.",
"One could then find $\\mathbf {y}(\\mathbf {x},\\mathbf {u})$ for the end point of a geodesic and use this to pull back any necessary functions from $\\mathcal {M}$ to $\\mathcal {M} ^{\\prime }$ .",
"In [13] a method of approximating a manifold by locally projecting down tangent spaces was developed.",
"Very little of the machinery is needed.",
"As stated in Theorem the tangent space $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ to $\\mathcal {M} ^\\prime $ at $\\mathbf {p} = [\\mathbf {x}\\ \\mathbf {u}] \\in \\mathcal {M} ^\\prime $ is given by $\\mathrm {null}(\\nabla \\mathbf {g}^\\star (\\mathbf {p}))$ .",
"Let $U^\\prime $ be an orthonormal basis for $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ .",
"Then the vector $\\mathbf {q}_0 = \\mathbf {p} + U^\\prime \\mathbf {w}$ lies in $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ when the latter is viewed as an affine subspace of $\\mathbb {R}^{2m+1}$ .",
"Set $N^\\prime = [\\nabla \\mathbf {g}^\\star (\\mathbf {p})]^+$ where $A^+$ is the Moore-Penrose pseudoinverse of the matrix $A$ .",
"The tangent vector $\\mathbf {q}_0$ is projected down onto $\\mathcal {M} ^\\prime $ by $\\mathbf {q}_{n+1} &= \\mathbf {q}_n - N^\\prime \\mathbf {g}^\\star (\\mathbf {q}_n)\\mbox{, $n=0,1,\\ldots $, with} \\\\\\mathbf {q}_0 &= \\mathbf {p} + U^\\prime \\mathbf {w}$ for some $\\mathbf {w} \\in \\mathbb {R}^{m} $ .",
"Computationally this is the most attractive of the procedures presented and is the one we will use for the remainder.",
"Since this is a numerical implementation of the implicit function theorem there will be a radius around the origin in $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ where it will be guaranteed to converge to a point on $\\mathcal {M} ^\\prime $ .",
"This is restated as a restriction on the length of the $\\mathbf {w}$ in ()."
],
[
"The Optimization Procedure",
"All of the pieces are in place to state the optimization procedure.",
"Let the optimization problem be stated as $\\min _{\\mathbf {z} \\in \\mathbb {R}^{m+k}} \\ f(\\mathbf {z})\\mbox{ subject to } \\tilde{\\mathbf {g}}(\\mathbf {z}) \\le \\mathbf {0}$ with $\\tilde{\\mathbf {g}}:\\mathbb {R}^{m+k} \\rightarrow \\mathbb {R}^{k} $ and $\\tilde{g}_i \\in \\mathbb {Q}[\\mathbf {z}]$ .",
"Triangularize $\\tilde{\\mathbf {g}}(\\mathbf {z})$ into $\\mathbf {g}(\\mathbf {z})$ which has the structure presented in ().",
"Take $\\mathbf {g}^{\\star }: \\mathbb {R}^{2m+1} \\rightarrow \\mathbb {R}^{m+1} $ as defining the manifold $\\mathcal {M} ^\\prime $ .",
"We will assume we have a black-box implementation $\\mathcal {B}(U^\\prime ,\\mathbf {w})$ of the mapping defined by ().",
"$\\mathcal {B}(U^\\prime ,\\mathbf {w})$ will return FALSE if the algorithm doesn't converge to a point on $\\mathcal {M} ^\\prime $ close to the base point $\\mathbf {p} \\in \\mathcal {M} ^\\prime $ .",
"That is, $\\mathcal {B}(U^\\prime ,\\mathbf {w})$ is also an oracle for when the implicit function theorem no longer holds around $\\mathbf {p}$ .",
"We will additionally assume a black-box implementation $\\Phi :\\mathcal {M} ^\\prime \\rightarrow \\mathcal {M} $ of the implicit function mapping.",
"For $\\mathbf {p} \\in \\mathcal {M} ^\\prime $ the point $\\Phi (\\mathbf {p}) = [\\mathbf {y}(\\mathbf {p})\\ \\mathbf {p}]$ is the unique point that lies on $\\mathcal {M}$ .",
"Now we will use the probabilistic descent method developed in [14].",
"A particular implementation is given by Procedure REF .",
"The Move step in Procedure REF picks a new base point on $\\mathcal {M} ^\\prime $ when the oracle $\\mathcal {B}(U^\\prime ,\\mathbf {w})$ tells us that the implicit function implementation is failing.",
"The base point defines the tangent space we are working in.",
"If the base point eventually becomes fixed then the convergence results are the same as those presented in [14].",
"If the base point never becomes fixed we say the algorithm didn't converge and no solution was found.",
"Because we are working with the vector space $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ we can also lay out a mesh.",
"This means that the MADS algorithm first developed in [15] is also viable.",
"The advantage here is the ability to handle inequality constraints as well."
],
[
"An Example",
"Here we look at a simple example that demonstrates how to find the pieces for Procedure REF .",
"Let the equality constraints be given by $\\mathbf {g}(y,x,u) &=\\left[\\begin{array}{c}u^2 + x^3 + y^5 \\\\u^4 + x^2 - 1\\end{array}\\right] \\\\&= \\mathbf {0}.$ See Figure REF for the solution set of $\\mathbf {g}(y,x,u)$ .",
"Figure: The manifolds ℳ\\mathcal {M} and ℳ ' \\mathcal {M} ^\\prime for the example in Section .ℳ ' \\mathcal {M} ^\\prime is only embedded in ℝ 2 \\mathbb {R}^{2} given by the coordinates [xu][x\\ u].This triangularized system of polynomials gives the pair $g^\\star (x,u) &= u^4 + x^2 - 1 \\mbox{ and }\\\\g^\\circ (y;x,u) &= u^2 + x^3 + y^5 \\mbox{.",
"}$ Using $g^\\circ (y;x,u)$ we can find $y$ as a function of $x$ and $u$ : $y(x,u) &= -\\@root 5 \\of {u^2 + x^3} \\mbox{.",
"}$ This is the implicit function we need to map from $\\mathcal {M} ^\\prime $ back to $\\mathcal {M}$ .",
"From this we have $\\Phi = [y(x,u)\\ x\\ u]$ .",
"We also need the gradient of $g^\\star (x,u)$ to find the tangent space of $\\mathcal {M} ^\\prime $ : $\\nabla g^\\star (x,u) &=\\left[\\begin{array}{c}2 x \\\\4 u^3\\end{array}\\right] \\mbox{.",
"}$ Looking at Figure REF we see that at least two different tangent spaces are required to implement the implicit function mapping $\\mathcal {T}_{\\mathbf {p}}\\mathcal {M} ^\\prime $ to $\\mathcal {M} ^\\prime $ .",
"For $\\mathcal {B}(U^\\prime ,\\mathbf {w})$ we could have it return FALSE whenever, say, $\\Vert \\mathbf {q}_n - \\mathbf {q}_0 \\Vert > 1/2$ , with the $\\mathbf {q}_i$ as in ().",
"This would require more than two different tangent spaces to cover $\\mathcal {M} ^\\prime $ but it's likely an acceptable criterion to function as an oracle."
],
[
"Conclusions",
"We have not implemented the procedure in Section numerically.",
"There are multiple components to the method each of which is a significant undertaking in itself to be done correctly and efficiently.",
"However this was not our main purpose.",
"Rather we wished to draw upon classic and recent results that work together in ways that are perhaps novel for nonlinear optimization problems.",
"Disclaimer Any opinions and conclusions expressed herein are those of the author and do not necessarily represent the views of the U.S. Census Bureau.",
"The research in this paper does not use any confidential Census Bureau information.",
"This was authored by an employee of the US national government.",
"As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only."
]
] | 1808.08548 |
[
[
"On the viability of Planck scale cosmology with quartessence"
],
[
"Abstract In this paper, using a qualitative theory of dynamical systems, we study the stability of a unified dark matter-dark energy framework known as quartessence Chaplygin model (QCM) with three different equation-of-states within ultraviolet (UV) deformed Friedmann-Robertson-Walker (FRW) cosmologies without Big-Bang singularity.",
"The UV deformation is inspired by the non-commutative (NC) Snyder spacetime approach in which by keeping the transformation groups and rotational symmetry there is a dimensionless, Planck scale characteristic parameter $\\mu_0$ with dual implications dependent on its sign that addresses the required invariant cutoffs for length and momentum in nature, in a separate manner.",
"Our stability analysis is done in the $(H,\\rho)$ phase space at a finite domain concerning the hyperbolic critical points.",
"According to our analysis, due to constraints imposed on the signs of $\\mu_0$ from the phenomenological parameters involved in quartessence models $(\\Omega_m^*, c_s^2, \\rho_*)$, for an expanding and accelerating late universe, all three QCMs can be stable in the vicinity of the critical points.",
"The requirement of stability for these quartessence models in case of admission of a minimum invariant length, can yield a flat as well as non-flat expanding and accelerating universe in which Big-Bang singularity is absent.",
"This feedback also phenomenologically credits to braneworld-like framework versus loop quantum cosmology-like one as two possible scenarios which can be NC Snyder spacetime generators (correspond to $\\mu_0<0$ and $\\mu_0>0$, respectively).",
"As a result, our analysis show that between quartessence models with Chaplygin gas equation-of-states and accelerating FRW backgrounds occupied by a minimum invariant length, there is a possibility of viability."
],
[
"Introduction",
"For decades particle physicists and cosmologists have focused on beyond standard model physics and modified gravity theories to achieve a clear understanding of the character of two mysteries and challenges of standard cosmology in our age i.e “dark matter” (DM henceforth) and “dark energy” (DE henceforth).",
"Despite the fact that none of them has any explicit evidence in laboratory physics, these two theories can separately provide a consistent explanation of surprising results indicated by the current astronomical observations [1]-[8].",
"Specifically, DM initially was suggested to explain the rotation curves of galaxies and cluster dynamics which it was not justifiable by standard baryonic matter.",
"Later, the proposal of DM extended to cosmology concerning on the issue of structure formation at large scales.",
"Also, to illustrate the accelerated expansion of our universe, the dominated existence of an unknown component called DE is essential.",
"In continuing this path, unlike the original assumption that these two theories are different from each other, an interesting idea proposed that DM and DE can be two manifestations of a single physical entity.",
"From the perspective of unification, it would be interesting to verify the possibility of a single unknown component (or field) rather than two ones which can explain the role of both.",
"Such a unified framework of DM-DE (with a density ratio of approximately $0.3-0.7$ ), in literatures was coined to name “quartessence\", see [9], [10], [11], [12], [13], [14], [15], [16] for instance.",
"The most interesting quartessence models studied so far are based on the Chaplygin gas model as well as its upgraded versions as an exotic background fluid with equation-of-state different from standard perfect fluid [17], [18], [19].",
"There are also other relevant equation-of-states which some authors [20], [21] have offered them as ansatzes which in asymptotic limit cases show the same behaviors for the background fluid.",
"Chaplygin gas models are constrained by cosmic microwave background (CMB) and other astrophysical experiments [22], [23], [24], [25].",
"However, there are some valid regions of parameter space which motivates us to consider them as consistent models with the current experimental data.",
"These regions for an equation of state like $p\\propto -1/\\rho ^n$ with some spacial values of $n$ are not excluded, so that it can still behave like a matter component at early eras and a cosmological constant at late times [26], [24], [25].",
"Additionally, these models as candidates for dark energy are free of fine-tuning problem that appears in the standard cosmology and quintessence models [22].",
"Also, they can elevate the cosmic coincidence problem that appears in the relatively constant ratio of relic density of cosmological constant and matter content of the universe [27].",
"These models can explain the formation of large structures in the universe and the halo of DM in galaxies [28], [29].",
"Even if the origin of DM and DE are different, such models can be used as a simplified model to study all matter content of the universe as a single fluid illustrating the cosmic evolution [30].",
"However, in theoretical physics community this paradigm is ruling to get a complete and coherent view from the early moments until late universe, a quantum description of early moments of cosmology in the absence of micro-level singularity, is required.",
"This is despite the fact that quartessence models as $\\Lambda $ CDM (cosmological constant and cold dark matter) are based on standard cosmology which suffers from initial conditions issue in particular past singularity.",
"Clearly, an initial singular state with infinite values of physical quantities, such as temperature or energy density, should be excluded from any cosmological model.",
"The prevailing belief is that quantum gravity (QG) settings as a framework which explores the universe at the micro-level spacetime (Planck scale), are natural solutions for solving this issue.",
"So far, many intellectual efforts done by QG community has led to the view that at micro-level, spacetime continuum breaks down into a discrete one.",
"So that even disjointed foam is very hard to peace with the GR principles due to the fact that it should endure a later transition to a spacetime continuum, [31].",
"Despite the fact that most known proposals related to QG such as loop quantum gravity [32], [33], string theory [34], [35], deformed special relativity [36], [37], are currently at a development stage, they predict qualitatively a different spacetime beyond some characteristic scales such as Planck length (energy) and momentum.",
"Therefore, in these models, Planck scale through separation of full quantum spacetime from classical one, acts as a natural border line or cutoffs which leads to the appearance of some corrections in the high energy physics.",
"Indeed, the above mentioned invariant scales induce some extensions of the standard uncertainty relation (Heisenberg uncertainty principle) so called “generalized uncertainty principle\" (GUP) [38], [39] which governs the motion of particles in micro-level spacetime.",
"Also, need for existence of GUP proposals at some concrete scales of distances and energies is highly confirmed via gedanken experiments [40].",
"However, as a more advanced alternative to GUP(s), there is a non-commutative (NC) spacetime [41], [42] idea arising from the results of string theory in which moreover discarding the point like concept of the structure of spacetime can also be viewed as NC by changing the nature of the spacetime coordinates.",
"Given that for each of the existing QG proposals, there is a relevant version of GUP.",
"So it is important to mention that some GUPs, particulary generalized algebras designed by Kempf et al.",
"[43], [44] via offering the possibility of space quantization, are compatible with NC spaces.",
"One of the outstanding achievements of NC spacetime idea which is required to get a consistent framework of QG is that it leads to the removal of the paradox appeared due to the creation of a black hole for an event that is sufficiently localized in spactime, [45] see also discussions displayed in [46], [47], [48], [49], [50].",
"Also, by attaching the NC space idea to standard quantum field theory some positive feedbacks have been extracted.",
"For instance, the singular behavior of the Einstein equations in very micro-level distances has cured within the NC space based on quantum field theory.",
"With this preface, in the present paper, through employing the methods of qualitative theory of dynamical systems [51], [52] we want to study the stability of a cosmology with GUP relevant to Snyder NC deformed Heisenberg algebra [53].",
"Moreover, we assume the background fluid is supported by quartessence Chaplygin models (QCMs).",
"We have selected the Snyder NC space approach since it can be connected to some “deformed special relativity” models released in [54], [55] as well as it has some incentives from loop quantum gravity [56].",
"Another advantage of the underlying QG proposal for extending it into cosmology setup is that it respects rotational symmetry, unlike some of its other counterparts.",
"Also within extended framework at hand the Big-Bang singularity can be absent due to bouncing mechanism induced by quantum correction terms in Friedmann dynamic equations [57].",
"The aforementioned positive feature concerning the resolving of initial condition problem from one side and valuable phenomenology functionality of the QCMs at large scales along with this fact that they are stable into standard cosmology [58], [59], from other side, motivates us to explore the response of this question: “Whether the quartessence Chaplygin cosmologies (QCCs) are still stable in a free initial singularity cosmological framework suggested by Snyder NC space approach to QG?\".",
"The result would be desirable in case of “yes\".",
"Since it means that the QCCs are also able to justify the current observations of a universe which has not been raised from a Big-Bang singularity.",
"Of course in light of study done in [60], [61] we know that a fundamental cutoff as minimal length can play the role of dark energy (especially cosmological constant) at late time cosmology.",
"Also recently, people shown that in the context of loop quantum cosmology, by taking an infrared natural cutoff within standard FRW cosmology, there is a possibility to explain the current acceleration of our universe tooIt should be noted that some people try to remove the need for a mysterious matter and energy in nature through modified gravitational theories.",
"However, the recent measurement of the speed of gravity with the gravitational wave ruled out many modified gravity theories as alternative explanation to dark energy [63].[62].",
"However, in what follows by admitting the existence of dark components in universe as the most common and challenging paradigm in modern cosmology which has been able to provide successful justifications of the the stability of galaxies and also observational data, then it turns out that the background fluid quartessence models can be consistent within a QG extended cosmological framework which is free of micro-level singularity.",
"This consistency can have a dual function.",
"First, it can be interpreted as a step towards providing a coherent theoretical picture from the beginning to the present day.",
"Second, it will be seen that the conditions of $H>0$ and $c_s^2>0$ which refer to expanding and accelerating universe lead us to admission and subsequently rejection of some theoretically possibilities for the Snyder dimensionless characteristic parameter $\\mu _0$ via its connection with phenomenological parameters involved in underlying QCMs.",
"More exactly, the present paper qualitatively suggests the possibility of control of the behavior of Planck scale characteristic parameter via the current astronomical signatures which indicates a down to up phenomenological view."
],
[
"Deformed FRW cosmologies from Snyder-deformed Heisenberg algebras ",
"Until the end of this section, we will derive Snyder deformed dynamics equations of the FRW cosmologies.",
"More exactly, we will regard the corrections appeared from the NC Snyder background within the standard HUP, on the classical trajectory of the universe.",
"Therefore, let us first start with a quick overview of the Snyder-deformed Heisenberg algebras by taking into account some details required."
],
[
"Snyder-deformed Heisenberg algebras",
"By concerning on an $n$ -dimensional NC deformed Euclidean space, the structure of the commutator between the coordinates can no longer be trivial rather it is deformed as follows $[\\tilde{q}_i,\\tilde{q}_j]=\\mu M_{ij}~~~~~\\lbrace i,j,...\\rbrace \\in \\lbrace 1,...,n\\rbrace ~,$ so that $\\tilde{q}_i$ 's denote the NC coordinates.",
"Here, $\\mu $ points to the NC Snyder deformation (or characteristic) parameter which its dimension and value is a squared length and a real number, respectively.",
"By demanding two conjectures, we will deal with the (Euclidean) Snyder space [53].",
"First, the rotation generators $M_{ij}=-M_{ji}=i(q_ip_j-q_jp_i)$ fulfill the usual $SO(n)$ algebra as well as the translation group remains undeformed (i.e.",
"$[p_i,p_j]=0$ ).",
"Secondly, under $SO(n)$ rotations the NC coordinates transform as vectors which results in keeping the rotational symmetry.",
"In the language of algebra the second assumption translates as follows $[M_{ij},\\tilde{q}_k]&=&\\tilde{q}_i\\delta _{jk}-\\tilde{q}_j\\delta _{ik}, \\\\\\nonumber [M_{ij},p_k]&=&p_i\\delta _{jk}-p_j\\delta _{ik}~.$ However, it is very important to stress that there are countless number of commutator relations between $\\tilde{q}_i$ and $p_j$ which all of them are unanimously adapted to the relations (REF ).",
"By rescaling of the NC coordinates $\\tilde{q}_i$ in terms of variables used in common phase space i.e.",
"($q_i,p_j$ ), one gets a deeper understanding of the subject.",
"By referring to works released in [64], [65], [66], we offer the most general $SO(n)$ covariant realization for $\\tilde{q}_i$ as follows $\\tilde{q}_i=q_i\\varphi _1(\\mu p^2)+\\mu (q_jp_j)p_i\\varphi _2(\\mu p^2)~,$ so that $\\varphi _1$ and $\\varphi _2$ represent two finite functions and also the convention $a_ib_i=\\sum _i a_ib_i$ is compatible.",
"It is trivial that to restore the standard Heisenberg algebra (i.e.",
"$\\mu =0$ ) the boundary condition $\\varphi _1(0)=1$ , should be administered.",
"Note here the two functions $\\varphi _1$ and $\\varphi _2$ are not unique, at all.",
"Indeed, for any given function $\\varphi _1$ which satisfies the boundary condition $\\varphi _1(0)=1$ , there is a relevant function as $\\varphi _2$ which is characterized via the relation $\\varphi _2=(1+2\\dot{\\varphi }_1\\varphi _1)/(\\varphi _1-2\\mu p^2\\dot{\\varphi }_1)$ so that $\\dot{\\varphi }_1=d\\varphi _1/d(\\alpha p^2)$ , see Ref.",
"[67].",
"So the aforementioned realization of $\\tilde{q}_i$ (i.e (REF )) addresses the following commutator relation between $\\tilde{q}_i$ and $p_j$ $[\\tilde{q}_i,p_j]=i\\left(\\delta _{ij}\\varphi _1+\\mu p_ip_j\\varphi _2\\right)~,$ where results in such a GUP model for the Snyder NC space at hand $\\Delta \\tilde{q}_i\\Delta p_j\\ge 2|\\delta _{ij}\\langle \\varphi _1\\rangle +\\mu \\langle p_ip_j\\varphi _2\\rangle |~.$ The above commutator relation along with inequality, obviously imposes that the standard framework can be recoverable by setting $\\mu \\rightarrow 0$ .",
"Interestingly, unlike three dimensional systems which we deal with countless realizations of the algebra and subsequently different GUPs (REF ), for one-dimensional systems, there is no such an issue.",
"By concerning on the one-dimensional systems the symmetry group is trivial i.e $SO(1)=\\text{Id}$ and the most general realization can be written as $\\tilde{q}=q\\varphi (\\mu p^2)=q\\sqrt{1-\\mu p^2}$ which makes the commutation relation (REF ) and inequality (REF ) to be re-expressed as $[\\tilde{q},p]=i\\sqrt{1-\\mu p^2}~,$ and $\\Delta \\tilde{q}\\Delta p\\ge 2|\\langle \\sqrt{1-\\mu p^2}\\rangle |~,$ respectively.",
"It should be noted that to fix the sign of the Snyder deformation parameter $\\mu $ , there is a freedom.",
"Precisely, in case of $\\mu >0$ a natural cut-off as $|p|<\\sqrt{1/\\mu }$ appears on the momentum while $\\mu <0$ derives an observable minimal length for $\\tilde{q}$ from the uncertainty relation (REF ).",
"As a noticable result, in case of negative sign for $\\mu $ at the first order, one gets the inequality $\\Delta q\\gtrsim (1/\\Delta p+l_s^2\\Delta p)$ which is the same thing predicted by string theory (here $l_s$ refers to string length which can be detected with $\\sqrt{-\\mu /2}$ ), [68], [69].",
"In conclusion, the Snyder-deformed commutator relation (REF ) addresses the existence of a fundamental cut-off as maximum momentum or minimal length if $\\mu >0$ or $\\mu <0$ , respectively."
],
[
"Snyder-deformed dynamical equations ",
"By turning to above review of the Snyder NC algebra, we are going to extract the relevant deformed dynamics of the FRW cosmological models.",
"Indeed, we want to derive classical dynamical equations ruling the universe which is affected by one of the possible initial corrections such as Snyder NC geometry (the corrections come from the algebra (REF )).",
"The classical Poisson bracket representation of the quantum-mechanical commutator (REF ) is $\\lbrace \\tilde{q},p\\rbrace =\\sqrt{1-\\mu p^2}.$ According to the above classical representation for any two-dimensional phase space function the Snyder deformed Poisson bracket can be re-expressed asDeformed Poisson bracket should meet some natural conditions which the quantum mechanical commutator possesses as anti-symmetricity, bilinearity and satisfies the Jacobi identity as well as the Leibniz rules.",
"$\\lbrace F,G\\rbrace =\\left({\\partial \\tilde{q}}{\\partial p}-{\\partial p}{\\partial \\tilde{q}}\\right)\\sqrt{1-\\mu p^2}~.$ It is thus expected that the time evolution of the coordinate and momentum with respect to Hamiltonian $\\mathcal {H}(\\tilde{q},p)$ can be deformed as $\\dot{\\tilde{q}}=\\lbrace \\tilde{q},\\mathcal {H}\\rbrace ={\\partial p}\\sqrt{1-\\mu p^2}, \\qquad \\dot{p}=\\lbrace p,\\mathcal {H}\\rbrace =-{\\partial \\tilde{q}}\\sqrt{1-\\mu p^2}~.$ Now, we expand the underlying framework to the cosmological context in particular FRW cosmological models with the following spatially isotropic metric $ds^2=-N^2dt^2+a^2\\left({1-kr^2}+r^2d\\theta ^2+r^2\\sin ^2\\theta d\\phi ^2\\right)~,$ where the lapse function $N=N(t)$ and scale factor $a=a(t)$ .",
"Also, its matter section obeys fluid energy conservation equation $\\dot{\\rho }+3H(\\rho +p)=0\\;,$ with a generic matter energy density $\\rho $ and pressure $p$ .",
"In line element (REF ), depending on the symmetry group, the curvature constant $k$ can be fixed to 0, $+1$ and $-1$ by pointing to the spatially flat, closed and open universe, respectively.",
"In order to compute the dynamic of the underlying FRW models the following scalar constraint should be satisfied $\\mathcal {H}=-{12a}-3ak+a^3\\rho =0~,~~~~~~8\\pi G\\equiv 1~,$ where its extended representation takes the following form $\\mathcal {H}_E=N̑{12}a+3Nak-Na^3\\rho +\\lambda \\pi ~.$ Here, $\\lambda $ and $\\pi $ denote a Lagrange multiplier and the momenta conjugate attributable to $N$ .",
"By turning to the Poisson bracket (REF ), we can assume that the commutator relation between the isotropic scale factor $a$ and relevant conjugate momentum $p_a$ in the underlying Snyder-deformed minisuperspace obeys the following from $\\lbrace a,p_a\\rbrace =\\sqrt{1-\\mu p_a^2}\\,,$ where if shutdowns the Snyder NC space deformation (i.e $\\mu =0$ ), it comes back to standard form $\\lbrace a,p_a\\rbrace =1$ , as expected from GR based mini superspace.",
"Now by having the extended Hamiltonian $\\mathcal {H}_E$ and Poisson bracket (REF ), one can obtain relevant deformed dynamics equations in two-dimensional phase space $(a,p_a)$ , as follows $\\dot{a}=\\lbrace a,\\mathcal {H}_E\\rbrace ={6a}\\sqrt{1-\\mu p_a^2}, \\qquad \\dot{p}_a=\\lbrace p_a,\\mathcal {H}_E\\rbrace =N\\left({12a^2}-3k+3a^2\\rho +a^3{da}\\right)\\sqrt{1-\\mu p_a^2}.$ Eventually, by solving the constraint (REF ) with respect to $p_a$ as well as considering the first case in equation (REF ) and also fixing $N=1$ , the first Friedmann equation modified by leading order Snyder NC space correction, reveals as $H^2=\\frac{\\rho }{3}-k̑{a^2}-4\\mu \\rho ^2 a^4+24\\mu ka^2\\rho -36\\mu k^2~~.$ Subsequently by taking time derivation of the expansion rate equation (REF ), we arrive at $\\dot{H}=-\\frac{\\rho +p}{2}+\\frac{k}{a^2}-8\\mu a^4\\rho ^2+12\\mu a^4\\rho (\\rho +p)+24\\mu ka^2\\rho -36\\mu k a^2(\\rho +p)~,$ as second order deformed Friedmann equation.",
"In above equations, the correction terms arisen from the Snyder NC geometry, are addressed with $\\mu $ parameter which is connected with Planck length $\\ell _{pl}$ via $\\mu \\equiv \\mu _0\\frac{\\ell _{pl}^{2}}{\\hbar ^{2}}$ .",
"To the end of this paper, to facilitate our calculations the natural unit is adopted i.e.",
"$ \\ell _{pl}$ ,$\\hbar $ and $c$ are fixed to unity (as before $8\\pi G\\equiv 1$ ).",
"Therefore, in the following we will work with dimensionless parameter $\\mu _0$ .",
"In the QG literatures it is thought that the value of this parameter as well as other counterparts suggested by other GUP models, must be constant of order unity.",
"Notably, by attaching the QG effects arisen from some common semi-classical approaches within different branches of physics (both theoretically and experimentally) so far for relevant dimensionless QG parameters released some explicit upper bounds (e.g.",
"can be mentioned to works as [70]-[75]).",
"As mentioned before, the above deformed dynamical equations explicitly show us that vanishing the $\\mu $ -terms leades to the restoration of Friedmann equations in their standard form.",
"At the end, by plugging the equation (REF ) to (REF ), we obtain $\\dot{H}=-H^2-\\frac{\\rho +3p}{6}+12\\mu _0\\rho p a^4+12\\mu _0 k a^2(\\rho -3p)-36\\mu _0 k^2~,$ where along with the continuity equation (REF ) are two out of three equations which forms a closed system for doing the Jacobian stability analysis of three different versions of Chaplygin quartessence models, within the quantum cosmological framework."
],
[
"Analysis Procedure",
"We begin the discussion of this section with a succinct and useful preview of our analysis method.",
"Overall, there are two paths to provide a dynamical analysis of a differential equation as $\\dot{y}=f(y)$ : first, finding the relevant straight solutions.",
"Second, reducing the analysis into a phase plane for all defined initial conditions.",
"The latter is the basis of “qualitative dynamic analysis” in which all possible solutions are considered rather than analyzing an individual solution.",
"More precisely, in this way one reduces dynamics into a two-dimensional (2D) phase space in which singular solutions $\\dot{y}= 0 $ and also nonsingular ones are displayed via critical points (CP) and phase curves, respectively.",
"Using the phase diagrams in a phase plane (2D space) we can clearly investigate some important issues such as “dynamical stability”.",
"Generally, we are able to reduce every conventional cosmological dynamics to the 2D phase plane with an autonomous system of equations similar to $\\dot{x} = Q_1(x, y),\\dot{y} = Q_2(x, y)$ , in which dot represents the differentiation with respect to cosmic time.",
"Through linearization of the Jacobian matrix at a given CP and extracting relevant eigenvalues Note that the mentioned eigenvalues are invariant creatures attributed to critical points since by changing the coordinates $x, y$ they remain unchanged [58].",
"$\\lambda _{1,2}$ , will be provided the possibility of categorizing the non-degenerated (or hyperbolic) CPs $(x_c,y_c)$ .",
"As a reminder, in case of the real part of both eigenvalues $\\lambda _{1,2}$ be nonvanishing at $(x_c,y_c)$ , the relevant CP is non-degenerated.",
"With a good approximation the dynamical behaviour of the above mentioned autonomous system in the vicinity of the CP $(x_c, y_c)$ is qualitatively traceable via the behaviour of its linear part $\\left(\\begin{array}{cc}\\end{array}\\dot{x} \\\\\\dot{y} \\\\\\right.=\\textbf {\\emph {M}}_{2\\times 2}|_{(x_c, y_c)}~.~\\left(\\begin{array}{cc}x-x_c \\\\y-y_c \\\\\\end{array}\\right)~~,~~~~~~~~\\textbf {\\emph {M}}=\\left(\\begin{array}{cc}\\acute{Q_{1,x}} & \\acute{Q_{1,y}}\\\\\\acute{Q_{2,x}} & \\acute{Q_{2,y}}\\\\\\end{array}\\right)\\,,$ where after integration, the above system gives the following solution $\\begin{array}{ll}x-x_c=Re\\Big (A_1 \\exp (\\lambda _1t)+A_2\\exp (\\lambda _2t)\\Big )~, \\\\\\\\y-y_c=Re\\Big (A_1 k_1 \\exp (\\lambda _1t)+A_2 k_2\\exp (\\lambda _2t)\\Big )~,\\end{array}$ with $k_1=\\frac{\\lambda _1}{\\acute{Q_{1,y}}(x_c, y_c)}-\\frac{\\acute{Q_{1,x}}(x_c, y_c)}{\\acute{Q_{1,y}}(x_c, y_c)}$ and $k_2=\\frac{\\lambda _2}{\\acute{Q_{1,y}}(x_c, y_c)}-\\frac{\\acute{Q_{1,x}}(x_c, y_c)}{\\acute{Q_{1,y}}(x_c, y_c)}$ .",
"Here, the prime sign refers to the derivative in terms of variables $x$ and $y$ .",
"By specifying the sign of the trace and the discriminant within the 2D flows dynamical systems then the possibility of a solution for stability analysis, will be available [76].",
"In case of $Det\\, \\textbf {M} > 0$ and the discriminant: $\\textbf {D} =(Tr\\, \\textbf {M})^2 -4\\,Det\\,\\textbf {M} >0$ , the eigenvalues are real with the same sign which addresses the critical point as a node.",
"If $Tr\\, \\textbf {M} > 0$ , the critical point is an unstable node i.e.",
"a repeller or source, while if $Tr\\, \\textbf {M} < 0$ it is a stable node i.e.",
"an attractor or sink.",
"In case of $Det\\, \\textbf {M} > 0$ and the discriminant: $\\textbf {D} =(Tr\\, \\textbf {M})^2 -4\\,Det\\,\\textbf {M} <0$ , the eigenvalues are complex conjugates which address the critical point as a focus.",
"If $Tr\\, \\textbf {M} > 0$ it is an unstable focus while if $Tr \\,\\textbf {M} <0$ it is a stable focus.",
"Also, note that if eigenvalues are purely imaginary then the critical point is a stable neutral center.",
"In case of $Det\\,\\textbf {M}< 0$ , the eigenvalues are real with opposite signs which address the critical point as a saddle point.",
"In case of $Tr\\,\\textbf {M}=0$ and $Det\\, \\textbf {M} > 0$ , the eigenvalues of the critical points have complex values which address the stable neutrally center type.",
"Otherwise, if $Det\\, \\textbf {M} < 0$ then the critical point represents a saddle point."
],
[
"Model I: Generalized Chaplygin Gas Quartessence (GCGQ)",
"Historically, the so called Chaplygin gas fluid model originally studied by Chaplygin [77] in the early twentieth century within the framework of aerodynamics via offering an exotic equation-of-state as $p=-\\frac{A}{\\rho }$ .",
"However, in recent years, this model with its upgraded versions, have been at the cosmology center of attention from phenomenological sense so that now we see them as one of the most popular candidates to DE-DM unified framework, [19], [78], [79], [80], [81].",
"In the first updated model of Chaplygin gas, the relevant negative pressure of underlying background fluid is connected to energy density via the following more general equation-of-state This equation-of-state and its original version (i.e.",
"$n=1$ ) can be thought as a perfect fluid which at high energy phase of universe behaves similar to a pressureless fluid while at low energy it indicates a cosmological constant.",
"$p=-A\\rho ^{-n}~,~~~~~A>0~~~~~\\mbox{and}~~~~0<n\\le 1 \\, ,$ In most literatures the above equation-of-state, describes a “generalized Chaplygin gas quartessence” (GCGQ) model and is intended as a starting point for investigations on the cosmological implication of Chaplygin gas models.",
"Putting above equation-of-state into the energy conservation fluid equation (REF ), one arrives at $\\rho =\\bigg (A+B a^{-3(n+1)}\\bigg )^{\\frac{1}{n+1}} \\, ,$ for the evolution of GCGQ energy density.",
"Here $a(t)$ represents the cosmic scale factor which for case of today universe can be fixed to unity.",
"By offering the new variables $\\Omega _m^*\\equiv \\frac{B}{A+B}~~~\\mbox{and}~~~\\rho _*\\equiv (A+B)^{\\frac{1}{n+1}} \\, ,$ then the equation (REF ) can be written as $\\rho (a)=\\rho _*\\bigg ((1-\\Omega _m^*)+\\Omega _m^* a^{-3(n+1)}\\bigg )^{\\frac{1}{n+1}} \\, .$ Here $\\rho _*$ can be interpreted as “today critical density” of universe since by fixing $a=1$ then $\\rho (1)=\\rho _*$ .",
"To provide a physical interpretation of variable $\\Omega _m^*$ it is necessary to compare the above equation with the following $\\Lambda CDM$ density energy $\\rho (a)=\\rho _*\\bigg ((1-\\Omega _m)a^{-3(\\omega ^*+1)}+\\Omega _m a^{-3}\\bigg )^{\\frac{1}{n+1}} \\, ,$ where $\\Omega _m$ and $(1-\\Omega _m)$ denote the current CDM density parameter and dark energy density, respectively.",
"It is clear that for spacial cases $n=0$ and $\\omega ^*=-1$ , these two models will meet each other which means that $\\Omega _m^*$ can be interpreted as ”effective matter density parameter” in relevant Chaplygin gas model.",
"Now let us follow our main aim i.e.",
"the stability analysis of GCGQ model within the context of Snyder NC deformed quantum cosmology.",
"By re-expressing the Eqs (REF ) and (REF ) as follows $\\dot{H}\\equiv \\frac{dH}{dt}=-H^2-\\frac{\\rho +3p}{6}+12\\mu _0\\rho p a^4+12\\mu _0 ka^2(\\rho -3p)-36\\mu _0 k^2=Q_1(H,\\rho )~,$ and $\\dot{\\rho }\\equiv \\frac{d\\rho }{dt}=-3H(\\rho +p)=Q_2(H,\\rho ) \\, ,$ we define our 2D dynamical system in which the quantities $(H,\\rho )$ play the role of the phase space variables.",
"More precisely, the evolution of the underlying system is traceable via trajectories into $(H,\\rho )$ -space uniquely specified by the initial conditions $(H_{cp},\\rho _{cp})$ .",
"Therefore, in this phase space the linearization matrix $\\textbf {\\emph {M}}$ of the system at the around of CP $(H_{cp},\\rho _{cp})$ , reads off as $\\textbf {\\emph {M}}=\\left(\\begin{array}{cc}\\acute{Q_{1,H}} & \\acute{Q_{1,\\rho }}\\\\\\acute{Q_{2,H}} & \\acute{Q_{2,\\rho }}\\\\\\end{array}\\right)_{(H_{cp},\\rho _{cp})} \\, ,$ where for non-static CPs $(H_{cp},\\rho _{cp})$ , the trace and the determinant are obtained as $Tr\\, \\textbf {M}=\\big (\\acute{Q_{1,H}}+ \\acute{Q_{2,\\rho }}\\big )_{(H_{cp},\\rho _{cp})}~~~~~,~~~~Det\\,\\textbf {M}=\\big (\\acute{Q_{1,H}}~.~ \\acute{Q_{2,\\rho }}-\\acute{Q_{1,\\rho }}~.~\\acute{Q_{2,H}}\\big )_{(H_{cp},\\rho _{cp})} \\, .$ Now by setting equations (REF ) and (REF ) to zero, non-static CPs are derived as $\\begin{array}{ll}H_{cp}=\\bigg [\\frac{\\rho _*}{3}(1-\\Omega _m^*)^{\\frac{\\omega }{\\omega -c_{s}^2}}-12\\mu _0\\rho _*^2(1-\\Omega _m^*)^{\\frac{2}{n+1}}\\bigg ((\\frac{\\omega +1}{\\omega })(\\frac{\\Omega _m^*-1}{\\Omega _m^*})\\bigg )^{-\\frac{4\\omega }{3(\\omega -c_{s}^2)}}+\\\\48\\mu _0 k\\rho _*(1-\\Omega _m^*)^{\\frac{\\omega }{\\omega -c_{s}^2}} \\bigg ((\\frac{\\omega +1}{\\omega })(\\frac{\\Omega _m^*-1}{\\Omega _m^*})\\bigg )^{-\\frac{2\\omega }{3(\\omega -c_{s}^2)}}-36\\mu _0k^2\\bigg ]^\\frac{1}{2} \\, ,\\\\\\rho _{cp}=\\rho _*(1-\\Omega _m^*)^{\\frac{\\omega }{\\omega -c_{s}^2}} \\,~~~\\mbox{with}~~~n=-\\frac{c_s^2}{\\omega } \\,,\\end{array}$ respectively.",
"Note that, expressions relevant to scale factor terms in (REF ) obtained from mixing the equation-of-state index $\\omega \\equiv \\frac{p}{\\rho }$ and the squared sound speed $c_{s}^{2}\\equiv \\frac{dp}{d\\rho }$ with (REF ).",
"Finally for the above non static CP, we have $Tr\\, \\textbf {M}=-H_{cp}(3n+5)~~~,~~~Det\\,\\textbf {M}=6H_{cp}^2(n+1)~~~,~~~D=H_{cp}^2(9n^2+6n+1) \\, ,$ At first look, one may think this is exactly what has already been achieved within standard cosmology.",
"Therefore, Planck scale corrections induced by Snyder NC space into FRW cosmologies does not affect standard results.",
"However, with a closer look one will find that the effect of UV natural cutoffs embeds into $H_{cp}$ term.",
"Expressions listed in (REF ) explicitly reflect this fact that determinant and discriminant are always positive so that to have a stable node CP there should be $Tr\\, \\textbf {M}>0$ i.e $H_{cp}>0$ .",
"In another words, in an expanding universe, the CP (REF ) behaves as an asymptotically stable node.",
"Despite that in the absence of underlying corrections, $H_{cp}$ is trivially positive, here it should be checked carefully.",
"Our consideration shows that concerning late time phase of the universe i.e.",
"fixing values close to $-1$ for equation-of-state parameter $\\omega $ and respect to standard constraints $\\Omega _m^*\\in (0.2,0.4)$ into flat as well as open spatial geometry model universe at hand, the condition of $H_{cp}>0$ holds only if $\\mu _0<0$ (i.e.",
"adoption of a minimum invariant length in fundamental level of nature), as displayed in Fig.",
"REF (left panel).",
"However, for case of closed universe ($k=+1$ ), we find that depending on the fixed values for present critical density of universe $\\rho _*$ , also there is the possibility of admitting the positive value (moreover negative values) for the dimensionless Snyder characteristic parameter $\\mu _0>0$ , as revealed in Fig.",
"REF (right panel).",
"Figure: Regions of existence H cp >0H_{cp}>0 (Eq.",
"()) within (μ 0 ,ρ * ,c s 2 )(\\mu _0, \\rho _*,c_s^2)parameter space, for flat, open (left panel) and closed (right panel) Snyder deformedquantum cosmology with equation-of-state parameter close to -1-1 (here ω=-0.98\\omega =-0.98) andany arbitrary value Ω m * ∈(0.2,0.5)\\Omega _m^*\\in (0.2,0.5).Note that although in language of perfect fluid, the equation-of-state (REF ) covers $-1\\le \\omega \\le 0$ , here for all three possible modes of spatial curvatures (i.e.",
"$k=0,~\\pm 1$ ), the condition of $H_{cp}>0$ does not support exactly $\\omega =-1$ .",
"It is not hard to prove that the Snyder NC space correction terms include scale factor $a$ into (REF ) are the main reason of the issue so that by rejecting them this issue could be disappeared.",
"It is also worthy to refer that the above parameter volume addresses interestingly the possibility of connection between two seemingly unrelated phases of the universe.",
"To say more exactly, the Snyder characteristic parameter $\\mu _0$ deals with the earliest phase of the universe linked to the two valuable quantities in current cosmology i.e.",
"today critical density of universe $\\rho _*$ and the squared sound speed As a reminder to highlight the role of this quantity in current cosmology, note that there is a close connection between the sign of $c_s^2$ with background dynamics of the universe.",
"The current accelerating phase of the universe strongly addresses a positive sign for $c_s^2$ .",
"$c_s^2$ .",
"As a consequence, based on the conventional approaches to cosmology which highly support this belief that the spatial geometry of the universe is exactly flat, the stability of the GCGQ model within the underlying QG extended cosmological framework will be possible only in case of admitting a lower bound for length in nature, $\\mu _0<0$ .",
"However, observational data (primarily the CMB) tells us that the curvature constant must be close to flat but not exactly flat.",
"Concerning the non-flat geometries, we see from Fig.",
"REF that the behavior of $\\mu _0$ for open universe is quite similar to flat one while the sign of $\\mu _0$ in closed universe is dependent on fixed values of $\\rho _*$ .",
"Also in Fig.",
"REF , it is displayed that the phase portraits in physical domain ($\\rho >0$ ) are equivalent to terms dictated by Fig.",
"REF .",
"As it is seen in the left panel, for each three curvature modes of the Snyder deformed-FRW model including a minimum length, there are two de Sitter nodes.",
"de Sitter node in the region $H>0$ is attractor and stable, while its counterpart in the region $H<0$ is repeller and represents an unstable CP.",
"Concerning closed curvature mode which includes the maximum momentum, the right panel shows circular trajectories around the static CP $(0,\\rho )$ which is affiliated to a center equilibrium CP and represents a static universe.",
"Note that in the left panel also one can see some static CPs associated to unstable saddle points which are located on the trajectories moving from the unstable de Sitter node ($H<0$ ) towards the stable de Sitter node ($H>0$ ).",
"Figure: The vector field portrait in phase space (H,ρ)(H,\\rho ) corresponding to Fig.",
".Left panel corresponds to any three curvature modes of Snyder deformed-FRW modelwith numerical values: μ 0 =-1\\mu _0=-1, ρ * =5\\rho _*=5, ω=-0.98,Ω m * ∈(0.2,0.5)\\omega =-0.98,~\\Omega _m^*\\in (0.2,0.5), c s 2 ∈(0,0.5]c_{s}^2\\in (0,0.5].Right panel only corresponds to closed curvature mode with the same numerical values except μ 0 =1\\mu _0=1, 0<ρ * <20<\\rho _*<2."
],
[
"Model II: Modified Chaplygin Gas Quartessence (MCGQ)",
"Over the years, for GCGQ models several modifications have been proposed.",
"If one regards the modified Chaplygin gas quartessence (MCGQ) in which pressure $p$ and energy density $\\rho $ are connected together via the following ansatz It is interesting to note that, equation-of-state (REF ) is wider than GCGQ model since it covers from radiation dominated era for small values of the scale factor in the early universe to large values of the scale factor in the late universe which cosmological constant prevails as the inducement of accelerated expansion of our universe.",
"[82], [83], [84] $p=C\\rho -D\\rho ^{-n}~,~~~~~~C,~D>0~~~~,~~~~~0<n\\le 1\\, ,$ then it results in $\\rho =\\bigg (\\frac{D}{C+1}+Ea^{-3(n+1)(C+1)}\\bigg )^{\\frac{1}{n+1}}~.$ By assuming the following new variables $\\Omega _m^*\\equiv \\frac{E(C+1)}{D+E(C+1)}~~~,~~~\\rho _*\\equiv \\bigg (\\frac{D+E(C+1)}{C+1}\\bigg )^\\frac{1}{n+1}\\, ,$ then Eq.",
"(REF ) can be expressed as follows $\\rho =\\rho _*\\bigg ((1-\\Omega _m^*)+\\Omega _m^*a^{-3(n+1)(C+1)}\\bigg )^{\\frac{1}{n+1}}~,$ where by merging it with Eq.",
"(REF ) in addition to equation-of-state index $\\omega \\equiv \\frac{p}{\\rho }$ and the squared sound speed $c_s^2\\equiv \\frac{dp}{d\\rho }$ , we get the following expression $a=\\bigg (\\frac{\\Omega _m^*-1}{\\Omega _m^*}-\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*(\\omega +1+\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*-1})}\\bigg )^{\\frac{(\\Omega _m^*-1)\\rho _*^{n+1}}{3D(n+1)}}~,~~~~n=\\frac{c_s^2-C}{C-\\omega } \\, ,$ for the scale factor $a$ appeared in (REF ).",
"With a simple calculation one can show that in the limit $C\\rightarrow 0$ , the above expression reduces to its counterpart in GCGQ model.",
"Note that, with the same argument mentioned in details previously, here also we can interpret variables $\\Omega ^*$ and $\\rho _*$ as effective matter density of MCGQ model and today energy density of the universe, respectively.",
"Finally, in the context of quartessence model at hand, the relevant expressions for non-static CPs, take the following form $\\begin{array}{ll}H_{cp}=\\bigg [\\frac{\\rho _*}{3}(1-\\Omega _m^*)^{\\frac{1}{n+1}}-12\\mu _0\\rho _*^2(1-\\Omega _m^*)^{\\frac{2}{n+1}}\\bigg (\\frac{\\Omega _m^*-1}{\\Omega _m^*}-\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*(\\omega +1+\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*-1})}\\bigg )^{\\frac{4(\\Omega _m^*-1)\\rho _*^{n+1}}{3D(n+1)}}+\\\\48\\mu _0 k\\rho _*(1-\\Omega _m^*)^{\\frac{1}{n+1}} \\bigg (\\frac{\\Omega _m^*-1}{\\Omega _m^*}-\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*(\\omega +1+\\frac{D\\rho _*^{-n-1}}{\\Omega _m^*-1})}\\bigg )^{\\frac{2(\\Omega _m^*-1)\\rho _*^{n+1}}{3D(n+1)}}-36\\mu _0k^2\\bigg ]^\\frac{1}{2}\\, ,\\\\\\rho _{cp}=\\rho _*(1-\\Omega _m^*)^{\\frac{1}{n+1}} \\,.\\end{array}$ Now, it is clear to show that, the result of (REF ) once again repeats.",
"Namely, in the presence of UV invariant cutoff raised within Snyder NC space road to QG, the underlying MCGQ cosmological model in case of $H_{cp}>0$ (expanding universe) is stable.",
"However, due to existence of some correction terms, the condition $H_{cp}>0$ is not trivial rather should be checked.",
"As before, concerning the flat as well as open universe in late time phase, for equation-of-state indices close to $\\omega \\approx -1$ (except -1) with $\\Omega _m^*\\in (0.2,0.4)$ , we find that independent of any arbitrary values $C,~D>0$ , the condition $H_{cp}>0$ holds only in case of $\\mu _0<0$ , as Fig.",
"REF (left panel).",
"However, by taking $k=+1$ into quantum cosmological model at hand then $(\\mu _0,\\rho _*,\\Omega ^*)$ parameter space addresses both possibilities i.e.",
"positive and negative signs for $\\mu _0$ , dependent on relevant values for $\\rho _*$ , as can be seen clearly in Fig.",
"(REF ).",
"Figure: Regions of existence H cp >0H_{cp}>0 (Eq.",
"()) within (μ 0 ,ρ * ,c s 2 )(\\mu _0, \\rho _*,c_s^2)parameter volume, for closed Snyder NC quantum cosmology with equation-of-state indexesclose to -1-1 (here ω=-0.95\\omega =-0.95) and any arbitrary values n∈(0,1]n\\in (0,1], Ω m * ∈(0.2,0.5)\\Omega _m^*\\in (0.2,0.5)and C,D>0C,~D>0.In similar to the former quartessence cosmology model which $H_{cp}$ has been divergent at $\\omega =-1$ , here also this issue can be seen.",
"Once again we mention that the root of this restriction is thanks to the Snyder NC correction terms include scale factor $a$ into (REF ).",
"In Fig.",
"(REF ) we show the phase portraits equivalent to terms dictated by Fig.",
"(REF ) in the physical domain, $\\rho >0$ .",
"As before, we see that in the presence of maximum momentum there are circular trajectories around static universe ($0,\\rho $ ) which is affiliated to a stable center equilibrium CP.",
"However, in the presence of minimum length there are two de Sitter nodes which in the case of an expanding universe, it is stable attractor while for its contracting counterpart, it is unstable repeller.",
"Here there is also the possibility of static universe which behaves as an unstable saddle CP.",
"Figure: The vector field portrait in phase space (H,ρ)(H,\\rho ) corresponding to Fig.",
".Left panel corresponds to closed curvature mode of Snyder deformed-FRW model in the presence ofmaximum momentum with numerical values: μ 0 =1\\mu _0=1, ρ * =3,ω=-0.95,Ω m * ∈(0.2,0.5)\\rho _*=3,~\\omega =-0.95,~\\Omega _m^*\\in (0.2,0.5), c s 2 ∈(0,0.5]c_{s}^2\\in (0,0.5]and C,D>0C,~D>0.",
"Right panel is in the presence of minimum length with the same numerical values exceptμ 0 =-1,ρ * =6\\mu _0=-1,~\\rho _*=6."
],
[
"Model III: Modified Generalized Chaplygin Gas Quartessence (MGCGQ)",
"The third proposed model for quartessence cosmology that we are interested in here to introduce is known as modified generalized Chaplygin gas (MGCG) with the following form of equation-of-state [85], [86] $p=\\beta \\rho -(\\beta +1)A\\rho ^{-n}\\, ,$ where $\\beta $ is an optional real constant so that in the absence of it (i.e $\\beta =0$ ) the GCGQ model will be recovered.",
"It is obvious that in the hot early universe the above equation-of-state reduces to $p=\\beta \\rho $ which by fixing $\\beta =1/3$ it addresses the radiation dominated epoch.",
"While in case of $\\beta =-1$ then $p=-\\rho $ , corresponding to the equation-of-state of a cosmological constant.",
"Here, the MGCG density evolves as $\\rho =\\bigg ((\\beta +1)A+Fa^{-3(\\beta +1)(n+1)}\\bigg )^{\\frac{1}{n+1}}\\, ,$ where using the following new variables $\\Omega _m^{*}\\equiv \\frac{F}{A+F},~~~~~\\rho _*\\equiv \\bigg (A+F\\bigg )^{\\frac{1}{n+1}} \\, ,$ then the above MGCG density takes the following form $\\rho =\\rho _*\\bigg ((1-\\Omega _m^*)+\\Omega _m^*a^{-3(\\beta +1)(n+1)}\\bigg )^{\\frac{1}{n+1}} \\, ,$ In line with previous routes, here we arrive at the following expressions $\\begin{array}{ll}H_{cp}=\\bigg [\\frac{\\rho _*}{3}(1-\\Omega _m^*)^{\\frac{\\beta -\\omega }{c_s^2-\\omega }}-12\\mu _0\\rho _{*}^{2}(1-\\Omega _m^*)^{\\frac{2(\\beta -\\omega )}{c_s^2-\\omega }}\\bigg ((\\frac{\\omega +1}{\\beta -\\omega })(\\frac{1-\\Omega _m^*}{\\Omega _m^*})\\bigg )^{-\\frac{4(\\beta -\\omega )}{3(\\beta +1)(c_s^2-\\omega )}}+\\\\48\\mu _0 \\rho _* k(1-\\Omega _m^*)^{\\frac{\\beta -\\omega }{c_s^2-\\omega }}\\bigg ((\\frac{\\omega +1}{\\beta -\\omega })(\\frac{1-\\Omega _m^*}{\\Omega _m^*})\\bigg )^{-\\frac{2(\\beta -\\omega )}{3(\\beta +1)(c_s^2-\\omega )}}-36\\mu _0k^2\\bigg ]^\\frac{1}{2} \\, ,\\\\\\rho _{cp}=\\rho _*(1-\\Omega _m^*)^{\\frac{\\beta -\\omega }{c_s^2-\\omega }} \\, .\\end{array}$ for the relevant non-static CPs, so that in the limit $\\beta \\rightarrow 0$ its counterpart in (REF ) can also be recovered, as expected.",
"Figure: The vector field portrait in phase space (H,ρ)(H,\\rho ) corresponding to MGCGQ model.Left panel corresponds to any three curvature modes of Snyder deformed-FRW modelwith numerical values: μ 0 =-1\\mu _0=-1, ρ * =5\\rho _*=5, ω=-0.98,Ω m * ∈(0.2,0.5)\\omega =-0.98,~\\Omega _m^*\\in (0.2,0.5), c s 2 ∈(0,0.5]c_{s}^2\\in (0,0.5] andβ>-1\\beta >-1.",
"Right panel only corresponds to closed curvature mode with the same numerical values exceptμ 0 =1\\mu _0=1, 0<ρ * <20<\\rho _*<2.Like the two previous models, we should follow the validity of the condition $H_{cp}>0$ which guarantees an expanding universe.",
"Our analysis interestingly shows that if the free parameter $\\beta $ is in range of $\\beta >-1$ , and $\\Omega _m^*\\in (0.2,0.4)$ then by fixing values close to $-1$ for $\\omega $ , we deal with $(\\mu _0, \\rho _*,\\Omega ^*)$ parameter volumes similar to Fig.",
"(REF ).",
"Namely, for cases of flat and open spatial geometry modes the condition of $H_{cp} > 0$ holds only in case of adoption of a minimum invariant length in fundamental level of nature, i.e.",
"$\\mu _0<0$ .",
"While for case of closed mode, depending on fixed values for present critical density of universe $\\rho _*$ there is the possibility of admitting the maximum momentum and minimal length.",
"The remarkable thing in above results is that for all three modes $k=0,\\pm 1$ , the condition $H_{cp}>0$ , will not be satisfied for values of $\\beta \\le -1$ .",
"In Fig.",
"(REF ) we draw the vector field portraits of dynamical system relevant to MGCGQ model.",
"Here also interpretation of the behavior of the trajectories in the neighborhood of the CPs is similar to the two previous models.",
"Quartessence as one of prevalent alternatives to $\\Lambda $ CDM, with a phenomenologically unified dark matter-energy framework, is based on past singular Friedmann-Robertson-Walker (FRW) cosmology.",
"However, in order to provide a complete picture from the beginning of the universe to today, some ingredients should be attached to the standard theory.",
"In this paper, we have focused on the stability of three quartessence models with generalized Chaplygin gas (GCG), modified Chaplygin gas (MCG) and generalized modified Chaplygin gas (GMCG) equation-of-state into a cosmology with generalized uncertainty principle arisen from non-commutative (NC) Snyder space leading to the absence of past singularity issue.",
"The relevant dynamical equations have been derived within a FRW minisuperspace in the presence of some invariant UV cutoffs given by Snyder NC geometry which address a road to quantum gravity.",
"The UV deformed Friedman equation governing our model includes an interesting feature.",
"Due to freedom in the sign of the Snyder characteristic parameter $\\mu $ (by setting the natural unites ($\\ell _{pl}=\\hbar =c=1$ ) it becomes equal to its dimensionless counterpart, i.e, $\\mu _0$ ), then the mentioned deformed Friedman equation can be linked to the cosmological dynamics of loop quantum gravity (LQG) by applying a cutoff on the momentum i.e $\\mu _0>0$ from one side and Randall-Sundrum braneworld in case of a cutoff on the length i.e.",
"$\\mu _0<0$ , from the other side.",
"Using the method of qualitative theory of dynamical systems, our stability analysis is performed within $(H,\\rho )$ phase plane at a finite domain by concerning the hyperbolic critical points.",
"Generally speaking, for all three GCG, MCG and GMCG cases, within expanding ($H>0$ ) and accelerating universe ($c_s^2>0)$ , the quartessence models are stable in the neighborhood of the critical points $(H_{cp},\\rho _{cp})$ , in the case of admitting one of theoretically possible signs for $\\mu _0$ .",
"The outstanding feature of our stability analysis is that it restricts freedom to accept the expected invariant UV cutoffs via the connection between QG free parameter $\\mu _0$ and the phenomenological parameters involved in quartessence models $(\\Omega _m^*, c_s^2, \\rho _*)$ .",
"In particular, our analysis explicitly shows that the requirement of stability for above mentioned quartessence models unanimously within a flat accelerating universe free of Big-Bang singularity, will be possible only in case of acceptance of a minimum invariant length in fundamental level (i.e.",
"$\\mu _0<0$ ).",
"Also, we have noticed that for all three of the above-mentioned background fluids within the underlying Snyder deformed cosmology with open spatial geometry, the possibility of stability in present time only exists in case of admitting a minimum length at the fundamental level.",
"While for closed one, depending on the fixed values for today critical density of universe $\\rho _*$ , one can accept one of possible cases for $\\mu _0$ .",
"For any three Chaplygin gas quartessence models, we have constructed the phase portraits in a 2D phase space ($H,\\rho $ ) separately and discussed on the behavior of trajectories in the neighborhood of the CPs.",
"As a result, it is common in all three quartessence models that in the presence of minimum length ($\\mu _0<0$ ), there is the possibility of a stable expanding and accelerating universe with all three possible curvature modes.",
"While, regarding the maximum momentum ($\\mu _0>0$ ) within the FRW background, only shows a stable static universe with closed spatially geometry.",
"As a consequence, our results are essentially independent of the free parameters of equation-of-states of Chaplygin gas models, which are constrained by experiments [22], [23], [24], [25], [26].",
"Briefly, this work contains the following important consequences.",
"First, the requirement of stability for three quartessence models can yield an expanding and accelerating universe compatible with current observational evidences in which Big-Bang singularity is absent.",
"To be more detailed, in case of setting the flat and open geometries for curvature constant modes within NC Snyder spacetime approach, it will be realized the braneword-like framework along with the relevant uncertainty relation of string theory.",
"While for the case of closed universe depending on $\\rho _*$ , also there is a chance to emerge of the LQG-like framework In light of our results, within the framework of flat universe which is accepted by the physics community, it seems that the quartessence Chaplygin gas models and LQG, can not be compatible with each other.. Secondly, by admitting a down to up phenomenological view, our analysis gives qualitatively a hint on the possibility of searching the micro-level spacetime via the control of the Planck scale characteristic parameter using the current astronomical observational signatures.",
"At the end, to emphasize on the importance of the latter as an incentive for proposing an upcoming project, we would like to refer to [87] in which via probing the effects of NC geometry using the latest CMB observations, authors have presented some positive feedbacks."
],
[
"Acknowledgment",
"The authors would like to thank an anonymous referee for insightful comments.",
"This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project number 1/5750-**."
]
] | 1808.08436 |
[
[
"Random Matrix Theory Model for Mean Notch Depth of the Diagonally Loaded\n MVDR Beamformer for a Single Interferer Case"
],
[
"Abstract Adaptive beamformers (ABFs) suppress interferers by placing a notch in the beampattern at the interferer direction.",
"This suppres- sion improves detection of a weaker signals in the presence of strong interferers.",
"Hence the notch depth plays a crucial role in determining the adaptive gain obtained from using ABF over conventional beam- forming.",
"This research derives models for the mean notch depth of a diagonally loaded MVDR ABF for a single interferer case.",
"The model describes the mean notch depth as a function of number of snapshots, the number of sensors in the array, the interferer to noise ratio (INR) level, the interferer direction and the diagonal loading level.",
"The derivation uses random matrix theory results on the be- havior of the eigenvectors of sample covariance matrix.",
"The notch depth predicted by the model is shown to be in close agreement with simulation results over a range of INRs and snapshots."
],
[
"Introduction",
"A common array processing problem is to detect a low power source in presence of high power interferers.",
"A conventional beamformer (CBF) produces a static beampattern which attenuates interferers by a fixed amount at each bearing.",
"At the output, the weaker signal of interest will be masked by the higher power interferer and undermines detection.",
"Alternatively, adaptive beamformers (ABF) can suppress interferers by placing deep notches in the beampattern in the interferer direction.",
"ABFs rely on the knowledge of the data covariance matrix to compute the beamformer weights.",
"In reality, the ensemble covariance matrix (ECM) for data is unknown a priori.",
"The traditional approach is to replace the ECM by the sample covariance matrix (SCM) to compute the beamformer weights.",
"A class of sample matrix inversion (SMI) ABFs involve inverting the SCM to compute the beamformer weights [1].",
"If the number of snapshots ($L$ ) is less than or approximately equal to the number of array sensors ($N$ ), the SCM is unstable or ill-conditioned for inversion.",
"A common approach is diagonally loading the SCM to make it invertible for computing the ABF weights.",
"The minimum variance distortionless response (MVDR) beamformer is one of the most extensively used SMI ABFs [2].",
"The main focus of this paper is to characterize the mean notch depth of a diagonally loaded (DL) MVDR ABF.",
"Prior work by Richmond [3] derived expressions for the mean and the variance of the SCM based MVDR beampattern.",
"However the derivation in [3] only considers the snapshot sufficient case ($N < L$ ) and does not include diagonal loading.",
"More recently, Buck and Wage [4] used Random Matrix Theory (RMT) results to develop a model for the mean notch depth of the dominant mode rejection (DMR) ABF.",
"DMR is a variant of the MVDR ABF that uses a constrained SCM instead of diagonal loading [5].",
"Mestre and Lagunas [6] have used RMT results to derive a deterministic expression for asymptotic output signal-to-interferer-plus-noise ratio (SINR) of a DL-MVDR ABF.",
"Their analysis is focused on deriving an estimator for the optimum loading factor ($\\delta $ ).",
"Similarly, Pajovic et al.",
"[7] used RMT results to derive an analytic expression for the output power of a DL minimum power distortionless response (MPDR) beamformer.",
"MPDR assumes source signal is present in the training data [1].",
"The results presented in this paper are similar in spirit to the work in [4], but for the DL-MVDR ABF also considered in [6].",
"Recent results from RMT are used to derive an approximate model for the mean notch depth of a the DL-MVDR.",
"The model will describe the notch depth as a function of the diagonal loading level ($\\delta $ ) in addition to the number of snapshots ($L$ ), number of sensors ($N$ ), the interferer to noise ratio (INR), and the interferer location ($\\theta _1$ ).",
"The rest of the paper is organized as follows: The next section describes the MVDR beamformer and defines related terminologies.",
"Sec.",
"summarizes the notch depth model derivation.",
"The simulation results are discussed in Sec.",
", followed by a brief conclusion in Sec."
],
[
"The MVDR Beamformer",
"The MVDR beamformer is one of the most extensively used ABFs [1], [2].",
"The weight vector for the MVDR ABF steered to bearing direction $\\theta _0$ is, ${\\bf w}= \\Sigma ^{-1}{\\bf v}_0/ \\left({\\bf v}_0^{\\rm H}\\Sigma ^{-1}{\\bf v}_0\\right)$ where $\\Sigma $ is the $N\\times N$ ECM and ${\\bf v}_0= {\\bf v}(\\theta _0)$ is the array steering vector corresponding to the look direction $\\theta _0$ .",
"Assuming a stationary narrowband interferer with power $\\sigma _1^2$ at bearing $\\theta _1$ and unit power white background noise, the ECM is $\\Sigma = \\sigma _1^2{\\bf v}_1{\\bf v}_1^{\\rm H}+ {\\bf I} = \\sum _{i=1}^{N}\\gamma _i\\xi _i\\xi _i^{\\rm H},$ where $\\gamma _1 > \\gamma _2 = \\ldots \\gamma _N = 1$ are the eigenvalues and $\\xi _i$ are the corresponding eigenvectors.",
"In the single interferer case $\\xi _1 = {\\bf v}_1/ \\sqrt{N}$ , i.e., the principal eigenvector is a scaled version of the interferer steering vector (${\\bf v}_1$ ).",
"The MVDR ABF places a notch in the direction corresponding to $\\xi _1$ .",
"In practice the ECM is estimated by computing the SCM (${\\bf S}$ ) from $L$ data snapshot vectors ${\\bf x}_l$ , ${\\bf S}= \\frac{1}{L}\\sum _{l=1}^{L}{\\bf x}_l{\\bf x}_l^{\\rm H}= \\sum _{i=1}^{N}g_i{\\bf e}_i{\\bf e}_i^{\\rm H}.$ where ${\\bf x}_l = a_{l}{\\bf v}_1+ {\\bf n}_l$ such that $a_{l} \\sim {\\cal CN}(0,\\sigma _1^2)$ and ${\\bf n}_l \\sim {\\cal CN}(0, {\\bf I})$ .",
"Since the noise power is unity, the INR is equal to $\\sigma _1^2$ .",
"Similarly $g_1 > g_2 \\ge \\ldots \\ge g_N$ are the eigenvalues and ${\\bf e}_i$ are the eigenvectors of the SCM.",
"The DL SCM is computed as ${\\bf S}_\\delta ={\\bf S}+ \\delta {\\bf I}$ where $\\delta > 0$ .",
"The eigenvectors are invariant to DL.",
"Hence the sample principal eigenvector ${\\bf e}_1$ estimates the interferer direction.",
"The DL-MVDR ABF weights ($\\hat{\\bf {\\bf w}}_\\delta $ ) are computed by replacing $\\Sigma $ with ${\\bf S}_\\delta $ in (REF )."
],
[
"Notch Depth",
"The notch depth is defined as the magnitude of the beampattern at true interferer direction, i.e., $\\mbox{ND}= |{\\bf w}^{\\rm H}{\\bf v}_1|^2$ .",
"The ensemble notch depth for the DL-MVDR is $\\mbox{ND}_{\\text{ens}_\\delta } = \\frac{\\cos ^2({\\bf v}_0,{\\bf v}_1)(1+\\delta )^2}{\\left|(1 + \\delta + N\\sigma _1^2\\sin ^2({\\bf v}_0,{\\bf v}_1)\\right|^2},$ where $\\cos ^2({\\bf v}_0,{\\bf v}_1)$ is the generalized cosine between ${\\bf v}_0$ and ${\\bf v}_1$ as defined in [8].",
"$\\mbox{ND}_{\\text{ens}_\\delta }$ is the ideally achievable notch depth assuming the ECM is known.",
"Computing the weights with the SCM results in a notch depth ${\\mbox{ND}}_\\delta $ which is shallower than the ensemble $\\mbox{ND}_{\\text{ens}_\\delta }$ .",
"The mismatch between the sample and ensemble principal eigenvectors is the main cause of this loss in notch depth [9].",
"RMT has results on bias of eigenvectors (${\\bf e}_i$ ) of the SCM, which will be used to derive the mean notch depth model in the next section."
],
[
"Model",
"This section derives two models for the DL-MVDR ABF notch depth.",
"The models characterize notch depth as a function of the number of sensors $N$ , the number of snapshots $L$ , the INR ($\\sigma _1^2$ ), the interferer direction ($\\theta _1$ ) and the diagonal loading level ($\\delta $ ).",
"The first model treats snapshots ($L$ ) as the independent variable and INR ($\\sigma _1^2$ ) as a parameter.",
"The second model characterizes the ND as a function of the INR ($\\sigma _1^2$ ) while treating the snapshots ($L$ ) as a parameter.",
"The derivation uses the RMT results on the fidelity of the sample principal eigenvectors.",
"The first part of the RMT result on the eigenvectors of SCM gives an expression for the magnitude of the projection between the sample and the ensemble principal eigenvector, [10][11][12], $|{\\bf e}_1^{\\rm H}\\xi _1|^2 \\overset{a.s.}{\\rightarrow }{\\left\\lbrace \\begin{array}{ll}\\frac{1 - c/(N\\sigma _1^2)^2}{1 + c/(N\\sigma _1^2)} & \\sigma _1^2 > \\sqrt{c}/N \\\\0 & \\sigma _1^2 \\le \\sqrt{c}/N\\end{array}\\right.",
"}$ where $c = N/L$ .",
"This result holds in the RMT asymptotic sense, i.e., $N,L \\rightarrow \\infty , N/L \\rightarrow c$ .",
"It implies that for a sufficiently strong interferer ($N\\sigma _1^2 > \\sqrt{c}$ ) the sample principal eigenvector ${\\bf e}_1$ is a biased estimate of its ensemble counterpart.",
"The second part of the result states that the noise eigenvectors are uniformly distributed over a unit sphere [10].",
"This implies the magnitude of projection of sample principal eigenvector on the orthogonal vector $\\xi _\\perp $ is, $|{\\bf e}_1^{\\rm H}\\xi _\\perp |^2 = (1 - |{\\bf e}_1^{\\rm H}\\xi _1|^2)/(N-1).$ Further, the look direction steering vector ${\\bf v}_0$ can be decomposed into two orthogonal unit vectors $\\xi _1$ and $\\xi _\\perp $ , ${\\bf v}_0= \\alpha \\xi _1 + \\beta \\xi _\\perp $ where $\\alpha = \\sqrt{N}\\cos ({\\bf v}_0,{\\bf v}_1)$ and $\\beta =\\sqrt{N}\\sin ({\\bf v}_0,{\\bf v}_1)$ ."
],
[
"Notch Depth vs Snapshots",
"The derivation of notch depth vs snapshots model begins from the expression for ${\\mbox{ND}}_\\delta $ by substituting for ${\\bf v}_0$ from (REF ) and setting ${\\bf v}_1= \\sqrt{N}\\xi _1$ .",
"This substitution results in expressions containing quadratic terms $|{\\bf e}_1^{\\rm H}\\xi _1|^2$ and $|{\\bf e}_1^{\\rm H}\\xi _\\perp |^2$ .",
"The two terms are then replaced using RMT results in Eq.",
"(REF ) and (REF ).",
"Collecting common terms in $L$ and factoring appropriately simplifies the notch depth expression to $\\mbox{ND}_{\\delta }\\approx \\cos ^2({\\bf v}_0,{\\bf v}_1)(1 + \\delta )^2\\frac{|f_3(L)f_2(L)|^2}{|f_1(L)|^2},$ where $\\begin{split}f_1(L) &= N + L(1 + \\delta + (N\\sigma _1^2)\\sin ^2({\\bf v}_0,{\\bf v}_1))\\\\f_2(L) &= \\sqrt{L} - \\sqrt{N}\\sigma _1^{-1}\\cot ({\\bf v}_0,{\\bf v}_1) \\\\f_3(L) &= \\sqrt{L} -\\frac{\\sqrt{N}\\sigma _1}{1 + \\delta }\\tan ({\\bf v}_0,{\\bf v}_1).\\end{split}$ This derivation assumes that the array is sufficiently long ($N \\gg 1$ ), the interferer power is strong enough ($N\\sigma _1^2 \\gg 1$ ) and the interferer lies outside the main lobe of CBF ($\\sigma _1^2\\tan ^2({\\bf v}_0,{\\bf v}_1)\\gg 1$ ).",
"For the snapshot sufficient case of $ c \\le 1$ DL is constrained to $\\delta > (1 - \\sqrt{c})^2 $ Figure: Notch depth vs snapshots modelThe model in Eq.",
"(REF ) can be visualized as a linear piecewise function of $L$ in a log-log scale.",
"This interpretation of the model is similar to Bode plot approach to interpret system transfer functions [13].",
"The same approach was used to interpret DMR notch depth model in [4].",
"As $L$ increases each factor in (REF ) becomes significant over a different range of values of $L$ .",
"The increase in magnitude of each factor dictates the slope of the linear piecewise function.",
"The values of $L$ for which the summands in each factor become equal predict the breakpoints, $L_1 & = N/\\left(\\delta + \\sigma _1^2 N\\sin ^2({\\bf v}_0,{\\bf v}_1)\\right)& \\mbox{(2nd order)}\\\\L_2 & = N \\cot ^2({\\bf v}_0,{\\bf v}_1)/\\sigma _1^2 & \\mbox{(1st order)}\\\\L_3 & = N \\sigma _1^2 \\tan ^2({\\bf v}_0,{\\bf v}_1)/(1 + \\delta )^2 & \\mbox{(1st order)}.$ The resulting model is shown in Fig.",
"REF .",
"The model predicts that for smaller values of $L$ , the DL-MVDR notch depth reduces to the CBF case.",
"Gathering more snapshots ($L > L_1$ ) results in increased nulling.",
"With a sufficiently large number of snapshots, notch depth converges to the ensemble value.",
"The breakpoint values $L_3$ suggests that increasing the diagonal loading ($\\delta $ ) reduces the snapshots required to achieve the DL ensemble notch depth ($\\text{ND}_{\\text{ens}_\\delta }$ )."
],
[
"Notch Depth vs INR",
"The notch depth vs INR model is developed following similar steps used in Sec.",
"REF .",
"This model collects the terms common in $\\sigma _1^2$ and factors appropriately to obtain $\\mbox{ND}_{\\delta } \\approx \\cos ^2({\\bf v}_0,{\\bf v}_1) \\frac{\\left|\\sigma _1 \\sqrt{c} \\tan ({\\bf v}_0,{\\bf v}_1) - (1 + c + \\delta )\\right|^2}{\\left| N \\sigma _1^2\\sin ^2({\\bf v}_0,{\\bf v}_1) + (1 + c + \\delta ) \\right|^2}$ The notch depth model in is once again interpreted using the same approached discussed in Sec.",
"REF .",
"This approach models the notch depth as a piecewise linear function of INR ($\\sigma _1^2$ ) in a log-log scale as shown in Fig.",
"REF .",
"Figure: Notch depth vs INR modelThe two dyadic factors in Eq.",
"(REF ) predict the breakpoint values of INR to be, $\\text{INR}_1 & = (1 + c +\\delta )/N\\sin ^2({\\bf v}_0,{\\bf v}_1)\\\\\\text{INR}_2 & = (1 + c +\\delta )^2/c \\tan ^2({\\bf v}_0,{\\bf v}_1).$"
],
[
"Simulation Results and Discussions",
"This section compares the estimated notch depth from computer simulations of DL-MVDR and the model predictions.",
"The simulations were performed for a uniform linear array with $N = 50$ sensors.",
"A single stationary interferer was assumed to be at bearing $u_1 = \\cos (\\theta _1) = 0.06$ , which is the location of the peak of the CBF first sidelobe.",
"Fig.",
"REF compares the notch depth as a function of snapshots $L$ , predicted by the RMT model in Eq.",
"(REF ) with the notch depth estimated from simulation for different INR levels.",
"The dashed lines represent the notch depth predicted by the model.",
"The discrete markers represent the average notch depth obtained from a 500 trial Monte Carlo experiment.",
"The black markers represent the ensemble notch depth ($\\text{ND}_{\\text{ens}_\\delta }$ ) at each INR ($\\sigma _1^2$ ) level.",
"The model predicted notch depth matches the averaged notch depth observed in the simulations.",
"Figure: Notch depth vs snapshots simulation results compared tomodel prediction for δ=0.5\\delta = 0.5.Figure: Notch depth vs snapshots for INR of 20 dB at differentdiagonal loading levelsFor most practical array sizes and strong interferers, the breakpoints predicted in (REF ) are such that $L_1 < 1$ , $L_2\\approx 1$ and $L_3$ is practically unattainable.",
"Hence, the first two breakpoints are limited to a theoretical interpretation of the model and are not observed in Fig.",
"REF .",
"In practice the region of operation lies between $L_2$ and $L_3$ where notch depth grows by 10 dB for every decade increase in $L$ .",
"Fig.",
"REF compares the notch depth as a function of snapshots at INR of 20 dB for different DL levels..",
"The figure indicates that higher DL allows the DL-MVDR ABF to approach ensemble ND with fewer snapshots as predicted by expression for $L_3$ .",
"However increasing the loading level also makes the ensemble notch depth $\\text{ND}_{\\text{ens}_\\delta }$ shallower.",
"Figure: Notch depth vs INR simulation results compared to modelprediction for δ=0.5\\delta = 0.5Figure: Notch depth vs INR for L = 2N at different diagonalloading levelsFig.",
"REF compares the notch depth as a function of INR, predicted by the RMT model in Eq.",
"(REF ) and the notch depth estimated from simulations for different snapshots $L$ .",
"The dashed lines represent notch depth predicted by the model.",
"The discrete markers represent the average notch depth obtained from 500 trial Monte Carlo experiments.",
"The solid line represents the ensemble behavior over the range of INR.",
"Again, the model predicted notch depth matches the averaged notch depth observed in the simulations.",
"The range of INRs between $\\text{INR}_1$ and $\\text{INR}_2$ is where the DL-MVDR ABF is adapting to the change in interferer power.",
"The notch depth grows by 20 dB for every 10 dB rise in interferer power once the interferer power is higher than $\\text{INR}_1$ .",
"The interferer is suppressed more, the stronger it becomes.",
"Once the interferer power exceeds $\\text{INR}_2$ , the notch depth growth merely keeps up with interferer power rise.",
"Consequently the interferer power in the beamformer output remains unchanged.",
"Hence there is no additional adaptive gain in using DL-MVDR ABF in this range of INR levels.",
"Fig.",
"REF compares the notch depth as function of INR for snapshots $L = 2N$ among different DL levels.",
"The figure shows that the DL-MVDR ABF with higher DL can suppress interferers with higher power $\\sigma _1^2$ .",
"On the other hand, increasing DL means that the INR must grow larger before the DL-MVDR ABF actually begins adapting.",
"$\\text{INR}_1$ is effectively the minimum interferer power required for the DL-MVDR ABF to depart from CBF performance and begin adapting."
],
[
"Conclusion",
"This paper presents RMT based models for the mean notch depth of a DL-MVDR ABF in a single interferer case.",
"The simulation results verify the accuracy of the notch depth predicted by the two models.",
"The derived models indicate that increasing the diagonal loading reduces snapshots required to converge to the ensemble notch depth.",
"Similarly, the ability to suppress an interferer with higher power increases with higher diagonal loading.",
"The improved suppression comes at a cost of shallower DL ensemble notch depth."
]
] | 1808.08352 |
[
[
"Gumbel Central Limit Theorem for Max-Min and Min-Max"
],
[
"Abstract The Max-Min and Min-Max of matrices arise prevalently in science and engineering.",
"However, in many real-world situations the computation of the Max-Min and Min-Max is challenging as matrices are large and full information about their entries is lacking.",
"Here we take a statistical-physics approach and establish limit-laws -- akin to the Central Limit Theorem -- for the Max-Min and Min-Max of large random matrices.",
"The limit-laws intertwine random-matrix theory and extreme-value theory, couple the matrix-dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix-entries' distribution.",
"Due to their generality and universality, as well as their practicality, these novel results are expected to have a host of applications in the physical sciences and beyond."
],
[
"=1 [1] Gumbel Central Limit Theorem for Max-Min and Min-Max Iddo Eliazar$^{1}$ eliazar@tauex.tau.ac.il Ralf Metzler$^2$ rmetzler@uni-potsdam.de Shlomi Reuveni$^{1},$ shlomire@tauex.tau.ac.il $^{1}$ School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, & The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel $^{2}$ University of Potsdam, Institute of Physics & Astronomy, 14476 Potsdam, Germany The Max-Min and Min-Max of matrices arise prevalently in science and engineering.",
"However, in many real-world situations the computation of the Max-Min and Min-Max is challenging as matrices are large and full information about their entries is lacking.",
"Here we take a statistical-physics approach and establish limit-laws – akin to the Central Limit Theorem – for the Max-Min and Min-Max of large random matrices.",
"The limit-laws intertwine random-matrix theory and extreme-value theory, couple the matrix-dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix-entries' distribution.",
"Due to their generality and universality, as well as their practicality, these novel results are expected to have a host of applications in the physical sciences and beyond.",
"The Central Limit Theorem (CLT) – a foundational cornerstone of statistical physics and probability theory – is of prime importance in science and engineering.",
"The CLT and its generalized version assert that the scaled sum of a large number of independent and identically distributed (IID) random variables is governed, asymptotically, by two limit-law statistics [1], [2]: Normal and Lévy-stable.",
"The CLT considers finite-variance IID random variables, and yields Normal statistics.",
"Departing the finite-variance dominion, the generalized CLT imposes sharp tail conditions on the distribution of the IID random variables [3], and yields both Normal and Lévy-stable statistics.",
"Extreme-value theory [4], [5] is applied whenever extreme behavior – rather than average behavior – is of relevance; e.g.",
"the prediction of rare events, and the safe design of critical systems such as dams, bridges, and power grids.",
"Extreme-value theory shifts the focus from sums to extrema, i.e.",
"maxima and minima.",
"The Fisher-Tippett-Gnedenko (FTG) theorem is the extreme-value counterpart of the above CLTs.",
"This theorem asserts that the scaled extrema of a large number of IID random variables are governed, asymptotically, by three limit-law statistics [6], [7]: Weibull, Frechet, and Gumbel.",
"As in the case of the generalized CLT, the FTG theorem imposes sharp tail conditions on the distribution of the IID random variables [3].",
"The limit-law statistics of the CLTs and the FTG theorem play key roles in physics, e.g.",
"in [8], [9], [10], [11], [12], [13], [14], [15], [16], [18], [17], [19], [20] and in [21], [23], [22], [24], [25], [26], [27], respectively.",
"Underlying these theorems is a common bedrock: a random-vector setting, with the IID random variables being the vector entries.",
"Elevating from one-dimensional to two-dimensional arrays, we arrive at a random-matrix setting: matrices whose entries are IID random variables.",
"Random matrices also play key roles in physics [28], [29], and much effort has been directed to the extreme-value analysis of their eigenvalues spectra [30], [31].",
"Here we focus on a different extreme-value analysis of random matrices: their Max-Min and Min-Max (see Fig.",
"1 for the Max-Min).",
"Figure: The Max-Min of a matrix is obtained by first taking theminimal entry of each row (depicted red), and then taking the maximum ofthese minimal entries (depicted blue).The Max-Min and Min-Max arise prevalently in science and engineering.",
"Perhaps the best known example is in game theory [32], a field which drew considerable attention from physicists [33], [34], [35], [36], [37], [38], [39].",
"There, a player seeks a strategy that will maximize gain, or minimize loss, in the worst-case scenario.",
"The player has a payoff matrix which specifies the gain/loss for each strategy taken vs. each scenario encountered; the player calculates the Max-Min in the case of gains, and the Min-Max in the case of losses.",
"However, in real-life situations the payoff matrix is often large and full information about its entries is lacking.",
"In turn, such situations call for a modeling approach employing large random matrices.",
"The Max-Min and Min-Max of large random matrices were investigated in mathematics [40], and in reliability engineering [41], [42], [43], [44].",
"In the pioneering work [40], Chernoff and Teicher established that the scaled Max-Min and Min-Max are governed, asymptotically, by the FTG statistics: Weibull, Frechet, and Gumbel.",
"In subsequent works [41], [42], [43], Kolowrocki further advanced the topic in the context of (so called) series-parallel and parallel-series systems.",
"In a more recent work [44], Reis and Castro obtained Gumbel limit-law statistics for the Max-Min via an iterative application of the FTG theorem: first to the minimum of each and every matrix row, and then to the maximum of the rows’ minima.",
"The results in [40], [41], [42], [43], [44] are notable and inspiring mathematical theorems.",
"However, from a practical perspective the application of these results is extremely challenging, even on a case by case basis.",
"More importantly, the results in [40], [41], [42], [43], [44] do not provide a clear-cut answer to the following focal question: is there a “Central Limit Theorem” for the Max-Min and Min-Max of random matrices?",
"The CLTs and the FTG theorem stand on two pillars: domain of attraction and scaling scheme.",
"The domain of attraction of the CLT is wide (encompassing all finite-variance distributions), and its scaling scheme is simple; the application of the CLT is thus straightforward, and its use is omnipresent.",
"For the generalized CLT and the FTG theorem matters are more intricate: the domains of attraction are narrow (characterized by the sharp conditions imposed on the distributions' tails [3]), and the scaling schemes are elaborate (they need to be carefully custom-tailored per each admissible distribution [3]).",
"Elevating from a random-vector setting to a random-matrix setting adds a third pillar to the two above: the asymptotic coupling between the matrix dimensions (as these are taken to infinity).",
"In [40], [41], [42], [43], [44] the intricacy of all three pillars is prohibitively high.",
"Consequently, the are no available Max-Min and Min-Max limit-laws with the following features: wide domain of attraction, simple scaling scheme, and simple asymptotic coupling.",
"Figure: Gumbel limit-law statistics for the scaled Max-Min oflarge random matrices.",
"Universality is demonstrated by data collapse for ninedifferent distributions from which the IID matrix entries are drawn: thecolored symbols depict the simulated data; the solid black line is theprobability density of the predicted Gumbel statistics, with its 95%confidence interval shaded in grey.Here we present “Central Limit Theorem” results for the Max-Min and Min-Max of large non-square random matrices.",
"Circumventing the use of the FTG theorem altogether, the results are based on novel Poisson-process limit-laws [45].",
"The results assert that the scaled Max-Min and Min-Max are governed, asymptotically, by Gumbel statistics.",
"The results' domain of attraction is vast, encompassing all distributions with a density.",
"The results' scaling schemes are similar to that of the CLT, and their asymptotic couplings are geometric.",
"The novel results established here are thus highly practical and applicable (see Fig.",
"2 for the Max-Min result).",
"Written for a general physics readership, this rapid communication offers a concise brief of the novel results and their implementation; for a comprehensive exposition, including detailed proofs, see [45].",
"The brief is organized as follows: we begin with an underlying setting, present Gumbel approximations for the Max-Min and Min-Max, and describe the implementation of these approximations; then, we present the Gumbel limit-laws (that yield the Gumbel approximations), discuss these limit-laws, and conclude with an outlook.",
"Setting.—Consider a random matrix with IID entries: $\\mathbf {M}=\\left(\\begin{array}{ccc}X_{1,1} & \\cdots & X_{1,n} \\\\\\vdots & \\ddots & \\vdots \\\\X_{m,1} & \\cdots & X_{m,n}\\end{array}\\right) \\text{ .}",
"$ Namely, the matrix is of dimensions $m\\times n$ , with rows labeled $i=1,\\cdots ,m$ , and columns labeled $j=1,\\cdots ,n$ .",
"The matrix entries are IID copies of a generic real-valued random variable $X$ , with probability density $f\\left( x\\right) $ ($-\\infty <x<\\infty $ ).",
"In what follows we denote by $F\\left( x\\right) =\\Pr \\left( X\\le x\\right) $ ($-\\infty <x<\\infty $ ) the corresponding distribution function, and by $\\bar{F}\\left( x\\right)=\\Pr \\left( X>x\\right) $ ($-\\infty <x<\\infty $ ) the corresponding survival function.",
"We set the focus on the Max-Min and Min-Max of the random matrix $\\mathbf {M}$ .",
"Denoting by $\\wedge _{i}=\\min \\left\\lbrace X_{i,1},\\cdots ,X_{i,n}\\right\\rbrace $ the minimum over the entries of row $i$ , the Max-Min is the maximum over the rows' minima: $\\wedge _{\\max }=\\max \\left\\lbrace \\wedge _{1},\\cdots ,\\wedge _{m}\\right\\rbrace \\text{ .",
"}$ Similarly, denoting by $\\vee _{j}=\\max \\left\\lbrace X_{1,j},\\cdots ,X_{m,j}\\right\\rbrace $ the maximum over the entries of column $j$ , the Min-Max is the minimum over the columns' maxima: $\\vee _{\\min }=\\min \\left\\lbrace \\vee _{1},\\cdots ,\\vee _{n}\\right\\rbrace \\text{ .",
"}$ To illustrate the setting, consider the aforementioned game-theory example.",
"If the matrix $\\mathbf {M}$ manifests gains then: the rows represent the player's strategies; the columns represent the scenarios the player is facing; $X_{i,j}$ is the player's gain when taking strategy $i$ and encountering scenario $j$ ; and $\\wedge _{\\max }$ is the player's Max-Min gain.",
"If the matrix $\\mathbf {M}$ manifests losses then the roles of its rows and columns are transposed, $X_{i,j}$ is the player's loss when encountering scenario $i$ and taking strategy $j$ , and $\\vee _{\\min }$ is the player's Min-Max loss.",
"From Eqs.",
"(REF ) and (REF ) it follows that the distribution/survival functions of the Max-Min and Min-Max are given, respectively, by $\\Pr \\left(\\wedge _{\\max }\\le x\\right)=[1-\\bar{F}\\left( x\\right)^n]^m$ and by $\\Pr \\left(\\vee _{\\min } > x\\right)=[1-F\\left( x\\right)^m]^n$ .",
"In the results to be presented here we scale the Max-Min and Min-Max appropriately, and establish their convergence (in law) to universal Gumbel statistics.",
"In what follows $Z$ denotes a `standard' Gumbel random variable, and $G(x)$ denotes the corresponding Gumbel distribution function [7]: $\\Pr \\left( Z\\le x\\right)=G(x)=\\exp \\left[ -\\exp \\left( -x\\right) \\right] $ ($-\\infty <x<\\infty $ ).",
"Our results involve an `anchor' $x_{\\ast }$ – an arbitrary value that can be realized by the generic random variable $X$ .",
"Specifically, the anchor meets two requirements: (i) $0<f\\left( x_{\\ast }\\right) <\\infty $ ; and (ii) $0<F\\left( x_{\\ast }\\right) <1$ , which is equivalent to $0<\\bar{F}\\left( x_{\\ast }\\right) <1$ .",
"For example, with regard to three of the distributions appearing in Fig.",
"2, the admissible values of the anchor are: $-\\infty <x_{\\ast }<\\infty $ for the Normal; $0<x_{\\ast }<\\infty $ for the Gamma; and $0<x_{\\ast }<1$ for the Beta.",
"Approximations.—We present Gumbel approximations for the Max-Min $\\wedge _{\\max }$ and the Min-Max $\\vee _{\\min }$ of a large random matrix $\\mathbf {M}$ with dimensions $m\\gg 1$ and $n\\gg 1$ .",
"The approximations are based on couplings between the matrix dimensions $m$ and $n$ , and the anchor $x_{\\ast }$ .",
"As we shall show hereinafter, these couplings are always implementable: given two of the triplet $\\lbrace m,n,x_{\\ast }\\rbrace $ we can always set the third to satisfy the couplings.",
"Also, in the approximations $Z$ is the `standard' Gumbel random variable of Eq.",
"(REF ).",
"Consider the coupling $m\\cdot \\bar{F}\\left( x_{\\ast }\\right) ^{n} \\simeq 1$ ; then, the Max-Min admits the approximation $\\wedge _{\\max }\\simeq Z_{\\max }:=x_{\\ast }+\\frac{1}{n}\\cdot \\frac{1}{\\alpha }Z{\\ ,} $ where $\\alpha =f\\left( x_{\\ast }\\right) /\\bar{F}\\left( x_{\\ast }\\right) $ .",
"Similarly, consider the coupling $n\\cdot F\\left(x_{\\ast }\\right) ^{m} \\simeq 1$ ; then, the Min-Max admits the approximation $\\vee _{\\min }\\simeq Z_{\\min }:=x_{\\ast }-\\frac{1}{m}\\cdot \\frac{1}{\\beta }Z\\text{ ,} $ where $\\beta =f\\left( x_{\\ast }\\right) /F\\left( x_{\\ast }\\right) $ .",
"Eqs.",
"(REF ) and (REF ) imply that: the deterministic approximation of the Max-Min $\\wedge _{\\max }$ and the Min-Max $\\vee _{\\min }$ is the anchor $x_{\\ast }$ ; the magnitude of the random fluctuations about $x_{\\ast }$ is $1/(n \\alpha )$ for the Max-Min, and is $1/(m \\beta )$ for the Min-Max; and the statistics of the random fluctuations about $x_{\\ast }$ are Gumbel.",
"Key statistical features of the Gumbel approximations $Z_{\\max }$ of Eq.",
"(REF ) and $Z_{\\min }$ of Eq.",
"(REF ) are detailed in Table 1: modes, medians, means, and standard deviations.",
"The probability densities of the Gumbel approximations $Z_{\\max }$ and $Z_{\\min }$ have a unimodal shape: monotone increasing below $x_{\\ast }$ , and monotone decreasing above $x_{\\ast }$ .",
"Implementation.—There are two ways of implementing the Gumbel approximations, which we now describe.",
"Both ways exploit the couplings underpinning the approximations.",
"The first way applies when the matrix dimensions $m$ and $n$ are given; in this case the dimensions determine the anchor $x_{\\ast }$ .",
"Specifically, for matrix $\\mathbf {M}$ with dimensions $m>n\\gg 1$ the approximation of Eq.",
"(REF ) holds with anchor $x_{\\ast }=\\bar{F}^{-1}[\\left(1/m\\right) ^{1/n}]$ .",
"Similarly, for matrix $\\mathbf {M}$ with dimensions $n>m\\gg 1$ the approximation of Eq.",
"(REF ) holds with anchor $x_{\\ast }= F^{-1}[\\left(1/n\\right) ^{1/m}]$ .",
"The second way applies when the anchor $x_{\\ast }$ is given; in this case the matrix dimensions $m$ and $n$ should be set accordingly.",
"Specifically, for the Max-Min setting $n\\gg 1$ and $m \\simeq 1 / \\bar{F}\\left( x_{\\ast }\\right) ^{n}$ yields the approximation of Eq.",
"(REF ).",
"And, for the Min-Max setting $m\\gg 1$ and $n \\simeq 1/ F\\left(x_{\\ast }\\right) ^{m} $ yields the approximation of Eq.",
"(REF ).",
"In this way the magnitudes of the random fluctuations about the anchor $x_{\\ast }$ are: of the order $O(1/n)$ in the approximation of Eq.",
"(REF ), and of the order $O(1/m)$ in the approximation of Eq.",
"(REF ).",
"The first way is a `scientific tool': given a matrix $\\mathbf {M}$ , it provides us with approximations for the Max-Min and Min-Max.",
"The second way is an `engineering tool': given a `target' anchor $x_{\\ast }$ , it tells us how to design the matrix $\\mathbf {M}$ so that $x_{\\ast }$ will be the deterministic approximation of the Max-Min and Min-Max; moreover, we can design the magnitudes of the random fluctuations about $x_{\\ast }$ to be as small as we wish [45].",
"Table: Key statistical features of the Gumbel approximations Z max Z_{\\max } of Eq.",
"() and Z min Z_{\\min } of Eq.",
"(): mode, median, mean, and standard deviation (SD); in the row for the mean, γ=0.577⋯\\protect \\gamma =0.577\\cdots is the Euler-Mascheroni constant.Limit-laws.—The Gumbel approximations of Eqs.",
"(REF ) and (REF ) emanate from corresponding Gumbel limit-laws which we now present.",
"In the limit-laws we fix the anchor $x_{\\ast }$ , and then grow the matrix dimensions infinitely large: $m,n\\rightarrow \\infty $ .",
"Also, in the limit-laws $G(x)$ is the `standard' Gumbel distribution function of Eq.",
"(REF ).",
"Grow the matrix dimensions via the coupled limit $\\lim _{m,n\\rightarrow \\infty }m\\cdot \\bar{F}\\left( x_{\\ast }\\right) ^{n}=1$ ; then, the Max-Min limit-law is $\\lim _{m,n\\rightarrow \\infty }\\Pr \\left[ \\alpha n \\left( \\ \\wedge _{\\max }-x_{\\ast }\\right) \\le x\\right] =G\\left( x\\right) $ ($-\\infty <x<\\infty $ ), where $\\alpha =f\\left( x_{\\ast }\\right) /\\bar{F}\\left( x_{\\ast }\\right) $ as above.",
"Similarly, grow the matrix dimensions via the coupled limit $\\lim _{m,n\\rightarrow \\infty }n\\cdot F\\left(x_{\\ast }\\right) ^{m}=1$ ; then, the Min-Max limit-law is $\\lim _{m,n\\rightarrow \\infty }\\Pr \\left[ \\beta m \\left( \\ x_{\\ast }-\\vee _{\\min }\\right) \\le x\\right] =G\\left( x\\right) $ ($-\\infty <x<\\infty $ ), where $\\beta =f\\left( x_{\\ast }\\right) /F\\left(x_{\\ast }\\right) $ as above.",
"Figure: The Gumbel limit-law of Eq.",
"(7) is tested for nine different distributions from which the IID matrix entries are drawn: (a) Beta; (b) Exponential; (c) Gamma; (d) Inverse Gaussian; (e) Log-Normal; (f) Normal; (g) Pareto; (h) Uniform; and (i) Weibull.",
"The statistics of the scaled Max-Min α·n∧ max -x * \\protect \\alpha \\cdot n \\left(\\ \\wedge _{\\max }-x_{\\ast }\\right) , with anchor x * =F ¯ -1 (0.8)x_{\\ast }=\\bar{F}^{-1} (0.8), were simulated by sampling 10 5 10^5 random matrices with the following dimensions: n=5,25,70n=5,25,70 rows and m≃1.25 n m \\simeq 1.25^n columns.",
"In all cases, the convergence of the simulations (colored symbols) to the probability density of the standard Gumbel law (solid black line, with its 95% confidence interval shaded in grey) is evident.Equations (REF ) and (REF ) imply that the scaled Max-Min $\\alpha n\\left( \\ \\wedge _{\\max }-x_{\\ast }\\right) $ and the scaled Min-Max $\\beta m\\left( \\ x_{\\ast }-\\vee _{\\min }\\right) $ converge – in law, as $m,n\\rightarrow \\infty $ – to a `standard' Gumbel random variable $Z$ (recall Eq.",
"(REF )).",
"Hence, the limit-laws of Eqs.",
"(REF ) and (REF ) yield, respectively, the approximations of Eqs.",
"(REF ) and (REF ).",
"The Gumbel limit-law of Eq.",
"(REF ) is tested for nine different distributions from which the IID matrix entries are drawn (Fig.",
"3); note that convergence is evident already for moderate values of the dimension $n$ .",
"The data collapse demonstrated in Fig.",
"2 corresponds to the nine distributions of Fig.",
"3 with dimension $n=70$ .",
"The Gumbel limit-laws of Eqs.",
"(REF ) and (REF ) stem from `bedrock' Poisson-process limit-laws.",
"Underlying the Max-Min $\\wedge _{\\max }$ is the ensemble of the rows' minima $\\left\\lbrace \\wedge _{1},\\cdots ,\\wedge _{m}\\right\\rbrace $ , and underlying the Min-Max $\\vee _{\\min }$ is the ensemble of the columns' maxima $\\left\\lbrace \\vee _{1},\\cdots ,\\vee _{n}\\right\\rbrace $ .",
"In [45] it is established that appropriately scaled versions of these ensembles converge – in law, as $m,n\\rightarrow \\infty $ – to a Poisson process that is characterized by the following exponential intensity function: $\\lambda (x)=\\exp (-x)$ ($-\\infty <x<\\infty $ ).",
"For the points of this Poisson process one can observe that: the maximal point is no larger than a real threshold $x$ if and only if there are no points above this threshold – an event whose probability is $\\exp \\left[ -\\int _{x}^{\\infty }\\lambda (x^{\\prime })dx^{\\prime }\\right] =G(x)$ [46].",
"Hence, the distribution function of the maximal point is $G(x)$ – which is the term that appears on the right-hand sides of Eqs.",
"(REF ) and (REF ) [45].",
"Discussion.—The limit-laws of Eqs.",
"(REF ) and (REF ) are highly invariant with respect to the IID entries of the random matrix $\\mathbf {M}$ .",
"Indeed, contrary to the CLT – no moment conditions are imposed on the entries' distribution.",
"And, contrary to the generalized CLT and to the FTG theorem – no tail conditions are imposed on the entries' distribution.",
"The Gumbel limit-laws merely require that the entries' distribution has a density.",
"In practice, this smoothness condition is widely satisfied.",
"The Gumbel limit-laws of Eqs.",
"(REF ) and (REF ) involve simple scaling schemes.",
"To appreciate their simplicity, we compare these schemes to that of the CLT.",
"Consider $A_k$ to be the average of $k$ IID random variables with common mean $ \\mu $ and standard deviation $ \\sigma $ .",
"The CLT asserts that the scaled average $\\sigma ^{-1}\\sqrt{k}(A_k - \\mu ) $ converges – in law, as $k\\rightarrow \\infty $ – to a `standard' Normal random variable (i.e.",
"with a zero mean and a unit standard deviation).",
"The scaled Max-Min $\\alpha n\\left( \\ \\wedge _{\\max }-x_{\\ast }\\right) $ of Eq.",
"(REF ) and the scaled Min-Max $\\beta m\\left( \\ x_{\\ast }-\\vee _{\\min }\\right) $ of Eq.",
"(REF ) are similar, in form, to the scaled average $ \\sigma ^{-1}\\sqrt{k} (A_k - \\mu ) $ .",
"Specifically: the anchor $x_{\\ast }$ is the counterpart of the mean $ \\mu $ ; and the scale terms $\\alpha n $ and $\\beta m $ are the counterparts of the scale term $\\sigma ^{-1}\\sqrt{k}$ .",
"Consequently, the scaling schemes of the limit-laws of Eqs.",
"(REF ) and (REF ) are as simple and straightforward as that of the CLT.",
"There are numerously many ways of setting the scaling schemes of the generalized CLT and of the FTG theorem, and each such way corresponds to specific distributions of the underlying IID random variables.",
"On the other hand, as detailed above, the scaling scheme of the CLT is set in a particular way.",
"This special CLT scaling scheme is universal in the following sense: it yields Normal limit-law statistics for all finite-variance distributions.",
"Addressing limit-laws for the Max-Min and Min-Max of random matrices [40], [41], [42], [43], [44]: there are numerously many ways of setting the scaling schemes; and there are also numerously many ways of asymptotically coupling the matrix dimensions, $m$ and $n$ , when growing them infinitely large ($m,n\\rightarrow \\infty $ ).",
"Similarly to the CLT, the Gumbel limit-laws of Eqs.",
"(REF ) and (REF ) employ particular scaling schemes, as well as particular asymptotic couplings.",
"In turn, as for the CLT, these special scaling schemes and asymptotic couplings are universal in the following sense: they yield Gumbel limit-law statistics for all distributions with a density.",
"The particular asymptotic couplings employed here are geometric, and they are parameterized by the anchor $x_{\\ast }$ .",
"Specifically, the geometric asymptotic couplings are given by: $\\lim _{m,n\\rightarrow \\infty }m\\cdot \\bar{F}\\left( x_{\\ast }\\right) ^{n}=1$ for the Gumbel limit-law of Eq.",
"(REF ), and $\\lim _{m,n\\rightarrow \\infty }n\\cdot F\\left(x_{\\ast }\\right) ^{m}=1$ for the Gumbel limit-law of Eq.",
"(REF ).",
"The couplings' parameterization is a degree-of-freedom that facilitates tunability.",
"Indeed, the anchor $x_{\\ast }$ – which is the counterpart of the mean $ \\mu $ in the CLT – can be tuned as we wish within its admissible values.",
"Outlook.—It has long been observed that seemingly identical pieces of matter happen to fail stochastically at different times and under different loads.",
"Consequently, one of the major original drivers for the development of extreme-value theory came from materials science – where statistical predictions for mechanical strength and fracture formation are of prime importance [47], [48].",
"The “weakest link hypothesis” is foundational in materials science [26], [27].",
"This hypothesis suggests that various mechanical systems can be modeled as having a chain-like structure – thus implying that such a system is only as strong as its weakest link.",
"The “weakest link hypothesis” naturally gives rise to the Max-Min: when statistically similar chain-like systems are compared – either by an evolutionary process or by industrial quality testing – the system with the strongest weakest link prevails.",
"The Min-Max also arises naturally from real-world applications.",
"Indeed, consider a back-up system in which critical files are stored on multiple separate hard drives.",
"If a file is damaged on one of the drives it could be retrieved from another; however, if all copies of a file are damaged then the file is lost forever.",
"The loss time of a given file is thus the maximum of its damage times over the different drives.",
"In turn, since all files are critical, system failure occurs at the first loss time of a file.",
"Thus, the system failure time is the Min-Max of the files' damage times.",
"Here we adopted the setting of random-matrix theory, considering large matrices with IID entries.",
"For the Max-Min and Min-Max of such matrices we established, respectively, the Gumbel approximations of Eqs.",
"(REF ) and (REF ); also, we showed how to apply these approximations as a `scientific tool' and as an `engineering tool'.",
"The approximations stem from the limit-laws of Eqs.",
"(REF ) and (REF ) – which assume the role of a “Gumbel Central Limit Theorem” for the Max-Min and Min-Max.",
"With their generality and universality, their easy practical implementation, and their many potential applications – e.g.",
"in game theory, in reliability engineering, in materials science, and in the design of back-up systems – the novel results presented herein are expected to serve diverse audiences in the physical sciences and beyond.",
"Acknowledgments.",
"R.M.",
"acknowledges Deutsche Forschungsgemeinschaft for funding (ME 1535/7-1) and support from the Foundation for Polish Science within an Alexander von Humboldt Polish Honorary Research Fellowship.",
"S.R.",
"gratefully acknowledges support from the Azrieli Foundation and the Sackler Center for Computational Molecular and Materials Science."
]
] | 1808.08423 |
[
[
"Deep-Learning Ensembles for Skin-Lesion Segmentation, Analysis,\n Classification: RECOD Titans at ISIC Challenge 2018"
],
[
"Abstract This extended abstract describes the participation of RECOD Titans in parts 1 to 3 of the ISIC Challenge 2018 \"Skin Lesion Analysis Towards Melanoma Detection\" (MICCAI 2018).",
"Although our team has a long experience with melanoma classification and moderate experience with lesion segmentation, the ISIC Challenge 2018 was the very first time we worked on lesion attribute detection.",
"For each task we submitted 3 different ensemble approaches, varying combinations of models and datasets.",
"Our best results on the official testing set, regarding the official metric of each task, were: 0.728 (segmentation), 0.344 (attribute detection) and 0.803 (classification).",
"Those submissions reached, respectively, the 56th, 14th and 9th places."
],
[
"History",
"Our team has worked on skin lesion analysis since early 2014 [1], and has employed deep learning with transfer learning for that task since 2015 [2].",
"From 2016 onwards, the community moved from traditional techniques towards deep learning, following the general trend of computer vision [3].",
"Deep learning poses a challenge for medical applications, as it needs very large training sets.",
"Thus, transfer learning becomes crucial for success in those applications, motivating our paper for ISBI 2017 [4].",
"Until 2017, the contribution of each factor of a deep learning solution (e.g., model choice, dataset size, data augmentation, image normalization, etc.)",
"to the performance of a skin lesion classifier was not evident.",
"We cleared such question by extensively analyzing several combinations of architectures, dataset sizes, and other eight relevant aspects [5].",
"We participated in the ISIC Challenge 2017, being ranked in 1st place for melanoma classification and 5th place for skin lesion segmentation [6].",
"In 2018, for the first time, we participated in all three tasks.Although our team has a long experience with skin-lesion classification (Task 3) and moderate experience with lesion segmentation (Task 1), this Challenge was the very first time we worked on attribute detection (Task 2).",
"We aimed, from the start, at deep learning solutions for all tasks.",
"We know from experience that the success factors for a competitive deep learning approach are data availability and model depth [6], [5].",
"To improve our chances, we also introduced two original contributions — synthetic lesions generation and stronger data augmentation approaches — to boost the models training.",
"Such contributions will be detailed next.",
"Participating in the Challenge brings our sportive desire to squeeze the models for their best performance — as always, we temper that goal with aesthetic considerations, avoiding as much as possible kludges and added complexity.",
"Added complexity has to bring proportional improvements over the metrics, or we will prefer the simpler model.",
"Each task allowed up to 3 distinct submissions.",
"We used them to contrast models trained with extra data with models trained with challenge-data only, or to compare different ways to ensemble the final solutions."
],
[
"Data",
"In previous work, we showed that the training set size responds by almost 50% of the variation on the prediction power of the classifier [5].",
"The freedom to use external sources enabled us to gather more data to boost our models.",
"First, we restricted ourselves to publicly available (for free, or for a fee) sources with high-quality images: ISIC 2018 Challenge [7], [8] the official challenge dataset, with 10,015 dermoscopic images.",
"ISIC ArchiveThe ISIC Archive: http://isdis.net/isic-project/ with over 13,000 dermoscopic images.",
"Interactive Atlas of Dermoscopy [9] with 1,000+ clinical cases, each with dermoscopic, and close-up clinical images.",
"Dermofit Image Library [10] with 1,300 images.",
"PH2 Dataset [11] with 200 dermoscopic images.",
"However, due to the extreme imbalance of the dataset, we decided to gather extra images for the severely underrepresented classes (namely Actinic keratosis, Basal cell carcinoma, Dermatofibroma, and Vascular lesion).",
"We found images browsing sources on the web, and asking for contributions from partner researchers in Medical Science (acknowledged in the final section).",
"The web sources were Dermatology Atlas (www.atlasdermatologico.com.br), Derm101 (www.derm101.com), DermIS (www.dermis.net/dermisroot).",
"With that extra effort, we acquired additional 631 images, being 414 BCC, 26 AKIEC, 132 DF and 59 VASC.",
"The final dataset continued seriously unbalanced, but the proportion of underrepresented classes grew considerably.",
"Our final dataset had 30,726 images.",
"We evaluated our extra data (full) on Task 1 (with 18,179 images, those with segmentation ground-truth), and Task 3 (with 30,324 images, those with diagnosis label).",
"We did not have ground truth for Task 2 — other than the 2017 Challenge data, which we briefly considered employing — so for this task, we did not use extra data.",
"For the three tasks we also made submissions using only Challenge data (only).",
"After picking a dataset, we divided it the into 3 splits, for each task: 10% for holdout (for our internal model selection) and the remaining 90% for training.",
"The training split was further divided into five 10%-validation/90%-training different splits (at random, not using cross-validation folds).",
"We considered case numbers, aliases, and near-duplicates in the split division, to minimize contaminations across splits.",
"We used the holdout sets to select the models.",
"We used the metrics observed in the holdout sets to identify strong release candidate models and/or good bets for a meta-learning phase.",
"Although the official validation data was very limited on this year's Challenge, we still used its scores as ancillary estimates.",
"The exact datasets and splits, for each task, will be listed, image by image, in our code repositoryhttps://github.com/learningtitans/isic2018-{part1,part2,part3} (available soon)."
],
[
"Experimental ",
"Our starting point was our last work on how to design powerful deep-learning classifiers for skin lesions [5].",
"We evaluated the main factors that vary on the approaches found in literature: use of transfer learning, model architecture, train dataset, image resolution, type of data augmentation, input normalization, use of segmentation, duration of training, additional use of SVM, and test data augmentation.",
"For the challenge, there was no time to perform such a detailed study — involving significance tests over a full-factorial design — but we wanted to make sound decisions along the way.",
"We decided to use our previous study to eliminate many choices and perform much-reduced designs, involving less than a handful of factors.",
"We will describe the factors (and their levels) in the task sections.",
"The team used the Slack collaboration tool as the main channel for communication.",
"We coordinated the tasks with Google Docs and shared the results of each intermediate experiment with Google Sheets.",
"We used code version control (with git) to facilitate future reproduction of intermediate steps."
],
[
"Notable Novelties",
"The models we proposed this year have several technical advances in comparison with the models we submitted last year: deeper architectures, changes in frameworks, better training craftsmanship, etc.",
"In this section, however, we showcase the most exciting scientific novelties.",
"For this year we took advantage of our recent results regarding new approaches for data augmentation: (a) image processing of real skin lesion images [12], and (b) synthetic skin lesions using GANs (Generative Adversarial Networks) [13].",
"In work (a), we investigated the impact of 13 image processing-based scenarios of data augmentation for melanoma classification.",
"Scenarios include traditional color and geometric transforms, and more unusual augmentations such as elastic transforms, random erasing and a novel augmentation that mix two different lesions.",
"Using our participation on ISIC Challenge 2017 (with Inception-v4) with as baseline, we observed similar performance using the new data augmentation methods, but without using external data.",
"That is, the image processing data augmentation methods were equal to the performance of the model trained with external data (which we know that has a huge impact on the classifier prediction power [5]).",
"Among all experiments and scenarios, scenario J (please refer to [12] for details) leads to better performance and was the one introduced in the experiments of the competition (only in Task 3).",
"In work (b), we created fake high-resolution (1024$\\times $ 512 pixels) skin lesion samples, aiming to extend the training set artificially.",
"To do that, we used GANs to teach the network the malignancy markers and also incorporating the specificities of a lesion border.",
"We inputted such information directly to the network, using a semantic map and an instance map.",
"Semantic maps are blobs that show the presence and the location of the five malignancy markers within the same lesions' segmentation masks.",
"Instance maps take information from superpixels, which group similar pixels creating visually meaningful blobs, limiting each unit regarding their meaning.",
"Please refer to [13] for details.",
"We used the synthetic images only on Task 3 (on the two submissions using external data).",
"We added the synthetic images to the training/training splits (never to the holdout or to the training/validation splits) keeping a 1:1 per class proportion (i.e., one synthetic image for each real image in each lesion class)."
],
[
"Computational Resources",
"To perform a large number of trials, we attempted to secure as much computational horsepower as possible.",
"For deep learning, that means large-memory CUDA-compatible GPUs.",
"For the experiments, we used NVIDIA GPUs available at RECOD Lab: two Titan X Pascal, six Titan Xp, one Tesla K40, and for Tesla P100.",
"We also used the NC6 (Tesla K80) and ND6 (Tesla P40) virtual machines provided by the Microsoft Azure Cloud platform.",
"Although there was a long phase of preliminary experiments, the training and testing of the final models that composed the submissions took only around ten days."
],
[
"Task 1: Lesion Boundary Segmentation",
"This is our second participation in the segmentation task.",
"Although we have some experience in this area, lesion segmentation is not the primary research line of our group.",
"From our previous participation, we decided to keep the U-shape networks and moderate training times.",
"Contrarily to our previous experiments, showing little difference between using low (128$\\times $ 128) or high (256$\\times $ 256) resolution, we opted risking for a possible small improvement given by the latter.",
"That was motivated by the knowledge the best networks are tied so closed to the inter-human agreement, and that even small contributions could help.",
"Observing the generated masks in preliminary experiments and the 2018 ground-truth annotations — together with the introduction of a threshold — we decided for a less fine-grained and more conservative approach concerning details of the final generated mask.",
"To enhance the results, we used a post-processing techniques, to fill the holes in the masks with a morphological operation."
],
[
"Experiments",
"We worked on two main models: the FusionNetUsing this implementation: github.com/GunhoChoi/FusionNet-Pytorch [14], a deep fully residual neural network designed for image segmentation in connectomics, and a U-Net-like modelCode based on github.com/ternaus/TernausNet [15], a convolutional neural network traditionally used for biomedical-image segmentation, with a VGG-16 [16] encoder pretrained on the ImageNet dataset.",
"We trained our models with two datasets: i) Challenge data; and ii) Challenge data plus external data.",
"Each model was trained using Adam optimizer, using the Cyclic Learning Rate technique [17], on which the learning rate cyclically vary within reasonable boundaries, improving the accuracy and reducing the training time by allowing the model to scape local minima faster.",
"For the cyclic learning rate technique, we used a base learning rate equal to $10^{-5}$ and a maximum learning rate of $10^{-4}$ with a step size of 500.",
"We trained the models for 100 epochs each, with early stopping with patience of 20 epochs.",
"For the loss function, we used the Binary Cross Entropy with soft Jaccard index, with Jaccard weight of 1.0.",
"We tested four main configurations for the competition: i) FusionNet using only the Challenge data; ii) FusionNet using the Challenge data and external data; iii) U-Net using only the Challenge data; and iv) U-Net using the Challenge data and external data.",
"From our experiments, we noticed that when using external data during the training phase, the results were significantly worse.",
"The large inter-human variability and the existence of several types of ground truths may explain why the task works best on a smaller, but better curated subset of training data.",
"After training all the desired networks, we designed the strategy for our submissions, which includes ensembling with our models and post processing the decision.",
"The chosen models were averages, we filled the holes with a morphological operation, and the segmentation mask was upsampled to the image original size.",
"Our three submissions were (1) average of FusionNet trained on Challenge only, and U-Net trained on Challenge only; (2) average of FusionNet trained on Challenge only, U-Net trained on Challenge only, and FusionNet trained on Challenge and external data; (3) U-Net trained on Challenge only.",
"Our final results on the official testing set were, respectively, 0.694, 0.686 and 0.728 for the threshold Jaccard index.",
"Also, our positions of each submission were, respectively, 88th, 93th and 56th among 112 submissions."
],
[
"Task 2: Lesion Attribute Detection",
"We addressed the task as a patch classification problem rather than a segmentation problem, since our team has a much stronger background in the former, including precoded and pretrained models."
],
[
"Experiments",
"Each image contains about 1,000 superpixels, identified with the same algorithm used in the Challenge to create the ground truths.",
"To address the extreme dataset imbalance, we train 750 balanced batches per epoch, for 30 epochs on two different internal splits for each model.",
"First, we crop the images into patches, each with a superpixel in its center.",
"The patch dimensions are one of the factors evaluated: 128$\\times $ 128 and 299$\\times $ 299.",
"By employing bigger patches, we expect the network to learn not only from the center superpixel, but also from its neighborhood.",
"We fine-tune an Inception-v4 [18] network pretrained on ImageNet.",
"We employ Stochastic Gradient Descent with a momentum of 0.9, batches of size 16, and starting learning rate of 0.001, decreasing it to 0.0001 after epoch 12.",
"Data augmentation is one of the factors we tested.",
"When applying augmentations on training (random flips, rotations, and color jitter) the result was significantly worse.",
"We suspect that these augmentations could displace the superpixel from the center of the patch.",
"Our final submission does not contain any data augmentation during training.",
"We keep the augmentation for test with 16 replicas at all models, with random flips and color jitter.",
"The network learns to classify each of the crops (which are linked to a superpixel of the image) as one of the six classes: absent, pigment network, negative network, streaks, milia-like cyst, and globules.",
"To generate the predictions, the test set also needs to be cropped into patches.",
"Each patch receives a prediction about its class.",
"We classify the patch by selecting the class with the highest score assigned by the network.",
"Next, we compose the masks from the predictions and apply a post-processing procedure to eliminate positive superpixels given a threshold (30 in our experiments showed the best result).",
"This is used to attenuate false positives that occurred especially in the most abundant class, absent.",
"Our three submissions were (1) average of 4 best deep learning model with final thresholding; (2) average of 4 deep learning models, without final thresholding; (3) the single best model on the training/validation split with post-processing.",
"Our final results on the official testing set were, respectively, 0.344, 0.337 and 0.323 for the Jaccard index.",
"Also, our positions of each submission were, respectively, 14th, 15th and 17th among 26 submissions.",
"Although the patch classification approach allowed our team to participate in this task, it is very time-consuming: testing takes longer than training!",
"As a consequence, we separated the holdout set but did not have time to evaluate the models on it.",
"We employed the less-than-ideal training/validation performance to select the models."
],
[
"Task 3: Lesion Diagnosis",
"Automated lesion classification is the most traditional research line in our group.",
"This year, our explorations started from our participation on ISIC Challenge 2017 [6] and our follow-up research [5].",
"We also look for novelties and insights from the Machine Learning community that could bring new competitive gains.",
"Although many of the experiments were performed systematically, simulating a factorial design, not all combinations were evaluated.",
"Also, the training of some models were limited due to time and computational resources."
],
[
"Experiments",
"We trained three different CNN architectures: Inception-v4 [18], ResNet-152 [19], and DenseNet-161 [20], all pretrained on ImageNet dataset.",
"We fine-tuned the networks on three datasets: full, only, and full+synthetic augmentation [13].",
"Each network was trained with Stochastic Gradient Descent with momentum of 0.9, batch size of 32, starting learning rate of $10^{-3}$ , multiplied by 0.1 whenever the validation loss fails to improve for 10 epochs, until it reaches $10^{-5}$ .",
"Images were resized online to 224$\\times $ 224 for ResNet and DenseNet, and to 299$\\times $ 299 for Inception-v4.",
"We normalized the images by subtracting the mean and dividing by the standard deviation channel-wise.",
"To deal with dataset imbalance, we set the optimization goal to a class-weighted cross-entropy, with the weights calculated by dividing the frequency of the most common class by the frequency of each class.",
"We applied early stopping with a patience of 22 epochs by monitoring the validation loss.",
"We performed online data augmentation as described in [12] (scenario J): random crops (preserving 0.4-1.0 of the original area, and 3/4-4/3 of the original aspect ratio); random vertical/horizontal flips; rotation (0-90°); shear (0-20°); area scaling (0.8-1.2); random color transformations on saturation, brightness, contrast, and hue.",
"We applied the transformations to the validation (single replica), holdout (32 replicas), and final test (128 replicas), taking the decision as the average of the replicas.",
"Our three submissions were (1) XGBoost ensemble of 43 deep learning models; (2) average of 8 best deep learning models (on the holdout set) augmented with synthetic imagesN.B.",
"that approach is wrongly named as an average of 15 models on the official leaderboard and (3) average of 15 deep learning models trained only with Challenge data.",
"Our final results on the official testing set were, respectively, 0.732, 0.725 and 0.803 for the normalized multi-class accuracy.",
"Also, our positions of each submission were, respectively, 32th, 39th and 9th among 141 submissions."
],
[
"Final Comments",
"We are very excited to see the ISIC Challenge as a continuing event, since we consider such initiative as pivotal for the development of our research area.",
"Until recently, making comparisons across different approaches for skin lesion analysis was essentially impossible, due to difficulties of code and data sharing, and lack of standardized evaluation metrics and datasets [3].",
"We also acknowledge the importance of keeping the testing set secret until all evaluations were over, preventing, thus, subtle methodological errors that inflate the performance evaluation of models [5], [21].",
"Despite the diversity of skin lesion types and their dermatological importance, we asked ourselves whether making the classification task (Task 3) so fine-grained was really necessary, especially given the huge class imbalance.",
"We hope to be surprised when the results become public, but we fear that confusion among very small classes (e.g., Benign Keratosis and Actinic Keratosis) will bring much noise to the evaluation.",
"In our current research, we are still focusing on coarse-grained melanoma/non-melanoma screening/triage classifiers — and we notice that real-world performances even for such coarse-grained procedures are still far from ideal.",
"We noticed the variability of the annotations as an important difficulty for Task 1.",
"While some lesions were very finely annotated, others are merely polygons around the lesion.",
"The performances on that task are — or at least were, in 2017 — close to the limit of inter-human agreement, and those different “definitions” of what is a segmentation bring extra fluctuations.",
"As a suggestion, maybe using the convex hulls of the human annotations is sufficient for location purposes, and provides a less noisy target for comparing algorithms.",
"Despite our lack of experience, we were excited to participate in Task 2.",
"It is a new problem for the automated skin lesion analysis community and poses several challenges: especially in terms of evaluating the models, given the hugely unbalanced annotations.",
"For us, however, the existence of that task had an additional importance: its ground-truth annotations allowed us to create, for the first time, realistic synthetic lesion images, with proper dermatologic configurations, using Generative Adversarial Networks [13].",
"As this opens a new frontier in dealing with the scarcity of annotated data, we hope the community will work to provide more of this valuable type of ground truth."
],
[
"Acknowledgements",
"A. Bissoto is funded by CNPq; M. Fornaciali and E. Valle are partially funded by Google Research Awards for Latin America 2017; E. Valle is also partially funded by a CNPq PQ-2 grant (311905/2017-0) and Universal grant (424958/2016-3).",
"RECOD Lab.",
"is partially supported by diverse projects and grants from FAPESP, CNPq, and CAPES.",
"We gratefully acknowledge NVIDIA Corporation for the donation of GPUs and Microsoft Azure for the GPU-powered cloud platform used in this work.",
"We are grateful to RECOD members — and in particular to Pedro Tabacof and Ramon Oliveira — for scientific and technical insights on machine learning.",
"We thank Prof. Flávia V. Bittencourt and Prof. Gabriela Salvio for kindly providing additional skin lesion images.",
"We also thank our partners from the School of Medical Sciences of UNICAMP for the dermoscopic discussions that enhanced our comprehension about skin lesion images."
]
] | 1808.08480 |
[
[
"Comparing CNN and LSTM character-level embeddings in BiLSTM-CRF models\n for chemical and disease named entity recognition"
],
[
"Abstract We compare the use of LSTM-based and CNN-based character-level word embeddings in BiLSTM-CRF models to approach chemical and disease named entity recognition (NER) tasks.",
"Empirical results over the BioCreative V CDR corpus show that the use of either type of character-level word embeddings in conjunction with the BiLSTM-CRF models leads to comparable state-of-the-art performance.",
"However, the models using CNN-based character-level word embeddings have a computational performance advantage, increasing training time over word-based models by 25% while the LSTM-based character-level word embeddings more than double the required training time."
],
[
"Introduction",
"Bi-directional Long-Short Term Memory Conditional Random Field models (BiLSTM-CRF), in which a BiLSTM is coupled with a CRF layer to connect output tags, have been shown to achieve state-of-art performance in sequence tagging tasks including part of speech (POS) tagging, chunking, and NER [5].",
"The combination of word embeddings and character-level word embeddings has been explored in this context, with Ma:2016 using Convolutional Neural Networks (CNNs) to construct character-level word embeddings and Lample:2016 applying LSTM networks.",
"This work showed that the use of character-level word embeddings improves the performance of the models, by contributing the ability to recognize unseen words.",
"Biomedical Named Entity Recognition (BNER) is a vital initial step for information extraction tasks in the biomedical domain, including the Chemical-Disease Relationship (CDR) extraction task where both chemical and disease entities must be identified [13].",
"Character-level word embeddings could be particularly significant in this context, given that new entity names are frequently created, and may follow consistent patterns including productive morphology such as common prefixes (e.g., di-) or suffixes (e.g., -ase).",
"Features that capture word-internal characteristics have been shown to be effective for BNER tasks in CRF models [7].",
"Lyu:2017 applied a BiLSTM-CRF model with LSTM-based character-level word embeddings to a gene and protein NER task, demonstrating state-of-art performance that outperformed traditional feature-based models.",
"Luo:2018 further improved on this result on a chemical NER task by adding an attention layer between the BiLSTM and CRF layers (Att-BiLSTM-CRF).",
"In an experiment by Nils:2017, optimal hyper-parameters for LSTM networks in sequence tagging tasks were explored, with the finding that incorporation of character-level word embeddings significantly improved performance on NER tasks on general datasets including CoNLL 2003 [21].",
"However, the choice of CNN-based [16] or LSTM-based character-level word embeddings [8] did not affect the performance significantly.",
"Since the CNN has fewer parameters to train than BiLSTM network, it is better in terms of training efficiency, and was recommended as the preferred approach.",
"In this paper, we implement and compare models with each type of word embedding to generate empirical results for the tasks of chemical and disease NER, using the BioCreative V CDR corpus [13].",
"These BNER categories are the most searched entities in the biomedical literature [6], and hence particularly important to study.",
"The results show that models with CNN-based character-level word embeddings achieve state-of-the-art results comparable to LSTM-based character-level word embeddings, while having the advantage of reduced training complexity, demonstrating that the prior results also hold for the BNER task.",
"Figure: Architecture of BiLSTM-CRF models with character-level word representations and additional features.",
"This figure is adapted from ReimersG17.Figure: Character-level word representations.",
"This figure is also adapted from ReimersG17."
],
[
"Experimental methodology",
"This section presents our empirical approach to comparing state-of-the-art neural network models for chemical and disease NER."
],
[
"Dataset",
"In our experiments, we use the BioCreative V CDR corpus [13].",
"This corpus provides a set of 1000 manually-annotated abstracts (9193 sentences) for training and development, and another set of 500 manually-annotated abstracts (4840 sentences) for test.",
"In particular, we used a pre-processed version of the CDR corpus from Luo:2018,https://github.com/lingluodlut/Att-ChemdNER which provides predicted POS-, chunking- and gazetteer-based tags: POS and chunking tags are predicted by the GENIA tagger [22].http://www.nactem.ac.uk/GENIA/tagger Gazetteer tags are encoded in BIO tagging scheme based on matching to the external Jochem chemical dictionary [3].",
"Following Luo:2018, we randomly sample 10% from the set of 1000 abstracts for development, and use the remaining for training."
],
[
"Models",
"We use the following BiLSTM-CRF-based sequence labeling models: Baseline BiLSTM model [20], [4] which uses a softmax layer to predict NER labels of input words.",
"BiLSTM-CRF [5] extends the BiLSTM model with a CRF layer which allows the model to use sentence-level tag information for sequence prediction.",
"BiLSTM-CRF + CNN-char [16] extends the BiLSTM-CRF model with character-level word embeddings.",
"For each word, its character-level word embedding is derived by applying a CNN to the character sequence in the word.",
"BiLSTM-CRF + LSTM-char also extends the BiLSTM-CRF model with character-level word embeddings which are derived by applying a BiLSTM to the character sequence in each word [8].",
"Following Luo:2018, we also consider the impact of extra features including syntactic features such as POS and chunking tags, and a chemical term feature based on matching to an external gazetteer.",
"Figure REF illustrates the general BiLSTM-CRF model architecture with character-level word embeddings and additional features, while Figure REF illustrates CNN-based and LSTM-based architectures for learning the character-level word embeddings.",
"Table: Fixed hyper-parameter configurations.Table: Hyper-parameters for learning character-level word embedding.",
"“charEmbedSize” and “# of Params.” denote the vector size of character embeddings and the total number of parameters, respectively."
],
[
"Implementation details",
"We used a well-known implementation of BiLSTM-CRF-based models from Nils:2017.https://github.com/UKPLab/emnlp2017-bilstm-cnn-crf We used the training set to learn model parameters, the development set to select optimal hyper-parameters, and the test set to report final results.",
"Here, we tune the model hyper-parameters using the performance across both NER categories (“Overall”) on the development set.",
"We employed pre-trained 50-dimensional word vectors from Luo:2018.",
"These pre-trained vectors were derived by training the Word2Vec skip-gram model [17] on a large text collection of 2 million MEDLINE abstracts.",
"Nils:2017 showed that the BiLSTM-CRF model achieved best performance with 2 BiLSTM layers.",
"Therefore, in our experiment, we only evaluated models up to 2 stacked BiLSTM layers.",
"The size of LSTM hidden states in each layer was selected from [100, 150, 200, 250].",
"We achieved the highest F1 score on the development set when using 250-dimensional LSTM hidden states for all models.",
"By default, each of the additional features (POS, chunking tags, gazetteer match tag) was incorporated into the model via a 10-dimensional embedding.",
"Other hyper parameters were also fixed as in Nils:2017 during initialization.",
"See tables REF and REF for more details.",
"In the training process, we used the score on development set to assess model improvement.",
"Early stopping was applied if there was no improvement after 10 epochs.",
"The threshold for a word that was not in the word embedding vocabulary to be added into the embedding was set to 5.",
"The average training time for each epoch was also recorded.",
"Table: Results (in %) on the test set.",
"[♠\\spadesuit ] denotes results reported on a 950/50 training/development split rather than our 900/100 split.",
"As indicated, Att-BiLSTM-CRFused LSTM-char word embeddings."
],
[
"Baseline results",
"Table REF presents our empirical results.",
"The first three rows show the performance of baseline models without the CRF layer, the next three rows show the performance of BiLSTM-CRF models without additional features, and then the next three rows show the results for BiLSTM-CRF models with additional gazetteer features.",
"As the empirical results in Table REF show, the model with CNN character-level embeddings (CNN-char) and the model with LSTM character-level embeddings (LSTM-char) achieved similar overall F1 scores (87.88% and 87.79%, respectively), outperforming BiLSTM-CRF by approximately 1% in absolute terms.",
"In particular, on chemical NER, both BiLSTM-CRF-based models with character-level word embeddings obtained the same F1 score (91.94%), while on disease NER the model with CNN-char obtained slightly higher performance (83.01%) than the model with LSTM-char (82.83%).",
"All models with the CRF layer outperformed their respective baseline BiLSTM models in F1 scores for all entity categories."
],
[
"Effect of additional features",
"When incorporating additional POS and chunking features into three baseline BiLSTM-CRF-based models, we found that no performance improvement based on the baseline models was observed.",
"On chemical NER, the additional gazetteer feature improved the baseline BiLSTM-CRF by about 0.8% while it only improved the baselines BiLSTM-CRF + CNN-char and BiLSTM-CRF + LSTM-char by about 0.3%, thus clearly indicating that character-level word embeddings can capture unseen word information.",
"Considering both NER categories together (“Overall”), the best performance was also obtained when the gazetteer feature was added, reaching overall F1 scores of 88.02% and 87.99%, respectively, for the two CNN-based and LSTM-based character-level embedding models."
],
[
"Comparison with prior work",
"The performance comparison between our BiLSTM-CRF-based models and other machine learning approaches to the two studied NER tasks is also shown in Table REF .",
"The pattern of chemical NER outperforming disease NER is consistent across all tools.",
"The Att-BiLSTM-CRF model [14] used a BiLSTM-CRF model with LSTM character-level word embedding and an additional attention layer.",
"It achieved an F1 score of 91.96% on chemical NER without additional features.",
"The positive effect of a gazetteer feature was also observed in their results; the model with syntactic and gazetteer features reached an F1 score of 92.57%.",
"Note that the datasets used in this paper might not be exactly the same as ours due to random sampling.",
"The last three rows of Table REF show the results presented in Leaman:2016, where 950 of the abstracts were used for training and 50 for development (cf.",
"our 900/100 split).",
"Dnorm [9] is a model based on pairwise learning to rank on disease name normalization, which achieved F1 score of 80.7% on disease NER.",
"The tmChem [11] is based on CRF; using numerous hand-crafted features it reached an F1 score of 88.4% on chemical entities.",
"As a semi-Markov model with a richer set of features for NER tasks, TaggerOne [10] achieved F1 score of 91.4% and 82.6% on chemical and disease entities, respectively.",
"Compared to previous non-deep-learning methods using CRFs, the BiLSTM-CRF models have significant advantage on F1 score of both chemical and disease entities, primarily due to improvement on recall."
],
[
"Discussion",
"In our experiment on the effect of additional features, we found that syntactic features such as POS and chunking information did not have clear positive effect on the performance.",
"In contrast, the match/partial match between words and entries in the chemical gazetteer is a good indicator for the presence of chemical entities.",
"Since the Jochem dictionary contains only chemical entities, it is not surprising that the performance on diseases was not substantially impacted by adding the gazetteer feature, although some small variations in performance can be observed, likely due to changed influences from neighboring terms.",
"The empirical results shown that models using either CNN-char or LSTM-char achieve a similar overall F1 score on chemical and disease NER.",
"The results are further comparable with other state-of-the-art models.",
"This indicates that these character-level models have sufficient complexity to learn the generalizable morphological and lexical patterns in biomedical named entity terms.",
"On the other hand, as shown by the substantial differences in the number of parameters in Table REF , CNN [12] has the advantage of reduced training complexity as compared to the LSTM models [4] under similar experimental settings.",
"In our experimental environment, the execution time of the model with LSTM-char increased 115% relative to the baseline BiLSTM-CRF model, while it only increased by 25% for with CNN-char, as detailed in Table REF .",
"Therefore, consistent with prior results on general NER, we conclude that CNN-based embeddings are preferable to LSTM-based embeddings for BNER.",
"Table: Training time of best performing models (2 BiLSTM layers and 250 LSTM units), computed on a Intel Core i5 2.9 GHz PC.We analyzed the error cases of the CNN-char and LSTM-char models without additional features: 3326 and 3271 words were incorrectly predicted using CNN-char and LSTM-char, respectively, with 2138 mistakes in common.",
"In errors which only was made by one of the two models, we found that CNN-char made more false positive predictions and fewer false negative predictions, while LSTM-char made approximately an even number of the two kinds of false predictions.",
"The relationship between the length of words and these errors was also explored.",
"For words less than 20 characters in length, the distribution of errors is almost identical for the two models.",
"However, for longer words, the model with LSTM-char tends to make more mistakes.",
"This supports prior observations that LSTM can be difficult to apply to long sequences of input [1].",
"In approximately 50% of error cases, the word length is short, less than 5 characters.",
"Short biomedical named entities are usually abbreviations and tend to be out-of-vocabulary terms, and are therefore particularly difficult for the character-level word embedding models to capture [2]."
],
[
"Conclusion",
"We compared the performance of BiLSTM-CRF models with CNN-based and LSTM-based character-level word embeddings for biomedical named entity recognition.",
"We confirmed previously published results on chemical and disease NER that demonstrate that character-level embeddings are helpful.",
"We further show empirically, generalizing prior results for general NER to the biomedical context, that there is little difference between the two approaches: both types of character-level word embeddings achieved identical F1 score on the chemical NER task, and similar performance on disease NER (with CNN-char showing a slight performance advantage).",
"However, the CNN embeddings show a substantial advantage in reduced training complexity."
],
[
"Acknowledgments",
"This work was supported by the ARC Discovery Project DP150101550 and ARC Linkage Project LP160101469."
]
] | 1808.08450 |
[
[
"Stellar $^{36,38}$Ar$(n,\\gamma)^{37,39}$Ar reactions and their effect on\n light neutron-rich nuclide synthesis"
],
[
"Abstract The $^{36}$Ar$(n,\\gamma)^{37}$Ar ($t_{1/2}$ = 35 d) and $^{38}$Ar$(n,\\gamma)^{39}$Ar (269 y) reactions were studied for the first time with a quasi-Maxwellian ($kT \\sim 47$ keV) neutron flux for Maxwellian Average Cross Section (MACS) measurements at stellar energies.",
"Gas samples were irradiated at the high-intensity Soreq applied research accelerator facility-liquid-lithium target neutron source and the $^{37}$Ar/$^{36}$Ar and $^{39}$Ar/$^{38}$Ar ratios in the activated samples were determined by accelerator mass spectrometry at the ATLAS facility (Argonne National Laboratory).",
"The $^{37}$Ar activity was also measured by low-level counting at the University of Bern.",
"Experimental MACS of $^{36}$Ar and $^{38}$Ar, corrected to the standard 30 keV thermal energy, are 1.9(3) mb and 1.3(2) mb, respectively, differing from the theoretical and evaluated values published to date by up to an order of magnitude.",
"The neutron capture cross sections of $^{36,38}$Ar are relevant to the stellar nucleosynthesis of light neutron-rich nuclides; the two experimental values are shown to affect the calculated mass fraction of nuclides in the region A=36-48 during the weak $s$-process.",
"The new production cross sections have implications also for the use of $^{37}$Ar and $^{39}$Ar as environmental tracers in the atmosphere and hydrosphere."
],
[
"unicode=true, colorlinks=true, citecolor = blue, filecolor = blue, linkcolor = blue, urlcolor = blue, Stellar $^{36,38}$ Ar$(n,\\gamma )^{37,39}$ Ar reactions and their effect on light neutron-rich nuclide synthesis M. Tessler Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel M. Paul [Corresponding author: ]paul@vms.huji.ac.il Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel S. Halfon Soreq NRC, Yavne 81800, Israel B. S. Meyer Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634, USA R. Pardo Argonne National Laboratory, Argonne, Illinois 60439, USA R. Purtschert Physics Institute, University of Bern, 3012 Bern, Switzerland K. E. Rehm Argonne National Laboratory, Argonne, Illinois 60439, USA R. Scott Argonne National Laboratory, Argonne, Illinois 60439, USA M. Weigand Goethe University Frankfurt, Frankfurt 60438, Germany L. Weissman Soreq NRC, Yavne 81800, Israel S. Almaraz-Calderon Argonne National Laboratory, Argonne, Illinois 60439, USA M. L. Avila Argonne National Laboratory, Argonne, Illinois 60439, USA D. Baggenstos Physics Institute, University of Bern, 3012 Bern, Switzerland P. Collon Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA N. Hazenshprung Soreq NRC, Yavne 81800, Israel Y. Kashiv Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA D. Kijel Soreq NRC, Yavne 81800, Israel A. Kreisel Soreq NRC, Yavne 81800, Israel R. Reifarth Goethe University Frankfurt, Frankfurt 60438, Germany D. Santiago-Gonzalez Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA A. Shor Soreq NRC, Yavne 81800, Israel I. Silverman Soreq NRC, Yavne 81800, Israel R. Talwar Argonne National Laboratory, Argonne, Illinois 60439, USA D. Veltum Goethe University Frankfurt, Frankfurt 60438, Germany R. Vondrasek Argonne National Laboratory, Argonne, Illinois 60439, USA The $^{36}$ Ar$(n,\\gamma )^{37}$ Ar ($t_{1/2}$ = 35 d) and $^{38}$ Ar$(n,\\gamma )^{39}$ Ar (269 y) reactions were studied for the first time with a quasi-Maxwellian ($kT \\sim 47$ keV) neutron flux for Maxwellian Average Cross Section (MACS) measurements at stellar energies.",
"Gas samples were irradiated at the high-intensity Soreq applied research accelerator facility-liquid-lithium target neutron source and the $^{37}$ Ar/$^{36}$ Ar and $^{39}$ Ar/$^{38}$ Ar ratios in the activated samples were determined by accelerator mass spectrometry at the ATLAS facility (Argonne National Laboratory).",
"The $^{37}$ Ar activity was also measured by low-level counting at the University of Bern.",
"Experimental MACS of $^{36}$ Ar and $^{38}$ Ar, corrected to the standard 30 keV thermal energy, are 1.9(3) mb and 1.3(2) mb, respectively, differing from the theoretical and evaluated values published to date by up to an order of magnitude.",
"The neutron capture cross sections of $^{36,38}$ Ar are relevant to the stellar nucleosynthesis of light neutron-rich nuclides; the two experimental values are shown to affect the calculated mass fraction of nuclides in the region A=36-48 during the weak $s$ -process.",
"The new production cross sections have implications also for the use of $^{37}$ Ar and $^{39}$ Ar as environmental tracers in the atmosphere and hydrosphere.",
"$^7$ Li$(p,n)$ , high-intensity neutron source, Maxwellian Averaged Cross Section (MACS), $^{36}$ Ar$(n,\\gamma )$ , $^{38}$ Ar$(n,\\gamma )$ , Accelerator Mass Spectrometry (AMS), Low Level Counting (LLC), dating tracers, nuclear explosion monitoring The argon isotopes $^{36}$ Ar and $^{38}$ Ar are among the rare stable nuclides for which no experimental neutron-capture cross sections exist above thermal energy.",
"While the abundances of $^{36,38}$ Ar in terrestrial atmospheric argon are very low relative to $^{40}$ Ar (produced mainly from $^{40}$ K decay [1], [2]), $^{36}$ Ar (84.59%) and $^{38}$ Ar (15.38%) are the major argon isotopes in the solar system [3] and likely so in stellar matter.",
"They are expected, together with the branching point $^{39}$ Ar, to play a role in nucleosynthesis of light neutron-rich nuclei (e.g.",
"$^{36}$ S, $^{40}$ Ar, $^{40}$ K), believed to be produced during the weak $s$ -process phase of stellar evolution [4], [5].",
"The $^{40}$ K ($t_{1/2}$ =1.248(3) Gy [6]) nuclide, in particular, is an important cosmo- or geochronometer and was used to estimate the age and duration of the $s$ -process as $\\sim $ 10 Gy [7], [8].",
"$^{40}$ K can be produced also in explosive oxygen burning [9] as a primary nucleosynthesis product in a massive star of initially pure hydrogen while the (secondary) $s$ -process production of $^{40}$ K requires initial abundances of heavy species.",
"A better understanding of Ar cross sections will help clarify the relative primary vs. secondary production of $^{40}$ K. In a different realm of study, the half-life of $^{37}$ Ar ($t_{1/2}$ =35.011(19) d [10]) makes this isotope an ideal chronometer for studying circulation and mixing [11], and that of $^{39}$ Ar (269(3) y [12]) for dating groundwater [13], [14] and ocean water up to about 1000 years [15].",
"The atmospheric steady state concentrations of $^{37}$ Ar and $^{39}$ Ar are mainly determined by the spallation reactions $^{40}$ Ar$(n,4n)^{37}$ Ar and $^{40}$ Ar$(n,2n)^{39}$ Ar and at lower neutron energies by the $^{36}$ Ar$(n,\\gamma )^{37}$ Ar and $^{38}$ Ar$(n,\\gamma )^{39}$ Ar reactions [11].",
"The latter are also relevant for the estimation of anthropogenic emissions from nuclear installations or for nuclear explosion monitoring [16].",
"We measured the $^{36}$ Ar and $^{38}$ Ar neutron capture cross sections by activation with quasi-Maxwellian neutrons produced by the $^7$ Li$(p,n)$ reaction at the superconducting linear accelerator of Soreq applied research accelerator facility (SARAF) [17], [18] and the Liquid-Lithium Target (LiLiT) [19], [20].",
"The activation products $^{37}$ Ar and $^{39}$ Ar were counted offline by accelerator mass spectrometry (AMS); $^{37}$ Ar production was also determined by Low-Level Counting (LLC).",
"Neutron irradiation of separate $^{36}$ Ar and $^{38}$ Ar samples was performed at the pneumatic transfer tube (rabbit) of the Soreq IRR-1 nuclear reactor in order to re-measure the respective thermal neutron capture cross sections.",
"Preliminary results of these experiments were reported in [21], [22].",
"Enriched $^{36}$ Ar, $^{38}$ Ar and mixed $^{38}$ Ar+$^{nat}$ Ar gas samples were filled into Ti spheres (10 mm outer diameter, 0.2 mm thick Ti shell) [23].",
"Due to the thermodynamical properties of Ar, the filling was made by successive compression with a custom-made piston and cryogenic pumping in order to achieve the required pressure ($\\sim $ 30 bar).",
"The samples used are listed in Table REF .",
"Table: Samples used and the results of the A+1 ^{A+1}Ar/ A ^AAr ratios.",
"36 ^{36}Ar and 38 ^{38}Ar gas samples were enriched to 99.935% and 99.957% for the respective isotopes.The final 37 ^{37}Ar/ 36 ^{36}Ar ratios were obtained by taking a weighted average of the AMS and LLC results (Fig.",
").Sphere #52a was irradiated with 1 mm thick Cd shield to estimate the epithermal neutron fraction.The 38 ^{38}Ar/ nat ^{nat}Ar ratio for sphere 54 (52b) is 11.7 (10.2).",
"For more details see the Supplemental Material .For the samples irradiated at SARAF-LiLiT (Table REF ), each gas sphere was placed with a 25 mm-diameter Au foil (12.5 $\\mu $ m thick), used as a neutron fluence monitor in an evacuated chamber downstream of LiLiT (Fig.",
"REF ).",
"LiLiT consists of a windowless film of liquid lithium (1.5 mm thick, 18 mm wide) flowing at 2-3 m/s, serving as both the neutron-producing target and the kW-power beam dump for the incident $\\sim $ 1.5 mA proton beam [19], [20].",
"The distance from the neutron source to the center of the Ar-filled sphere was 11.3 mm, intercepting $\\sim $ 30% of the outgoing neutrons.",
"Figure: (Color online) (right) Diagram of the Liquid-Lithium Target (LiLiT) and activation target assembly.The (∼\\sim 1.5 mA, ∼\\sim 9 mm full width) proton beam (open red arrow) impinges on the free-surface lithium film (cyan) (see for details).The Ar-filled sphere and Au foil are positioned in the outgoing neutron cone (green dotted lines)in a vacuum chamber separated from the LiLiT chamber.",
"(top left) Simulated neutron spectrum incident on the 36 ^{36}Ar sample (black) and a fit in the range E n ∼0-110E_n\\sim 0-110 keV with a Maxwell-Boltzmann flux (red) at kT∼47kT \\sim 47 keV.",
"(bottom left) Count rate (left y-axis) of fission chamber (see text) and calibration to proton current (right y-axis) during the 36 ^{36}Ar run.The proton beam energy, measured by Rutherford back scattering off a Au target after the acceleration module, was found to be $1932\\pm 3$ ($1940\\pm 3$ ) keV for the $^{36}$ Ar ($^{38}$ Ar) irradiation.",
"A proton beam energy spread of $\\sim $ 15 keV, estimated from beam dynamics calculations, was verified experimentally [24].",
"Auto-radiographic scans [25] of the Au foils were conducted to determine proton beam centering.",
"An offset of 2.0 (2.2) mm for the $^{36}$ Ar ($^{38}$ Ar) irradiation was found; this offset was accounted for in our simulations.",
"The neutron yield was continuously monitored with a fission-product ionization chamber [26], located $\\sim $ 80 cm downstream the target at 0$$ .",
"The fission chamber count rate was calibrated to beam current (at low intensity) using a Faraday cup located $\\sim $ 1 m upstream of the Li target.",
"The total integrated current was $\\sim $ 10.8 (7.35) milliampere hour for the $^{36}$ Ar ($^{38}$ Ar) irradiation (Fig.",
"REF ).",
"The $^{37}$ Ar nuclide decays by pure electron capture with no $\\gamma $ -ray emission; $^{37}$ Ar is notable for its role in Davis' solar neutrino experiment [29] where its production via $^{37}$ Cl$(\\nu _e,e^-)^{37}$ Ar was detected by Auger electron counting.",
"We detected and counted for the first time $^{37}$ Ar by Accelerator Mass Spectrometry (AMS) at the ATLAS facility of Argonne National Laboratory to measure the $^{37}$ Ar/$^{36}$ Ar ratio of the irradiated samples.",
"Ar gas was directly fed from the sphere container into an Electron Cyclotron Resonance (ECR) ion source through a remote-controlled sapphire leak valve.",
"$^{36,37}$ Ar$^{8+}$ ions were extracted from the ion source and accelerated alternately through ATLAS at an energy of 6 MeV/$u$ by appropriate scaling of all accelerator elements.",
"It was found necessary to strip the $^{37}$ Ar$^{8+}$ ions and count $^{37}$ Ar$^{18+}$ (fully stripped) in order to suppress the $^{37}$ Cl (Z=17) background.",
"Stripping was done with a 200 $\\mu $ g/cm$^2$ C foil at an intermediate stage of the ATLAS linear accelerator.",
"The stripping process (normally not used in AMS measurements at ATLAS) however produced an isotope fractionation and the effective beam transmission efficiency ($1.84(18)\\times 10^{-2}$ ) was determined by interpolation between the measured $^{36}$ Ar and $^{38}$ Ar transmissions.",
"The $^{37}$ Ar$^{18+}$ ions were counted using a $\\Delta $ E-E telescope of Si detectors, 50 and 300 $\\mu $ m thick, respectively, showing background-free spectra; the detection sensitivity in the present experiment was $^{37}$ Ar/Ar $\\sim 10^{-15}$ (see Supplemental Material [28]).",
"The $^{37}$ Ar activity of the same samples was also determined by ultra-low-level counting (LLC) in a second stage.",
"Stainless steel vials containing $\\sim $ 1 cm$^3$ aliquots of the same activated samples were shipped to the University of Bern.",
"Each gas was quantitatively transferred into a 100 cm$^3$ copper proportional counter which was then filled with P6 gas (6% methane + 94% commercial $^{37}$ Ar-free argon) to a pressure of $\\sim $ 6 bars.",
"The $^{37}$ Ar activity was measured by detecting Auger electrons in an underground LLC laboratory during 1-2 days [30], [16].",
"Energy calibration was performed with copper K-shell X-rays (E=8.133 keV) induced by an external $^{241}$ Am $\\gamma $ source.",
"The $^{37}$ Ar peak was identified at the K-capture decay energy of 2.82 keV [31] and integrated by means of a Gaussian fit [28].",
"The amount of $^{36}$ Ar in the sample was determined, after $^{37}$ Ar counting, from the filling pressure of the detector and the $^{40}$ Ar/$^{36}$ Ar ratio measured by mass spectrometry [32] using established procedures.",
"The overall uncertainty of 8% of the final $^{37}$ Ar/$^{36}$ Ar ratio is dominated by counting statistics and the uncertainties of counting yield (5%) [28].",
"A comparison of the $^{37}$ Ar/$^{36}$ Ar ratios measured by AMS and LLC is illustrated in Fig.",
"REF .",
"Figure: (Color online) Comparison of the 37 ^{37}Ar/ 36 ^{36}Ar ratio (at the end of irradiation) measured by AMS (black) and LLC (red).Accelerator Mass Spectrometry of $^{39}$ Ar has been previously performed at ATLAS [33], [34].",
"A high ion energy is essential for the separation and discrimination of $^{39}$ Ar from the extremely intense source background of the stable $^{39}$ K isobar.",
"In our experiment, the ECR was operated at low power to reduce as much as possible impinging of the plasma onto the chamber walls, believed to be a source of $^{39}$ K contamination.",
"$^{38,39,40}$ Ar$^{8+}$ ions were accelerated to 6 MeV/$u$ , similarly as described before and $^{39}$ Ar$^{8+}$ ions were analyzed in the Enge gas-filled magnetic spectrograph [35], which physically separates $^{39}$ Ar from beam contaminants, e.g.",
"$^{39}$ K$^{8+}$ and $^{34}$ S$^{7+}$ , which have close-by m/q values (Fig.",
"REF ).",
"The accelerator transmission efficiency for $^{39}$ Ar$^{8+}$ (0.40(3)) was interpolated between those of $^{38}$ Ar$^{8+}$ and $^{40}$ Ar$^{8+}$ [28].",
"Figure: Identification spectrum of 39 ^{39}Ar ions in the detector measured for the LiLiT irradiated 38 ^{38}Ar gas (top) and for non-irradiated 38 ^{38}Ar gas (bottom).The horizontal axis represents dispersion along the focal plane and the vertical axis a differential energy loss signal measured in the fourth anode of the focal-plane ionization chamber .The ratios $r{\\ }={\\ }^{A+1}$ Ar/$^{A}$ Ar at the end of irradiation are determined by $r = \\frac{N_{A+1}}{\\epsilon {\\ }t}\\frac{q e}{10^{-9}{\\ } i_{A}} e^{\\lambda t_{cool}}$ where $N_{A+1}$ is the number of $^{A+1}$ Ar detected, $\\epsilon $ is the detector efficiency (measured to be 0.91(3) for $^{38}$ Ar due to grid shadowing in the spectrograph focal-plane detector), $t$ the counting time, $q$ is the ion charge state (18 for $^{37}$ Ar and 8 for $^{39}$ Ar), $e$ is the electronic charge in coulomb, and $i_{A}$ the $^A$ Ar$^{q+}$ beam intensity (nanoampere); $\\lambda =\\frac{ln(2)}{t_{1/2}}$ is the $^{A+1}$ Ar decay constant and $t_{cool}$ is the time between the end of irradiation and counting.",
"The final results of the $^{A+1}$ Ar/$^A$ Ar ratios for all gas samples are presented in Table REF .",
"In the reactor irradiations, two small Au samples were attached to the $^{36}$ Ar and $^{38}$ Ar spheres for neutron monitoring, using 98.65(9) b [37] for the $^{197}$ Au thermal neutron capture cross section.",
"A minor correction for the epithermal activation of Au was applied, using the $^{198}$ Au activity measured for a gas sphere entirely shielded with 1 mm thick Cd.",
"In contrast to the $^{36}$ Ar sample, two $^{38}$ Ar samples irradiated at the reactor (Table REF ) were mixed with $^{nat}$ Ar to use $^{41}$ Ar ($\\sigma _{th}(^{40}$ Ar)=0.66(1) b [37]) as an internal neutron monitor in addition to the Au monitors; excellent agreement was obtained between the two neutron fluence calibrations [28].",
"The $^{36,38}$ Ar measured thermal capture cross sections are listed in Table REF .",
"Uncertainties (1$\\sigma $ ) for $^{36}$ Ar ($^{38}$ Ar) are 3% (2%) and 7% (11%) from the neutron fluence and atom ratio determinations, respectively.",
"For the LiLiT irradiated samples, the average experimental cross section, $\\sigma _{exp}$ , is obtained by $\\sigma _{exp} = \\frac{r}{\\Phi _{n}}$ , where $\\Phi _n$ is the effective neutron fluence (n/cm$^2$ ).",
"In view of the complex geometry of the gas sphere irradiation, $\\Phi _n$ is calculated as $\\Phi _n = \\frac{\\sum l_n}{V}$ where $V$ (0.46 cm$^3$ ) is the gas sphere's volume, $l_n$ is the length a neutron travels inside the Ar gas and $\\sum l_n$ is the sum of the lengths traveled by all the neutrons inside the Ar gas sphere during the irradiation.",
"$\\sum l_n$ is calculated by a detailed simulation (see below), taking a statistically representative sample of neutrons and scaling by the Au activity.",
"The validity of the expression $\\frac{\\sum l_n}{V}$ for the neutron fluence, $\\Phi _n$ , was confirmed by comparing the value calculated in this way for the Au (planar) monitor with its measured activity; experimental and calculated values agree within 0.5%.",
"The values of $\\Phi _n$ (n/cm$^2$ ) and $\\sigma _{exp}$ for $^{36}$ Ar ($^{38}$ Ar) are 6.2(1)$\\times 10^{14}$ (4.22(9)$\\times 10^{14}$ ) and 1.4(1) mb (0.95(10) mb), respectively.",
"Uncertainties (1$\\sigma $ ) for the $^{36}$ Ar ($^{38}$ Ar) experimental cross section $\\sigma _{exp}$ are 2% (2%) and 7% (11%) from the neutron fluence and atom ratio determinations, respectively.",
"The experimental cross section measured in our experiments is an energy-averaged value over the neutron spectrum and interpretation in terms of a Maxwellian Averaged Cross Section (MACS) requires knowledge of the shape of the spectrum.",
"The integral neutron spectrum seen by the targets under the irradiation conditions of the experiment is however not measurable.",
"Instead we rely on detailed simulations using the codes SimLiT [50] for the thick-target $^7$ Li$(p,n)$ neutron yield, and GEANT4 [51] for neutron transport (Fig.",
"REF ) [52].",
"The SimLiT-GEANT4 simulations have been carefully benchmarked in separate experiments and excellent agreement with experimental time-of-flight and (differential and integral) energy spectra was obtained [52], [50], [53].",
"The simulated neutron spectrum,$\\frac{dn_{sim}}{dE_n}$ is well fitted in the range $E_n\\sim 0-110$ keV ($\\sim $ 90% of the incident neutrons) by a Maxwell-Boltzmann (MB) flux $v\\frac{dn_{MB}}{dE_n} \\propto E_n exp(-E_n/kT)$ with $kT\\sim 47$ keV (Fig.",
"REF ).",
"The quantitative normalization of the neutron spectrum, $\\frac{dn_{sim}}{dE_n}$ , was obtained by comparing the experimental number of $^{198}$ Au nuclei (measured by gamma activity with a high-purity germanium detector) in the Au foil monitor with the number of $^{198}$ Au nuclei calculated in the detailed simulation of the entire setup (see [52] for details).",
"We calculate the MACS at a given thermal energy $kT$ with the procedure developed in [52], [54], using the expression $MACS(kT) = \\frac{2}{\\sqrt{\\pi }} C_{H\\mbox{-}F}(kT) \\sigma _{exp}$ where the correction factor $C_{H\\mbox{-}F}(kT)$ is given by $C_{H\\mbox{-}F}(kT) = \\frac{\\int _0 ^{\\infty } \\sigma (E_n) E_n e^{-\\frac{E_n}{kT}}dE_n}{\\int _0 ^{\\infty } E_n e^{-\\frac{E_n}{kT}}dE_n} / \\frac{\\int _0 ^{\\infty } \\sigma (E_n) \\frac{dn_{sim}}{dE_n} dE_n}{\\int _0 ^{\\infty } \\frac{dn_{sim}}{dE_n} dE_n}.",
"$ $\\sigma (E_n)$ may have coherent contributions from compound-resonances and (weakly energy dependent) direct captures (DC).",
"We note here that $\\sigma _{exp}$ includes all contributions in the experimental energy range; we use in Eq.",
"(REF ) the Hauser-Feshbach model for the energy dependence of $\\sigma (E_n)$ in the wider MB range and estimate the additional uncertainties associated with direct capture.",
"In order to account for the sensitivity to the low density of available compound states in $^{37,39}$ Ar, we apply different codes [28]: TENDL-2014 [55], -2015 [48], -2017 [56] and TALYS-1.8 [57] with a microscopic level density and average the $C_{H\\mbox{-}F}(kT)$ values obtained; the greater of 20% of the correction or their standard deviation is attributed to the MACS corrections.",
"It should however be noted that the extrapolation of the MACS to different thermal energies and determination of their uncertainties were made using a limited number of theoretical models, due to the total absence of experimental knowledge of resonances in the $^{37,39}$ Ar compound nuclei.",
"We also add an estimated independent 15% uncertainty from s-wave and p-wave DC contributions.",
"Detailed calculations of the correction factor and its uncertainties will be included in an expanded version of this Letter.",
"Our MACS values and uncertainties are listed in Table REF and compared to existing theoretical values.",
"Table: Comparison of the experimental thermal cross sections and MACS(30 keV) obtained in this work to theoretical and evaluated data.The experimental MACS values (Table REF ) obtained in this work are notably different from previous calculations.",
"Fig.",
"REF shows the $^{36,38}$ Ar$(n,\\gamma )$ reaction rates ($N_A\\left<\\sigma v\\right>$ ) based on our measurements and extrapolation to different temperatures, compared to the rates adopted so far [49].",
"Figure: (Color online) Comparison of the 36 ^{36}Ar (top) and 38 ^{38}Ar (bottom) (n,γ)(n,\\gamma ) reaction rates (N A σvN_A\\left<\\sigma v\\right>) extracted from this work (red)to the Kadonis recommended values (black).",
"The dashed curves encompass the estimated 1σ\\sigma uncertainty.In order to show the potential effect of these experimental rates on stellar nucleosynthesis, we performed a single-zone network calculation using physical conditions appropriate for the He core burning phase of a massive star in which the new $^{36,38}$ Ar$(n,\\gamma )$ rates are used, leaving all others unchanged [49].",
"The calculations are done using the single-zone NucNet Tools reaction network code [58] starting at the H-burning phase with solar abundances [3] and continuing into a single-zone He core burning (T= 300 MK, density of 1 kg/cm$^3$ ).",
"Substantial (10-50%) changes in the calculated mass fractions for neutron-rich light nuclides between $^{34}$ S and $^{58}$ Fe are observed (Fig.",
"REF ), reminiscent of the sensitivity observed in the weak $s$ -process region (A$\\sim $ 56-70) due to the change of a single cross section [59].",
"The mass fraction of $^{36}$ Ar itself is observed to increase by a factor of $\\sim $ 10 due to its lower measured capture-cross section.",
"Especially interesting is the $\\sim $ 45% decrease in the calculated mass fraction of the important cosmo/geo-chronometer $^{40}$ K implying a weaker contribution of the secondary $s$ -process relative to primary production.",
"As shown in Frank et al.",
"[60], the mass fraction of $^{40}$ K differs considerably over time whether it is primary only or secondary only.",
"For example, with a larger primary production of $^{40}$ K which is the dominant initial heat generator in Earth-like exoplanets, considerable heating would occur in these worlds even early in the Galaxy history.",
"Figure: (top) Comparison of the mass fractions calculated for stable nuclei between 34 ^{34}S and 58 ^{58}Fe changing by >>10% at the end of a single-zonecalculation modeling He burning in a massive star, using literature rates (solid circles) or replacing the 36,38 ^{36,38}Ar(n,γ)(n,\\gamma ) rates with the experimental values from this work(solid squares).",
"We observe a smoother distribution of mass fractions in the vicinity of 36 ^{36}Ar when using the experimental cross sections.",
"(bottom) Ratio of the mass fractions usingexperimental and literature reaction rates as above.The measurements of the $^{36}$ Ar$(n,\\gamma )$ cross sections affect also the calculation of the natural $^{37}$ Ar background activity in the atmosphere, the interpretation of $^{37}$ Ar emission rates in underground nuclear explosion monitoring [16] and the investigation of atmospheric air circulation [11].",
"The detection of $^{37}$ Ar by AMS demonstrated here opens the way to an alternative method for the monitoring of environmental samples [22].",
"Similarly to $^{37}$ Ar, the $^{38}$ Ar$(n,\\gamma )^{39}$ Ar reaction contributes to the $^{39}$ Ar production rate in the atmosphere [11] and determines the initial value for the use of $^{39}$ Ar as a groundwater dating chronometer [13], [14], [61].",
"In summary, first measurements of the neutron capture cross sections of $^{36}$ Ar and $^{38}$ Ar at stellar energies were performed.",
"The experimental value for $^{36}$ Ar, in particular, is smaller than the one adopted so far from theoretical calculations and evaluations by a factor of $\\sim $ 10.",
"Nucleosynthesis calculations for the weak $s$ -process regime using the measured cross sections are shown to increase the mass fraction of $^{36}$ Ar by a factor of $\\sim $ 10 and lower the residual mass fraction of neutron-rich nuclides in the region A=36-48 by 10 to 50%.",
"The $^{36,38}$ Ar$(n,\\gamma )$ cross sections affect the interpretation of environmental monitoring using $^{37}$ Ar or $^{39}$ Ar as geophysical tracers.",
"We would like to thank the SARAF and LiLiT (Soreq NRC) and the ATLAS operation staffs for their dedicated help during the experiments.",
"This work was supported in part by the Israel Science Foundation (Grant No.",
"1387/15), by the Pazy Foundation (Israel), the Israel Ministry of Science (Eshkol Grant No.",
"18145), the US Department of Energy, Office of Nuclear Physics, under Award No.",
"DE-AC02-06CH11357.",
"D.S.G.",
"acknowledges the support by the U.S. Department of Energy, Office of Nuclear Physics, under Award No.",
"DE-FG02-96ER40978.",
"This research has received funding from the European Research Council under the European Unions's Seventh Framework Program (FP/2007-2013)/ERC Grant Agreement No.",
"615126.",
"Supplementary material We present here supplementary material referred to in the paper.",
"Figure: Ti sphere used as container for the pressurized A ^{A}Ar gas for irradiation (left), and the sphere target holder for irradiation at SARAF-LiLIT (right).Table: Samples used and the results of the A+1 ^{A+1}Ar/ A ^AAr ratio after the irradiation for all gas samples.",
"36 ^{36}Ar and 38 ^{38}Ar gas samples were enriched to 99.935% and 99.957% for the respective isotopes.The final 37 ^{37}Ar/ 36 ^{36}Ar ratios were obtained by taking a weighted average of the AMS and LLC results (Fig.",
").Sphere #52a was irradiated with 1 mm thick Cd shield to estimate the epithermal neutron fraction.Figure: Identification spectra of the 37 ^{37}Ar counts with the Δ\\Delta E-E telescope of Si detectors for the LiLiT irradiated sphere #39.Figure: Same as Fig.",
"for the reactor irradiated sphere with Cd #52a.Figure: Same as Fig.",
"for the reactor irradiated sphere (without Cd) #60.Figure: Same as Fig.",
"for a non irradiated 36 ^{36}Ar (blank).",
"One 37 ^{37}Ar count detected for this sample over 6.5 hours,likely due to a memory effect in the ion source, corresponds to a concentration 37 ^{37}Ar/ 36 ^{36}Ar=9×10 -16 9\\times 10^{-16}.Figure: Two-dimensional spectra of Δ\\Delta E 4 _4 vs. focal plane position gated on the 39 ^{39}Ar region-of-interest in the time-of-flight, Δ\\Delta E 2 _2 and Δ\\Delta E 3 _3 detector parameters.Table: AMS results obtained for the LiLiT irradiated sphere (#39) 37 ^{37}Ar/ 36 ^{36}Ar ratio.",
"Statistical uncertainties are given in ().",
"An additional systematic uncertaintyof 10% is due to the 37 ^{37}Ar transmission.Figure: Repeated measurements of the 37 ^{37}Ar/ 36 ^{36}Ar ratio for the LiLiT irradiated sphere (#39).",
"See Table for numerical results.Table: Same as Table for the Cd shielded sphere irradiated at the reactor (#52a).Table: Same as Table for the sphere irradiated at the reactor (#60).Figure: Same as Fig.",
"for the sphere irradiated at the reactor (#60).",
"See Table for numerical results.Figure: Energy spectra (in counts per minute) and gauss fit of 37 ^{37}Ar measurements by low-level counting (LLC).The peak energy correspond to the 2.82 keV K-capture decay energy of 37 ^{37}Ar.Table: LLC results of the 37 ^{37}Ar/ 36 ^{36}Ar ratios for all three samples.Figure: Transmission efficiency of 38 ^{38}Ar 8+ ^{8+} and 38 ^{38}Ar 8+ ^{8+}.Table: Comparison of the 36 ^{36}Ar (top) and 38 ^{38}Ar (bottom) thermal and Maxwellian Average (30 keV) cross sections measured in this work (red squares)to measured (thermal) and theoretical data (black circles).",
"Numerical values are listed in Table (main paper)."
]
] | 1808.08556 |
[
[
"Ranked Schr\\\"oder Trees"
],
[
"Abstract In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species.",
"Unfortunately, the classical Schr\\\"oder tree model is not adapted to take into account the chronology between the branching nodes.",
"In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there?",
"In this paper, we enrich this model in two distinct ways in order to obtain two ranked tree models for phylogenetics, i.e.",
"models coding chronology.",
"For that purpose, we first develop a model of (strongly) increasing Schr\\\"oder trees, symbolically described in the classical context of increasing labeling.",
"Then we introduce a generalization for the labeling with some unusual order constraint in Analytic Combinatorics (namely the weakly increasing trees).",
"Although these models are direct extensions of the Schr\\\"oder tree model, it appears that they are also in one-to-one correspondence with several classical combinatorial objects.",
"Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers."
],
[
"Introduction",
"In biology a phylogenetic tree is a classical tool to represent the evolutionary relationship among species.",
"At each bifurcation, or multifurcation, of the tree, the descendant species from distinct branches have distinguished themselves in some manner.",
"One of the first illustrations of an evolutionary tree was made by Darwin in his book On The Origin of Species [6].",
"His idea was to represent the divergence of characters and species.",
"Multifurcations represent a well-marked variety of a certain kind and this process then continues on the new varieties and so on.",
"Interest grew in tree evolutions as these models give insight on how species evolved.",
"Different tree models were proposed with the idea of finding trees that fits best nowadays observations and data sets.",
"These models of graphs include rooted, unrooted, labeled, unlabeled, bifurcating or multifurcating trees or networks.",
"By defining some metrics between these models, people develop algorithms focusing on state space exploration or on tree inference.",
"For details on tree models in phylogenetics and inference algorithms see the book of Felsenstein [10] and the one of Steel [18] for a more recent survey with combinatorial aspects also.",
"Thanks to the development of bioinformatics many tools have thus emerged, in order to build automatically such tree diagrams.",
"Some examples of programs are PHYLIP, a tool for inferring phylogenetic trees [9] or PAML that is phylogenetic analyser based on the maximum likelihood [21].",
"In order to develop these new tools several structural studies have been realized to model correctly the fundamental parameters defined by biologists.",
"In 1870 Schröder presented an original model published into the paper Vier combinatorische Probleme [17].",
"The fourth problem presents a phylogenetic tree model enumerating trees by their number of leaves.",
"See for example [8] for the phylogenetic interpretation.",
"While it has been highlighted that this first model is not adapted to take into account the chronology between branching nodes belonging to two distinct fringe subtrees, other approaches have been developed to consider such a history of the evolution process.",
"In particular in the context of binary trees, we can mention the stochastic model of Yule [22] and its generalization by Aldous [1].",
"Such tree models, including history evolution, are usually called ranked tree models in phylogenetics.",
"But these new models are not based on the original Schröder tree model.",
"To the best of our knowledge, there seems to have been no attempt to enrich Schröder's original model so as to encode the chronology of evolution.",
"Figure: A Schröder tree: without chronological evolution (on the left handside),and with chronological evolution (on the right handside)So, the main goal of this paper consists in designing ranked tree models based on the classical Schröder structure.",
"In Figure REF we have represented the same phylogenetic tree on the left handside as a classical Schröder tree, and on the right handside as a strongly increasing Schröder tree, the first model we develop in this paper.",
"A first natural idea in this direction consists in considering the model of a recursive tree.",
"Such a structure is a rooted labeled tree, whose root is labeled by 1 and the successors of a given node, with label $\\nu $ , have a label greater than $\\nu $ .",
"Each integer between 1 and the total number of nodes is present once in the tree.",
"Many variations of this model have been presented in the literature: see [7] and the references therein.",
"In this context, we are able to define a simple evolution process that allows to build very efficiently large trees with simple iterative rules.",
"Furthermore, usually the history of construction is naturally kept in the final large tree through the increasing labeling.",
"It is also important to note that apparently minor changes on the growth rules induce drastic differences in the typical properties of the considered models.",
"See for example the book of Drmota [7] that presents many extensions of the classical model (e.g.",
"plane oriented recursive trees, fixed arity – or out-degree – recursive trees) and details several quantitative studies for different fundamental parameters like the profile of such tree models.",
"Let us recall the sample of a recursive tree (uniformly for all trees of the same size, i.e.",
"the same number of nodes): start with the single size-1 tree, reduced to a root, and iterate: at step $n\\in \\lbrace 2,3,\\dots \\rbrace $ choose uniformly a node in the tree under construction (labeled with an integer between 1 and $n-1$ ) and attach to it a new node labeled by $n$.",
"While many variations on these models have been studied, it is very interesting to note that the increasing version of Schröder trees seems not to have been analyzed.",
"Our model is also very natural due to its similarities to the probabilistic model of Yule trees (cf.",
"e.g.",
"[19]) that take into account the chronological mutations of species.",
"We develop in this paper two distinct models for phylogenetic trees satisfying in priority two new constraints: (1) to take into account the chronological evolution and (2) to be efficient to simulate.",
"Both models are based on some increasingly labeling of Schröder tree structures.",
"In this paper, we are focusing on the distinct histories possible for a fixed number of final species.",
"From a graph model point of view, it consists in the quantitative study of the number of structures of a given size.",
"Furthermore, beyond some characteristics shared by our model and recursive trees, or increasing fixed arity trees, we will point out several relations to other classical combinatorial objects, in particular permutations, Stirling numbers.",
"Due to the many links to combinatorial objects, increasing Schröder trees are thus interesting in themselves as combinatorial structures.",
"The paper is organized as follows.",
"In Section we introduce formally our first ranked phylogenetic tree model and introduce a non classical point of view for the tree specification.",
"We present the enumeration of the trees and relate them to permutations.",
"Then we compute important parameters of the model.",
"We conclude this section with the presentation of a linear algorithm for the uniform sampling of trees.",
"Section is devoted to our second model for ranked phylogenetic trees.",
"It is based on a non-classical way of increasingly labeling a tree structure.",
"The section is composed like the first one: after the enumeration of the trees, we relate them to classical combinatorial objects, derive some tree parameters and we finally conclude the section with an efficient unranking algorithm for the uniform sampling of our trees.",
"Some technical proofs are detailed in the appendix due to obtain a clear paper structure."
],
[
"Strongly Increasing Schröder trees",
"The first model we develop is based on a almost classical notion of increasing labeling in Analytic Combinatorics."
],
[
"The model and its context",
"The tree structure associated to strongly increasing Schröder tree corresponds to Schröder trees, i.e.",
"the combinatorial class of rooted planeA plane tree is such that the children of a node are ordered.",
"trees whose internal nodes have arity at least 2.",
"The reader can refer to [11] for some details.",
"The size of a Schröder tree is the number of leaves in the tree.",
"Note that in the tree structure neither the internal nodes, nor the leaves are labeled.",
"The combinatorial class $\\mathcal {S}$ of Schröder trees is specified as $\\mathcal {S}= \\mathcal {Z}\\cup \\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 2}}\\mathcal {S},$ that translates, via the classical symbolic method presented by Flajolet and Sedgewick [11], into the following equation, $S(z) = z + \\frac{S(z)^2}{1-S(z)}$ , satisfied by its ordinary generating function $S(z) = \\sum _{n\\ge 1} s_n z^n$ where $s_n$ is the number of structures of size $n$ (i.e.",
"with $n$ leaves).",
"In this section, we are interested in an increasingly labeled variation of Schröder trees.",
"Definition 1 A strongly increasing Schröder tree has a tree structure that is a Schröder tree and moreover its internal nodes are labeled with the integers between 1 and $\\ell $ (where $\\ell $ is the number of internal nodes), in such a way that all labels are distinct and the sequence of labels in each path from the root to a leaf is increasing.",
"Note, in the Analytic Combinatorics context, such a labeling of trees is called increasing labeling (without the term strongly).",
"In order to distinguish clearly this first model from the second one presented in Section we have added this term.",
"But from here, inside this section we will use the classical denotation increasing tree.",
"Figure: A strongly increasing Schröder treeTrees that are increasingly labeled can be in a certain extent specified with the Greene's operator $\\;^{\\square } \\star $ (cf.",
"for example [11]).",
"Then the specification is translated into an equation satisfied by the exponential generating function.",
"But in our context, the size of a tree is the number of leaves (which corresponds to the final number of species), and the increasingly labeling constraint is related to the internal nodes.",
"We specify this class by using a second variable $u$ to mark the internal nodes.",
"$S^*(z,u) = \\sum _{n,\\ell } s_{n,\\ell } z^n \\frac{u^\\ell }{\\ell !",
"}= z + \\int _{v=0}^{u} \\frac{S^*(z,v)^2}{1-S^*(z,v)}\\mathrm {d}v.$ While the integral equation could be analyzed further, we prefer, in the following, to introduce an alternative way to define our objects.",
"This new approach is easier to handle and it also naturally extends to define our second model of trees developed in Section , namely the weakly increasing Schröder trees.",
"In Figure REF we have represented an increasing Schröder tree of size 30 with 27 internal nodes.",
"This increasing tree is the same tree as the one represented in Figure REF with the chronological evolution, where the internal node labeled by $\\ell $ is laid on level $\\ell -1$ , for all $\\ell \\in \\lbrace 1, \\dots , 27\\rbrace $ .",
"In order to describe the building of increasing Schröder trees, we introduce an evolution process.",
"It consists in an iterative way that substitutes a leaf by an internal node attached to several leaves.",
"More formally: Start with a single (unlabeled) leaf; Iterate the following process: at step $\\ell $ (for $\\ell \\ge 1$ ), select a leaf and replace it by an internal node with label $\\ell $ attached to a sequence of at least two leaves.",
"Remark that the increasing labeling corresponds to the chronology of the tree building."
],
[
"Exact enumeration and relationship with permutations",
"Let us denote by $ the class of increasing Schröder trees.By using the evolution processwe exhibit a specification for $ as follows: $ \\mathcal {Z}\\cup \\left(\\Theta \\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 1}}(\\mathcal {Z}) \\right).$ In this specification, $\\mathcal {Z}$ stands for the leaves, and the operator $\\Theta $ is the classical pointing operator (cf.",
"in [11] for details).",
"The specification is a direct rewriting of the evolution process.",
"A tree is either reduced to a leaf or at each step an atom (i.e.",
"a leaf) is pointed in the tree under construction and is replaced by an internal node (whose labeling is deterministic: it corresponds to the step number) attached to a sequence of at least two leaves (the one that has been pointed is reused as the leftmost child, it is the reason why the operator $\\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 1}}$ does not contain the empty sequence and starts with sequences containing one element).",
"The symbolic method translates this specification into a functional equation satisfied by the generating series associated to the combinatorial class.",
"Note that the functional equation is satisfied by the ordinary generating series associated to $: $ T(z) = n1 tn zn$.The increasing labeling is here transparent and thus the objects seems not labeled(in fact, the leaves, marked by $ Z$ are really unlabeled):\\begin{equation} T(z) = z+ \\frac{z^2}{1-z} \\; T^{\\prime }(z).\\end{equation}By extracting the coefficients of the series, we derive the two following recurrences.\\begin{equation} \\left\\lbrace \\begin{array}{l}t_1 = 1, \\quad t_2 = 1,\\\\\\text{and for } n>2,\\\\t_n = n \\cdot t_{n-1}.\\end{array}\\right.\\qquad \\left\\lbrace \\begin{array}{l}t_1 = 1,\\\\\\text{and for } n>1,\\\\t_n = \\sum _{k=1}^{n-1} k\\cdot t_k .\\end{array}\\right.\\end{equation}Both recurrences are computed thanks to equation~(\\ref {eq:funeq}).The direct extraction $ [zn] T(z)$ exhibits the rightmost recurrence.This recurrence exhibits that the calculation of the $ n$-th term is of quadratic complexity(in the number of arithmetic operations).The leftmost recurrence is obtained byextracting $ [zn] (1-z)T(z)$ and then by simplifying the resulted equation.Here the calculation of the $ n$-th term is of linear complexity.$ Thus we directly prove $t_n = n!",
"/ 2$ for all $n\\ge 2$ .",
"The sequence $(t_n)_n$ appears under the reference OEIS A001710Throughout this paper, a reference OEIS A$\\cdots $ points to Sloane’s Online Encyclopedia of Integer Sequences www.oeis.org..",
"Observing the growth rate of $(t_n)_n$ proves that the ordinary generating series $T(z)$ is formal: its radius of convergence is 0."
],
[
"Analysis of typical parameters",
"Here we are interested in the quantitative study of four distinct parameters of increasing Schröder trees.",
"The first one corresponds to the number of internal (labeled) nodes of a size-$n$ tree.",
"This fundamental parameter corresponds to the number of steps in the evolution process that are necessary to build the given tree.",
"Recall the arity of internal nodes is at least two, thus this parameter is not deterministic.",
"The second and the third parameters are related to the root node.",
"We study its arity and the number of leaves attached to it in a typical tree of size $n$ .",
"But in a tree of size $n$ (tending to infinity) all internal nodes whose labels are independent from $n$ have the same characteristics than the root: thus these two parameters are also important for the global quantitative aspects of a large tree.",
"Finally, the fourth parameter corresponds to the typical number of binary nodes in a large tree.",
"This study becomes natural once we have seen the typical value of the number of internal nodes of a large tree."
],
[
"Quantitative analysis of the number of iteration steps",
"A fundamental parameter characterizing the increasing Schröder trees is their number of internal nodes.",
"This parameter is interesting in itself, but furthermore it corresponds to the maximal label value in the tree, and thus it is also the number of steps of the building process.",
"To study both the number of internal nodes and the number of leaves, we enrich the specification (REF ) with an additional parameter $\\mathcal {U}$ marking the internal nodes.",
"$& \\mathcal {Z}\\cup \\left(\\mathcal {U}\\times \\Theta _\\mathcal {Z}\\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 1}}(\\mathcal {Z}) \\right); \\nonumber \\\\&T(z,u) = z + \\frac{u\\; z^2}{1-z} \\; \\partial _z T(z,u).$ The operator $\\Theta _\\mathcal {Z}$ consists in pointing an element marked by $\\mathcal {Z}$ .",
"The partial differentiation according to $z$ is written as $\\partial _z \\cdot $ .",
"With the notation $T(z,u) = \\sum _{n\\ge 1} t_n(u) z^n$ , the equation (REF ) gives two recurrences satisfied either by $(t_n(u))$ , or by $(t_{n,k})$ , where $t_{n,k}$ is the number of trees with $n$ leaves and $k$ internal nodes (that are increasingly labeled): $ &\\left\\lbrace \\begin{array}{l}t_1(u) = 1, \\qquad t_2(u) = u, \\\\\\text{and if } n> 2,\\\\t_{n}(u) = (1+(n-1)u) t_{n-1}(u);\\end{array}\\right.",
"\\nonumber \\\\&\\left\\lbrace \\begin{array}{l l}t_{n,k} = t_{n-1,k} + (n-1) \\; t_{n-1,k-1} & \\text{ if } 0 < k < n\\\\t_{1,0} = 1, \\qquad t_{n,1} = 1 &\\text{ if } n > 1 \\text{ and} \\\\t_{i,j} = 0 & \\text{ otherwise}.\\end{array}\\right.$ Figure: Distribution of t n,k t_{n,k} for size-nn trees, n∈{1,2,⋯,6}n \\in \\lbrace 1, 2, \\dots , 6\\rbrace ,of the number of internal nodes k∈{0,1,⋯,n-1}k \\in \\lbrace 0,1,\\dots , n-1\\rbrace Remark that the extremal conditions are trivially obtained through our construction in particular the sequence $(t_{n,n-1})_n$ is enumerating increasing binary trees.",
"Once again, these efficient recurrences are obtained thanks to the extraction of $[z^n] (1-z)\\cdot T(z,u)$ .",
"In Figure REF , for the tree size-$n$ from 1 to 6, we present the distribution of the number of trees according to their number $k$ of internal nodes.",
"The Borel transform, denoted as $\\mathcal {B}\\cdot $ , translates an ordinary generating series into its analog exponential generating series.",
"For example, we obtain $\\mathcal {B}T(z) = \\sum _{n\\ge 1} t_n \\; \\frac{z^n}{n!",
"}$ .",
"In particular, due to the growth of the coefficients $(t_n)_n$ we directly observe that $\\mathcal {B}T(z)$ is analytic around 0 (with radius of convergence 1).",
"Proposition 2 The Borel transform on $T(z,u)$ relatively to the variable $z$ gives $\\mathcal {B}T(z,u)= \\sum _{n\\ge 1} \\sum _{k=0}^{n-1} t_{n,k} \\; u^k \\; \\frac{z^n}{n!",
"}=\\frac{u(1-zu)^{-\\frac{1}{u}} -u +z}{1+u}.$ Here we just present the key-ideas of the proof, but details are given in Appendix .",
"[Key-ideas] Applying the Borel transform on equation (REF ) and then classical properties of the Borel transform for the function $z\\cdot f(z)$ and for the derivative $f^{\\prime }(z)$ yields the result.",
"Let us come back to the polynomial $t_n(u)=\\sum _{k=0}^{n-1} t_{n,k} \\; u^k$ .",
"It corresponds almost to the sequence OEIS A145324 related to Stirling numbers.",
"Corollary 3 Let $n\\ge 2$ .",
"The distribution of the number of internal nodes in increasing Schröder trees of size $n$ is $t_n(u)=\\sum _{k=0}^{n-1} t_{n,k} \\; u^k = u \\; \\prod _{\\ell =2}^{n-1} (1+\\ell u).$ The proof relies on a direct rewriting of the first recurrence in equation (REF ).",
"The generating function corresponds to the $n$ -th row in the triangle presented in Figure REF .",
"Although the sequence $(t_n(u))$ is stored in OEIS we exhibit here another link with a very classical triangle.",
"By reading each row of the triangle from right to left, we obtain a shifted version of the triangles OEIS A136124,A143491.",
"It corresponds almost to the generating function of Stirling Cycle numbers [11]: $SC_n(u)=\\prod \\limits _{i=1}^{n-1}(u+i)$ .",
"The associated sequence enumerates size-$n$ permutations that decompose into $k$ cycles, defined as Stirling numbers of the first kind.",
"More formally we prove: Proposition 4 Defining $\\hat{t}_n(u)=\\sum _{k=1}^n t_{n,k} \\; u^{n-k}$ , we obtain $\\displaystyle {\\hat{t}_n(u) = \\frac{u}{1+u}SC_n(u)}$ .",
"Let $\\mathcal {X}_n$ be the random variable that maps increasing Schröder trees of size $n$ to their numbers of internal nodes.",
"We want to establish a limit law for the distribution $\\left(\\mathbb {P}_{n}(\\mathcal {X}_n=k)\\right)_k$ .",
"But let us first compute its mean and its standard deviation so that we will then study the convergence of the normalized random variable $\\mathcal {X}_n^\\star =\\frac{\\mathcal {X}_n -\\mathbb {E}(\\mathcal {X}_n)}{\\sqrt{\\mathbb {V}(\\mathcal {X}_n)}}$ .",
"We follow here the classical approach presented, for example, in [11].",
"Since we consider the uniform distribution among trees of a given size $n$ , we obviously get $\\mathbb {P}_{n}(\\mathcal {X}_n=k)=\\frac{t_{n,k}}{t_{n}}$ .",
"Proposition 5 Let $n\\ge 2$ , the mean value of $\\mathcal {X}_n$ is equal to $\\mathbb {E}_{n}(\\mathcal {X}_n) = n-H_n+\\frac{1}{2} = n- \\ln n-\\gamma +\\frac{1}{2}+ O\\left(\\frac{1}{n}\\right),$ with $H_n$ the $n$ -th harmonic number and $\\gamma $ the Euler constant ($\\gamma \\approx 0.57721\\dots $ ).",
"Furthermore, $\\mathbb {V}_{n}[\\mathcal {X}_n] = \\ln n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4} + O\\left(\\frac{\\log n}{n}\\right).$ Recall that the ordinary generating function for the Harmonic numbers sequence is $H(z)=\\frac{1}{1-z}\\; \\ln \\frac{1}{1-z}$ (see e.g.",
"[11]), then the result is proved by a direct computation.",
"The proof is presented in Appendix .",
"This proposition allows us to exhibit the limit law of the distribution $(X^\\star _n)$ and proves then that the sequence $(\\mathcal {X}_n)$ converges in distribution to a Gaussian law.",
"Theorem 6 Let $\\mathcal {X}_n$ be the random variable describing the distribution of the number of internal nodes in increasing Schröder trees of size $n$ , or equivalently the number of building steps to get a size-$n$ tree, we have $\\displaystyle {\\frac{\\mathcal {X}_n -\\mathbb {E}_{n}(\\mathcal {X}_n)}{\\sqrt{\\mathbb {V}_{n}(\\mathcal {X}_n)}} \\xrightarrow{} \\mathcal {N}(0,1).",
"}$ The proof is obtained via an adaptation of Flajolet and Sedgewick's approach for the limit Gaussian law of Stirling Cycle numbers [11]: see Appendix .",
"Observing the mean value $\\mathbb {E}_{n}(\\mathcal {X}_n)$ we remark that only the second order in the asymptotic behavior permits to conclude that some internal nodes are not binary.",
"The next parameters we are interested in are related to the root of the increasing Schröder trees.",
"Concerning this particular node, we want to understand first its typical arity, and then the number of leaves attached to it in a large tree.",
"To avoid the description of several notations in the paper, we have chosen to reuse the previous notations for this new sequence.",
"Thus here the variable $\\mathcal {U}$ marks the arity of the root.",
"The specification is direct: either the root-leaf is modified in the evolution process, or it is not the root that is substituted.",
"$ &\\mathcal {Z}\\cup \\left(\\mathcal {U}\\times \\Theta _\\mathcal {Z}(\\mathcal {Z}) \\times \\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 1}}(\\mathcal {U}\\times \\mathcal {Z}) \\right)\\\\&\\cup \\left(\\Theta _\\mathcal {Z}(\\mathcal {Z}) \\times \\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 1}}(\\mathcal {Z}) \\right).$ We directly obtain the translation $T(z,u) = z + \\frac{u^2\\; z^2}{1-uz} + \\frac{z^2}{1-z} \\; \\partial _z \\left(T(z,u) - z\\right).$ In the same way as before we prove $ &\\left\\lbrace \\begin{array}{l}t_1(u) = 1, \\qquad t_2(u) = u^2, \\\\\\text{and if } n> 2,\\\\t_{n}(u) = u^{n-1}(u-1) + n\\; t_{n-1}(u);\\end{array}\\right.",
"\\nonumber \\\\&\\left\\lbrace \\begin{array}{l c l}t_{n,k} = n\\; t_{n-1,k} & & \\text{ if } 1 < k < n-1\\\\t_{1,0} = 1, \\quad t_{2,2} = 1 & & \\\\t_{n,n-1} = n-1 \\quad t_{n,n}=1 & &\\text{ if } n>2 \\text{ and} \\\\t_{i,j} = 0 & & \\text{ otherwise}.\\end{array}\\right.$ These sequences are related to OEIS A094112,A092582, that define properties on permutations (either some avoiding pattern, or with some fixed size initial run).",
"Corollary 7 For $n\\ge 2$ and $2 \\le k \\le n-1$ , we get $\\displaystyle {t_{n,k}=n!",
"\\frac{k}{(k+1)!",
"}}$ .",
"A proof by induction is direct.",
"Theorem 8 Let $\\mathcal {X}_n$ be the random variable describing the distribution of the number of children of the root in increasing Schröder trees of size $n$ , we have, for $n\\ge 2$ and $2 \\le k \\le n-1$ , $\\mathbb {P}_{n}(\\mathcal {X}_n=k) = \\frac{2k}{(k+1)!",
"}, \\quad \\text{and} \\quad \\mathbb {P}_{n}(\\mathcal {X}_n=n) = \\frac{2}{n!",
"}.$ The second characteristics for the root node is the number of leaves that are attached to it.",
"Here the specification and thus the ordinary differential equation are more involved.",
"In particular, the operators needed for the specification are not so classical so we prefer to explain directly the differential equation.",
"Once again, let $T(z,u) = \\sum _{n,k} t_{n,k} \\; u^k z^n$ be the bivariate generating with $t_{n,k}$ the number of size-$n$ increasing Schröder trees with $k$ leaves as children of the root.",
"Then, $T\\left( z,u \\right) = z + \\frac{u^2\\; z^2}{1-uz} + \\frac{z^2}{1-z}\\; \\partial _z T\\left( z,u \\right)+ \\frac{ z\\left( 1-u \\right)}{1-z} \\; \\partial _u T\\left( z,u \\right).$ Let us give the details to understand the construction.",
"A tree is either reduced to a leaf or a single internal node with some leaves: $z + \\frac{u^2\\; z^2}{1-uz}$ .",
"Or in the iterative process a leaf attached to the root is selected, then replaced by an internal node with at least two leaves (that are not anymore attached to the root of the whole tree): $\\frac{z}{1-z} \\; \\partial _u T\\left( z,u \\right)$ .",
"Or, during the iterative process, a leaf that is not attached to the root is selected and replaced by an internal node attached to at least two leaves: $\\frac{z^2}{1-z}\\; \\partial _z T\\left( z,u \\right)- \\frac{ u\\; z}{1-z} \\; \\partial _u T\\left( z,u \\right)$ .",
"The second term removes the trees built in the first one where we have selected a leaf attached to the root (and also marked by $z$ ).",
"Again by denoting $T(z,u) = \\sum _n t_n(u) z^n$ , we can extract the following recurrence, for all $n\\ge 4$ , $t_n(u) = &(n+u) \\; t_{n-1}(u) + u(1-n) \\; t_{n-2}(u) \\\\&+ (1-u) \\; t^{\\prime }_{n-1}(u) + (u^2-u) \\; t^{\\prime }_{n-2}(u),$ with $t_1(u)=1$ , $t_2(u)=u^2$ and $t_3(u)=u^3+2u$ .",
"Theorem 9 Asymptotically, the mean and the variance of the number $\\mathcal {X}_n$ of leaves attached to the root are $\\mathbb {E}_{n}(\\mathcal {X}_n)=\\frac{2e}{n} + O\\left(\\frac{1}{n!",
"}\\right)$ and $\\mathbb {V}_{n}(\\mathcal {X}_n)=\\frac{2e}{n} + O\\left(\\frac{1}{n^2}\\right)$ .",
"Let us remark the second term in the expansion of $\\mathbb {E}_{n}(\\mathcal {X}_n)$ is extremely small in front of the main term.",
"Here the specification is easier to exhibit, and its translation via the symbolic method is direct ($\\mathcal {U}$ is marking the binary nodes): $& \\mathcal {Z}\\cup \\left(\\Theta _{\\mathcal {Z}} \\left( \\mathcal {U}\\times \\mathcal {Z}\\cup \\operatornamewithlimits{\\mbox{\\sc Seq}_{\\ge 2}}(\\mathcal {Z}) \\right) \\right);\\\\&T(z,u) = z + \\left( u\\; z^2 + \\frac{z^3}{1-z} \\right) \\partial _z T(z,u).$ Let us again extract the recurrence $t_n(u)$ , for all $n\\ge 4$ : $t_n(u)= (1+u(n-1)) t_{n-1}(u) + (1-u)(n-2) t_{n-2},$ with $t_1(u)=1, t_2(u)=u$ and $t_3(u)=1+2u^2$ .",
"Note that due to this recurrence, the probability distribution $p_n(u)=\\frac{2}{n!}",
"t_n(u)$ also exhibits a simple recurrence (cf. [11]).",
"Thus we easily compute the mean and the second factorial moment of the number of binary nodes in size-$(n\\ge 3)$ trees: $&\\mathbb {E}_{n}(\\mathcal {X}_n) = \\frac{7}{3} + n - 2\\cdot \\sum _{k=1}^{n} \\frac{1}{k} - \\frac{1}{n},\\quad \\text{and} \\\\&\\mathbb {E}_{n}(\\mathcal {X}_n(\\mathcal {X}_n-1)) = \\sum _{k=3}^{n} \\frac{2k-2}{k} \\mathbb {E}_{{k-1}}(\\mathcal {X}_{k-1}) \\\\& \\hspace*{85.35826pt} - \\frac{2k-4}{k (k-1) } \\mathbb {E}_{{k-2}}(\\mathcal {X}_{k-2}).$ Theorem 10 Asymptotically, the mean and the variance of the number $\\mathcal {X}_n$ of binary internal nodes are $\\displaystyle {\\mathbb {E}_{n}(\\mathcal {X}_n) = n -2\\ln (n) + \\frac{7}{3} -2\\gamma + O\\left(\\frac{1}{n}\\right)}$ and $\\displaystyle {\\mathbb {V}_{n}(\\mathcal {X}_n) = 4\\ln (n)+O(1)}$ .",
"Furthermore there is a limiting distribution satisfying $\\displaystyle {\\frac{\\mathcal {X}_n -\\mathbb {E}_{n}(\\mathcal {X}_n)}{\\sqrt{\\mathbb {V}_{n}(\\mathcal {X}_n)}} \\xrightarrow{} \\mathcal {N}(0,1).",
"}$ [Key-ideas] The recurrences give the closed form formulas for $\\mathbb {E}_{n}(\\mathcal {X}_n)$ and $\\mathbb {V}_{n}(\\mathcal {X}_n)$ .",
"Then it is important to notice that $t_n(u)$ can be approximated by $\\tilde{t}_n(u)$ verifying $\\tilde{t}_n(u)=(1+u(n-1))\\tilde{t}_{n-1}(u)$ .",
"The latter recurrence is the same recurrence as the one exhibited for the number of internal nodes (equation (REF )).",
"Thus, by the same arguments, the sequence of distributions $(t_n(u))$ converges in distribution to a Gaussian law."
],
[
"Bijection with permutations",
"Observing the exact value $t_n = n!",
"/ 2$ enhances the chances of finding some relation between our model of increasing trees and a subclass of permutations.",
"Let us start with this goal.",
"First, for a size-$n$ permutation $\\sigma $ denoted by $(\\sigma _1, \\dots , \\sigma _n)$ , we define $\\sigma _i$ to be its $i$ -th element (the image of $i$ ), and $\\sigma ^{-1}(k)$ to be the preimage of $k$ (the position of $k$ in the permutation).",
"We are now ready to define the recursive map $\\mathcal {M}$ between $\\mathcal {HP}$ , the class of permutations such that 1 appears before 2 and the class $of increasing Schröder trees.The base case is the permutation $ (1,2)$ which corresponds to the root labeled by~$ 1$attached to two unlabeled leaves.",
"Let $$ be a size-$ n$ permutation in $ HP$,with $ n3$.",
"We observe its the greatest element: if $ n=n$ then we add a new rightmostleaf to the last added internal node (the one with the largest label);otherwise let $ k=-1(n)$, we create a new binary node $$ labeled with a new integer(the smallest as possible) and attached to two new leaves, then we replace the $ k$-thleaf by this new tree rooted at $$, in the tree under construction based on $ n$,i.e.",
"that is $$ without the greatest element $ n$.",
"Remark that duringthe tree construction we must traverse the leaves, we can take an arbitrary traversal.$ Theorem 11 The map $\\mathcal {M}$ is a one-to-one correspondence between $\\mathcal {HP}$ and $.$ The mapping is size preserving: at each iteration we remove exactly one element from the permutation and add exactly one leaf to the tree by either adding a leaf to the last exiting node or by killing one leaf and adding to new ones.",
"The mapping is injective since by induction at each iteration we remove the greatest element of the permutation and its following its index the actions are performed on the resulting tree in a non-ambiguous manner.",
"Finally the mapping is based on two classes with the same number of elements (of each size).",
"Figure: A size-8 example of the mapping ℳ\\mathcal {M}In the Figure REF we present the mapping on an example.",
"Remark that we have ordered the steps reversely to understand the process in a easiest way."
],
[
"Uniform random sampling",
"Obviously, through the latter bijection we are able to obtain an uniform random sampler.",
"It suffices to uniformly sample permutations and to use the bijection to build the associated increasing Schröder tree.",
"While there exists fast algorithms to sample permutations, see for example [2], using the bijection efficiently is not obvious.",
"However, through the bijection a direct probabilistic construction of increasing Schröder trees can be obtained.",
"Such a probabilistic construction presents two main advantages.",
"Firstly, it simplifies the implementation of a sampler and, secondly, and more importantly, it gives a purely probabilistic approach to the original combinatorial class, which we are interesting in.",
"This probabilistic approach can be used to compare to other probabilistic tree models or to exhibit important characteristics of trees on average using probabilities rather than combinatorics.",
"We introduce in this section an algorithm to uniformly sample increasing Schröder trees of a given size $n$ .",
"A first remark is that the uniform sampling of structures with increasing labeling constraints is not so classical in the context of Analytic Combinatorics.",
"There are some studies by Martínez and Molinero [13], [14] in the context of the recursive method and some other about Boltzmann sampling either directly for the method [5] or focusing on a specific application [3].",
"For the uniform sampling of our evolution process, we are focusing on two goals.",
"Our fundamental goal consists in controlling the probability distribution used for the sampling.",
"In fact, we may extract some statistical information based on the samplings, thus the probability distribution is central.",
"We choose to sample uniformly trees of the same size, because then we can bias our generator (and tune the bias) to construct other probability distributions.",
"Secondly our algorithmic framework must be very efficient to sample large trees (with several thousands of leaves).",
"Thus a detailed complexity analysis is necessary to be sure that the algorithm cannot be easily improved.",
"Our approach is based on the combinatorics underlying the very efficient recurrence $t_n = n\\cdot t_{n-1}$ : a tree of size $n$ can be built from a tree of size $n-1$ in $n$ different ways.",
"We exhibit a construction based on this recurrence.",
"This leads to the following iterative algorithm of random sampling.",
"[h] Increasing Schröder Tree Builder [1] TreeBuilder$n$ $n=1$ return the single leaf $T :=$ the root labeled by 1 and attached to two leaves $\\ell := 2$ $i$ from 3 to $n$ $k := rand\\_int(1,i)$ $k=i$ Add a new leaf to the last added internal node in $T$ Create a new binary node at position $k$ in $T$ Create with label $\\ell $ and attached to two leaves $\\ell :=\\ell +1$ return T The function $rand\\_int(a,b)$ returns uniformly at random an integer in $\\lbrace a, a+1, \\dots , b\\rbrace $ .",
"Theorem 12 The function TreeBuilder($n$ ) in Algorithm REF is a uniform sampling algorithm for size-$n$ trees.",
"Asymptotically, it operates in $O(n)$ operations on trees and necessitates $O(n \\ln n)$ random bits.",
"The correctness of the algorithm is a direct consequence of the mapping $\\mathcal {M}$ .",
"it gives the probabilistic construction of trees of $.Using the adequate data structures, as for example by keeping an array of pointersto all leaves and another one to the last inserted internal node,each insertion in the tree under construction is done in constant time.$"
],
[
"Weakly Increasing Schröder trees",
"In this section we aim at developing another model for ranked trees based on Schröder structures.",
"In fact we relax somehow the labeling constraint."
],
[
"The model and its context",
"Weakly increasing Schröder trees are a generalization of strongly increasing Schröder trees.",
"The tree structure is still an unlabeled Schröder tree.",
"But the labeling is different.",
"Internal nodes are labeled between 1 to $\\ell $ in such a way that the sequence of labels in each path from the root to a leaf is also increasing.",
"The difference here is that different nodes can have the same label.",
"This model is also built iteratively.",
"Start with a single (unlabeled) leaf; Iterate the following process: at step $\\ell $ (for $\\ell \\ge 1$ ), select a subset of leaves and replace each of them by an internal node with label $\\ell $ attached to a sequence of at least two leaves.",
"Figure: A weakly increasing Schröder treeIn Figure REF we present a weakly increasing tree of size 30 with 16 distinct labels.."
],
[
"Exact enumeration and relationship with ordered Bell numbers",
"We can specify the process through the symbolic method.",
"But once again the labeling is transparent and does not appear in the specification.",
"$G(z) = z+ G\\left(\\frac{z^2}{1-z}+z\\right) - G(z).$ At each iteration and for each leaf we can either leave it as it is or expand it into a new internal node with at least 2 leaves.",
"The configuration where no leaf is expanded is forbidden, thus we remove $G(z)$ in equation (REF ).",
"From this equation we extract the recurrence $g_n = \\left\\lbrace \\begin{array}{l c l}1 & \\quad & \\text{if } n=1 \\\\\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\; g_k.",
"& & \\text{otherwise.",
"}\\end{array}\\right.$ The first coefficients correspond to a shift of the sequence of Ordered Bell numbers (also called Fubini numbers) referenced as OEIS A000670.",
"$g_n = 0,1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261,\\dots $ By following the approach developed by Pippenger in [16] for the derivation of the exponential generating function for ordered Bell numbers we obtain, by starting from our equation (REF ), $(\\mathcal {B}G)^{\\prime }(z) = 1/(2-e^z)$ .",
"Thus, after integration $ \\displaystyle {\\mathcal {B}G(z) = \\frac{1}{2}\\left( z - \\ln {(2-e^z)}\\right)}$ .",
"Usually ordered Bell numbers are specified by $B=\\operatornamewithlimits{\\mbox{\\sc Seq}}(\\operatornamewithlimits{\\mbox{\\sc Set}}_{\\ge 1}(z))$ .",
"Obviously this gives the exponential generating function $B(z)= 1/ (2-e^z)$ .",
"Thus, we have proved that our sequence is a shift of the one of ordered Bell numbers.",
"As a by-product, we have exhibited a new way for specifying ordered Bell numbers.",
"Recall the $n$ -th ordered Bell number, denoted by $B_n$ , counts the total number of partitions of a set of size $n$ where additionally we consider an order over the subsets of the partition.",
"$B_n = \\sum \\limits _{k=0}^n k!",
"\\; \\genfrac\\lbrace \\rbrace {0.0pt}{}{n}{k} \\sim \\frac{n!",
"}{2 \\left(\\ln 2\\right)^{n+1}},$ where $\\genfrac\\lbrace \\rbrace {0.0pt}{}{n}{k}$ stands for the Stirling partition numbers (also called Stirling numbers of the second kind).",
"The number $B_n$ corresponds to the number $g_{n+1}$ of weakly increasing Schröder trees of size $n+1$ ."
],
[
"Bijection between ordered Bell numbers and weakly increasing\nSchröder trees",
"In ordered partitions, the subsets are ordered but the elements inside a subset are not.",
"In the following let us denote by $p=[p_1,p_2,\\dots ,p_\\ell ]$ an ordered partition such that $p_i$ is the subset of the partition at position $i$ .",
"For example if $p=[ \\lbrace 3,4 \\rbrace , \\lbrace 1,5,7 \\rbrace , \\lbrace 2,6\\rbrace ] $ , then $p_1=\\lbrace 3,4 \\rbrace $ .",
"We denote by $|p_i|$ the size of the $i$ -th subset: $|p_1| = 2$ .",
"The total size (i.e.",
"number of elements) of the partition is denoted by $|p|$ Thus the elements of an ordered partition range from 1 to $|p|$ .",
"For the exhibition of the correspondence we will use a canonical order inside the subsets, consisting in enumerating the elements increasingly.",
"Let $p = [p_1, p_2,\\dots , p_\\ell ]$ be an ordered partition, and $p_i = \\lbrace \\alpha _1, \\alpha _2, \\dots , \\alpha _r\\rbrace $ (with $r\\ge 1$ ), such that $\\alpha _1< \\alpha _2< \\dots < \\alpha _r$ .",
"We define a run in $p_i$ to be a maximal sequence $\\alpha _i, \\alpha _{i+1},\\dots , \\alpha _j$ equal to $\\alpha _i, \\alpha _{i}+1,\\dots , \\alpha _i+j-i$ .",
"It is maximal in the sense that $\\alpha _{i-1} < \\alpha _i - 1$ and $\\alpha _{j+1} > \\alpha _j+1$ .",
"We define the map $runs$ that lists all the runs of a subset.",
"For instance, in our example $p$ , in $p_1$ there is a single run: $3,4$ and in $p_2$ , there are 3 runs.",
"The mapping deals with incomplete ordered partitions (in the sense that some integers are not present in the partition).",
"We define a normalization of a partition, denoted by $norm$ , that maps an incomplete ordered partition of size $k$ into the corresponding ordered partition of size $k$ whose elements are $\\lbrace 1,\\dots ,k\\rbrace $ and that keeps the relative order between the elements.",
"For example by taking the first two subsets from $p$ as $p^{\\prime }=[ p_1,p_2 ]$ , then $p^{\\prime }$ is an incomplete ordered partition of size 5 and we get $norm(p^{\\prime })=[ \\lbrace 2,3 \\rbrace , \\lbrace 1,4,5 \\rbrace ]$ .",
"From the ordered partition $p$ , the mapping $\\mathcal {M}^{\\prime }$ builds the corresponding tree by processing the subsets of the ordered partition successively.",
"We start by creating a new ordered partition that contains only $p_1$ , $p^{\\prime }=norm([p_1])$ .",
"The size of $p_1$ determines the arity of the root: it equals $|p_1|+1$ .",
"The root label is 1.",
"Then at each step $i$ with $i \\in \\lbrace 2,\\dots ,\\ell \\rbrace $ , we process the subset $p_i$ as follows.",
"Normalize the incomplete ordered partition $[p_1, p_2,\\dots , p_i]$ .",
"In the normalized ordered partition the corresponding subset of $p_i$ is denoted by $p^{\\prime }_i$ .",
"The number of new internal nodes is $|runs(p^{\\prime }_i)|$ , all labeled by $i$ .",
"Suppose $runs(p^{\\prime }_i)=[r_1,r_2,\\dots ,r_j]$ (with each $r_\\ell $ a set of successive integers and possibly a single one).",
"Take an order for the leaves in the tree under construction (the postorder one for example) and iterate the process: For $\\ell $ from 1 to $j$ , take the leaf whose index is the first element of $r_\\ell $ and replace it with an internal node with label $i$ of arity $|r_\\ell | +1$ .",
"Figure: Weakly increasing tree of size 8In Figure REF the mapping $\\mathcal {M}^{\\prime }$ is applied on our example $p=[ \\lbrace 3,4 \\rbrace , \\lbrace 1,5,7 \\rbrace , \\lbrace 2,6\\rbrace ] $ .",
"The resulting weakly increasing tree is of size 9."
],
[
"Quantitative analysis of the number of iteration steps",
"In our classical iterative equation, we add a new variable $u$ to mark each iteration.",
"$ G(z,u) = z+ u\\; G \\left( \\frac{z^2}{1-z}+z, u \\right) - u \\; G(z,u).$ Which leads to the following recurrence, $ g_{n,k} = \\left\\lbrace \\begin{array}{l c l}1 & & \\text{if } n=1, k=0 ,\\\\\\sum \\limits _{j=1}^{n-1}\\binom{n-1}{j-1} \\; g_{j,k-1} & & \\text{otherwise } .\\\\\\end{array}\\right.$ Figure: Distribution of g n,k g_{n,k} for n∈{0,1,2,⋯,6}n \\in \\lbrace 0,1,2,\\dots , 6\\rbrace This recurrence is analogous to the one relating ordered Bell numbers and Stirling partition numbers.",
"Theorem 13 The distribution of the the number of building steps in weakly increasing Schröder trees of size $n$ satisfies $g_{n,k} = k!",
"\\; \\genfrac\\lbrace \\rbrace {0.0pt}{}{n+1}{k}.$ Let $\\mathcal {X}_n$ be the random variable describing this distribution, we have $\\frac{\\mathcal {X}_n -\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n)}{\\sqrt{\\mathbb {V}_{\\mathcal {G}_n}(\\mathcal {X}_n)}} \\xrightarrow{} \\mathcal {N}(0,1),$ with $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) \\sim \\frac{1}{2\\ln 2} \\; n$ and $\\mathbb {V}_{\\mathcal {G}_n}(\\mathcal {X}_n) \\sim \\frac{1-\\ln 2}{(2\\ln 2)^2} \\; n$ .",
"The one-to-one correspondence between weakly increasing Schröder trees and ordered Bell numbers gives the combinatorial proof of the distribution for $(g_{n,k})$ .",
"The analysis of the limiting distribution is classical in the quasi-powers framework.",
"See for example [11].",
"In this model the number of iteration steps does not correspond to the number of the internal nodes as at each iteration any subset of leaves can be expanded into internal nodes with new leaves.",
"The specification marking both internal nodes and leaves is $ G(z,u) = z+ G\\left(\\frac{uz^2}{1-z}+z, u \\right) - G(z,u).$ We recall that the substitution $G(\\frac{uz^2}{1-z}+z)$ means that for each iteration each leaf can be left as it is $z$ or expanded into an internal node of unbounded arity with new leaves $\\frac{z^2}{1-z}$ .",
"It is in the second part that an internal node will be created and thus we mark it with $u$ .",
"Theorem 14 The average number of internal nodes $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n)$ in size $n$ weakly increasing trees verifies $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) = n-\\ln {n} + o(1).$ The main ideas of the proof are in Appendix ."
],
[
"Uniform random sampling",
"We introduce in this section an algorithm to uniformly sample weakly increasing Schröder trees of a given size $n$ directly, without an intermediate step of generating uniformly an ordered partition.",
"The global approach for our algorithmic framework deals with the recursive generation method adapted to the Analytic Combinatorics point of view in [12].",
"But in our context, we note that we can obtain for free (from a complexity view) an unranking algorithm.",
"This kind of algorithm has been developed in the 70’s by Nijenhuis and Wilf [15] and then has been introduced to the context of Analytic Combinatorics by Martínez and Molinero [13].",
"Here the idea is not to draw uniformly an object, but first to define a total order over the objects under consideration (here weakly increasing Schröder trees) and then an integer (named the rank) is sampled to build deterministically the associated object.",
"Such an approach gives also a way to do exhaustive generation (refer to the paper [4] for an example of both methods: recursive generation and unranking).",
"For both types of algorithms (recursive generators and unranking ones), there is a first step of pre-computations (done only once before the sampling of many objects).",
"We must compute (and store) the numbers of trees of sizes from 1 to $n$ .",
"Here this phase can be done with a quadratic complexity (in the number of arithmetic operations) because of the recursive formula for $g_n$ (cf.",
"equation (REF )).",
"The second (and last) step for the sampling consists in the recursive construction of the tree of rank $r$ that corresponds to an uniformly sampled integer in $\\lbrace 0, 1,\\dots , g_n-1\\rbrace $ .",
"For this purpose, we come back to the original recursive equation (REF ), and in particular, we look at the sum over decreasing $k$ : $g_n = \\binom{n-1}{n-2} g_{n-1} + \\binom{n-1}{n-3} g_{n-2}+ \\dots + \\binom{n-1}{0}g_1.$ The latter recurrence is combinatorially easy to understand.",
"Through the evolution process, to build a size $n$ tree, we take a size $k\\in \\lbrace 1,\\dots , n-1\\rbrace $ tree constructed with exactly one less iteration.",
"The binomial coefficient $\\binom{n-1}{k-1}$ corresponds to the number of composition of $n$ in $k$ parts.",
"Then we traverse the tree, and each time we see a leaf, we do the following rule: if the next part is of value 1, we leave the leaf unchanged otherwise for a value $\\ell >1$ , we replace the leaf by an internal node (well labeled with the single new value valid for this step) and attached to it $\\ell $ leaves.",
"We then take the next part of the composition into consideration and continue the tree traversal.",
"In the latter sum, the first term is much bigger than the second one, that is must bigger than the third one and so on.",
"This approach, focusing first on the dominant terms corresponds to the Boustrophedonic order presented in [12].",
"It allows to improve essentially the average complexity of the random sampling algorithm.",
"In our case of weakly increasing Schröder trees that do not follow a standard specification (cf.",
"[12]), the complexity gain is even better.",
"[h] Weakly increasing Tree Unranking [1] UnrankTree$n, s$ $n=1$ return the tree reduced to a single leaf $k := n-1$ $r := s$ $r >= 0$ $r := r - \\binom{n-1}{k-1}\\cdot g_k$ $k := k - 1$ $k := k+1$ $r := r + \\binom{n-1}{k-1}\\cdot g_k$ $s^{\\prime } := r \\mod {g}_k$ $T := $UnrankTree($k,s^{\\prime }$ ) $C := $UnrankComposition($n, k, r // g_k$ ) Substitute in $T$ some leaves according to $C$ return the tree $T$ The sequence $(g_k)_{k\\le n}$ and $(\\ell !",
")_{\\ell \\in \\lbrace 1,\\dots ,n\\rbrace }$ have been precomputed and stored.",
"Line 13: The operation $//$ is the Euclidean division.",
"[1] UnrankComposition$n, k, s$ $n=k$ and $s=0$ return $[1, 1, \\dots , 1]$ $s^{\\prime } := s$ $s^{\\prime } < \\binom{n-2}{k-1}$ $C := $UnrankComposition($n-1, k, s^{\\prime }$ ) $C[len(C)] := C[len(C)]+1$ return $C$ $s^{\\prime } := s^{\\prime } - \\binom{n-2}{k-1}$ $ C := $UnrankComposition($n-1, k-1, s^{\\prime }$ ) $\\cup [1]$ return $C$ Theorem 15 The function UnrankTree is an unranking algorithm and calling it with the parameters $n$ and an uniformly sampled integer in $\\lbrace 0,\\dots , g_n-1\\rbrace $ , it is an uniform sampler for size-$n$ weakly increasing Schröder trees.",
"[Key-ideas] The total order for weakly increasing Schröder trees is the following.",
"Let $\\alpha $ and $\\beta $ be two trees.",
"If the size of $\\alpha $ is strictly smaller than the one of $\\beta $ , we define $\\alpha < \\beta $ .",
"Let us suppose that both sizes are equal to $n$ .",
"In the recursive construction, let $\\tilde{\\alpha }$ (and $\\tilde{\\gamma }_1$ be the tree (resp.",
"the composition for the leaf substitution) building the tree $\\alpha $ (and respectively $\\tilde{\\beta }$ and $\\tilde{\\gamma }_2$ the ones associated to $\\beta $ ).",
"If the size of $\\tilde{\\alpha }$ is strictly greater than the one of $\\tilde{\\beta }$ , we define $\\alpha < \\beta $ .",
"Let us now suppose that both sizes of $\\tilde{\\alpha }$ and $\\tilde{\\beta }$ are equal.",
"By using an arbitrary order for the composition unranking, we can order $\\alpha $ and $\\beta $ .",
"This total order over the trees is satisfied by our algorithm: thus this latter is correct.",
"Theorem 16 Once the pre-computations have been done, the function UnrankTree necessitates $O(n^2)$ arithmetic operations to construct any tree of size $n$ .",
"Due to the fact that usually the difference between $n$ and $k$ is very small, a detailed analysis of the average case, or an more adapted composition unranking should give a better complexity analysis.",
"In fact as we have seen before, in a large typical tree, there are in average $n-\\ln n$ internal nodes and thus most of them must be of arity 2 and are given by the first term in the latter sum defining $g_n$ .",
"[Proof-ideas] The main idea is the following: during a call to UnrankTree, there are exactly the same number of new leaves in the tree under construction to the number of loops in the while instruction on Line 6.",
"Outside this while block, the number of arithmetic operations is essentially due to the unranking algorithm for compositions.",
"The actual version of this algorithm induces a quadratic complexity in the number of arithmetic operations.",
"The unranking algorithm for the composition is based on the classical result about the composition of $n>0$ in $k\\in \\lbrace 1, \\dots , n\\rbrace $ parts: $C_{n,k} &= \\binom{n-1}{k-1} \\\\&= C_{n-1,k} + C_{n-1,k-1}.$"
],
[
"Appendix related strongly increasing Schröder trees: Section ",
"The Borel transform consists in the following transformation on ordinary generating series: $\\mathcal {B}\\left(\\sum _{n\\ge 0} a_n z^n\\right) = \\sum _{n\\ge 0} a_n \\; \\frac{z^n}{n!",
"}.$ Lemma 17 Using Borel transform formula on formal series, we easily derive the following identities: $\\mathcal {B}(zf) (z) = \\int _{0}^z \\mathcal {B}f \\mathrm {d}t$ ; $\\mathcal {B}(f^{\\prime }) (z) = (\\mathcal {B}f)^{\\prime }(z) + z (\\mathcal {B}f)^{\\prime \\prime }(z)$ .",
"We are now ready to prove Proposition REF .",
"[Proof of Proposition REF ] Applying Borel on equation (REF ) and using properties (i) and (ii) we obtain $T(z,u)=zuT(z,u)+(1-u)\\cdot \\int _{0}^z T(z,u) dz -\\frac{z^2}{2}+z.$ Then by differentiating by z $\\frac{\\partial (1-zu)T(z,u)}{\\partial z}=\\frac{\\partial (1-u)\\cdot \\int _{0}^z T(z,u) dz -\\frac{z^2}{2}+z}{\\partial z}.$ Thus, after simplifications $(1-zu)\\frac{\\partial T(z,u)}{\\partial z}=T(z,u) -z+1 \\quad \\text{with } T(0,0)=1.$ Solving the differential equation gives the stated result.",
"Let us denote by $\\mathcal {X}_n$ the random variable corresponding to the to number of internal nodes in increasing Schröder trees of size $n$ .",
"Proposition REF aims at proving the mean value and the variance of $\\mathcal {X}_n$ .",
"[Proof of Proposition REF ] Recall that the mean and variance can be computed mechanically from the bivariate generating function $\\mathbb {E}_{n}(\\mathcal {X}_n) &= \\frac{[z^n] \\partial _{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)}, \\text{ and}\\\\\\mathbb {E}_{n}(\\mathcal {X}_n^2) &= \\frac{[z^n] \\partial ^2_{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)}+ \\frac{[z^n] \\partial _{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)},$ where $\\cdot _{\\mid _{u=1}}$ stands for the substitution of $u$ by 1.",
"$\\mathbf {E}_{T_n}[\\mathcal {X}] & = \\frac{[z^n] \\partial _{u} T(z,u) \\mid _{u=1} }{[z^n]T(z,1)} \\\\& = [z^n]\\left( \\frac{z}{(1-z)^2}-\\frac{1}{1-z}\\ln \\left(\\frac{1}{1-z}\\right) \\right) \\\\&\\hspace*{28.45274pt} + \\frac{1}{2}\\left(\\frac{1}{1-z}-z-1\\right) \\\\& = n-H_n+\\frac{1}{2}$ Let $n\\ge 2$ , the mean value of $\\mathcal {X}_n$ is equal to $\\mathbb {E}_{n}[\\mathcal {X}_n^2] = n(n-1) -2n(H_n-1) + \\sum _{k=1}^{n-1} \\frac{1}{n-k} H_k,$ and thus $\\mathbb {E}_{n}[\\mathcal {X}_n^2] = &n(n-1) -2n\\; \\ln n -2(\\gamma -1)n + \\ln ^2 n \\\\&+2\\gamma \\; \\ln n +\\gamma ^2 - \\frac{\\pi ^2}{6} + O\\left(\\frac{\\log n}{n}\\right).$ In the same vein, when $n$ tends to infinity, we get $\\mathbb {V}_{T_n}[\\mathcal {X}] = \\ln n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4} + O\\left(\\frac{\\log n}{n}\\right).$ We are now ready to prove the limit distribution for $\\mathcal {X}_n$ .",
"[Proof of Theorem REF ] This proof is an adaptation on Flajolet and Sedgewick's proof on the limit Gaussian law of Stirling Cycle numbers [11] We take the probability generating function of $\\hat{T}_n(u)$ it is obvious that if $\\frac{t_{n,k}}{t_n}$ is a limit Gaussian law then so is $\\frac{\\hat{t}_{n,k}}{t_n}=\\frac{t_{n,n-k}}{t_n}$ .",
"We will just get the mirror of the probability the standard deviation will not change $\\hat{\\sigma _n}=\\sigma _n$ and the mean will be the mirror mean so $\\hat{\\mu }_n=n-\\mu _n$ .",
"$ \\hat{p}_n(u) = \\frac{2u(u+2)(u+3)\\dots (u+n-1)}{n!",
"}.$ Thus we have $ p_n(u) = \\frac{2\\Gamma (u+n)}{(u+1)\\Gamma (u)\\Gamma (n+1)}.$ Near $u=1$ we find an estimate of $p_n(u)$ using Stirling formula for the Gamma function $ p_n(u) = \\frac{ n^{u-1}}{\\Gamma (u)}\\left(1+O\\left(\\frac{1}{n}\\right)\\right)=\\frac{ \\left( e^{u-1} \\right)^{\\log n} }{\\Gamma (u)}\\left(1+O\\left(\\frac{1}{n}\\right)\\right).$ Now we can study the standardized random variable $\\hat{X}^\\star _n=\\frac{\\hat{\\mathcal {X}}-\\hat{\\mu }_n }{\\hat{\\sigma }_n}$ .",
"The standardization of a random variable can be translated directly on the characteristic function.",
"$ \\phi _{X^\\star _n}(t) = e^{-it\\frac{\\mu }{\\sigma }} \\phi _{X_n}(\\frac{t}{\\sigma }).$ $\\phi _{\\hat{X}^\\star _n}(t) = &e^{-it\\frac{\\log n-\\gamma +\\frac{1}{2}}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}} } } \\\\& \\cdot \\frac{ \\left( exp(\\log n (e^{\\frac{it}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}}}}-1)) \\right) }{\\Gamma (u)} \\left(1+O\\left(\\frac{1}{n}\\right)\\right).$ For a fixed $t$ and as $n \\rightarrow \\infty $ , $ \\log \\phi _{\\hat{X}^\\star _n}(t) = -\\frac{t^2}{2} + O(\\frac{1}{\\log n}).$ This last result is obtained by limited development of $\\log n \\cos {\\frac{t}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}}}}$ .",
"Finally we have $ \\phi _{\\hat{X}^\\star _n}(t) \\sim e^{-\\frac{t^2}{2}},$ which is the characteristic function of the Gaussian law."
],
[
"Appendix related to weakly increasing Schröder trees:\nSection ",
"Derivation for the exponential generating function for $(g_n)$ : We have, $ g_n = \\delta _n + \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1}g_k.$ Where $\\delta _l$ is 1 for $l=1$ and 0 otherwise.",
"Adding $g_n$ to both sides gives $ 2 g_n = \\delta _n + \\sum \\limits _{k=1}^{n}\\binom{n-1}{k-1}g_k.$ Finally multiplying both sides by $\\frac{z^l}{l!",
"}$ and summing over all $l\\ge 0$ $2G(z)=z+ \\sum \\limits _{l \\ge 0} \\frac{z^l}{l!",
"}\\sum \\limits _{k=1}^{n}\\binom{n-1}{k-1}g_k$ Deriving this last equation yields to the equation of Ordered Bell number.",
"which has been studied by different authors.",
"See [16] for a derivation of the exponential generating function, $G(z)^{\\prime } = \\frac{1}{2-e^z}.$ Finally we have, $ G(z) = \\frac{1}{2}\\left( z - \\ln {(2-e^z)}\\right).$ Proof of the theorem REF We define $f_n =\\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1)g_{k+1}$ Lemma 18 $\\frac{f_n}{g_n} \\sim C$ where $C \\approx 1.38$ is a constant Wilf has given an approximation of the error term of Ordered Bell numbers in [20] which we can use, $ g_n = \\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
"),$ with $\\gamma = \\frac{1}{\\sqrt{ln(2)^2 + 4\\pi ^2}} \\approx 0.16\\dots $ .",
"Remark that $\\frac{1}{ln(2)} \\approx 1.44\\dots > \\gamma $ .",
"Now, $\\frac{f_n}{g_n} & = \\frac{ \\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1) \\left( \\frac{k!",
"}{2\\ln (2)^{k+1}}+O(\\gamma ^{k}k!)",
"\\right)}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\& = \\frac{ \\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1) \\left( \\frac{k!",
"}{2\\ln (2)^{k+1}} \\right)}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ \\sum \\limits _{k=0}^{n-2}(k+1) \\left( \\frac{1}{\\ln (2)^{k+1}} \\right)}{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ n(\\sum \\limits _{k=1}^{n-1}\\frac{1}{(n-1-k)!\\ln 2^k)}) - ( \\sum \\limits _{k=1}^{n-1}\\frac{n-k}{(n-1-k)!\\ln 2^k}) }{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ n(\\sum \\limits _{k=1}^{n-1}\\frac{1}{(n-1-k)!\\ln 2^k)} - ( \\sum \\limits _{k=1}^{n-1}\\frac{n-k}{(n-1-k)!\\ln 2^k}) }{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = c + O(\\frac{1}{n})$ Then for large $n$ , we can show the result by induction.",
"Taking $Gu_n=\\frac{(n-1)!",
"}{2\\ln (2)^n}(n-\\ln n) +O(\\gamma ^{n-1}(n-1)!",
")$ .",
"$&\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) = \\frac{Gu_n}{g_n} & \\\\&= c+ O(\\frac{1}{n})+\\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\left( \\frac{(k-1)!",
"}{2\\ln (2)^k}(k-\\ln k +O(\\gamma ^{k-1}(k-1)!)",
"\\right) }{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\& = c+ O(\\frac{1}{n})+\\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}k - \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}\\ln k}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
")}\\\\& \\quad + \\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\$ Thus we get: $&\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) =\\\\& = c+ O(\\frac{1}{n})+ n +c^{\\prime } +\\frac{-\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}\\ln n}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{k-1}(k-1)!)}",
"\\\\& \\quad + \\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{k-1}(k-1)!",
")}\\\\& = c+ O(\\frac{1}{n})+ n +c^{\\prime }- \\ln n +\\frac{ \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
")}\\\\& = c+ n +c^{\\prime }- \\ln n +O(\\frac{1}{n})\\\\& \\sim n - \\ln n.$"
],
[
"Weakly Increasing Schröder trees",
"In this section we aim at developing another model for ranked trees based on Schröder structures.",
"In fact we relax somehow the labeling constraint."
],
[
"The model and its context",
"Weakly increasing Schröder trees are a generalization of strongly increasing Schröder trees.",
"The tree structure is still an unlabeled Schröder tree.",
"But the labeling is different.",
"Internal nodes are labeled between 1 to $\\ell $ in such a way that the sequence of labels in each path from the root to a leaf is also increasing.",
"The difference here is that different nodes can have the same label.",
"This model is also built iteratively.",
"Start with a single (unlabeled) leaf; Iterate the following process: at step $\\ell $ (for $\\ell \\ge 1$ ), select a subset of leaves and replace each of them by an internal node with label $\\ell $ attached to a sequence of at least two leaves.",
"Figure: A weakly increasing Schröder treeIn Figure REF we present a weakly increasing tree of size 30 with 16 distinct labels.."
],
[
"Exact enumeration and relationship with ordered Bell numbers",
"We can specify the process through the symbolic method.",
"But once again the labeling is transparent and does not appear in the specification.",
"$G(z) = z+ G\\left(\\frac{z^2}{1-z}+z\\right) - G(z).$ At each iteration and for each leaf we can either leave it as it is or expand it into a new internal node with at least 2 leaves.",
"The configuration where no leaf is expanded is forbidden, thus we remove $G(z)$ in equation (REF ).",
"From this equation we extract the recurrence $g_n = \\left\\lbrace \\begin{array}{l c l}1 & \\quad & \\text{if } n=1 \\\\\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\; g_k.",
"& & \\text{otherwise.",
"}\\end{array}\\right.$ The first coefficients correspond to a shift of the sequence of Ordered Bell numbers (also called Fubini numbers) referenced as OEIS A000670.",
"$g_n = 0,1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261,\\dots $ By following the approach developed by Pippenger in [16] for the derivation of the exponential generating function for ordered Bell numbers we obtain, by starting from our equation (REF ), $(\\mathcal {B}G)^{\\prime }(z) = 1/(2-e^z)$ .",
"Thus, after integration $ \\displaystyle {\\mathcal {B}G(z) = \\frac{1}{2}\\left( z - \\ln {(2-e^z)}\\right)}$ .",
"Usually ordered Bell numbers are specified by $B=\\operatornamewithlimits{\\mbox{\\sc Seq}}(\\operatornamewithlimits{\\mbox{\\sc Set}}_{\\ge 1}(z))$ .",
"Obviously this gives the exponential generating function $B(z)= 1/ (2-e^z)$ .",
"Thus, we have proved that our sequence is a shift of the one of ordered Bell numbers.",
"As a by-product, we have exhibited a new way for specifying ordered Bell numbers.",
"Recall the $n$ -th ordered Bell number, denoted by $B_n$ , counts the total number of partitions of a set of size $n$ where additionally we consider an order over the subsets of the partition.",
"$B_n = \\sum \\limits _{k=0}^n k!",
"\\; \\genfrac\\lbrace \\rbrace {0.0pt}{}{n}{k} \\sim \\frac{n!",
"}{2 \\left(\\ln 2\\right)^{n+1}},$ where $\\genfrac\\lbrace \\rbrace {0.0pt}{}{n}{k}$ stands for the Stirling partition numbers (also called Stirling numbers of the second kind).",
"The number $B_n$ corresponds to the number $g_{n+1}$ of weakly increasing Schröder trees of size $n+1$ ."
],
[
"Bijection between ordered Bell numbers and weakly increasing\nSchröder trees",
"In ordered partitions, the subsets are ordered but the elements inside a subset are not.",
"In the following let us denote by $p=[p_1,p_2,\\dots ,p_\\ell ]$ an ordered partition such that $p_i$ is the subset of the partition at position $i$ .",
"For example if $p=[ \\lbrace 3,4 \\rbrace , \\lbrace 1,5,7 \\rbrace , \\lbrace 2,6\\rbrace ] $ , then $p_1=\\lbrace 3,4 \\rbrace $ .",
"We denote by $|p_i|$ the size of the $i$ -th subset: $|p_1| = 2$ .",
"The total size (i.e.",
"number of elements) of the partition is denoted by $|p|$ Thus the elements of an ordered partition range from 1 to $|p|$ .",
"For the exhibition of the correspondence we will use a canonical order inside the subsets, consisting in enumerating the elements increasingly.",
"Let $p = [p_1, p_2,\\dots , p_\\ell ]$ be an ordered partition, and $p_i = \\lbrace \\alpha _1, \\alpha _2, \\dots , \\alpha _r\\rbrace $ (with $r\\ge 1$ ), such that $\\alpha _1< \\alpha _2< \\dots < \\alpha _r$ .",
"We define a run in $p_i$ to be a maximal sequence $\\alpha _i, \\alpha _{i+1},\\dots , \\alpha _j$ equal to $\\alpha _i, \\alpha _{i}+1,\\dots , \\alpha _i+j-i$ .",
"It is maximal in the sense that $\\alpha _{i-1} < \\alpha _i - 1$ and $\\alpha _{j+1} > \\alpha _j+1$ .",
"We define the map $runs$ that lists all the runs of a subset.",
"For instance, in our example $p$ , in $p_1$ there is a single run: $3,4$ and in $p_2$ , there are 3 runs.",
"The mapping deals with incomplete ordered partitions (in the sense that some integers are not present in the partition).",
"We define a normalization of a partition, denoted by $norm$ , that maps an incomplete ordered partition of size $k$ into the corresponding ordered partition of size $k$ whose elements are $\\lbrace 1,\\dots ,k\\rbrace $ and that keeps the relative order between the elements.",
"For example by taking the first two subsets from $p$ as $p^{\\prime }=[ p_1,p_2 ]$ , then $p^{\\prime }$ is an incomplete ordered partition of size 5 and we get $norm(p^{\\prime })=[ \\lbrace 2,3 \\rbrace , \\lbrace 1,4,5 \\rbrace ]$ .",
"From the ordered partition $p$ , the mapping $\\mathcal {M}^{\\prime }$ builds the corresponding tree by processing the subsets of the ordered partition successively.",
"We start by creating a new ordered partition that contains only $p_1$ , $p^{\\prime }=norm([p_1])$ .",
"The size of $p_1$ determines the arity of the root: it equals $|p_1|+1$ .",
"The root label is 1.",
"Then at each step $i$ with $i \\in \\lbrace 2,\\dots ,\\ell \\rbrace $ , we process the subset $p_i$ as follows.",
"Normalize the incomplete ordered partition $[p_1, p_2,\\dots , p_i]$ .",
"In the normalized ordered partition the corresponding subset of $p_i$ is denoted by $p^{\\prime }_i$ .",
"The number of new internal nodes is $|runs(p^{\\prime }_i)|$ , all labeled by $i$ .",
"Suppose $runs(p^{\\prime }_i)=[r_1,r_2,\\dots ,r_j]$ (with each $r_\\ell $ a set of successive integers and possibly a single one).",
"Take an order for the leaves in the tree under construction (the postorder one for example) and iterate the process: For $\\ell $ from 1 to $j$ , take the leaf whose index is the first element of $r_\\ell $ and replace it with an internal node with label $i$ of arity $|r_\\ell | +1$ .",
"Figure: Weakly increasing tree of size 8In Figure REF the mapping $\\mathcal {M}^{\\prime }$ is applied on our example $p=[ \\lbrace 3,4 \\rbrace , \\lbrace 1,5,7 \\rbrace , \\lbrace 2,6\\rbrace ] $ .",
"The resulting weakly increasing tree is of size 9."
],
[
"Quantitative analysis of the number of iteration steps",
"In our classical iterative equation, we add a new variable $u$ to mark each iteration.",
"$ G(z,u) = z+ u\\; G \\left( \\frac{z^2}{1-z}+z, u \\right) - u \\; G(z,u).$ Which leads to the following recurrence, $ g_{n,k} = \\left\\lbrace \\begin{array}{l c l}1 & & \\text{if } n=1, k=0 ,\\\\\\sum \\limits _{j=1}^{n-1}\\binom{n-1}{j-1} \\; g_{j,k-1} & & \\text{otherwise } .\\\\\\end{array}\\right.$ Figure: Distribution of g n,k g_{n,k} for n∈{0,1,2,⋯,6}n \\in \\lbrace 0,1,2,\\dots , 6\\rbrace This recurrence is analogous to the one relating ordered Bell numbers and Stirling partition numbers.",
"Theorem 13 The distribution of the the number of building steps in weakly increasing Schröder trees of size $n$ satisfies $g_{n,k} = k!",
"\\; \\genfrac\\lbrace \\rbrace {0.0pt}{}{n+1}{k}.$ Let $\\mathcal {X}_n$ be the random variable describing this distribution, we have $\\frac{\\mathcal {X}_n -\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n)}{\\sqrt{\\mathbb {V}_{\\mathcal {G}_n}(\\mathcal {X}_n)}} \\xrightarrow{} \\mathcal {N}(0,1),$ with $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) \\sim \\frac{1}{2\\ln 2} \\; n$ and $\\mathbb {V}_{\\mathcal {G}_n}(\\mathcal {X}_n) \\sim \\frac{1-\\ln 2}{(2\\ln 2)^2} \\; n$ .",
"The one-to-one correspondence between weakly increasing Schröder trees and ordered Bell numbers gives the combinatorial proof of the distribution for $(g_{n,k})$ .",
"The analysis of the limiting distribution is classical in the quasi-powers framework.",
"See for example [11].",
"In this model the number of iteration steps does not correspond to the number of the internal nodes as at each iteration any subset of leaves can be expanded into internal nodes with new leaves.",
"The specification marking both internal nodes and leaves is $ G(z,u) = z+ G\\left(\\frac{uz^2}{1-z}+z, u \\right) - G(z,u).$ We recall that the substitution $G(\\frac{uz^2}{1-z}+z)$ means that for each iteration each leaf can be left as it is $z$ or expanded into an internal node of unbounded arity with new leaves $\\frac{z^2}{1-z}$ .",
"It is in the second part that an internal node will be created and thus we mark it with $u$ .",
"Theorem 14 The average number of internal nodes $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n)$ in size $n$ weakly increasing trees verifies $\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) = n-\\ln {n} + o(1).$ The main ideas of the proof are in Appendix ."
],
[
"Uniform random sampling",
"We introduce in this section an algorithm to uniformly sample weakly increasing Schröder trees of a given size $n$ directly, without an intermediate step of generating uniformly an ordered partition.",
"The global approach for our algorithmic framework deals with the recursive generation method adapted to the Analytic Combinatorics point of view in [12].",
"But in our context, we note that we can obtain for free (from a complexity view) an unranking algorithm.",
"This kind of algorithm has been developed in the 70’s by Nijenhuis and Wilf [15] and then has been introduced to the context of Analytic Combinatorics by Martínez and Molinero [13].",
"Here the idea is not to draw uniformly an object, but first to define a total order over the objects under consideration (here weakly increasing Schröder trees) and then an integer (named the rank) is sampled to build deterministically the associated object.",
"Such an approach gives also a way to do exhaustive generation (refer to the paper [4] for an example of both methods: recursive generation and unranking).",
"For both types of algorithms (recursive generators and unranking ones), there is a first step of pre-computations (done only once before the sampling of many objects).",
"We must compute (and store) the numbers of trees of sizes from 1 to $n$ .",
"Here this phase can be done with a quadratic complexity (in the number of arithmetic operations) because of the recursive formula for $g_n$ (cf.",
"equation (REF )).",
"The second (and last) step for the sampling consists in the recursive construction of the tree of rank $r$ that corresponds to an uniformly sampled integer in $\\lbrace 0, 1,\\dots , g_n-1\\rbrace $ .",
"For this purpose, we come back to the original recursive equation (REF ), and in particular, we look at the sum over decreasing $k$ : $g_n = \\binom{n-1}{n-2} g_{n-1} + \\binom{n-1}{n-3} g_{n-2}+ \\dots + \\binom{n-1}{0}g_1.$ The latter recurrence is combinatorially easy to understand.",
"Through the evolution process, to build a size $n$ tree, we take a size $k\\in \\lbrace 1,\\dots , n-1\\rbrace $ tree constructed with exactly one less iteration.",
"The binomial coefficient $\\binom{n-1}{k-1}$ corresponds to the number of composition of $n$ in $k$ parts.",
"Then we traverse the tree, and each time we see a leaf, we do the following rule: if the next part is of value 1, we leave the leaf unchanged otherwise for a value $\\ell >1$ , we replace the leaf by an internal node (well labeled with the single new value valid for this step) and attached to it $\\ell $ leaves.",
"We then take the next part of the composition into consideration and continue the tree traversal.",
"In the latter sum, the first term is much bigger than the second one, that is must bigger than the third one and so on.",
"This approach, focusing first on the dominant terms corresponds to the Boustrophedonic order presented in [12].",
"It allows to improve essentially the average complexity of the random sampling algorithm.",
"In our case of weakly increasing Schröder trees that do not follow a standard specification (cf.",
"[12]), the complexity gain is even better.",
"[h] Weakly increasing Tree Unranking [1] UnrankTree$n, s$ $n=1$ return the tree reduced to a single leaf $k := n-1$ $r := s$ $r >= 0$ $r := r - \\binom{n-1}{k-1}\\cdot g_k$ $k := k - 1$ $k := k+1$ $r := r + \\binom{n-1}{k-1}\\cdot g_k$ $s^{\\prime } := r \\mod {g}_k$ $T := $UnrankTree($k,s^{\\prime }$ ) $C := $UnrankComposition($n, k, r // g_k$ ) Substitute in $T$ some leaves according to $C$ return the tree $T$ The sequence $(g_k)_{k\\le n}$ and $(\\ell !",
")_{\\ell \\in \\lbrace 1,\\dots ,n\\rbrace }$ have been precomputed and stored.",
"Line 13: The operation $//$ is the Euclidean division.",
"[1] UnrankComposition$n, k, s$ $n=k$ and $s=0$ return $[1, 1, \\dots , 1]$ $s^{\\prime } := s$ $s^{\\prime } < \\binom{n-2}{k-1}$ $C := $UnrankComposition($n-1, k, s^{\\prime }$ ) $C[len(C)] := C[len(C)]+1$ return $C$ $s^{\\prime } := s^{\\prime } - \\binom{n-2}{k-1}$ $ C := $UnrankComposition($n-1, k-1, s^{\\prime }$ ) $\\cup [1]$ return $C$ Theorem 15 The function UnrankTree is an unranking algorithm and calling it with the parameters $n$ and an uniformly sampled integer in $\\lbrace 0,\\dots , g_n-1\\rbrace $ , it is an uniform sampler for size-$n$ weakly increasing Schröder trees.",
"[Key-ideas] The total order for weakly increasing Schröder trees is the following.",
"Let $\\alpha $ and $\\beta $ be two trees.",
"If the size of $\\alpha $ is strictly smaller than the one of $\\beta $ , we define $\\alpha < \\beta $ .",
"Let us suppose that both sizes are equal to $n$ .",
"In the recursive construction, let $\\tilde{\\alpha }$ (and $\\tilde{\\gamma }_1$ be the tree (resp.",
"the composition for the leaf substitution) building the tree $\\alpha $ (and respectively $\\tilde{\\beta }$ and $\\tilde{\\gamma }_2$ the ones associated to $\\beta $ ).",
"If the size of $\\tilde{\\alpha }$ is strictly greater than the one of $\\tilde{\\beta }$ , we define $\\alpha < \\beta $ .",
"Let us now suppose that both sizes of $\\tilde{\\alpha }$ and $\\tilde{\\beta }$ are equal.",
"By using an arbitrary order for the composition unranking, we can order $\\alpha $ and $\\beta $ .",
"This total order over the trees is satisfied by our algorithm: thus this latter is correct.",
"Theorem 16 Once the pre-computations have been done, the function UnrankTree necessitates $O(n^2)$ arithmetic operations to construct any tree of size $n$ .",
"Due to the fact that usually the difference between $n$ and $k$ is very small, a detailed analysis of the average case, or an more adapted composition unranking should give a better complexity analysis.",
"In fact as we have seen before, in a large typical tree, there are in average $n-\\ln n$ internal nodes and thus most of them must be of arity 2 and are given by the first term in the latter sum defining $g_n$ .",
"[Proof-ideas] The main idea is the following: during a call to UnrankTree, there are exactly the same number of new leaves in the tree under construction to the number of loops in the while instruction on Line 6.",
"Outside this while block, the number of arithmetic operations is essentially due to the unranking algorithm for compositions.",
"The actual version of this algorithm induces a quadratic complexity in the number of arithmetic operations.",
"The unranking algorithm for the composition is based on the classical result about the composition of $n>0$ in $k\\in \\lbrace 1, \\dots , n\\rbrace $ parts: $C_{n,k} &= \\binom{n-1}{k-1} \\\\&= C_{n-1,k} + C_{n-1,k-1}.$"
],
[
"Appendix related strongly increasing Schröder trees: Section ",
"The Borel transform consists in the following transformation on ordinary generating series: $\\mathcal {B}\\left(\\sum _{n\\ge 0} a_n z^n\\right) = \\sum _{n\\ge 0} a_n \\; \\frac{z^n}{n!",
"}.$ Lemma 17 Using Borel transform formula on formal series, we easily derive the following identities: $\\mathcal {B}(zf) (z) = \\int _{0}^z \\mathcal {B}f \\mathrm {d}t$ ; $\\mathcal {B}(f^{\\prime }) (z) = (\\mathcal {B}f)^{\\prime }(z) + z (\\mathcal {B}f)^{\\prime \\prime }(z)$ .",
"We are now ready to prove Proposition REF .",
"[Proof of Proposition REF ] Applying Borel on equation (REF ) and using properties (i) and (ii) we obtain $T(z,u)=zuT(z,u)+(1-u)\\cdot \\int _{0}^z T(z,u) dz -\\frac{z^2}{2}+z.$ Then by differentiating by z $\\frac{\\partial (1-zu)T(z,u)}{\\partial z}=\\frac{\\partial (1-u)\\cdot \\int _{0}^z T(z,u) dz -\\frac{z^2}{2}+z}{\\partial z}.$ Thus, after simplifications $(1-zu)\\frac{\\partial T(z,u)}{\\partial z}=T(z,u) -z+1 \\quad \\text{with } T(0,0)=1.$ Solving the differential equation gives the stated result.",
"Let us denote by $\\mathcal {X}_n$ the random variable corresponding to the to number of internal nodes in increasing Schröder trees of size $n$ .",
"Proposition REF aims at proving the mean value and the variance of $\\mathcal {X}_n$ .",
"[Proof of Proposition REF ] Recall that the mean and variance can be computed mechanically from the bivariate generating function $\\mathbb {E}_{n}(\\mathcal {X}_n) &= \\frac{[z^n] \\partial _{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)}, \\text{ and}\\\\\\mathbb {E}_{n}(\\mathcal {X}_n^2) &= \\frac{[z^n] \\partial ^2_{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)}+ \\frac{[z^n] \\partial _{u} T(z,u)_{\\mid _{u=1}} }{[z^n]T(z,1)},$ where $\\cdot _{\\mid _{u=1}}$ stands for the substitution of $u$ by 1.",
"$\\mathbf {E}_{T_n}[\\mathcal {X}] & = \\frac{[z^n] \\partial _{u} T(z,u) \\mid _{u=1} }{[z^n]T(z,1)} \\\\& = [z^n]\\left( \\frac{z}{(1-z)^2}-\\frac{1}{1-z}\\ln \\left(\\frac{1}{1-z}\\right) \\right) \\\\&\\hspace*{28.45274pt} + \\frac{1}{2}\\left(\\frac{1}{1-z}-z-1\\right) \\\\& = n-H_n+\\frac{1}{2}$ Let $n\\ge 2$ , the mean value of $\\mathcal {X}_n$ is equal to $\\mathbb {E}_{n}[\\mathcal {X}_n^2] = n(n-1) -2n(H_n-1) + \\sum _{k=1}^{n-1} \\frac{1}{n-k} H_k,$ and thus $\\mathbb {E}_{n}[\\mathcal {X}_n^2] = &n(n-1) -2n\\; \\ln n -2(\\gamma -1)n + \\ln ^2 n \\\\&+2\\gamma \\; \\ln n +\\gamma ^2 - \\frac{\\pi ^2}{6} + O\\left(\\frac{\\log n}{n}\\right).$ In the same vein, when $n$ tends to infinity, we get $\\mathbb {V}_{T_n}[\\mathcal {X}] = \\ln n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4} + O\\left(\\frac{\\log n}{n}\\right).$ We are now ready to prove the limit distribution for $\\mathcal {X}_n$ .",
"[Proof of Theorem REF ] This proof is an adaptation on Flajolet and Sedgewick's proof on the limit Gaussian law of Stirling Cycle numbers [11] We take the probability generating function of $\\hat{T}_n(u)$ it is obvious that if $\\frac{t_{n,k}}{t_n}$ is a limit Gaussian law then so is $\\frac{\\hat{t}_{n,k}}{t_n}=\\frac{t_{n,n-k}}{t_n}$ .",
"We will just get the mirror of the probability the standard deviation will not change $\\hat{\\sigma _n}=\\sigma _n$ and the mean will be the mirror mean so $\\hat{\\mu }_n=n-\\mu _n$ .",
"$ \\hat{p}_n(u) = \\frac{2u(u+2)(u+3)\\dots (u+n-1)}{n!",
"}.$ Thus we have $ p_n(u) = \\frac{2\\Gamma (u+n)}{(u+1)\\Gamma (u)\\Gamma (n+1)}.$ Near $u=1$ we find an estimate of $p_n(u)$ using Stirling formula for the Gamma function $ p_n(u) = \\frac{ n^{u-1}}{\\Gamma (u)}\\left(1+O\\left(\\frac{1}{n}\\right)\\right)=\\frac{ \\left( e^{u-1} \\right)^{\\log n} }{\\Gamma (u)}\\left(1+O\\left(\\frac{1}{n}\\right)\\right).$ Now we can study the standardized random variable $\\hat{X}^\\star _n=\\frac{\\hat{\\mathcal {X}}-\\hat{\\mu }_n }{\\hat{\\sigma }_n}$ .",
"The standardization of a random variable can be translated directly on the characteristic function.",
"$ \\phi _{X^\\star _n}(t) = e^{-it\\frac{\\mu }{\\sigma }} \\phi _{X_n}(\\frac{t}{\\sigma }).$ $\\phi _{\\hat{X}^\\star _n}(t) = &e^{-it\\frac{\\log n-\\gamma +\\frac{1}{2}}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}} } } \\\\& \\cdot \\frac{ \\left( exp(\\log n (e^{\\frac{it}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}}}}-1)) \\right) }{\\Gamma (u)} \\left(1+O\\left(\\frac{1}{n}\\right)\\right).$ For a fixed $t$ and as $n \\rightarrow \\infty $ , $ \\log \\phi _{\\hat{X}^\\star _n}(t) = -\\frac{t^2}{2} + O(\\frac{1}{\\log n}).$ This last result is obtained by limited development of $\\log n \\cos {\\frac{t}{\\sqrt{\\log n + \\gamma -\\frac{\\pi ^2}{6}-\\frac{5}{4}}}}$ .",
"Finally we have $ \\phi _{\\hat{X}^\\star _n}(t) \\sim e^{-\\frac{t^2}{2}},$ which is the characteristic function of the Gaussian law."
],
[
"Appendix related to weakly increasing Schröder trees:\nSection ",
"Derivation for the exponential generating function for $(g_n)$ : We have, $ g_n = \\delta _n + \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1}g_k.$ Where $\\delta _l$ is 1 for $l=1$ and 0 otherwise.",
"Adding $g_n$ to both sides gives $ 2 g_n = \\delta _n + \\sum \\limits _{k=1}^{n}\\binom{n-1}{k-1}g_k.$ Finally multiplying both sides by $\\frac{z^l}{l!",
"}$ and summing over all $l\\ge 0$ $2G(z)=z+ \\sum \\limits _{l \\ge 0} \\frac{z^l}{l!",
"}\\sum \\limits _{k=1}^{n}\\binom{n-1}{k-1}g_k$ Deriving this last equation yields to the equation of Ordered Bell number.",
"which has been studied by different authors.",
"See [16] for a derivation of the exponential generating function, $G(z)^{\\prime } = \\frac{1}{2-e^z}.$ Finally we have, $ G(z) = \\frac{1}{2}\\left( z - \\ln {(2-e^z)}\\right).$ Proof of the theorem REF We define $f_n =\\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1)g_{k+1}$ Lemma 18 $\\frac{f_n}{g_n} \\sim C$ where $C \\approx 1.38$ is a constant Wilf has given an approximation of the error term of Ordered Bell numbers in [20] which we can use, $ g_n = \\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
"),$ with $\\gamma = \\frac{1}{\\sqrt{ln(2)^2 + 4\\pi ^2}} \\approx 0.16\\dots $ .",
"Remark that $\\frac{1}{ln(2)} \\approx 1.44\\dots > \\gamma $ .",
"Now, $\\frac{f_n}{g_n} & = \\frac{ \\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1) \\left( \\frac{k!",
"}{2\\ln (2)^{k+1}}+O(\\gamma ^{k}k!)",
"\\right)}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\& = \\frac{ \\sum \\limits _{k=0}^{n-2}\\binom{n-2}{k}(k+1) \\left( \\frac{k!",
"}{2\\ln (2)^{k+1}} \\right)}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ \\sum \\limits _{k=0}^{n-2}(k+1) \\left( \\frac{1}{\\ln (2)^{k+1}} \\right)}{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ n(\\sum \\limits _{k=1}^{n-1}\\frac{1}{(n-1-k)!\\ln 2^k)}) - ( \\sum \\limits _{k=1}^{n-1}\\frac{n-k}{(n-1-k)!\\ln 2^k}) }{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = \\frac{ n(\\sum \\limits _{k=1}^{n-1}\\frac{1}{(n-1-k)!\\ln 2^k)} - ( \\sum \\limits _{k=1}^{n-1}\\frac{n-k}{(n-1-k)!\\ln 2^k}) }{\\frac{(n-1)}{\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"+ O(\\frac{1}{n})\\\\& = c + O(\\frac{1}{n})$ Then for large $n$ , we can show the result by induction.",
"Taking $Gu_n=\\frac{(n-1)!",
"}{2\\ln (2)^n}(n-\\ln n) +O(\\gamma ^{n-1}(n-1)!",
")$ .",
"$&\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) = \\frac{Gu_n}{g_n} & \\\\&= c+ O(\\frac{1}{n})+\\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\left( \\frac{(k-1)!",
"}{2\\ln (2)^k}(k-\\ln k +O(\\gamma ^{k-1}(k-1)!)",
"\\right) }{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\& = c+ O(\\frac{1}{n})+\\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}k - \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}\\ln k}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
")}\\\\& \\quad + \\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!)}",
"\\\\$ Thus we get: $&\\mathbb {E}_{\\mathcal {G}_n}(\\mathcal {X}_n) =\\\\& = c+ O(\\frac{1}{n})+ n +c^{\\prime } +\\frac{-\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} \\frac{(k-1)!",
"}{2\\ln (2)^k}\\ln n}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{k-1}(k-1)!)}",
"\\\\& \\quad + \\frac{\\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{k-1}(k-1)!",
")}\\\\& = c+ O(\\frac{1}{n})+ n +c^{\\prime }- \\ln n +\\frac{ \\sum \\limits _{k=1}^{n-1}\\binom{n-1}{k-1} O(\\gamma ^{k-1}(k-1)!)",
"}{\\frac{(n-1)!",
"}{2\\ln (2)^n}+O(\\gamma ^{n-1}(n-1)!",
")}\\\\& = c+ n +c^{\\prime }- \\ln n +O(\\frac{1}{n})\\\\& \\sim n - \\ln n.$"
]
] | 1808.08376 |
[
[
"Quantum Reversibility Is Relative, Or Do Quantum Measurements Reset\n Initial Conditions?"
],
[
"Abstract I compare the role of the information in the classical and quantum dynamics by examining the relation between information flows in measurements and the ability of observers to reverse evolutions.",
"I show that in the Newtonian dynamics reversibility is unaffected by the observer's retention of the information about the measurement outcome.",
"By contrast -- even though quantum dynamics is unitary, hence, reversible -- reversing quantum evolution that led to a measurement becomes in principle impossible for an observer who keeps the record of its outcome.",
"Thus, quantum irreversibility can result from the information gain rather than just its loss -- rather than just an increase of the (von Neumann) entropy.",
"Recording of the outcome of the measurement resets, in effect, initial conditions within the observer's (branch of) the Universe.",
"Nevertheless, I also show that observer's friend -- an agent who knows what measurement was successfully carried out and can confirm that the observer knows the outcome but resists his curiosity and does not find out the result -- can, in principle, undo the measurement.",
"This relativity of quantum reversibility sheds new light on the origin of the arrow of time and elucidates the role of information in classical and quantum physics.",
"Quantum discord appears as a natural measure of the extent to which dissemination of information about the outcome affects the ability to reverse the measurement."
],
[
"Introduction",
"Quantum as well as classical equations of motion are reversible.",
"Yet, irreversibility we, observers, perceive is an undeniable “fact of life”.",
"In particular, quantum measurements are famously regarded as irreversible [1].",
"This irreversibility is a reason why modeling of quantum measurements using unitary dynamics is sometimes viewed as controversial.",
"Of course, decoherence [2], [3], [4], [5] (now usually included as an essential ingredient of a fully consummated measurement process) is rightly regarded as effectively irreversible.",
"The arrow of time it dictates can be tied to the dynamical second law [6], [7].",
"Our aim here is to point out that, over and above the familiar irreversibility exemplified by decoherence that stems from the second law, and in contrast to the classical physics, irreversibility of an even more fundamental kind arises in quantum physics in course of measurements.",
"We shall explore it by turning “reversibility” from an abstract concept that characterizes equations of motion to an operationally defined property: We shall investigate when the evolution of a measured system and a measuring apparatus can be, at least in principle, reversed even if the information gained in course of the measurement is preserved (e.g., the record imprinted on the state of the apparatus pointer is copied).",
"This operational view of reversibility yields new insights: We shall see that reversing quantum measurements becomes impossible for an observer who retains record of the measurement outcome.",
"This is because the state of the measured quantum system revealed and recorded by the observer assumes—for that observer—the role reserved for the initial state in the classical, Newtonian physics.",
"Consequently, clear distinction between the initial conditions and dynamics—the basis of classical physics [8]—is lost in a quantum setting.",
"Indeed, quantum measurements can be reversed only when the record of the outcome is no longer preserved anywhere else in the Universe.",
"By contrast, classical measurement can be reversed even if the record of the outcome is retained.",
"Irreversibility caused by the acquisition of information in a quantum measurement has a different origin and a different character from irreversibility that follows from the second law [6].",
"There, the arrow of time – the impossibility of reversal – is tied to the increase of entropy, and, hence, to the loss of information.",
"In quantum measurements irreversibility can be a consequence of the acquisition (rather than loss) of information.",
"This loss of the ability to reverse is relative—it depends on the information in possession of the agent attempting reversal.",
"Thus, a friend of the observer, an agent who refrains from finding out the outcome (but can control the dynamics that led to that measurement) can, at least in principle (and in a setup reminiscent of “Wigner's friend” [9]) undo the evolution that resulted in that measurement even after he confirms that the observer had—prior to reversal—perfect record of the outcome.",
"Measurements re-set initial conditions relevant for observer's evolution in a manner that is tied to the choice of what is measured (as emphasized by John Wheeler [10], see Fig.",
"1).",
"Quantum measurements (more generally, “quantum jumps”) undermine one of the foundational principles of the classical, Newtonian dynamics: There, consecutive measurements just narrowed down the bundle of the possible past trajectories consistent with observer's knowledge.",
"Thus, in a classical, deterministic Universe it was always possible to imagine a single actual trajectory that fit within this bundle, and was traceable to the point marking the initial condition.",
"This meant that evolution was reversible, an that it could be retraced—hence, reversed—using the present state of the system as a starting point into the dynamical laws and “running the evolution backwards”.",
"Figure: An agent—an observer—within the evolving and expanding Universe carries out measurements that help define initial conditions of that Universe .",
"Thus, initial conditions (at Big Bang) are determined in part by measurements carried out at present.",
"This dramatic image (due to John Wheeler) is illustrated by the study of the ability to reverse an act of acquisition of information in this paper.This idealization of a single starting point of “my Universe”—i.e., the unique Universe consistent with the outcomes of all the past measurements at observer's disposal—is no longer tenable in the quantum setting.",
"Quantum measurement derails evolution, resetting it onto the track consistent with its outcome.",
"The loss of distinction between initial conditions and dynamical laws is tied to the enhanced role of information in the quantum Universe: Information is not just a passive reflection of the deterministic trajectory dictated by the dynamics (as was imagined in the classical, Newtonian settings) but it is acquired in a measurement process that changes the state of both the measured object and of the measuring apparatus (or of an agent / observer).",
"We start in the next section by comparing information-theoretic prerequisites of a successful reversal in the quantum and classical case.",
"In Section III we discuss the use of quantum discord to quantify the inability to reverse measurements.",
"Section IV shows that another agent, a friend of the observer, can confirm that the observer is in possession of the information about the outcome in a way that does not preclude the reversal and does not reveal the outcome.",
"This leads us to conclude that in a quantum world reversibility is indeed relative—it depends on the information in possession of the agent.",
"Discussion and summary are offered in Section V. We note that much of the technical content of the paper amounts to the proverbial “beating around the bush”.",
"This is because the key point is “personal” and simple—an agent who is in possession of the information about the outcome is incapable of undoing the measurement that led to that outcome.",
"Yet, the tools at our disposal—state vectors, density matrices, unitary evolution operators—constrain us to discuss the measurement process “from the outside”.",
"And, from that external vantage point, information retained by the observer or copied into his record-keeping device plays the same role as the information acquired by the environment in course of decoherence or (especially) quantum Darwinism [3], [5], [11].",
"One could even say that we are stuck in the shoes of Wigner's friend [9], looking at the observer “from the outside”.",
"The ultimate message of this paper is that the observer / agent is incapable of undoing the acts of the acquisition of information, and that this inability to reverse reveals an origin of the arrow of time that is uniquely quantum and that is not dependent on the entropy increase mandated by the second law.",
"There is of course no contradiction between the resulting arrows of time, and (as decoherence accompanies quantum measurements [2], [3], [4], [5], [6]) they generally appear together and point in the same direction, but they are nevertheless distinct.",
"One way to express this difference is to note that, while our discussion is phrased in the language that presumes unitarity of evolutions, this inability to reverse may be easier to express using Bohr's “collapse” imagery [12]."
],
[
"Records and Reversibility",
"We study operational reversibility—the ability of an observer to reverse evolution—in the classical and quantum setting.",
"Our goal is to show that, in the quantum world, information has physical consequences that go far beyond its role in the classical, Newtonian dynamics.",
"This illustrates the difference between the nature and function of information in quantum and classical physics.",
"The key gedankenexperiment involves a measured quantum (or classical) system ${\\mathcal {S}}$ ($\\bf S$ ), and an agent / apparatus ${\\mathcal {A}}$ ($\\bf A$ ).",
"The records from ${\\mathcal {A}}$ ($\\bf A$ ) can be further copied into the memory device $\\cal D$ ($\\bf D$ ).",
"We shall now show that presence of the copy of the record of the measurement outcomes has no bearing on the (in principle) ability to reverse a classical measurement, but precludes reversal of a quantum measurement.",
"Thus, the pre-measurement state of the classical $\\bf SA$ can be restored even when $\\bf D$ knows the outcome.",
"Such reversal is not possible for a quantum $\\cal SA$ as long as $\\cal D$ retains a copy of the measurement result.",
"It is important to emphasize the distinction between the usual discussions of reversibility (that focus on the reversibility of the equations that generate the dynamics) and our aims: Here we take for granted that it is possible to implement operators that can undo dynamical evolutions (including these leading to measurements) in the absence of any leaks of information.",
"Thus, in a sense, we are siding with Loschmidt in his debate with Boltzmann.",
"For instance, we assume observer can switch the sign of the Hamiltonian that resulted in the measurement.",
"Our aim is to shift the focus of attention from the dynamics to the role of the information observer has in implementing reversals."
],
[
"Reversing classical measurement (while keeping record of its outcome)",
"We start by examining measurements carried out by a classical agent / apparatus $\\bf A$ on a classical system $\\bf S$ .",
"The state $\\tt s$ of $\\bf S$ (e.g., location of $\\bf S$ in phase space) is measured (with some accuracy, but we do not need to assume perfection) by a classical $\\bf A$ that starts in the “ready to measure” state $\\tt A_0$ : $ { \\tt s A_0} \\ { \\stackrel{\\mathfrak {E}_{\\bf SA}}{\\Longrightarrow } } \\ { \\tt s A_s} \\qquad \\mathrm {(1a)}$ The question we address is whether the combined state of $\\bf SA$ can be restored to the pre-measurement ${ \\tt s A_0}$ even when the information about the outcome is retained somewhere, e.g.",
"copied into the memory device $\\bf D$ .",
"The dynamics $\\mathfrak {E}_{\\bf SA}$ responsible for the measurement is assumed to be reversible and, in Eq.",
"(1a), it is classical.",
"Therefore, classical measurement can be undone simply by implementing ${\\mathfrak {E}_{\\bf SA}^{-1}}$ that is assumed to be at the disposal of the observer.",
"And example of ${\\mathfrak {E}_{\\bf SA}^{-1}}$ is (Loschmidt inspired) instantaneous reversal of all velocities.",
"Our main point is that the reversal ${ \\tt s A_s} \\ { \\stackrel{\\mathfrak {E}_{\\bf SA}^{-1}}{\\Longrightarrow } } \\ { \\tt s A_0} \\qquad \\mathrm {(1a´)}$ can be accomplished even after the measurement outcome is copied onto the memory device $\\bf D$ : ${ \\tt s A_s D_0} \\ { \\stackrel{\\mathfrak {E}_{\\bf AD}}{\\Longrightarrow } } \\ { \\tt s A_s D_s} \\qquad \\mathrm {(2a)}$ so that the pre-measurement state of $\\bf S$ is recorded elsewhere (here, in $\\bf D$ ).",
"Above, ${\\mathfrak {E}_{\\bf AD}}$ plays the same role as ${\\mathfrak {E}_{\\bf SA}}$ in Eq.",
"(1a).",
"That is, the examination of $\\bf S$ and $\\bf A$ separately, or of the combined $\\bf SA$ will not reveal any evidence of irreversibility.",
"After the reversal; ${ \\tt s A_s D_s} \\ { \\stackrel{\\mathfrak {E}_{\\bf SA}^{-1}}{\\Longrightarrow } } \\ { \\tt s A_0 D_s} \\qquad \\mathrm {(3a)}$ the state of $\\bf SA$ is identical to the pre-measurement state, even though recording device retains the copy of the outcome.",
"Classical controlled-not gates provide a simple example of the claims above, as one can readily verify.",
"Starting with a partly known state of the system does not change this conclusion.",
"Thus, initial information transfer from $\\bf S$ to $\\bf A$ : $ { \\tt (w_s s + w_r r) A_0} \\ { \\stackrel{\\mathfrak {E}_{\\bf SA}}{\\Longrightarrow } } \\ { \\tt w_s s ~A_s+ w_r r ~A_r} \\qquad \\mathrm {(4a)}$ when the system is beforehand in a classical mixture of two states $\\tt {r,~s}$ with the respective probabilities $\\tt {w_r,~w_s}$ can be undone—$\\bf S$ and $\\bf A$ will return to the initial state—even if an intermediate information transfer from $\\bf A$ to $\\bf D$ has occurred: $ { \\tt ({ \\tt w_s s ~A_s+ w_r r ~A_r}) D_0} \\ { \\stackrel{\\mathfrak {E}_{\\bf AD}}{\\Longrightarrow } } \\ { \\tt w_s s ~A_s D_s+ w_r r ~A_r D_r} \\ .",
"\\qquad \\mathrm {(5a)}$ This is easily seen: $ { \\tt w_s s ~A_s D_s+ w_r r ~A_r D_r} \\ {\\stackrel{\\mathfrak {E}_{\\bf SA}^{-1}}{\\Longrightarrow } } \\ ({\\tt w_s s ~D_s+ w_r r ~D_r}) {\\tt {A_0}} \\ .",
"\\qquad \\mathrm {(6a)}$ In the end $\\bf S$ is still correlated with $\\bf D$ —that is $\\bf D$ has the record of the outcome of the measurement of $\\bf S$ by $\\bf A$ .",
"However, anyone who measures the combined state of $\\bf S$ and $\\bf A$ will confirm that the evolution that resulted in the measurement of $\\bf S$ by $\\bf A$ has been reversed.",
"That is, the apparatus / agent ${\\mathcal {A}}$ is back in the pre-measurement state, and the system ${\\mathcal {S}}$ has the pre-measurement probability distribution over the classical microstates r,s (even if they are still correlated with the states of the memory device D).",
"Thus, in classical dynamics retention of records—presence of information about the outcome of the measurement—does not preclude the ability to reverse evolutions."
],
[
"Reversing quantum measurement (can't keep the record of the outcome)",
"Consider now a measurement of a quantum system ${\\mathcal {S}}$ by a quantum ${\\mathcal {A}}$ : $ \\bigl ( \\sum _s \\alpha _s | s \\rangle \\bigr ) | A_0 \\rangle { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ \\sum _s \\alpha _s | s \\rangle | A_s \\rangle \\qquad \\mathrm {(1b)}$ The evolution operator ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}$ is unitary (for example, ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}=\\sum _{s,k} | s \\rangle \\!",
"\\langle s | | A_{k+s} \\rangle \\!",
"\\langle A_k |$ with orthogonal $\\lbrace | s \\rangle \\rbrace $ , $\\lbrace | A_k \\rangle \\rbrace $ would do the job).",
"Therefore, evolution that leads to a measurement is in principle reversible.",
"Reversal implemented by ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}^\\dagger $ is possible, and will restore the pre-measurement state of ${\\mathcal {S}}{\\mathcal {A}}$ : $\\sum _s \\alpha _s | s \\rangle | A_s \\rangle { \\stackrel{{\\mathfrak {U}}^\\dagger _{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ \\bigl ( \\sum _s \\alpha _s | s \\rangle \\bigr ) | A_0 \\rangle \\qquad \\mathrm {(1b´)}$ Let us however assume that the outcome of the measurement is copied before reversal is attempted: $ \\bigl ( \\sum _s \\alpha _s | s \\rangle | A_s \\rangle \\bigr ) | D_0 \\rangle { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {A}}{\\cal D}}}{\\Longrightarrow } } \\sum _s \\alpha _s | s \\rangle | A_s \\rangle | D_s \\rangle \\ .",
"\\qquad \\mathrm {(2b)}$ Here ${\\mathfrak {U}}_{{\\mathcal {A}}{\\cal D}}$ plays the same role and can have the same structure as ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}$ .",
"Note that Eqs.",
"(2a,b) implement repeatable measurement / copying on the states $\\lbrace | s \\rangle \\rbrace $ , $\\lbrace | A_s \\rangle \\rbrace $ of the system and of the apparatus, respectively.",
"That is, these states of ${\\mathcal {S}}$ and ${\\mathcal {A}}$ remain untouched by the measurement and copying processes.",
"Repeatability implies that the outcome states $\\lbrace | s \\rangle \\rbrace $ as well as the record states $\\lbrace | A_s \\rangle \\rbrace $ are orthogonal [13], [14].",
"This will matter in our discussion of measurements involving mixtures.",
"When the information about the outcome is copied, the combined pre-measurement state $\\bigl ( \\sum _s \\alpha _s | s \\rangle \\bigr ) | A_0 \\rangle $ of ${\\mathcal {S}}{\\mathcal {A}}$ pair cannot be restored by ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}^\\dagger $ .",
"That is: $ {\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}^\\dagger \\bigl ( \\sum _s \\alpha _s | s \\rangle | A_s \\rangle | D_s \\rangle \\bigr ) \\ = \\ | A_0 \\rangle \\bigl (\\sum _s \\alpha _s | s \\rangle | D_s \\rangle \\bigr ) \\qquad \\mathrm {(3b)}$ The apparatus is restored to the pre-measurement $| A_0 \\rangle $ , but the system remains entangled with the memory device.",
"On its own, its state is represented by the mixture: $\\varrho ^{{\\mathcal {S}}} = \\sum _s w_{ss} | s \\rangle \\!",
"\\langle s | \\qquad \\mathrm {(7)}$ where $w_{ss}=|\\alpha _s|^2$ .",
"Reversing quantum measurement of a state that corresponds to a superposition of the potential outcomes is possible only providing the memory of the outcome is no longer preserved anywhere else in the Universe."
],
[
"Quasiclassical case",
"The special (measure zero) case when the quantum system is, prior to the measurement, in the eigenstate of the measured observable, constitutes an interesting exception to the above “impossibility to reverse”.",
"Then the measurement outcome: $ | s \\rangle | A_0 \\rangle | D_0 \\rangle { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ | s \\rangle | A_s \\rangle | D_0 \\rangle \\qquad \\mathrm {(1c)}$ can be copied $ | s \\rangle | A_s \\rangle | D_0 \\rangle {\\stackrel{{\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}}{\\Longrightarrow } } | s \\rangle | A_s \\rangle | D_s \\rangle \\qquad \\mathrm {(2c)}$ and yet the evolution of ${\\mathcal {S}}{\\mathcal {A}}$ can be reversed.",
"$ | s \\rangle | A_s \\rangle | D_s \\rangle \\ { \\stackrel{{\\mathfrak {U}}^\\dagger _{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ | s \\rangle | A_0 \\rangle | D_s \\rangle \\qquad \\mathrm {(3c)}$ The above three equations describe evolution of quantum systems, yet they have the same structure and allow for the reversal in spite of the record retained by $\\cal D$ in the same way as for the classical case (motivating the use of “quasiclassical” in the title of this subsection).",
"It is straightforward to show that the same conclusion holds for mixed states that are diagonal in the basis in which the system is measured.",
"That is, the pre-measurement $\\rho ^{{\\mathcal {S}}} = \\sum _s w_{ss} | s \\rangle \\!",
"\\langle s | $ is then identical as the post-measurement $\\varrho ^{{\\mathcal {S}}}$ where the \"pre\" and \"post\" are indicated in using different version of Greek “rho”.",
"This mixed quasiclassical case parallels classical Eqs.",
"(4a-6a)."
],
[
"Superpositions of Outcomes and Measurement Reversal",
"We have now demonstrated the difference between the in principle ability to reverse quantum and classical measurements.",
"Information flows do not matter for classical, Newtonian dynamics.",
"However, when information about a quantum measurement outcome is communicated—copied and retained by any other system—the evolution that led to that measurement cannot be reversed.",
"Thus, from the point of view of the measurer, information retention about an outcome of a quantum measurement implies irreversibility.",
"We have also examined the quasiclassical case and concluded that the presence of arbitrary superpositions in quantum theory is responsible for the irreversibility of measurements: When the considerations are restricted to such a quasiclassical set of orthogonal states, reversibility of measurements is restored.",
"Physical significance of the phases between the potential outcomes makes quantum states vulnerable to the information leakage and prevents reversal of the evolution that led to the measurement.",
"This significance of arbitrary superposition was illustrated by the example of a mixture diagonal in the set of states that is left unperturbed by measurements.",
"Measurement on a mixture that is diagonal in the same basis with which measurements correlate the state of the apparatus remains in principle reversible.",
"Thus, in a quantum Universe where measurements are carried out only on pre-decohered systems (e.g., macroscopic systems in our Universe) and observers acquire information only about the decoherence-resistant states, one may come to believe that reversible dynamics is all there is.",
"Of course, decoherence is an irreversible procsess, so in a sense, in our Universe, the price for this illusion of Newtonian reversibility is a massive irreversibility which is paid “up front”, extracted by decoherence.",
"Presence of superpositions in correlated states of quantum systems can be quantified by quantum discord [15], [17], [16].",
"We shall now examine the relation between quantum discord and the ability to reverse measurements."
],
[
"Measurements of quantum mixtures, reversibility, and discord",
"The above conclusion about the impossibility to reverse quantum measurements (except for the quasiclassical case) continues to apply when the pre-measurement state of the system is a mixture diagonal in a basis that is different from the measurement basis $\\lbrace | s \\rangle \\rbrace $ defined by ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}=\\sum _{s,k} | s \\rangle \\!",
"\\langle s | | A_{k+s} \\rangle \\!",
"\\langle A_k |$ .",
"Thus, when the pre-measurement density matrix of the system is given by: $ \\rho ^{\\mathcal {S}}=\\sum _{r,s} w_{rs} | r \\rangle \\!",
"\\langle s | \\ , \\qquad \\mathrm {(8)}$ measurement by ${\\mathcal {A}}$ results in a combined state: $\\bigl ( \\sum _{r,s} w_{rs} | r \\rangle \\!",
"\\langle s | \\bigr ) | A_0 \\rangle \\!",
"\\langle A_0 | { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\sum _{r,s} w_{rs} | r {A_r} \\rangle \\!",
"\\langle s {A_s} | \\ .\\qquad \\mathrm {(9)}$ Copying: $ \\bigl (\\sum _{r,s} w_{rs} | r {A_r} \\rangle \\!",
"\\langle s {A_s} |\\bigr ) | D_0 \\rangle \\!",
"\\langle D_0 | { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {A}}{\\cal D}}}{\\Longrightarrow } } \\sum _{r,s} w_{rs} | r {A_r}{D_r} \\rangle \\!",
"\\langle s {A_s}{D_s} | \\qquad \\mathrm {(10)}$ leads to a state that exhibits quantum correlations between all three systems.",
"Reversal of the evolution: $ \\sum _{r,s} w_{rs} | r {A_r}{D_r} \\rangle \\!",
"\\langle s {A_s}{D_s} | { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}^\\dagger }{\\Longrightarrow } } | A_0 \\rangle \\!",
"\\langle A_0 | \\sum _{r,s} w_{rs} | r {D_r} \\rangle \\!",
"\\langle s {D_s} | \\qquad \\mathrm {(11)}$ that acts purely on the ${\\mathcal {S}}{\\mathcal {A}}$ pair restores only the pre-measurement state of the apparatus, but not the state of the system, $\\varrho ^{{\\mathcal {S}}} = \\sum _s w_{ss} | s \\rangle \\!",
"\\langle s | =\\mathrm {Tr}_{\\cal D} (\\sum _{r,s} w_{rs} | r {D_r} \\rangle \\!",
"\\langle s {D_s} |) \\ ,\\qquad \\mathrm {(12)}$ as the reduced density matrix of the system is now—unlike the pre-measurement $\\rho ^{\\mathcal {S}}$ , Eq.",
"(8)—diagonal in the measurement basis $| s \\rangle $ .",
"Thus, in contrast to the classical case, acquiring and communicating information about quantum systems matters: Reversibility of the global dynamics is not enough.",
"Presence of a copy of the information (that did not matter in the classical case) precludes the possibility of implementing local reversals.",
"The information-theoretic price—the extent of irreversibility—can be quantified by $ \\Delta H$ , the difference in entropy between the pre-measurement and post-measurement density matrices; $ \\Delta H = -\\sum _s w_{ss}\\lg w_{ss} + \\mathrm {Tr}\\rho _{\\mathcal {S}}\\lg \\rho _{\\mathcal {S}}= H( \\varrho _{\\mathcal {S}}) - H(\\rho _{{\\mathcal {S}}}) \\ .",
"\\qquad \\mathrm {(13)}$ We shall now show that this entropy increase caused by copying coincides with the quantum discord [15], [17], [16] in the correlated post-measurement state $\\sum _{r,s} w_{rs} | r {A_r} \\rangle \\!",
"\\langle s {A_s} |$ of the system and the apparatus.",
"This suggests that vanishing of discord may be a condition for the reversibility undisturbed by copying."
],
[
"Introducing quantum discord",
"Discord is the difference between the mutual information defined by the symmetric equation that involves von Neumann entropies of the two systems separately and jointly: $ I({\\mathcal {S}}:{\\mathcal {A}}) = H_{\\mathcal {S}}+ H _{\\mathcal {A}}- H_{{\\mathcal {S}}{\\mathcal {A}}} \\ , \\qquad \\mathrm {(14)}$ where $H_X=-\\mathrm {Tr}\\rho _X \\lg \\rho _X$ , and the asymmetric definition of mutual information $J({\\mathcal {S}};{\\mathcal {A}})_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace } }$ .",
"The asymmetric version of mutual information obtains from the joint entropy when it is expressed in terms of the conditional entropy: $H_{{{\\mathcal {S}}{\\mathcal {A}}}|{\\lbrace | A_k \\rangle \\rbrace }} = H_{{\\mathcal {S}}|{\\mathcal {A}}{\\lbrace | A_k \\rangle \\rbrace }}+ H_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }} \\ , \\qquad \\mathrm {(15)}$ where we have assumed that the measurements were performed on ${\\mathcal {A}}$ in the basis $\\lbrace | A_k \\rangle \\rbrace $ .",
"Thus, $H_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }}$ is the entropy computed using probabilities of states $\\lbrace | A_k \\rangle \\rbrace $ , and $H_{{\\mathcal {S}}|{\\mathcal {A}}{\\lbrace | A_k \\rangle \\rbrace }}$ is the conditional entropy one still has after the outcomes of measurement on ${\\mathcal {A}}$ in the basis $\\lbrace | A_k \\rangle \\rbrace $ are known.",
"In the classical setting, when Shannon entropies are computed from classical probabilities, analogous two expressions for the joint entropy coincide [18].",
"However, in the quantum setting, possible post-measurement states—hence, conditional information—have to be defined with respect to the basis set characterizing the measurement that is carried out on one of the two systems (here ${\\mathcal {A}}$ ) in order to gain partial information about the other (here ${\\mathcal {S}}$ ).",
"Using this basis-dependent joint entropy $H_{{{\\mathcal {S}}{\\mathcal {A}}}|{\\lbrace | A_k \\rangle \\rbrace }}$ in Eq.",
"(14) instead of $H_{{\\mathcal {S}}{\\mathcal {A}}} $ one gets an asymmetric expression for mutual information: $ J({\\mathcal {S}};{\\mathcal {A}})_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }}=H_{\\mathcal {S}}+ H_{\\mathcal {A}}- (H_{{\\mathcal {S}}|{{\\mathcal {A}}\\lbrace | A_k \\rangle \\rbrace }}+ H_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }}) \\ .",
"\\qquad \\mathrm {(16)}$ Discord is the difference between the symmetric and asymmetric formulae for mutual informationThere are subtleties in the definition of the discord.",
"Definition given here is the so-called thermal discord or one-way deficit.",
"It differs from the “original” discord defined in [15], [17], [16].",
"A brief discussion in the context of Maxwell's demon can be found in [19].",
"More extensive discussions of discord and related measures are also available [20], [21].",
"We note that appearance of discord in the correlated ${\\mathcal {S}}{\\mathcal {A}}$ state can be traced [22] to the presence of quantum coherence in the states of ${\\mathcal {S}}$ .",
": $\\delta ({\\mathcal {S}}:{\\mathcal {A}})_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }}=I({\\mathcal {S}}:{\\mathcal {A}})-J({\\mathcal {S}};{\\mathcal {A}})_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }} \\ , \\qquad \\mathrm {(17a)}$ or; $\\delta ({\\mathcal {S}}:{\\mathcal {A}})_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }}=(H_{{\\mathcal {S}}|{{\\mathcal {A}}\\lbrace | A_k \\rangle \\rbrace }} + H_{{\\mathcal {A}}|{\\lbrace | A_k \\rangle \\rbrace }})-H_{{\\mathcal {S}}{\\mathcal {A}}} \\ .",
"\\qquad \\mathrm {(17b)}$ When the two systems are classical (so that their states can be completely described by probabilities) the two definitions of the mutual information coincide, and quantum discord disappears—it is identically equal to zero.",
"In the quantum domain probabilities usually do not suffice, and the two expressions for the mutual information differ.",
"In the case we have considered above the system was in a mixed state, but the initial state of the apparatus was pure, and the measurement that correlated ${\\mathcal {S}}$ with ${\\mathcal {A}}$ was unitary, so that $H_{{\\mathcal {S}}{\\mathcal {A}}}=H(\\rho _{\\mathcal {S}})$ .",
"Moreover, $H_{{\\mathcal {S}}|{{\\mathcal {A}}\\lbrace | A_s \\rangle \\rbrace }}=0$ (as a measurement of ${\\mathcal {A}}$ with the result $| A_k \\rangle $ reveals the corresponding pure states of ${\\mathcal {S}}$ ) and $H_{{\\mathcal {A}}|{\\lbrace | A_s \\rangle \\rbrace }}=H(\\varrho _{\\mathcal {S}})$ (as the entropy of ${\\mathcal {A}}$ is, after it correlates with ${\\mathcal {S}}$ computed from the probabilities $w_{ss}$ and equals $-\\sum _s w_{ss}\\lg w_{ss}$ ).",
"Consequently, the entropy increase $\\Delta H$ of Eq.",
"(13) is indeed equal to the discord in the post-measurement (but pre-copying) state of ${\\mathcal {S}}{\\mathcal {A}}$ ."
],
[
"Reversibility and quantum discord",
"We now consider a general case, where the pre-measurement density matrices $\\rho ^{\\mathcal {S}}$ , $\\rho ^{\\mathcal {A}}$ and the post-measurement $\\rho ^{{\\mathcal {S}}{\\mathcal {A}}}$ can all be mixed.",
"The evolution that leads to the measurement is still unitary ${{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}$ .",
"And we still assume that the apparatus should obtain and retain at least an imperfect record of the system.",
"That is, there should be states $\\lbrace \\rho _s^{\\mathcal {S}}\\rbrace $ of the systems that leave imprints on the state of the apparatus: $ \\rho _s^{\\mathcal {S}}\\rho _0^{\\mathcal {A}}\\ { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} \\ .",
"\\qquad \\mathrm {(18)}$ An initial mixture of $\\lbrace \\rho _s^{\\mathcal {S}}\\rbrace $ will evolve, by linearity, into the corresponding mixture of the outcomes.",
"$ \\sum _s p_s \\rho _s^{\\mathcal {S}}\\rho _0^{\\mathcal {A}}\\ { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}}{\\Longrightarrow } } \\ \\sum _s p_s \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} = \\rho ^{{\\mathcal {S}}{\\mathcal {A}}} \\ .",
"\\qquad \\mathrm {(19)}$ The correlation could be imperfect (i.e., one might only be able only infer some information about some of the $\\lbrace \\rho _s^{\\mathcal {S}}\\rbrace $ from ${\\mathcal {A}}$ ).",
"Copying involves interaction of ${\\mathcal {A}}$ and $\\cal D$ .",
"As before, we enquire under what circumstances transfer of information about ${\\mathcal {S}}$ via ${\\mathcal {A}}$ to $\\cal D$ does not preclude reversal, so that the evolution generated by ${\\mathfrak {U}}_{{\\mathcal {S}}{\\mathcal {A}}}^\\dagger $ restores the pre-measurement state of ${\\mathcal {S}}{\\mathcal {A}}$ in spite of the correlation with $\\cal D$ established by: $ \\sum _s p_s \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} | D_0 \\rangle \\!",
"\\langle D_0 | \\ { \\stackrel{{\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}}{\\Longrightarrow } } \\ \\sum _s p_s \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} | D_s \\rangle \\!",
"\\langle D_s | = \\rho ^{{\\mathcal {S}}{\\mathcal {A}}\\cal D} \\qquad \\mathrm {(20)}$ To allow for reversal the state of ${{\\mathcal {S}}{\\mathcal {A}}}$ must not be affected by the copying.",
"That is, $\\varrho ^{{\\mathcal {S}}{\\mathcal {A}}}=\\mathrm {Tr}_{\\cal D} \\rho ^{{\\mathcal {S}}{\\mathcal {A}}\\cal D}=\\rho ^{{\\mathcal {S}}{\\mathcal {A}}}\\ , \\qquad \\mathrm {(21)}$ where $\\rho ^{{\\mathcal {S}}{\\mathcal {A}}}$ and $\\varrho ^{{\\mathcal {S}}{\\mathcal {A}}}$ are the density matrices before and after the copying operation.",
"This is a density matrix version version of the “repeatability condition” (see [13], [14]): Copying can be repeated (since the “original” remains unchanged), and we shall see that this repeatability leads to similar consequences—to the orthogonality of the records that can be copied.",
"Unitarity of ${\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}$ is responsible for our next result.",
"Unitary evolutions preserve Hilbert-Schmidt norm.",
"Therefore, $\\sum _{r,s} p_r p_s \\mathrm {Tr}\\rho _r^{{\\mathcal {S}}{\\mathcal {A}}} \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} = \\sum _{r,s} p_r p_s \\mathrm {Tr}\\rho _r^{{\\mathcal {S}}{\\mathcal {A}}} \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}} |\\langle D_r | D_s \\rangle |^2 \\ .",
"\\qquad \\mathrm {(22)}$ The overlap of the copy states in $\\cal D$ is non-negative and bounded, $0 < |\\langle D_r | D_s \\rangle |^2 \\le 1$ .",
"Therefore, there are only two ways to satisfy this equality: Either $|\\langle D_r | D_s \\rangle |^2=1$ (i.e., there is no copy!",
"), or $p_r p_s \\mathrm {Tr}\\rho _r^{{\\mathcal {S}}{\\mathcal {A}}} \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}}=0 \\ .",
"\\qquad \\mathrm {(23)}$ For the non-trivial case when $p_r p_s>0$ and $r \\ne s$ this leads to; $\\mathrm {Tr}\\rho _r^{{\\mathcal {S}}{\\mathcal {A}}} \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}}=0 \\qquad \\mathrm {(24)}$ as a necessary condition to allow for copying that does not interfere with the possibility of the reversal.",
"Indeed, when (as we have assumed) copying evolution operator ${\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}$ involves only ${\\mathcal {A}}$ and $\\cal D$ , we can repeat the above reasoning starting with the reduced density matrix of ${\\mathcal {A}}$ alone and demanding that it is untouched by the copying operation: $\\varrho ^{{\\mathcal {A}}}=\\mathrm {Tr}_{{\\mathcal {S}}\\cal D} \\rho ^{{\\mathcal {S}}{\\mathcal {A}}\\cal D}=\\rho ^{{\\mathcal {A}}}\\ .",
"\\qquad \\mathrm {(25)}$ (Clearly, if copying were to affect density matrix of ${\\mathcal {A}}$ , it would affect also $\\rho ^{{\\mathcal {S}}{\\mathcal {A}}}$ , so Eq.",
"(21) cannot not be satisfied unless Eq.",
"(25) holds.)",
"In the end we will conclude that repeatability is not ruled out by retention of the copies of the outcomes providing that: $p_r p_s \\mathrm {Tr}\\rho _r^{{\\mathcal {A}}} \\rho _s^{{\\mathcal {A}}}=0 \\ .",
"\\qquad \\mathrm {(26)}$ For the non-trivial case when $p_r p_s>0$ this implies orthogonality of the records: $\\mathrm {Tr}\\rho _r^{{\\mathcal {A}}} \\rho _s^{{\\mathcal {A}}}=0 \\qquad \\mathrm {(27)}$ as a necessary condition to allow for copying of the information from ${\\mathcal {A}}$ that does not interfere with the possibility of the reversal.",
"To assure that copying will indeed leave $\\varrho ^{{\\mathcal {S}}{\\mathcal {A}}}$ unchanged, we need satisfy the same condition that selects pointer states [23], [24]: The unitary ${\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}$ that produces copies must commute with the pre-copying $\\varrho ^{{\\mathcal {S}}{\\mathcal {A}}}$ to leave it unaffected.",
"This will be the case when the Hamiltonian $\\bf H_{{\\mathcal {A}}\\cal D}$ that generates ${\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}$ commutes with the pointer observable of ${\\mathcal {A}}$ —with the apparatus observable that keeps the records of the state of the system.",
"This pointer observable will have in general degenerate eigenstates—eigenspaces that serve (within the apparatus Hilbert space) as a “one leg” of the support of the density matrices $\\rho _s^{{\\mathcal {S}}{\\mathcal {A}}}$ .",
"Orthogonality of the record states of ${\\mathcal {A}}$ implies zero “one way” discord in the basis corresponding to these pointer eigenspaces.",
"We note that there is an important difference between Eqs.",
"(23, 24) and Eqs.",
"(26, 27) we have derived.",
"They rely on different assumptions: Eqs (26, 27) are “local” – they focus on the content of the records in the apparatus alone, and demand distinguishabiility (orthogonality) of its states.",
"This focus is justified by the nature of the copying interaction—it involves only ${\\mathcal {A}}$ and $\\cal D$ , so only the records in ${\\mathcal {A}}$ are relevant.",
"By contrast, Eqs.",
"(23, 24) could be satisfied equally well by orthogonality of local state of ${\\mathcal {S}}$ alone or, indeed, of the global states of ${\\mathcal {S}}{\\mathcal {A}}$ .",
"In other words, when one can access the composite system ${\\mathcal {A}}{\\mathcal {S}}$ , the condition that allows for reversible copying can be satisfied by the global state even when it is not met by the record states of ${\\mathcal {A}}$ alone [14].",
"Our next goal is to consider effects of such more global copying operations."
],
[
"Knowing of the record but not the outcome",
"Immediately above, in Eqs.",
"(25-27), we have insisted that the orthogonality condition $\\mathrm {Tr}\\rho _r^{{\\mathcal {S}}{\\mathcal {A}}} \\rho _s^{{\\mathcal {S}}{\\mathcal {A}}}=0$ should be satisfied “in the apparatus”, that is, that the apparatus eigenspaces that correspond to the records should be orthogonal.",
"This insistence stemmed from the fact that the copying evolution ${\\mathfrak {U}}_{{\\mathcal {A}}\\cal D}$ coupled only to ${\\mathcal {A}}$ .",
"However, one can imagine a situation where ${\\mathfrak {U}}_{({\\mathcal {S}}{\\mathcal {A}})\\cal D}$ couples $\\cal D$ to a global observable of ${\\mathcal {S}}{\\mathcal {A}}$ .",
"In that case, one might be able to find out that ${\\mathcal {A}}$ “knows” the outcome – the state of ${\\mathcal {S}}$ – without actually finding out the outcome.",
"The simplest such example is afforded by a one qubit apparatus that measures a one qubit system.",
"The correlated—entangled—state of the two is then simply: $ | \\psi _{{\\mathcal {S}}{\\mathcal {A}}} \\rangle = a_\\uparrow | \\uparrow A_\\uparrow \\rangle + a_\\downarrow | \\downarrow A_\\downarrow \\rangle \\qquad \\mathrm {(28)}$ in obvious notation.",
"Agent $\\cal D$ can then detect presence of the correlations established when ${\\mathcal {S}}$ and ${\\mathcal {A}}$ interacted.",
"We now consider two operators that can confirm the existence of the correlation between ${\\mathcal {S}}$ and ${\\mathcal {A}}$ .",
"The first such operator, when measured, would establish whether the states of ${\\mathcal {S}}$ and ${\\mathcal {A}}$ are correlated in the basis (here $\\lbrace | \\uparrow \\rangle , | \\downarrow \\rangle \\rbrace $ ) in which the measurement was carried out: ${\\bf \\hat{A}} = y_\\uparrow | \\uparrow A_\\uparrow \\rangle \\!",
"\\langle \\uparrow A_\\uparrow | + y_\\downarrow | \\downarrow A_\\downarrow \\rangle \\!",
"\\langle \\downarrow A_\\downarrow | $ $ + n | \\uparrow A_\\downarrow \\rangle \\!",
"\\langle \\uparrow A_\\downarrow | + n^{\\prime } | \\downarrow A_\\uparrow \\rangle \\!",
"\\langle \\downarrow A_\\uparrow | \\ .",
"\\qquad \\mathrm {(29)}$ The detection of either of the $y$ eigenvalues would imply a successful measurement (while either of $n$ eigenvalues would signify error).",
"Moreover, when $y_\\uparrow = y_\\downarrow =y$ , such measurement would reveal consensus without betraying the actual outcome.",
"Thus, agent $\\cal D$ —friend of the observer—could confirm the success of the measurement, but the evolution that led to the measurement can be be still undone.",
"This is “relative reversibility”—the evolution that led to measurement can be at least in principle undone by an agent who can confirm that the measurement was successful providing he does this without finding out the outcome.",
"When $y_\\uparrow \\ne y_\\downarrow $ , the measurement by $\\cal D$ would correlate his state with the outcome, and the reversal would become impossible.",
"An alternative confirmation of a successful ${\\mathcal {S}}{\\mathcal {A}}$ measurement can be accomplished by detecting entanglement in $| \\psi _{{\\mathcal {S}}{\\mathcal {A}}} \\rangle $ .",
"Bell operator: ${\\bf \\hat{B}} = b^+_= | \\beta ^+_= \\rangle \\!",
"\\langle \\beta ^+_= | + b^-_= | \\beta ^-_= \\rangle \\!",
"\\langle \\beta ^-_= | +b^+_{\\ne } | \\beta ^+_{\\ne } \\rangle \\!",
"\\langle \\beta ^+_{\\ne } | + b^-_{\\ne } | \\beta ^-_{\\ne } \\rangle \\!",
"\\langle \\beta ^-_{\\ne } | \\qquad \\mathrm {(30)}$ can be used for this purpose.",
"Above, subscripts “=” and “$\\ne $ ” stand for “parallel” and “antiparallel”, and the Bell eigenstates are; $| \\beta ^\\pm _= \\rangle = | \\uparrow A_\\uparrow \\rangle \\pm | \\downarrow A_\\downarrow \\rangle \\ ; \\qquad \\mathrm {(31a)}$ $| \\beta ^\\pm _{\\ne } \\rangle = | \\uparrow A_\\downarrow \\rangle \\pm | \\downarrow A_\\uparrow \\rangle \\ .",
"\\qquad \\mathrm {(31b)}$ Detection of either $b^+_=$ or $b^-_=$ implies successful measurement.",
"However, unless $b^+_= = b^-_=$ , measurement will also reveal phases between the outcome states, and (unless $| \\psi _{{\\mathcal {S}}{\\mathcal {A}}} \\rangle $ happens to be one of the above Bell states) it will result in decoherence in the Bell basis (and, hence, prevent reversal).",
"It is interesting to note that when one imposes degeneracy that enables reversal on either $\\bf \\hat{A}$ or $\\bf \\hat{B}$ , eigenstates of these two operators coincide.",
"The resulting consensus operator is given by: ${\\bf \\hat{C}} = y (| \\uparrow A_\\uparrow \\rangle \\!",
"\\langle \\uparrow A_\\uparrow | + | \\downarrow A_\\downarrow \\rangle \\!",
"\\langle \\downarrow A_\\downarrow |)$ $ + n (| \\uparrow A_\\downarrow \\rangle \\!",
"\\langle \\uparrow A_\\downarrow | + | \\downarrow A_\\uparrow \\rangle \\!",
"\\langle \\downarrow A_\\uparrow |) \\qquad \\mathrm {(32)}$ Thus, by measuring $\\bf \\hat{C}$ one can confirm that ${\\mathcal {A}}$ “knows the outcome” without impairing the possibility of reversal."
],
[
"Discussion",
"Our results shed new light both on the relation between quantum and classical and on the role of information in measurements.",
"So far we have mainly emphasized their relevance for the distinction between quantum and classical physics.",
"To re-state briefly the main conclusion, retention of information about classical states has no bearing on the in principle ability to reverse classical evolution that leads to measurement, but it precludes reversing quantum measurements (with the exception of the quasiclassical case).",
"Thus, information plays a far more important role in quantum Universe than it used to play in classical physics.",
"This operational view of reversibility yields new insights: (i) In quantum physics irreversibility in course of measurements need not be blamed solely on decoherence, but is caused by observer's acquisition of the data about the system.",
"Observer who retains record of the outcome cannot restore the pre-measurement states of both the system ${\\mathcal {S}}$ and the apparatus ${\\mathcal {A}}$ .",
"So, from observer's point of view, while classical measurements can be undone, quantum measurements are fundamentally irreversible.",
"(ii) Acquisition of information results in decrease of the von Neumann entropy of the system.",
"Therefore, this aspect of irreversibility of measurements is not a consequence of the second law.",
"Yet, while observer can take advantage of this (apparent!)",
"violation of the second law, he cannot reverse measurement on his own.",
"(iii) However, observer's friend (who knows about the measurement, but not its outcome) can, in principle, induce such a reversal providing there is no copy of the record of the outcome left anywhere.",
"Our discussion calls for a re-consideration of the nature and origin of the initial conditions in quantum physics.",
"Distinction between the laws that dictate evolution of the state of a system and initial conditions the define its starting point dates back to Newton [8].",
"This clean separation is challenged by quantum measurements.",
"Seen from the inside, by the observer, measurement re-sets initial conditions.",
"Acquisition of information simultaneously redefines the state of the observer and observer's branch of the universal state vector.",
"From then on, observer will exist within the Universe he helped define (see Fig.",
"1).",
"On the other hand, observer's friend will—for as long as he does not find out what the observer found out—live in a Universe where the initial condition is the pre-measurement state with a coherent superposition of all the potential outcomes.",
"Familiar “paradox” of Wigner's friend offers an interesting setting for this discussion.",
"Wigner speculated [9] (following to some extent von Neumann [1]) that “collapse of the wavepacket” may be ultimately precipitated by consciousness.",
"The obvious question is, of course, “how conscious should the observer be”.",
"The answer suggested by our discussion is that—if the evidence of collapse is the irreversibility of the evolution that caused it—retention of the information suffices.",
"Thus, there is no need for “consciousness” (whatever that means): Record of the outcome is enough.",
"On the other hand, observer conscious of the outcome certainly retains its record, so being conscious of the result suffices to preclude the reversal—to make the “collapse” irreversible.",
"Quantum Darwinism [11] traces emergence of the objective classical reality to the proliferation of information throughout the environment.",
"Our discussion of the consequences of retention of information for reversibility is clearly relevant in this context, although its detailed study is beyond the scope of this paper.",
"This research was supported by the Department of Energy via LDRD program in Los Alamos, and, in part, by the Foundational Questions Institute grant “Physics of What Happens”."
]
] | 1808.08598 |
[
[
"Vector Approximate Message Passing Algorithm for Structured Perturbed\n Sensing Matrix"
],
[
"Abstract In this paper, we consider a general form of noisy compressive sensing (CS) where the sensing matrix is not precisely known.",
"Such cases exist when there are imperfections or unknown calibration parameters during the measurement process.",
"Particularly, the sensing matrix may have some structure, which makes the perturbation follow a fixed pattern.",
"While previous work has focused on extending the approximate message passing (AMP) and LASSO algorithm to deal with the independent and identically distributed (i.i.d.)",
"perturbation, we propose the robust variant vector approximate message passing (VAMP) algorithm with the perturbation being structured, based on the recent VAMP algorithm.",
"The performance of the robust version of VAMP is demonstrated numerically."
],
[
"Introduction",
"Compressed Sensing (CS) aims to reconstruct an $N$ -dimensional sparse signal from $M$ underdetermined linear measurements ${\\mathbf {y}}={\\mathbf {A}}{\\mathbf {x}}+{\\mathbf {w}}$ , where $M<N$ and $\\mathbf {w}$ is additive noise.",
"It has been shown that in the absence of noise, perfect reconstruction is possible given that the signal is exactly $K$ sparse and the measurement matrix satisfies certain properties (e.g., restricted isometry, spark, null space).",
"In practical applications, the measurement matrix $\\mathbf {A}$ may not be known exactly due to, e.g., model mismatch, imperfect calibration and imperfections in the signal acquisition hardware.",
"Consequently, several works have studied the recovery algorithm and performance bounds for the general signal with independent and identically distributed (i.i.d.)",
"perturbation [1].",
"In addition, the measurement matrix uncertainty in quantized settings has also been studied [2].",
"For the sparse signal recovery under i.i.d.",
"perturbation, the recovery performance of algorithms such as basis pursuit (BP) and orthogonal matching pursuit (OMP) algorithm are analyzed [3], [4].",
"While the above works study the effect of perturbation on established algorithms, there also exist some algorithms which take the measurement matrix uncertainty into account.",
"In [5], the Sparsity-cognizant Total Least Squares (S-TLS) approach is developed.",
"A modified version of the Dantzig selector dealing with the matrix uncertainty is proposed in [6].",
"To taking the structure of perturbation into account, a weighted S-TLS (WS-TLS) is proposed, and numerical results demonstrate that WS-TLS performs significantly better than S-TLS [5].",
"Approximate message passing (AMP) algorithm is a popular method for performing high dimensional inference, due to its low computational complexity and good performance [7].",
"In [8], a generalized AMP (GAMP) algorithm is proposed to cope with the generalized linear model [8].",
"Since then, AMP and GAMP algorithm has been applied in various signal processing applications, such as data detection and channel estimation [9].",
"Recently, orthogonal AMP [10] and vector AMP (VAMP) algorithms [11] are proposed, which can deal with a larger ensemble of measurement matrix set, compared to the AMP algorithm.",
"Given that some statistical parameters are unknown, expectation maximization approximate message passing (EM-AMP) and expectation maximization vector approximate message passing (EM-VAMP) are proposed to jointly recover the unknown signal and learn the statistical parameters [12], [13].",
"In [14], an AMP algorithm is extended to deal with the sparse signal recovery problem under matrix uncertainty.",
"The perturbation is treated as an additive white Gaussian noise, and the matrix uncertainty GAMP (MU-GAMP) is proposed.",
"Provided that the perturbation has some additional structure, an alternating MU-GAMP is proposed to jointly estimate the measurement matrix and signal, in contrast with this paper where the structured perturbation is also treated as the random variables.",
"In [15], the robust approximate message passing algorithm is proposed, and the mean square error of the Bayes-optimal reconstruction of sparse signals under matrix uncertainty is calculated via replica method.",
"In this paper, we consider a kind of general structured perturbation.",
"This structure arises because the sensing matrix has known structure such that its elements can not be chosen arbitrarily.",
"For example, in signal and communication problems, the convolving operation between channel and data can be reformulated as a linear regression problem.",
"For the zero boundary conditions, the sensing matrix has a Toeplitz structure, while a circulant structure appears for periodic boundary conditions [16].",
"As a result, the structure of model uncertainty has to be taken into account to improve the reconstruction performance.",
"Since the equivalent noise (perturbation plus additive noise) is coloured and related to the unknown signal, in contrast with the white Gaussian noise in [14], conventional AMP and VAMP algorithm can not be applied in this scenario.",
"Here we propose to approximate the likelihood function in each iteration, and numerical results demonstrate the effectiveness of the proposed method."
],
[
"Algorithm",
"The mathematical model we consider in this paper is [17] ${\\mathbf {y}}=\\left({\\mathbf {A}}+{\\sum _{i=1}^qe_i{\\mathbf {E}}_i}\\right){\\mathbf {x}}+{\\mathbf {w}}.$ where we assume that ${\\mathbf {y}}\\in {\\mathbb {R}}^M$ , ${\\mathbf {A}}\\in {\\mathbb {R}}^{M\\times N}$ denotes the random known sensing matrix and $\\Vert {\\mathbf {A}}\\Vert _{\\rm F}^2=N$ , where $\\Vert \\cdot \\Vert _{\\rm F}$ denotes the Frobenius norm, ${\\mathbf {E}}_i\\in {\\mathbb {R}}^{M\\times N}$ denotes the known structure of the perturbation, $e_i,~i=1,\\cdots ,q$ are i.i.d.",
"random variables and satisfying $e_i\\sim {\\mathcal {N}}(e_i;0,\\gamma _e^{-1})$ Here ${\\mathcal {N}}(e_i;0,\\gamma _e^{-1})$ means that $e_i$ follows Gaussian distribution with mean zero and variance $\\gamma _e^{-1}$ .",
"Sometimes we use ${\\mathcal {N}}(0,\\gamma _e^{-1})$ instead when the random variable is clear., ${\\mathbf {x}}\\in {\\mathbb {R}}^N$ .",
"The prior distribution of signal $\\mathbf {x}$ follows ${\\mathbf {x}}\\sim \\prod \\limits _{i=1}^{N}p(x_i)$ , where $p(x_i)$ is a sparsity-inducing prior, ${\\mathbf {w}}\\sim {\\mathcal {N}}({\\mathbf {0}},\\gamma _w^{-1}{\\mathbf {I}}_M)$ .",
"Note that [14] and [17] study model (REF ).",
"However, [14] treats $\\lbrace a_i\\rbrace _{i=1}^{q}$ as unknown deterministic parameters in contrast to [17] as random parameters, which correspond to two classical ways to model measurement uncertainty.",
"As shown in [17], the strategy of modeling measurement uncertainty as random parameters yields accurate results.",
"The drawback is that one needs to estimate the statistics of the random parameters.",
"Compared to [17] which assumes an unknown deterministic vector $\\mathbf {x}$ , this paper enforces prior distribution of $\\mathbf {x}$ .",
"The perturbation model in (REF ) is very general and we list some specific structure of perturbation as follows: i.i.d perturbation, where the perturbation takes the form $\\sum _{i=1}^M{\\sum _{j=1}^Ne_{ij}{\\mathbf {E}}_{ij}}$ , $e_{ij}\\sim {\\mathcal {N}}(0,\\gamma _e^{-1})$ and ${\\mathbf {E}}_{ij}$ is a all zero matrix except that the $(i,j)$ -th element is one.",
"Matrix-restricted structured perturbation where the perturbation takes the form ${\\mathbf {D}}{\\mathbf {E}}{\\mathbf {C}}$ with $\\mathbf {D}$ and $\\mathbf {C}$ being known matrices.",
"This structure can model the scenario in which the coefficients of the sensing matrix have unequal uncertainties, as shown in [17].",
"Circulant structure perturbation.",
"Here the $N\\times N$ circulant matrix ${\\mathbf {A}}$ is of the form ${\\mathbf {A}}=\\left[\\begin{array}{cccc}a_1 & a_2 & \\cdots & a_N \\\\a_N & a_1 & \\cdots & a_{N-1} \\\\\\vdots & \\vdots & \\vdots & \\vdots \\\\a_2 & a_3 & \\cdots & a_1 \\\\\\end{array}\\right].$ As a result, the perturbation also takes this form [17].",
"By defining ${\\mathbf {z}}={\\sum _{i=1}^qe_i{\\mathbf {E}}_i}{\\mathbf {x}}+{\\mathbf {w}}$ , model (REF ) is equivalent to ${\\mathbf {y}}={\\mathbf {A}}{\\mathbf {x}}+{\\mathbf {z}},$ where ${\\mathbf {z}}\\sim {\\mathcal {N}}({\\mathbf {0}},{\\sum _{i=1}^q}\\gamma _e^{-1}{\\mathbf {E}}_i{\\mathbf {x}}{\\mathbf {x}}^{\\rm T}{\\mathbf {E}}_i^{\\rm T}+\\gamma _w^{-1}{\\mathbf {I}}_M)\\triangleq {\\mathcal {N}}({\\mathbf {0}},{\\Gamma }({\\mathbf {x}}))$ .",
"Figure: The factor graph used for the derivation of the robust VAMP algorithm.",
"The circles represent variable nodes and the squares represent factor nodes from ().In the following text, we introduce the VAMP briefly For the detailed derivation of VAMP utilizing expectation propagation, please refer to [11].. We start with the joint probability density function of $\\mathbf {x}$ and $\\mathbf {y}$ as $p({\\mathbf {y}},{\\mathbf {x}})=p({\\mathbf {x}})p({\\mathbf {y}}|{\\mathbf {x}})=p({\\mathbf {x}}){\\mathcal {N}}({\\mathbf {y}};{\\mathbf {A}}{\\mathbf {x}},{\\Gamma }({\\mathbf {x}})).$ By splitting x into two identical variables ${\\mathbf {x}}_1$ and ${\\mathbf {x}}_2$ , we obtain an equivalent factorization $p({\\mathbf {y}},{\\mathbf {x}}_1,{\\mathbf {x}}_2)=p({\\mathbf {x}}_1)\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2){\\mathcal {N}}({\\mathbf {y}};{\\mathbf {A}}{\\mathbf {x}}_2,{\\Gamma }({\\mathbf {x}}_2)).$ The factor graph corresponding to the above factorization (REF ) is presented in Fig.",
"REF .",
"We then pass messages on this factor graph.",
"We initialize the message of the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ to the variable node ${\\mathbf {x}}_1$ with $\\mu _{\\delta \\rightarrow {\\mathbf {x}}_1}({\\mathbf {x}}_1)={\\mathcal {N}}({\\mathbf {x}}_1;{\\mathbf {r}}_{1k},\\gamma _{1k}^{-1}{\\mathbf {I}}_N)$ where $k=0$ .",
"Combing the factor node $p({\\mathbf {x}}_1)$ , the sum product (SP) belief on variable node ${\\mathbf {x}}_1$ is $b_{\\rm sp}({\\mathbf {x}}_1)\\propto p({\\mathbf {x}}_1){\\mathcal {N}}({\\mathbf {x}}_1;{\\mathbf {r}}_{1k},\\gamma _{1k}^{-1}{\\mathbf {I}}_N).$ where $\\propto $ means proportional to.",
"We calculate the posterior means and variances as $&\\hat{\\mathbf {x}}_{1k}={\\rm E}[{\\mathbf {x}}_1|b_{\\rm sp}({\\mathbf {x}}_1)],\\\\&\\eta _{1k}^{-1}=<{\\rm diag}({\\rm Cov}[{\\mathbf {x}}_1|b_{\\rm sp}({\\mathbf {x}}_1)])>,$ where $<{\\mathbf {x}}>=(\\sum \\limits _{i=1}^Nx_i)/N$ , ${\\rm Cov}[\\cdot |b_{\\rm sp}({\\mathbf {x}}_1)]$ is the covariance matrix with respect to the belief estimate $b_{\\rm sp}({\\mathbf {x}}_1)$ and ${\\rm diag}({\\mathbf {A}})$ returns a column vector whose elements are the main diagonal of ${\\mathbf {A}}$ .",
"Exploiting the expectation propagation, the above belief $b_{\\rm sp}({\\mathbf {x}}_1)$ is approximated as a Gaussian distribution $b_{\\rm app}({\\mathbf {x}}_1)$ given by $b_{\\rm app}({\\mathbf {x}}_1)={\\mathcal {N}}({\\mathbf {x}}_1;\\hat{\\mathbf {x}}_{1k},\\eta _{1k}^{-1}{\\mathbf {I}}_N).$ Then we calculate the message from the variable node ${\\mathbf {x}}_1$ to the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ , which is the ratio of the most recent approximate belief $b_{\\rm app}({\\mathbf {x}}_1)$ to the most recent message from $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ to ${\\mathbf {x}}_1$ , i.e., $&\\mu _{{\\mathbf {x}}_1\\rightarrow \\delta }={\\mathcal {N}}({\\mathbf {x}}_1;{\\mathbf {r}}_{2k},\\gamma _{2k}{\\mathbf {I}}_N)\\\\&\\propto {\\mathcal {N}}({\\mathbf {x}}_1;\\hat{\\mathbf {x}}_{1k},\\eta _{1k}^{-1}{\\mathbf {I}}_N)/{\\mathcal {N}}({\\mathbf {x}}_1;{\\mathbf {r}}_{1k},\\gamma _{1k}^{-1}{\\mathbf {I}}_N),$ where ${\\mathbf {r}}_{2k}$ and $\\gamma _{2k}$ are calculated according to line 5 in Algorithm 1.",
"For the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ , the message from the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ to the variable node ${\\mathbf {x}}_2$ can be calculated directly as $\\mu _{\\delta \\rightarrow {\\mathbf {x}}_2}({\\mathbf {x}}_2)={\\mathcal {N}}({\\mathbf {x}}_2;{\\mathbf {r}}_{2k},\\gamma _{2k}{\\mathbf {I}}_N)$ which can be viewed as the prior of the variable node ${\\mathbf {x}}_2$ .",
"For the rightmost factor node ${\\mathcal {N}}({\\mathbf {y}};{\\mathbf {A}}{\\mathbf {x}}_2;{\\Gamma }({\\mathbf {x}}_2))$ , its covariance matrix depends on the unknown $\\mathbf {x}$ .",
"As a result, we approximate ${\\Gamma }({\\mathbf {x}}_2)$ as $&{\\Gamma }({\\mathbf {x}}_2)\\approx {\\rm E}_{{\\mathbf {x}}_2\\sim {\\mathcal {N}}({\\mathbf {r}}_{2k},\\gamma _{2k}^{-1}{\\mathbf {I}})}\\left[{\\Gamma }({\\mathbf {x}}_{2k})\\right]\\\\&={\\sum _{i=1}^q}\\gamma _a^{-1}{\\mathbf {A}}_i({\\mathbf {r}}_{2k}{\\mathbf {r}}_{2k}^{\\rm T}+\\gamma _{2k}^{-1}{\\mathbf {I}}){\\mathbf {A}}_i^{\\rm T}+\\gamma _w^{-1}{\\mathbf {I}}_m\\triangleq {\\Gamma }_{2k}.$ As a consequence, we obtain an approximate model with the likelihood ${\\mathcal {N}}({\\mathbf {y}}_{2k};{\\mathbf {A}}_{2k}{\\mathbf {x}}_2; \\gamma _{w,2k}^{-1}{\\mathbf {I}}_M)$ , where ${\\mathbf {y}}_{2k} = \\gamma _{w,2k}^{-\\frac{1}{2}}{\\Gamma }_{2k}^{-\\frac{1}{2}}{\\mathbf {y}},\\\\{\\mathbf {A}}_{2k} = \\gamma _{w,2k}^{-\\frac{1}{2}}{\\Gamma }_{2k}^{-\\frac{1}{2}}{\\mathbf {A}},$ where $\\gamma _{w,2k}^{-\\frac{1}{2}}$ is to ensure $\\Vert {\\mathbf {A}}_{2k}\\Vert _{\\rm F}^2=N$ and $\\gamma _{w,2k}=\\Vert {\\Gamma }_{2k}^{-\\frac{1}{2}}{\\mathbf {A}}\\Vert _{\\rm F}^2/N$ .",
"With such an approximation, the SP belief on variable ${\\mathbf {x}}_2$ is $b_{\\rm sp}({\\mathbf {x}}_2)\\propto {\\mathcal {N}}({\\mathbf {y}}_{2k};{\\mathbf {A}}_{2k}{\\mathbf {x}}_2; \\gamma _{w,2k}^{-1}{\\mathbf {I}}_M){\\mathcal {N}}({\\mathbf {x}}_2,{\\mathbf {r}}_{2k},\\gamma _{2k}{\\mathbf {I}}_N).$ Utilizing the expectation propagation, the SP belief $b_{\\rm sp}({\\mathbf {x}}_2)$ on variable ${\\mathbf {x}}_2$ can be further approximated as $b_{\\rm app}({\\mathbf {x}}_2)={\\mathcal {N}}({\\mathbf {x}}_2;\\hat{\\mathbf {x}}_{2k},\\eta _{2k}^{-1}{\\mathbf {I}}_N),$ where $&\\hat{\\mathbf {x}}_{2k}=(\\gamma _{w,2k}{\\mathbf {A}}_{2k}^{\\rm T}{\\mathbf {A}}_{2k}+\\gamma _{2k}{\\mathbf {I}})^{-1}(\\gamma _{w,2k}{\\mathbf {A}}_{2k}^{\\rm T}{\\mathbf {y}}_{2k}+\\gamma _{2k}{\\mathbf {r}}_{2k}),\\\\&\\eta _{2k}^{-1}=\\frac{1}{N}{\\rm Tr}\\left[(\\gamma _{w,2k}{\\mathbf {A}}_{2k}^{\\rm T}{\\mathbf {A}}_{2k}+\\gamma _{2k}{\\mathbf {I}})^{-1}\\right].$ We then obtain the message from the variable node ${\\mathbf {x}}_2$ to the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ with $&\\mu _{{\\mathbf {x}}_2\\rightarrow \\delta }({\\mathbf {x}}_2)\\propto b_{\\rm app}({\\mathbf {x}}_2)/{\\mathcal {N}}({\\mathbf {x}}_2,{\\mathbf {r}}_{2k},\\gamma _{2k}{\\mathbf {I}}_N)\\\\=&{\\mathcal {N}}({\\mathbf {x}}_2;{\\mathbf {r}}_{1,k+1},\\gamma _{1,k+1}^{-1}{\\mathbf {I}}_n),$ where ${\\mathbf {r}}_{1,k+1}$ and $\\gamma _{1,k+1}^{-1}$ are given in line 10 in Algorithm 1.",
"Similarly, we calculate the message from the variable node ${\\mathbf {x}}_2$ to the factor node $\\delta ({\\mathbf {x}}_1-{\\mathbf {x}}_2)$ as $&\\mu _{\\delta \\rightarrow {\\mathbf {x}}_1}({\\mathbf {x}}_1)=\\mu _{{\\mathbf {x}}_2\\rightarrow \\delta }({\\mathbf {x}}_2),$ which closes the loop of the proposed VAMP algorithm and is shown in Algorithm 1.",
"[h] Vector AMP in perturbed setting [1] Initialize ${\\mathbf {r}}_{10}$ and ${\\gamma }_{10}\\ge 0$ and set the maximum number of iterations $K_{\\rm it}$ ; $k=0,1,\\cdots ,K_{\\rm it}$ // Denoising Calculate $\\hat{\\mathbf {x}}_{1k}$ and ${\\eta }_{1k}$ according to ().",
"${\\gamma }_{2k}={\\eta }_{1k}-\\gamma _{1k}$ , ${\\mathbf {r}}_{2k}=({\\eta }_{1k}\\hat{\\mathbf {x}}_{1k}-{\\gamma }_{1k}{\\mathbf {r}}_{1k})/{\\gamma }_{2k}$ // Whitening Approximate ${\\Gamma }({\\mathbf {x}}_2)$ with ${\\Gamma }_{2k}$ and obtain the equivalent ${\\mathbf {y}}_{2k}$ (REF ), ${\\mathbf {A}}_{2k}$ () and $\\gamma _{w,2k}$ .",
"// LMMSE estimation Calculate $\\hat{\\mathbf {x}}_{2k}$ and ${\\eta }_{2k}$ according to (REF ).",
"${\\gamma }_{1,k+1}={\\eta }_{2k}-\\gamma _{2k}$ , ${\\mathbf {r}}_{1,k+1}=({\\eta }_{2k}\\hat{\\mathbf {x}}_{2k}-{\\gamma }_{2k}{\\mathbf {r}}_{2k})/{\\gamma }_{1,k+1}$ Return $\\hat{\\mathbf {x}}_{1K_{\\rm it}}$ .",
"Now we discuss the computation complexity of Algorithm 1.",
"For the VAMP presented in [11], the main computational burden lies in the SVD of the sensing matrix, which performs only once.",
"For Algorithm 1, the main additional computational burden lies in line 7, which involves in calculating the eigenvalue decomposition of ${\\Gamma }_{2k}$ and the singular value decomposition of ${\\mathbf {A}}_{2k}$ for each iteration.",
"For some cases, the computation complexity of Algorithm 1 is comparable to that of VAMP.",
"Given that the model is ${\\mathbf {y}}=({\\mathbf {A}}+{\\mathbf {E}}){\\mathbf {x}}+{\\mathbf {n}}$ and the elements of perturbation ${\\mathbf {E}}$ are i.i.d., where $E_{ij}\\sim {\\mathcal {N}}(0,\\gamma _e^{-1})$ , one can see that ${\\Gamma }({\\mathbf {x}})=(\\gamma _w^{-1}+\\gamma _e^{-1}\\Vert {\\mathbf {x}}\\Vert _2^2){\\mathbf {I}}_M$ and the whitening operation in line 7 is unnecessary."
],
[
"Numerical Results",
"In this section, numerical results are performed to verify the effectiveness of the proposed robust VAMP.",
"The performance of the following algorithms are evaluated: The AMP-oracle algorithm with precisely known sensing matrix The PI-AMP algorithm which does not take the perturbation into account.",
"The MU-GAMP algorithm presented in [14].",
"The VAMP-oracle algorithm with precisely known sensing matrix The VAMP-PI algorithm which ignores perturbation The VAMP-PC algorithm shown in Algorithm 1 which considers model perturbation.",
"In the numerical simulation, we assume a Bernoulli Gaussian prior, i.e., $p(x_i)=(1-\\rho )\\delta (x_i)+\\rho {\\mathcal {N}}(x_i,\\mu _x,\\sigma _x^2)$ , where $\\rho =0.2$ , $\\mu _x=0$ and $\\sigma _x^2=1$ .",
"For the first two numerical experiments, the elements of matrix $\\mathbf {A}$ are i.i.d.",
"drawn from Gaussian distribution.",
"We assume that each deterministic ${\\mathbf {E}}_i$ is also drawn from Gaussian distribution, and we set $M=0.5N$ and $q=N$ .",
"The normalized mean square error (NMSE) is defined as ${\\rm NMSE}(\\hat{\\mathbf {x}})=10\\log \\frac{\\Vert {\\mathbf {x}}-\\hat{\\mathbf {x}}\\Vert _2^2}{\\Vert {\\mathbf {x}}\\Vert _2^2}$ , where ${\\mathbf {x}}$ denotes the true value.",
"We also define ${\\rm SNR}_{\\rm w}\\triangleq 10\\log \\frac{\\Vert \\mathbf {Ax}\\Vert ^2}{\\Vert {\\mathbf {w}}\\Vert ^2}$ and ${\\rm SNR}_{\\rm e}\\triangleq 10\\log \\frac{\\Vert {\\sum _{i=1}^qe_i{\\mathbf {E}}_i{\\mathbf {x}}}\\Vert ^2}{\\Vert {\\mathbf {w}}\\Vert ^2}$ .",
"The maximum number of iterations is $K_{\\rm it}=60$ .",
"In the first numerical simulation, the NMSE versus iteration is presented.",
"We set $\\rm {SNR_w} = 30dB$ and $\\rm {SNR_e} = 20dB$ .",
"From Fig.",
"REF , one can see that the oracle VAMP and GAMP algorithm achieves the lowest NMSE, and the oracle VAMP achieves the fastest speed of convergence.",
"For unknown structured perturbation, PC-VAMP works better than MU-GAMP.",
"Figure: NMSE versus algorithm iteration in a single realizationIn the second simulation, all the parameters are the same as that in the first simulation and $\\rm {SNR_w} = 30dB$ .",
"In Fig.",
"REF , we see that there exists obvious performance gap between PI-VAMP algorithm and MU-GAMP given ${\\rm SNR}_e\\le 30{\\rm dB}$ .",
"Compared to the MU-GAMP algorithm, PC-VAMP algorithm works better.",
"When the perturbation is small such that ${\\rm SNR}_e\\ge 35{\\rm dB}$ , the performances of all the AMP and VAMP algorithms are similar.",
"Figure: Mean NMSE versus SNR e \\rm {SNR_e}.",
"The reported NMSE is averaged over 50 realizations.The last experiment investigates the performance of PC-VAMP algorithm for real image recovery.",
"We threshold the wavelet coefficients such that the sparsity is $\\rho =0.4133$ .",
"We set $\\mu _x=3.0\\times 10^{-4}$ , $\\sigma _x^2=1.7\\times 10^{-2}$ .",
"We use a $N\\times N$ circulant matrix (REF ), set $a_i=0.3^i,~i=0,\\cdots ,N-1$ and $N=64^2$ .",
"The perturbation also has the circulant structure.",
"We use a random matrix to compress the observations such that the measurement ratio is $0.9$ .",
"For this compressed observation model, we set $\\rm {SNR_w}=40dB$ and $\\rm {SNR_e}=20dB$ .",
"From Fig.",
"REF , it can be seen that PC-VAMP yields the best recovery results with the perturbation being unknown, and the PSNR is $25{\\rm dB}$ .",
"Figure: 64×6464\\times 64 image recovery results."
],
[
"Conclusion",
"In this paper, we propose a matrix-uncertainty extension of the VAMP algorithm, when some structured perturbation is added on the sensing matrix.",
"By iteratively approximating the original likelihood function with constant covariance matrix, we obtain a modified VAMP algorithm.",
"Numerical results demonstrate the effectiveness of the proposed algorithm."
]
] | 1808.08579 |
[
[
"Optimal Nonparametric Inference with Two-Scale Distributional Nearest\n Neighbors"
],
[
"Abstract The weighted nearest neighbors (WNN) estimator has been popularly used as a flexible and easy-to-implement nonparametric tool for mean regression estimation.",
"The bagging technique is an elegant way to form WNN estimators with weights automatically generated to the nearest neighbors; we name the resulting estimator as the distributional nearest neighbors (DNN) for easy reference.",
"Yet, there is a lack of distributional results for such estimator, limiting its application to statistical inference.",
"Moreover, when the mean regression function has higher-order smoothness, DNN does not achieve the optimal nonparametric convergence rate, mainly because of the bias issue.",
"In this work, we provide an in-depth technical analysis of the DNN, based on which we suggest a bias reduction approach for the DNN estimator by linearly combining two DNN estimators with different subsampling scales, resulting in the novel two-scale DNN (TDNN) estimator.",
"The two-scale DNN estimator has an equivalent representation of WNN with weights admitting explicit forms and some being negative.",
"We prove that, thanks to the use of negative weights, the two-scale DNN estimator enjoys the optimal nonparametric rate of convergence in estimating the regression function under the fourth-order smoothness condition.",
"We further go beyond estimation and establish that the DNN and two-scale DNN are both asymptotically normal as the subsampling scales and sample size diverge to infinity.",
"For the practical implementation, we also provide variance estimators and a distribution estimator using the jackknife and bootstrap techniques for the two-scale DNN.",
"These estimators can be exploited for constructing valid confidence intervals for nonparametric inference of the regression function.",
"The theoretical results and appealing finite-sample performance of the suggested two-scale DNN method are illustrated with several numerical examples."
],
[
"Introduction",
"In nearly all economic applications, economists aspire to infer causal relationships.",
"Have international trades really polarized domestic income inequality?",
"Will metro access deteriorate neighborhood housing values?",
"And are food stamps actually alleviating poverty for the poor?",
"The introduction of the potential outcomes framework, or the Rubin causal model [56], [48] has revolutionized the way how these economic questions are answered.",
"Beyond regressions, the potential outcomes framework perceives outcomes as in parallel universes, one with the event happened and another without.",
"The usual interest is thus the mean difference between the two conceptual outcomes, which becomes the classic idea of average treatment effects (ATE).",
"Traditionally for the computation of average treatment effects, the unit of analysis is usually the average treatment effects conditional on an individual's fixed feature vector.",
"This unit has been under the name of conditional average treatment effects (CATE) [51] or heterogeneous treatment effects (HTE) [39], [20].",
"Heterogeneous treatment effects are treatment effects at the individual level as opposed to the average treatment effects at the population level.",
"With the insights from the potential outcomes framework, some classical tools, such as instrumental variables estimation, are given new interpretations and many popular tools, such as difference-in-difference, matching, inverse probability weighting, discontinuity design, synthetic control, and program evaluation methods in panels, have been developed [40], [47].",
"Together with structure models, they become the weapons that researchers and practitioners employ to hunt for causality in economic observational studies.",
"Observational studies usually do not have the luxury of random treatment assignments as laboratory or social experiments.",
"To address this fundamental difficulty, the potential outcomes framework assumes the unconfoundedness condition [56].",
"In plain words, unconfoundedness basically implies that by conditioning on a set of observed features, treatment assignments can be perceived as random.",
"While this condition greatly simplifies theoretical cumbersomeness, practitioners usually find it hard to reach consensus on what control variables they should condition on.",
"This is a dilemma that has existed since the introduction of the potential outcomes framework.",
"One approach is confounder selection and various criterions have been proposed [63], [64], [35].",
"However, the increasing availability of high-dimensional data [29], [26], [27], [31] may offer us a new perspective.",
"It is more plausible for the unconfoundedness condition to hold if the conditioning set is allowed to include all the relevant information at hand, that is, to be high-dimensional.",
"In the language of directed acyclic graphs (DAG) [53], it means that all the backdoors confounding causality are blocked.",
"A short review about some recent developments about causal inference with high-dimensional data is given in Section 2.",
"However, these developments have focused exclusively on the average treatment effects.",
"The unit of analysis, individual level heterogeneous treatment effects, has received much less attention.",
"In this paper, we argue that in the big data era the concept of heterogeneous treatment effects has advantages over the concept of average treatment effects in three aspects.",
"First, there is an identification concern.",
"The identification of average treatment effects comes from the unconfoundedness condition and the overlap condition, or known as the common support condition.",
"The overlap condition implies that given any covariate value, this observation has the chance to be in the treatment group.",
"[21] has shown that while it can be more plausible for unconfoundedness to hold with high-dimensional covariates, it can be the opposite for the common support.",
"Second, heterogeneous treatment effects can help further explore the mechanism behind treatment effects.",
"Traditional average treatment effect estimations usually act like a black box and further analysis about the mechanism behind the treatment effects is often restrained [43], [23].",
"However, heterogeneous treatment effects can be conveniently incorporated into the mediation analysis framework [44], [61].",
"The mediation analysis aims to explore the mechanism behind treatment effects and has been popular in political science, epidemiology, and biomedical studies.",
"Third, heterogeneous treatment effects are actually the center of gravity in many modern causal inference applications, such as program evaluations, personalized medicine, and customized marketing [46], [33], [54].",
"To conclude, heterogeneous treatment effects can provide much richer information than average treatment effects and this information can be invaluable in a wide range of modern big data applications including economics, business, and healthcare.",
"However, related research about their estimation is still limited.",
"To our knowledge, [65] are the first to address the need of a useful estimator for heterogeneous treatment effects in the high-dimensional setting.",
"In their seminal paper, they establish the asymptotic theory for the random forests algorithm and creatively introduce this machine learning method into the estimation of heterogeneous treatment effects.",
"In Monte Carlo simulations, their method outperforms the classical nonparametric $k$ -NN estimator in biases and mean squared errors.",
"The intuition for their groundbreaking result is that the random forests algorithm can be perceived as a variant of nearest neighbor methods, but the algorithm is able to fully exploit data information and consequently assigns adaptive weights to nearest neighbors.",
"With the extra information from data, their method can achieve improved precision with high-dimensional covariates in finite samples.",
"However, further improvement is still possible.",
"While the random forests algorithm is extremely powerful, it still lies within the nearest neighbor framework and the nature of its finite sample bias does not improve fundamentally.",
"It is true that algorithms on data can help find weights to nearby points and balance the trade-off between the bias and variance for the estimated sample.",
"Nevertheless, the limits born with the theory are not relaxed.",
"Another difficulty of the random forests approach is its estimation of variances.",
"Since the derivation of the variance of random forests estimators is complicated, the practical efforts to estimate their variances are not trivial.",
"This difficulty can potentially sacrifice its working precision and add extra complications in real applications.",
"In this paper, we revisit and enhance the classical $k$ -NN estimator with a simple yet powerful algorithm, extend it to the distributional setting, and propose an estimator for heterogeneous treatment effects with relaxed theoretical constraints as well as alleviated operational difficulty.",
"Our recipe is to subsample the data and average the 1-nearest neighbor estimators from each subsample.",
"This turns out to be equivalent to assigning a monotone weight to the nearest neighbors in a distributional fashion.",
"We name the new estimator distributional nearest neighbors (DNN) and prove it to be asymptotically unbiased and normal if the subsampling scale diverges with sample size $n$ .",
"A nice feature about DNN estimators is that we can further reduce finite sample bias by combining DNN estimators with different subsampling scales.",
"This bias reduction turns out to improve the estimation performance of heterogeneous treatment effects to a new level.",
"In our Monte Carlo simulations, two-scale DNN estimators outperform the random forests approach.",
"Compared to the random forests approach, our two-scale DNN estimators are also implementation friendly.",
"First, the bootstrap method [24] can be directly used to estimate the variance of two-scale DNN estimators.",
"This feature of DNN significantly increases its working performance compared to plug-in estimations of variances.",
"Second, the algorithm for two-scale DNN is simple and scalable for big data.",
"Without sample splitting and random partitioning, the process of assigning two-scale DNN weights only consists of steps of calculating and sorting distances.",
"The rest of this paper is organized as follows.",
"Related literature is reviewed in Section 2.",
"Section 3 introduces the DNN procedure and investigates its asymptotic properties.",
"We formally suggest our two-scale DNN framework and establish the asymptotic properties of the new method in Section 4.",
"Section 5 presents several Monte Carlo simulation examples to demonstrate the advantages of DNN.",
"We provide an application of the two-scale DNN to a real-life data set to study the heterogeneity of treatment effects of smoking on children's birth weights across mothers' ages in Section 6.",
"Section 7 discusses some implications and extensions of our work.",
"The proofs of the main results are relegated to the Appendix."
],
[
"Related literature",
"For the estimation of average treatment effects in high-dimensional settings, [10], [11] first used Lasso as a dimensionality reduction tool to select pre-treatment variables that are important to the outcomes or treatment assignments and then conducted conventional instrumental variable estimation after selection.",
"[8] proposed an approximate residual balancing estimator to combine balancing and regression adjustment.",
"They added a penalty term to the covariate balancing process and then removed the remaining bias.",
"The above methods are based on linear parametric settings, while [18], [17] provided a powerful general semi-parametric framework where variable selection and parameter estimation can be separated to some degree.",
"Their method is named debiased/double machine learning.",
"[45] found that before inverse probability weighting, covariate balancing propensity score can help with the estimation.",
"Later, [28] proved this finding in theory and proposed an efficient and doubly robust estimator by generalizing the covariate balancing propensity score method.",
"For reviews, see, for example, [10], [19], [12], [7], [6], [52].",
"Traditionally, the estimation of heterogeneous treatment effects relies on a fully parametric setting.",
"In this case, the heterogeneous treatment effect is just a projection with fitted parameters.",
"The validity of this approach depends heavily on the specification and extrapolation.",
"A more flexible but crude method is subgroup analysis.",
"In subgroup analysis, one would first divide the sample into subsamples according to the values of some variables.",
"Then comparisons are made between average treatment effects estimated from these subsamples.",
"This approach relies on arbitrary splitting rules and risks missing key distributional information.",
"Another approach to estimating heterogeneous treatment effects is to perceive it as a nonparametric regression problem.",
"As a result, classical nonparametric methods such as kernel methods and $k$ -nearest neighbor methods are the first ones that have been considered in the analysis of heterogeneity [51], [3].",
"Compared to parametric methods and subgroup analysis, nonparametric methods are immune to model misspecification and can well reflect distributional changes.",
"However, classical nonparametric methods suffer from the curse of dimensionality.",
"The classical $k$ -nearest neighbor methods, as [65] put it, often work fine with a small number of covariates, but lose precision quickly as the number of covariates increases.",
"To improve performance with many covariates, modern machine learning algorithms have been proposed and adapted to enhance the performance of classical nonparametric methods.",
"In estimating heterogeneous treatment effects, boosting, multivariate adaptive regression splines, and Bayesian additive regression trees [32], [33], [54] have been proposed recently.",
"The issue with these machine learning methods is that while they are intuitively plausible algorithms, their asymptotic properties have not yet been fully established.",
"One encouraging and distinctive exception is the random forests approach [65].",
"Resonating with [5]'s idea of recursive partitioning, they creatively proved the asymptotic properties of the random forests algorithm [16] and introduced it for the estimation of heterogeneous treatment effects.",
"Our paper is deeply rooted in the classical $k$ -NN framework.",
"For classical and recent results, for instance, see [50], [34], [57], [14], [13].",
"$k$ -NN methods can be used for density estimation, entropy estimation, classification tasks, and regression problems.",
"As [65]'s random forest approach, our two-scale DNN framework is an enhancement of classical nonparametric regressions and we apply it for causal inference.",
"However, our algorithm can be applied to several other cases such as density estimation.",
"The $k$ -NN idea is also related to the matching literature in economics and statistics [1], [55].",
"Our work relates to the panel data analysis [42] although it seems less obvious at first sight.",
"If each subsample of size $s$ is treated as repeated observations for each individual, the joint analysis of all subsamples thus becomes a panel data problem.",
"The subsample size $s$ in this paper is then the counterpart of the time-series length $T$ in panel analysis.",
"Therefore this paper in spirit connects to the bias reduction literature in panels [36], [37], [4], [22], where the source of bias is the time-series length $T$ ."
],
[
"Model setting",
"For the estimation of heterogeneous treatment effects, we exploit the conventional potential outcomes framework.",
"While our results can be conveniently extended to the multi-valued treatment settings, we focus on the binary case without loss of generality.",
"Suppose we have observations on $(\\mbox{\\bf X}, W, Y)$ in which $Y$ is a scalar response, $\\mbox{\\bf X}$ is the pre-treatment feature vector with dimensionality $d$ , and $W$ is a binary treatment assignment indicator, with $W = 1$ treated and $W = 0$ untreated.",
"It is assumed that there is one potential outcome associated with each treatment assignment status, only $Y(0)$ and $Y(1)$ for the binary case, where $Y(0)$ and $Y(1)$ are two random variables corresponding to the outcome without and with treatment, respectively.",
"The dilemma is that they are not observed simultaneously.",
"The observed response $Y = Y(0) (1 - W) + Y(1) W$ .",
"Traditionally, the interest is on the (super-population) average treatment effect (ATE) $\\tau $ of $W$ on $Y$ , which is defined as $ \\tau = \\mathbb {E} \\, [Y(1) - Y(0)].$ Given a fixed feature vector $\\mbox{\\bf x}$ , the heterogeneous treatment effect (HTE) of $W$ on $Y$ at the point $\\mbox{\\bf x}$ is given by $\\tau (\\mbox{\\bf x}) = \\mathbb {E} \\, [Y(1) - Y(0) | \\mbox{\\bf X}= \\mbox{\\bf x}].$ The estimation and inference of HTE $\\tau (\\mbox{\\bf x})$ is our goal in this paper.",
"Ideally, if both $Y(1)$ and $Y(0)$ were observable, the problem would reduce to a classical nonparametric regression problem.",
"However, it is a luxury that we do not have in observational studies.",
"Instead we assume the unconfoundedness condition [56].",
"Condition 1 The treatment assignment is unconfounded in that it does not depend on the potential outcomes conditioning on $\\mbox{\\bf X}$ , that is, $ Y(0), Y(1) \\protect \\mathchoice{\\protect \\mathrel {\\displaystyle \\perp }\\copy 0\\hspace{0.0pt}\\hspace{2.22214pt}\\box 0}{}{}{}$ $\\textstyle \\perp $$\\textstyle \\perp $$\\scriptstyle \\perp $$\\scriptstyle \\perp $$\\scriptscriptstyle \\perp $$\\scriptscriptstyle \\perp $ W | X.",
"The unconfoundedness condition entails that treatment assignments can be regarded as random for observations with the same feature vector.",
"Under this condition, it holds that $\\tau (\\mbox{\\bf x}) = \\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 1] - \\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 0].$ Thus the estimation of $\\tau (\\mbox{\\bf x})$ can be decomposed into the estimation problems of $\\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 1]$ in the treated group and $\\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 0]$ in the control group, respectively, which are classical nonparametric regression problems.",
"In this paper, we take the approach of estimating $\\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 1]$ and $\\mathbb {E} \\, [Y| \\mbox{\\bf X}= \\mbox{\\bf x}, W = 0]$ separately and then combine them to estimate the heterogeneous treatment effect $\\tau (\\mbox{\\bf x})$ .",
"Since this approach does not involve the estimation of propensity scores, the semi-parametric efficiency issue will be discussed in Section 7.",
"To ease our presentation, we will demonstrate our estimation method for the treated group with $W=1$ .",
"For the control group with $W=0$ , the method and theory can be applied in the same fashion.",
"Another way to put it is that we assume that the untreated response to be a constant of zero for demonstration.",
"In real applications, we run separate regressions on both arms.",
"Similar to nonparametric regression models, we assume that for the treated group, $Y = \\mu (\\mbox{\\bf X}) + \\epsilon ,$ with $\\epsilon $ an independent noise with mean zero and $\\mu (\\cdot )$ the unknown relationship between $\\mbox{\\bf X}$ and $Y$ .",
"Moreover, an independent and identically distributed ($i.i.d.$ ) sample of size $n$ , $\\lbrace (\\mbox{\\bf X}_i, Y_i)_{i=1}^n\\rbrace $ , is observed for the treated group.",
"In the following, we assume (REF ) is our working model and our target is to estimate $\\mu (\\mbox{\\bf x})$ for some given $\\mbox{\\bf x}$ .",
"Here, $\\mbox{\\bf x}$ can be beyond the $\\mbox{\\bf X}_i$ 's appeared in the sample.",
"And any future assumptions are with respect to model (REF )."
],
[
"Distributional nearest neighbors estimator",
"We first revisit the classical $k$ -nearest neighbors ($k$ -NN) procedure for nonparametric regression.",
"Given a fixed point $\\mbox{\\bf x}$ , we can compute the Euclidean distance of each sample point to $\\mbox{\\bf x}$ and then reorder the sample with this distance.",
"The sample can be relabeled using the order of distances, $\\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert \\le \\Vert \\mbox{\\bf X}_{(2)} - \\mbox{\\bf x}\\Vert \\le \\cdots \\le \\Vert \\mbox{\\bf X}_{(n)} - \\mbox{\\bf x}\\Vert ,$ where $\\Vert \\cdot \\Vert $ denotes the Euclidean distance, and the ties are broken by maintaining the original order of labels.",
"Other distance measures can be used.",
"Yet such a generalization is not the focus of this paper.",
"Here $\\mbox{\\bf X}_{(1)}$ is the closest point in the sample to point $\\mbox{\\bf x}$ , and $Y_{(1)}$ associated with $\\mbox{\\bf X}_{(1)}$ is thus the 1-nearest neighbor estimate of $\\mu (\\mbox{\\bf x})$ .",
"In general, the $k$ -nearest neighbors ($k$ -NN) estimator uses the first $k$ nearest neighbors for estimation $\\hat{\\mu }_\\text{$k$-NN} = \\frac{1}{k} \\; \\sum _{i=1}^k Y_{(i)}.$ The closest $k$ nearest neighbors have equal weights $\\frac{1}{k}$ while the other observations have zero weights.",
"It can be seen as a special case of the more general weighted nearest neighbors approach [50] $\\hat{\\mu }_\\text{$w$-NN} = \\sum _{i=1}^n w_i \\, Y_{(i)},$ where $\\lbrace w_i\\rbrace _{i=1}^n$ are subject to further choice.",
"It is well known that the classical nonparametric methods, such as kernel methods and nearest neighbors methods, suffer the curse of dimensionality.",
"They have bias terms that asymptotically vanish but can compromise finite sample precision.",
"Moreover, the precision becomes worse with increasing dimension of covariates.",
"To be adapted to high-dimensional data, the challenge for nearest neighbor methods is thus how to choose the $\\lbrace w_i\\rbrace _{i=1}^n$ and remove the bias terms.",
"As far as we know, except for recent developments in [57] for classification and [13] for entropy estimation, there is not yet a relatively complete answer for regression models.",
"The only result for regressions we are aware is from page 181, [14].",
"Based on the weighted idea, they have a result that can asymptotically remove the first order bias.",
"However, since there are too many parameters to choose, the weights they give are not intuitively straightforward and perhaps have more theoretical values.",
"In contrast, in this paper we propose a new estimator which makes the bias reduction practical and straightforward.",
"Before we proceed, a formal notation is given to the 1-nearest neighbor estimator with subsampled observations, which is the building block of our new DNN estimator.",
"Let $\\lbrace i_1,\\cdots , i_s\\rbrace $ with $i_1< i_2 <\\cdots < i_s$ and $s\\le n$ be a subset of $\\lbrace 1,\\cdots , n\\rbrace $ .",
"With $\\mbox{\\bf Z}_{i_j}$ as a shorthand for $(\\mbox{\\bf X}_{i_j}, Y_{i_j})$ , we define $\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s})$ as the 1-nearest neighbor estimator to $\\mu (\\mbox{\\bf x})$ in the subsample $\\lbrace (\\mbox{\\bf Z}_{i_j})_{j=1}^s\\rbrace $ , $\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}) = Y_{(1)}(\\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}).$ Our recipe for DNN estimator is thus to average the 1-nearest neighbor estimators from all the subsamples of size $s$ , where $1 \\le s \\le n$ .",
"When $s = n$ , it is just the conventional 1-nearest neighbor estimator $Y_{(1)}$ .",
"When $s = 1$ , it reduces to the simple sample average.",
"This setup happens to coincide with the classical idea of a U-statistic with $\\Phi $ as its kernel [41], [38], [49].",
"Our formal definition for a DNN estimator with subsampling scale $s$ is $D_n(s)(\\mbox{\\bf x}) = \\binom{n}{s}^{-1} \\sum _{1 \\le i_1 < i_2 < \\ldots < i_s \\le n} \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}).$ It also has an equivalent L-statistic representation [58], $D_n(s)(\\mbox{\\bf x}) = \\binom{n}{s}^{-1} \\Big \\lbrace \\binom{n-1}{s-1} Y_{(1)} + \\binom{n-2}{s-1} Y_{(2)} + \\cdots + \\binom{s-1}{s-1} Y_{(n-s+1)} \\Big \\rbrace .$ This representation becomes intuitive when it is connected to the U-statistic definition.",
"It is that out of the total $\\binom{n}{s}$ subsample combinations, $(\\mbox{\\bf X}_{(1)}, Y_{(1)})$ will be in $\\binom{n-1}{s-1}$ of them, and each time $(\\mbox{\\bf X}_{(1)}, Y_{(1)})$ appears, $Y_{(1)}$ will be the 1-nearest neighbor estimator from this subsample.",
"Since the weights are assigned in a distributional fashion on the entire sample, we name this new estimator distributional nearest neighbors (DNN).",
"This distributional view is novel compared to the conventional idea of seeking weights for nearby nearest neighbors with a relatively small neighborhood size.",
"The problem of assigning weights from Euclidean spaces shifts to a problem of looking for distributions in some functional space.",
"A surprising payoff of the DNN estimator is that we can construct a two-scale DNN estimator that has reduced estimation bias, making the adaptation of high-dimensional data practical.",
"One more interesting fact is that the distribution of weights is characterized only by the sample size $n$ and the subsampling scale $s$ .",
"Its insensitivity to the realized sample is appealing compared to the idea of data exploitation in many machine learning algorithms.",
"The reduction in estimation bias to be presented in Section 4 is exclusively enabled through neat statistical theory.",
"Before that, we first introduce some regularity conditions and establish the asymptotic theory for DNN."
],
[
"Technical conditions",
"We start with imposing some regularity conditions.",
"It is assumed that the underlying data generating process of the response $Y$ is the sum of a deterministic function $\\mu (\\cdot )$ of features $\\mbox{\\bf X}$ and an independent random noise $\\epsilon $ .",
"We also assume that the feature vector $\\mbox{\\bf X}$ lies within its support, $\\mathrm {supp}(\\mbox{\\bf X}) \\subset \\mathbb {R}^d$ with fixed dimensionality $d$ .",
"The feature vector $\\mbox{\\bf X}$ follows some unknown distribution that has a density $f(\\cdot )$ with respect to the Lebesgue measure $\\lambda $ on Euclidean space $\\mathbb {R}^d$ .",
"Beyond this setting, some mild and commonly used regularity conditions are also imposed as below.",
"Condition 2 The density $f(\\cdot )$ is bounded away from 0 and $\\infty $ , $f$ and $\\mu (\\cdot )$ are twice continuously differentiable with bounded second derivatives in a neighborhood of $\\mbox{\\bf x}$ , and $Y$ has finite second moment, $\\mathbb {E} \\, Y^2 < \\infty $ .",
"The random noise $\\epsilon $ has zero mean and finite variance $\\sigma _\\epsilon ^2 > 0$ .",
"With the underlying data generating process specified above, we further assume that we have $i.i.d.$ data.",
"Condition 3 We have an $i.i.d.$ sample of size $n$ , $(\\mbox{\\bf X}_1, Y_1), (\\mbox{\\bf X}_2, Y_2), \\ldots , (\\mbox{\\bf X}_n, Y_n)$ , from model (REF ).",
"In summary, we have three assumptions in total.",
"Condition 1 is the unconfoundedness assumption that is the fundamental setup in the potential outcomes framework for causal inference.",
"With this setup, our strategy in this paper is to reduce the causal inference problem to nonparametric regression problems.",
"Condition REF is regular and commonly imposed in nonparametric regressions.",
"Condition REF specifies the data we have.",
"Although it appears idealized at first sight, the assumption of $i.i.d.$ data is commonly made in modern machine learning.",
"It enables researchers to provide key insights with simplified technical presentation."
],
[
"Asymptotic results for DNN",
"We are ready to present the asymptotic properties for DNN.",
"Our first theorem establishes the bias of the DNN estimator $D_n(s)(\\mbox{\\bf x})$ , and the second theorem shows that $D_n(s)(\\mbox{\\bf x})$ can be asymptotically normal with appropriately chosen subsampling scale $s$ .",
"Theorem 1 Given $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X})$ , under Conditions 2–3 we have $\\mathbb {E} \\, D_n(s) (\\mbox{\\bf x}) = \\mu (\\mbox{\\bf x}) + B(s),$ where $B(s) = \\Gamma (2/d + 1) \\frac{f(\\mbox{\\bf x}) \\, \\mathrm {tr}(\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) + 2 \\, \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x})}{2 \\, d \\, V_d^{2/d} \\, f(\\mbox{\\bf x})^{1+2/d}} \\, s^{-2/d} + o(s^{-2/d}),$ $V_d = \\frac{\\pi ^{d/2}}{\\Gamma (1+d/2)},$ $\\Gamma (\\cdot )$ denotes the Gamma function, $f^{\\prime }(\\mbox{\\bf x})$ and $\\mu ^{\\prime }(\\mbox{\\bf x})$ are the first order gradients at $\\mbox{\\bf x}$ for $f(\\mbox{\\bf x})$ and $\\mu (\\mbox{\\bf x})$ , respectively, $\\mu ^{\\prime \\prime }(\\mbox{\\bf x})$ is the Hessian matrix of $\\mu (\\cdot )$ at $\\mbox{\\bf x}$ , and $\\mathrm {tr}(\\cdot )$ gives the trace.",
"Theorem 1 gives us the form of the finite sample bias of the DNN estimator.",
"The key idea of the proof comes from [14] for the case of $k$ -nearest neighbors.",
"Details of our proof are provided in Appendix A.",
"We can see that the leading order of bias converges to 0 at rate $s^{-2/d}$ .",
"Thus when subsampling scale $s \\rightarrow \\infty $ , the DNN estimator is asymptotically unbiased.",
"The interesting and surprising result from Theorem 1 is that the coefficient in the leading order of bias term $B(s)$ does not depend on subsampling scale $s$ .",
"This is the key feature that opens doors to our two-scale DNN framework, which is to be presented in Section 4.",
"When the dimensionality of features $d$ is large, the rate of convergence of the main bias term is slow.",
"In situations like this, it can be more beneficial to get rid of the first order finite sample bias.",
"Theorem 2 Given $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X})$ , under Conditions 2–3, and assuming in addition, $s \\rightarrow \\infty $ and $s \\big / n \\rightarrow 0$ , we have for some positive $\\sigma _n^2 = O(\\frac{s}{n})$ , $\\frac{D_n(s)(\\mbox{\\bf x}) - \\mu (\\mbox{\\bf x}) - B(s)}{\\sigma _n^2} \\overset{{D}}{\\longrightarrow }N(0,1).$ Details of the proof are given in Appendix A. Theorem 2 establishes the asymptotic normality of the DNN estimator.",
"Since the DNN estimator is a U-statistic, our proof builds on the traditional U-statistic framework in [58], [49].",
"The major difference of the approach in our paper with the traditional U-statistic framework is that the classical U-statistic framework only allows subsampling scale $s$ to be finite.",
"Some of the classical results do not yet apply naturally to the case when $s \\rightarrow \\infty $ .",
"Theorem 2 tells us that the DNN estimator is asymptotically normal, which is natural since it is a U-statistic.",
"An interesting difference with the random forest approach is that the convergence rate of the DNN estimator can be derived to be $\\sqrt{n/s}$ .",
"When $s = O(n^{\\frac{d}{d+4}})$ , the optimal rate of convergence in terms of mean squared errors is obtained.",
"Another feature is that the DNN estimator is a simple L-statistic.",
"The bootstrap [24] can be directly used for variance estimation [62], [59].",
"Consequently we do not need to know the exact form of the asymptotic variance as long as it is bounded.",
"The exact form of the asymptotic variance depends on the unknown error variance $\\sigma _\\epsilon ^2$ , the underlying unknown distribution $f$ , and the evaluated point $\\mbox{\\bf x}$ .",
"If the exact form were available, the plug-in estimation of variance here can still be challenging with the intermediate unknowns.",
"Now we have established the asymptotic properties for the distributional nearest neighbors estimator we proposed.",
"It is not surprising that these results are related to the classical $k$ -NN [50] since in essence they can be perceived as two different choices of weights.",
"The reason that we take the detour to make this variation is that our DNN is more adapted to the two-scale bias reduction procedure, which is to be presented shortly.",
"The difference is shown in Monte Carlo simulations in Section 5.1.",
"We give two explanations here.",
"First, unlike DNN, $k$ -NN has two main bias terms whose orders change when $k$ increases.",
"This increases the difficulty for two-scale bias reduction to be directly applied to $k$ -NN.",
"Second, DNN uses sample means for two-scale bias reduction while $k$ -NN uses individual realizations.",
"In finite samples there can be a substantial difference.",
"However, we do not rule out the possible existence of some other debiasing algorithms for $k$ -NN.",
"Nevertheless the DNN estimator is simple and can work well with the two-scale framework.",
"We are skeptical whether similar algorithms for $k$ -NN could in general outperform the two-scale DNN."
],
[
"Two-scale DNN Estimator",
"As seen in Theorem REF , the DNN estimator can be asymptotically unbiased and normal for $s$ appropriately chosen.",
"In fact, we see in Theorem REF that $\\mathbb {E} \\, D_n(s) (\\mbox{\\bf x}) = \\mu (\\mbox{\\bf x}) + c \\, s^{-2/d} + o(s^{-2/d}).$ Here the positive constant $c$ is specified in Theorem REF .",
"It depends only on the underlying data generating process and does not change when we choose a different subsampling scale $s$ .",
"Such an appealing property gives us an effective way to completely remove the first order finite sample bias.",
"Consider two DNN estimators of different subsampling scales $s_1$ and $s_2$ .",
"Their finite sample biases have the following forms $D_n(s_1)(\\mbox{\\bf x}) = \\mu (\\mbox{\\bf x}) + c \\, s_1^{-2/d} + o(s_1^{-2/d}), \\\\$ $D_n(s_2)(\\mbox{\\bf x}) = \\mu (\\mbox{\\bf x}) + c \\, s_2^{-2/d} + o(s_2^{-2/d}).$ We then proceed with solving the following system of linear equations $w_1 + w_2 & = 1, \\\\w_1 \\; s_1^{-2/d} + w_2 \\; s_2^{-2/d} & = 0, $ yielding the weights $w_1^* = w_1^*(s_1,s_2) = s_2^{-2/d}/(s_2^{-2/d} - s_1^{-2/d})$ and $ w_2^* = - s_1^{-2/d}/(s_2^{-2/d} - s_1^{-2/d})$ .",
"We propose the two-scale DNN estimator as $D_n(s_1, s_2)(\\mbox{\\bf x}) = w_1^*D_n(s_1)(\\mbox{\\bf x}) + w_2^*D_n(s_2)(\\mbox{\\bf x}).$ Equation (REF ) ensures that the two-scale DNN is still unbiased for $\\mu (\\mbox{\\bf x})$ .",
"Equation () gives the constraint to remove the first order bias.",
"Compared to the simple DNN estimator $D_n(s)(\\mbox{\\bf x})$ , the introduced two-scale DNN estimator $D_n(s_1, s_2)(\\mbox{\\bf x})$ is free of the first order finite sample bias.",
"As a result, the trade-off between bias and variance can be made on a new level.",
"From our extensive simulation studies in Section 4, this simple result will substantively improve precision in estimation and mean squared errors.",
"The construction of confidence intervals also becomes more meaningful with reduced bias.",
"From our derivations above or as further numerical examples in Section 4.3 will demonstrate, we can see either $w_1^*$ or $w_2^*$ is negative.",
"It implies that two-scale DNN can assign negative weights to distant nearest neighbors.",
"Although [65] points it as a promising future direction, assigning negative weights is beyond the scope of current random forests algorithms.",
"The least present-day random forests can assign is zero weight.",
"This interesting feature may partially provide an intuitive rationale behind the performance of two-scale DNN."
],
[
"Asymptotic normality",
"We next give a formal theorem for the two-scale DNN estimator.",
"Theorem 3 Given $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X})$ , under Conditions 2–3, and assuming in addition, $s_1 \\rightarrow \\infty $ with $s_1 \\big / n \\rightarrow 0$ and $s_2 \\rightarrow \\infty $ with $s_2 \\big / n \\rightarrow 0$ , for some positive $\\sigma _n^2 = O(\\frac{s_1}{n} + \\frac{s_2}{n})$ we have $\\frac{D_n(s_1, s_2)(\\mbox{\\bf x}) - \\mu (\\mbox{\\bf x}) - \\Lambda }{\\sigma _n^2} \\overset{{D}}{\\longrightarrow }N(0,1),$ where $(w_1^*, w_2^*)$ is the solution to Equations REF and , and $\\Lambda = o(s_1^{-2/d} + s_2^{-2/d})$ .",
"Finally, we present a theorem for the case when the control group is not degenerate.",
"The naughty notations of subscripts would come back temporarily for the next few paragraphs.",
"For the treated group and the control group, respectively, let $n_1$ and $n_0$ denote the i.i.d.",
"sample size, $s_{\\cdot }^{(1)}$ and $s_{\\cdot }^{0}$ denote the subsampling scale, $\\mathrm {supp}(\\mbox{\\bf X}_1)$ and $\\mathrm {supp}(\\mbox{\\bf X}_0)$ denote the support, $\\mathrm {supp}(\\mbox{\\bf X}_1)$ and $\\mathrm {supp}(\\mbox{\\bf X}_0)$ denote the underlying density function, $\\mu _1(\\cdot )$ and $\\mu _0(\\cdot )$ denote the regression function, $\\epsilon _1$ and $\\epsilon _0$ denote the random noise, $D_n^{(1)}(s_1)(\\mbox{\\bf x})$ and $D_n^{(0)}(s_0)(\\mbox{\\bf x})$ denote the DNN estimator, $\\Lambda $ denote the bias, and $Y_1$ and $Y_0$ denote the response.",
"The heterogeneous treatment effect at point $\\mbox{\\bf x}$ is $\\tau (\\mbox{\\bf x})$ .",
"Theorem 4 Given $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X}_1) \\cap \\mathrm {supp}(\\mbox{\\bf X}_0)$ , under Condition 1–3 for both the treated and control group with their subscripts, and assuming in addition, $s_i^{(1)} \\rightarrow \\infty $ with $s_i^{(1)} \\big / n_1 \\rightarrow 0$ for $i =1, 2$ and $s_i^{(0)} \\rightarrow \\infty $ with $s_i^{(0)} \\big / n_0 \\rightarrow 0$ for $i =1, 2$ , for some positive $\\sigma _n^2 = O(\\frac{s_1^{(1)}}{n_1} + \\frac{s_2^{(1)}}{n_1} + \\frac{s_1^{(0)}}{n_0} + \\frac{s_2^{(0)}}{n_0})$ we have $\\frac{[D_n^{(1)}(s_1^{(1)}, s_2^{(1)})(\\mbox{\\bf x}) - D_n^{(0)}(s_1^{(0)}, s_2^{(0)})(\\mbox{\\bf x})]- \\tau (\\mbox{\\bf x}) - \\Lambda }{\\sigma _n^2} \\overset{{D}}{\\longrightarrow }N(0,1),$ where $\\Lambda = o({s_1^{(1)}}^{-2/d} + {s_2^{(1)}}^{-2/d} + {s_1^{(0)}}^{-2/d} + {s_2^{(0)}}^{-2/d})$ .",
"The evaluated point $\\mbox{\\bf x}$ is required to be in the support of both the treated and control groups.",
"This condition is easier to check and verify than the common support condition in estimating average treatment effects (ATE), while the latter requires that supports fully overlap.",
"We can see that the final rate of convergence and the resulting precision are dominated by the slower side.",
"This is true since the approach of the estimation of heterogeneous treatment effects we adopted is inevitably a two-sided game.",
"A potential caveat of this approach can be the semi-parametric efficiency, which will be discussed in Section 7.",
"However, an interesting feature of the separated regression approach is that the whole sample does not need to represent the proper portions of the treated and the control in the population.",
"This can provide flexibility and robustness in observational studies when the sampling cost is very different between the treated group and the control group.",
"For example, it is easier to survey those who come to vote than the ones who do not.",
"Our presentation until now deals only with the binary treatment case.",
"It is also clear that the two-scale DNN can be flexibly extended to multi-valued treatment settings."
],
[
"Implementation",
"For the remaining of the paper we exploit $D_n(s)$ and $D_n(2s)$ to form the two-scale DNN estimator.",
"We use this combination for simplicity.",
"It is possible that there are better choices.",
"Solving Equations (REF ) and () for $s$ and $2s$ , $\\frac{1}{1 - 2^{2/d}} D_n(s) + \\frac{-2^{2/d}}{1 - 2^{2/d}} D_n(2s)$ is our choice.",
"When $d = 3$ , the two-scale DNN estimator is approximately $-1.70 \\, D_n(s) + 2.70 \\, D_n(2s)$ .",
"When $d = 10$ , the two-scale DNN estimator is approximately $-6.73 \\, D_n(s) + 7.73 \\, D_n(2s)$ .",
"The coefficients for the case of $d = 10$ are large and can sacrifice variance estimation.",
"We make a compromise in situations like this and choose the weights for $d = 3$ , that is, $-1.70 \\, D_n(s) + 2.70 \\, D_n(2s)$ .",
"It is also a trade off between the bias and variance.",
"[5], [65] proposed a honest rule, that is, the response of each observation is only used to estimate the treatment effects or to decide where to place the splits, but not both.",
"As a consequence, they split the sample into two subsamples.",
"One is used for deciding partitions, and the other for estimation.",
"Our approach is different.",
"When we apply the two-scale DNN estimator to a specific choice of the subsampling scale $s$ , the weight distribution is deterministic.",
"Besides when we compute distances and obtain rankings, the information on $\\mbox{\\bf X}$ but not $Y$ is used.",
"Our framework agrees with the honest rule so there is no need to split the data in the implementation of DNN, which makes our algorithm scalable.",
"Certainly a small proportion of the whole sample can always be used for tuning separately.",
"In this paper, we also provide a simple and straightforward tuning algorithm as an example.",
"Our tuning procedure is to compute two-scale DNN estimators for $s = 1, 2, \\ldots $ , and go on until the difference in absolute differences in two-scale DNN estimators changes the sign.",
"It is the point where the curvature of two-scale DNN estimator changes.",
"The intuition comes from the curve structure in Figure 1.",
"This algorithm works fast and well in our simulations.",
"It is worth noting that the information of responses is used in tuning the subsampling scale $s$ .",
"However, the tuning process does not compromise the validity of the two-scale DNN estimators.",
"Cross validation methods can also be exploited.",
"But the related technical details are not the focus of this paper.",
"Since the DNN estimator is an L-statistic, the two-scale DNN estimator, as a linear combination of the DNN estimators, is still an L-statistic.",
"The bootstrap can be directly employed to estimate variance for the two-scale DNN estimators in our paper [62], [59].",
"When the untreated group is not degenerate as in our application in Section 5, we run two-scale DNN on the treated group and the control group, respectively, and then take a difference.",
"We then bootstrap this difference by resampling within each group strata to provide precise inference for the HTE estimate."
],
[
"Simulation studies",
"This section presents two simulation studies to demonstrate the advantages of the two-scale DNN framework.",
"The first one studies the newly suggested DNN in a setting where we ignore the problem of choosing the subsampling scale $s$ , that is, we will go through all choices for $s$ and plot the resulting biases and mean squared errors (MSEs) as functions of $s$ in one graph.",
"In the second study, we compare the performance of our two-scale DNN estimators with the random forest approach in [65] over various settings."
],
[
"Two-scale DNN",
"We present the finite-sample properties of the DNN estimator and the substantial improvement in precision coming from the two-scale framework.",
"We run a Monte Carlo simulation and the data generating process is $ y = (x_1-1)^2 + (x_2 + 1)^3 - 3 \\, x_3 + \\epsilon $ with vector $(x_1, x_2, x_3, \\epsilon )^T \\sim \\mbox{\\bf N}(0,\\mbox{\\bf I}_4)$ , and sample size $n = 1000$ .",
"The evaluated target point is $(0.5, -0.5, 0.5)^T$ .",
"The coordinates are chosen to place the target point well in the interior and avoid irregular border cases.",
"We estimate the heterogeneous treatment effects at this point with subsampling scales $s$ running from 1 to 250.",
"The two-scale DNN is implemented using subsampling scales $s$ and $2s$ for simplicity.",
"The simulation results are presented in Figure 1.",
"The subsampling scale $s$ is on the horizontal axis while the resulting biases and mean squared errors are on the vertical axis.",
"The upper left region of Figure 1 depicts the bias from DNN estimation, while the upper right depicts the mean squared error from DNN estimation.",
"The lower two regions are devoted to the two-scale DNN estimation in the same fashion.",
"Figure: The choice of subsampling scale ss and the resulting bias and mean squared errorFrom Figure 1, we see that as the subsampling scale $s$ increases, the bias of the DNN estimator shrinks toward zero.",
"But the marginal benefit on bias reduction becomes smaller and smaller.",
"The classical $U$ -shaped pattern of bias and variance trade-off shows up in the mean squared errors.",
"Compared to the simple DNN, the two-scale DNN speeds up both the processes of bias reduction and bias-variance trade-off and squeezes the curves toward lower levels of subsampling scale $s$ .",
"The most celebrated fact is that the best mean squared error achieved drops by over half.",
"This significant improvement takes only an extra step of weighted averaging.",
"Figure: The choice of neighborhood size kk and the resulting bias and mean squared errorWe repeat the same exercise to classical $k$ -NN estimators and try out different two-scale strategies.",
"The result is in Figure 2.",
"It is very interesting that the best mean squared error achieved does not improve as much for the two-scale $k$ -NN estimators.",
"We offered two explanations in Section 3.4."
],
[
"Comparisons with random forest",
"We also compare the two-scale DNN framework with the causal forest (CF) approach in [65].",
"The comparisons are made in eleven settings.",
"All simulation settings are run 1000 times and the sample size is 1000.",
"The first setting has the same data generating process as in Equation REF .",
"We first make point estimations and estimate the variances of the estimations on the same test point $\\mbox{\\bf x}= (0.5, -0.5, 0.5)^T$ as in Section 4.1.",
"Then point estimations are made on a random point with its coordinates independently drawn from the uniform distribution on $[0, 1]$ .",
"The second setting is built on the first setting by adding irrelevant noise variables, that is, $y = (x_3 - 1)^2 + (x_5 + 1)^3 - 3 \\, x_7 + \\epsilon $ with $(x_1, x_2, \\ldots , x_{10}, \\epsilon )^T \\sim \\mbox{\\bf N}(0, \\mbox{\\bf I}_{11})$ .",
"We also have both a fixed and a random test point.",
"The fixed test point is at the point $\\mbox{\\bf x}$ with $x_3 = 0.5$ , $x_5 = -0.5$ , $x_7 = 0.5$ , and other coordinates zero.",
"The random test point has the above nonzero coordinates independently drawn from the uniform distribution on $[0, 1]$ .",
"From the third setting, we slowly increase the dimensionality $d$ .",
"To be more specific, for $j = 10, 15, 20, 25, 30, 35, 40, 45$ , and 50, respectively, the data generating process is $y = \\log \\left[\\sum _{i=1}^{j} (x_i^3 - 2\\,x_i^2 + 2\\,x_i) \\right]^2 + \\epsilon $ with $(x_1, x_2, \\ldots , x_{j}, \\epsilon )^T \\sim \\mbox{\\bf N}(0, \\mbox{\\bf I}_{j+1})$ .",
"The fixed test point is at the $\\mbox{\\bf x}$ with the first $[\\frac{j-1}{2}]$ coordinates being $0.5$ , and the other coordinates zero.",
"$[\\frac{j-1}{2}]$ denotes the largest integer smaller or less than $\\frac{j-1}{2}$ .",
"The random test point has again the nonzero coordinates independently drawn from the uniform distribution on $[0, 1]$ .",
"This design is to mimic the situations where researchers are interested in varying some variables of interest while maintaining the remaining at mean levels.",
"This is particularly useful in mediation analysis.",
"A real-life application is provided in Section 6.",
"For the two-scale DNN estimation, we first tune the subsampling scale $s$ by the strategy which was explained in Section 4.3.",
"Then with the chosen subsampling scale $s$ and $2s$ , the weights in the L-statistic representation of the two-scale DNN estimator can be conveniently computed.",
"When the weights are ready, we simply sort the observations by Euclidean distance and then take a weighted average.",
"The L-statistic representation avoids the computation cost of running through all subsample combinations.",
"We further estimate the variance by bootstrapping the data set and repeatedly obtaining two-scale DNN estimators for the bootstrap data.",
"Since our DNN estimator in this paper does not yet have the function of feature screening [29], [26], [30], one extra step of screening can be added for the second setting.",
"For screening, we compute the pairwise distance correlations [60] between each candidate feature variable and the outcome.",
"Feature variables with close to zero correlations can be screened out.",
"[60] provided the R package of energy for the distance correlation computation.",
"To train random forests in our simulation, we use the R package of grf from [9], which is a generalization of [65].",
"The simulation results are shown in Table 1.",
"The first column is the number of settings.",
"The method used is in the second column.",
"Columns 3–6 present simulation results on the fixed test point.",
"In particular, Column 3 is the mean bias for the fixed point and Column 4 the mean squared error.",
"The variance for the fixed point is computed from simulations and presented in Column 5.",
"Column 6 gives the estimated variance for the fixed point.",
"The last two columns devote to the random test point.",
"The mean bias for random points is in Column 7 and Column 8 presents the mean squared error.",
"As we can see in all the settings, the two-scale DNN estimator demonstrates a consistent pattern of smaller biases and lower mean squared errors.",
"Although two-scale DNN estimators have slightly larger variances in some cases, the estimations of variances are more accurate compared to the random forest approach.",
"It is a consequence of the fact that the bootstrap can be applied to the two-scale DNN estimators directly.",
"Table: Two-scale DNN and random forest"
],
[
"Empirical analysis",
"In this section we work on real-life data and study the heterogeneity of treatment effects of smoking on children's birth weights across mothers' ages.",
"This is the empirical application that [3] studied with their kernel-based estimator.",
"We will work on the same data setThe data set used in this section is obtained from the research webpage of Robert P. Lieli, https://sites.google.com/site/robertplieli/research.",
"A related data set [2] has also been used as the touchstone of heterogeneity study in panel data models.",
"in order to give a relatively complete picture of current approaches on estimating heterogeneous treatment effects.",
"While our economic interests are both heterogeneity, our empirical targets are slightly different.",
"This is also why we make a comparison in this section rather than in simulations.",
"In our study, the feature vector $\\mbox{\\bf X}$ includes variables such as mother's age, mother's education, father's education, gestation length in weeks, and the number of prenatal visits.",
"The response variable $Y$ is child's birth weight.",
"The binary treatment $W$ is whether the mother smoked or not during pregnancy.",
"The data set consists of 475,506 observations in total, in which there are 85,062 black mothers with 4,926 smoked during pregnancy and 390,444 white mothers with 58,977 smoked during pregnancy.",
"For more details, see [3].",
"Since this section is mainly for demonstration purpose, we ignore the panel structure in the data and directly make the unconfoundedness assumption and $i.i.d$ assumption for simplicity.",
"We estimate the heterogeneous treatment effects with the feature variables, except for age, fixed at the average levels of the corresponding treated group.",
"For the age variable, we deliberately choose its level and vary it from 16 to 35.",
"Our purpose is to estimate the treatment effect heterogeneity across ages with the other variables fixed at average levels.",
"Since [3] also conduct a subgroup analysis with black and white mothers, we do the same exercise here.",
"In effect, it is like we have added an extra control variable of race and are exploring treatment effect heterogeneity in both dimensions of race and age.",
"Figure: Heterogeneous Treatment Effects of Smoking on Child Birth WeightsFigure 3 shows the empirical results with two-scale DNN estimators.",
"We can see a clear downward sloping curve for both black and white mothers.",
"It implies that as age increases, the loss in new born's weights associated with mother's smoking behavior becomes larger.",
"Our result is in general consistent with [3] but with significantly improved precision.",
"The confidence intervals in Figure 3 are generated by bootstrapping the difference between the treated and control group.",
"Our estimations are significant with the $95\\%$ confidence intervals staying away from zero.",
"There is also a difference between black and white mothers.",
"For black mothers, there is an interesting hump around the age of 26.",
"It is an interesting unknown pattern.",
"Without further assumptions, we can infer that the mechanism of how smoking affects birth weights may be through some factors associated with age."
],
[
"Discussions",
"In this paper we have built the two-scale DNN framework for the estimation of heterogeneous treatment effects.",
"The framework encompasses both theory and practice.",
"We further test the new two-scale DNN estimator in Monte Carlo simulations and real-life application.",
"In both cases, the two-scale DNN estimator achieves improved performance in precision.",
"However, it is inevitable that the two-scale DNN estimator has imperfections.",
"First, the dimensionality of features $d$ has been fixed in this paper.",
"We reserve the question to allow $d \\rightarrow \\infty $ for future studies.",
"This generality connects to the dimension reduction literature and makes the fundamental unconfoundedness condition more plausible.",
"Second, we have assumed $i.i.d.$ data for analysis, which is more plausible in a cross-sectional setting.",
"It is thus interesting to see how the two-scale DNN estimator can adapt to other data structures, such as time series and panel data.",
"The third issue is the semi-parametric efficiency.",
"Our approach consists of only the estimations of regression functions while leaves some other relevant functions out of the picture, such as the propensity score function.",
"The separated regression approach has mostly ignored the information between treatment status and control variables.",
"The potential consequence is thus the loss of semi-parametric efficiency.",
"It will be appealing if the efficient influence function can be derived and utilized for HTE estimation.",
"Fourth, the two-scale DNN framework mitigates the curse of dimensionality but still suffers from it.",
"We are curious whether data exploiting algorithms and theoretical derivations can be combined to further push boundaries."
],
[
"Proof of Theorem 1",
"Our road map for the proof of Theorem 1 consists of three steps.",
"First, in Lemma 1 we derive the bias term of the 1-nearest neighbor feature vector to the target point $\\mbox{\\bf x}$ in a general case with i.i.d.",
"sample of size $n$ .",
"Second, in Lemma 2 a bridge is built to link the discrepancy in the features to the discrepancy in the response.",
"Third, we employ the two facilities developed above for the analysis of the DNN estimator $D_n(s)$ with subsampling scale $s$ .",
"There are several ways to prove it.",
"In this appendix, we present a clean proof using the Lebesgue differentiation theorem.",
"Lemma 2 is the key to the proof of Theorem 1.",
"In this appendix, with the powerful idea of projecting onto a half line from [14], we present a neat proof with spherical integration for Lemma 2.",
"As we will see, Theorem 1 is just one step away from Lemmas 1–2."
],
[
"Lemma 1 and its proof",
"Lemma 1 Given $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X})$ , under Conditions 2–3, and when $n \\rightarrow \\infty $ , the 1-nearest neighbor to $\\mbox{\\bf x}$ in the i.i.d.",
"sample $\\lbrace (\\mbox{\\bf X}_i, Y_i)\\rbrace _{i=1}^n$ has its 1-nearest neighbor feature $\\mbox{\\bf X}_{(1)}$ satisfying $\\mathbb {E} \\; \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ^2 = \\frac{\\Gamma (2/d + 1)}{(f(\\mbox{\\bf x}) V_d)^{2/d}} n^{-2/d} + o(n^{-2/d}),$ where $V_d = \\frac{\\pi ^{d/2}}{\\Gamma (1 + d/2)}$ and $\\Gamma (\\cdot )$ is the Gamma function.",
"Proof of Lemma 1: We will use the following two results in the proof of Lemma 1.",
"Since they are well known, their proofs are not presented in this appendix.",
"By the Lebesgue differentiation theorem, when $r \\rightarrow 0$ , $\\varphi (B(\\mbox{\\bf x}, r)) = f(\\mbox{\\bf x}) V_d \\, r^d + o(r^d)$ , where $\\varphi $ is some measure on $\\mbox{\\bf X}$ , $B(\\mbox{\\bf x}, r)$ is the Euclidean ball in $\\mathbb {R}^d$ , $V_d$ is the volume of unit ball, $f$ is the density of measure $\\varphi $ with respect to the Lebesgue measure $\\lambda $ , and $f$ is continuous at $\\mbox{\\bf x}$ .",
"For $a>0$ and $b>0$ , we have $\\int _0^\\infty x^{a- 1} \\exp ( - b x^p ) \\, d x = \\frac{1}{p} b^{-a/p} \\Gamma (\\frac{a}{p}).$ For our target, $\\mathbb {E} \\, \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ^2& = \\int _0^\\infty \\mathbb {P} \\, (\\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ^2 > t) \\; d t \\\\& = \\int _0^\\infty \\mathbb {P} \\, (\\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert > \\sqrt{t}) \\; d t \\\\& = \\int _0^\\infty [1 - \\varphi (B(\\mbox{\\bf x}, \\sqrt{t}))]^n \\; d t\\\\& = n^{-2/d} \\, \\int _0^\\infty [1 - \\varphi (B(\\mbox{\\bf x}, \\frac{\\sqrt{t}}{n^{1/d}}))]^n \\; d t.$ We then take the limit when $n \\rightarrow \\infty $ , $\\lim _{n \\rightarrow \\infty } \\, n^{2/d} \\, \\mathbb {E} \\, \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ^2& = \\lim _{n \\rightarrow \\infty } \\, \\int _0^\\infty [1 - \\varphi (B(\\mbox{\\bf x}, \\frac{\\sqrt{t}}{n^{1/d}}))]^n \\; d t \\\\& = \\lim _{n \\rightarrow \\infty } \\, \\int _0^\\infty [1 - (f(\\mbox{\\bf x}) V_d + o(1)) \\frac{t^{d/2}}{n}) ]^n \\; d t \\\\& = \\int _0^\\infty \\, \\lim _{n \\rightarrow \\infty } [1 - (f(\\mbox{\\bf x}) V_d + o(1)) \\frac{t^{d/2}}{n} ]^n \\; dt \\\\& = \\int _0^\\infty \\exp \\lbrace - (f(\\mbox{\\bf x}) \\, V_d + o(1)) \\, t^{d/2} \\rbrace \\; d t \\\\& = \\frac{\\Gamma (2/d + 1)}{(f(\\mbox{\\bf x})V_d)^{2/d}} + o(1),$ which completes the proof of Lemma 1.",
"Since we have assumed $f(\\mbox{\\bf x})V_d$ to be bounded everywhere, the resulting $o(1)$ is uniform for $t$ in the second equality and in the third equality the integral and limit can interchange.",
"From Lemma 1, we know that the first order of the squared distance of the closest feature vector from an $i.i.d.$ random sample of size $n$ to the target point $\\mbox{\\bf x}$ is $n^{-2/d}$ .",
"If the $i.i.d.$ sample has size $s$ , as a consequence, the first order of squared distance is $s^{-2/d}$ .",
"When size $s \\rightarrow \\infty $ , the squared distance goes to zero.",
"When the dimensionality of features $d$ is large, the rate of convergence is slow.",
"The coefficient is also explicitly derived.",
"It is intuitive that when the density $f(\\mbox{\\bf x})$ at $\\mbox{\\bf x}$ is small, the squared distance is large."
],
[
"Lemma 2 and its proof",
"As in [14], we first define the projection onto the half line $\\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert $ , $m(r) = \\lim _{\\delta \\rightarrow 0} \\mathbb {E} \\, [\\mu (\\mbox{\\bf X}) \\; | \\; r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta ] = \\mathbb {E} \\; [Y \\; | \\; \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert = r].$ An immediate consequence of this definition is that $m(0) = \\mathbb {E} \\, [Y \\, | \\, \\mbox{\\bf X}= \\mbox{\\bf x}] = \\mu (\\mbox{\\bf x})$ .",
"Lemma 2 When $r \\rightarrow 0$ , we have for $\\mbox{\\bf x}\\in \\mathrm {supp}(\\mbox{\\bf X})$ , $m(r) = m(0) + \\frac{f(\\mbox{\\bf x}) \\, tr (\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) + 2 \\, \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x})}{2 \\, d \\, f(\\mbox{\\bf x})} \\, r^2 + o(r^2).$ Proof of Lemma 2: The spherical coordinate integration is to be used in our proof.",
"We first introduce the notation.",
"$B(\\mbox{\\bf 0},r)$ denotes the ball with radius $r$ centered at $\\mbox{\\bf 0}$ in Euclidean space $\\mathbb {R}^d$ , $\\mathbb {S}^{d-1}$ denotes the unit sphere in $\\mathbb {R}^d$ , $\\nu $ denotes a measure constructed on the sphere $\\mathbb {S}^{d-1}$ , and $\\mbox{$\\xi $}\\in \\mathbb {S}^{d-1}$ denotes a point on the sphere.",
"We omit other details.",
"Integration with spherical coordinates is equivalent to the standard integration, $\\int _{\\tiny B(\\mbox{\\bf 0},r)} \\, f(\\mbox{\\bf x}) \\, d \\mbox{\\bf x}= \\int _0^r u^{d-1} \\int _{\\mathbb {S}^{d-1}} \\, f(u \\, \\mbox{$\\xi $}) \\, \\nu (d \\, \\mbox{$\\xi $}) \\, du.$ Some integration formulas are here.",
"It holds that $\\int _{\\mathbb {S}^{d-1}} \\; \\nu (d \\, \\mbox{$\\xi $}) = d \\, V_d,$ $\\int _{\\mathbb {S}^{d-1}} \\mbox{$\\xi $}\\; \\nu (d \\, \\mbox{$\\xi $}) = \\mbox{\\bf 0},$ $\\int _{S^{d-1}} \\mbox{$\\xi $}^T M \\, \\mbox{$\\xi $}\\; \\nu (d \\, \\mbox{$\\xi $}) = \\mathrm {tr}(M) \\, V_d,$ where $M$ is a $d \\times d$ matrix.",
"First, we decompose the $m(r)$ into two components which we will analyze separately, $m(r)& = \\lim _{\\delta \\rightarrow 0} \\mathbb {E} \\, [\\mu (\\mbox{\\bf X}) \\; | \\; r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta ] \\\\& = \\lim _{\\delta \\rightarrow 0} \\frac{\\mathbb {E} \\, [\\mu (\\mbox{\\bf X}) \\mathbb {1}(r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta )]}{\\mathbb {P} \\, (r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta )}.$ Before we proceed, we use the spherical coordinate representation for the denominator and numerator, $\\mathbb {P} \\, (r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta ) = \\int _r^{r+\\delta } u^{d-1} \\int _{\\mathbb {S}^{d-1}} \\, f(\\mbox{\\bf x}+ u \\, \\mbox{$\\xi $}) \\, \\nu (d \\, \\mbox{$\\xi $}) \\, du$ $\\mathbb {E} \\, [\\mu (\\mbox{\\bf X}) \\mathbb {1}(r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta )] = \\int _r^{r+\\delta } u^{d-1} \\int _{\\mathbb {S}^{d-1}} \\, \\mu (\\mbox{\\bf x}+ u \\, \\mbox{$\\xi $}) f(\\mbox{\\bf x}+ u \\, \\mbox{$\\xi $}) \\, \\nu (d \\, \\mbox{$\\xi $}) \\, du.$ Using spherical coordinate integration, we also have $\\int _{\\mathbb {S}^{d-1}} f(\\mbox{\\bf x}+ r \\, \\mbox{$\\xi $}) \\; \\nu (d \\, \\mbox{$\\xi $})& = \\int _{\\mathbb {S}^{d-1}} (f(\\mbox{\\bf x}) + f^{\\prime }(\\mbox{\\bf x})^T\\, \\mbox{$\\xi $}\\, r+ \\frac{1}{2} \\, \\mbox{$\\xi $}^T \\, f^{\\prime \\prime }(\\mbox{\\bf x}) \\, \\mbox{$\\xi $}\\, r^2 + o(r^2))\\; \\nu (d \\, \\mbox{$\\xi $}) \\\\& = f(\\mbox{\\bf x})\\, d \\, V_d + \\frac{1}{2} \\, \\mathrm {tr}(f^{\\prime \\prime }(\\mbox{\\bf x})) V_d \\, r^2 + o(r^2),$ and $& \\int _{\\mathbb {S}^{d-1}} \\mu (\\mbox{\\bf x}+ r \\, \\mbox{$\\xi $})f(\\mbox{\\bf x}+ r \\, \\mbox{$\\xi $}) \\; \\nu (d \\, \\mbox{$\\xi $}) \\\\= & \\int _{\\mathbb {S}^{d-1}} [\\mu (\\mbox{\\bf x}) + \\mu ^{\\prime }(\\mbox{\\bf x})^T\\mbox{$\\xi $}\\, r + \\frac{1}{2} \\, \\mbox{$\\xi $}^T \\mu ^{\\prime \\prime }(\\mbox{\\bf x})\\, \\mbox{$\\xi $}\\, r^2 + o(r^2)][f(\\mbox{\\bf x}) + f^{\\prime }(\\mbox{\\bf x})^T\\mbox{$\\xi $}\\, r + \\frac{1}{2} \\mbox{$\\xi $}^T f^{\\prime \\prime }(\\mbox{\\bf x}) \\, \\mbox{$\\xi $}\\, r^2 + o(r^2)]\\; \\nu (d \\, \\mbox{$\\xi $})\\\\= & \\int _{\\mathbb {S}^{d-1}} \\Big \\lbrace \\mu (\\mbox{\\bf x})f(\\mbox{\\bf x}) + [f(\\mbox{\\bf x})\\mu ^{\\prime }(\\mbox{\\bf x})^T \\mbox{$\\xi $}+ \\mu (\\mbox{\\bf x})f^{\\prime }(\\mbox{\\bf x})^T \\mbox{$\\xi $}] \\, r \\\\& \\qquad + [\\frac{1}{2} \\, f(\\mbox{\\bf x}) \\, \\mbox{$\\xi $}^T \\mu ^{\\prime \\prime }(\\mbox{\\bf x}) \\, \\mbox{$\\xi $}+ \\frac{1}{2} \\, \\mu (\\mbox{\\bf x})\\, \\mbox{$\\xi $}^T f^{\\prime \\prime }(\\mbox{\\bf x}) \\, \\mbox{$\\xi $}+ \\mbox{$\\xi $}^T \\,\\mu ^{\\prime }(\\mbox{\\bf x})f^{\\prime }(\\mbox{\\bf x})^T \\, \\mbox{$\\xi $}] \\, r^2 + o(r^2) \\Big \\rbrace \\; \\nu (d \\, \\mbox{$\\xi $})\\\\= & \\mu (\\mbox{\\bf x})f(\\mbox{\\bf x}) \\, d \\, V_d + \\frac{1}{2} \\, [f(\\mbox{\\bf x})\\, \\mathrm {tr}(\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) + \\mu (\\mbox{\\bf x})\\, \\, \\mathrm {tr}(f^{\\prime \\prime }(\\mbox{\\bf x}))] V_d \\, r^2 + \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x}) V_d \\, r^2 + o(r^2).$ With the above results, by L'Hôpital's rule $m(r)& = \\lim _{\\delta \\rightarrow 0} \\frac{\\mathbb {E} \\, [\\mu (\\mbox{\\bf X}) \\mathbb {1}(r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta )]}{\\mathbb {P} \\, (r \\le \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert \\le r + \\delta )} \\\\& = \\frac{\\mu (\\mbox{\\bf x})f(\\mbox{\\bf x}) \\, d \\, V_d + \\frac{1}{2} \\, [f(\\mbox{\\bf x})\\, \\mathrm {tr}(\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) + \\mu (\\mbox{\\bf x})\\, \\, \\mathrm {tr}(f^{\\prime \\prime }(\\mbox{\\bf x}))] V_d \\, r^2 + \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x}) V_d \\, r^2 + o(r^2)}{f(\\mbox{\\bf x})\\, d \\, V_d + \\frac{1}{2} \\, \\mathrm {tr}(f^{\\prime \\prime }(\\mbox{\\bf x})) V_d \\, r^2 + o(r^2)}\\\\& = \\mu (\\mbox{\\bf x}) + \\frac{f(\\mbox{\\bf x})\\mathrm {tr}(\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) + 2 \\, \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x})}{2 \\, d \\, f(\\mbox{\\bf x})} \\; r^2 + o(r^2).$ This completes the proof of Lemma 2.",
"Lemma 2 tells us that when the distance $r \\rightarrow 0$ , the difference in projections onto the half line $\\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert $ can be approximated by the squared distance $r^2$ .",
"We already know the order of squared distance from Lemma 1.",
"The only remaining question is the discrepancy between the points and their projections."
],
[
"Proof of Theorem 1",
"Suppose $\\mbox{\\bf X}_{(1)}$ is known.",
"Then $\\mathbb {E} \\, [Y \\, | \\, \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert = \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ] \\ne \\mathbb {E} \\, [Y \\, | \\, \\mbox{\\bf X}= \\mbox{\\bf X}_{(1)}]$ in general.",
"The former is the mean on a sphere while the latter is the mean at a point of the sphere.",
"However, for the iterative expectation, the unconditional mean of the response of the 1-nearest neighbor $\\mathbb {E} \\,Y_{(1)} = \\mathbb {E}_{\\tiny \\mbox{\\bf X}_{(1)}} \\, \\mathbb {E} \\, [Y \\, | \\, \\mbox{\\bf X}= \\mbox{\\bf X}_{(1)}] = \\mathbb {E}_{\\tiny \\mbox{\\bf X}_{(1)}} \\, \\mathbb {E} \\, [Y \\, | \\, \\Vert \\mbox{\\bf X}- \\mbox{\\bf x}\\Vert = \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert ]$ .",
"Letting $r_{(1)} = \\Vert \\mbox{\\bf X}_{(1)} - \\mbox{\\bf x}\\Vert $ , then $\\mathbb {E} \\, m(r_{(1)}) = \\mathbb {E} \\, Y_{(1)}$ .",
"One interesting fact is that their variances are different in general.",
"This fact will be revisited in Lemma 3.",
"We are now able to finish the proof of Theorem 1, $\\mathbb {E} \\, D_n(s) (\\mbox{\\bf x})& = \\mathbb {E} \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}) \\\\& = \\mathbb {E} \\, [Y_{(1)}(\\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s})] \\\\& = \\mathbb {E} \\, [m(r_{(1)})(\\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s})] \\\\& = \\mu (\\mbox{\\bf x}) + \\Gamma (2/d + 1) \\frac{f(\\mbox{\\bf x}) \\, tr (\\mu ^{\\prime \\prime }(\\mbox{\\bf x})) +2 \\, \\mu ^{\\prime }(\\mbox{\\bf x})^Tf^{\\prime }(\\mbox{\\bf x})}{2 \\, d \\, V_d^{2/d} \\, f(\\mbox{\\bf x})^{1+2/d}} \\, s^{-2/d} + o(s^{-2/d}).$ The result follows directly from Lemmas 1–2.",
"To summarize, the projection idea [14] enables us to derive the finite sample bias of the DNN estimator.",
"Moreover, the coefficient here does not depend on the subsample size $s$ .",
"This interesting fact opens the door to the new two-scale framework."
],
[
"Proof of Theorem 2",
"We prove the asymptotic normality of the DNN estimator in Theorem 2.",
"Our proof builds on the U-statistic framework.",
"The classical results of the asymptotic normality of U-statistics are not yet directly applicable since the subsampling scale $s \\rightarrow \\infty $ .",
"However, we follow the routine of proving the classical U-statistics [38] and then give a sufficient condition that makes the transition fluent.",
"The condition we give in this paper is $s/n \\rightarrow 0$ .",
"This condition is so intuitive a posteriori that it simply means that the subsample size $s$ is relatively small compared to the whole sample size $n$ and can be treated as a constant as in classical U-statistics.",
"A related approach is [65]'s approach which is built on the ANOVA framework [25]."
],
[
"Lemma 3 and its proof",
"Lemma 3 prepares us the order of variance of the first order Hájek projections.",
"This interpretation will soon explain itself in the formal proof of Theorem 2.",
"As always, we introduce the new notation first.",
"Given $\\mbox{\\bf x}$ , the projection of $\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s)$ onto $\\mbox{\\bf Z}_1$ is denoted as $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1)$ , $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1) = \\mathbb {E} \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) | \\mbox{\\bf Z}_1 = \\mbox{\\bf z}_1] = \\mathbb {E} \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s)].$ In this section, let $\\mathbb {E}_1$ ($\\mathrm {var}_1$ ) and $\\mathbb {E}_2$ ($\\mathrm {var}_2$ ) denote expectation (variance) with respect to $\\mbox{\\bf Z}_1$ and $\\lbrace \\mbox{\\bf Z}_2, \\mbox{\\bf Z}_3, \\ldots , \\mbox{\\bf Z}_s\\rbrace $ , respectively, and $\\widetilde{\\mbox{\\bf X}}_{(1)}$ the closest $\\mbox{\\bf X}$ to $\\mbox{\\bf x}$ among $\\lbrace \\mbox{\\bf X}_2, \\mbox{\\bf X}_3, \\ldots , \\mbox{\\bf X}_s\\rbrace $ .",
"Lemma 3 Given $\\mbox{\\bf x}$ , under Conditions 2 and 3, for the variance of $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1)$ , denoted as $\\eta _1$ , when $s \\rightarrow \\infty $ , we have $\\eta _1 = \\mathrm {var}_1 \\, \\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1) = \\frac{\\sigma ^2}{2s-1} + o(s^{-2}).$ Proof of Lemma 3: The strategy is first to decompose $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1)$ into several terms.",
"We then analyze the terms one by one and try to understand them intuitively.",
"Their properties will be carefully studied.",
"Finally, we use these results to derive the order of the variance of $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1)$ .",
"Observe that $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1)& = \\mathbb {E}_2 \\, \\Phi (x; \\mbox{\\bf z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\\\& = \\mathbb {E}_2 \\, [\\Phi (x; \\mbox{\\bf z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert )] \\\\& \\qquad \\qquad + \\mathbb {E}_2 \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert > \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert )] \\\\& = \\mathbb {E}_2 \\, y_1 \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert ) \\\\& \\qquad \\qquad + \\mathbb {E}_2 \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert > \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}|)] \\\\& = y_1 \\, \\mathbb {E}_2 \\, \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert ) \\\\& \\qquad \\qquad + \\mathbb {E}_2 \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\\\& \\qquad \\qquad \\qquad \\qquad - \\mathbb {E}_2 \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\mathbb {1}(\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert )] \\\\& = \\mathbb {E}_2 \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) + \\mbox{\\bf A}\\mbox{\\bf B},$ where $\\mbox{\\bf A}= y_1 - \\mathbb {E}_2 \\, [\\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\; | \\: \\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert ],$ $\\mbox{\\bf B}= \\mathbb {E}_2 \\, \\mathbb {1} (\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert ).$ Despite its complicated appearance, the representation above is actually very intuitive.",
"It means that to the value of the response of the 1-nearest neighbor, the marginal contribution of knowing the first observation $\\mbox{\\bf z}_1$ , $\\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1) - \\mathbb {E}_2 \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s)$ , equals the product of the marginal contribution from observation $y_1$ when $\\mbox{\\bf x}_1$ is actually closer than the rest, $\\mbox{\\bf A}$ , and the probability of the scenario, $\\mbox{\\bf B}$ .",
"For $\\mbox{\\bf A}$ and $\\mbox{\\bf B}$ , there are some useful facts.",
"$\\mathbb {E}_1 \\, \\mbox{\\bf A}\\mbox{\\bf B}= \\mathbb {E} \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) - \\mathbb {E}_2 \\, \\Phi (\\mbox{\\bf x}; \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s) \\\\ = B(s) - B(s-1) = O(s^{-2/d}) - O((s-1)^{-2/d}) = o(s^{-1}) $ .",
"$\\mbox{\\bf A}$ and $\\mbox{\\bf B}$ are asymptotically independent when $s \\rightarrow \\infty $ since $\\mbox{\\bf B}$ degenerates.",
"$\\mathbb {E}_1 \\; \\mbox{\\bf B}= \\mathbb {E}_1 \\, \\mathbb {E}_2 \\, \\mathbb {1} (\\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert \\le \\Vert \\widetilde{\\mbox{\\bf X}}_{(1)} - \\mbox{\\bf x}\\Vert ) = \\frac{1}{s}$ , by symmetry.",
"$\\mathbb {E}_1 \\; \\mbox{\\bf B}= \\mathbb {E}_1 \\, [1 - \\rho (B(\\mbox{\\bf x}, \\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert ))]^{s-1}$ .",
"$\\mathbb {E}_1 \\; \\mbox{\\bf B}^2 = \\mathbb {E}_1 \\, [1 - \\rho (B(\\mbox{\\bf x}, \\Vert \\mbox{\\bf x}_1 - \\mbox{\\bf x}\\Vert ))]^{2s-2} = \\frac{1}{2s-1}$ .",
"Combining the results together, if we denote $\\sigma ^2 = \\mathbb {E}_1 \\; \\mbox{\\bf A}^2 < \\infty $ , then as $s \\rightarrow \\infty $ , $\\eta _1 = \\mathrm {var}_1 \\, \\Phi _1(\\mbox{\\bf x}, \\mbox{\\bf z}_1) = \\mathrm {var}_1 (\\mbox{\\bf A}\\mbox{\\bf B}) = \\mathbb {E}_1 \\mbox{\\bf A}^2 \\mbox{\\bf B}^2 - (\\mathbb {E}_1 \\, \\mbox{\\bf A}\\mbox{\\bf B})^2 = \\frac{\\sigma ^2}{2s-1} + o(s^{-2}).$ We emphasize that $\\sigma ^2$ is different from $\\sigma _\\epsilon ^2$ in general, as mentioned earlier in the proof of Theorem 1.",
"Besides the noise variance $\\sigma _\\epsilon ^2$ , $\\sigma ^2$ also consists of the variation arising from landing at different points on the sphere with the same radius.",
"Thus, $\\sigma ^2$ also depends on the target point $\\mbox{\\bf x}$ and the density function $f$ .",
"We have also restricted the response $y$ to be bounded, thus $\\sigma ^2 = O(1)$ .",
"The proof of Lemma 3 completes."
],
[
"Proof of Theorem 2",
"In this section, we omit $\\mbox{\\bf x}$ for simplicity whenever there is no confusion.",
"We will first introduce Hoeffding's canonical decomposition [41].",
"It is an extension of the projection idea.",
"Then we will find that the Hájek projection can be seen as the first order part of the decomposition.",
"And since the Hájek projection is the sum of $i.i.d.$ terms, it can be asymptotically normal.",
"Finally, we explicitly compare the orders between different terms and give a sufficient condition for our DNN estimator to achieve asymptotic normality.",
"We first demonstrate Hoeffding's canonical decomposition [41].",
"To ease notation, we use $\\mbox{\\bf Z}_i$ as a shorthand for $(\\mbox{\\bf X}_i,Y_i)$ .",
"The following definitions can be made as natural extensions of the $\\Phi _1$ projection in Lemma 3.",
"Define $\\Phi _1(\\mbox{\\bf z}_1) & = \\mathbb {E} \\, \\Phi (\\mbox{\\bf z}_1, \\mbox{\\bf Z}_2, \\ldots , \\mbox{\\bf Z}_s),\\\\\\Phi _2(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2) & = \\mathbb {E} \\, \\Phi (\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf Z}_3, \\ldots , \\mbox{\\bf Z}_s), \\\\& \\; \\; \\vdots \\\\\\Phi _s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3, \\ldots , \\mbox{\\bf z}_s) & = \\mathbb {E} \\, \\Phi (\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3, \\ldots , \\mbox{\\bf z}_s), \\\\\\widetilde{\\Phi }_1(\\mbox{\\bf z}_1) & = \\Phi _1(\\mbox{\\bf z}_1) - \\mathbb {E} \\, \\Phi , \\\\\\widetilde{\\Phi }_2(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2) & = \\Phi _2(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2) - \\mathbb {E} \\, \\Phi , \\\\& \\; \\; \\vdots \\\\\\widetilde{\\Phi }_s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3, \\ldots , \\mbox{\\bf z}_s) & = \\Phi _s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3, \\ldots , \\mbox{\\bf z}_s) - \\mathbb {E} \\, \\Phi .$ The canonical terms are defined as $g_1(\\mbox{\\bf z}_1) & = \\widetilde{\\Phi }_1(\\mbox{\\bf z}_1), \\\\g_2(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2) & = \\widetilde{\\Phi }_2(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2) - g_1(\\mbox{\\bf z}_1) - g_2(\\mbox{\\bf z}_2), \\\\g_3(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3) & = \\widetilde{\\Phi }_3(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3) - \\sum _{i=1}^3 g_1(\\mbox{\\bf z}_i) - \\sum _{1 \\le i < j \\le 3} g_2(\\mbox{\\bf z}_i, \\mbox{\\bf z}_j) , \\\\& \\; \\; \\vdots \\\\g_s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\ldots , \\mbox{\\bf z}_s) & = \\widetilde{\\Phi }_s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\mbox{\\bf z}_3, \\ldots , \\mbox{\\bf z}_s) - \\sum _{i=1}^s g_1(\\mbox{\\bf z}_i) - \\sum _{1 \\le i < j \\le s} g_2(\\mbox{\\bf z}_i, \\mbox{\\bf z}_j) - \\cdots \\\\& \\qquad - \\cdots - \\sum _{1 \\le i_1 \\le i_2 \\le \\cdots \\le i_{s-1} \\le s} g_{s-1}(\\mbox{\\bf z}_{i_1},\\mbox{\\bf z}_{i_2}, \\ldots , \\mbox{\\bf z}_{i_{s-1}}).$ So the kernel $\\Phi $ can then be written as the sum of canonical terms, $ \\Phi (\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\ldots , \\mbox{\\bf z}_s) - \\mathbb {E} \\, \\Phi = \\sum _{i=1}^{s} g_1(\\mbox{\\bf z}_i) + \\sum _{1 \\le i < j \\le s} g_2(\\mbox{\\bf z}_i, \\mbox{\\bf z}_j) + \\ldots + g_s(\\mbox{\\bf z}_1, \\mbox{\\bf z}_2, \\ldots , \\mbox{\\bf z}_s).$ Another perspective to look at the above equation is the Efron-Stein ANOVA decomposition [25].",
"They found that a symmetric kernel can be decomposed into $2^n - 1$ random variables which are all with zero mean and uncorrelated.",
"From either perspective, we have $\\mathrm {var}\\; \\Phi (\\mbox{\\bf Z}_1, \\ldots , \\mbox{\\bf Z}_s) = \\binom{s}{1} \\,\\mathbb {E} \\, g_1^2 + \\binom{s}{2} \\, \\mathbb {E} \\, g_2^2 + \\ldots + \\binom{s}{s} \\, \\mathbb {E} \\, g_s^2.$ We now use the Equation REF to decompose $D_n(s)$ , $& D_n(s) - \\mathbb {E} \\, D_n(s) \\\\& = \\binom{n}{s}^{-1} \\sum _{1 \\le i_1<i_2< \\ldots < i_s \\le n} \\widetilde{\\Phi }(\\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}) \\\\& = \\binom{n}{s}^{-1} \\Big \\lbrace \\binom{n-1}{s-1} \\sum _{i=1}^n g_1(\\mbox{\\bf Z}_i) + \\binom{n-2}{s-2} \\sum _{1 \\le i < j \\le n} g_2(\\mbox{\\bf Z}_i, \\mbox{\\bf Z}_j) + \\ldots \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad + \\binom{n-s}{s-s} \\sum _{1 \\le i_1 < i_2 < \\ldots < i_s \\le n} g_s(\\mbox{\\bf Z}_{i_1}, \\mbox{\\bf Z}_{i_2}, \\ldots , \\mbox{\\bf Z}_{i_s}) \\Big \\rbrace .$ The Hájek projection [38] can be seen as part of Hoeffding's canonical decomposition.",
"In this paper, the Hájek projection $\\widehat{D}_n(s)$ of $D_n(s) - \\mathbb {E} D_n(s)$ , is $\\widehat{D}_n(s) = \\binom{n}{s}^{-1} \\binom{n-1}{s-1} \\sum _{i=1}^{n} g_1(\\mbox{\\bf Z}_i).$ The variance of the Hájek projection, denoted as $\\sigma _n^2$ , is $\\sigma _n^2 = \\mathrm {var}\\; \\widehat{D}_n(s) = \\frac{s^2}{n} \\, \\mathrm {var}\\, \\Phi _1(\\mbox{\\bf x}; \\mbox{\\bf z}_1) = \\frac{s^2}{n} \\, \\eta _1.$ Since the Hájek projection is the average of independent and identically distributed terms, by the Lindeberg–Lévy Central Limit Theorem [15], we can have $\\frac{\\widehat{D}_n(s)}{\\sigma _n} \\overset{{D}}{\\longrightarrow }N(0,1).$ Next we will find the appropriate condition to ensure $\\frac{D_n(s) - \\mathbb {E} \\, D_n(s)}{\\sigma _n} \\overset{{D}}{\\longrightarrow }N(0,1)$ .",
"One immediate choice is $\\frac{D_n(s) - \\mathbb {E}D_n(s) - \\widehat{D}_n(s)}{\\sigma _n} \\overset{\\mathrm {P}}{\\longrightarrow }0$ .",
"Moreover, we also have this inequality as in [65], $[D_n(s) - \\mathbb {E}D_n(s) - \\widehat{D}_n(s)]^2& = \\binom{n}{s}^{-2} \\Big \\lbrace \\binom{n-2}{s-2}^2 \\binom{n}{2} \\mathbb {E} \\, g_2^2 + \\ldots + \\binom{n-s}{s-s}^2 \\binom{n}{s} \\mathbb {E} \\, g_s^2 \\Big \\rbrace \\\\& = \\sum _{r=2}^s \\Big \\lbrace \\binom{n}{s}^{-2} \\binom{n-r}{s-r}^2 \\binom{n}{r} \\mathbb {E} \\, g_r^2 \\Big \\rbrace \\\\& = \\sum _{r=2}^s \\Big \\lbrace \\frac{s!(n-r)!}{n!(s-r)!}",
"\\binom{s}{r} \\mathbb {E} \\, g_r^2 \\Big \\rbrace \\\\& \\le \\frac{s(s-1)}{n(n-1)} \\; \\sum _{r=2}^s \\binom{s}{r} \\mathbb {E} \\, g_r^2 \\\\& \\le \\frac{s^2}{n^2} \\; \\mathrm {var}\\; \\Phi .$ Combining the results we have obtained, $\\mathrm {var}\\, \\Phi $ is bounded and $\\eta _1 = O(\\frac{1}{s})$ , one sufficient condition is convergence in mean square and when $s/n \\rightarrow 0$ , $\\frac{[D_n(s) - \\mathbb {E} D_n(s) - \\hat{D}_n(s)]^2}{\\sigma _n^2} \\le \\frac{\\frac{s^2}{n^2} \\, \\mathrm {var}\\; \\Phi }{\\frac{s^2}{n} \\; \\eta _1} \\rightarrow 0.$ This completes the proof of Theorem 2."
],
[
"Proofs of Theorem 3 and 4",
"They follow naturally from the proof of Theorem 2 in Section A.2."
]
] | 1808.08469 |
[
[
"Semi-Supervised Event Extraction with Paraphrase Clusters"
],
[
"Abstract Supervised event extraction systems are limited in their accuracy due to the lack of available training data.",
"We present a method for self-training event extraction systems by bootstrapping additional training data.",
"This is done by taking advantage of the occurrence of multiple mentions of the same event instances across newswire articles from multiple sources.",
"If our system can make a highconfidence extraction of some mentions in such a cluster, it can then acquire diverse training examples by adding the other mentions as well.",
"Our experiments show significant performance improvements on multiple event extractors over ACE 2005 and TAC-KBP 2015 datasets."
],
[
"Credits",
"This document has been adapted from the instructions for earlier ACL and NAACL proceedings, including those for ACL 2017 by Dan Gildea and Min-Yen Kan, NAACL 2017 by Margaret Mitchell, ACL 2012 by Maggie Li and Michael White, those from ACL 2010 by Jing-Shing Chang and Philipp Koehn, those for ACL 2008 by Johanna D. Moore, Simone Teufel, James Allan, and Sadaoki Furui, those for ACL 2005 by Hwee Tou Ng and Kemal Oflazer, those for ACL 2002 by Eugene Charniak and Dekang Lin, and earlier ACL and EACL formats.",
"Those versions were written by several people, including John Chen, Henry S. Thompson and Donald Walker.",
"Additional elements were taken from the formatting instructions of the International Joint Conference on Artificial Intelligence and the Conference on Computer Vision and Pattern Recognition."
],
[
"Introduction",
"The following instructions are directed to authors of papers submitted to NAACL-HLT 2018 or accepted for publication in its proceedings.",
"All authors are required to adhere to these specifications.",
"Authors are required to provide a Portable Document Format (PDF) version of their papers.",
"The proceedings are designed for printing on A4 paper."
],
[
"General Instructions",
"Manuscripts must be in two-column format.",
"Exceptions to the two-column format include the title, authors' names and complete addresses, which must be centered at the top of the first page, and any full-width figures or tables (see the guidelines in Subsection REF ).",
"Type single-spaced.",
"Start all pages directly under the top margin.",
"See the guidelines later regarding formatting the first page.",
"The manuscript should be printed single-sided and its length should not exceed the maximum page limit described in Section .",
"Pages are numbered for initial submission.",
"However, do not number the pages in the camera-ready version.",
"By uncommenting \\aclfinalcopy at the top of this document, it will compile to produce an example of the camera-ready formatting; by leaving it commented out, the document will be anonymized for initial submission.",
"When you first create your submission on softconf, please fill in your submitted paper ID where *** appears in the \\def\\aclpaperid{***} definition at the top.",
"The review process is double-blind, so do not include any author information (names, addresses) when submitting a paper for review.",
"However, you should maintain space for names and addresses so that they will fit in the final (accepted) version.",
"The NAACL-HLT 2018 style will create a titlebox space of 2.5in for you when \\aclfinalcopy is commented out.",
"The author list for submissions should include all (and only) individuals who made substantial contributions to the work presented.",
"Each author listed on a submission to NAACL-HLT 2018 will be notified of submissions, revisions and the final decision.",
"No authors may be added to or removed from submissions to NAACL-HLT 2018 after the submission deadline."
],
[
"The Ruler",
"The NAACL-HLT 2018 style defines a printed ruler which should be presented in the version submitted for review.",
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"The camera ready copy should not contain a ruler.",
"( users may uncomment the \\aclfinalcopy command in the document preamble.)",
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"In most cases one would expect that the approximate location will be adequate, although you can also use fractional references (e.g., the first paragraph on this page ends at mark $108.5$ )."
],
[
"Electronically-available resources",
"NAACL-HLT provides this description in 2e (naaclhlt2018.tex) and PDF format (naaclhlt2018.pdf), along with the 2e style file used to format it (naaclhlt2018.sty) and an ACL bibliography style (acl_natbib.bst) and example bibliography (naaclhlt2018.bib).",
"These files are all available at http://naacl2018.org/downloads/ naaclhlt2018-latex.zip.",
"We strongly recommend the use of these style files, which have been appropriately tailored for the NAACL-HLT 2018 proceedings."
],
[
"Format of Electronic Manuscript",
"For the production of the electronic manuscript you must use Adobe's Portable Document Format (PDF).",
"PDF files are usually produced from using the pdflatex command.",
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"In this case, try alternative ways to obtain the PDF.",
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"It is of utmost importance to specify the A4 format (21 cm x 29.7 cm) when formatting the paper.",
"When working with dvips, for instance, one should specify -t a4.",
"Or using the command \\special{papersize=210mm,297mm} in the latex preamble (directly below the \\usepackage commands).",
"Then using dvipdf and/or pdflatex which would make it easier for some.",
"Print-outs of the PDF file on A4 paper should be identical to the hardcopy version.",
"If you cannot meet the above requirements about the production of your electronic submission, please contact the publication chairs as soon as possible."
],
[
"Layout",
"Format manuscripts two columns to a page, in the manner these instructions are formatted.",
"The exact dimensions for a page on A4 paper are: Left and right margins: 2.5 cm Top margin: 2.5 cm Bottom margin: 2.5 cm Column width: 7.7 cm Column height: 24.7 cm Gap between columns: 0.6 cm Papers should not be submitted on any other paper size.",
"If you cannot meet the above requirements about the production of your electronic submission, please contact the publication chairs above as soon as possible."
],
[
"Fonts",
"For reasons of uniformity, Adobe's Times Roman font should be used.",
"In 2e this is accomplished by putting \\usepackage{times} \\usepackage{latexsym} in the preamble.",
"If Times Roman is unavailable, use Computer Modern Roman (2e's default).",
"Note that the latter is about 10% less dense than Adobe's Times Roman font.",
"Table: Font guide."
],
[
"The First Page",
"Center the title, author's name(s) and affiliation(s) across both columns.",
"Do not use footnotes for affiliations.",
"Do not include the paper ID number assigned during the submission process.",
"Use the two-column format only when you begin the abstract.",
"Title: Place the title centered at the top of the first page, in a 15-point bold font.",
"(For a complete guide to font sizes and styles, see Table REF ) Long titles should be typed on two lines without a blank line intervening.",
"Approximately, put the title at 2.5 cm from the top of the page, followed by a blank line, then the author's names(s), and the affiliation on the following line.",
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"Do not format title and section headings in all capitals as well except for proper names (such as “BLEU”) that are conventionally in all capitals.",
"The affiliation should contain the author's complete address, and if possible, an electronic mail address.",
"Start the body of the first page 7.5 cm from the top of the page.",
"The title, author names and addresses should be completely identical to those entered to the electronical paper submission website in order to maintain the consistency of author information among all publications of the conference.",
"If they are different, the publication chairs may resolve the difference without consulting with you; so it is in your own interest to double-check that the information is consistent.",
"Abstract: Type the abstract at the beginning of the first column.",
"The width of the abstract text should be smaller than the width of the columns for the text in the body of the paper by about 0.6 cm on each side.",
"Center the word Abstract in a 12 point bold font above the body of the abstract.",
"The abstract should be a concise summary of the general thesis and conclusions of the paper.",
"It should be no longer than 200 words.",
"The abstract text should be in 10 point font.",
"Text: Begin typing the main body of the text immediately after the abstract, observing the two-column format as shown in the present document.",
"Do not include page numbers.",
"Indent: Indent when starting a new paragraph, about 0.4 cm.",
"Use 11 points for text and subsection headings, 12 points for section headings and 15 points for the title.",
"Table: Example commands for accented characters, to be used in, e.g., Bib names."
],
[
"Sections",
"Headings: Type and label section and subsection headings in the style shown on the present document.",
"Use numbered sections (Arabic numerals) in order to facilitate cross references.",
"Number subsections with the section number and the subsection number separated by a dot, in Arabic numerals.",
"Do not number subsubsections.",
"Table: Citation commands supported by the style file.The citation style is based on the natbib package andsupports all natbib citation commands.It also supports commands defined in previous ACL style filesfor compatibility.Citations: Citations within the text appear in parentheses as or, if the author's name appears in the text itself, as Gusfield Gusfield:97.",
"Using the provided style, the former is accomplished using \\cite and the latter with \\shortcite or \\newcite.",
"Collapse multiple citations as in , ; this is accomplished with the provided style using commas within the \\cite command, e.g., \\cite{Gusfield:97,Aho:72}.",
"Append lowercase letters to the year in cases of ambiguities.",
"Treat double authors as in , but write as in when more than two authors are involved.",
"Collapse multiple citations as in , .",
"Also refrain from using full citations as sentence constituents.",
"We suggest that instead of “ showed that ...” you use “Gusfield Gusfield:97 showed that ...” If you are using the provided and Bib style files, you can use the command \\citet (cite in text) to get “author (year)” citations.",
"If the Bib file contains DOI fields, the paper title in the references section will appear as a hyperlink to the DOI, using the hyperref package.",
"To disable the hyperref package, load the style file with the nohyperref option: \\usepackage[nohyperref]{naaclhlt2018} Digital Object Identifiers: As part of our work to make ACL materials more widely used and cited outside of our discipline, ACL has registered as a CrossRef member, as a registrant of Digital Object Identifiers (DOIs), the standard for registering permanent URNs for referencing scholarly materials.",
"As of 2017, we are requiring all camera-ready references to contain the appropriate DOIs (or as a second resort, the hyperlinked ACL Anthology Identifier) to all cited works.",
"Thus, please ensure that you use Bib records that contain DOI or URLs for any of the ACL materials that you reference.",
"Appropriate records should be found for most materials in the current ACL Anthology at http://aclanthology.info/.",
"As examples, we cite to show you how papers with a DOI will appear in the bibliography.",
"We cite to show how papers without a DOI but with an ACL Anthology Identifier will appear in the bibliography.",
"As reviewing will be double-blind, the submitted version of the papers should not include the authors' names and affiliations.",
"Furthermore, self-references that reveal the author's identity, e.g., “We previously showed ...” should be avoided.",
"Instead, use citations such as “ Gusfield:97 previously showed ... ” Any preliminary non-archival versions of submitted papers should be listed in the submission form but not in the review version of the paper.",
"NAACL-HLT 2018 reviewers are generally aware that authors may present preliminary versions of their work in other venues, but will not be provided the list of previous presentations from the submission form.",
"Please do not use anonymous citations and do not include when submitting your papers.",
"Papers that do not conform to these requirements may be rejected without review.",
"References: Gather the full set of references together under the heading References; place the section before any Appendices, unless they contain references.",
"Arrange the references alphabetically by first author, rather than by order of occurrence in the text.",
"By using a .bib file, as in this template, this will be automatically handled for you.",
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"The and Bib style files provided roughly fit the American Psychological Association format, allowing regular citations, short citations and multiple citations as described above.",
"Example citing an arxiv paper: .",
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"See corresponding .bib file for further details.",
"Submissions should accurately reference prior and related work, including code and data.",
"If a piece of prior work appeared in multiple venues, the version that appeared in a refereed, archival venue should be referenced.",
"If multiple versions of a piece of prior work exist, the one used by the authors should be referenced.",
"Authors should not rely on automated citation indices to provide accurate references for prior and related work.",
"Appendices: Appendices, if any, directly follow the text and the references (but see above).",
"Letter them in sequence and provide an informative title: Appendix A.",
"Title of Appendix."
],
[
"Footnotes",
"Footnotes: Put footnotes at the bottom of the page and use 9 point font.",
"They may be numbered or referred to by asterisks or other symbols.This is how a footnote should appear.",
"Footnotes should be separated from the text by a line.Note the line separating the footnotes from the text."
],
[
"Graphics",
"Illustrations: Place figures, tables, and photographs in the paper near where they are first discussed, rather than at the end, if possible.",
"Wide illustrations may run across both columns.",
"Color illustrations are discouraged, unless you have verified that they will be understandable when printed in black ink.",
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"Caption of the Figure.” “Table 1.",
"Caption of the Table.” Type the captions of the figures and tables below the body, using 11 point text."
],
[
"Accessibility",
"In an effort to accommodate people who are color-blind (as well as those printing to paper), grayscale readability for all accepted papers will be encouraged.",
"Color is not forbidden, but authors should ensure that tables and figures do not rely solely on color to convey critical distinctions.",
"A simple criterion: All curves and points in your figures should be clearly distinguishable without color."
],
[
"Translation of non-English Terms",
"It is also advised to supplement non-English characters and terms with appropriate transliterations and/or translations since not all readers understand all such characters and terms.",
"Inline transliteration or translation can be represented in the order of: original-form transliteration “translation”."
],
[
"Length of Submission",
"The NAACL-HLT 2018 main conference accepts submissions of long papers and short papers.",
"Long papers may consist of up to eight (8) pages of content plus unlimited pages for references.",
"Upon acceptance, final versions of long papers will be given one additional page – up to nine (9) pages of content plus unlimited pages for references – so that reviewers' comments can be taken into account.",
"Short papers may consist of up to four (4) pages of content, plus unlimited pages for references.",
"Upon acceptance, short papers will be given five (5) pages in the proceedings and unlimited pages for references.",
"For both long and short papers, all illustrations and tables that are part of the main text must be accommodated within these page limits, observing the formatting instructions given in the present document.",
"Supplementary material in the form of appendices does not count towards the page limit; see appendix A for further information.",
"However, note that supplementary material should be supplementary (rather than central) to the paper, and that reviewers may ignore supplementary material when reviewing the paper (see Appendix ).",
"Papers that do not conform to the specified length and formatting requirements are subject to be rejected without review.",
"Workshop chairs may have different rules for allowed length and whether supplemental material is welcome.",
"As always, the respective call for papers is the authoritative source."
],
[
"Acknowledgments",
"The acknowledgments should go immediately before the references.",
"Do not number the acknowledgments section.",
"Do not include this section when submitting your paper for review.",
"Preparing References: Include your own bib file like this: \\bibliographystyle{acl_natbib} \\bibliography{naaclhlt2018} Where naaclhlt2018 corresponds to a naaclhlt2018.bib file."
],
[
"Supplemental Material",
"Submissions may include resources (software and/or data) used in in the work and described in the paper.",
"Papers that are submitted with accompanying software and/or data may receive additional credit toward the overall evaluation score, and the potential impact of the software and data will be taken into account when making the acceptance/rejection decisions.",
"Any accompanying software and/or data should include licenses and documentation of research review as appropriate.",
"NAACL-HLT 2018 also encourages the submission of supplementary material to report preprocessing decisions, model parameters, and other details necessary for the replication of the experiments reported in the paper.",
"Seemingly small preprocessing decisions can sometimes make a large difference in performance, so it is crucial to record such decisions to precisely characterize state-of-the-art methods.",
"Nonetheless, supplementary material should be supplementary (rather than central) to the paper.",
"Submissions that misuse the supplementary material may be rejected without review.",
"Essentially, supplementary material may include explanations or details of proofs or derivations that do not fit into the paper, lists of features or feature templates, sample inputs and outputs for a system, pseudo-code or source code, and data.",
"(Source code and data should be separate uploads, rather than part of the paper).",
"The paper should not rely on the supplementary material: while the paper may refer to and cite the supplementary material and the supplementary material will be available to the reviewers, they will not be asked to review the supplementary material.",
"Appendices (i.e.",
"supplementary material in the form of proofs, tables, or pseudo-code) should be uploaded as supplementary material when submitting the paper for review.",
"Upon acceptance, the appendices come after the references, as shown here.",
"Use \\appendix before any appendix section to switch the section numbering over to letters."
],
[
"Multiple Appendices",
"...can be gotten by using more than one section.",
"We hope you won't need that."
]
] | 1808.08622 |
[
[
"Painting Outside the Box: Image Outpainting with GANs"
],
[
"Abstract The challenging task of image outpainting (extrapolation) has received comparatively little attention in relation to its cousin, image inpainting (completion).",
"Accordingly, we present a deep learning approach based on Iizuka et al.",
"for adversarially training a network to hallucinate past image boundaries.",
"We use a three-phase training schedule to stably train a DCGAN architecture on a subset of the Places365 dataset.",
"In line with Iizuka et al., we also use local discriminators to enhance the quality of our output.",
"Once trained, our model is able to outpaint $128 \\times 128$ color images relatively realistically, thus allowing for recursive outpainting.",
"Our results show that deep learning approaches to image outpainting are both feasible and promising."
],
[
"Introduction",
"The advent of adversarial training has led to a surge of new generative applications within computer vision.",
"Given this, we aim to apply GANs to the task of image outpainting (also known as image extrapolation).",
"In this task, we are given an $m \\times n$ source image $I_s$ , and we must generate an $m \\times n + 2k$ image $I_o$ such that: $I_s$ appears in the center of $I_o$ $I_o$ looks realistic and natural Image outpainting has been relatively unexplored in literature, but a similar task called image inpainting has been widely studied.",
"In contrast to image outpainting, image inpainting aims to restore deleted portions in the interiors of images.",
"Although image inpainting and outpainting appear to be closely related, it is not immediately obvious whether techniques for the former can be directly applied to the latter.",
"Image outpainting is a challenging task, as it requires extrapolation to unknown areas in the image with less neighboring information.",
"In addition, the output images must appear realistic to the human eye.",
"One common method for achieving this in image inpainting involves using GANs [4], which we aim to repurpose for image outpainting.",
"As GANs can be difficult to train, we may need to modify the archetypal training procedure to increase stability.",
"Despite the challenges involved in its implementation, image outpainting has many novel and exciting applications.",
"For example, we can use image outpainting for panorama creation, vertically-filmed video expansion, and texture creation.",
"In this project, we focus on achieving image outpainting with $m = 128$ , $n = 64$ , and $k = 32$ ."
],
[
"Related Work",
"One of the first papers to address image outpainting used a data-driven approach combined with a graph representation of the source image [8].",
"Although the researchers were able to achieve realistic results, we hope to apply adversarial training for potentially even better results.",
"A key implementation of image inpainting using deep learning by Pathak et al.",
"introduced the notion of a Context Encoder, a CNN trained adversarially to reconstruct missing image regions based on surrounding pixels [7].",
"The results presented were relatively realistic, but still had room for visual improvement.",
"Iizuka et al.",
"improved on the Context Encoder approach for image inpainting by adding a second discriminator that only took as input the inpainted patch and its immediate surroundings [4].",
"This “local\" discriminator, combined with the already-present “global\" discriminator, allowed for the researchers to achieve very visually convincing results.",
"As a result, this approach is a promising starting point for achieving image outpainting.",
"Finally, recent work in image inpainting by Liu et al.",
"[6] utilized partial convolutions in conjunction with the perceptual and style loss first introduced by Gatys et al.",
"[3].",
"Using these techniques, the researchers were able to achieve highly realistic results with a fraction of the training required by [4].",
"In addition, the researchers provided various quantitative metrics that will be useful for evaluating our models' performance."
],
[
"Dataset",
"As a sanity check for the outpainting model architecture, we expect our model to be able to overfit on a single $128 \\times 128$ color image of a city.",
"We use a $128 \\times 128$ image as opposed to the $512 \\times 512$ image size from [4] to speed up training.",
"For this experiment, we use the same single image for training and testing.",
"Our primary dataset for image outpainting is composed of $36,500$ $256 \\times 256$ images from the Places 365 dataset [9].",
"We downsampled these images to $128 \\times 128$ .",
"This dataset is composed of a diverse set of landscapes, buildings, rooms, and other scenes from everyday life, as shown in Figure REF .",
"For training, we held out 100 images for validation, and trained on the remaining $36,400$ images.",
"Figure: City image and sample images from Places365 dataset."
],
[
"Preprocessing",
"In order to prepare our images for training, we use the following preprocessing pipeline.",
"Given a training image $I_{\\mathrm {tr}}$ , we first normalize the images to $I_{n} \\in [0,1]^{128 \\times 128 \\times 3}$ .",
"We define a mask $M \\in \\lbrace 0,1\\rbrace ^{128 \\times 128}$ such that $M_{ij} = 1 - \\mathbf {1} [ 32 \\le j < 96]$ in order to mask out the center portion of the image.",
"Next, we compute the mean pixel intensity $\\mu $ , over the unmasked region $I_{n} \\odot (1-M)$ .",
"Afterwards, we set the outer pixels of each channel to the average value $\\mu $ .",
"Formally, we define $I_{m} = \\mu \\cdot M + I_{n} \\odot (1-M)$ .",
"In the last step of preprocessing, we concatenate $I_m$ with $M$ to produce $I_p \\in [0, 1]^{128 \\times 128 \\times 4}$ .",
"Thus, as the result of preprocessing $I_{\\mathrm {tr}}$ , we output $(I_n, I_p)$ ."
],
[
"Training Pipeline",
"We adopt a DCGAN architecture $(G, D)$ similar to that used by Iizuka et al.",
"Here, the generator $G$ takes the form of an encoder-decoder CNN, while the discriminator $D$ uses strided convolutions to repeatedly downsample an image for binary classification [4].",
"In each iteration of training, we randomly sample a minibatch of training data.",
"As shown in Figure REF , for each training image $I_\\mathrm {tr}$ , we preprocess $I_\\mathrm {tr}$ to get $I_n$ and $I_p$ , as previously described.",
"We run the generator on $I_p$ to get the outpainted image $I_o = G(I_p) \\in [0, 1]^{128\\times 128\\times 3}$ .",
"Afterwards, we run the discriminator to classify the ground truth ($I_n$ ) and outpainted image ($I_o$ ).",
"We compute losses and update parameters according to our training schedule, which will be discussed next.",
"Figure: Training pipeline"
],
[
"Training Schedule",
"In order to facilitate and stabilize training, we utilize the three-phase training procedure presented by [4].",
"In this scheme, we define three loss functions: $\\mathcal {L}_{\\mathrm {MSE}}(I_n, I_p) = \\Vert M \\odot (G(I_p) - I_n)\\Vert ^2_2$ $\\mathcal {L}_D(I_n, I_p) = -\\left[\\log D(I_n) + \\log (1 - D(G(I_p)))\\right]$ $\\mathcal {L}_G(I_n, I_p) = \\mathcal {L}_{\\mathrm {MSE}}(I_n, I_p) -\\alpha \\cdot \\log D(G(I_p))$ The first phase of training (termed Phase 1) conditions the generator by updating the generator weights according to $\\mathcal {L}_{\\mathrm {MSE}}$ for $T_1$ iterations.",
"The next phase (termed Phase 2) similarly conditions the discriminator by updating the discriminator weights according to $\\mathcal {L}_D$ for $T_2$ iterations.",
"The rest of training (termed Phase 3) proceeds for $T_3$ iterations, in which the discriminator and generator are trained adversarially according to $\\mathcal {L}_D$ and $\\mathcal {L}_G$ , respectively.",
"In $\\mathcal {L}_G$ , $\\alpha $ is a tunable hyperparameter trading off the MSE loss with the standard generator GAN loss."
],
[
"Network Architecture",
"Due to computational restrictions, we propose an architecture for outpainting on Places365 that is shallower but still conceptually similar to that by Iizuka et al.",
"[4].",
"For the generator $G$ , we still maintain the encoder-decoder structure from [4], as well as dilated convolutions to increase the receptive field of neurons and improve realism.",
"For the discriminator $D$ , we still utilize local discriminators as in [4], albeit modified for image outpainting.",
"Specifically, say the discriminator is run on an input image $I_d$ (equivalent to either $I_n$ or $I_o$ during training).",
"In addition, define $I_\\ell $ to be the left half of $I_d$ , and $I_r^{\\prime }$ to be the right half of $I_d$ , flipped along the vertical axis.",
"This helps to ensure that the input to $D_\\ell $ always has the outpainted region on the left.",
"Then, to generate a prediction on $I_d$ , the discriminator computes $D_g(I_d)$ , $D_\\ell (I_\\ell )$ , and $D_\\ell (I_r^{\\prime })$ .",
"These three outputs are then fed into the concatenator $C$ , which produces the final discriminator output $p = C(D_g(I_d) \\parallel D_\\ell (I_\\ell ) \\parallel D_\\ell (I_r^{\\prime }))$ .",
"We describe the layers of our architecture in Figures REF and REF .",
"Here, $f$ is the filter size, $\\eta $ is the dilation rate, $s$ is the stride, and $n$ is the number of outputs.",
"In all networks, every layer is followed by a ReLU activation, except for the final output layer of the generator and concatenator: these are followed by a sigmoid activation.",
"Figure: Generator, GGFigure: Discriminator, DD"
],
[
"Evaluation Metrics",
"Although the output of the generator is best evaluated qualitatively, we still utilize RMSE as our primary quantitative metric.",
"Given a ground truth image $I_{\\mathrm {tr}} \\in [0, 255]^{128\\times 128\\times 3}$ and a normalized generator output image $I_o^{\\prime } = 255 \\cdot I_o \\in [0, 255]^{128\\times 128\\times 3}$ , we define the RMSE as: $\\mathrm {RMSE}(I_{\\mathrm {tr}}, I_o^{\\prime }) = \\sqrt{\\frac{1}{|\\mathrm {supp}(M)|}\\sum \\limits _{i, j, k}(M \\odot (I_{\\mathrm {tr}} - I_o^{\\prime }))^2_{ijk}}$"
],
[
"Postprocessing",
"In order to improve the quality of the final outpainted image, we apply slight postprocessing to the generator's output $I_o$ .",
"Namely, after renormalizing $I_o$ via $I_o^{\\prime } = 255 \\cdot I_o$ , we blend the unmasked portion of $I_{\\mathrm {tr}}$ with $I_o^{\\prime }$ using OpenCV's seamless cloning, and output the blended outpainted image $I_{\\mathrm {op}}$ ."
],
[
"Overfitting on a Single Image",
"In order to test our architecture and training pipeline, we ran an initial baseline on the single city image.",
"The network was able to overfit to the image, achieving a final RMSE of only $0.885$ .",
"This suggests that the architecture is sufficiently complex, and likely able to be used for general image outpainting."
],
[
"Outpainting on Places365",
"We trained our architecture using only a global discriminator on Places365.",
"As in [4], we used a $18/2/80$ split for the different phases of training, and ran three-phase training with $T_1 = 40,950$ , $T_2 = 4,550$ , $T_3 = 182,000$ , and $\\alpha = 0.0004$ .",
"As we utilized a batch size of 16, this schedule corresponded to approximately 100 epochs and 40 hours of training on a K80 GPU.Animations of the generator's output during the course of training are available at http://marksabini.com/cs230-outpainting/.",
"At the end of training, we fed images from the validation set through the outpainting pipeline.",
"The final results, along with their RMSE values, are shown in Figure REF .",
"As seen in the third example of Figure REF , the network does not merely copy visual features and lines during outpainting.",
"Rather, it learns to hallucinate new features, as shown by the appearance of a house on the left hand side.",
"Figure: Outpainting results for a sample of held-out images in the validation set, shown alongside the original ground truth.",
"These results were produced without the aid of local discriminators.",
"The RMSE between each ( groundtruth , output )(\\textrm {ground truth}, \\textrm {output}) pair is displayed below each column.Figure REF shows the training and dev MSE loss of this full run.",
"In Phase 1, the MSE loss decreases quickly as it is directly optimized.",
"On the other hand, in Phase 3, the MSE loss increases slightly as we optimize the joint loss (3).",
"Figure: Training and dev MSE loss for training on Places365 with only a global discriminator.",
"The orange, blue, and green sections represent Phase 1, 2, and 3 of training, respectively."
],
[
"Local Discriminator",
"In order to attempt to improve the quality of our results, we also trained our architecture using a local discriminator on Places365.",
"Due to slower training, we trained this augmented architecture using three-phase training with $T_1 = 20,000$ , $T_2 = 4,000$ , $T_3 = 95,000$ , and $\\alpha = 0.0004$ .",
"With a batch size of 16, this schedule corresponded to approximately 50 epochs and 26 hours of training on a K80 GPU.",
"After training, we compared the visual quality and the RMSE of images outpainted with and without the aid of a local discriminator.",
"As seen in Figure REF , we observed that training with a local discriminator reduced vertical banding, improved color fidelity, and achieved a lower RMSE.",
"This is likely because the local discriminator is able to focus on only the outpainted regions.",
"However, the local discriminator caused training to proceed roughly $60\\%$ slower, and tended to introduce more point artifacts.",
"Figure: Training with local discriminators (LD) reduced vertical banding and improved color fidelity, but increased point artifacts and training time."
],
[
"Significance of Dilated Convolutions",
"We tuned the architecture by experimenting with different dilation rates for the dilated convolution layers of the generator.",
"We attempted to overfit our architecture on the single city image with various layer hyperparameters.",
"As shown in Figure REF , with insufficient dilation, the network fails to outpaint due to a limited receptive field of the neurons.",
"With increased dilations, the network is able to reconstruct the original image.",
"Figure: Effect of dilated convolutions.",
"[i,j,k][i, j, k] represent η\\eta for layers 4, 5, and 6 in the generator, respectively."
],
[
"Recursive Outpainting",
"An outpainted image $I_o$ can be fed again as input to the network after expanding and padding with the mean pixel value.",
"In Figure REF , we repeat this process recursively five times, expanding an image's width up to a factor of $3.5$ .",
"As expected, the noise tends to compound with successive iterations.",
"Despite this, the model successfully learns the general textures of the image and extrapolates the sky and landscape relatively realistically.",
"Figure: Each right image is the result of recursively-outpainting the corresponding left image five times."
],
[
"Conclusions",
"We were able to successfully realize image outpainting using a deep learning approach.",
"Three-phase training proved to be crucial for stability during GAN training.",
"In addition, dilated convolutions were necessary to provide sufficient receptive field to perform outpainting.",
"The results from training with only a global discriminator were fairly realistic, but augmenting the network with a local discriminator generally improved quality.",
"Finally, we investigated recursive outpainting as a means of arbitrarily extending an image.",
"Although image noise compounded with successive iterations, the recursively-outpainted image remained relatively realistic."
],
[
"Future Work",
"Going forward, there are numerous potential improvements for our image outpainting pipeline.",
"To boost the performance of the model, the generator loss could be augmented with perceptual, style, and total variation losses [3].",
"In addition, the architecture could be modified to utilize partial convolutions, as explored in [6].",
"To further stabilize training, the Wasserstein GAN algorithm could be incorporated into three-phase training [2].",
"With the aid of sequence models, the image outpainting pipeline could even be conceivably extended to perform video outpainting."
],
[
"Acknowledgements",
"We would like to thank Jay Whang for his continued mentorship throughout the course of this project.",
"We would also like to thank Stanford University's CS 230 (Deep Learning), taught by Professor Andrew Ng and Kian Katanforoosh, for offering a great learning opportunity.",
"In addition, we would like to thank Amazon Web Services for providing GPU credits."
],
[
"Supplementary Material",
"The code and accompanying poster for our project can be found at https://github.com/ShinyCode/image-outpainting."
]
] | 1808.08483 |
[
[
"Scientific Relation Extraction with Selectively Incorporated Concept\n Embeddings"
],
[
"Abstract This paper describes our submission for the SemEval 2018 Task 7 shared task on semantic relation extraction and classification in scientific papers.",
"We extend the end-to-end relation extraction model of (Miwa and Bansal) with enhancements such as a character-level encoding attention mechanism on selecting pretrained concept candidate embeddings.",
"Our official submission ranked the second in relation classification task (Subtask 1.1 and Subtask 2 Senerio 2), and the first in the relation extraction task (Subtask 2 Scenario 1)."
],
[
"Task Overview",
"The SemEval 2018 Task 7 Shared Task [8] focuses on the task of recognizing the semantic relation that holds between scientific concepts.",
"The task involves semantic relation extraction and classification into six categories specific to scientific literature: usage, result, model-feature, part_whole, topic, compare.",
"Two types of tasks are proposed: 1) identifying pairs of entities that are instances of any of the six semantic relations (extraction task), and 2) classifying instances into one of the specific relation types (classification task).",
"Consider the following input sentence: “[Unsupervised training] is first used to train a [phone n-gram model] for a particular domain.\"",
"Given the concept pair [Unsupervised training] and [phone n-gram model], the relation extraction task is to identify whether there is a relation between the concepts, while the the relation classification task is to identity the relation as usage.",
"Relation directionality is not taken into account for the evaluation of the extraction task.",
"Directionality is taken into account when relevant for the classification task (5 out of the 6 semantic relations are asymmetrical).",
"We will use this example throughout the paper to illustrate various parts of our system.",
"The SemEval 2018 Task 7 dataset contains 350 abstracts from the ACL Anthology for training and validation, and 150 abstracts for testing each subtask.",
"Since the scale of the data is small for supervised training of neural systems, we introduce several strategies to leverage a large quantity of unlabeled scientific articles.",
"In addition to initializing a neural system with pre-trained word embeddings, as in [16], we also try to incorporate embeddings of concepts that span multiple words.",
"In neural models such as [19], phrases are often represented by an average (or weighted average) of the token's sequential LSTM representation.",
"The intuition behind explicit modeling of multi-word concept embeddings is that the concept use may be different from that of its individual words.",
"Due to the size of the dataset and the nature of scientific literature, a large number of the scientific terms in the test set have never appeared in the training set, so supervised learning of the phrase embeddings is not feasible.",
"Therefore, we pre-trained scientific term embeddings on a large scientific corpus and provide a strategy to selectively incorporate the pre-trained embeddings into the relation extraction system."
],
[
"Neural Architecture Model",
"Our system is an extension of [16] and [19] with LSTM RNNs that represent both word sequences and dependency tree structures, and perform relation extraction between concepts on top of these RNNs.",
"As illustrated in Figure REF , it is composed of a 5 types of layers in a hierarchical neural model to encode context information.",
"The first two layers (token, token LSTM) use the neural modeling framework in [16].",
"The forward and backward dependency layers and the relation classification layer are based on [19].",
"The concept selection layer is novel, to the best of our knowledge.",
"The different layers are described in more detail below.",
"Figure: Neural relation extraction model with bidirectional sequentialand dependency path LSTMs."
],
[
"Token Layer.",
"The token layer concatenates three types of vector space embeddings.",
"Word embeddings are learned for words from a fixed vocabulary (plus the unknown word token), initialized using Word2vec pre-training with large scholarly corpora.",
"The character-based embedding for a token is derived from its characters as the concatenation of forward and backward representations from a bidirectional LSTM.",
"The character look-up table is initialized at random.",
"The advantage of building a character-based embedding layer is that it can handle out-of-vocabulary words and equations, which are frequent in this data, all of which are mapped to “UNK\" tokens in the Word Embedding Layer.",
"Word embeddings are learned for words from a fixed vocabulary (plus the unknown word token), initialized using Word2vec pre-training with large scholarly corpora.",
"A feature embedding is learned as a mapping from features associated with capitalization (all capital, first capital, all lower, any capital but first letter) and part-of-speech tags.",
"The embeddings are randomly initialized and trained jointly with other parameters during supervised training."
],
[
"Token LSTM Layer",
"We apply a bidirectional LSTM at the token level taking the concatenated character-word-feature embedding as input.",
"An LSTM hidden state generated in this layer is denoted as $h^S$ ."
],
[
"Forward & Backward Dependency Layers",
"Given the concept pair $(C_l,C_r)$ , the Forward Dependency Layer (generating $h^F$ ) traces from the closest common ancestor $w_a$ (for example the word “used\" in Fig.",
"REF ) to the headword $w_j$ (word “model\") of the right target concept $C_r$ ( “phone n-gram model\").",
"The Backward Dependency Layer (generating $h^B$ ) traces from the ancestor to the headword $w_i$ of the left concept $C_l$ .",
"We map the dependency relation into vector space and concatenate the resulting embedding to the embedding ($h^S$ ) of the headword of the concepts $C_l$ or $C_r$ for the backward and forward dependency layers, respectively.",
"We concatenate the resulting bi-directional LSTM vector for the headwords together with the common ancestor in both Forward & Backward Dependency Layer as input to Relation Classification Layer $h^{DP}=[\\overleftarrow{h_{w_i}^B};\\overrightarrow{h_{w_i}^B},\\overleftarrow{h_{w_j}^F};\\overrightarrow{h_{w_j}^F};\\overleftarrow{h_{w_a}^B};\\overrightarrow{h_{w_a}^B};\\overleftarrow{h_{w_a}^F};\\overrightarrow{h^F_{w_a}}]$ ."
],
[
"Concept Selection Layer",
"The concepts in the task are mostly phrases rather than single words, in the SemEval Task 7.",
"We therefore seek ways to obtain prior knowledge for those terms.",
"We train a scientific concept extraction model using the state-of-the-art scientific neural tagging technique in [16], given the scientific concept annotation in the SemEval 2018 Task7 training data.",
"We were able to achieve 79.8% F1 score (span level) to identify the scientific concepts.",
"We then use the model to extract all scientific concepts in the ACL anthology and AI2 dataset (refer to Sec. ).",
"We keep all the concepts that occur more than 10 times in the whole corpus, which results in around 15k concepts.",
"We treat each of the 15k concepts as an individual token and retrain word2vec embeddings $v_k$ together with all other single words.",
"At training time, given a scientific concept pair $(C_l,C_r)$ , we search through the 15k concepts to get all the concept candidates that have n-gram string match with $C_l$ and $C_r$ respectively (n is from 1 to the length of the target concept $C$ ).",
"For example, for the concept phone n-gram model, the candidate concepts we get are {phone n-gram, n-gram model, n-gram, model, phone}.",
"Since there may exist cases where no match could be found in the 15k concepts, we introduce a null vector $v_{\\varnothing }$ .",
"$v_{\\varnothing }$ is learned with other neural network parameters.",
"Assume there are K concept candidates in the candidate list, we denote the embeddings for the concept candidates to be $V = \\lbrace v_1 \\dots v_K, v_\\varnothing \\rbrace $ .",
"The attention weights are calculated by $\\alpha _{lk} \\propto \\exp (h_{C_l}^S W_{ATT} v_k)$ , where $v_k\\in V$ .",
"$h_{C_l}^S$ is the concatenation of bidirectional LSTM hidden states of the first and last word in $C_l$ .We also tried using the weighted average of all LSTM word embeddings in the span to calculate $h_{C_l}^S$ ; this yields a slightly worse result.",
"$W_{ATT}$ is a parameter matrix for the bilinear score for $h_{C_l}^S$ and $v_k$ .",
"The final concept embedding $v_{C_l}$ is $v_{C_l} = \\sum _{v_k \\in V} \\alpha _{lk} v_k$ .",
"For a target concept C, if exact match exists in the 15K concepts, we set the pre-trained concept embedding to be $v_{C_l}$ .",
"We concatenate the resulting embedding for both concepts in the concept pair as input to the final classification layer ($v_C = [v_{C_l};v_{C_r}]$ )."
],
[
"Relation Classification Layer",
"We concatenate the output of Forward & Backward Dependency Layer $h^{DP}$ and Concept Embedding Selection Layer $v_C$ as input to Relation Classification Layer.",
"Besides, we also introduce a distance feature between the two concepts which indicates how many other concepts there are in between the target concept pairs.",
"We concatenate the distance embedding with all the other features.",
"The concatenated features are then projected down to a lower dimension through $tanh$ function and make the final prediction through a $softmax$ function."
],
[
"External Data",
"We use two external resources for pretraining word embeddings: i) the Semantic Scholar Corpus,http://labs.semanticscholar.org/corpus/ a collection of over 20 million research papers from which we extract a subset of 110k abstracts of publications in the artificial intelligence area; and ii) the ACL Anothology Reference Corpus, which contains 22k full papers published in the ACL Anothology [6]."
],
[
"Baseline",
"We compare our model with a baseline that removes the Concept Selection Layer and replaces it with a weighted sum (using attention) of hidden states (from the Sequential LSTM Layer) for all words in a concept."
],
[
"Implementation details",
"All parameters are tuned based on dev set performance; the best parameters are selected and used for final evaluation.",
"For all experiments, we explore tuning with two different evaluation metrics: macro-F1 score and micro-F1 score.The official evaluation is macro-F1, but since the number of instances in each class is highly unbalanced, the observed macro-F1 scores were unstable.",
"We therefore introduce micro-F1 score for tuning and evaluation as well.",
"We keep the pre-trained concept embedding fixed as additional input feature.",
"The word embedding dimension is 250; the LSTM hidden dimension is 100 (for both sequential and dependency layer); the character-level hidden dimension is 25; and the optimization algorithm is SGD with a learning rate of 0.05.",
"For Subtask 2, since 5 out of 6 relation types have directionality, we add relation label “_REVERSE\" to all the 5 directional relations together with a “NONE\" type, which result in 12 labels in total.",
"For each epoch, we also randomly filter out some “NONE\" samples with probability $p$ during training, since the “NONE\" type relation dominates the training set and would bias the model towards predicting “NONE\" types.",
"We tune $p$ according to dev set, and use $p=0.4$ for the final evaluation.",
"Table: Ablation study showing the impact of neural network configurations on system performance on the dev set for the relation classification task (Subtask 2, senerio 2).-DepFeat removes the input dependency relation embeddings from the Backward & Forward Dependency Layers.",
"-DistFeat and -Concept omit the distance and concept selection features, respectively, from the final classification layer.",
"-DepLSTM removes the Backward & Forward Dependency Layers entirely (using the LSTM embeddings in the weighted token average)."
],
[
"Ablation Study",
"Table REF provides the results of an ablation study on the dev set showing the impact of removing different components of our system.",
"Looking at micro F1 scores, dependency path information is very important (performance dropped 11.5% without it), and the Concept Selection Layer is also important as it gives 2.5 absolute improvement.",
"The Dependency relation feature and the distance feature also show 1-2 points gain.",
"It is worth noticing that removing the Concept Layer (-Concept) does better than replacing it with the weighted sequential LSTM sum (Baseline).",
"With the small amount of training data, it is difficult for the baseline system to learn a good transformation from word to phrase."
],
[
"Competition Result",
"The results of our system is in Table REF .",
"We submit two sets of results, one tuned with micro F1 and the other with macro F1.",
"It turns out that even though the official evaluation metric is macro F1 score, our model tuned by micro F1 gets better results in the final competition.",
"In Subtask 1.1 and Subtask 2 scenario 2, we were the second place team with F1 score of 78.9% and 39.1% respectively.",
"We were the first place in Subtask 2 scenario 1 with 50.0% F1.",
"Table: Competition result for the top 3 teams.",
"The official evaluation metric is macro F1 score.",
"T1.1 means Subtask 1.1, T2-E means Subtask 2 senerio 1 (extraction task), T2-C means Subtask 2 senerio 2 (classification task)."
],
[
"Related Work",
"There has been growing interest in research on automatic methods to help researchers search and extract information from scientific literature.",
"Past research has addressed citation sentiment [4], [3], citation networks [13], [9], [23], [7], [12], summarization [1] and some analysis of research community [25], [2].",
"However, due to scarce hand-annotated data resources, previous work on information extraction (IE) for scientific literature is very limited.",
"Most previous work focuses on unsupervised methods for extracting scientific terms such as bootstrapping [11], [24], or extracting relations [10].",
"[16], [5], [18], [17], [15] applied semi-supervised learning and multi-task learning to neural based models to leverage large unannotated scholarly datasets for a scientific term extraction task [5].",
"Although not much supervised relation extraction work has been done on scientific literature, neural network techniqueshave obtained the state of the art for general domain relation extraction.",
"Both convolutional [22] and RNN-based architectures [26], [19], [20], [21], [14] have been successfully applied to the task and significantly improve performance."
],
[
"Conclusion",
"This paper describes the system of the UWNLP team submitted to SemEval 2018 Task 7.",
"We extend state-of-the-art neural models for information extraction by proposing a Concept Selection module which can leverage the semantic information of concepts pre-trained from a large scholarly dataset.",
"Our system ranked second in the relation classification task (subtask 1.1 and subtask 2 senerio 2), and first in the relation extraction task (subtask 2 scenario 1)."
],
[
"Acknowledgments",
"This research was supported by the NSF (IIS 1616112), Allen Distinguished Investigator Award, and gifts from Allen Institute of AI, Google, Amazon, Samsung, and Bloomberg.",
"We thank the anonymous reviewers for their helpful comments"
]
] | 1808.08643 |
[
[
"On $J$-Colouring of Chithra Graphs"
],
[
"Abstract The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one.",
"Chithra graphs serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within various biological networks found in living systems.",
"In this paper, we discuss recently introduced $J$-colouring of the family of Chithra graphs."
],
[
"Introduction",
"For general notations and concepts in graphs and digraphs see [1], [2], [6].",
"Unless mentioned otherwise, all graphs $G$ mentioned in this paper are simple and finite graphs.",
"Note that the order and size of a graph $G$ are denoted by $\\nu (G)=n$ and $\\varepsilon (G)=p$ .",
"The minimum and maximum degrees of $G$ are respectively denoted bby $\\delta (G)$ and $\\Delta (G)$ .",
"The degree of a vertex $v \\in V(G)$ is denoted $d_G(v)$ or simply by $d(v)$ , when the context is clear.",
"We recall that if $\\mathcal {C}= \\lbrace c_1,c_2,c_3,\\dots ,c_\\ell \\rbrace $ and $\\ell $ sufficiently large, is a set of distinct colours, a proper vertex colouring of a graph $G$ denoted $\\varphi :V(G) \\mapsto \\mathcal {C}$ is a vertex colouring such that no two distinct adjacent vertices have the same colour.",
"The cardinality of a minimum set of colours which allows a proper vertex colouring of $G$ is called the chromatic number of $G$ and is denoted $\\chi (G)$ .",
"When a vertex colouring is considered with colours of minimum subscripts the colouring is called a minimum parameter colouring.",
"Unless stated otherwise we consider minimum parameter colour sets throughout this paper.",
"The number of times a colour $c_i$ is allocated to vertices of a graph $G$ is denoted by $\\theta (c_i)$ and $\\varphi :v_i \\mapsto c_j$ is abbreviated, $c(v_i) = c_j$ .",
"Furthermore, if $c(v_i) = c_j$ then $\\iota (v_i) = j$ .",
"We shall also colour a graph in accordance with the Rainbow Neighbourhood Convention [3]."
],
[
"Rainbow Neighbourhood Convention",
"The rainbow neighbourhood convention has been introduced in [3] as follows: Consider a proper minimal colouring $\\mathcal {C} =\\lbrace c_1,c_2,c_3,\\dots ,c_\\ell \\rbrace $ of a graph $G$ under consideration, where $\\ell =\\chi (G)$ .",
"Colour maximum possible number of vertices of $G$ with the colour $c_1$ , then colour the maximum possible number of remaining uncoloured vertices with colour $c_2$ and proceeding like this, at the final stage colour the remaining uncoloured vertices by the colour $c_\\ell $ .",
"Such a colouring may be called a $\\chi ^-$ -colouring of a graph.",
"The inverse to the convention requires the mapping $c_j\\mapsto c_{\\ell -(j-1)}$ .",
"Corresponding to the inverse colouring we define $\\iota ^{\\prime }(v_i) = \\ell -(j-1)$ if $c(v_i) = c_j$ .",
"The inverse of a $\\chi ^-$ -colouring is called a $\\chi ^+$ -colouring of $G$ ."
],
[
"Chithra Graphs",
"The family of Chithra graphs has been defined in [4] as follows: Consider the set $\\mathcal {V}(G)$ of all subsets of the vertex set, $V(G)$ .",
"Let $\\mathcal {W}$ be a collection of any finite number, say $k$ , subsets $W_i$ , with repetition of selection allowed.",
"That is, $\\mathcal {W}= \\lbrace W_i: 1 \\le i \\le k, W_i \\in \\mathcal {V}(G)\\rbrace $ , with repetition allowed.",
"(This means that, contrary to conventional set theory, we have $\\lbrace W_i,\\ldots ,W_i\\rbrace \\ne \\lbrace W_i\\rbrace $ ).",
"For all non-empty subsets $W_i,\\ 1 \\le \\ell \\le k$ , the corresponding additional vertices $u_1, u_2, u_3, \\ldots , u_\\ell $ , add the edges $u_iv_j$ , $\\forall v_j \\in W_j \\subseteq V(G)$ .",
"This new graph is called a Chithra graph of the given graph $G$ and is denoted by $\\mathbb {C}_{\\mathcal {W}}(G)$ .",
"The family of Chithra graphs of the graph $G$ is denoted by $\\mathfrak {C}(G)$ .",
"Note in general that the empty subset of $V(G)$ may be selected.",
"However, the empty subset does not represent an additional vertex.",
"Hence, the vertex $u_\\emptyset $ is empty.",
"This argument implies that for a set of empty sets, $\\emptyset = \\bigcup \\limits _{i=1}^{k}\\emptyset _i$ , we have $\\mathbb {C}_\\emptyset (G)=G \\in \\mathfrak {C}(G)$ as well.",
"Also, note that the Chithra graphs can be constructed from non-connected (disjoint) graphs and from edgeless (null) graphs.",
"It means that for the edgeless graph on $n$ vertices, denoted by $\\mathfrak {N}_n$ , the complete bi-partite graph, $K_{n,m}\\in \\mathfrak {C}(\\mathfrak {N}_n)$ .",
"Equally so, $K_{n,m}\\in \\mathfrak {C}(\\mathfrak {N}_m)$ .",
"We note that the family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one.",
"Compared to split graphs the important relaxation is that a complete component (clique) is not a necessary requirement.",
"A split graph is isomorphic to a Chithra graph on condition that each vertex in the independent set is mapped onto a subset of the set of vertices of the clique component.",
"We say that a split graph is an explicit Chithra graph of a complete graph.",
"A number of well-known classes of graphs are indeed explicit Chithra graphs of some graph $G$ .",
"Small graphs such as sun graphs, sunlet graphs, crown graphs and helm graphs are all explicit Chithra graphs of some cycle $C_n$ .",
"Chithra graphs can serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within various biological networks found in living systems."
],
[
"$J$ -Colouring of Chithra graphs",
"The closed neighbourhood $N[v]$ of a vertex $v \\in V(G)$ which contains at least one coloured vertex of each colour in the chromatic colouring, is called a rainbow neighbourhood.",
"We say that vertex $v$ yields a rainbow neighbourhood.",
"We also say that vertex $u \\in N[v]$ belongs to the rainbow neighbourhood $N[v]$ .",
"Definition 2.1 [5] A maximal proper colouring of a graph $G$ is a Johan colouring or a $J$ -colouring if and only if every vertex of $G$ belongs to a rainbow neighbourhood of $G$ .",
"The maximum number of colours in a $J$ -colouring is denoted by $J(G)$ .",
"Definition 2.2 [5] A maximal proper colouring of a graph $G$ is a modified Johan colouring or a $J^*$ -colouring if and only if every internal vertex of $G$ belongs to a rainbow neighbourhood of $G$ .",
"The maximum number of colours in a $J^*$ -colouring is denoted by $J^*(G)$ .",
"It follows that $J(\\mathfrak {N}_n) = J^*(\\mathfrak {N}_n) = 1,\\ n \\in \\mathbb {N}$ and for any graph $G$ which admits a $J$ -colouring, $\\chi (G) \\le J(G)$ .",
"Furthermore, if a graph $G$ admits a $J$ -colouring it admits a $J^*$ -colouring.",
"However, the converse is not always true.",
"Unless mentioned otherwise all subsets $W_i \\subseteq V(G)$ will be non-empty.",
"Theorem 2.3 If a graph $G$ admits a $J$ -colouring and if for each $W_i \\in \\mathcal {W}$ , $W_i =N(v)$ for some $v \\in V(G)$ , then $J(\\mathbb {C}_{\\mathcal {W}}(G))= J(G)$ .",
"It is obvious that if $G$ does not admit a $J$ -colouring then $\\mathbb {C}_{\\mathcal {W}}(G)$ cannot admit a $J$ -colouring for all $\\mathcal {W}\\subseteq \\mathcal {V}(G)$ .",
"Hence, assume that $G$ admits a $J$ -colouring.",
"Consider any $W_i = N(v)$ for some $v \\in V(G)$ .",
"From the definition of the corresponding Chithra graph, $v$ and $u_i$ are non-adjacent and hence $c(u_i)=c(v)$ and this colouring remains as a maximal proper colouring.",
"Also, $N[u_i]$ is a rainbow neighbourhood of $u_i$ .",
"By mathematical induction, the same argument follows for all $W_i \\in \\mathcal {W}$ .",
"Therefore, $\\mathbb {C}_{\\mathcal {W}}(G)$ admits a $J$ -colouring.",
"Corollary 2.4 If for a graph $G$ each $W_i \\in \\mathcal {W}$ , $W_i =N(v)$ for some $v \\in V(G)$ , then $\\chi (\\mathbb {C}_{\\mathcal {W}}(G))= \\chi (G)$ .",
"The result follows by similar reasoning in the proof of Theorem REF .",
"Theorem 2.5 If a graph $G$ admits a $J$ -colouring then, $J(\\mathbb {C}_{\\mathcal {W}}(G))= J(G)+1$ if for each $v \\in V(G)$ , then there is at least one $W_i \\in \\mathcal {W}$ such that $N[v] \\subseteq W_i $ .",
"It is obvious that if $G$ does not admit a $J$ -colouring, then $\\mathbb {C}_{\\mathcal {W}}(G)$ cannot admit a $J$ -colouring for all $\\mathcal {W}\\subseteq \\mathcal {V}(G)$ .",
"Hence, assume that $G$ admits a $J$ -colouring.",
"Consider any $W_i$ for which $N[v] \\subseteq W_i$ for some $v \\in V(G)$ .",
"From the definition of the corresponding Chithra graph, $u_i$ is adjacent to at least $N[v]$ and hence $c(u_i) \\ne c(v^{\\prime }),\\ v^{\\prime } \\in N[v]$ .",
"Therefore, $u_i$ requires a new colour say, $c_t$ to ensure a maximal proper colouring.",
"Also, $N[u_i]$ is a rainbow neighbourhood of $u_i$ .",
"Since $c(u_i) = c(u_j)=c_t$ , $\\forall i,j$ , all vertices in $V(G)$ yield the new rainbow neighbourhood in $\\mathbb {C}_{\\mathcal {W}}(G)$ .",
"Therefore, $\\mathbb {C}_{\\mathcal {W}}(G)$ admits a $J$ -colouring and $J(\\mathbb {C}_{\\mathcal {W}}(G))= J(G)+1$ .",
"Corollary 2.6 For a graph $G$ , $\\chi (\\mathbb {C}_{\\mathcal {W}}(G))= \\chi (G)+1$ if for each $v \\in V(G)$ , then there exists at least one $W_i \\in \\mathcal {W}$ such that $N[v] \\subseteq W_i $ .",
"The result follows by similar reasoning in the proof of Theorem REF .",
"Corollary 2.7 If all $W_i \\in \\mathcal {W}$ , is such that for all $J$ -colourings of $G$ , at least one coloured vertex of each colour is an element of $W_i$ and each $v \\in V(G)$ is an element of some $W_i$ then, $J(\\mathbb {C}_{\\mathcal {W}}(G))= J(G)+1$ .",
"The result follows by similar reasoning in the proof of Theorem REF ."
],
[
"Special Class of Chithra Graphs",
"For $X \\subseteq V(G)$ , recall that $\\langle X\\rangle $ denotes the subgraph induced by $X$ .",
"Consider a graph $G$ of even order $n$ with the properties that $V(G)$ can be partitioned into $X=\\lbrace v_i:1\\le i\\le \\frac{n}{2}\\rbrace $ and $Y=\\lbrace u_i: 1\\le i \\le \\frac{n}{2}\\rbrace $ such that $N(u_i) = N(v_i),\\ \\forall \\, i$ .",
"Then, $\\mathbb {C}_Y(G)=\\mu (\\langle X\\rangle )$ is the Mycielskian of $\\langle X\\rangle $ .",
"Then, we have Theorem 2.8 Whether or not a graph $G$ admits a $J$ -colouring, its Mycielskian graph $\\mu (G)$ cannot.",
"Consider a graph $G$ of order $n$ which admits a proper colouring which satisfies some general condition denoted, $\\chi _{gen}(G)$ .",
"Let $V(G) = \\lbrace v_i:1\\le i \\le n\\rbrace $ .",
"Construct the graph $G^{\\prime }$ by adding $n$ additional vertices $Y =\\lbrace u_i:1\\le i \\le i\\rbrace $ and edges $u_iv_j$ such that $N(u_i) = N(v_i)$ .",
"Clearly, $\\mathbb {C}_Y(G^{\\prime }) = \\mu (G)$ .",
"By colouring $c(u_i)=c(v_i)$ the graph $G^{\\prime }$ admits a proper colouring which satisfies the same general condition and $\\chi _{gen}(G^{\\prime })=\\chi _{gen}(G)=|c(N[v_i])|=|c(N[u_i])|$ .",
"We recall that $\\chi (\\mu (G))=\\chi (G)+1$ .",
"The aforesaid can be generalise and we can then state that for any graph $G$ , we have $\\chi _{gen}(\\mu (G))=\\chi _{gen}(G)+1$ .",
"It implies that the vertex say $x$ which corresponds to $Y$ in $\\mathbb {C}_Y(G^{\\prime })$ has $|c(N[x])|=\\chi _{gen}(G)+1$ .",
"Therefore, irrespective of whether a graph $G$ admits a $J$ -colouring or not, its Mycielskian graph $\\mu (G)$ does not admit a $J$ -coloring."
],
[
"Conclusion",
"Chithra graphs are extremely general in that any graph with at least one edge is indeed a Chithra graph.",
"To clarify the aforesaid statement, note that if $d_G(v) \\ge 1$ , then $G^{\\prime }=G-v$ implies that $\\mathbb {C}_{N(v)}(G^{\\prime }) \\simeq G$ .",
"Complication in studying Chithra graph is the fact that a graph $G$ can generally be described isomorphic to different Chithra graphs.",
"Admittedly, the description for some classes of graphs up to isomorphism, is unique.",
"For example, for $K_n$ , $n \\ge 3$ it follows that $K_n = \\mathbb {C}_{\\lbrace v_1,v_2,v_3,\\dots ,v_{n-1}\\rbrace }(K_{n-1})$ .",
"Identifying the appropriate description of a Chithra graph as an explicit Chithra graph of some known graph class to settle particular results is a hard problem.",
"Corollary REF and Corollary REF signal the ease with which most graph invariants can be defined in terms of explicit Chithra graphs.",
"For example, for a minimum dominating set $D$ of a graph $G$ and $W_i \\subseteq D$ , for each $i$ we have that, $\\gamma (\\mathbb {C}_{\\mathcal {W}}(G)) = \\gamma (G)$ .",
"All these remarks show there is a wide scope for further research."
]
] | 1808.08661 |
[
[
"Energy Efficient and Fair Resource Allocation for LTE-Unlicensed Uplink\n Networks: A Two-sided Matching Approach with Partial Information"
],
[
"Abstract LTE-Unlicensed (LTE-U) has recently attracted worldwide interest to meet the explosion in cellular traffic data.",
"By using carrier aggregation (CA), licensed and unlicensed bands are integrated to enhance transmission capacity while maintaining reliable and predictable performance.",
"As there may exist other conventional unlicensed band users, such as Wi-Fi users, LTE-U users have to share the same unlicensed bands with them.",
"Thus, an optimized resource allocation scheme to ensure the fairness between LTE-U users and conventional unlicensed band users is critical for the deployment of LTE-U networks.",
"In this paper, we investigate an energy efficient resource allocation problem in LTE-U coexisting with other wireless networks, which aims at guaranteeing fairness among the users of different radio access networks (RANs).",
"We formulate the problem as a multi-objective optimization problem and propose a semi-distributed matching framework with a partial information-based algorithm to solve it.",
"We demonstrate our contributions with simulations in which various network densities and traffic load levels are considered."
],
[
"Introduction",
"1000x data requirement is a major challenge for cellular networks in 5G networks [1].",
"To overcome the challenge, exploiting more spectrums for reliable communication is regarded as a promising solution.",
"Industrial scientific and medical (ISM) radio bands, in particular, 5.8 GHz have attracted wide interest [2].",
"The overall available spectrum bandwidth in the unlicensed bands in major markets (e.g.",
"US, Europe, China, Japan) is several hundred megahertz (MHz)[2].",
"LTE-unlicensed (LTE-U) is deployed to allow cellular user equipment (UE) to utilize ISM radio bands, in particular, 5.8 GHz.",
"To enhance system capacity, unlicensed carriers are integrated into a cellular network by using the carrier aggregation (CA).",
"The CA enables the aggregation of two or more component carriers into a combined bandwidth with one carrier serving as the Primary Component Carrier (PCC) and others serving as Secondary Component Carriers (SCCs) [3], [4], [5].",
"For LTE-U, licensed carrier serves as the PCC, while the unlicensed bands work as the SCCs in Time-Division-Duplexed (TDD) or Supplemental DL (SDL) only [2].",
"Furthermore, in [6], the authors proposed a mechanism that allowed device-to-device (D2D) communications operating in unlicensed bands utilizing LTE-U technologies.",
"Wi-Fi networks, with low cost and high data rates, have been the dominant players on all unlicensed bands in 2.4 and 5 GHz.",
"However, spectrum efficiency in Wi-Fi systems is low, especially given the overloaded conditions.",
"In contrast, LTE works more efficiently in terms of resource management and error control.",
"Therefore, the deployment of LTE-U not only alleviates the spectrum scarcity of the cellular system, but also improves the spectrum efficiency on the unlicensed bands.",
"Despite the huge potential to meet cellular traffic surges, LTE-U is still in its infancy; several deployment challenges remain to be overcome.",
"First, Wi-Fi systems would experience significant performance degradation in the presence of LTE-U systems without a proper coexistence scheme [7], [8].",
"Wi-Fi systems employ carrier sense multiple access with collision avoidance (CSMA/CA) to access the unlicensed bands, and a Wi-Fi user will back off if the co-channel LTE-U signals is above the energy detection threshold (e.g., -62dBm over 20MHz) [9].",
"Therefore, a suitable coexistence mechanism is required in the LTE-U channel access scheme design.",
"Secondly, LTE-U users may fail to meet its quality of service (QoS) requirement due to Wi-Fi transmission.",
"What's more, the interference between LTE-U users of multiple operators would also lead to performance degradation of LTE-U users.",
"Such unplanned and unmanaged deployment would result in severe performance degradation for both Wi-Fi and LTE-U networks and poor spectrum efficiency.",
"LTE-U calls for coexistence schemes to enable harmonious resource sharing between Wi-Fi and LTE-U.",
"Thus, coexistence mechanisms have attracted substantial interest.",
"Fair spectrum sharing between Wi-Fi and LTE-U can be ensured by using either non-coordinated or coordinated network managements.",
"Non-coordinated schemes, such as LTE blank subframe allocation [10], listen-before talk (LBT) scheme [11], the carrier sense adaptive transmission (CSAT) by LTE-U forum [9], and 3 LBT schemes (Category (Cat) 2, 3, 4) by European Telecommunications Standards Institute (ETSI) [12], require modifications on the LTE-U side only, while coordinated schemes require information sharing about network operations and spectral resources using centralized network interconnections, including cooperative control for spectrum access and managing coexistence using an X2 interface [13].",
"Research on the optimal resource allocation of the unlicensed spectrum has also been undertaken.",
"Geometric programming [14] has been widely used in wireless communication to solve network resource allocation problems, which has been often used in LTE-U scenarios.",
"In [15], the optimization performance of a hybrid method to perform both traffic offloading and resource sharing based on a duty cycle scheme is revealed.",
"A fair-LBT (F-LBT) scheme is proposed by considering both the throughput and fairness of an LTE-U and a Wi-Fi system [16].",
"In [17], a matching-based student-project model is developed to guarantee unlicensed users¡¯ QoS, together with the system-wide stability.",
"Contention window size for both Wi-Fi and LTE-U users are jointly adapted to maximize LTE-U throughput while guaranteeing the Wi-Fi throughput threshold [18].",
"In [19], power allocation problem of the small base stations is formulated as a non-cooperative game by using a multi-framework.",
"Fair proportional allocation is developed to optimize both Wi-Fi and LTE-U throughput [20].",
"A centralized joint power optimization and joint time division channel access optimization scheme is proposed to achieve significant gains for both Wi-Fi and LTE-U throughput [21].",
"A Nash bargaining game theoretic framework is also employed to solve the joint channel and power allocation problem in [22].",
"In [23], the unlicensed spectrum is divided into a contention period, for Wi-Fi users only, and a contention-free period, for LTE-U users.",
"The optimal contention period is obtained by using the Nash bargaining solution.",
"In [24], a joint user association and power allocation for licensed and unlicensed spectrum algorithm is proposed to maximize maximize sum rate of LTE-U/Wi-Fi heterogeneous networks.",
"Fair coexistence has not been defined clearly, and one of the definitions is that the deployment of an LTE-U system should not affect one Wi-Fi system more than another Wi-Fi system with respect to throughput and latency [25], [2].",
"Throughput fairness is explored by means of both $\\alpha $ -fairness and max-min approach and time division access and channel sharing between Wi-Fi and LTE-U are found to be effective coexistence schemes.",
"Moreover, a criterion for switching between these two schemes is also established in [26], subject to different network scenarios.",
"We hold that a fair coexistence should consider both Wi-Fi and LTE-U users' QoS, such as throughput threshold and power consumption.",
"Due to the limitations of power in end-user devices, if a user's throughput requirement were fulfilled by consuming an excessive amount of power, user's satisfaction would be affected.",
"The ratio of the achievable user throughput to the consumed user energy, i.e., energy efficiency (EE), is an important indicator for wireless communications especially from a user's perspective, which has been widely explored in a 5G ultra-dense networks [27], cognitive radio [28], and OFDMA networks [29].",
"Therefore, it is interesting and critical to study the EE minimization problem in Wi-Fi and LTE-U coexistence scenarios while meeting their QoS requirements."
],
[
"Matching Theory Framework",
"Matching theory is a mathematical framework for forming mutually beneficial relations, which was first applied in economics.",
"It can be easily adapted to study resource allocation problems of a wireless communication system.",
"Matching theory can model the interactions between two distinct sets of players with different or even conflicting interests [30].",
"For example, in an LTE uplink network, UE aims to achieve its QoS (mainly throughput) with minimal energy consumption while the objectives of small cell base stations (SCBSs) are serving users with certain QoS requirements and maximizing its capacity.",
"Compared with game theory, a UE does not need other UEs' actions to make decisions.",
"A preference list in terms of performance matrix, such as throughput and EE, is set up based on the local information including channel conditions.",
"UEs made proposals according to this list.",
"The only global information required from a centralized agent is the rejection/acceptance decision of each UE's proposal and blocking pair.",
"Recently, matching theory has emerged as a promising tool to cope with future wireless resource allocation problems.",
"In a full duplex OFDMA network, UL and DL user pairing and sub-channel allocations are modelled as a one-to-one three-sided matching to maximize the sum system rate [31].",
"In [32], the decoupled uplink-downlink user association problem in multi-tier full-duplex cellular networks is formulated as two-sided many-to-one matching.",
"An algorithm, based on a stable marriage algorithm is developed to find a near optimum with much lower complexity compared to a conventional coupled and decoupled user association scheme.",
"A resource allocation problem for device-to-device (D2D) communications underlaying cellular networks is studied in [33]; a two-sided many-to-many matching scheme with an externality is proposed to find the sub-optimality.",
"A matching based algorithm to study the resource allocation problem in an LTE-U scenario is proposed in [17].",
"The student-project model is used, in which students (cellular users) propose projects (unlicensed bands), and the decisions are made by lectures (base stations) to achieve maximal system (both LTE-U and Wi-Fi) throughput.",
"Based on this paper, the same goal is studied by considering user mobility in [34].",
"However, all of the above work considers optimal system performance as a whole, instead of QoS (such as throughput) for each user.",
"In addition, another limitation of the above works is that the matching is with complete preference lists.",
"This is not always the case in the real world, for example, some bands may fail to achieve a user's QoS requirement, due to its availability and channel variation, which means that some bands are not acceptable to certain users, making the preference list incomplete."
],
[
"Contributions",
"The major contributions of this paper are summarized as follows: Figure: System architecture of a LTE-U and Wi-Fi system Different from existing works, which typically consider only the fairness problem or overall EE (defined as the ratio of the overall data rate and the total energy consumption), we propose an optimized shared scheme for LTE-U networks coexisting with Wi-Fi in ISM bands, which aims at maximizing the EE of independent LTE-U users while guaranteeing fairness among different users.",
"That is, the proposed algorithm would guarantee the QoS requirement for each user (including CUs and Wi-Fi users).",
"The optimization problem is formulated as a multi-objective optimization problem, in which typically a set of Pareto solutions can be achieved.",
"We utilize the weighted sum method to transform the multi-objective optimization problem into a single-objective optimization problem and find the Pareto optimal solution.",
"The single-objective optimization problem can be further modelled as a one-to-many matching game with partial information.",
"Here partial information means incomplete preference lists, which is due to the fact that some UBs fail to fulfil a user's minimal throughput requirement and are not acceptable to that user.",
"Such problem has not yet been solved.",
"We propose a semi-distributed two-step stable algorithm to solve it.",
"Numerical results demonstrate that the proposed algorithm can achieve good performance with fast convergence speed.",
"The rest of the paper is organized as follows.",
"The system model is described in Section III.",
"The problem formulation from a multi-objective optimization to a single-objective formulation is developed in IV.",
"To solve the optimization problem, a two-step matching-based resource allocation and user association algorithm are proposed in Section V. In Section VI, numerical results are presented and analysed.",
"Section VII concludes the paper."
],
[
"System Model",
"As shown in Fig.",
"REF , we consider a LTE-U network coexisting with a Wi-Fi network in ISM bands (e.g.",
"2.4 and 5.8 GHz), composed of $M$ independently uniformly distributed small-cell base stations (SCBSs), $SCBS=\\lbrace SCBS_1,...SCBS_m,...SCBS_M\\rbrace $ , and $N$ independently uniformly distributed Wi-Fi access points (APs), $AP=\\lbrace AP_1,...AP_n,...AP_N\\rbrace $ .",
"All the SCBSs are deployed by the same cellular network operator.",
"$K$ cellular users (CUs), $CU=\\lbrace CU_1,...CU_k,...CU_K\\rbrace $ and $N^{\\prime }$ Wi-Fi users (WU), $WU=\\lbrace WU_1,...WU_{n^{\\prime }},...WU_{N^{\\prime }}\\rbrace $ are uniformly distributed in the area of interest.",
"As shown in Fig.",
"REF , the whole unlicensed spectrum is divided into $U$ orthogonal UBs.",
"Then in the time domain, each UB is divided into slots; the period of a slot is $T$ .",
"Each slot consists of several sub-frames, the duration of a subframe is $t$ , which is smaller than the coherence time of the signal channel.",
"Thus, during the transmission period of a sub-frame, the power attenuation caused by Rayleigh fading in each link can be regarded as a fixed parameter.",
"Moreover, each sub-frame is considered strictly independent.",
"Figure: TDD sharing of unlicensed bands between Wi-Fi and LTE-U usersWUs communicate with Wi-Fi APs under a standard carrier sense multiple access protocol with collision avoidance (CSMA/CA).",
"CUs are served by SCBSs by using a licensed band for both uplink and downlink transmission, while they seek to aggregate unlicensed bands for a supplementary uplink transmission.",
"Table: General NotationA CU can access its local SCBS for uplink transmission with one of $U$ UBs.",
"We consider LTE-U using a duty cycle scheme to manage the coexistence in the unlicensed spectrum in the time domain.",
"By using this duty cycle method, CUs will use a almost blank subframe (ABS) pattern [10] to guarantee Wi-Fi QoS by muting a fraction of time for $UB_u$ .",
"The fraction $l_u$ will be adaptively adjusted based on the Wi-Fi data requirement.",
"Here, we consider the static synchronous muting pattern.",
"The notations in this paper can be found in Table REF ."
],
[
"LTE-U Throughput",
"During the transmission slot of LTE-U, we denote the uplink capacity $C_{k,m,u}^{C}$ of $k$ -th CU $CU_k$ associating with $SCBS_m$ on unlicensed band $UB_u$ .",
"Thus, the uplink throughput on $UB_u$ is given by: $R_{k,m,u}^{CU}=\\sum _{i=1}^{I_{k,m,u}}C_{k,m,u,i}^{CU},$ where $I_{k,m,u}$ is the number of sub-frames in $UB_U$ allocated to $CU_k$ served by $SCBS_m$ .",
"$C_{k,m,u,i}$ is the achievable data rate of $CU_k$ served by $SCBS_m$ the $u$ -th sub-frame of $UB_u$ , given as: $C_{k,m,u,i}^{CU}= {t_i}B_{u}log_{2}(1+\\frac{\\chi _{k,m,u}P_{k,m}^{CU}g_{k,m,u}}{\\sigma _{N}^{2}+\\sum _{j\\ne k}^{K}\\sum _{m}^{M}\\rho _{j,m,u}P_{j,m}^{CU}g_{j,m,u}})$ where, $\\chi _{k,m,u}$ is an indicator function, defined as: $\\chi _{k,m,u}={\\left\\lbrace \\begin{array}{ll}& \\text{1, if } CU_k \\text{ is served by } SCBS_m \\text{ using } UB_u,\\\\& \\text{0, otherwise.}\\end{array}\\right.",
"}$ $P_{k,m}^{CU}$ represents the transmission power from $CU_k$ to $SCBS_m$ .",
"$g_{k,m,u}$ is the channel power gain between $CU_k$ and $SCBS_m$ on $UB_u$ , and $g_{j,m,u}$ is the channel gain between $CU_j$ and $SCBS_m$ on $UB_u$ .",
"$\\sigma _{N}^{2}$ is the thermal noise."
],
[
"Wi-Fi Throughput",
"For each WU $WU_{n^{\\prime }}$ , there is equal probability of accessing one of the unlicensed bands.",
"We regard the WUs sharing the same UB as one WU, the interactions between co-channel CUs and WUs can be simplified to the interactions between co-channel CUs and a WU[17], [34].",
"The WU that occupies $UB_u$ is denoted as $WU_u$ .",
"Thus, the throughput of $Th_u$ can be expressed by [35]: $Th_u=\\frac{\\overline{E(p)}P_{tr}^{u}P_{s}^{u}}{(1-P_{tr}^{u})\\delta +P_{tr}^{u}P_{s}^{u}{T_{s}}+P_{tr}^{u}(1-P_{s}^{u}){T_{c}}},$ where $\\overline{E(p)}$ is the average packet size of Wi-Fi transmission, $P_{tr}^{u}$ is the probability that $UB_u$ is occupied, and $P_{s}^{u}$ is the successful transmission probability in $UB_u$ .",
"$\\delta $ is the slot time defined in 802.11.",
"${T_{s}}$ and ${T_{c}}$ are the average time consumed by a successful transmission and a collision in $UB_u$ , respectively.",
"Based on the ABS scheme, the fraction of time slots $l_u$ of $UB_u$ will be allocated to the $WU_u$ using $UB_u$ .",
"To guarantee throughput requirement $R^W_u$ of $WU_u$ , $l_u$ is given as£º $Th_ul_{u}T\\ge R_u^W.$"
],
[
"Problem Formulation",
"We define the EE of $CU_k$ , i.e., the throughput of $CU_k$ obtained per unit power consumption with the unit of '$bits-per_Joule$ ' [28] as follows: ${PE}_k^{CU}=\\frac{\\sum _m^M\\sum _u^U\\chi _{k,m,u}R_{k,m,u}}{{\\sum _m^M\\sum _u^U\\chi _{k,m,u}I_{k,m,u}P_{k,m}^{CU}}}$ We formulate the following EE maximization problem for each CU as a multi-objective optimization problem: $&{min}(-{PE}_{1}^{CU},...,-{PE}_{K}^{CU}),\\\\s.t \\nonumber \\\\&\\sum _{k}^{K}\\sum _{u}^{U}\\chi _{k,m,u}\\le 1,\\, m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u} I_{k,m,u} t\\le T{l_u},\\,\\,\\,\\, k \\in \\lbrace 1, ..., K\\rbrace ,\\\\&\\chi _{k,m,u}\\in \\left\\lbrace 0,1 \\right\\rbrace ,\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace , \\nonumber \\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&P_{k,m}^{CU}\\le P_{max},\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&Th_u(l_{u})T\\ge R_u^W,\\, u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u}R_{k,m,u}\\ge R_k^{L},\\, k \\in \\lbrace 1, ..., K\\rbrace .$ where, constraint () indicates that a CU can be allocated up to 1 UB at a time.",
"() is the limitation of the available resource of each UB for LTE-U transmission.",
"In (), $\\chi _{k,m,u}$ is a binary number, equal to 1 if $CU_k$ served by $SCBS_m$ on $UB_u$ , or 0 otherwise.",
"The transmission power limit of each CU is set in ().",
"The throughput minimum requirement of each Wi-Fi user and CU is shown in () and (), respectively.",
"The general technique used to solve the multi-objective optimization is a weighted-sum or scalarization method by transforming a multi-objective function into a single-objective function [36] as: $&{min}({-\\sum _{k=1}^K}\\gamma _k {PE}_k^{CU}),\\\\s.t \\nonumber \\\\&\\sum _{k=1}^K\\gamma _k=K,\\\\&\\sum _{k}^{K}\\sum _{u}^{U}\\chi _{k,m,u}\\le 1,\\, m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u} I_{k,m,u} t\\le T{l_u},\\,\\,\\,\\, k \\in \\lbrace 1, ..., K\\rbrace ,\\\\&\\chi _{k,m,u}\\in \\left\\lbrace 0,1 \\right\\rbrace ,\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace , \\nonumber \\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&P_{k,m}^{CU}\\le P_{max},\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&Th_u(l_{u})T\\ge R_u^W,\\, u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u}R_{k,m,u}\\ge R_k^{L},\\, k \\in \\lbrace 1, ..., K\\rbrace .$ The effectiveness of the transformations is given in $Lemma$ REF [36] as: Lemma 1 The single-objective minimizer is an effective solution for the original multi-objective problem.",
"If the $\\gamma _k$ weight vector is strictly greater than zero, then the single-objective minimizer is a strict Pareto optimum.",
"where strict Pareto optimum is defined as follows: Definition 1 Strict Pareto Optimum: A solution Matrix M is said to be a strict Pareto optimum or a strict efficient solution for the multi-objective problem (REF ) if and only if there is no $m\\subseteq S$ such that ${PE}_k^{CU}(m)\\le {PE}_k^{CU}(m^{\\prime })$ for all $k \\in {1, .",
".",
".",
",K}$ , with at least one strict inequality.",
"$S$ is the constraints (-).",
"If all the CUs are of the same priority, i.e., $\\gamma _k=1, k \\in \\lbrace 1, ..., K\\rbrace .$ The EE optimization is finally transformed into: $&{min}(-\\sum _{k=1}^K {PE}_k^{CU}),\\\\s.t \\nonumber \\\\&\\sum _{k}^{K}\\sum _{u}^{U}\\chi _{k,m,u}\\le 1,\\, m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u} I_{k,m,u} t\\le T{l_u},\\,\\,\\,\\, k \\in \\lbrace 1, ..., K\\rbrace ,\\\\&\\chi _{k,m,u}\\in \\left\\lbrace 0,1 \\right\\rbrace ,\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace , \\nonumber \\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&P_{k,m}^{CU}\\le P_{max},\\,k \\in \\lbrace 1, ..., K\\rbrace , m \\in \\lbrace 1, ..., M\\rbrace ,\\\\&Th_u(l_{u})T\\ge R_u^W,\\, u \\in \\lbrace 1, ..., U\\rbrace ,\\\\&\\sum _{m}^{M}\\sum _{u}^{U}\\chi _{k,m,u}R_{k,m,u}\\ge R_k^{L},\\, k \\in \\lbrace 1, ..., K\\rbrace .$ We denote the solution for optimization problem (REF ) as Matrix M, which, according to Lemma.",
"REF , is an strict Pareto optimum for the multi-objective optimization problem (REF ).",
"In the expression of $PE_k^{CU}$ , which is nonlinear, $I_{k,m,u}$ and $\\chi _{k,m,u}$ are integers, while $R_{k,m,u}$ and $P_{k,m}^{CU}$ are continuous variables.",
"The objective function (10) is a summation of $PE_k^{CU}, k \\in \\lbrace 1, ..., K\\rbrace $ , thus, it is a mixed integer nonlinear programming (MINLP) problem, which is typically NP-hard.",
"Thus, to reduce the computation complexities, we developed a matching-based solution, which will be discussed in the following section."
],
[
"Introduction to Matching Theory and Student-Project-Allocation Problem",
"Student project allocation (SPA) is a one-to-many matching game, where each student has a preference list of the projects that they can choose from, while the lecturers have a preference list of students for each project or a preference list of student-project pairs.",
"There is an upper bound, also known as the quota, on the number of students that can be assigned to each particular project [37].",
"Inspired by the SPA problem, we model the resource allocation problem in (REF ) as an SPA game, where the CUs, UBs and SCBSs are considered equivalent to students, projects and lecturers, respectively.",
"Similarly, SCBSs offer the set of available UBs and maintain a preference list for each UB, and each CU has a preference list of UBs that they can use for uplink transmission.",
"SCBSs allocate UBs to CUs based on the achievable EE on UBs.",
"Meanwhile, our resource allocation problem differs from the SPA game in the following aspects: Maximum throughput: The quota in the SPA problem is replaced by the maximum achievable throughput of a UB.",
"The maximum achievable throughput of a UB determines the maximum number of CUs that it can be allocated to while meeting the minimum required Wi-Fi throughput in the TDD mode.",
"Incompleteness of preference lists: The SCBSs sense the availabilities of and keep the CUs updated.",
"Any UB that is not able to fulfil a CU's minimal throughput requirement will be deleted from the preference list of the CU and the CU will be removed from the preference list of that UB.",
"Only a subset of UBs (CUs) are in the preference list of a CU (UB), i.e., the preference lists are incomplete.",
"The kth CU preferring the uth UB over the u'th UB is denoted by $pri(CU_k, UB_u)> pri(CU_k, UB_{u^{\\prime }})$ .",
"Similarly, $pri(UB_u, CU_k)> pri(UB_u, CU_{k^{\\prime }})$ indicates that the uth UB prefers the kth CU over k'th CU.",
"The one-to-many matching is defined as follows: Definition 2 Let $\\mu $ denote the one-to-many matching between two disjoint sets CU and UB.",
"$\\mu (CU_k)=UB_u$ indicates that the kth CU is matched to the uth UB, $\\mu (UB_u)=\\lbrace CU_k,..., CU_{k^{\\prime }}\\rbrace $ indicates that the uth UB is matched to $\\lbrace CU_k,..., CU_{k^{\\prime }}\\rbrace $ , $\\mu (CU_k)=CU_k$ indicates that the kth CU is not really matched to any UB.",
"The stability implies the robustness of the matching against deviations caused by the individual rationality of players, i.e., the CUs in our resource allocation problem.",
"In an unstable matching, two CUs may swap their matched UBs to maximize their own EE, leading to an undesirable and unstable resource allocation.",
"The definition of stability of the one-to-many matching is given as follows: Definition 3 Stability of One-to-Many Matching.",
"The one-to-many matching $\\mu $ between two disjoint sets CU and UB is stable, only if it is not blocked by any blocking individual or blocking pair, where the blocking individual and the blocking pair are defined in the following.",
"Blocking individual in the EE optimization problem is defined as: Definition 4 Blocking Individual.",
"A CU is a blocking individual if it prefers to stay unmatched rather than being matched to any available UB.",
"The blocking pair in the EE optimization problem is defined as: Definition 5 Blocking Pair.",
"A pair $(CU_k, UB_u)$ is a blocking pair if all the following 3 conditions are satisfied: (1) $\\mu (CU_k)$$\\ne $$UB_u$ and $pri(CU_k,UB_u)$$>$$pri(CU_k,\\mu (CU_k))$ ; (2) $\\mu (UB_u)$$\\ne $$CU_k$ and $pri(UB_u,CU_k)$$>$$pri(UB_u,\\mu (UB_u))$ ; (3) There is enough spectrum in $UB_u$ to meet the minimum throughput requirement of $CU_k$ ."
],
[
"Preference Lists of CUs Over UBs",
"We assume that the preference of $CU_k$ over $UB_u$ is based on EE $PE_{k,m,u}^{CU}$ achieved by $CU_k$ served by $SCBS_m$ using $UB_u$ to guarantee its QoS threshold, which is written as follows: $ PE_{k,m,u}^{CU}=\\frac{\\sum _m^M\\sum _u^U\\chi _{k,m,u}R_{k,m,u}}{{\\sum _m^M\\sum _u^U\\chi _{k,m,u}P_{k,m}^{CU}I_{k,m,u}t}}$ $CU_k$ prefers $UB_{u}$ over $UB_{u^{\\prime }}$ if $CU_k$ can achieve higher EE using $UB_{u}$ than $UB_{u^{\\prime }}$ , which is stated as follows: $pri(CU_k,UB_{u})>pri(CU_k,UB_{u^{\\prime }}) \\Leftrightarrow PE_{k,m,u}^{CU}>PE_{k,m,u^{\\prime }}^{CU}$ None of the CUs have any knowledge about other co-channel coexisting CUs, before the final band allocation is performed at SCBSs.",
"Thus, the preference lists are set up based on local channel sensing information and unlicensed band availability alone."
],
[
"Preference Lists of SCBS Over ($CU_k$ , {{formula:3a651ade-dd8b-4dee-a26c-e230dcf636a8}} ) Pair",
"However the preference of $SCBS_m$ over the user-band pair ($CU_k$ , $UB_u$ ) is based on the EE achieved by allocating $UB_u$ to $CU_k$ to fulfil the QoS threshold of $CU_k$ .",
"It is written as $SCBS_m$ prefers $CU_k$ over $CU_{k^{\\prime }}$ to occupy $UB_{u}$ if $CU_k$ can achieve higher EE than $CU_{k^{\\prime }}$ by using $UB_{u}$ , which is stated as follows: $pri(UB_{u},CU_k)>pri(UB_{u},CU_{k^{\\prime }}) \\Leftrightarrow PE_{k,m,u}^{CU}>PE_{k^{\\prime },m,u}^{CU}$"
],
[
"Step 1: Modified GS Algorithm for One-to-Many Game",
"To solve the above matching game, a 2-step algorithm is proposed.",
"The first step is an extension of the GS algorithm applied for a one-to-many matching with incomplete preference lists.",
"Each iteration begins with the unmatched CUs proposing their favourite (i.e., the first UB) UB on their current preference lists.",
"The UBs which have been proposed to will be removed from the CUs' preference lists.",
"For each $UB_u$ , SCBSs decide whether to accept or reject the CU's proposal $UB_u$ based on SCBSs' preference lists over ($CU_k$ , $UB_u$ ) pairs.",
"SCBSs choose to keep the most preferred CUs as long as these CUs do not occupy more resources than the UB could offer; the remaining CUs are rejected.",
"Such a procedure runs until every CU is either matched or its preference list is empty.",
"The implementation detail of Step 1 of the algorithm is stated in REF as follows: One-to-Many Matching [1] Input: $CU$ , $UB$ , $PL^{CU}$ , $PL^{UB}$ Output: Matching $\\mu _1$ Step 1£º Proposing£º All free $CU_k$ propose their favourite $UB_u$ in their preference lists, and remove $UB_u$ from the list.",
"Step 2£º Accepting/rejecting£º $UB_u$ accept the most preferred $n$ proposers based on its preference list, the rest are rejected.",
"The sum of the slot time of the accepted proposers does not exceed its available resource time.",
"None of the accepted proposers are free.",
"All the rejected proposers are free.",
"Criterion£º If every CUs is either allocated with a UB or its preference list is empty, this algorithm is terminated with an output $M_1$ .",
"Otherwise, Step 1 and Step 2 are performed again.",
"Theorem 1 Stability of $\\mu _1$.",
"In any instance of one-to-many matching, stable matching is achieved by using REF .",
"We prove this theorem by contradiction and assume that for an instance of one-to-many matching, REF terminates with an instable matching $\\mu _1$, i.e., there exists at least one blocking pair ($CU_k$ , $UB_u$ ) or one blocking individual $CU_k$ .",
"If there exists one blocking pair ($CU_k$ , $UB_u$ ) in $\\mu _1$: Case 1: In $\\mu _1$, ${UB}_u$ is unmatched and ${CU}_k$ is matched with ${UB}_u^{\\prime }$ .",
"If ${UB}_u$ is not on the preference list of ${CU}_k$ , then, ${CU}_k$ does not have an incentive to match with ${UB}_u$ ; If $pri(CU_{k},UB_{u^{\\prime }})>pri(CU_k,UB_{u})$ , and ${CU}_k$ is matched with ${UB}_u^{\\prime }$ in $\\mu $, then ${CU}_k$ does not have an incentive to match with ${UB}_u$ ; If $pri(CU_{k},UB_{u})>pri(CU_k,UB_{u^{\\prime }})$ , then ${CU}_k$ proposes to ${UB}_u$ before ${UB}_{u^{\\prime }}$ .",
"${CU}_k$ is rejected during the proposal stage or is accepted by ${UB}_u$ first, then is rejected.",
"In conclusion, in any situation in which ${CU}_k$ is matched and ${UB}_u$ is unmatched, a blocking pair does not exist.",
"Case 2: In $\\mu _1$, ${UB}_u$ being unmatched and ${CU}_k$ unmatched.",
"${UB}_u$ is unmatched means that it receives no proposal from CU, including ${CU}_k$ .",
"This means that ${UB}_u$ is not on ${CU}_k^{\\prime }s$ preference list, then ${CU}_k$ does not have incentive to match with ${UB}_u$ .",
"In conclusion, in any situation in which both ${CU}_k$ and ${UB}_u$ are unmatched, blocking pair does not exist.",
"Case 3: In $\\mu _1$, ${UB}_u$ being matched with ${CU}_k^{\\prime }$ and ${CU}_k$ unmatched.",
"${CU}_k$ is unmatched means that either it has no ${UB}_u$ in its preference list, or all its proposals have been rejected.",
"For the former, ${CU}_k$ does not have an incentive to match with ${UB}_u$ .",
"For the latter, ${UB}_u$ rejects ${CU}_k$ because it prefers other proposer(s).",
"Thus, ${UB}_u$ does not have an incentive to match with ${CU}_k$ .",
"In conclusion, in any situation in which both ${CU}_k$ is unmatched and ${UB}_u$ is matched, blocking pair does not exist.",
"Case 4: In $\\mu _1$, ${UB}_u$ is matched with ${CU}_k^{\\prime }$ and ${CU}_k$ with ${UB}_u^{\\prime }$ .",
"${UB}_u$ must be on ${CU}_k^{\\prime }$ s preference list, and vice versa, otherwise, there is no incentive to form the (${CU}_k$ , ${UB}_u$ ) pair.",
"If $pri(CU_{k},UB_{u^{\\prime }})>pri(CU_k,UB_{u})$ , then, ${CU}_k$ does not have an incentive to match with ${UB}_u$ if it is matching with ${UB}_{u^{\\prime }}$ .",
"If $pri(CU_{k},UB_{u})>pri(CU_k,UB_{u^{\\prime }})$ , then, ${CU}_k$ proposes to ${UB}_u$ first and is rejected, because ${UB}_u$ prefers ${CU}_k^{\\prime }$ to ${CU}_k$ , then ${UB}_u$ does not have an incentive to match with ${CU}_k^{\\prime }$ .",
"In conclusion, in any situation in which both ${CU}_k$ and ${UB}_u$ are matched, a blocking pair does not exist.",
"Contradictions, as (${CU}_k$ , ${UB}_u$ ) is any pair, thus, it could be said that there is no blocking pair in matching $\\mu _1$.",
"If one blocking individual $CU_k$ or $UB_u$ exists in $\\mu _1$: for blocking individual $CU_k$ : Case 1: In $\\mu _1$, ${CU}_k$ is matched with ${UB}_u$ , i.e., ${UB}_u$ is on ${CU}_k$ 's preference list, as such ${CU}_k$ does not have incentive be unmatched.",
"In conclusion, in any situation in which both ${CU}_k$ and ${UB}_u$ are unmatched, blocking individual $CU_k$ does not exist.",
"The proof that blocking individual $UB_u$ does not exist is similar to that blocking individual $CU_k$ does not exist.",
"As the above blocking pair ($CU_k$ , $UB_u$ ), blocking individuals $CU_k$ or $UB_u$ can be any pair or individual, thus, we could prove that there is no blocking pair or blocking individual in matching $\\mu _1$.",
"Theorem 2 Praeto optimality of $\\mu _1$.",
"In any instance of one-to-many matching, stable matching $\\mu _1$ achieved by REF is Praeto optimal, i.e., no player(s) can better off, whilst no players are worse off.",
"In stable matching $\\mu _1$: Case 1: There exists an unmatched ${CU}_k$ , which can be matched to ${UB}_u$ to increase the achievable EE of both ${CU}_k$ and ${UB}_u$ , meaning that (${CU}_k$ , ${UB}_u$ ) is the blocking pair of matching $\\mu _1$, contracting Theorem REF.",
"Case 2: There exists a (${CU}_k$ , ${UB}_u$ ) pair.",
"Obviously, ${CU}_k$ does not have an incentive to be unmatched; ${CU}_k$ has the incentive to change partner from ${UB}_u$ to ${UB}_{u^{\\prime }}$ to increase its achievable EE, meaning that (${CU}_k$ , ${UB}_{u^{\\prime }}$ ) is a blocking pair of matching $\\mu _1$, contracting Theorem REF.",
"It is impossible to increase the EE of some CUs' without decreasing that of the remaining of the CUs.",
"The state stands for UB, which can be proven similarly as above.",
"We define the computational complexity of REF as the number of accepting/rejecting decisions required to output a stable matching $\\mu _1$.",
"The complexity of REF , i.e., the convergence of REF is given in Theorem REF.",
"Theorem 3 Complexity of REF (Convergence of REF ).",
"In any instance of many-to-one matching, a matching $\\mu _1$ can be obtained by using REF within $\\mathcal {O}(KU)$ iterations.",
"In each iteration, a CU proposes to its most favourite UB in its current preference list, and SCBS accepts/rejects the proposal.",
"The maximum number of elements in the preference list of $CU_k$ equals the number of UBs, i.e., $U$ .",
"Thus, stable matching $\\mu _1$ can be obtained in $\\mathcal {O}(KU)$ overall time, where $K$ is the number of CUs and $U$ is the number of UBs."
],
[
"Step 2: EE Optimization",
"As proven above, stability and Pareto optimality have been guaranteed by using algorithm REF , meaning that there are no incentives for any CUs and UBs to form new matching.",
"However, the preference lists of CUs could to be incomplete, some CUs may be unmatched [38], [39].",
"To further maximize system's EE by increasing the number of CUs matched by algorithm REF , an iteration of algorithm REF begins with an unmatched $CU_k$ proposing to its most favourite $UB_u$ , and $UB_u$ would be deleted from the preference list of $CU_k$ .",
"An SCBS would consider this proposal acceptable if the following criteria are fulfilled: After deleting several non-favourites or all CUs matched with $UB_u$ in $\\mu _1$ obtained via algorithm REF , the minimal throughput of $CU_k$ can be achieved by using $UB_u$ All the deleted CUs could be served by other UBs to fulfil their minimal throughput requirement.",
"The EE of the new matching $\\mu _k$ is greater than that of the previous matching $\\mu _1$.",
"Such matching $\\mu _k$ would be considered as a profitable reallocation, and would be updated as the new matching, if only one profitable reallocation exists.",
"Should there be multiple profitable reallocations, the one that enhances the overall EE the most would be the new matching.",
"The iterations would run several times, until every CU is either allocated with a UB or its preference list is empty.",
"The detail of algorithm REF is described as follows: System EE Maximization [1] Input: $CU$ , $UB$ , $PL^{CU}$ , $PL^{UB}$ , $\\mu _1$ Output: Matching $\\mu _2$ Step 1£º Proposing£º Every free $CU_k$ proposes to their favourite $UB_u$ in their preference lists, and removes $UB_u$ from the list.",
"Step 2£º Reallocation£º Each $CU_k$ is accommodated in $UB_u$ by deleting its non-favorite partners in $\\mu _2$, to ensure that the occupying slot time does not exceed the available slot time All the deleted CUs can be accommodated by other UBs.",
"A matching $\\mu _k$ is formed.",
"EE increases from matching $\\mu _1$ to $\\mu _k$.",
"$\\mu _k$ is stored if all the above three criteria are fulfilled.",
"Step 2 is performed until all free CUs have gone through Step 2.",
"Step 3£º Accepting/rejecting£º The $\\mu _k$ that increases the system's EE most is updated; $CU_k$ is set to be served.",
"The rest $\\mu _{k^{\\prime }}$ are rejected, and $CU_{k^{\\prime }}$ are rejected and set to be free.",
"Criterion£º Each CUs is either allocated with a UB or its preference list is empty, this algorithm is terminated with an output $\\mu _2$.",
"Otherwise, step 1, step 2 and step 3 are performed again.",
"Theorem 4 Stability of $\\mu _2$.",
"In any instance of one-to-many matching, stability is achieved by using REF in $\\mu _2$.",
"We prove this theorem by contradiction and assume that for an instance of one-to-many matching, REF terminates with an instable matching $\\mu _2$, i.e., there exists at least one blocking pair ($CU_k$ , $UB_u$ ) or one blocking individual $CU_k$ or $UB_u$ .",
"If there exists one blocking pair ($CU_k$ , $UB_u$ ) in $\\mu _2$: Case 1: In $\\mu _2$, ${UB}_u$ is unmatched and ${CU}_k$ is matched with ${UB}_u^{\\prime }$ .",
"If ${UB}_u$ is not on the preference list of ${CU}_k$ , then, ${CU}_k$ does not have an incentive to match with ${UB}_u$ ; If $pri(CU_{k},UB_{u^{\\prime }})>pri(CU_k,UB_{u})$ , and ${CU}_k$ is matched with ${UB}_u^{\\prime }$ in $\\mu _2$, then ${CU}_k$ does not have an incentive to match with ${UB}_u$ ; If $pri(CU_{k},UB_{u})>pri(CU_k,UB_{u^{\\prime }})$ , then ${CU}_k$ proposes ${UB}_u$ before ${UB}_{u^{\\prime }}$ in REF , or re-matches to ${UB}_u$ before ${UB}_{u^{\\prime }}$ in REF .",
"The result is that ${CU}_k$ matches to ${UB}_{u^{\\prime }}$ , meaning that ${CU}_k$ is rejected at some stage in REF or REF .",
"In conclusion, in any situation in which ${CU}_k$ is matched and ${UB}_u$ is unmatched, a blocking pair does not exist.",
"Case 2: In $\\mu _1$, ${UB}_u$ being unmatched and ${CU}_k$ unmatched.",
"${UB}_u$ is unmatched means that it receives no proposal from CU, including ${CU}_k$ in both REF and REF .",
"As both REF and REF terminate when every CU is matched or its preference list is empty.",
"${UB}_u$ being unmatched means that either its preference list is empty or does not contain ${UB}_u$ .",
"Then ${CU}_k$ does not have an incentive to match with ${UB}_u$ .",
"In conclusion, in any situation in which both ${CU}_k$ and ${UB}_u$ are unmatched, a blocking pair does not exist.",
"Case 3: In $\\mu _1$, ${UB}_u$ being matched with ${CU}_k^{\\prime }$ and ${CU}_k$ unmatched.",
"${CU}_k$ is unmatched means that either it has no ${UB}_u$ in its preference list, or all its proposal have been rejected in both REF , and ${CU}_k$ can not be matched to any UBs in the reallocation stage inREF .",
"For the former case, ${CU}_k$ does not have an incentive to match with ${UB}_u$ .",
"For the latter case, ${UB}_u$ rejects ${CU}_k$ because it prefers other proposer(s), and there are not enough spectrum resources in ${UB}_u$ to serve ${CU}_k$ .",
"Thus, ${UB}_u$ does not have incentive to match with ${CU}_k$ .",
"In conclusion, in any situation in which both ${CU}_k$ is unmatched and ${UB}_u$ is matched, a blocking pair does not exist.",
"Case 4: In $\\mu _1$, ${UB}_u$ is matched with ${CU}_k^{\\prime }$ and ${CU}_k$ with ${UB}_u^{\\prime }$ .",
"${UB}_u$ must be on ${CU}_k^{\\prime }$ s preference list, and vice versa, otherwise, there is no incentive to form the (${CU}_k$ , ${UB}_u$ ) pair.",
"If $pri(CU_{k},UB_{u^{\\prime }})>pri(CU_k,UB_{u})$ , then, ${CU}_k$ does not have an incentive to match with ${UB}_u$ if it is matched with ${UB}_{u^{\\prime }}$ .",
"If $pri(CU_{k},UB_{u})>pri(CU_k,UB_{u^{\\prime }})$ , then, ${CU}_k$ proposes to ${UB}_u$ first and is rejected, either because ${UB}_u$ prefers ${CU}_k^{\\prime }$ to ${CU}_k$ , or $({UB}_u, {CU}_k^{\\prime })$ is formed in the re-allocation stage.",
"For the former, ${UB}_u$ does not have an incentive to match with ${CU}_k^{\\prime }$ .",
"For the latter, ${UB}_u$ does not have sufficient spectrum resource to serve ${CU}_k$ , otherwise, the $({CU}_k, {UB}_u)$ pair has been formed in $\\mu _2$.",
"In conclusion, in any situation in which both ${CU}_k$ and ${UB}_u$ are matched, a blocking pair does not exist.",
"Contradictions, as (${CU}_k$ , ${UB}_u$ ) is any pair, thus, we could say that there is no blocking pair in matching $\\mu _1$.",
"If there exists one blocking individual $CU_k$ or $UB_u$ in $\\mu _1$: for blocking individual $CU_k$ : Case 1: In $\\mu _1$, ${CU}_k$ is matched with ${UB}_u$ , i.e., ${UB}_u$ is on ${CU}_k$ 's preference list, then ${CU}_k$ does not have an incentive to be unmatched.",
"In conclusion, in any situation in which both ${CU}_k$ is matched and blocking individual $CU_k$ does not exist.",
"the proof that blocking individual $UB_u$ does not exist is similar to that blocking individual $CU_k$ does not exist.",
"In the above proof, blocking pair ($CU_k$ , $UB_u$ ), blocking individual $CU_k$ or $UB_u$ can be any pair or individual, thus, we could prove that there is no blocking pair or blocking individual in matching $\\mu _1$.",
"Theorem 5 Praeto optimality of $\\mu _2$.",
"In any instance of one-to-many matching, Praeto optimality is achieved by using REF in $\\mu _2$.",
"In stable matching $\\mu _1$: Case 1: An unmatched ${CU}_k$ exists, which can be matched to ${UB}_u$ to increase the achievable EE of both ${CU}_k$ and ${UB}_u$ , meaning that (${CU}_k$ , ${UB}_u$ ) is the blocking pair of matching $\\mu _1$, contracting Theorem REF.",
"Case 2: An existing a (${CU}_k$ exists, ${UB}_u$ ) pair.",
"Obviously, ${CU}_k$ does not have an incentive to be unmatched; ${CU}_k$ has the incentive to change partner from ${UB}_u$ to ${UB}_{u^{\\prime }}$ to increase its achievable EE, meaning that (${UB}_u$ , ${UB}_{u^{\\prime }}$ ) is a blocking pair of matching $\\mu _1$, contracting Theorem REF.",
"It is impossible to increase the EE of a CU without decreasing that of the remaining CUs.",
"The statement stands for UB, which can be proven similarly as above.",
"Theorem 6 Complexity of REF (Convergence of REF ).",
"In any instance of many-to-one matching, a matching $\\mu _2$ can be obtained by using REF based on matching $\\mu _1$ within $\\mathcal {O}(mU(K-m)(U-1))$ iterations, where $m$ is the number of unmatched CUs in $\\mu _1$.",
"At every step in REF , each one of $m$ unmatched proposes to favourite UB, such as $UB_u$ , in its current preference list.",
"The maximum number of CUs being matched to $UB_u$ in $mu_1$ is $(K-m)$ .",
"Then, the matched CUs of $UB_u$ will be deleted from $mu_1$ and re-matched to the rest of UBs in their preference lists.",
"The maximum number of CUs that are deleted is $(K-m)$ .",
"For each deleted CU, the maximum number of UBs in its preference list is $(U-1)$ .",
"Thus the maximum number of accepting/rejecting decisions made is $(K-m)(U-1)$ for each proposal of an unmatched CU.",
"As there $m$ unmatched CUs, the total number of accepting/rejecting decisions made is $(K-m)(U-1)*mU$ ."
],
[
"Simulation Setting",
"We perform a Monte Carlo simulation in a circle with a radius of 100m, with CUs randomly and uniformly distributed being served by a SCBS.",
"The throughput requirements of Wi-Fi users and CUs are both random values between the range of [0, $TR^W$ ] and [0, $TR^C$ ], respectively.",
"We evaluate the performance of the proposed algorithm in the network with the number of CUs.",
"We assume the total number of UB to be 9.",
"We set the slot time $T$ to be 10 $\\mu $ s, and the sub-frame duration $t$ to be 1 $\\mu $ s, which is much smaller than the channel coherence time.",
"For each scenario with a certain network density and traffic load level, simulation is run 10,000 times.",
"CUs are randomly located in the area of interest 100 times, and in each time channel fading is performed 100 times.",
"All other parameters can be referred to in Table.",
"REF .",
"Figure: System Energy Efficiency for Scenarios with Different Number of CUsWe first analyse the system EE obtained by the proposed matching-based scheme in scenarios with a different number of CUs and traffic load level in Fig.",
"REF .",
"Our proposed algorithm outperforms the greedy algorithm and random allocation under both low-density (6 CUs) and high-density networks (18 CUs) with a light traffic load from 10 Mbps per CU and heavy traffic load at 40 Mbps per CU.",
"The system EE improves 30% and 50% obtained by our proposed method as compared with that obtained by the greedy algorithm, under the light and the heavy traffic load scenarios respectively.",
"For the same number of CUs, with the increasing of traffic load per CU, the system EE decreases because more CUs remain unserved in the heavy traffic load scenario, as shown in Fig.",
"REF .",
"This is because more resources are occupied to serve a CU with a high traffic demand, leading to a drop in the number of CUs that can served in the network, i.e., more CUs fail to achieve their throughput requirement.",
"Figure: The Number of CUs ServedFigure: System Energy Efficiency in Different Traffic Load LevelOn the contrary, with the same traffic load level, more CUs tend to be served in the dense scenarios, leading to an increase of system EE as shown in Fig.",
"REF .",
"In dense scenario, more CUs have the chance to meet their throughput requirement, due to many factors, such as the distance between CU and SCBS and channel condition between CU and SCBS.",
"Although the number of CUs served increases with the number of CUs in the network, except for the low traffic demand scenario, the percentage of CUs that have their throughput requirement fulfilled drops, as shown in Fig.",
"REF .",
"In a low traffic demand scenario, where the spectrum resource is sufficient to serve every CU with their required throughput demand, almost 100% of CUs' being served rate is achieved by the proposed algorithm, compared with less than 90% achieved by the greedy algorithm and the even lower served rate when using a random algorithm.",
"In medium and high traffic demand scenario, the percentage of CUs served decreases with the increase of CUs in the network by using any one of the three algorithms.",
"However, the proposed algorithm still outperforms the greedy algorithm and random algorithm by around 35% and 50% 120%, respectively.",
"Thus, we could say that the proposed algorithm works more effectively in CUs' fairness compared with the greedy algorithm or the random allocation scheme.",
"Figure: The Percentage of CUs Served Comparison"
],
[
"Throughput Analysis",
"Throughput is another performance matrix for both the system and an individual CU.",
"As shown in Fig.",
"REF , in the 6 CUs scenarios with low traffic demand, three algorithms achieve similar results.",
"This is because the unlicensed spectrum resource is sufficient to serve every CU with their relatively low traffic demands.",
"In low traffic demand, system throughput increases with the number of CUs almost linearly as shown by using the proposed algorithm and the greedy algorithm, because the spectrum resource is still sufficient.",
"The proposed algorithm outperforms the greedy algorithm.",
"However, there is another aspect in heavy traffic load.",
"In the network with 6 CUs, the proposed algorithm achieves 66% more than the greedy algorithm, and more than 100% more than the random scheme.",
"With the increase of the number of CUs in the network, the overall throughput achieved by using the proposed algorithm tends to saturate in heavy traffic load scenarios.",
"This is because the capacity is limited by the available unlicensed spectrum resources.",
"Figure: System Throughput In Different Traffic load Level"
],
[
"Computational Complexity",
"The theoretical upper bound of the computation complexity of REF and REF have been given in Theorem REF, and Theorem REF.",
"Here we show the actual computation complexity of the proposed algorithm in typical traffic load scenarios in Fig.",
"REF .",
"There are positive correlations between the complexity and network density at the same traffic load level.",
"Specifically, at the lowest traffic load (10 Mbps), complexity is slightly more than the number of CUs in the network.",
"This means that almost all the CUs' first proposal are accepted, due to the low traffic demand of each CU.",
"In a low traffic case, most CUs are matched by using $\\ref {alg:step1}$ ; $\\ref {alg:step2}$ is seldom performed.",
"The complexity increases with the traffic load level from 10 to 30 Mbps.",
"This is because with the increase of traffic load level, increasing CUs are unmatched in $\\mu _1$ by using $\\ref {alg:step1}$ ; the number of iterations that $\\ref {alg:step2}$ performs is increasing.",
"The complexity of an iteration in $\\ref {alg:step2}$ ($\\mathcal {O}((K-m)(U-1))$ ) is much larger than that in $\\ref {alg:step1}$ ($\\mathcal {O}(U)$ ), leading to an increase of complexity.",
"At an even higher traffic load level, the complexity begins to drop.",
"At this stage, the number of UBs in a CU's preference lists is much smaller than that in a medium traffic load level.",
"The complexity of obtaining matching $\\mu _1$ is much smaller.",
"Although the number of unmatched CUs rises in the scenario with the same network density, elements in their preference lists are much smaller, the complexity in an iteration drops significantly, leading to the decrease of computational complexity at a high traffic load level.",
"Figure: Computational Complexity in Different Scenario"
],
[
"Conclusion",
"In this work, we have studied the uplink resource allocation problem in a LTE-U and Wi-Fi coexistence scenario to maximize each CU's EE.",
"We formulated the problem as a multi-objective optimization, and transformed it into a single-objective optimization by using the weighted-sum method.",
"We proposed a semi-distributed 2-step matching with partial information based algorithm to solve the problem.",
"Compared with the greedy algorithm based resource allocation scheme, our proposed scheme achieves improvements of up to $50\\%$ in terms of EE and up to $66\\%$ in terms of throughput.",
"Furthermore, we have analysed the computational complexity of the proposed algorithm theoretically and by simulations, thereby showing the complexity is reasonable for real-world deployment.",
"In the future, work will be extended into the heterogeneous LTE-U networks, where hyper-dense deployment of LTE-U cells may exist.",
"We will also consider a comprehensive optimized resource allocation scheme for LTE-U taking into account that CU can choose between licensed and unlicensed bands.",
"In such scenarios, a multi-side matching model should be considered, which poses new challenges in achieving the solutions."
],
[
"Acknowledgment",
"This paper acknowledges the support of the MOST of China for the \"Small Cell and Heterogeneous Network Planning and Deployment\" project under grant No.",
"2015DFE12820, and H2020 DECADE project."
]
] | 1808.08508 |
[
[
"Masses and ages for metal-poor stars: a pilot program combining\n asteroseismology and high-resolution spectroscopic follow-up of RAVE halo\n stars"
],
[
"Abstract Very metal-poor halo stars are the best candidates for being among the oldest objects in our Galaxy.",
"Samples of halo stars with age determination and detailed chemical composition measurements provide key information for constraining the nature of the first stellar generations and the nucleosynthesis in the metal-poor regime.}",
"Age estimates are very uncertain and are available for only a small number of metal-poor stars.",
"Here we present the first results of a pilot program aimed at deriving precise masses, ages and chemical abundances for metal-poor halo giants using asteroseismology, and high-resolution spectroscopy.",
"We obtained high-resolution UVES spectra for four metal-poor RAVE stars observed by the K2 satellite.",
"Seismic data obtained from K2 light curves helped improving spectroscopic temperatures, metallicities and individual chemical abundances.",
"Mass and ages were derived using the code PARAM, investigating the effects of different assumptions (e.g.",
"mass loss, [alpha/Fe]-enhancement).",
"Orbits were computed using Gaia DR2 data.",
"{The stars are found to be \"normal\" metal-poor halo stars (i.e.",
"non C-enhanced), with an abundance pattern typical of old stars (i.e.",
"alpha and Eu-enhanced), and with masses in the 0.80-1.0 M_sun range.",
"The inferred model-dependent stellar ages are found to range from 7.4 to 13.0 Gyr, with uncertainties of ~ 30%-35%.",
"We also provide revised masses and ages for metal-poor stars with Kepler seismic data from APOGEE survey and a set of M4 stars.",
"{The present work shows that the combination of asteroseismology and high-resolution spectroscopy provides precise ages in the metal-poor regime.",
"Most of the stars analysed in the present work (covering the metallicity range of [Fe/H] ~ -0.8 to -2 dex), are very old >9 Gyr (14 out of 19 stars ), and all of them are older than > 5 Gyr (within the 68 percentile confidence level)."
],
[
"Introduction",
"The Milky Way halo is a key component to understand the assembly history of our Galaxy.",
"The halo is composed by stars that were accreted during mergers as well as stars that formed in-situ , , and is suggested to be one of the oldest component of our Galaxy, (e.g., , , ).",
"In addition, metal-poor halo giant stars enshrine information on when star formation began, on the nature of the first stellar generation and on the chemical enrichment time-scale in the Galactic halo (, , ).",
"A comprehensive understanding of the Galactic halo can be obtained only when combining precise stellar chemical abundances, kinematics, and ages.",
"While detailed chemical information can be obtained via high-resolution spectroscopic analysis and precise kinematics is being provided by astrometric missions like Gaia, the determination of reliable stellar ages (i.e.",
"ages that are precise and unbiased), is still a challenging task, especially in the case of red giants.",
"Before the confirmation of solar-like oscillations in red-giant stars , ages had been estimated only for a limited sample of nearby field stars, either by model-dependent techniques such as isochrone fitting, or empirical methods such as nucleo-cosmo-chronometry.",
"The age determination via the classic isochrone-fitting method has always been hampered by the fact that in the red-giant locus the isochrones clump together, which leads to a large degeneracy.",
"This degeneracy leads to age uncertainties easily above 80% for the oldest stars (e.g., , ).",
"The few metal-poor field halo stars with a better age determination than the isochrone fitting uses the nucleo-cosmo-chronometry technique (mostly derived using the Th-232 and U-238 ratio), and these indicate old ages (, , , , , , , ).",
"These old ages seem to confirm the expectations that metal-poor halo objects are among the oldest objects in our Galaxy.",
"Although the nucleo-cosmo-chronometry method is more precise than isochrone fitting in the case of red giants, it is not a viable solution for all stars.",
"The method requires high-resolution and high signal-to-noise (SNR) spectra in the blue region of the spectrum (SNR$>$ 300 at $\\sim $ 390 nm), and high r-process enhancement in order to allow for the presence of strong, and sufficiently measurable, U and Th lines.",
"Asteroseismology of red giant stars has, in recent years, demonstrated to provide precise masses for such stars, and therefore ages (, , ).",
"Solar-like oscillations are commonly summarised by two parameters: $\\Delta \\nu $ (average frequency separation) and $\\nu _{\\rm max}$ (frequency of maximum oscillation power).",
"These two quantities provide precise mass (precision of about 10%) and radius (precision of about 3%), using the so-called seismic scaling relations, and an additional information on stellar temperature ($T_{\\rm eff}$ ) (, , ).",
"Since for red giants the stellar masses are a good proxy for stellar age, it is possible to determine a model-dependent age with a precision that can be better than 30% depending on the quality of the seismic information .",
"More precise ages, error $\\sim $ 15%, can be obtained via Bayesian methods combining seismic information with Gaia data and information on the stellar evolutionary stage .",
"Since the age determination using asteroseismology relies on the mass-age relation that red giants follow, this means that the method is biased by any event that changes the stellar mass, as, for example, mass accretion from a companion or stellar mergers or mass-loss.",
"One way to look for mass accretion events from a companion is to look for radial velocity, photometric variations, or chemical signs of accretion (e.g.",
"high carbon and s-process enhancements - , ).",
"The effect of mass-loss can be minimised by looking at stars in the low-RGB phase, where the effect of mass loss are smaller compared to red-clump stars .",
"Figure: RAVE spectra of the 4 metal-poor stars presented in this paper.",
"Spectra are normalised and corrected for radial velocity, the Fe content labeled comes from the analysis of RAVE spectra using the same method as in .The first study to determine masses for a sample of metal-poor halo giants with both seismic information (from Kepler, ) and chemistry from high-resolution APOGEE spectra, was the one of .",
"The authors used scaling relations at face value and reported masses larger (M$>$ 1 M$_\\odot $ ) than what would be expected for a typical old population.",
"Similar results were obtained by , also using scaling relations for three metal-poor stars.",
"These findings led to the need for further tests of the use of asteroseismology in the low metallicity regime.",
", analysed a group of red giants in the globular cluster M4 (${\\rm [Fe/H]}$ $= -$ 1.10 dex and [$\\alpha $ /Fe]=0.4 dex) with seismic data from K2 mission , and found low seismic masses compatible with the old age of the cluster, hence suggesting that seismic masses and radii estimates would be reliable in the metal-poor regime provided a correction to the $\\Delta \\nu $ scaling relation is taken into account for red giant branch (hereafter RGB) stars.",
"The correction presented in is a correction theoretically motivated, based on the computation of radial mode frequencies of stellar modes.",
"In this work, we present a first set of four stars, identified as metal poor ([Fe/H] $\\sim -2$ dex) in the RAVE survey, for which we have seismic information from the K2 mission and high-resolution spectra.",
"The paper is organised as follows: in Sec.",
"we describe how the stars have been selected and observed.",
"The seismic light curve analysis and the determination of atmospheric parameters and abundances from stellar spectra are described in Sec. .",
"In Sec.",
"we derive radii, masses and ages for our stars using both PARAM and scaling relations.",
"We recompute masses for the and M4 samples.",
"We also analyse the offsets and uncertainties introduced by different seismic pipelines, erroneous assumptions in temperature, [$\\alpha $ /Fe]-enhancements, and mass loss.",
"Distances and orbits of the stars are derived in Sec , using Gaia-DR2 parallaxes and proper-motions.",
"In Sec.",
"we discuss each of the four RAVE stars in light of their chemistry, age and orbital properties.",
"Sec.",
"summarises our conclusions and provide an outlook.",
"Figure: t-SNE projection of ∼\\sim 420,000 RAVE spectra.",
"The scaling in both direction is arbitrary,therefore the units on the axes are omitted.",
"The colour scale corresponds to the gravity of thestars as computed by .",
"Giants are shown in red and dwarfs in blue.",
"Lighter shaded hexagons include fewer stars than darker ones.",
"Over-plotted black dots indicate locations of RAVE stars in K2 Campaigns 1,3.",
"Illustrated as stars are the RAVE-K2 objects studied in the present work, which fall in the metal-poor locus of the diagram.Targets analysed in this works belong to K2 mission campaigns 1 and 3.",
"The K2 Campaign 1 field (C1), centred at RA 11 h 35 m 46 s DEC $+$ 01$^\\circ $ 25' 00” (l=265, b=$+$ 58), was observed from 30 May 2014 to 21 August 2014, and contains one metal-poor RAVE star.",
"The K2 Campaign 3 field (C3), centred at RA 22 h 26 m 40 s DEC $-$ 11$^\\circ $ 25' 02” (l=51, b=$-$ 52), was observed from 14 November 2014 to 03 February 2015, contains three RAVE metal-poor giants.",
"RAVE targets were observed as part of the “The K2 Galactic Archaeology Program Campaign”(C1-C3 proposal GO1059, and described in ).",
"Light curves were obtained using the same approach as described in Section 3 of ."
],
[
"Target selection",
"In C1 and C3 K2 fields there are a total of 376 RAVE targets for which solar-like oscillations have been detected.",
"Following the joint spectroscopic and seismic analysis described in we identified four stars expected to have metallicities ${\\rm [Fe/H]}$ $\\le $ $-$ 1.5 dex.",
"The spectra of the metal-poor targets are visible in Fig.REF .",
"RAVE spectra cover a narrow spectral interval (8410-8795 Å) at intermediate resolution (R$\\sim $ 7,500), that combined with the low metallicity of the targets (few detectable lines, as visible in Fig.",
"REF ) make the traditional spectroscopic analysis challenging: the atmospheric parameters may suffer of degeneracies and offsets.",
"Using the t-SNE projection (, we confirmed that the four stars were, indeed, metal poor.",
"The t-SNE projection is an algorithm that, when applied to spectra, provides a low-dimensional projection of the spectrum space and isolates objects that present similar morphology.",
"In our case, as visible in Fig.",
"REF , metal-poor stars clump in the upper-left region of the projection.",
"In the figure $\\sim $ 420,000 RAVE spectra with SNR $>$ 10 are projected, with the RAVE stars in K2 C1 and C3 represented as empty circles.",
"The four stars that fall into the very metal-poor island (top right) are the metal poor giants analysed in the present work.",
"Figure: Δν\\Delta \\nu and ν max \\nu _{\\rm max}as measured by different pipelines.",
"Each colour (blue, magenta, red and green) corresponds to a different pipeline (COR, GRD, A2Z and YE - see Appendix).",
"The values plotted in black correspond to a further test using COR with inflated uncertainties (BM_\\_N)."
],
[
"Gaia DR2",
"The four stars are in Gaia DR2 (, ).",
"Parallaxes, proper motions and flags are listed in Table REF .",
"The duplicated_source flag is listed as Dup.",
"Table: Gaia DR2 data and seismic data for the 4 RAVE metal-poor stars studied here.",
"In this work we adopted Δν\\Delta \\nu and ν max \\nu _{\\rm max} from COR pipeline and investigated the effect of adopting errors computed considering the dispersion among four different seismic pipelines (COR, GRD, YE, A2Z), here identified as BM_N seismic values.Star S1 (Epic ID: 201359581) has a duplicated$\\_$ source flag = true, meaning that this source presented more than one detection and only one entry was kept.",
"This means that the star had observational or processing problems, leading to possible erroneous astrometric or photometric solution.",
"This same star has an astrometric$\\_$ excess$\\_$ sigma $\\ge $ 2 that, combined with astrometric$\\_$ excess$\\_$ noise flag $>$ 0, indicates large astrometric errors and an untrustworthy solution.",
"For this same star the Gaia DR2 radial velocity has an error of 5.17 km/s, hence larger than the $\\sim $ 0.8 km/s expected for a star of that temperature and brightness.",
"Star S2 (Epic ID: 205997746) has a Priam$\\_$ flag indicating a silver photometry quality and a lower quality in the temperature, radius and luminosity solutions (while the rest of the stars in the sample have a better, golden, photometry quality).",
"For S1 (201359581), we did not consider the ages and masses derived by taking into account the Gaia DR2 information.",
"In addition, we consider the solutions for S2 (205997746) of lower quality respect to the other 2 stars, S3 (206034668) and S4 (206443679).",
"We will use the Gaia DR2 proper motions when computing orbits for our stars in Section , with the exception of S1, for which we will use UCAC-5 proper motions.",
"Gaia DR2 parallax, $\\varpi $ , can be used for deriving the surface gravity: $\\begin{split}\\log (g) _{\\varpi }= \\log (g)_\\odot + 4 \\log \\left(\\frac{T_{\\rm eff}}{{\\rm T}_{{\\rm eff},\\odot }}\\right)+\\log \\left( \\dfrac{m}{m_{\\odot }}\\right) +\\\\+0.4 \\left( m_{V}+5 -5 \\log (1/\\varpi )-3.2 (E(B-V)) + BC - M_{{\\rm bol},\\odot } \\right)\\end{split}$ We derived $\\log (g)$$_\\varpi $ for the stars of our sample, assuming the bolometric correction (BC) as in and , using Ks magnitudes and assuming stellar masses of 0.9 M$_\\odot $ and spectroscopic (UVES) temperatures.",
"Errors were calculated via propagation of uncertainties and varying stellar masses from 0.8 to 2.2 M$_\\odot $ (a typical red giant star mass range).",
"We also took into account the effect of the different offsets in the $\\varpi $ , considering the zero point correction and the offset pointed out by : thus we considered an offset effect that varies $\\varpi $ within ($\\varpi -$ 0.3) and ($\\varpi +$ 0.2).",
"Resulting gravities and their uncertainties are listed together with the stellar parameters obtained from spectroscopy (see next Sections) in Table REF ."
],
[
"High-resolution spectra",
"UVES high resolution spectra of our targets were collected in the period 99D, using UVES-CD 3 set-up , program ID: 099.D-0913(A).",
"Spectra have a resolving power of $\\sim $ 110,000 and cover $\\sim $ 4170-6200Åspectral range.",
"Observing date, exposure time and SNR of spectra are listed in Table REF .",
"Table: Coordinates and set-up of the ESO-UVES observations of the stars.",
"The SNR listed is the one calculated in the all spectral range."
],
[
"Seismic Data",
"Very metal poor stars typically have large radial velocities that induce a Doppler shift of observed frequencies.",
"Although small, this shift can be larger than the precision on asteroseismic frequencies.",
"In this work we use the average seismic parameters $\\Delta \\nu $ and $\\nu _{\\rm max}$ .",
"Because $\\Delta \\nu $ is a frequency difference and because the precision on $\\nu _{\\rm max}$ is much lower than for individual mode frequencies the Doppler correction does not need to be applied to asteroseismic average parameters .",
"In order to quantify the impact of the different seismic inputs on the estimates of the mass and age of our stars, we first considered the $\\Delta \\nu $ and $\\nu _{\\rm max}$ measurements coming from four different seismic pipelines: COR: It is the method adopted for CoRoT and Kepler stars (, ).",
"In a first step, the average frequency separation $\\Delta \\nu $ , is measured from the autocorrelation of the time series computed as the Fourier spectrum of the filtered Fourier spectrum of the signal.",
"The significance of the result is checked using a statistical test based on the H0 hypothesis.",
"GRD: This pipeline is based on fitting a background model to the data .",
"The model is a model H , comprised of two Harvey profiles, a Gaussian oscillation envelope, and an instrumental noise background.",
"For the estimate of $\\nu _{\\rm max}$ the central frequency of the Gaussian component is considered.",
"The median and the standard deviations are used to summarise the normal-like posterior probability density for $\\nu _{\\rm max}$ .",
"To estimate the average frequency separation a model was fitted to the power spectrum .",
"YE: This is a three stages approach.",
"First, a signal-to-noise ratio spectrum (SNR) in function of frequency is created by dividing the power spectrum by a heavily smoothed version of the raw power spectrum.",
"The second step consists in using a combination of H0 and H1 hypothesis for detecting oscillation power in segments of the SNR spectrum.",
"If a segment shows detection of oscillations power, then $\\nu _{\\rm max}$ and $\\Delta \\nu $ are detected as a third step (, ).",
"A2Z: A first estimate of $\\Delta \\nu $ was done using the same method as COR.",
"$\\nu _{\\rm max}$ is measured by fitting a Gaussian on top of the background to the power spectrum.",
"Then $\\Delta \\nu $ is recomputed from the power spectrum of the power spectrum and by considering only the central orders of the spectrum centred on the highest radial mode (, ).",
"Differently from the previous pipelines, this one measured a value for $\\Delta \\nu $ only for 2 of the 4 targets and provided significantly larger error bars for $\\nu _{\\rm max}$ .",
"We then checked that the different pipelines were in agreement for the four stars, as showed in Fig.",
"REF .",
"As we are dealing with a small number of stars and since the four pipelines are in agreement, we can perform a star-by-star analysis of the goodness of the seismic values.",
"From a visual inspection, as visible in Appendix A, it appears that: i) A2Z pipeline is providing $\\Delta \\nu $ with very large uncertainties; ii) YE and GRD pipelines provide a $\\nu _{\\rm max}$ value that appears shifted respect to the expected value, for star S1 and S2 respectively (see Appenfix Fig.",
"REF ).",
"As shown by previous works (e.g.",
", ), using individual frequencies for deriving $\\Delta \\nu $ is more precise than the method presented above.",
"The individual frequencies fitting exercise is difficult to perform for K2 light-curves, because of the short duration of the K2 runs.",
"For this reason the use of the universal pattern is preferred, as in , which uses the detailed information of the whole oscillation pattern .",
"This dedicated analysis provides refined values of the global seismic parameters, with smaller uncertainties.",
"This choice is justified also by the tests performed in .",
"For these reasons we have therefore adopted $\\Delta \\nu $ from the COR pipeline as our preferred value.",
"An additional test has been performed, for RGB stars in the $\\alpha $ -rich APOGEE-Kepler sample: individual mode frequencies has been measured for $\\approx $ 1,000 stars and then a comparison between $\\Delta \\nu $ measured from individual frequencies with the $\\Delta \\nu $ measured by COR pipeline had been performed.",
"A small ($\\lesssim $ 1%) difference between $\\Delta \\nu $ as determined by COR, and $\\Delta \\nu $ determined from individual radial-mode frequencies is found (Davies et al., in preparation), supporting our choice for COR values.",
"This is also relevant because the $\\Delta \\nu $ determined from individual mode frequencies is closer to the $\\Delta \\nu $ given in the stellar models adopted in PARAM, the tool used in this work for deriving mass, radii, and ages.",
"We additionally considered the seismic values from GRD pipeline, which has error bars in $\\Delta \\nu $ and $\\nu _{\\rm max}$ compatible with the COR pipeline and with the data quality (see more details in Appendix ).",
"For having a better comprehension of the impact of the use of a global error coming from considering all the pipelines we also adopted a fifth set of $\\Delta \\nu $ and $\\nu _{\\rm max}$ (BM_N), where the $\\Delta \\nu $ and $\\nu _{\\rm max}$ are from the COR pipeline but with inflated errors that consider the dispersion of the pipelines respect to COR values: $\\sigma ^2_{x,{\\rm BM\\_N}}=\\sigma ^2_{x,{\\rm COR}}+\\frac{\\sum _{\\rm i=GRD,YE,A2Z}(x_i - x_{\\rm COR})^2}{3}$ where $x$ =$\\Delta \\nu $ or $\\nu _{\\rm max}$ .",
"The adopted seismic values, COR and BM_N, are listed in Table REF (the complete set of seismic values are in Appendix Table REF ) and a comparison of the different sets of $\\Delta \\nu $ and $\\nu _{\\rm max}$ is shown Fig.",
"REF .",
"Figure: Δν\\Delta \\nu and ν max \\nu _{\\rm max}distribution of the 4 stars studied in this work.",
"On the background the Δν\\Delta \\nu -ν max \\nu _{\\rm max}distribution distribution of the APOKASC sample, colour coded following the mass.We compared the $\\Delta \\nu $ and $\\nu _{\\rm max}$ of our sample with the $\\Delta \\nu $ and $\\nu _{\\rm max}$ distribution of the APOKASC sample.",
"The high quality of the APOKASC sample makes it the perfect benchmark to provide a first glance on the masses expected for our objects.",
"Fig.",
"REF shows that our four stars fall in the region where the less massive stars are located."
],
[
"RAVE spectra analysis",
"The analysis of the RAVE spectra has been performed following the method described in .",
"We iteratively derived atmospheric parameters by fixing the gravity to the seismic value, $\\log (g)$$_{\\rm S}$ .",
"As a starting point for deriving $T_{\\rm eff}$ , we used the Infra-Red Flux Method (IRFM) temperature published in RAVE-DR5 , allowing for variations as large as 250 K. This analysis was performed using the GAUFRE pipeline .",
"The seismic gravity we used is defined as: $\\log (g)_{\\rm S}= \\log (g)_\\odot + \\log \\left(\\dfrac{\\nu _{\\rm max}}{\\nu _{{\\rm max}, \\, \\odot }}\\right)+ \\frac{1}{2} \\log \\left(\\frac{T_{\\rm eff}}{{\\rm T}_{{\\rm eff},\\odot }}\\right)$ with the adoption of the following solar values: $\\nu _{\\rm max}$$_{,\\odot }$ = 3090 $\\mu $ Hz, $\\Delta \\nu $ $_\\odot $ = 135.1 $\\mu $ Hz, $\\log (g)$$_\\odot $ =4.44 dex, and $T_{\\rm eff}$$_{,\\odot }$ = 5777 K .",
"Table: Radial velocity, atmospheric parameters and abundances of the metal-poor RAVE stars in K2 Campaigns 1 and 3, as derived from RAVE spectra.",
"Temperature and abundances have been derived by fixing the gravity to the seismic value (following the method described in ) and using RAVE spectra.",
"Abundances were determined under LTE assumptions.Atmospheric parameters and abundances derived from RAVE spectra are listed in Table REF .",
"Abundances were derived under Local Thermodynamic Equilibrium (LTE).",
"The chemical abundances obtained from RAVE spectra suggest that the four stars are $\\alpha $ -enhanced with [$\\alpha $ /Fe]$\\sim $ 0.3 dex.",
"On the other hand the individual abundance ratios of [Mg/Fe], [Si/Fe] and [Ti/Fe] are significantly discrepant for the different stars.",
"Notice that the [Fe/H] values reported in Table REF are not corrected for non-local thermodynamic equilibrium (NLTE) effects.",
"We will return to this point when discussing the abundance ratios obtained from high-resolution spectra."
],
[
"UVES spectra analysis",
"We analysed the high-resolution UVES spectra using the GAUFRE pipeline for retrieving $T_{\\rm eff}$ , $\\log (g)$ , and ${\\rm [Fe/H]}$ iteratively using the seismic information on $\\log (g)$ , using Eq.",
"REF .",
"The analysis was performed with the GAUFRE module GAUFRE$\\_$ EW, that derives atmospheric parameters via ionisation and excitation equilibrium using the equivalent widths (EW) of FeI and FeII lines, MARCS model atmospheres and the silent version of MOOG 2017http://www.as.utexas.edu/$\\sim $ chris/moog.html.",
"For sake of comparison we derived atmospheric parameters also using the classical method (imposing excitation and ionisation equilibrium using FeI and FeII lines), results are listed as $T_{\\rm eff}$$_{,\\rm Cl}$ and $\\log (g)$$_{\\rm Cl}$ in Table REF .",
"The error in $T_{\\rm eff}~$ was calculated considering the range of $T_{\\rm eff}~$ within the Fe I abundances were independent from the line excitation potential (slope equal to zero) and by varying $\\log (g)$ and v$_{\\rm mic}$ within errors.",
"The error in $\\log (g)$ was calculated via propagation of uncertainty when the adopted $\\log (g)$ was derived using asteroseismology (Eq.",
"REF ).",
"When $\\log (g)$ was measured via the classic method (ionization equilibrium of Fe I and Fe II), the uncertainty was derived by varying $T_{\\rm eff}~$ , v$_{\\rm mic}$ , and ${\\rm [Fe/H]}$ by their uncertainty, since the values are interdependent.",
"Abundances of different chemical elements were derived using MOOG 2017, in the updated version properly treating Rayleigh scattering Code available at: https://github.com/alexji/moog17scat.",
"For the abundances analysis an ad-hoc model atmosphere with the same atmospheric parameters found by GAUFRE, was created via interpolation using MARCS models.",
"The linelist was constructed using the linelists in , , implemented, when necessary, with line parameters retrieved from VALD DR4 database (, , , , ).",
"The C abundance was derived via fitting the A-X CH band-head at $\\sim $ 4000-4300 Å.",
"Line parameters were taken from .",
"We measured the abundances of the following alpha-elements: Mg, Si, Ca, and Ti.",
"NLTE corrections for Ti are taken from the work of .",
"In addition we measured the abundances of several iron peak elements (Cr, Mn, Fe, Ni, Cu, Zn, and Ga).",
"For Fe we adopted the line-by-line NLTE corrections provided by .",
"NLTE corrections for Mn are taken from .",
"Line-by-line corrections for Fe and Mn are taken from a user-friendly interface available online Available at the website http://nlte.mpia.de/.",
"As indicator of r-process enrichment we measured abundances of Eu and Gd.",
"As s-process markers we measured Sr and Ba.",
"Final abundances are listed Table REF (for more details see Appendix ).",
"The uncertainties on abundances provided in Table REF (and in REF ) were calculated considering: the internal error of the fit, the errors on $T_{\\rm eff}~$ and $\\log (g)$ , and the error on continuum normalisation.",
"The error on the fit is provided by MOOG itself.",
"We computed the impact of $T_{\\rm eff}~$ and $\\log (g)$ uncertainties by creating different model atmospheres by varying atmospheric parameters within the errors.",
"Error on continuum normalisation has been taken into account by creating, for each stellar spectrum, ten different continuum normalisations and then analysing them.",
"The error listed in Table REF is the sum in quadrature of these three different errors.",
"In Figure REF we compare the abundance pattern of the four RAVE stars with that of CS 31082-001 (dotted grey curve) which is considered to be a typical pure r-process enriched star .",
"The abundances for CS 31082-001 were taken from .",
"Figure: Atmospheric parameters of the sample of metal-poor stars, as taken from literature and this work: RAVE spectra and seismic parameters (red squares), RAVE-DR5 (blue triangles), RAVE-on (cyan triangles) and ESO high-resolution spectra and seismic parameters (black circles).The four stars are clearly enhanced in core collapse (SN type II) nucleosynthetic products (such as Mg, Si, and Eu), as one would expected to be the case for old stars.",
"However, the range in $\\alpha $ -enhancement is very large, and it is not correlated with metallicity.",
"S1, S2 and S4 can be classified as r-I stars (i.e.",
"stars with 0.3 $\\le $ [Eu/Fe] $\\le $ 1 and [Ba/Eu]$<$ , ), while S3 is clearly Ba-enhanced.",
"The low C-enhancement, and the low [Ba/Fe] ratios (with only the exceptional case of S3), suggest minor contribution from AGB-mass transfer (if any).",
"The values obtained from our HR analysis for [Mg/Fe], [Si/Fe], and [Ti/Fe] can now be compared with those reported in Table REF obtained from the RAVE spectra.",
"In most of the cases the discrepancies are above the quoted error bars, and it is probably due to the combination of the lower resolution and shorter spectral coverage of RAVE spectra, that leads to undetected line blends and the presence of very few lines per element.",
"The [$\\alpha $ /Fe] ratios coming from high-resolution UVES spectra show a large variation.",
"Enhancements for S2 and S4 seem systematically larger than the ones of S1 and S3.",
"In Fig.",
"REF the atmospheric parameters in this work (from RAVE and UVES spectra) are compared with the literature values presented in RAVE-DR5 (calibrated values), RAVE-on (, where the stellar parameters were obtained by using a data-driven approach).",
"It is worth noticing that the RAVE-on catalogue misplaced these red giants in metallicity and/or gravity.",
"This misclassification might be due to the training sample adopted in , consisting mostly of APOGEE red giants, that are mostly metal rich.",
"In Fig.",
"REF is visible also that for the star 201359581 the temperature obtained with the method is $\\sim $ 350 K higher than the one measured from the high-resolution spectrum.",
"This is a consequence of the fact that the starting $T_{\\rm eff}~$ adopted was erroneous.",
"For stars S2, S3 and S4, there is a good agreement between the temperatures estimated from the RAVE and high-resolution analysis spectra, upon the use of the seismic gravity.",
"The agreement is also seen in metallicity, where the most discrepant case, S4, is our most metal-poor star for which the non-local thermodynamic equilibrium (NLTE) corrections are more important (we took into account NLTE effects, when analysing UVES spectra).",
"Two important results can be extracted from Fig.",
"REF : i) by combining the RAVE spectra with seismic gravities it is possible to reach precise stellar parameters, similar to what is obtained from high-resolution spectra (see the agreement between the black dots (UVES) and red points (RAVE) for 3 out of the 4 stars); ii) the high-resolution analysis has confirmed that one of the stars has metallicity [Fe/H] $<-$ 2.",
"The difficulty in determining the metallicity of such metal-poor objects from moderate resolution spectra covering a rather short wavelength range, not having the extra seismic information, is clearly illustrated by the discrepant metallicities found by RAVE DR5 and RAVE-on, versus the good agreement with the value published in Valentini et al.",
"(2017) upon the use of K2 information, where the temperatures and gravities are consistent."
],
[
"Mass and age determination",
"Mass determinations have been performed using two different methods: i) a direct method, using scaling relations, and ii) a Bayesian fitting using the PARAM code .",
"Masses derived using scaling relation differ from the ones from PARAM (see discussion in Rodrigues et al.",
"2017).",
"We now illustrate this difference for the case of our four metal-poor stars.",
"The resulting masses from the two methods are summarized in Table REF ."
],
[
"Mass estimate using scaling relations:",
"For our computations using the scaling relations we adopt as input $\\Delta \\nu $ and $\\nu _{\\rm max}$ from the COR pipeline and the $T_{\\rm eff}~$ measured from the UVES spectra.",
"The scaling relations are in the form: $\\frac{M}{M_\\odot } &\\simeq & \\left(\\frac{\\nu _{\\rm max}}{\\nu _{\\rm max, \\odot }}\\right)^{3}\\left(\\frac{\\Delta \\nu }{\\Delta \\nu _{\\odot }}\\right)^{-4}\\left(\\frac{T_{\\rm eff}}{T_{\\rm eff, \\odot }}\\right)^{3/2}\\\\\\frac{R}{R_\\odot } &\\simeq & \\left(\\frac{\\nu _{\\rm max}}{\\nu _{\\rm max, \\odot }}\\right)\\left(\\frac{\\Delta \\nu }{\\Delta \\nu _{\\odot }}\\right)^{-2}\\left(\\frac{T_{\\rm eff}}{T_{\\rm eff, \\odot }}\\right)^{1/2}$ were the solar values adopted are the same ones listed in Section , and $\\Delta \\nu $ =135.1 $\\mu $ Hz.",
"The uncertainties on the masses and radii are calculated using propagation of uncertainties, under the assumption of uncorrelated errors.",
".",
"For deriving ages and masses via Bayesian inference we adopted the latest version of the PARAM code.",
"The new version of the code uses $\\Delta \\nu $ that has been computed along MESA evolutionary tracks, plus $\\nu _{\\rm max}$ computed using the scaling relation.",
"The following modifications were implemented with respect to the version described in , namely: i) we extended the grid towards the metal poor end, down to [Fe/H]=$-$ 3 dex, by calculating evolutionary tracks for ${\\rm [Fe/H]}$ = $-$ 2.0 and $-$ 3.0 dex, with He enrichment computed according ; and ii) we took $\\alpha $ -elements enrichment into account, by converting the observed chemical composition into a solar-scaled equivalent metallicity.",
"We investigated the solutions provided by PARAM when setting an upper limit to the age at 14 Gyr and without age upper limit (the latter helps in understanding the shape of the PDF of mass and age)."
],
[
"Mass-loss and alpha-enhanced tracks",
"PARAM provides also an estimate for stellar distance and luminosity, $L$ (listed in Table REF ).",
"The luminosities provided by PARAM were used to construct Fig.",
"REF , where we placed our stars in the temperature-luminosity diagram.",
"The figure shows a set of MESA evolutionary tracks for masses 0.8 and 1.0 M$_\\odot $ , at two different metallicities Z=0.00060 and Z=0.00197.",
"In the same figure the four stars are also plotted, together with the track, in the $\\nu _{\\rm max}$ -$T_{\\rm eff}$ (middle panel) and $\\Delta \\nu $ -$T_{\\rm eff}~$ planes.",
"The stars of our sample are most likely low-luminosity RGB stars which are not expected to undergo significant mass loss.",
"The evolutionary state of star S1 (201359581), on the other hand, is more uncertain, since it is locate close to the RGB bump (dashed line), following also Fig.",
"1 of , it can be core-He burning, RGB, early AGB.",
"The evolutionary status of this star becomes relevant when it comes to discussing the reliability of age estimates, since stars in the red-clump or early AGB phases suffer of significant mass loss, that hampers the mass (and hence age) determination.",
"Finally, since our stars are well located below the bump (with a flag on S1 that is an borderline case), we consider their abundances not affected by extra-mixing process that happens at the bump and early AGB stage.",
"Because the adopted MESA stellar tracks in assume the solar mixture for the metals, we adopt the $\\alpha $ -enhancement correction to convert [Fe/H] into [M/H] by using the formula from , updated using the relative mass fraction of elements from OPAL tables https://opalopacity.llnl.gov/pub/opal/type1data/GN93/ascii/GN93hz : ${\\rm [M/H]}^{\\rm chem}= {\\rm [Fe/H]}+ \\log {\\left( C \\times 10^{[\\alpha /{\\rm Fe}]}+(1-C)\\right)}$ where $C$ =0.684.",
"This is a necessary step, given that all our stars are $\\alpha $ -enhanced.",
"We tested the effectiveness of this assumption by comparing two PARSEC track sets (from MS to RGB tip), which are also provided for $\\alpha $ -enhanced cases.",
"In Appendix ) we compare one track computed for [$\\alpha $ /Fe]=+0.4 dex and ${\\rm [Fe/H]}$ =$-$ 2.15, and one not alpha enhanced, but with the corresponding metallicity following Equation REF (${\\rm [Fe/H]}$ =$-$ 1.86).",
"The test shows that the deviation in age between the tracks has its maximum at the RGB tip, in the mass regime of our stars.",
"This deviation is in the order of 1-2%, a smaller effect respect to the typical age uncertainty.",
"Figure: Top panel: Position in the temperature - luminosity diagram of the four RAVE stars of this work (nomenclature following Table 1).",
"Evolutionary tracks at masses M= 0.8 and 1.0 M ⊙ _\\odot , at two different metallicities (Z=0.00060 and 0.00197) are plotted.",
"Middle panel: Position in the temperature - ν max \\nu _{\\rm max} diagram of the four RAVE stars of this work, same tracks as top panel.",
"Bottom panel: Position in the temperature - Δν\\Delta \\nu diagram of the four RAVE stars of this work, same tracks as top panel.",
"Error bars of the plotted quantities are of the size of the points.We derived mass and ages by adopting first the atmospheric parameters derived from RAVE spectra and then for the atmospheric parameters obtained from UVES spectra.",
"We also computed mass and ages using the different seismic inputs discussed in Section (COR and BM$\\_$ N).",
"This strategy allows us to see the impact of different precision in the atmospheric parameters and seismic parameters.",
"Results are summarised in Appendix Table REF .",
"Results obtained with the high-resolution input for temperature, metallicity, and an averaged [$\\alpha $ /Fe] (computed as ([Mg/Fe]+[Si/Fe]+[Ca/Fe])/3) are in Fig.",
"REF and in Appendix Fig.",
"REF .",
"In these figures it is visible that the PDF of masses and ages obtained with the seismic values with BM_N seismic values are broad and, in the case of 205997746, double peaked.",
"This is a consequence of the inflated error in BM_N, caused by blindly combining all the spectroscopic pipelines.",
"This shows that, when dealing with a detailed analysis of individual stars, a star-by-star approach for testing the performances of each seismic pipeline is a necessary step for increasing the precision of mass and age determination.",
"Table: Seismic mass and radius calculated using scaling relations (T eff T_{\\rm eff}~measured from UVES spectra), and mass, radius, and age derived using PARAM, for the 4 metal-poor RAVE stars in K2 Campaigns 1 and 3.",
"The last column lists the stellar radius provided by Gaia DR2.Figure: Left column: violin plot of the PDFs of mass (top) and age (bottom).",
"The right magenta shaded PDF is derived using the seismic parameters from BM_N seismic set of parameters, with the new errors that take into account dispersion between pipelines, the PDFs on the left of the violin are calculated using seismic parameters from COR pipeline (black line, gray shaded) and varying the T eff T_{\\rm eff}~of +100 K (dashed blue line) or --100 K (dotted red line).",
"Right column: modes and 68 percentile errorbar of masses (top) and ages (bottom) of the 4 stars of this work.",
"Magenta points are values computed using BM_\\_N seismic values, black diamonds are the values derived using COR seismic values, and red and blue triangles are values obtained using COR seismic values and varying the T eff T_{\\rm eff}~of --100 and +100 K respectively.Our adopted final values of stellar mass and radius, derived using COR seismic input and UVES spectra, are shown in Tab.",
"REF , where we also show, for comparison, the results obtained directly from the scaling relations.",
"The mass and ages of PARAM are obtained adopting a mass-loss value derived from law with an efficiency parameter of $\\eta $ =0.2.",
"We adopted this value since it is in agreement to what measured in by comparing the asteroseismic masses of Red Clump stars and Red Giants in the old open clusters NGC6791 and NGC6819.",
"The error associated to the mode value of radius, mass and age derived using PARAM is calculated as the shortest credible interval with 68 per cent of the probability density function (PDF).",
"Masses derived with scaling relations (Eq.",
"REF ) are larger than those derived using PARAM by circa 30%.",
"This is due to the correction needed to $\\Delta \\nu $ (see ), that leads to a more accurate mass estimation for red giants.",
"In PARAM this correction is not necessary.",
"The code can, in fact, derive the theoretical $\\Delta \\nu $ directly by interpolation, since this quantity has been estimated along each evolutionary track."
],
[
"Using Luminosities from Gaia DR2 to further constrain PARAM",
"In the work of the adoption of the intrinsic stellar luminosity, $L$ , derived using Gaia parallaxes, leads to a significant improvement into the mass and age determination (from an error of 5% in mass and 19% in age to 3% and 10% respectively).",
"These estimates were based on high-quality Kepler seismic data and very precise atmospheric parameters.",
"In addition, the uncertainties on luminosity were assumed to be 3%, from Gaia end-of-the-mission performances.",
"Gaia DR2 does not still reach this precision and offsets in $\\varpi $ have to be taken into account.",
"Nevertheless we calculated mass, radius and age using the additional information on $L$ , calculated from parallax and find out the shape of the PDFs were affected, suggesting some tension with the input luminosities.",
"Instead of using the luminosities tabulated in Gaia DR2, we considered the weighted mean of the $L$ calculated from K$s$ , I, and V magnitudes, considering BC provided by and Codes available at https://github.com/casaluca/bolometric-corrections and the reddening derived from maps.",
"Errors on $L$ were calculated via error propagation, with the error on BC calculated via Monte-Carlo simulation of 100 points for each star.",
"Luminosities are listed in Table REF and show $\\sim $ 15% uncertainties, and not the 3% end of mission expectation.",
"We thus opted for not using luminosities as an extra constraint in our calculations of mass and radius.",
"Table: Masses (left column) and ages (right column) of the red giants in M4 analysed in this work.",
"The thick red vertical line identifies the literature value, while the fine lines identifies the upper an lower values.",
"On the top row are reported individual masses (left) and ages (right), with the errorbar indicating the 68 percentile of the PDF.",
"At the bottom the individual PDF for mass and ages are plotted."
]
] | 1808.08569 |
[
[
"Gauge dependence of the one-loop divergences in $6D$, ${\\cal N} = (1,0)$\n abelian theory"
],
[
"Abstract We study the gauge dependence of the one-loop effective action for the abelian $6D$, ${\\cal N}=(1,0)$ supersymmetric gauge theory formulated in harmonic superspace.",
"We introduce the superfield $\\xi$-gauge, construct the corresponding gauge superfield propagator, and calculate the one-loop two-and three-point Green functions with two external hypermultiplet legs.",
"We demonstrate that in the general $\\xi$-gauge the two-point Green function of the hypermultiplet is divergent, as opposed to the Feynman gauge $\\xi =1$.",
"The three-point Green function with two external hypermultiplet legs and one leg of the gauge superfield is also divergent.",
"We verified that the Green functions considered satisfy the Ward identity formulated in ${\\cal N}=(1,0)$ harmonic superspace and that their gauge dependence vanishes on shell.",
"Using the result for the two- and three-point Green functions and arguments based on the gauge invariance, we present the complete divergent part of the one-loop effective action in the general $\\xi$-gauge."
],
[
"Introduction",
"Gauge theories with extended supersymmetries in higher dimensions attract a considerable attention for a long time [1], [2], [3], [4], [5], [6], [7], [8].",
"On the one hand, such theories are non-renormalizable due to the dimensionful coupling constant (see, e.g., [9], [10]).",
"On the other hand, one can expect an improvement of the ultraviolet behavior due to the extended supersymmetry.",
"It is very interesting to check this conjecture on the explicit examples of higher-dimensional supersymmetric theories.",
"To be more realistic, one can expect that the full canceling of divergences is presumably possible only in the lowest loops even in the maximally extended theories (see, e.g., [11]).",
"The problem reveals clear analogies with the most interesting case of gravity.",
"However, the analysis in supersymmetric gauge theories is much simpler.",
"In order to fully display the underlying properties of theories with some symmetries it is highly desirable to be aware of the regularization and quantization schemes which do not break these symmetries.",
"In the case of extended supersymmetries these purposes can be achieved within the harmonic superspace approach [12], [13], [14], [15], [16], [17].",
"For $6D$ supersymmetric gauge theories (which will be the subject of the present paper) this formalism [18], [19], [20], [21], [22], [23] ensures manifest ${\\cal N}=(1,0)$ supersymmetry.",
"With the use of the background field method in harmonic superspace [16], [24], gauge symmetry can also be made manifest.",
"For these reasons the harmonic superspace formalism seems to be most suitable for quantum calculations in $6D$ supersymmetric theories (note that $6D\\,,$ ${\\cal N}=(1,0)$ theories are in general anomalous, see, e.g., [25], [26], [27], [28]).",
"Recently, some explicit calculations based on the harmonic superspace method were done for ${\\cal N}=(1,0)$ and ${\\cal N}=(1,1)$ gauge theories [29], [30], [31], [32], [33], following the general pattern of Ref.",
"[4].",
"These calculations were basically performed in the Feynman gauge $\\xi =1$ , which ensures the simplest form of the propagator of the gauge superfield.",
"This considerably simplifies the calculation of quantum corrections.",
"However, the gauge dependence of the results obtained by the harmonic superspace technique has not been yet analyzed.",
"Meanwhile, the calculations in the non-minimal gauges are frequently rather useful as compared to those in the Feynman gauge, because they are capable to make manifest divergences in the lower loops.",
"For example, for ${\\cal N}=1$ supersymmetric gauge theories in the one-loop approximation ghosts are not renormalized in the Feynman gauge, while divergences appear for $\\xi \\ne 1$ [35].",
"For calculations in higher orders, the knowledge of gauge dependence in the lower-order approximations is also essential, see, e.g., [36].",
"These are the reasons why a vast literature is devoted to calculations in non-minimal gauges.",
"As a characteristic example, let us mention a recent paper [37].",
"In the present paper we consider the simplest $6D\\,,$ ${\\cal N}=(1,0)$ supersymmetric gauge theory, namely, ${\\cal N}=(1,0)$ supersymmetric electrodynamics, and investigate the structure of the gauge-dependent contributions to the effective action by the harmonic superspace technique.",
"In particular, we demonstrate that (unlike the case of the Feynman gauge considered, e.g., in [29]) the two-point Green function of hypermultiplets is divergent already at the one-loop level.",
"The gauge-dependent divergences are also present in the gauge multiplet - hypermultiplet Green functions.",
"In this paper we explicitly calculate the one-loop three-point Green function and find its divergent part.",
"Moreover, we derive the Ward identity in the harmonic superspace and verify that the Green functions obtained by calculating harmonic supergraphs satisfy this identity, as expected.",
"This result is a non-trivial verification of the correctness of our calculations.",
"One more test, which has also been done in this paper, is the demonstration of the property that the gauge dependence of the effective action vanishes on shell (this is a consequence of the general theorem, see Refs.",
"[38], [39], [40], [41], [42], [43]).",
"Using the results for the two- and three-point Green functions, we also restore the complete result for the one-loop divergences, based on the gauge invariance of the theory under consideration.",
"The paper is organized as follows: In Sect.",
"we recall some basic points of the formulation of $6D\\,,$ ${\\cal N}=(1,0)$ supersymmetric electrodynamics in harmonic superspace.",
"We present the superfield action for this theory, write down the Ward identity, and formulate the harmonic superspace Feynman rules.",
"In particular, we construct the propagator of the gauge superfield in the non-minimal gauges which are analogs of the $\\xi $ -gauges in the usual electrodynamics.",
"In Sect.",
", using these Feynman rules, we investigate the gauge dependence of the one-loop two-point Green functions of the gauge superfield and the hypermultiplet.",
"We also calculate the one-loop three-point gauge superfield - hypermultiplet Green function.",
"Checking the Ward identities for these Green functions is the subject of Sect.",
".",
"The vanishing of the gauge dependence on shell in the approximation we are considering is demonstrated in Sect.",
".",
"The total divergent part of the one-loop effective action (which is an infinite series in $V^{++}$ ) is constructed in Sect.",
", by invoking the arguments based on the gauge invariance.",
"Also we verify that the gauge dependence of the expression obtained vanishes on shell.",
"Some technical details are collected in two Appendices."
],
[
"The harmonic superspace action",
"The harmonic superspace is very convenient for formulating $6D$ , ${\\cal N}=(1,0)$ supersymmetric theories, because it ensures manifest supersymmetry at all steps of quantum calculations.",
"It is parametrized by the coordinate set $(x^M,\\theta ^{ai}, u_i^\\pm )$ which will be referred to as the central basis.",
"Here $x^M$ with $M=0,\\ldots 5$ are the usual coordinates of the six-dimensional Minkowski space.",
"The Grassmann anticommuting coordinates $\\theta ^{ai}$ with $a=1,\\ldots 4$ and $i=1,2$ form a left-handed $6D$ spinor.",
"The harmonic variables $u_i^\\pm $ satisfy the condition $u^{+i} u_i^- = 1$ , with $u_i^- = (u^{+i})^*$ .",
"The analytic basis of the harmonic superspace is parametrized by the coordinates $x^M_A = x^M + \\frac{i}{2}\\theta ^{-}\\gamma ^M \\theta ^+;\\qquad \\theta ^{\\pm a} = u^\\pm _i \\theta ^{ai}, \\quad u^{\\pm }_i\\,,$ where $\\gamma ^M$ are $6D$ $\\gamma $ -matrices.",
"The coordinate subset $(x^M_A, \\theta ^{+ a}, u^\\pm _i)$ parametrizes the analytic harmonic subspace which is closed on its own under $6D\\,, {\\cal N}=(1,0)$ supersymmetry transformations.",
"It is convenient to define the spinor covariant derivatives $D^+_a = u^{+}_i D_{a}^i;\\qquad D^-_a = u^{-}_i D_{a}^i,$ such that $\\lbrace D^+_a, D^{-}_b\\rbrace = i(\\gamma ^M)_{ab}\\partial _M$ , and to introduce the notation $(D^+)^4 = -\\frac{1}{24}\\varepsilon ^{abcd} D_a^+ D_b^+ D_c^+ D_d^+.$ Also we will need the harmonics derivatives in the central basis $D^{++} = u^{+i} \\frac{\\partial }{\\partial u^{-i}};\\qquad D^{--} = u^{-i} \\frac{\\partial }{\\partial u^{+i}};\\qquad D^0 = u^{+i} \\frac{\\partial }{\\partial u^{+i}} - u^{-i} \\frac{\\partial }{\\partial u^{-i}}.$ They satisfy the commutation relations of the $SU(2)$ algebra.",
"The analytic basis form of these derivatives can be easily found and is given, e.g., in [34].",
"For constructing the ${\\cal N}=(1,0)$ invariants we need the invariant superspace integration measures: $&& \\int d^{14}z = \\int d^6x\\,d^8\\theta ;\\qquad \\int d\\zeta ^{(-4)} = \\int d^6x\\, d^4\\theta ^+;\\\\&& \\int d^6x\\,d^8\\theta = \\int d^6x\\,d^4\\theta ^{+} (D^+)^4.$ In this paper we consider ${\\cal N}=(1,0)$ supersymmetric electrodynamics, which is a particular abelian case of ${\\cal N}=(1,0)$ supersymmetric Yang–Mills theory with hypermultiplets.",
"The harmonic superspace form of the action of $6D$ , ${\\cal N}=(1,0)$ supersymmetric Yang–Mills theory was pioneered in Ref.",
"[20].",
"As opposed to the analogous $4D$ , ${\\cal N}=2$ construction, the gauge theory coupling constant $f_0$ in $6D$ has the dimension $m^{-1}\\,$ .",
"In the harmonic superspace approach the gauge superfield $V^{++}(z,u)$ satisfies the analyticity condition $D^+_a V^{++} = 0$ and is real with respect to the special conjugation denoted by $\\widetilde{}$ , i.e.",
"$\\widetilde{V^{++}} = V^{++}$ .",
"The hypermultiplets are described by the analytic superfield $q^+$ and its $\\widetilde{}$ -conjugate $\\widetilde{q}^+$ .",
"Like in the non-supersymmetric case, the action of ${\\cal N}=(1,0)$ electrodynamics is quadratic in the gauge superfield.",
"It can be written as $S = \\frac{1}{4f_0^2} \\int d^{14}z\\,\\frac{du_1 du_2}{(u_1^+ u_2^+)^2} V^{++}(z,u_1) V^{++}(z,u_2) - \\int d\\zeta ^{(-4)} du\\,\\widetilde{q}^+ \\nabla ^{++} q^+.$ where $\\nabla ^{++} = D^{++} + i V^{++}$ and $D^{++}$ is taken in the analytic basis.",
"The gauge transformations has the form $V^{++} \\rightarrow V^{++} - D^{++} \\lambda ; \\qquad q^+ \\rightarrow e^{i\\lambda } q^+; \\qquad \\widetilde{q}^+ \\rightarrow e^{-i\\lambda } \\widetilde{q}^+,$ where $\\lambda $ is an analytic superfield parameter which is real with respect to the $\\widetilde{}$ -conjugation.",
"It is useful to introduce the non-analytic superfield $V^{--}(z,u) = \\int du_1\\,\\frac{V^{++}(z,u_1)}{(u^+ u_1^+)^2}.$ It satisfies the conditions $D^{++} V^{--} = D^{--} V^{++}$ and transforms as $V^{--} \\rightarrow V^{--} - D^{--}\\lambda $ under the gauge transformations.",
"Starting from this superfield, it is possible to construct the analytic superfield $F^{++} = (D^+)^4 V^{--}$ , which is gauge invariant in the abelian case.",
"For further use, we also define the non-analytic superfield $q^-$ as a solution of the equation $q^+ = \\nabla ^{++} q^- = (D^{++} + i V^{++}) q^-.$ From this definition one can derive that the gauge transformations act on $q^-$ as $q^- \\rightarrow e^{i\\lambda } q^-.$ In the explicit form the solution of Eq.",
"(REF ) can be expressed as the series $&&\\hspace*{-22.76219pt} q^- = \\int \\frac{du_1}{(u^+ u_1^+)} q_1^+ -i \\int \\frac{du_1\\,du_2}{(u^+ u_1^+)(u_1^+ u_2^+)} V^{++}_1 q_2^+ - \\int \\frac{du_1\\,du_2\\,du_3}{(u^+ u_1^+)(u_1^+ u_2^+)(u_2^+ u_3^+)} V^{++}_1 V^{++}_2 q_3^+ + \\ldots \\nonumber \\\\&&\\hspace*{-22.76219pt} =(-i)^{n-1} \\sum \\limits _{n=1}^\\infty \\int du_1 \\ldots du_n\\, \\frac{V^{++}_1 \\ldots V^{++}_{n-1}}{(u^+ u_1^+) \\ldots (u_{n-1}^+ u_n^+)} q_n^+,$ where subscripts numerate the harmonic “points”.",
"For quantizing the theory (REF ) it is necessary to fix the gauge.",
"This can be done by adding the gauge-fixing term to the action, $S_{\\mbox{\\scriptsize gf}} = - \\frac{1}{4f_0^2\\xi _0} \\int d^{14}z\\, du_1 du_2 \\frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^3} D_1^{++} V^{++}(z,u_1) D_2^{++} V^{++}(z,u_2),$ where $\\xi _0$ is the bare gauge-fixing parameter.",
"This term corresponds to the $\\xi $ -gauge in the usual electrodynamics.",
"In particular, the Feynman gauge amounts to the choice $\\xi _0=1$ .",
"In the abelian case we are considering it is not necessary to introduce the ghosts superfields.",
"Therefore, the generating functional for our theory can be written as $Z = \\int DV^{++}\\,D\\widetilde{q}^+\\, Dq^+\\, \\exp \\Big \\lbrace i(S+S_{\\mbox{\\scriptsize gf}} + S_{\\mbox{\\scriptsize sources}})\\Big \\rbrace ,$ where $S_{\\mbox{\\scriptsize sources}}$ is a sum of the source terms, $\\int d\\zeta ^{(-4)}\\,du\\, \\Big [V^{++} J^{(+2)} + j^{(+3)} q^+ + \\widetilde{j}^{(+3)} \\widetilde{q}^+\\Big ].$ Here $J^{(+2)}$ is the analytic source for the gauge superfield, while $j^{(+3)}$ and $\\widetilde{j}^{(+3)}$ denote sources for the hypermultiplet superfields.",
"The effective action is constructed from the generating functional for the connected Green functions $W = -i\\ln Z$ by making the Legendre transformation, $\\Gamma = W - S_{\\mbox{\\scriptsize sources}},$ where it is necessary to express the sources in terms of the fields with the help of the equations $V^{++} = \\frac{\\delta W}{\\delta J^{(+2)}};\\qquad q^+ = \\frac{\\delta W}{\\delta j^{(+3)}};\\qquad \\widetilde{q}^+ = \\frac{\\delta W}{\\delta \\widetilde{j}^{(+3)}}.$"
],
[
"Ward identity",
"In the abelian gauge theory at the quantum level the gauge invariance is encoded in the Ward identity [44], which is a particular case of the Slavnov–Taylor identities [45], [46].",
"The harmonic superspace analog of this identity can be formulated, using the standard technique.",
"For this purpose we make the transformation (REF ) in the generating functional (REF ) which evidently remains invariant.",
"Taking into account that the classical action is gauge invariant, in the lowest order in $\\lambda $ we obtain $0 = \\Big \\langle \\int d\\zeta ^{(-4)}\\,du\\, \\Big [ - \\frac{\\delta S_{\\mbox{\\scriptsize gf}}}{\\delta V^{++}} D^{++}\\lambda - J^{(+2)} D^{++}\\lambda + i j^{(+3)} \\lambda q^+ - i \\widetilde{j}^{(+3)} \\lambda \\widetilde{q}^+\\Big ]\\Big \\rangle ,$ where we used the notation $\\Big \\langle A(V^{++}, q^+, \\widetilde{q}^+)\\Big \\rangle = \\frac{1}{Z} \\int DV^{++}\\,D\\widetilde{q}^+\\, Dq^+\\, A(V^{++}, q^+, \\widetilde{q}^+) \\exp \\Big \\lbrace i(S+S_{\\mbox{\\scriptsize gf}} + S_{\\mbox{\\scriptsize sources}})\\Big \\rbrace .$ Integrating in Eq.",
"(REF ) by parts with respect to the derivatives $D^{++}$ , using an arbitrariness of $\\lambda $ , and expressing the result in terms of superfields, we obtain $0 = D^{++} \\frac{\\delta S_{\\mbox{\\scriptsize gf}}}{\\delta V^{++}}-D^{++}\\frac{\\delta \\Gamma }{\\delta V^{++}} - i q^+ \\frac{\\delta \\Gamma }{\\delta q^+} + i \\widetilde{q}^+ \\frac{\\delta \\Gamma }{\\delta \\widetilde{q}^+},$ where $\\Gamma $ is the effective action defined by Eq.",
"(REF ), and we also took into account that the gauge-fixing term is quadratic in the gauge superfield.",
"Introducing $\\Delta \\Gamma = \\Gamma - S_{\\mbox{\\scriptsize gf}},$ the Ward identity can be written in a more compact form, $D^{++}\\frac{\\delta \\Delta \\Gamma }{\\delta V^{++}} = - i q^+ \\frac{\\delta \\Delta \\Gamma }{\\delta q^+} + i \\widetilde{q}^+ \\frac{\\delta \\Delta \\Gamma }{\\delta \\widetilde{q}^+}.$ It is important that this equation is valid for arbitrary non-zero values of the involved superfields.",
"Differentiating Eq.",
"(REF ) with respect to various superfields we derive an infinite set of identities relating the longitudinal part of the $(n+1)$ -point Green functions to the $n$ -point Green functions.",
"For example, differentiating with respect to $V^{++}_2$ and setting all fields equal to zero at the end, we obtain that quantum corrections to the two-point Green function of the gauge superfield are transversal, $D^{++}_1 \\frac{\\delta ^2\\Delta \\Gamma }{\\delta V^{++}_1 \\delta V^{++}_2} = 0.$ Differentiating Eq.",
"(REF ) with respect to $q^+_2$ and $\\widetilde{q}^+_3$ and setting the fields equal to zero at the end give an analog of the usual Ward identity relating three- and two-point Green functions: $&& D^{++}_1 \\frac{\\delta ^3\\Delta \\Gamma }{\\delta V^{++}_1 \\delta q^+_2 \\delta \\widetilde{q}^+_3} = - i (D_1^+)^4 \\delta ^{14}(z_1-z_2) \\delta ^{(-3,3)}(u_1,u_2) \\frac{\\delta ^2\\Delta \\Gamma }{\\delta q^+_1 \\delta \\widetilde{q}^+_3}\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + i (D^+_1)^4\\delta ^{14}(z_1-z_3)\\delta ^{(-3,3)}(u_1,u_3) \\frac{\\delta ^2\\Delta \\Gamma }{\\delta q^+_2 \\delta \\widetilde{q}^+_1}.\\qquad $ When deriving this equation, we have taken into account the property implied by the Grassmann analyticity $\\frac{\\delta q^+_1}{\\delta q^+_2} = (D^+_1)^4 \\delta ^{14}(z_1-z_2) \\delta ^{(-3,3)}(u_1,u_2)\\,,$ where $\\delta ^{14}(z_1-z_2) = \\delta ^6(x_1-x_2) \\delta ^8(\\theta _1-\\theta _2).$ It is convenient to multiply the identity (REF ) with the analytic superfields $\\lambda _1$ , $q^+_2$ , and $\\widetilde{q}^+_3$ , and integrate the expression obtained over both analytic arguments, $&& \\int d\\mu \\, \\widetilde{q}^+_3 D^{++}\\lambda _1 q^+_2\\, \\frac{\\delta ^3\\Delta \\Gamma }{\\delta V^{++}_1 \\delta q^+_2 \\delta \\widetilde{q}^+_3} = i \\int d \\zeta ^{(-4)}_1 du_1\\, d\\zeta ^{(-4)}_3\\, du_3\\, \\widetilde{q}^+_3 \\lambda _1 q^+_1\\, \\frac{\\delta ^2\\Delta \\Gamma }{\\delta q^+_1 \\delta \\widetilde{q}^+_3}\\qquad \\nonumber \\\\&& - i \\int d \\zeta ^{(-4)}_1 du_1\\, d\\zeta ^{(-4)}_2\\, du_2\\, \\widetilde{q}^+_1 \\lambda _1 q^+_2\\, \\frac{\\delta ^2\\Delta \\Gamma }{\\delta q^+_2 \\delta \\widetilde{q}^+_1},\\qquad $ where $\\int d\\mu = \\int d\\zeta ^{(-4)}_1\\, du_1\\, d\\zeta ^{(-4)}_2\\, du_2\\, d\\zeta ^{(-4)}_3\\, du_3.$ This form of the Ward identity is most convenient, when checking it for one or another particular class of diagrams."
],
[
"The Feynman rules",
"For the explicit calculation of quantum correction it is necessary to formulate the relevant Feynman rules.",
"This can be accomplished quite similarly to the $4D$ , ${\\cal N}=2$ case considered in detail in Refs.",
"[13], [14].",
"To find the propagator of the gauge superfield in the $\\xi $ -gauge, we consider the sum of the gauge superfield action and the gauge-fixing term $&& S_{\\mbox{\\scriptsize gauge}} + S_{\\mbox{\\scriptsize gf}} = \\frac{1}{4f_0^2}\\Big (1-\\frac{1}{\\xi _0}\\Big ) \\int d^{14}z\\, du_1 du_2 \\frac{1}{(u_1^+ u_2^+)^2} V^{++}(z,u_1) V^{++}(z,u_2) \\nonumber \\\\&& + \\frac{1}{4f_0^2\\xi _0} \\int d\\zeta ^{(-4)}\\, du\\, V^{++}(z,u) \\partial ^2 V^{++}(z,u), \\qquad $ where we made use of the identity $D_1^{++} \\frac{1}{(u_1^+ u_2^+)^3} = \\frac{1}{2} (D_1^{--})^2 \\delta ^{(3,-3)}(u_1,u_2)$ and took into account that, when acting on the analytic superfields, $\\frac{1}{2} (D^+)^4 (D^{--})^2 \\Rightarrow \\partial ^2.$ Following Ref.",
"[31], we consider the free theory and solve the equation of motion for the superfield $V^{++}$ in the presence of the source term, $\\frac{1}{2\\xi _0 f_0^2} \\partial ^2 V^{++}(z,u_1) + \\frac{1}{2f_0^2}\\Big (1-\\frac{1}{\\xi _0}\\Big ) \\int du_2 \\frac{1}{(u_1^+ u_2^+)^2} (D_1^+)^4 V^{++}(z,u_2) + J^{(+2)}(z,u_1) = 0.$ The solution can be presented as $V^{++}(z,u_1) = -\\frac{2\\xi _0 f_0^2}{\\partial ^2} J^{(+2)}(z,u_1) + \\frac{2f_0^2(\\xi _0-1)}{\\partial ^4} \\int du_2 \\frac{1}{(u_1^+ u_2^+)^2} (D_1^+)^4 J^{(+2)}(z,u_2),$ whence one extracts the $\\xi $ -gauge form of the propagator of the gauge superfield $&& G_V^{(2,2)}(z_1,u_1;z_2,u_2) = - 2 f_0^2 \\Big (\\frac{\\xi _0}{\\partial ^2} (D_1^+)^4 \\delta ^{(2,-2)}(u_2,u_1)\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad - \\frac{\\xi _0-1}{\\partial ^4} (D_1^+)^4 (D_2^+)^4 \\frac{1}{(u_1^+ u_2^+)^2}\\Big ) \\delta ^{14}(z_1-z_2).\\qquad $ The second term vanishes in the Feynman gauge $\\xi _0=1$ .",
"Such a choice considerably simplifies calculation of quantum corrections.",
"However, the purpose of the present paper is to investigate the $\\xi _0$ -dependence of various Green functions for the generic choice of $\\xi _0$ .",
"In left part of Fig.",
"REF , the propagator (REF ) is depicted by the wavy line ending on the points 1 and 2.",
"For completeness, we also present the expression for the hypermultiplet propagator, $G_q^{(1,1)}(z_1,u_1;z_2,u_2) = (D_1^+)^4 (D_2^+)^4 \\frac{1}{\\partial ^2} \\delta ^{14}(z_1-z_2) \\frac{1}{(u_1^+ u_2^+)^3},$ which is denoted by the solid line on the right.",
"Figure: The propagators of the gauge superfield V ++ V^{++} and of the hypermultiplets.The only vertex of the theory (REF ) is presented in Fig.",
"REF and stands for the interaction of the hypermultiplet with the gauge superfield $S_I = - i \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^{+} V^{++} q^+.$ Figure: The only vertex comes from the interaction of the hypermultiplet with the gauge superfield.The superficial degree of divergence in the theory under consideration has been calculated in Ref.",
"[29]: $\\omega = 2L - N_q - \\frac{1}{2} N_D.$ Here $L$ is a number of loops, $N_q$ is a number of external hypermultiplet legs, and $N_D$ is a number of spinor supersymmetric covariant derivatives acting on external legs.",
"This formula implies that in the one-loop approximation only diagrams without external hypermultiplet legs or with two such legs can be divergent."
],
[
"Two-point function of the gauge superfield",
"In the one-loop approximation the two-point function of the gauge superfield $V^{++}$ is divergent.",
"In the abelian case this divergence comes only from the diagram pictured in Fig.",
"REF .",
"However, this diagram does not contain propagators of the gauge superfield and is therefore gauge-independent.",
"Figure: The diagram representing the one-loop two-point Green function in the abelian case.Thus, in the one-loop approximation this Green function in the $\\xi $ -gauge is the same as in the Feynman gauge.",
"It is given by the expression [29] $\\int \\frac{d^6p}{(2\\pi )^6} \\int d^8\\theta \\, du_1\\, du_2\\, V^{++}(p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) \\frac{1}{(u_1^+ u_2^+)^2} \\Big [\\frac{1}{4f_0^2} - \\frac{i}{2} \\int \\frac{d^6k}{(2\\pi )^6} \\frac{1}{k^2 (k+p)^2}\\Big ].\\qquad $ The corresponding divergent part of the effective action is gauge-independent and in the dimensional reduction schemeHere we use the regularization by dimensional reduction [47].",
"However, for calculating power divergences one should use another regularization, e.g., some modifications of the higher covariant derivative regularization [48], [49].",
"At least for $4D$ , ${\\cal N}=2$ supersymmetric theories such a regularization can be formulated in the harmonic superspace [50].",
"can be written as $-\\frac{1}{6\\varepsilon (4\\pi )^3}\\int d\\zeta ^{(-4)}\\, du\\, (F^{++})^2,$ where $\\varepsilon = 6-D$ ."
],
[
"Two-point hypermultiplet Green function",
"In the one-loop approximation the two-point Green function of the hypermultiplet is contributed to by the single logarithmically divergent diagram presented in Fig.",
"REF .",
"Figure: The two-point Green function of the hypermultiplet in the one-loop approximationIn the Feynman gauge this superdiagram vanishes.",
"However, it includes the propagator of the gauge superfield, for which reason we can expect that the result for it is in fact gauge-dependent.",
"Using the Feynman rules defined above, the expression for this diagram in the generic $\\xi $ -gauge can be written as $&& -2if_0^2 \\int d\\zeta ^{(-4)}_1\\, du_1\\, d\\zeta ^{(-4)}_2\\, du_2\\, \\widetilde{q}^+(z_1,u_1) q^+(z_2,u_2) \\frac{1}{(u_1^+ u_2^+)^3} \\frac{(D_1^+)^4 (D_2^+)^4}{\\partial ^2} \\delta ^{14}(z_1-z_2)\\qquad \\nonumber \\\\&& \\times \\Big (\\frac{\\xi _0}{\\partial ^2} (D_1^+)^4 \\delta ^{(2,-2)}(u_2,u_1) - \\frac{\\xi _0-1}{\\partial ^4} (D_1^+)^4 (D_2^+)^4 \\frac{1}{(u_1^+ u_2^+)^2}\\Big ) \\delta ^{14}(z_1-z_2).$ The derivatives $(D_1^+)^4 (D_2^+)^4$ in the hypermultiplet propagator can be used to convert the integrations over $d\\zeta ^{(-4)}$ into those over $d^{14}z$ , $&& -2if_0^2 \\int d^{14}z_1\\, du_1\\, d^{14}z_2\\, du_2\\, \\widetilde{q}^+(z_1,u_1) q^+(z_2,u_2) \\frac{1}{(u_1^+ u_2^+)^3}\\,\\frac{1}{\\partial ^2} \\delta ^{14}(z_1-z_2)\\qquad \\nonumber \\\\&& \\times \\Big (\\frac{\\xi _0}{\\partial ^2} (D_1^+)^4 \\delta ^{(2,-2)}(u_2,u_1) - \\frac{\\xi _0-1}{\\partial ^4} (D_1^+)^4 (D_2^+)^4 \\frac{1}{(u_1^+ u_2^+)^2}\\Big ) \\delta ^{14}(z_1-z_2).$ Taking into account the identities $&& \\delta ^{8}(\\theta _1-\\theta _2)\\, (D^{+}_1)^4 \\delta ^{8}(\\theta _1-\\theta _2) =0,\\vphantom{\\Big (}\\\\&& \\delta ^{8}(\\theta _1-\\theta _2)\\, (D^{+}_1)^4 (D^{+}_2)^4 \\delta ^{8}(\\theta _1-\\theta _2) = (u_1^+ u_2^+)^4\\, \\delta ^{8}(\\theta _1-\\theta _2)\\vphantom{\\Big (}\\,,\\qquad $ we find that the first term in this expression vanishes, reducing (REF ) to the form $2if_0^2 \\int d^{6}x_1\\, d^{6}x_2\\,d^8\\theta \\, du_1\\, du_2\\, \\widetilde{q}^+(x_1,\\theta , u_1) q^+(x_2,\\theta ,u_2)\\frac{(\\xi _0-1)}{(u_1^+ u_2^+)}\\, \\frac{1}{\\partial ^2} \\delta ^{6}(x_1-x_2)\\, \\frac{1}{\\partial ^4} \\delta ^{6}(x_1-x_2).$ This expression can be rewritten in the momentum representation as $- 2if_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{1}{k^4 (k+p)^2} \\int d^8\\theta \\, du_1\\, du_2\\, \\frac{(\\xi _0-1)}{(u_1^+ u_2^+)} \\widetilde{q}^+(p,\\theta , u_1) q^+(-p,\\theta \\,u_2).$ We observe that this expression is logarithmically divergent and does not vanish, unless the Feynman gauge is chosen.",
"If the theory is regularized by dimensional reduction, the corresponding contribution to the divergent part takes the form $-\\frac{2f_0^2}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du_1\\, du_2\\, \\frac{(\\xi _0-1)}{(u_1^+ u_2^+)} \\widetilde{q}^+(z, u_1) q^+(z, u_2).$"
],
[
"Three-point gauge-hypermultiplet Green function",
"According to Eq.",
"(REF ), all diagrams containing two external hypermultiplet legs are logarithmically divergent, irrespective of the number of the external gauge legs.",
"That is why in calculating the one-loop divergences it is necessary to take into account such Green functions.",
"The simplest of them is the three-point gauge superfield - hypermultiplet.",
"In the one-loop approximation, it is contributed to by the single supergraph depicted in Fig.",
"REF .",
"Figure: The diagram representing the three-point gauge-hypermultiplet function in the one-loop approximation.Calculating this diagram by Feynman rules in the general $\\xi $ -gauge, we obtain $&& - 2f_0^2 \\int d\\zeta ^{(-4)}_1\\,du_1\\,d\\zeta ^{(-4)}_2\\,du_2\\,d\\zeta ^{(-4)}_3\\,du_3\\,\\widetilde{q}^+(z_1,u_1)q^+(z_3,u_3) V^{++}(z_2,u_2) \\Big (\\frac{\\xi _0}{\\partial ^2} (D_1^+)^4 \\qquad \\nonumber \\\\&&\\times \\delta ^{(2,-2)}(u_3,u_1) - \\frac{(\\xi _0-1)}{\\partial ^4} (D_1^+)^4 (D_3^+)^4 \\frac{1}{(u_1^+ u_3^+)^2}\\Big ) \\delta ^{14}(z_1-z_3)\\, \\frac{1}{(u_1^+ u_2^+)^3}\\frac{(D_1^+)^4 (D_2^+)^4}{\\partial ^2}\\qquad \\nonumber \\\\&&\\times \\delta ^{14}(z_1-z_2)\\, \\frac{1}{(u_2^+ u_3^+)^3}\\frac{(D_2^+)^4 (D_3^+)^4}{\\partial ^2} \\delta ^{14}(z_2-z_3).$ To work out this expression, we, first, convert the integrals over $d\\zeta ^{(-4)}$ in it into integrals over $d^{14}z$ using Eq.",
"(): $&& - 2f_0^2 \\int d^{14}z_1\\,d^{14}z_2\\,d^{14}z_3\\,du_1\\,du_2\\,du_3\\,\\widetilde{q}^+(z_1,u_1) q^+(z_3,u_3) V^{++}(z_2,u_2)\\Big (\\frac{\\xi _0}{\\partial ^2} (D_1^+)^4 \\qquad \\nonumber \\\\&& \\times \\delta ^{(2,-2)}(u_3,u_1) - \\frac{(\\xi _0-1)}{\\partial ^4} (D_1^+)^4 (D_3^+)^4 \\frac{1}{(u_1^+ u_3^+)^2}\\Big )\\delta ^{14}(z_1-z_3)\\, \\frac{1}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}\\qquad \\nonumber \\\\&& \\times \\frac{(D_2^+)^4}{\\partial ^2} \\delta ^{14}(z_1-z_2)\\, \\frac{1}{\\partial ^2} \\delta ^{14}(z_2-z_3).\\vphantom{\\frac{1}{2}}$ Next, we integrate by parts with respect to $(D^+_2)^4$ (assuming that $D_2^+$ acts on $z_1$ ), taking into account that $\\delta ^8(\\theta _1-\\theta _2)\\prod \\limits _{n=1}^N D^+_{i_n a_n} \\delta ^8(\\theta _1-\\theta _2) = 0 \\qquad \\mbox{for arbitrary odd $N$}.$ In the term containing the harmonic $\\delta $ -function we further integrate over $du_3$ .",
"Integrating also over $\\theta _2$ , we finally obtain for (REF ): $&& 2f_0^2 \\int d^{6}x_1\\,d^{6}x_2\\,d^{6}x_3\\,d^8\\theta _1\\,d^8\\theta _3\\,\\delta ^8(\\theta _1-\\theta _3)\\, \\Bigg \\lbrace \\int du_1\\, du_2\\, \\widetilde{q}^+(x_1,\\theta _1,u_1) q^+(x_3,\\theta _3,u_1) \\nonumber \\\\&& \\times V^{++}(x_2,\\theta _1, u_2) \\frac{\\xi _0}{(u_1^+ u_2^+)^6}\\, \\frac{(D_1^+)^4 (D_2^+)^4}{\\partial ^2} \\delta ^{14}(z_1-z_3)\\,\\frac{1}{\\partial ^2} \\delta ^{6}(x_1-x_2)\\,\\,\\frac{1}{\\partial ^2} \\delta ^{6}(x_2-x_3)\\nonumber \\\\&& + \\int du_1\\,du_2\\,du_3\\, V^{++}(x_2,\\theta _1, u_2) q^+(x_3,\\theta _3,u_3) \\frac{(\\xi _0-1)}{(u_1^+ u_3^+)^2 (u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}\\frac{1}{\\partial ^2} \\delta ^{6}(x_1-x_2) \\qquad \\nonumber \\\\&&\\times \\frac{1}{\\partial ^2} \\delta ^{6}(x_2-x_3)\\, \\Bigg [(D_2^+)^4 \\widetilde{q}^+(x_1,\\theta _1,u_1)\\, \\frac{(D_1^+)^4 (D_3^+)^4}{\\partial ^4} \\delta ^{14}(z_1-z_3)+ \\widetilde{q}^+(x_1,\\theta _1,u_1)\\nonumber \\\\&&\\times \\frac{(D_2^+)^4 (D_1^+)^4 (D_3^+)^4}{\\partial ^4} \\delta ^{14}(z_1-z_3) - \\frac{1}{4}\\varepsilon ^{abcd} D_{2a}^+ D_{2b}^+\\, \\widetilde{q}^+(x_1,\\theta _1,u_1)\\, \\frac{D_{2c}^+ D_{2d}^+ (D_1^+)^4 (D_3^+)^4}{\\partial ^4} \\nonumber \\\\&&\\times \\delta ^{14}(z_1-z_3)\\Bigg ] \\Bigg \\rbrace .$ As the further step, we use the identities (REF ), () together with $&& \\delta ^{8}(\\theta _1-\\theta _2)\\, D^{+}_{2a} D^{+}_{2b} (D^{+}_1)^4 (D^{+}_3)^4\\delta ^{8}(\\theta _1-\\theta _2)\\vphantom{\\Big (}\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad = -i(\\gamma ^M)_{ab} (u_2^+ u_1^+)\\, (u_2^+ u_3^+)\\, (u_1^+ u_3^+)^3 \\,\\delta ^{8}(\\theta _1-\\theta _2) \\partial _M;\\vphantom{\\Big (}\\qquad \\\\&& \\delta ^{8}(\\theta _1-\\theta _2)\\, (D^{+}_2)^4 (D^{+}_1)^4 (D^{+}_3)^4\\delta ^{8}(\\theta _1-\\theta _2)\\vphantom{\\Big (}\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad = (u_1^+ u_2^+)^2\\, (u_1^+ u_3^+)^2\\, (u_2^+ u_3^+)^2\\,\\delta ^{8}(\\theta _1-\\theta _2) \\partial ^2 \\vphantom{\\Big (}\\qquad $ in order to do the integrals over the Grassmann coordinate $\\theta _2$ .",
"After renaming $\\theta _1 \\rightarrow \\theta $ , the expression for the diagram in question in the momentum representation is written as $&& 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\,d^8\\theta \\,\\Bigg \\lbrace -\\int du_1\\,du_2\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) q^+(-q,\\theta ,u_1)\\nonumber \\\\&& \\times \\frac{\\xi _0}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_1^+ u_2^+)^2}+ \\int du_1\\,du_2\\,du_3\\,\\Bigg [ (D^+_{2})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_1)\\,V^{++}(-p,\\theta , u_2)\\nonumber \\\\&&\\times q^+(-q,\\theta ,u_3)\\,\\frac{(\\xi _0-1)}{k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}- \\widetilde{q}^+(q+p,\\theta ,u_1)\\, V^{++}(-p,\\theta , u_2)\\nonumber \\\\&& \\times q^+(-q,\\theta ,u_3)\\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}- D^+_{2a} D^+_{2b}\\,\\widetilde{q}^+(q+p,\\theta ,u_1)\\nonumber \\\\&&\\times V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\,\\frac{(\\xi _0-1)(\\widetilde{\\gamma }^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2}\\Bigg ]\\Bigg \\rbrace ,$ where $(\\widetilde{\\gamma }^M)^{ab} = \\varepsilon ^{abcd} (\\gamma ^M)_{cd}/2$ .",
"The divergent part of this expression can now be found after the Wick rotation.",
"There remains only one divergent integral $\\int \\frac{d^6k}{(2\\pi )^6}\\frac{1}{k^2 (k+q)^2 (k+q+p)^2}\\,,$ which, after regularizing it by dimensional reduction, is reduced to $-i\\int \\frac{d^DK}{(2\\pi )^6}\\frac{1}{K^2 (K+Q)^2 (K+Q+P)^2} = -\\frac{i}{\\varepsilon (4\\pi )^3} + \\mbox{finite terms},$ where the capital letters denote Euclidean momentums.",
"Thus, the divergent part of the diagram in Fig.",
"5 can be presented as $\\frac{2if_0^2}{\\varepsilon (4\\pi )^3} \\int d^{14}z \\Bigg \\lbrace \\int du_1\\,du_2\\,\\widetilde{q}^+_1 V^{++}_2 q^+_1\\frac{\\xi _0}{(u_1^+ u_2^+)^2}+ \\int du_1\\,du_2\\,du_3\\, \\widetilde{q}^+_1\\, V^{++}_2 q^+_3 \\frac{(\\xi _0-1)}{(u_1^+ u_2^+) (u_2^+ u_3^+)}\\Bigg \\rbrace ,$ where the subscripts on the superfields refer to the relevant harmonic arguments."
],
[
"Verification of the Ward identities",
"To be convinced of the correctness of the results obtained in the previous sections, let us check that the two- and three-point Green functions derived above satisfy the Ward identities.",
"First, for completeness, we verify the Ward identity (REF ).",
"The two-point Green function of the gauge superfield is obtained by differentiating Eq.",
"(REF ) with respect to $V^{++}$ , using Eq.",
"(REF ).",
"This gives $\\frac{\\delta ^2\\Delta \\Gamma }{\\delta V^{++}_1 \\delta V^{++}_2} = G_V(i\\partial _M) \\frac{1}{(u_1^+ u_2^+)^2} (D_1^{+})^4 (D_2^+)^4 \\delta ^{14}(z_1-z_2),$ where $G_V(p_M) = \\frac{1}{2f_0^2} - i \\int \\frac{d^6k}{(2\\pi )^6}\\frac{1}{k^2 (k+p)^2} + \\ldots $ Therefore, $&&\\hspace*{-17.07164pt} D_1^{++}\\frac{\\delta ^2\\Delta \\Gamma }{\\delta V^{++}_1 \\delta V^{++}_2} = G_V(i\\partial _M) D_1^{--} \\delta ^{(2,-2)}(u_1,u_2)\\cdot (D_1^{+})^4 (D_2^+)^4 \\delta ^{14}(z_1-z_2) = G_V(i\\partial _M)\\nonumber \\\\&&\\hspace*{-17.07164pt} \\times \\Big [D_1^{--} \\Big (\\delta ^{(2,-2)}(u_1,u_2) (D_1^{+})^4 (D_2^+)^4 \\Big ) - \\delta ^{(2,-2)}(u_1,u_2) \\Big (D_1^{--} (D_1^{+})^4\\Big ) (D_2^+)^4\\Big ]\\delta ^{14}(z_1-z_2) = 0.\\nonumber \\\\$ Thus, we have verified that the Ward identity (REF ) is indeed satisfied.",
"The two-point Green function of the hypermultiplet is obtained by differentiating Eq.",
"(REF ) with respect to $q^+$ and $\\widetilde{q}^+$ .",
"These derivatives are calculated with the help of Eq.",
"(REF ).",
"We obtain $\\frac{\\delta ^2\\Gamma }{\\delta q_2^+\\,\\delta \\widetilde{q}_1^+} = G_q(i\\partial _M) \\frac{1}{(u_1^+ u_2^+)} (D^+_1)^4 (D^+_2)^4 \\delta ^{14}(z_1-z_2),$ where $G_q(p_M) = - 2if_0^2 \\int \\frac{d^6k}{(2\\pi )^6} \\frac{(\\xi _0-1)}{k^4 (k+p)^2} +\\ldots $ The three-point gauge superfield - hypermultiplet Green function can be constructed quite similarly, starting from Eq.",
"(REF ), but we prefer not to present the expression for it explicitly.",
"Instead, we will check for it the Ward identity in the form (REF ).",
"From Eq.",
"(REF ) we obtain $&& \\int d\\zeta ^{(-4)}_1\\,du_1\\, d\\zeta ^{(-4)}_2\\,du_2\\, d\\zeta ^{(-4)}_3\\,du_3\\, \\widetilde{q}^+_3 D^{++}\\lambda _1 q^+_2\\, \\frac{\\delta ^3\\Delta \\Gamma }{\\delta V^{++}_1 \\delta q^+_2 \\delta \\widetilde{q}^+_3} \\nonumber \\\\&& = 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\,d^8\\theta \\,\\Bigg \\lbrace -\\int du_1\\,du_2\\,\\widetilde{q}^+(q+p,\\theta ,u_2) D^{++}_1\\lambda (-p,\\theta ,u_1) q^+(-q,\\theta ,u_2)\\nonumber \\\\&& \\times \\frac{\\xi _0}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_1^+ u_2^+)^2}+ \\int du_1\\,du_2\\,du_3\\,\\Bigg [ (D^+_{1})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_3)\\,D^{++}_1\\lambda (-p,\\theta , u_1)\\nonumber \\\\&&\\times q^+(-q,\\theta ,u_2)\\,\\frac{(\\xi _0-1)}{k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_3^+ u_2^+)^2}{(u_3^+ u_1^+)^3 (u_1^+ u_2^+)^3}- \\widetilde{q}^+(q+p,\\theta ,u_3)\\, D^{++}_1\\lambda (-p,\\theta , u_1)\\nonumber \\\\&& \\times q^+(-q,\\theta ,u_2)\\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_3^+ u_1^+) (u_1^+ u_2^+)}- D^+_{1a} D^+_{1b}\\,\\widetilde{q}^+(q+p,\\theta ,u_3)\\nonumber \\\\&&\\times D^{++}_1\\lambda (-p,\\theta , u_1)\\, q^+(-q,\\theta ,u_2)\\,\\frac{(\\xi _0-1)(\\widetilde{\\gamma }^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_3^+ u_2^+)}{(u_3^+ u_1^+)^2 (u_1^+ u_2^+)^2}\\Bigg ]\\Bigg \\rbrace .$ Next, we integrate by parts with respect to the harmonic derivatives $D^{++}_1$ , taking into account the identity $D^{++}_1 \\frac{1}{(u_1^+ u_2^+)^n} = \\frac{1}{(n-1)!}",
"(D_1^{--})^{n-1}\\delta ^{(n,-n)}(u_1,u_2) = \\frac{(-1)^{n-1}}{(n-1)!}",
"(D_2^{--})^{n-1}\\delta ^{(2-n,n-2)}(u_1,u_2) .$ After some algebra (described in Appendix ), this gives $&&\\hspace*{-17.07164pt} \\int d\\mu \\, \\widetilde{q}^+_3 D^{++}\\lambda _1 q^+_2\\, \\frac{\\delta ^3\\Delta \\Gamma }{\\delta V^{++}_1 \\delta q^+_2 \\delta \\widetilde{q}^+_3} = - 2f_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6q}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{1}{k^4 (k+q+p)^2} \\int d^8\\theta \\, du_1\\, du_3\\,\\qquad \\nonumber \\\\&&\\hspace*{-17.07164pt} \\times \\frac{(\\xi _0-1)}{(u_1^+ u_3^+)} \\widetilde{q}^+(q+p,\\theta , u_3) \\lambda (-p,\\theta ,u_1) q^+(-q,\\theta \\,u_1) - 2f_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6q}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{1}{k^4 (k+q)^2} \\nonumber \\\\&&\\hspace*{-17.07164pt} \\times \\int d^8\\theta \\, du_1\\, du_2\\, \\frac{(\\xi _0-1)}{(u_1^+ u_2^+)} \\widetilde{q}^+(q+p,\\theta , u_1) \\lambda (-p,\\theta ,u_1) q^+(-q,\\theta \\,u_2).$ The right-hand side of this equation can be rewritten as $&& i \\int d \\zeta ^{(-4)}_1 du_1\\, d\\zeta ^{(-4)}_3\\, du_3\\, \\widetilde{q}^+_3 \\lambda _1 q^+_1\\, \\frac{\\delta ^2\\Gamma }{\\delta q^+_1 \\delta \\widetilde{q}^+_3}\\nonumber \\\\&& - i \\int d \\zeta ^{(-4)}_1 du_1\\, d\\zeta ^{(-4)}_2\\, du_2\\, \\widetilde{q}^+_1 \\lambda _1 q^+_2\\, \\frac{\\delta ^2\\Gamma }{\\delta q^+_2 \\delta \\widetilde{q}^+_1},\\qquad $ thus demonstrating that the Green functions (REF ) and (REF ) satisfy the Ward identity (REF ), as it should be.",
"Obviously, they also satisfy the Ward identity in the original form (REF ).",
"This completes checking the correctness of our calculation."
],
[
"The vanishing of the gauge dependence on shell",
"According to the general theorem of Refs.",
"[38], [39], [40], [41], [42], [43], the gauge-dependent terms should disappear on shell.",
"Let us verify that our results are in agreement with this statement.",
"It is convenient to represent the effective action in the form $\\Gamma = \\Gamma _{\\xi _0=1} + \\widetilde{\\Gamma },$ where $&& \\Gamma _{\\xi _0=1} = S + S_{\\mbox{\\scriptsize gf}} -\\frac{i}{2} \\int \\frac{d^6p}{(2\\pi )^6} \\int d^8\\theta \\, du_1\\, du_2\\, V^{++}(p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) \\frac{1}{(u_1^+ u_2^+)^2} \\nonumber \\\\&&\\times \\int \\frac{d^6k}{(2\\pi )^6}\\frac{1}{k^2 (k+p)^2}- \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,d^8\\theta \\, du_1\\,du_2\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) \\qquad \\nonumber \\\\&& \\times q^+(-q,\\theta ,u_1) \\frac{1}{(u_1^+ u_2^+)^2}\\int \\frac{d^{6}k}{(2\\pi )^6}\\, \\frac{2f_0^2}{k^2 (q+k)^2 (q+k+p)^2} + \\ldots $ is the effective action in the Feynman gauge and $&& \\widetilde{\\Gamma }=- 2if_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{1}{k^4 (k+p)^2} \\int d^8\\theta \\, du_1\\, du_2\\, \\frac{(\\xi _0-1)}{(u_1^+ u_2^+)} \\widetilde{q}^+(p,\\theta , u_1)\\, q^+(-p,\\theta \\,u_2)\\nonumber \\\\&& + 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\,d^8\\theta \\,\\Bigg \\lbrace -\\int du_1\\,du_2\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) q^+(-q,\\theta ,u_1)\\nonumber \\\\&& \\times \\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_1^+ u_2^+)^2}+ \\int du_1\\,du_2\\,du_3\\,\\Bigg [ (D^+_{2})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_1)\\,V^{++}(-p,\\theta , u_2)\\nonumber \\\\&&\\times q^+(-q,\\theta ,u_3)\\,\\frac{(\\xi _0-1)}{k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}- \\widetilde{q}^+(q+p,\\theta ,u_1)\\, V^{++}(-p,\\theta , u_2)\\nonumber \\\\&& \\times q^+(-q,\\theta ,u_3)\\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} \\frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}- D^+_{2a} D^+_{2b}\\,\\widetilde{q}^+(q+p,\\theta ,u_1)\\nonumber \\\\&&\\times V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\,\\frac{(\\xi _0-1)(\\widetilde{\\gamma }^M)^{ab} k_M}{2k^4 (q+k)^2 (q+k+p)^2}\\, \\frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2}\\Bigg ]\\Bigg \\rbrace + \\ldots $ stands for the gauge-dependent remainder of the effective action.",
"The purpose of this section is to demonstrate, by an explicit calculation, that in the approximation considered, $\\widetilde{\\Gamma }$ indeed vanishes on shell.",
"To this end, we use the equations of motion for the hypermultiplets following from the action (REF ), $0 = \\nabla ^{++} q^+ = D^{++} q^+ + iV^{++} q^+;\\qquad 0 = \\nabla ^{++} \\widetilde{q}^+ = D^{++} \\widetilde{q}^+ - iV^{++} \\widetilde{q}^+.$ In Appendix (after some lengthy calculations) we demonstrate that, with these equations taken into account, the gauge-dependent part of the one-loop effective action can be cast in the form $&& \\widetilde{\\Gamma }= 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,d^4\\theta ^+\\,du\\, \\widetilde{q}^+(q+p,\\theta ,u) V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u)\\nonumber \\\\&& \\times \\Big ((q+p)^2+q^2\\Big ) \\int \\frac{d^{6}k}{(2\\pi )^6}\\, \\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} + O\\left((V^{++})^2\\right).\\qquad $ On shell, where $q^2=0$ and $(q+p)^2=0$ ,These equations can be derived directly from the hypermultiplet free equation of motion, see Ref.",
"[34] for details.",
"this expression vanishes.",
"Thereby we have proved that the gauge dependence is vanishing on shell.",
"Note that, while deriving this result, we ignored all terms proportional to $(V^{++})^k$ for $k\\ge 2$ , because in this paper we limit our attention only to the diagrams without external gauge superfield legs at all, and to those having a single gauge superfield leg.",
"In this approximation, terms of higher orders in $V^{++}$ are irrelevant."
],
[
"The total divergent part of the one-loop effective action",
"So far we investigated gauge dependence of the two- and three-point Green functions only.",
"In particular, we demonstrated that the corresponding one-loop divergences are gauge-dependent.",
"However, according to Eq.",
"(REF ), the Green functions with an arbitrary number of external gauge legs (and two external hypermultiplet legs) are also divergent.",
"Nevertheless, the total divergent part of the one-loop effective action can be found using the reasoning based on the gauge invariance.",
"Actually, the one-loop divergences corresponding to the two- and three-point Green functions (see Eqs.",
"(REF ), (REF ), and (REF )) have the form $&& \\Gamma ^{(1)}_\\infty = -\\frac{1}{6\\varepsilon (4\\pi )^3}\\int d\\zeta ^{(-4)}\\, du\\, (F^{++})^2-\\frac{2f_0^2}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du_1\\, du_2\\, \\frac{(\\xi _0-1)}{(u_1^+ u_2^+)} \\widetilde{q}^+_1 q^+_2+ \\frac{2if_0^2}{\\varepsilon (4\\pi )^3}\\nonumber \\\\&& \\times \\int d^{14}z \\Bigg \\lbrace \\int du_1\\,du_2\\,\\widetilde{q}^+_1 V^{++}_2 q^+_1\\frac{\\xi _0}{(u_1^+ u_2^+)^2}+ \\int du_1\\,du_2\\,du_3\\, \\widetilde{q}^+_1\\, V^{++}_2 q^+_3 \\frac{(\\xi _0-1)}{(u_1^+ u_2^+) (u_2^+ u_3^+)}\\Bigg \\rbrace \\qquad \\nonumber \\\\&& + O\\Big (\\widetilde{q}^+ (V^{++})^2 q^+\\Big ).\\vphantom{\\frac{1}{2}}$ The first term in this equation is gauge invariant.",
"The expression corresponding to the first term in the curly brackets can also be rewritten in the explicitly gauge invariant form, $&& \\frac{2if_0^2}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du_1\\,du_2\\,\\widetilde{q}^+_1 V^{++}_2 q^+_1\\frac{\\xi _0}{(u_1^+ u_2^+)^2} = \\xi _0\\, \\frac{2if_0^2}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du\\, \\widetilde{q}^+ V^{--} q^+\\nonumber \\\\&& = \\xi _0\\,\\frac{2if_0^2}{\\varepsilon (4\\pi )^3} \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^+ F^{++} q^+.$ According to Eq.",
"(REF ), the remaining two terms in Eq.",
"(REF ) are the lowest terms in the series expansion of the gauge invariant expression $- \\frac{2 f_0^2 (\\xi _0-1)}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du\\, \\widetilde{q}^+\\, q^-$ in powers of $V^{++}$ .",
"Thus, the divergent part of the one-loop effective action can be written as $&& \\Gamma ^{(1)}_\\infty = -\\frac{1}{6\\varepsilon (4\\pi )^3}\\int d\\zeta ^{(-4)}\\, du\\, (F^{++})^2 + \\frac{2if_0^2 \\xi _0}{\\varepsilon (4\\pi )^3} \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^+ F^{++} q^+\\nonumber \\\\&& - \\frac{2 f_0^2 (\\xi _0-1)}{\\varepsilon (4\\pi )^3} \\int d^{14}z\\, du\\, \\widetilde{q}^+\\, q^-.$ Note that this expression does not include $O\\Big (\\widetilde{q}^+ (V^{++})^2 q^+\\Big )$ , because for obtaining the gauge invariant expression such terms should contain $F^{++}$ in which the number $N_D=4$ of spinor derivatives acts on $V^{--}$ .",
"However, according to Eq.",
"(REF ) these terms are finite and do not contribute to the divergent part of the one-loop effective action.",
"Therefore, Eq.",
"(REF ) provides the exact result for the divergent part of the effective action of the theory in question.",
"Note that on shell the gauge dependence of Eq.",
"(REF ) vanishes.",
"Actually, on shell, as the consequence of the equation of motion $\\nabla ^{++}q^+ = 0\\,$ , we have the chain of relations $(\\nabla ^{++})^2 q^- = 0 \\;\\Rightarrow \\; (\\nabla ^{++})^2 \\nabla ^{--}q^- = 0 \\; \\Rightarrow \\; \\nabla ^{++} \\nabla ^{--} q^- = 0 \\; \\Rightarrow \\; \\nabla ^{--} q^- = 0\\,.$ Acting on the latter equation by $\\nabla ^{++}$ it is easy to find $q^- = \\nabla ^{--} q^+.",
"$ In deriving these relations, we made use of the well known properties $D^{++}\\omega ^{-n} = 0 \\rightarrow \\omega ^{-n} = 0,\\; D^{--}\\omega ^{+ m} = 0 \\rightarrow \\omega ^{+ m} = 0$ for $n\\ge 1, m\\ge 1$ .",
"As a consequence of (REF ), we obtain that on shell $&& \\int d^{14}z\\, du\\, \\widetilde{q}^+\\, q^- = \\int d\\zeta ^{(-4)}\\, du\\, (D^+)^4\\Big (\\widetilde{q}^+\\, \\nabla ^{--} q^+\\Big )\\nonumber \\\\&&\\qquad \\qquad = \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^+\\, (D^+)^4\\Big ( (D^{--} +iV^{--}) q^+\\Big ) = i \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^+\\, F^{++} q^+.\\qquad $ Thus, on shell, the one-loop divergence (REF ) takes the form $\\Gamma ^{(1)}_\\infty = -\\frac{1}{6\\varepsilon (4\\pi )^3}\\int d\\zeta ^{(-4)}\\, du\\, (F^{++})^2 + \\frac{2if_0^2}{\\varepsilon (4\\pi )^3} \\int d\\zeta ^{(-4)}\\, du\\, \\widetilde{q}^+ F^{++} q^+.$ We see that this expression does not depend on the parameter $\\xi $ and, hence, on the gauge choice."
],
[
"Summary",
"In this paper, using the $6D\\,,$ ${\\cal N}=(1,0)$ harmonic superspace formalism, we studied the gauge dependence of the one-loop effective action for ${\\cal N}=(1,0)$ supersymmetric quantum electrodynamics.",
"As compared to the case of the Feynman gauge, in the general $\\xi $ -gauge some new divergences appear.",
"In particular, we demonstrated that in the general case the hypermultiplet Green function is divergent already in the one-loop approximation, as opposed to the case of the Feynman gauge, in which this divergence vanishes.",
"Moreover, we calculated the three-point gauge - hypermultiplet Green function in the general $\\xi $ -gauge.",
"To check the correctness of the calculation, we have verified the relevant Ward identity.",
"Also it was checked that the gauge dependence vanishes on shell.",
"Taking into account the gauge invariance, we also restored the divergent part of the one-loop effective action with terms of higher orders in the gauge superfield $V^{++}$ .",
"It is given by Eq.",
"(REF ) and contains a new term which is absent in the Feynman gauge.",
"We demonstrated that the gauge dependence of this general expression also vanishes on shell.",
"It would be interesting to investigate the gauge dependence in the non-abelian case.",
"In particular, from the results of this paper we can expect that in the general $\\xi $ -gauge the $6D\\,,$ ${\\cal N}=(1,1)$ sypersymmetric Yang–Mills theory is not finite even in the one-loop approximation, while the divergent terms are vanishing on shell."
],
[
"Acknowledgements",
"This work was supported by the grant of Russian Science Foundation, project No.",
"16-12-10306."
],
[
"Ward identity in harmonic superspace",
"Let us show how to pass from Eq.",
"(REF ) to its equivalent form (REF ).",
"After integrating by parts with respect to the derivatives $D^{++}_1$ and using the identity (REF ), we obtain $&& \\int d\\mu \\, \\widetilde{q}^+_3 D^{++}\\lambda _1 q^+_2\\, \\frac{\\delta ^3\\Delta \\Gamma }{\\delta V^{++}_1 \\delta q^+_2 \\delta \\widetilde{q}^+_3}\\nonumber \\\\&& = 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\,d^8\\theta \\,\\Bigg \\lbrace -\\int du_1\\,du_2\\,\\widetilde{q}^+(q+p,\\theta ,u_2) \\lambda (-p,\\theta ,u_1) q^+(-q,\\theta ,u_2)\\nonumber \\\\&& \\times \\frac{\\xi _0}{k^2 (q+k)^2 (q+k+p)^2} D_2^{--} \\delta ^{(0,0)}(u_1,u_2)+ \\int du_1\\,du_2\\,du_3\\,\\Bigg [ (D^+_{1})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_3)\\nonumber \\\\&&\\times \\lambda (-p,\\theta , u_1) q^+(-q,\\theta ,u_2)\\,\\frac{(\\xi _0-1)(u_3^+ u_2^+)^2}{k^4 (q+k)^2 (q+k+p)^2}\\, \\Bigg ( \\frac{1}{2 (u_1^+ u_2^+)^3} (D^{--}_3)^2 \\delta ^{(-1,1)}(u_1,u_3)\\nonumber \\\\&& + \\frac{1}{2 (u_1^+ u_3^+)^3} (D^{--}_2)^2 \\delta ^{(-1,1)}(u_1,u_2) \\Bigg )- \\widetilde{q}^+(q+p,\\theta ,u_3)\\, \\lambda (-p,\\theta , u_1) q^+(-q,\\theta ,u_2) \\nonumber \\\\&&\\times \\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2}\\Bigg (\\frac{1}{(u_1^+ u_3^+)} \\delta ^{(1,-1)}(u_1,u_2) + \\delta ^{(1,-1)}(u_1,u_3)\\frac{1}{(u_1^+ u_2^+)}\\Bigg )\\nonumber \\\\&& - D^+_{1a} D^+_{1b}\\,\\widetilde{q}^+(q+p,\\theta ,u_3) \\lambda (-p,\\theta , u_1)\\, q^+(-q,\\theta ,u_2)\\,\\frac{(\\xi _0-1)(\\widetilde{\\gamma }^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\\,(u_3^+ u_2^+)\\nonumber \\\\&&\\times \\Bigg (\\frac{1}{(u_1^+ u_2^+)^2} D^{--}_3 \\delta ^{(0,0)}(u_1,u_3) + \\frac{1}{(u_1^+ u_3^+)^2} D^{--}_2 \\delta ^{(0,0)}(u_1,u_2)\\Bigg )\\Bigg ]\\Bigg \\rbrace .",
"$ Then we integrate by parts with respect to the derivatives $D^{--}$ and take off one harmonic integral with the help of the delta functions.",
"Taking into account that the first term vanishes as a consequence of the analyticity of the superfields $\\lambda $ , $\\widetilde{q}^+$ , and $q^+$ , the expression (REF ) can be further rewritten as $&&\\hspace*{-22.76219pt} 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\, \\frac{(\\xi _0-1)}{k^4 (q+k)^2 (q+k+p)^2} \\int d^8\\theta \\, du_1\\,\\lambda (-p,\\theta , u_1) \\Bigg \\lbrace \\int du_2\\, \\frac{1}{(u_1^+ u_2^+)}\\nonumber \\\\&&\\hspace*{-22.76219pt}\\times q^+(-q,\\theta ,u_2)\\, \\Bigg [ \\frac{1}{2}(D^+_{1})^4\\, (D_1^{--})^2 \\widetilde{q}^+(q+p,\\theta ,u_1) - k^2 \\widetilde{q}^+(q+p,\\theta ,u_1)+ \\frac{1}{2} (\\widetilde{\\gamma }^M)^{ab} k_M\\, D^+_{1a} \\nonumber \\\\&&\\hspace*{-22.76219pt}\\times \\, D^+_{1b} D_1^{--}\\,\\widetilde{q}^+(q+p,\\theta ,u_1) \\Bigg ]+ \\int du_3\\, \\frac{1}{(u_1^+ u_3^+)} \\Bigg [ \\frac{1}{2} (D^+_{1})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_3)\\, (D_1^{--})^2 q^+(-q,\\theta ,u_1) \\nonumber \\\\&&\\hspace*{-22.76219pt} - k^2 \\widetilde{q}^+(q+p,\\theta ,u_3)\\, q^+(-q,\\theta _1,u_1)- \\frac{1}{2} (\\widetilde{\\gamma }^M)^{ab} k_M\\, D^+_{1a} D^+_{1b}\\,\\widetilde{q}^+(q+p,\\theta ,u_3)\\, D_1^{--} q^+(-q,\\theta ,u_1) \\Bigg ] \\Bigg \\rbrace .\\nonumber \\\\$ Once again, integrating by parts and taking into account that $\\frac{1}{2} (D^{+})^4 (D^{--})^2 = \\partial ^2;\\qquad (\\widetilde{\\gamma }^M)^{ab} D^+_{1a} D^+_{1b} D_1^{--} = - 4i\\partial ^M$ on the analytic superfields, this expression can be cast in the form $&&\\hspace*{-17.07164pt} - 2f_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6q}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{(\\xi _0-1)}{k^4 (k+q)^2} \\int d^8\\theta \\, du_1\\, du_2\\, \\frac{1}{(u_1^+ u_2^+)}\\widetilde{q}^+(q+p,\\theta , u_1) \\lambda (-p,\\theta ,u_1) \\qquad \\nonumber \\\\&&\\hspace*{-17.07164pt} \\times q^+(-q,\\theta \\,u_2) - 2f_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6q}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{(\\xi _0-1)}{k^4 (k+q+p)^2} \\int d^8\\theta \\, du_1\\, du_3\\, \\frac{1}{(u_1^+ u_3^+)} \\widetilde{q}^+(q+p,\\theta , u_3) \\nonumber \\\\&&\\hspace*{-17.07164pt} \\times \\lambda (-p,\\theta ,u_1) q^+(-q,\\theta \\,u_1),\\vphantom{\\frac{1}{2}}$ where we have also used the relations $(q+p)^2 + k^2 + 2 k_M (q+p)^M = (q+k+p)^2\\,, \\qquad q^2 + k^2 + 2 k_M q^M = (q+k)^2.$"
],
[
"Gauge-dependent part of the effective action and the hypermultiplet equations of motion",
"In this appendix we verify that the gauge-dependent part of the effective action vanishes on shell.",
"This is an important non-trivial check of the correctness of our calculations.",
"First, we consider the two-point Green function of the hypermultiplet given by Eq.",
"(REF ).",
"Using the identity $\\qquad \\frac{1}{(u_1^+ u_2^+)} = D^{++}_1 \\frac{(u_1^- u_2^+)}{(u_1^+ u_2^+)^2} + D^{--}_1 \\delta ^{(1,-1)}(u_1,u_2) = D^{++}_1 D^{++}_2 \\frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} + D^{--}_1 \\delta ^{(1,-1)}(u_1,u_2),\\qquad $ we rewrite it as $&& \\widetilde{\\Gamma }^{(2)} = - 2if_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\,\\frac{d^6k}{(2\\pi )^6} \\frac{(\\xi _0-1)}{k^4 (k+p)^2} \\int d^8\\theta \\, du_1\\, du_2\\, \\Big ( D^{++}_1 D^{++}_2\\frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} \\qquad \\nonumber \\\\&& + D^{--}_1 \\delta ^{(1,-1)}(u_1,u_2) \\Big ) \\widetilde{q}^+(p,\\theta , u_1) q^+(-p,\\theta \\,u_2).\\qquad $ The second term in this expression vanishes due to the analyticity of the hypermultiplet superfield, $\\int d^8\\theta \\, du D^{--} \\widetilde{q}^+(p,\\theta ,u) q^+(-p,\\theta ,u) = \\int d^4\\theta ^+\\, du\\, (D^+)^4 \\Big (D^{--} \\widetilde{q}^+(p,\\theta ,u) q^+(-p,\\theta ,u)\\Big ) = 0.$ After integrating by parts with respect to the harmonic derivatives, the considered contribution to the effective action can be represented as $- 2if_0^2 \\int \\frac{d^6p}{(2\\pi )^6}\\, \\frac{d^6k}{(2\\pi )^6} \\frac{(\\xi _0-1)}{k^4 (k+p)^2}\\int d^8\\theta \\, du_1\\, du_2\\, \\frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} D^{++} \\widetilde{q}^+(p,\\theta , u_1)\\, D^{++} q^+(-p,\\theta \\,u_2).$ Using the equations of motion for the hypermultiplets $0 = \\nabla ^{++} q^+ = \\big (D^{++} + i V^{++}\\big )q^+;\\qquad 0 = \\nabla ^{++} \\widetilde{q}^+ = \\big (D^{++} - i V^{++}\\big )\\widetilde{q}^+,$ we see that on shell the expression (REF ) is proportional to $\\widetilde{q}^+ (V^{++})^2 q^+$ .",
"However, in this paper we do not consider terms quadratic in the gauge superfield $V^{++}$ .",
"This implies that, within the accuracy of our approximation, the part of the one-loop effective action corresponding to the hypermultiplet two-point function vanishes on shell.",
"Next, we consider the gauge dependent part of the three-point gauge superfield - hypermultiplet Green function.",
"It corresponds to the terms proportional to $\\widetilde{q}^+ V^{++} q^+$ in the expression (REF ).",
"We will demonstrate that $\\widetilde{\\Gamma }^{(3)}$ vanishes on shell (in the approximation when all terms with more than one $V^{++}$ are omitted).",
"Using the identity $\\frac{1}{(u_1^+ u_2^+)^2} = D_2^{++} \\frac{(u_2^- u_1^+)}{(u_2^+ u_1^+)^3} + \\frac{1}{2} (D_2^{--})^2 \\delta ^{(2,-2)}(u_2,u_1)$ and discarding terms quadratic in $V^{++}$ (coming from $D^{++} q^+$ and $D^{++}\\widetilde{q}^+$ after using the equations of motion), we obtain $&&\\int d^8\\theta \\, du_1\\, du_2\\, \\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta ,u_2) q^+(-q,\\theta ,u_1) \\frac{1}{(u_1^+ u_2^+)^2}\\nonumber \\\\&& \\longrightarrow \\frac{1}{2} \\int d^8\\theta \\, du\\, \\widetilde{q}^+(q+p,\\theta ,u) (D^{--})^2 V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u) \\nonumber \\\\&& = \\frac{1}{2} \\int d^4\\theta ^+\\, du\\, \\widetilde{q}^+(q+p,\\theta ,u) (D^+)^4 (D^{--})^2 V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u) \\qquad \\nonumber \\\\&& = -p^2 \\int d^4\\theta ^+\\, du\\, \\widetilde{q}^+(q+p,\\theta ,u) V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u),$ where the arrow indicates that we omitted some terms vanishing on shell, as well as $O((V^{++})^2)$ terms.",
"Using Eq.",
"(REF ) twice, we have $&& \\int d^8\\theta \\, du_1\\, du_2\\, du_3\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3) \\frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}\\qquad \\nonumber \\\\&& \\longrightarrow - \\int d^8\\theta \\, du\\, D^{--}\\widetilde{q}^+(q+p,\\theta ,u) V^{++}(-p,\\theta ,u) D^{--} q^+(-q,\\theta ,u) \\nonumber \\\\&& = - 2 q^M (q+p)_M \\int d^4\\theta ^+\\, du\\, \\widetilde{q}^+(q+p,\\theta ,u) V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u).$ The remaining terms vanish.",
"Indeed, let us consider the expression $\\int du_1\\, du_2\\, du_3\\, D^+_{2a} D^+_{2b}\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\,\\frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2} \\nonumber \\\\$ and make use of the relation $(u_1^+ u_3^+) = D^{++}_1 D^{++}_3 (u_1^- u_3^-)$ .",
"Then, after integrating by parts with respect to the harmonic derivatives $D^{++}_1$ and $D^{++}_3$ , up to the terms quadratic in $V^{++}$ , we observe that on shell the resulting expression is proportional to $(u_1^- u_1^-) = 0$ , $&& (92) \\longrightarrow \\int du_1\\, du_2\\, du_3\\, D^+_{2a} D^+_{2b}\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\,(u_1^- u_3^-)\\nonumber \\\\&&\\times D^{--}_1 \\delta ^{(2,-2)}(u_1,u_2) D^{--}_3 \\delta ^{(2,-2)}(u_3,u_2) = 0.\\vphantom{\\frac{1}{2}}$ Similarly, using the identity $(u_1^+ u_3^+)^2 = D^{++}_1 D^{++}_3\\Big ((u_1^- u_3^-)(u_1^+ u_3^+)\\Big )$ , we obtain $&& \\int du_1\\, du_2\\, du_3\\,(D^+_{2})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\,\\frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3} \\nonumber \\\\&& \\longrightarrow \\frac{1}{4} \\int du_1\\, du_2\\, du_3\\,(D^+_{2})^4\\,\\widetilde{q}^+(q+p,\\theta ,u_1) V^{++}(-p,\\theta , u_2)\\, q^+(-q,\\theta ,u_3)\\, (u_1^- u_3^-)(u_1^+ u_3^+)\\qquad \\nonumber \\\\&& \\times (D^{--}_1)^2 \\delta ^{(2,-2)}(u_1,u_2) (D^{--}_3)^2 \\delta ^{(2,-2)}(u_3,u_2) = 0.\\vphantom{\\frac{1}{2}}$ Finally, collecting all terms, we conclude that the exploiting of the hypermultiplet equations of motion allows us to rewrite the part of $\\widetilde{\\Gamma }$ corresponding to the three-point gauge superfield - hypermultiplet Green function in the form $&& \\widetilde{\\Gamma }^{(3)} = 2f_0^2 \\int \\frac{d^{6}p}{(2\\pi )^6}\\,\\frac{d^{6}q}{(2\\pi )^6}\\,\\frac{d^{6}k}{(2\\pi )^6}\\,\\frac{(\\xi _0-1)}{k^2 (q+k)^2 (q+k+p)^2} \\Big ((q+p)^2 + q^2\\Big )\\nonumber \\\\&&\\times \\int d^4\\theta ^+\\, du\\, \\widetilde{q}^+(q+p,\\theta ,u) V^{++}(-p,\\theta ,u) q^+(-q,\\theta ,u).$ For the on-shell hypermultiplets the relations $q^2 =0$ and $(q+p)^2=0$ are valid, so this expression vanishes.",
"The conclusion is that the gauge-dependent contributions to the effective action are indeed canceled on shell in the approximation we stick to."
]
] | 1808.08446 |
[
[
"You are what you eat: A social media study of food identity"
],
[
"Abstract Food preferences not only originate from a person's dietary habits, but also reflect personal values and consumer awareness.",
"This study addresses `food identity' or the relationship between food preferences and personal attributes based on the concept of `food left-wing' (e.g., vegetarians) and `food right-wing' (e.g., fast-food lovers) by analyzing social data using information entropy and networks.",
"The results show that food identity extends beyond the domain of food: The food left-wing has a strong interest in socio-environmental issues, while the food right-wing has a higher interest in large-scale shopping malls and politically conservative issues.",
"Furthermore, the social interactions of food left-wing and right-wing factions show segregated structures, indicating different information consumption patterns.",
"These findings suggest that food identity may be applicable as a proxy for personal attributes and offer insights into potential buying patterns."
],
[
"Introduction",
"In today's age of gluttony, we are overwhelmed with information about food.",
"Food is available at a moment's notice in supermarkets, and we continually see advertisements for food online.",
"With almost unlimited options, the choice of what to eat and what not to eat depends less on biological aspects, such as individual survival or likes and dislikes, but rather it reflects the values of the individual [1].",
"Brillat-Savarin, a French gastronome famously stated, “Tell me what you eat and I will tell you what you are” [2].",
"This indicates that food was believed to be linked with identity.",
"Although several studies have examined this aspect (e.g., [3]), much remains understudied.",
"There are terms typifying the relationship between food and political ideologies that appear in sociological discourse in American culture, such as “Starbucks people” and “Coors beer people” [4].",
"These terms are meant as ironic signifiers of the metropolitan intelligentsia (liberal and Democratic Party-supporting people), who buy expensive coffee and read the New York Times at Starbucks versus rural people (conservative and Republican Party-supporting people), who drink inexpensive canned Coors beer while watching live broadcasts of American football.",
"Other terms that similarly parody these two camps' lifestyles and political ideologies are “latte liberals” and “bird-hunting conservatives” [5].",
"In Japan, it remains to be seen whether such stark stereotypes would apply, but there is prior work describing the political and dietary sensibilities of Japanese people using the terms “food left-wing” and “food right-wing” [6].",
"Proponents of “food left-wing” are those who pursue natural food and are health-conscious, typically vegetarians and vegans.",
"In contrast, the “food right-wing” group generally consumes available food products and enjoys eating fast food.",
"Hayamizu (2013) describes how food preferences (food left-wing and right-wing) speak to political attitudes in Japan.",
"If food preferences reflect the values of people, then these preferences are also likely closely tied not only to political ideology but also to other personal attributes.",
"If that were the case, food preferences could be a “mirror” reflecting latent consumption preferences and attitudes.",
"We study “food identity” based on the concept of the food left-wing and right-wing in order to determine whether this concept is useful in gaining insights into personal attributes.",
"To this end, we analyze social data from Twitter, in which a large amount of food-related information is spontaneously posted and shared.",
"Twitter is a better data source for our purse, because its use is better motivated in Japan given the high level of penetration in the country [7].",
"Although there are many previous researches about food-related social data analyses [8], [9], [10], little attention has been paid to food identity as a proxy for personal values and consumer awareness.",
"This is the main focus of this study.",
"The official Twitter Search application programming interface (API) [11] was used to create a crawler to harvest social data from the site and collected two datasets (Fig.",
"1) Alternative choice could be the Twitter Streaming API, but it has several issues: e.g., It often does not work properly for non-space separated languages (e.g., Japanese) and it is limited to 1% of the full Twitter Firehose.",
"Thus, we decided to use the Twitter Search API.. We ran the crawler three times a day and thus obtained almost all of the tweets that contained keywords of interest described below.",
"Figure: Schematic illustration of data collection and analysis.We used 18 food-related keywords from Shimizu (2013) to identify the food left-wing and right-wing (see Table 1).",
"Tweets containing these keywords were collected for a period of approximately one month beginning in December 2016.",
"This process yielded 650,900 Japanese tweets containing keywords referencing food left-wing tendencies, and 3,141,527 Japanese tweets containing keywords referencing food right-wing tendencies (Dataset 1).",
"Dataset 1 was then used to identify users making 30 or more total tweets containing any food left-wing keywords (using any of the keywords once or more per day, on average).",
"The same process was performed for those making tweets containing food right-wing keywords.",
"We excluded users making 30 or more total posts containing both food left-wing and right-wing keywords because they were deemed to not have a specific food preference.",
"This process yielded 1,233 users making food left-wing tweets and 5,010 users making food right-wing tweets.",
"This research treated the former group as food left-wing and the latter as food right-wing.",
"Table: Keywords used in collecting Dataset 1Timelines for all users identified with Dataset 1 were obtained: 3,655,936 tweets from the food left-wing and 15,091,255 tweets from the food right-wing (Dataset 2).",
"We computed the frequencies of keywords listed in Table 1 and later using texts from Dataset 2.",
"To compare the food left-wing and right-wing groups, we constructed 10 bootstrapped samples in each group with 1000 resamplings, used for the measurements described below.",
"The frequency of a keyword is often used as an indicator of the aggregate level of interest in a topic on social media, but this poses one major problem.",
"It does not consider how many individuals actually used the keyword.",
"For example, in an extreme case, the most frequently used keyword could be used by a single user.",
"To avoid this, we measure keyword entropy ($H_{k}$ ) below as an indicator of collective interest in a keyword by using normalized entropy [12]: $H_{k} = - \\sum _{x \\in U}^{}P_{k}(x)\\log _{2}P_{k}(x)/\\log _{2}N. $ Here, $U$ refers to a set of users, with $P_{k}(x)$ being the probability of a post with the keyword $k$ made by user $x$ , and $N$ is the total number of users, with $0 \\le H_{k} \\le 1$ .",
"As is clear in the definition, when many users make posts including keyword $k$ , then $H_{k}$ increases.",
"To compare keyword entropy between the food left-wing and right-wing groups, we employ the following formula, which is often called the laterality index ($LI$ ): $LI = \\frac{H_{k}^{R} - H_{k}^{L}}{H_{k}^{R} + H_{k}^{L}} $ where $R$ represents the food right-wing, and $L$ represents the food left-wing with $- 1 \\le LI \\le 1$ .",
"For keyword $k$ , a larger $LI$ implies a greater degree of collective interest among food right-wing users; a smaller $LI$ implies a greater degree of collective interest among food left-wing users.",
"We use association networks [13] to examine how the food left-wing and right-wing groups are aware of various keywords.",
"First, the tweet texts from Dataset 2 are segmented into words using MeCab [14] and the mecab-ipadic-NEologd (a Japanese dictionary) [15], and then cleaned by removing stopwords (defined in SlothLib [16]), symbols (e.g., !",
"and @), and URLs.",
"Next, the cleaned texts are used as the input to a word embedding method called word2vec [17], [18].",
"Word2vec can construct lower dimensional vectors that reflect word meanings based on word usage.",
"Using the trained word2vec model, we can convert words used within a corpus into vectors, by which semantically similar words would become similar vectors.",
"We used the Gensim library [19] for the word2vec modeling with the default parameter setting.",
"The resulting word vectors are visualized using association networks in the following manner.",
"Given a seed word, we list words whose cosine similarity to the seed word vector is greater than the similarity threshold (0.4); we then selected the top 20 most similar words.",
"Using the selected words as new seeds, we listed words in a same manner.",
"Words obtained by such “association chains” are used as nodes.",
"If term $w_2$ is selected when term $w_1$ is a seed, $w_1$ and $w_2$ are linked.",
"If multiple words are selected when $w_1$ is a seed, then $w_1$ is connected to all these words.",
"In this way, we visualize consumer awareness as word associations in tweets.",
"Twitter has the functionality of reposting or retweeting a friend's posts to own followers.",
"This leaves a record of how information spreads among users.",
"Given that user B retweets a post from user A, A and B can be treated as a node, with the directed link A$\\rightarrow $ B.",
"All of the retweets in Dataset 2 can be turned into directed links in this way, allowing for the reconstruction of social interactions in retweets among food left-wing and right-wing users.",
"We refer to this as a retweet network, described as $G = (V,E)$ [20].",
"Here, $V$ is a set of the users listed using the above method, and $E$ is a set of links describing retweet transmissions.",
"$V$ includes users other than the left-wing and right-wing seed users in Dataset 2, but those never retweeted were not included.",
"Analyzing the retweet network ($G$ ) allows us to examine structural patterns of information transmissions within and between food left-wing and right-wing groups.",
"First, we confirmed whether food left-wing and right-wing users had different food preferences and whether they had other preferences outside of food.",
"We computed the frequencies of the keywords in Table 2 for the categories of food, health, socio-environmental issues, and politics that were featured in the Nikkei newspaper from 2015 to 2016 (except for meat, fish, and vegetables) Note that a health freak is a person extremely enthusiastic about health..",
"The resulting word frequencies were then used to compute keyword entropy to compare the degree of interest in each group.",
"Table: Keywords related to food, health, socio-environment, and politicsFigure 2 shows keyword entropy across different categories.",
"In Fig.",
"2A, there is a high degree of interest in major foods, such as “meat,” “fish,” and “vegetables.” Comparing the food left-wing and right-wing groups, there is no marked difference in meat and fish, but the right-wing group shows a higher degree of interest in vegetables.",
"Of considerable interest here is that the food left-wing showed a greater interest in keywords like “trans fatty acid” that can lead to heart disease and “GM (genetically modified)” crops such as soybeans and corn in which the produce is manipulated by humans.",
"In contrast, the food right-wing showed a higher interest in “instant noodles,” the standard-bearer for junk food.",
"To confirm in what context these keywords were used, we randomly sampled 200 tweets that included each of these keywords, and then manually checked whether these keywords were used in a positive or negative context.",
"It transpired that the food left-wing users mostly made negative statements about trans fatty acid (positive: 1, negative: 199) and genetically modified (GM) crops (positive: 4, negative: 176); instant noodles were mostly used in a positive context by the food-right (positive: 131, negative: 29).",
"These results align with the image of the food left-wing as liking natural food and the right-wing as liking fast food.",
"Figure: Degree of collective interest in keywords related to food, health, socio-environment, and politics (means and SDs computed from the bootstrapped samples).Next, we examined the two groups to determine their preferences for categories other than food.",
"This can be confirmed in Figs.",
"2B-D when looking at categories of health, socio-environment, and politics.",
"In the health category, the food left-wing had a strong interest in “jogging,” “agrochemical-free,” and “low fat” while the food right-wing had a strong interest in “high calorie,” and “health freak.” This brings to mind the image of food right-wing users as eating fast food while worrying about the calories associated with it.",
"Given that food left-wing users show a greater interest in most of the keywords in the socio-environmental category, we can see how this concurs with the image of food left-wing users as highly conscious about socio-environmental issues.",
"Interestingly, the food right-wing showed a strong interest in keywords like “conservative” and “online right-wingers” and “Abe” (the current prime minister of Japan, seen as a right-wing politician).",
"Thus, there is some degree of correlation between food and politics.",
"In contrast, such correlation was not observed in the food left-wing.",
"Figure 3 summarizes the results of the keyword entropy when plotted by the laterality index ($LI$ ), showing specific differences between the two groups in categories that go beyond food.",
"Again, a greater $LI$ implies a greater degree of interest among food right-wing users, while a smaller $LI$ implies a greater degree of interest among food left-wing users.",
"We will not repeat the same observations shown in Fig.",
"2.",
"Rather, we emphasize that the food left-wing group had a marked interest in socio-environmental issues and the food-right showed some interest in political issues.",
"Figure: Collective interest in all keywords (means and SDs computed from the bootstrapped samples).",
"In the PDF version of this paper, colors correspond to categories (Food: red, Health: blue, Socio-environment: green, Politics: orange).It is notable that the term “animal experiment” is of high interest among the food left-wing.",
"Using association networks [13], we examined whether this is indeed the case.",
"Figure 4 is an association network of the words obtained when using “animal experiment” as the seed word when mapping against food left-wing users.",
"The links show word associations with a cosine similarity of 0.4 or higher.",
"The term “animal experiment” directly ties to “animal cruelty” and then to word clusters comprising terms like “cruel,” “torture,” “fur,” and “mink.” Other notable connections seeded from “animal experiment” include “boycott,” “cosmetics,” and “military research.” This association network reveals how food left-wing users harbor a strong negative image toward animal experiments.",
"We conducted the same process on the food right-wing data, but there was no markedly negative association with the term “animal experiment.” Figure: Association network of the term “animal experiment” for food left-wing users.These findings indicate that the food left-wing has strong feelings about animal welfare and strongly shirks from the idea of artificially manipulating animals.",
"Therefore, we can plainly see the way in which consumer attitudes, such as refusing to buy cosmetics that have been tested on animals or refusing to buy mink coats that are the result of killing and skinning animals, are manifested in tweets.",
"Next, we present an investigation into consumer interests in specific brands.",
"We used a 2016 Forbes survey [21] and a 2016 Nikkei survey [22] to identify 24 global brands across four industries (Table 3).",
"The keyword entropy ($H_{k}$ ) is measured based on the co-occurrence of brand names and positive words (Table 4) in Dataset 2.",
"In this case, a larger $H_{k}$ implies more positive interest in the brand.",
"Table: Industries and major brands usedTable: Positive keywordsFigure 5 shows the degree of positive interest in brands across the categories of technology, retail, beverage, and automobile, comparing between the food left-wing and food right-wing.",
"Figure 6 summarizes these results in terms of laterality index (LI).",
"Figure: Positive interest in brands related to technology, retail, beverage, and automobile (means and SDs computed from the bootstrapped samples).Figure: Positive interest in brands (means and SDs computed from the bootstrapped samples).The food right-wing group preferred most of the technology firms except for Apple and Sony Given that most posts about Amazon concern online shopping, Amazon might be more appropriately interpreted belonging to the retail category rather than technology..",
"In the retail space, the food right-wing showed a markedly higher interest in IKEA and Costco.",
"Incidentally, Starbucks—which we mentioned in the introduction—showed an expected high level of interest among the food left-wing.",
"Further differences were seen in the beverage space where the food left-wing showed strong interest in overseas beer brands like Corona, while the food right-wing showed high interest in energy drink brands like Red Bull.",
"A counter intuitive finding is that the food left-wing's positive interest in soda, especially in Pepsi.",
"We observed positive tweets, such as “Pepsi is way tastier than Coke.",
"As for Guarana, it's a bit in the middle...” The food-left wing showed a higher interest prefers in most of the automobile brands except Toyota.",
"The differences in positive interest between the food left-wing and right-wing groups are a useful metric when determining which products to promote through advertising based on the unique preferences of each group.",
"As described previously, we created a retweet network ($G$ ) based on the propagation of retweets ($|V|$ =165,609, $|E|$ =382,899).",
"Figure 7 visualizes the largest connected components with a maximum of 50 or more orders of connection ($|V|$ =1,113, $|E|$ =33,231).",
"Nodes represent users with blue being the food left-wing and red being the food right-wing.",
"Yellow represents those whose orientation could not be ascertained (i.e., users other than the left-wing and right-wing users in Dataset 2).",
"The links represent retweet transmissions.",
"One salient feature of these networks is the formation of two distinct left and right clusters.",
"Unlike the food left-wing users scattered on the right of Fig.",
"7, those on the left side of the graph suggest that the food-left users actively communicate about organic food-related information via retweets, thereby forming a cohesive cluster.",
"In contrast, the food right-wing does not manifest as densely as the food left-wing, suggesting retweet communications are less cohesive.",
"Figure: Retweet networks of food left-wing and right-wing users.",
"Nodes denote users.",
"Blue corresponds to the food left-wing, and red corresponds to the food right-wing; yellow is unknown.",
"The node size is proportional to overall degree.",
"The links denote retweet transmissions.Another notable feature in Fig.",
"7 is the existence of a “bridge” (the large yellow node) that links two separate clusters.",
"The bridge has major potential influence because this class of users can convery information to both the food left-wing and right-wing users.",
"This account belongs to livedoor News (@livedoornews) that manually curate and reposte articles posted to the livedoor News website.",
"Since livedoor News has a food section, this account makes many posts about food and cuisine; some are preferred by the food left-wing and others by the food right-wing.",
"We have demonstrated differences in personal values and consumer awareness for left-wing and right-wing users via social data analysis.",
"Food preferences are not a binary of left or right but rather a continuum, as personal attributes are multi-dimensional in nature.",
"This research, however, implies that mapping the plurality of personal attributes across the single dimension of left and right and looking at these two extremes does offer useful insights.",
"Measuring the degree of collective interest in certain keywords and brand names revealed that beyond the domain of food, the food left-wing has a strong interest in socio-environmental issues (notably, animal experiments) and the food right-wing has a higher interest in large-scale shopping malls offering discounts and volume sales of goods, such as IKEA and Costco.",
"In addition, we observed differences between the food left-wing and right-wing groups in word usage.",
"Table 5 shows the top 100 popular keywords ranked by the average TF-IDF score [23].",
"Note that the word's TF-IDF score is its term frequency divided by its document (tweet) frequency.",
"Many food left-wing related words are ranked in the table in the food left-wing group (e.g., beauty, health, and nature), while several food-right wing related words are ranked in the table in the food right-wing group (e.g., supermarket and ice cream).",
"Table: Top 100 popular keywords ranked by the average TF-IFD socre.The segregated network structures in retweet interactions revealed that food-related information is also consumed differently in the food left-wing and right-wing groups.",
"One point of interest in these networks is that clusters of food left-wing and right-wing users are linked via the conduit of a news outlets (livedoor News).",
"Given that online social networks are mechanisms for disseminating information, identifying users' food preferences might be efficient in the distribution of a specific piece of information, an advertisement, or a campaign.",
"Looking solely at these results, one could conjecture as to the differences in consumer attitudes among these groups—the food-left wing rejects industrialization and prefers a return to nature, and the food right-wing does the opposite.",
"However, reality is not quite so simple.",
"Indeed, the food left-wing had a higher degree of interest in Apple and Sony as technology brands, as well as in most automobile brands.",
"Therefore, we cannot say conclusively that the food left-wing comprehensively rejects industrialization and prefers a return to nature in contexts other than food.",
"This study has implications for social sciences and applications.",
"The quantification of personal attributes is an important procedure in many areas of social sciences.",
"Personal attributes, however, are difficult to accurately measure by survey research alone.",
"Given the fact that food identity could be a concise yet useful proxy for various personal attributes and that there are numerous reports about food on social media, mining food identities from online digital traces can contribute to social science research.",
"Furthermore, the idea of food identity could be applicable in social media marketing and other applications.",
"For example, rather than relying on influencers ([24]), which represent a small and limited population, or on simply trafficking large quantities of advertisements at random, selectively running advertisements based on the underlying values around food is much more likely to hit the mark.",
"However, one should remember that simply extrapolating the concepts of the food left-wing and right-wing without conducting prior research would prove detrimental when doing social sciences or engaging in marketing.",
"We recognize the limitations of social media analysis for studying food identity, because social media users are biased toward age, gender, geolocation, etc.",
"To supplement social data analysis, we have to incorporate survey research, which is one of our future research directions.",
"Moreover, the keyword-based food preference identification done here has potential concerns with misclassification.",
"For example, food left-wing users who often post criticisms about fast food could be classified into food right-wing users; similar errors could occur for food right-wing users.",
"In addition, this approach cannot capture criticisms and cynicisms, potentially conflating very different types of contexts.",
"To resolve these issues, contextual information needs to be considered to improve the classification accuracy.",
"Although several limitations remain, our social media analysis has demonstrated that food identity can be a useful concept to address personal attributes and consumer awareness.",
"K.S.",
"thanks to M. Karasawa and K. Hioki for discussions.",
"This research was supported in part by Yoshida Hideo Memorial Foundation, JSPS/MEXT KAKENHI Grant Numbers JP16K16112 and JP17H06383 in #4903, JST PRESTO Grant Number JPMJPR16D6, and JST CREST Grant Number JPMJCR17A4."
]
] | 1808.08428 |
[
[
"Kapranov's construction of sh Leibniz algebras"
],
[
"Abstract Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a K\\\"{a}hler manifold and Chen-Sti\\'{e}non-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to construct sh Leibniz algebras.",
"Let $\\mathcal{A}$ be a commutative dg algebra.",
"Given a derivation of $\\mathcal{A}$ valued in a dg module $\\Omega$, we show that there exist sh Leibniz algebra structures on the dual module of $\\Omega$.",
"Moreover, we prove that this process establishes a functor from the category of dg module valued derivations to the category of sh Leibniz algebras over $\\mathcal{A}$."
],
[
"Introduction",
"Higher homotopies and higher structures are playing important roles in mathematics and some branches of theoretical physics, such as gauge theory and topological field theory (see Huebschmann ).",
"Higher homotopies, as explained by Huebschmann in , often arise from the process of transferring certain strict geometric or algebraic structure on a huge chain complex to a smaller but chain homotopic complex.",
"For instance, an sh Lie algebra (also known as $L_\\infty $ -algebra ) yields from a dg Lie algebra by applying homological perturbation theory .",
"Here and in the sequel, sh is short for strongly homotopy and dg is short for differential graded.",
"Sh Leibniz algebras, also known as sh Loday algebras, Loday infinity algebras or Leibniz$_\\infty $ algebras , are also examples of higher structures.",
"In fact, the notion of Leibniz$_\\infty $ algebras is a generalization of $L_\\infty $ algebras where the skew-symmetricity constraint on multibrackets is discarded.",
"In this note, we use the notion of Leibniz$_\\infty [1]$ algebras (see Definition REF ), which is equivalent to the notion of sh Leibniz algebras, and study a particular method to construct Leibniz$_\\infty [1]$ algebras.",
"This method first appeared in Kapranov's approach to Rozansky-Witten theory : Given a Kähler manifold $X$ , Kapranov discovered an $L_\\infty $ algebra structure on $\\Omega ^{0,\\bullet -1}_X(T_X)$ via the Atiyah class $\\alpha _X$ .",
"More precisely, let $\\nabla $ be the Chern connection on the holomorphic tangent bundle $T_X$ .",
"Then the curvature $R_\\nabla \\in \\Omega _X^{0,1}(\\operatorname{Hom}(S^2(T_X), T_X))$ is a Dolbeault representative of the Atiyah class $\\alpha _X$ .",
"The $L_\\infty $ brackets $\\lbrace \\lambda _k\\rbrace _{k\\ge 1}$ on $\\Omega ^{0,\\bullet -1}_X(T_X)$ are defined by $\\lambda _1 = \\bar{\\partial }$ .",
"$\\lambda _2 = R_\\nabla $ .",
"$\\lambda _{k+1} = \\nabla ^{1,0}(\\lambda _k) \\in \\Omega _X^{0,1}(\\operatorname{Hom}(S^{k+1}(T_X), T_X))$ , for $k = 2,3,4,...$ .",
"Kapranov's construction of $L_\\infty $ algebras is generalized in Chen, Stiénon and Xu's work where the setting is a Lie algebroid pair (Lie pair, for short) $(L,A)$ .",
"It is shown that the graded vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ admits a Leibniz$_\\infty [1]$ algebra structure (*Theorem 3.13) via the Atiyah class of the Lie pair $(L,A)$ .",
"The Atiyah class of Lie pairs encompasses the original Atiyah class of holomorphic vector bundles and the Molino class of foliations as special cases.",
"This construction of Leibniz$_\\infty [1]$ algebra structures is similar to that of Kapranov — First, we choose a splitting $j: L/A \\rightarrow L$ of vector bundles so that $L \\cong A\\oplus L/A$ .",
"Second, choose an $L$ -connection $\\nabla $ on $L/A$ extending the $A$ -module structure.",
"Then the Leibniz$_\\infty [1]$ brackets $\\lbrace \\lambda _k\\rbrace _{k\\ge 1}$ on $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ are determined as follows: $\\lambda _1 = d_{\\mathrm {CE}}$ is the Chevalley-Eilenberg differential of the Bott representation of $A$ on $L/A$ .",
"Define a bundle map $R_2: L/A \\otimes L/A \\rightarrow A^\\vee \\otimes L/A$ via the Atiyah cocycle $\\alpha _{L/A}^{\\nabla }$ (see Section REF ): $R_2(b_1,b_2) = \\alpha _{L/A}^{\\nabla }(-,b_1)b_2,\\;\\;\\forall b_1,b_2 \\in \\Gamma (L/A).$ The second structure map $\\lambda _2$ is specified by $\\lambda _2(\\xi _1 \\otimes b_1, \\xi _2 \\otimes b_2) = (-1)^{\\vert \\xi _1\\vert +\\vert \\xi _2\\vert } \\xi _1\\wedge \\xi _2 \\wedge R_2(b_1,b_2),$ for all $\\xi _1,\\xi _2 \\in \\Gamma (\\wedge ^\\bullet A^\\vee )$ and $b_1,b_2 \\in \\Gamma (L/A)$ .",
"Define a sequence of bundle maps $R_k: (L/A)^{\\otimes k} \\rightarrow A^\\vee \\otimes L/A, k \\ge 3$ recursively by $R_{n+1}=\\nabla R_n$ , i.e., $R_{n+1}(b_0 \\otimes \\cdots \\otimes b_n) = R_n(\\nabla _{j(b_0)}(b_1 \\otimes \\cdots \\otimes b_n)) - \\nabla _{j(b_0)}R_n(b_1 \\otimes \\cdots \\otimes b_n).$ The $k$ -th structure map is specified by $\\lambda _k(\\xi _1 \\otimes b_1,\\cdots ,\\xi _k\\otimes b_k) = (-1)^{\\vert \\xi _1\\vert +\\cdots +\\vert \\xi _k\\vert }\\xi _1\\wedge \\cdots \\wedge \\xi _k\\wedge R_k(b_1,\\cdots ,b_k),$ for all $\\xi _i \\in \\Gamma (\\wedge ^\\bullet A^\\vee ), b_i \\in \\Gamma (L/A), 1 \\le i \\le k$ .",
"We call $(\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A),\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ a Kapranov Leibniz$_\\infty [1]$ algebra.",
"Its construction needs, a priori, some extra choices (a splitting $j$ and an $L$ -connection $\\nabla $ on $L/A$ ).",
"Then one asks a natural question (*Remark 3.19) — how does the Leibniz$_\\infty [1]$ algebra structure on $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ depend on the choice of splitting data and connections?",
"The main goal of this note is to answer this question—Kapranov Leibniz$_\\infty [1]$ algebra structures on $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ associated with different choices of $j$ and $\\nabla $ , are mutually isomorphic in the category of Leibniz$_\\infty [1]$ algebras over $\\Gamma (\\wedge ^\\bullet A^\\vee )$ (see Theorem 1.3 or Theorem REF ).",
"We adopt an algebraic approach to achieve this goal.",
"The algebraic notion we need is a dg module valued derivation of a commutative differential graded algebra (cdga for short) ${A}$ (see Definition REF ).",
"As an immediate example from complex geometry, consider a complex manifold $X$ .",
"The Dolbeault dg algebra ${A}=(\\Omega _X^{0,\\bullet },\\bar{\\partial })$ is a cdga.",
"Let $\\Omega =(\\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee ), \\bar{\\partial })$ be the dg ${A}$ -module generated by the section space of holomorphic cotangent bundle $(T^{1,0}X)^\\vee $ .",
"Then $\\partial : {A}\\rightarrow \\Omega $ is an $\\Omega $ -valued derivation of ${A}$ .",
"We now explain how Kapranov's original method and Chen-Stiénon-Xu's construction can be further generalized in the setting of a dg module valued derivation ${A}\\xrightarrow{} \\Omega $ .",
"Consider the dual dg ${A}$ -module $\\mathcal {B}=\\Omega ^\\vee $ of $\\Omega $ .",
"First, one chooses a $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ , i.e., a map $\\nabla :\\mathcal {B}\\rightarrow \\Omega \\otimes _{A}\\mathcal {B}$ that extends the $\\delta $ -map (see Definition REF ).",
"Then one can define a sequence of degree 1 maps $\\lbrace \\mathcal {R}^\\nabla _k:~\\otimes ^k\\mathcal {B}\\rightarrow \\mathcal {B}\\rbrace _{k \\ge 1}$ as follows: $\\mathcal {R}_1^\\nabla = \\partial _{A}$ is the differential on $\\mathcal {B}$ .",
"$\\mathcal {R}^\\nabla _2=\\operatorname{At}_\\mathcal {B}^\\nabla : ~\\mathcal {B}\\otimes _{{A}} \\mathcal {B}\\rightarrow \\mathcal {B}$ is the twisted Atiyah cocycle (see Definition REF ).",
"$\\mathcal {R}_k^\\nabla $ for $ k \\ge 3$ are defined recursively by $\\mathcal {R}_k^\\nabla =\\nabla \\mathcal {R}_{k-1}^\\nabla $ (see Equation (REF )).",
"Our first result is the following Theorem 1.1 When endowed with structure maps $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k \\ge 1}$ , the dg ${A}$ -module $\\mathcal {B}$ becomes a Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"Here by saying that $\\mathcal {B}$ is a Leibniz$_\\infty [1]$ ${A}$ -algebra, we mean that its higher structure maps $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 2}$ are all ${A}$ -multilinear.",
"We emphasise that the Kapranov Leibniz$_\\infty [1]$ algebra $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ should be treated as an object in the category of Leibniz$_\\infty [1]$ ${A}$ -algebras.",
"In fact, if we treat $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ merely as a Leibniz$_\\infty [1]$ algebra over $\\mathbb {K}$ , it is always isomorphic to a trivial one (see Remark REF ).",
"We call $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ the Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra associated with the dg module valued derivation ${A}\\xrightarrow{} \\Omega $ and the $\\delta $ -connection $\\nabla $ .",
"Our second result is that the above construction is functorial: Theorem 1.2 The above construction defines a functor $\\operatorname{Kap}$ , called Kapranov functor, from the category of dg module valued derivations of a cdga ${A}$ to the category of Leibniz$_\\infty $ ${A}$ -algebras.",
"Moreover, the Kapranov functor $\\operatorname{Kap}$ is homotopy invariant, i.e., if $\\delta $ and $\\delta ^\\prime $ are two homotopic derivations of ${A}$ valued in the same dg module, then $\\operatorname{Kap}(\\delta )$ is isomorphic to $\\operatorname{Kap}(\\delta ^\\prime )$ .",
"Applying Theorem REF to dg module valued derivations arising from Lie pairs, we obtain the answer of our motivating question: Theorem 1.3 Let $(L,A)$ be a Lie pair.",
"The Kapranov Leibniz$_\\infty [1]$ algebra structure on the graded vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ is unique up to isomorphisms in the category of Leibniz$_\\infty [1]$ $\\Gamma (\\wedge ^\\bullet A^\\vee )$ -algebras.",
"This note is organized as follows: Section consists of our conventions, notations, and the notion of twisted Atiyah classes.",
"We will see that twisted Atiyah classes encompass Atiyah classes of Lie pairs and dg Lie algebroids as special cases.",
"Section contains a brief summary of sh Leibniz algebras, the construction of the Kapranov functor, and its applications.",
"Finally, we present some relevant remarks and open questions in Section .",
"Acknowledgements.",
"We would like to thank Bangming Deng, Wei Hong, Kai Jiang, Honglei Lang, Camille Laurent-Gengoux, Mathieu Stiénon, Jim Stasheff, Yannick Voglaire, and Ping Xu for useful discussions and comments.",
"Xiang is grateful to Penn State University, Peking University and Tsinghua University, for their hospitality during his visits."
],
[
"Atiyah classes of commutative dg algebras and their twists",
"In , Atiyah introduced a cohomology class, which has come to be known as Atiyah class, to characterize the obstruction to the existence of holomorphic connections on a holomorphic vector bundle.",
"The notion of Atiyah classes have been developed in the past decades for diverse purposes (see Bottacin,CV,CLX,Costello,CSX,LaurentSX-CR,LaurentSX,MSX).",
"In this section, we recall Atiyah classes of commutative dg algebras defined by Costello and introduce a version of twisted Atiyah classes."
],
[
"Atiyah classes of commutative dg algebras",
"Throughout this paper, $\\mathbb {K}$ denotes a field of characteristic zero and graded means $\\mathbb {Z}$ -graded.",
"A commutative differential graded algebra (cdga for short) over $\\mathbb {K}$ is a pair $({A},d_{A})$ , where ${A}$ is a commutative graded $\\mathbb {K}$ -algebra, and $d_{A}:~{A}\\rightarrow {A}$ , usually called the differential, is a homogeneous degree one derivation of square zero.",
"We also write ${A}$ for a cdga without making its differential explicitly.",
"An ${A}$ -module is a representation of the underlying commutative graded algebra of ${A}$ by forgetting the differential $d_{A}$ .",
"By a dg ${A}$ -module, we mean an ${A}$ -module ${E}$ , together with a degree one and square zero endomorphism $\\partial _{A}^{E}$ of the graded $\\mathbb {K}$ -vector space ${E}$ , called the differential, such that $\\partial ^{E}_{A}(a e) = (d_{A}a ) e + (-1)^{\\vert a\\vert }a \\partial _{A}^{E}(e),$ for all $a \\in {A}, e \\in {E}$ .",
"To work with various different dg ${A}$ -modules, the differential $\\partial _{A}^{E}$ of any dg ${A}$ -module ${E}$ will be denoted by the same notation $\\partial _{A}$ .",
"A dg ${A}$ -module $({E},\\partial _{A})$ will also be simply denoted by ${E}$ .",
"The dg ${A}$ -module of Kähler differentials is the graded ${A}$ -module $\\Omega _{{A}\\mid \\mathbb {K}}^1 = \\operatorname{span}\\lbrace d_{dR}a:~ a \\in {A}\\rbrace /\\lbrace d_{dR}(ab)-(d_{dR}a)b - (-1)^{\\vert a\\vert }ad_{dR}b:~ a,b \\in {A}\\rbrace ,$ together with the differential $\\partial _{A}$ such that the algebraic de Rham operator $d_{dR}:~{A}\\rightarrow \\Omega _{{A}\\mid \\mathbb {K}}^1$ is a cochain map, i.e., $\\partial _{A}(d_{dR}a) = d_{dR}(d_{A}a)$ for all $a \\in {A}$ .",
"In the sequel, we assume that $\\Omega ^1_{{A}\\mid \\mathbb {K}}$ is projective as an ${A}$ -module.",
"A degree $r$ morphism of dg ${A}$ -modules, denoted by $\\alpha \\in \\operatorname{Hom}^r_{\\mathrm {dg}{A}}({E},{F})$ , is a degree $r$ ${A}$ -module morphism $\\alpha :~{E}\\rightarrow {F}$ , which is also compatible with differentials: $\\partial _{A}(\\alpha ) := \\partial _{A}\\circ \\alpha -(-1)^{r}\\alpha \\circ \\partial _{A}= 0: {E}\\rightarrow {F}.$ Definition 2.1 (Costello ) Let ${A}$ be a cdga and ${E}$ an ${A}$ -module.",
"A connection on ${E}$ is a (degree 0) map of graded $\\mathbb {K}$ -vector spaces $\\blacktriangledown :~ {E}\\rightarrow \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{E},$ satisfying the Leibniz rule $\\blacktriangledown (ae) = (d_{dR}a) \\otimes e + a\\blacktriangledown (e),\\;\\;\\forall a \\in {A}, e \\in {E}.$ Assume that $({E},\\partial _{A})$ is a dg ${A}$ -module.",
"Given a connection $\\blacktriangledown $ on ${E}$ , $\\operatorname{At}_{E}^\\blacktriangledown :=[\\blacktriangledown ,\\partial _{A}] = \\blacktriangledown \\circ \\partial _{A}- \\partial _{A}\\circ \\blacktriangledown \\in \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}\\operatorname{End}_{A}({E})$ is a closed element of degree 1 which measures the failure of $\\blacktriangledown $ to be a cochain map.",
"Its cohomology class $\\operatorname{At}_{E}\\in H^1({A},\\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}\\operatorname{End}_{A}({E}))$ is independent of the choice of connections, and is called the Atiyah class of the dg ${A}$ -module ${E}$ .",
"The existence of connections on ${E}$ is guaranteed if ${E}$ is a projective ${A}$ -module.",
"Hence we make the following Convention 2.2 In this note, all ${A}$ -modules are assumed to be projective.",
"Example 2.3 (Mehta-Sténon-Xu ) Let $(\\mathcal {M},Q_\\mathcal {M})$ be a smooth dg manifold, where $\\mathcal {M}= (M,\\mathcal {O}_\\mathcal {M})$ is a smooth $\\mathbb {Z}$ -graded manifold, and $Q_\\mathcal {M}$ is a homological vector field on $\\mathcal {M}$ .",
"Then ${A}= (C^\\infty (\\mathcal {M}),Q_\\mathcal {M})$ is a cdga.",
"For each dg vector bundle $(\\mathcal {E},Q_{\\mathcal {E}})$ over $(\\mathcal {M},Q_\\mathcal {M})$ , its space of sections ${E} = (\\Gamma (\\mathcal {E}),Q_{\\mathcal {E}})$ is a dg ${A}$ -module.",
"The Atiyah class $\\operatorname{At}_{E}$ of ${E}$ coincides, up to a minus sign, with the Atiyah class $\\operatorname{At}_{\\mathcal {E}}$ of the dg vector bundle $\\mathcal {E}$ with respect to the dg Lie algebroid $T\\mathcal {M}$ defined by Mehta-Sténon and Xu.",
"This is a particular instance of Atiyah classes of dg vector bundles with respect to a general dg Lie algebroid (see Section REF ).",
"Dg module valued derivations and twisted Atiyah classes A key notion in this note is dg module valued derivation (dg derivation for short): Definition 2.4 Let $({A},d_{A})$ be a cdga and $(\\Omega ,\\partial _{A})$ a dg ${A}$ -module.",
"A dg derivation of ${A}$ valued in $(\\Omega ,\\partial _{A})$ is a degree 0 derivation $\\delta :~ {A}\\rightarrow \\Omega $ of the commutative graded algebra ${A}$ valued in the ${A}$ -module $\\Omega $ , $\\delta (ab) = \\delta (a)b + a\\delta (b),\\;\\;\\;\\forall a,b \\in {A},$ which commutes with the differentials as well: $\\delta \\circ d_{A}= \\partial _{A}\\circ \\delta :~ \\; {A}\\rightarrow \\Omega .$ Such a dg derivation is simply denoted by ${A}\\xrightarrow{} \\Omega $ .",
"Let $\\delta $ and $\\delta ^\\prime $ be two $(\\Omega ,\\partial _{A})$ -valued dg derivations of ${A}$ .",
"They are said to be homotopic, written as $\\delta \\sim \\delta ^{\\prime }$ , if there exists a degree $(-1)$ derivation $h$ of ${A}$ valued in the ${A}$ -module $\\Omega $ such that $\\delta ^\\prime - \\delta = [\\partial _{A},h] = \\partial _{A}\\circ h + h \\circ d_{A}: {A}\\rightarrow \\Omega .$ An immediate example of dg derivations is $ \\Omega _X^{0,\\bullet }\\xrightarrow{} \\Omega _X^{0,\\bullet } ((T^{1,0}X)^\\vee )$ arising from a complex manifold $X$ , which has already been explained in Section .",
"Another fundamental example is the dg derivation ${A}\\xrightarrow{} \\Omega _{{A}\\mid \\mathbb {K}}^1$ , which is universal in the following sense: For any generic dg derivation ${A}\\xrightarrow{} \\Omega $ , there exists a unique dg ${A}$ -module morphism $\\bar{\\delta }:~\\Omega _{{A}\\mid \\mathbb {K}}^1 \\rightarrow \\Omega $ such that the following diagram commutes: ${{A}[d]_-{d_{dR}} [r]^-{\\delta } & \\Omega \\\\\\Omega _{{A}\\mid \\mathbb {K}}^1.",
"[ur]_-{\\exists !",
"~\\bar{\\delta }} &}$ Thus, $\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})}:~ \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}\\operatorname{End}_{A}({E}) \\rightarrow \\Omega \\otimes _{A}\\operatorname{End}_{A}({E})$ is a dg ${A}$ -module morphism as well.",
"Definition 2.7 (Twisted Atiyah class) Let ${E}$ be a dg ${A}$ -module and ${A}\\xrightarrow{} \\Omega $ a dg derivation.",
"The dg ${A}$ -module morphism $\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})}$ in Equation (REF ) sends the Atiyah class $\\operatorname{At}_{E}\\in H^1({A},\\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}\\operatorname{End}_{A}({E}))$ of ${E}$ to a cohomology class $\\operatorname{At}_{E}^\\delta \\in H^1({A}, \\Omega \\otimes _{A}\\operatorname{End}_{A}({E})),$ which is called the $\\delta $ -twisted Atiyah class of ${E}$ .",
"It follows immediately that twisted Atiyah classes are homotopic invariant: Proposition 2.8 If $\\delta \\sim \\delta ^\\prime $ , then for any dg ${A}$ -module ${E}$ , $\\operatorname{At}_{E}^\\delta = \\operatorname{At}_{E}^{\\delta ^\\prime } \\in H^1({A}, \\Omega \\otimes _{A}\\operatorname{End}_{A}({E})).$ Below we give a different characterization of the twisted Atiyah class $\\operatorname{At}_{E}^\\delta $ .",
"We need another key notion in this note — $\\delta $ -connections, which can be thought of as operations extending $\\delta $ .",
"Definition 2.9 Let ${A}\\xrightarrow{} \\Omega $ be a dg derivation and ${E}$ an ${A}$ -module.",
"A $\\delta $ -connection on ${E}$ is a degree 0, $\\mathbb {K}$ -linear map of graded $\\mathbb {K}$ -vector spaces $\\nabla :~ \\;{E}\\rightarrow \\Omega \\otimes _{A}{E}$ satisfying the following Leibniz rule: $\\nabla (ae) = \\delta (a) \\otimes e + a\\nabla (e),\\;\\;\\forall a \\in {A}, e \\in {E}.$ Remark 2.11 A connection $\\blacktriangledown $ as in Definition REF induces a $\\delta $ -connection $\\nabla $ as in Definition REF via the following triangle ${{E}[r]^-{\\blacktriangledown } [dr]_-{\\nabla } & \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{E}[d]^-{\\bar{\\delta }\\otimes _{A}\\operatorname{id}_{{E}}} \\\\& \\Omega \\otimes _{A}{E}.",
"}$ It follows that $\\delta $ -connections always exist on projective ${A}$ -modules.",
"However, $\\delta $ -connections do not necessarily arise in this manner.",
"Proposition 2.13 Let ${E}= ({E}, \\partial _{A})$ be a dg ${A}$ -module.",
"1) For any $\\delta $ -connection $\\nabla $ on ${E}$ , the degree 1 element $\\operatorname{At}_{E}^{\\nabla }:=[\\nabla ,\\partial _{A}] = \\nabla \\circ \\partial _{A}- \\partial _{A}\\circ \\nabla \\in \\Omega \\otimes _{A}\\operatorname{End}_{A}({E})$ is a cocycle.",
"2) The cohomology class $[\\operatorname{At}^{\\nabla }_{E}] \\in H^1({A}, \\Omega \\otimes _{A}\\operatorname{End}_{A}({E}))$ coincides with the $\\delta $ -twisted Atiyah class $\\operatorname{At}_{E}^\\delta $ of ${E}$ .",
"The first statement is clear.",
"It only suffices to prove the second one: Observe that the difference of two $\\delta $ -connections is a degree zero element in $ \\Omega \\otimes _{A}\\operatorname{End}_{A}({E})$ .",
"Hence, the cohomology class $[\\operatorname{At}_{E}^{\\nabla }]$ is independent of the choice of $\\delta $ -connections.",
"We choose a particular $\\delta $ -connection $\\nabla $ induced by a connection $\\blacktriangledown $ on ${E}$ as in the commutative triangle (REF ).",
"Since the map $\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})}$ defined in Equation (REF ) is a dg ${A}$ -module morphism, it follows that $\\operatorname{At}_{E}^{\\nabla } &= [\\nabla ,\\partial _{A}] = [(\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})}) \\circ \\blacktriangledown ,\\partial _{A}] = (\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})})[\\blacktriangledown ,\\partial _{A}] = (\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})})(\\operatorname{At}_{E}^\\blacktriangledown ).$ Passing to the cohomology, we have $[\\operatorname{At}_{E}^{\\nabla }] = (\\bar{\\delta } \\otimes _{A}\\operatorname{id}_{\\operatorname{End}_{A}({E})})(\\operatorname{At}_{E}) = \\operatorname{At}_{E}^\\delta .$ Definition 2.14 We call $\\operatorname{At}_{E}^{\\nabla }$ the $\\delta $ -twisted Atiyah cocycle of ${E}$ with respect to the $\\delta $ -connection $\\nabla $ on ${E}$ .",
"Denote the ${A}$ -dual $\\Omega ^\\vee $ of $\\Omega $ by $B$ , which is also a dg ${A}$ -module.",
"Given a $\\delta $ -connection $\\nabla :\\;{E}\\rightarrow \\Omega \\otimes _{A}{E}$ of an ${A}$ -module ${E}$ , the covariant derivation along $b \\in \\mathcal {B}$ is $\\nabla _b:~{E}\\rightarrow {E},\\qquad \\nabla _b(e):= \\iota _b\\nabla (e),\\quad ~\\forall e\\in {E}.$ The $\\delta $ -twisted Atiyah cocycle $\\operatorname{At}_{E}^{\\nabla }$ could be viewed as a degree 1 element in $\\operatorname{Hom}_{A}(\\mathcal {B}\\otimes _{A}{E},{E})$ by setting $\\operatorname{At}_{E}^{\\nabla }(b,e) &= (-1)^{\\vert b\\vert }\\iota _b\\operatorname{At}_{E}^{\\nabla }(e) = (-1)^{\\vert b\\vert }\\iota _b(\\nabla (\\partial _{A}(e)) - \\partial _{A}(\\nabla (e))) \\\\&= -\\partial _{A}(\\nabla _b e) + \\nabla _{\\partial _{A}(b)}e + (-1)^{\\vert b\\vert }\\nabla _b\\partial _{A}(e) \\\\&= \\nabla _{\\partial _{A}(b)}e - [\\partial _{A},\\nabla _b](e),$ for all $b \\in \\mathcal {B}$ and $e \\in {E}$ .",
"Moreover, as $\\operatorname{At}_{E}^{\\nabla }$ is a 1-cocycle, it is a morphism of dg ${A}$ -modules, i.e., $\\operatorname{At}_{E}^{\\nabla }\\in \\operatorname{Hom}^1_{\\mathrm {dg}{A}}(\\mathcal {B}\\otimes _{A}{E},{E})$ .",
"As an immediate consequence of Proposition REF and Equation (REF ), we have the following Proposition 2.14 Let ${A}\\xrightarrow{} \\Omega $ be a dg derivation and ${E}$ a dg ${A}$ -module.",
"Then the $\\delta $ -twisted Atiyah class $\\operatorname{At}^\\delta _{E}$ vanishes if and only if there exists a $\\delta $ -connection $\\nabla $ on ${E}$ such that the associated twisted Atiyah cocycle $\\operatorname{At}_{E}^{\\nabla }$ vanishes, i.e., the map $\\nabla :~{E}\\rightarrow \\Omega \\otimes _{A}{E}$ is compatible with the differentials.",
"In this case, for all $\\partial _{A}$ -closed elements $b\\in \\mathcal {B}$ and $ e\\in {E}$ , $\\nabla _b e$ is also $\\partial _{A}$ -closed.",
"Atiyah classes of dg Lie algebroids and Lie pairs In this section, we briefly recall Atiyah classes of dg vector bundles with respect to a dg Lie algebroid defined in and Atiyah classes of Lie pairs defined in (see for the equivalence between the two types of Atiyah classes arising from integrable distributions), and show that both of them can be viewed as twisted Atiyah classes.",
"Dg Lie algebroids A dg Lie algebroid can be thought of as a Lie algebroid object in the category of smooth dg manifolds.",
"The precise description is as follows.",
"Definition 2.15 A dg Lie algebroid over a dg manifold $(\\mathcal {M},Q_\\mathcal {M})$ is a quadruple $(\\mathcal {L}, Q_\\mathcal {L},\\rho _\\mathcal {L}, [-,-]_{\\mathcal {L}}),$ where 1) $(\\mathcal {L},Q_\\mathcal {L})$ is a dg vector bundle over $(\\mathcal {M},Q_\\mathcal {M})$ ; 2) $(\\mathcal {L},\\rho _\\mathcal {L}, [-,-]_\\mathcal {L})$ is a graded Lie algebroid over $\\mathcal {M}$ ; 3) The anchor map $\\rho _\\mathcal {L}:~ (\\mathcal {L},Q_\\mathcal {L}) \\rightarrow (T\\mathcal {M},L_{Q_\\mathcal {M}})$ is a morphism of dg vector bundles; 4) $Q_\\mathcal {L}:~ \\Gamma (\\mathcal {L}) \\rightarrow \\Gamma (\\mathcal {L})$ is a derivation with respect to the bracket $[-,-]_\\mathcal {L}$ , i.e., $Q_\\mathcal {L}([X,Y]_\\mathcal {L}) = [Q_\\mathcal {L}(X),Y]_\\mathcal {L}+ (-1)^{\\vert X\\vert }[X,Q_\\mathcal {L}(Y)]_\\mathcal {L},\\;\\;\\forall X,Y \\in \\Gamma (\\mathcal {L}).$ Given a dg Lie algebroid $(\\mathcal {L}, Q_\\mathcal {L},\\rho _\\mathcal {L}, [-,-]_{\\mathcal {L}})$ and a dg vector bundle $(\\mathcal {E},Q_{\\mathcal {E}})$ over $(\\mathcal {M},Q_\\mathcal {M})$ , Mehta, Stiénon and Xu constructed the Atiyah class $\\operatorname{At}_{\\mathcal {E}}$ of $\\mathcal {E}$ with respect to $\\mathcal {L}$ as follows: Choose a Lie algebroid $\\mathcal {L}$ -connection $\\nabla ^{\\mathcal {E}}$ on the vector bundle $\\mathcal {E}$ , i.e., a degree 0 $\\mathbb {K}$ -bilinear map $\\nabla ^{\\mathcal {E}}:~ \\Gamma (\\mathcal {L}) \\times \\Gamma (\\mathcal {E}) \\rightarrow \\Gamma (\\mathcal {E})$ subject to the relations $\\nabla ^{\\mathcal {E}}_{fX}e &= f\\nabla ^{\\mathcal {E}}_{X}e, & \\nabla ^{\\mathcal {E}}_{X}(fe) &= (\\rho _\\mathcal {L}(X)f)e + (-1)^{\\vert f\\vert \\vert X\\vert } f\\nabla ^{\\mathcal {E}}_{X}e,$ for all $f \\in C^\\infty (\\mathcal {M}), X \\in \\Gamma (\\mathcal {L})$ and $e \\in \\Gamma (\\mathcal {E})$ .",
"There associates a degree 1 cocycle $\\operatorname{At}_{\\mathcal {E}}^{\\nabla ^{\\mathcal {E}}} \\in \\Gamma (\\mathcal {L}^\\vee \\otimes \\operatorname{End}(\\mathcal {E}))$ defined by $\\operatorname{At}_{\\mathcal {E}}^{\\nabla ^{\\mathcal {E}}}(X,e) = Q_{\\mathcal {E}}(\\nabla ^{\\mathcal {E}}_X e) - \\nabla ^{\\mathcal {E}}_{Q_\\mathcal {L}(X)}e - (-1)^{\\vert X\\vert }\\nabla ^{\\mathcal {E}}_X (Q_{\\mathcal {E}}e), \\;\\;\\forall X \\in \\Gamma (\\mathcal {L}), e \\in \\Gamma (\\mathcal {E}).$ Its cohomology class $\\operatorname{At}_{\\mathcal {E}} \\in H^1(\\Gamma (\\mathcal {L}^\\vee \\otimes \\operatorname{End}(\\mathcal {E})))$ , which is independent of the choice of $\\mathcal {L}$ -connections, is called the Atiyah class of the dg vector bundle $\\mathcal {E}$ with respect to the dg Lie algebroid $\\mathcal {L}$ .",
"Meanwhile, there associates a $(\\Gamma (\\mathcal {L}^\\vee ),Q_{\\mathcal {L}^\\vee })$ -valued derivation of the cdga $(C^\\infty (\\mathcal {M}),Q_\\mathcal {M})$ defined by $\\delta _\\mathcal {L}: \\; C^\\infty (\\mathcal {M}) \\xrightarrow{} \\Omega ^1(\\mathcal {M})\\xrightarrow{} \\Gamma (\\mathcal {L}^\\vee ),$ where $Q_{\\mathcal {L}^\\vee }$ is induced from the differential $Q_\\mathcal {L}$ on $\\mathcal {L}$ .",
"The fact that $\\delta _\\mathcal {L}$ commutes with the two differentials $Q_{\\mathcal {M}}$ and $Q_{\\mathcal {L}^\\vee }$ follows from (3) of Definition REF .",
"The section space of a dg vector bundle $\\mathcal {E}$ gives rise to a dg $(C^\\infty (\\mathcal {M}),Q_\\mathcal {M})$ -module ${E}:=(\\Gamma (\\mathcal {E}),Q_\\mathcal {E})$ .",
"It is obvious that a $\\delta _\\mathcal {L}$ -connection $\\nabla ^{\\delta _\\mathcal {L}}$ on ${E}= \\Gamma (\\mathcal {E})$ is equivalently to a Lie algebroid $\\mathcal {L}$ -connection $\\nabla ^\\mathcal {L}$ on the graded vector bundle $\\mathcal {E}$ .",
"Comparing Equations (REF ) and (REF ), we have the following Proposition 2.18 The Atiyah class $\\operatorname{At}_{\\mathcal {E}}$ of the dg vector bundle $\\mathcal {E}$ with respect to the dg Lie algebroid $\\mathcal {L}$ coincides, up to a minus sign, with the twisted Atiyah class $\\operatorname{At}_{E}^{\\delta _\\mathcal {L}}$ of the dg $(C^\\infty (\\mathcal {M}),Q_\\mathcal {M})$ -module ${E}=(\\Gamma (\\mathcal {E}),Q_\\mathcal {E})$ , where the dg derivation $\\delta _\\mathcal {L}$ is given by Equation (REF ).",
"Lie pairs By a Lie pair $(L,A)$ , we mean two Lie algebroids $L$ and $A$ over the same smooth manifold $M$ such that $A \\subset L$ is a Lie subalgebroid.",
"The quotient bundle $B=L/A$ carries a natural flat (Lie algebroid) $A$ -connection, called the Bott $A$ -module structure.",
"Let us recall the Atiyah class of the Lie pair $(L,A)$ defined in .",
"First of all, there is a short exact sequence of vector bundles over $M$ , $0 \\rightarrow A \\xrightarrow{} L \\xrightarrow{} B \\rightarrow 0.$ Choose a splitting of Sequence (REF ), i.e., a vector bundle injection $j:~ B \\rightarrow L$ , which determines a bundle projection $\\operatorname{pr}_A:~ L \\rightarrow A$ such that $\\operatorname{pr}_A \\circ i &= \\operatorname{id}_A, & \\operatorname{pr}_B \\circ j &= \\operatorname{id}_B, & i \\circ \\operatorname{pr}_A + j \\circ \\operatorname{pr}_B &= \\operatorname{id}_L.$ Using this splitting, one could identify $L$ with $ A\\oplus B$ .",
"Meanwhile, for each $A$ -module $(E,\\partial _A^E)$ , where $E$ is a vector bundle over $M$ and $\\partial _A^E$ is a flat $A$ -connection on $E$ , choose an $L$ -connection $\\nabla ^L$ on $E$ extending the given flat $A$ -connection.",
"Then there associates a 1-cocycle $\\alpha _E^{\\nabla ^L} \\in \\Gamma (A^\\vee \\otimes B^\\vee \\otimes \\operatorname{End}(E))$ , called the Atiyah cocycle, of the Lie algebroid $A$ valued in the $A$ -module $B^\\vee \\otimes \\operatorname{End}(E)$ : $\\alpha _E^{\\nabla ^L}(a,b)e &:= \\nabla _{a}\\nabla ^L_{j(b)}e - \\nabla ^L_{j(b)}\\nabla _{a}e - \\nabla ^L_{[a,j(b)]}e,$ for all $a \\in \\Gamma (A), b \\in \\Gamma (B)$ and $e \\in \\Gamma (E)$ .",
"The cohomology class $\\alpha _{E} = [\\alpha _E^{\\nabla ^L}] \\in H_{\\mathrm {CE}}^1(A,B^\\vee \\otimes \\operatorname{End}(E))$ does not depend on the choice of $j$ and $\\nabla ^L$ , and is called the Atiyah class of the $A$ -module $E$ with respect to the Lie pair $(L,A)$ .",
"From the Lie pair $(L,A)$ , we get a cdga $\\Omega ^\\bullet _A = (\\Gamma (\\wedge ^\\bullet A^\\vee ),d_{A})$ , and a dg $\\Omega ^\\bullet _A$ -module $\\Omega _A^\\bullet (B^\\vee ):= (\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B^\\vee ),\\partial _A )$ , where $\\partial _A$ is the $A$ -module structure dual to the Bott $A$ -module structure on $B$ .",
"Here the degree convention is that $\\Gamma ({B^\\vee })$ concentrates in degree zero.",
"Fixing a splitting $j$ of Sequence (REF ), we construct an $\\Omega _A^\\bullet (B^\\vee )$ -valued derivation $\\delta _j$ of $\\Omega ^\\bullet _A$ , i.e., a map $\\delta _j:~\\Gamma (\\wedge ^\\bullet A^\\vee ) \\rightarrow \\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B^\\vee ).$ As a degree zero derivation of the graded $\\mathbb {K}$ -algebra $\\Gamma (\\wedge ^\\bullet A^\\vee )$ , $\\delta _j$ is fully determined by its action on its generators, i.e.",
"elements in $C^\\infty (M)$ and $\\Gamma (A^\\vee )$ — Define $\\delta _j:~ \\;&C^\\infty (M) \\xrightarrow{} \\Gamma (L^\\vee ) \\xrightarrow{} \\Gamma (B^\\vee ), \\\\\\delta _j:~ \\;&\\Gamma ( A^\\vee ) \\xrightarrow{} \\Gamma ( L^\\vee ) \\xrightarrow{} \\Gamma (\\wedge ^2 L^\\vee ) \\xrightarrow{} \\Gamma ( L^\\vee \\otimes L^\\vee ) \\xrightarrow{} \\Gamma ( A^\\vee \\otimes B^\\vee ),$ where $d_L:~ \\Gamma (\\wedge ^\\bullet L^\\vee ) \\rightarrow \\Gamma (\\wedge ^{\\bullet +1}L^\\vee )$ is the Chevalley-Eilenberg differential of the Lie algebroid $L$ .",
"A straightforward verification shows that $\\delta _j$ is compatible with the differentials and thus is an $\\Omega ^\\bullet _A(B^\\vee )$ -valued dg derivation of $\\Omega ^\\bullet _A$ .",
"Note that $\\delta _j$ depends on a choice of a splitting $j$ of Sequence (REF ).",
"However, we have Proposition 2.21 The $\\Omega _A^\\bullet (B^\\vee )$ -valued dg derivations $\\delta _j$ of $\\Omega _A^\\bullet $ associated with different splittings of Sequence (REF ) are homotopic to each other.",
"Given two splittings $j$ and $j^\\prime $ of Sequence (REF ), their difference is a bundle map $j^\\prime -j:~ B \\rightarrow A$ .",
"Define a degree $(-1)$ derivation $h:~\\Gamma (\\wedge ^{\\bullet }A^\\vee ) \\rightarrow \\Gamma (\\wedge ^{\\bullet -1}A^\\vee \\otimes B^\\vee )$ by setting $h|_{C^\\infty (M)}=0,\\qquad h|_{\\Gamma (A^\\vee )}=(j^\\prime -j)^\\vee .$ It follows from direct verifications that $\\delta _{j^\\prime } - \\delta _j = [\\partial _A,h]: \\Gamma (\\wedge ^{\\bullet }A^\\vee ) \\rightarrow \\Gamma (\\wedge ^{\\bullet }A^\\vee \\otimes B^\\vee ).",
"$ This proves that $\\delta _j\\sim \\delta _{j^\\prime }$ .",
"Let $(E,\\partial _A^E)$ be an $A$ -module.",
"There induces a dg $\\Omega ^\\bullet _A$ -module ${E}:= (\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes E),\\partial _A^E)$ .",
"Proposition 2.22 The Atiyah class $\\alpha _E$ of the $A$ -module $E$ with respect to the Lie pair $(L,A)$ coincides with the twisted Atiyah class $\\operatorname{At}^{\\delta _j}_{E}$ of the dg $\\Omega _A^\\bullet $ -module ${E}$ , where the dg derivation $\\delta _j$ is given as in Equation (REF ).",
"First of all, the spaces where the two Atiyah classes live are exactly the same, i.e., $(\\alpha _E \\in )~ H^1_{\\mathrm {CE}}(A,B^\\vee \\otimes \\operatorname{End}(E)) = H^1(\\Omega _A^\\bullet , \\Omega _A^\\bullet (B^\\vee ) \\otimes _{\\Omega _A^\\bullet } \\operatorname{End}_{\\Omega _A^\\bullet }({E})) ~(\\ni \\operatorname{At}^{\\delta _j}_{E}).$ According to Proposition REF , to find the twisted Atiyah class $\\operatorname{At}^{\\delta _j}_{E}$ , one may use a $\\delta _j$ -connection $\\nabla ^{\\delta _j}$ on ${E}$ , which is determined by its restriction on $\\Gamma (E)$ : $\\nabla ^{\\delta _j}\\!\\mid _E:~ \\Gamma (E) \\rightarrow \\Gamma (B^\\vee ) \\otimes \\Gamma (E).$ This is equivalent to an $L$ -connection $\\nabla ^L$ on $E$ extending the given flat $A$ -connection by setting $\\nabla ^L_{a+b} = \\nabla _a + \\nabla ^{\\delta _j}_b\\!\\mid _E,\\qquad \\forall a+b\\in L\\cong A\\oplus B.$ The two associated Atiyah cocycles coincide by straightforward computations, i.e., $\\operatorname{At}_{E}^{\\nabla ^{\\delta _j}} = \\alpha _E^{\\nabla ^L}$ .",
"As a consequence of Propositions REF and REF , both Atiyah classes of dg Lie algebroids and those of Lie pairs arise from Atiyah classes of cdgas.",
"In particular, we have Corollary 2.24 Let $A$ be a Lie algebroid and $E$ an $A$ -module.",
"Denote by $\\mathcal {E}$ the corresponding dg vector bundle over $(A[1],d_A)$ .",
"If the Atiyah class of the dg vector bundle $\\mathcal {E}$ with respect to the dg Lie algebroid $T(A[1])$ vanishes, then the Atiyah class of $E$ with respect to any Lie pair $(L,A)$ vanishes.",
"Functoriality We now study functorial properties of twisted Atiyah classes.",
"Let $\\operatorname{H}(\\operatorname{dg}{A})$ denote the homology category of dg ${A}$ -modules: Objects in $\\operatorname{H}(\\operatorname{dg}{A})$ are dg ${A}$ -modules, and morphisms in $\\operatorname{H}(\\operatorname{dg}{A})$ are dg ${A}$ -module morphisms modulo homotopy .",
"Let ${A}\\xrightarrow{} \\Omega $ be a dg derivation.",
"For each object ${E}$ in $\\operatorname{H}(\\operatorname{dg}{A})$ , by Definition REF , the twisted Atiyah class $\\operatorname{At}^\\delta _{E}\\in H^1({A}, \\Omega \\otimes _{A}\\operatorname{End}_{A}({E}))\\cong \\operatorname{Hom}^{1}_{\\operatorname{H}(\\operatorname{dg}{A})}({E},\\Omega \\otimes _{A}{E})$ is a degree 1 morphism in the category $\\operatorname{H}(\\operatorname{dg}{A})$ .",
"This identification defines a functorial transformation.",
"In fact, when the dg derivation $\\delta $ is fixed, the $\\delta $ -twisted Atiyah class is a functorial transformation on $\\operatorname{H}(\\operatorname{dg}{A})$ from the identity functor $\\operatorname{id}$ to the tensor functor $\\Omega \\otimes _{A}-$ : Proposition 2.25 Let ${E}$ and ${F}$ be dg ${A}$ -modules, $\\lambda \\in \\operatorname{Hom}_{\\operatorname{H}(\\operatorname{dg}{A})}({E},{F})$ .",
"The following diagram commutes in the category $\\operatorname{H}(\\operatorname{dg}{A})$ : ${{E}[r]^-{\\operatorname{At}^\\delta _{E}} [d]_-{(-1)^{\\vert \\lambda \\vert }\\lambda } & \\Omega \\otimes _{A}{E}[d]^-{\\operatorname{id}_\\Omega \\otimes _{A}\\lambda } \\\\{F}[r]^-{\\operatorname{At}^\\delta _{F}} & \\Omega \\otimes _{A}{F}.",
"}$ Let us first show the non-twisted case.",
"Namely, the following diagram commutes in $\\operatorname{H}(\\operatorname{dg}{A})$ : ${{E}[r]^-{\\operatorname{At}_{E}} [d]_-{(-1)^{\\vert \\lambda \\vert }\\lambda } & \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{E}[d]^-{\\operatorname{id}\\otimes _{A}\\lambda } \\\\{F}[r]^-{\\operatorname{At}_{F}} & \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{F}.",
"}$ In fact, this can be directly verified.",
"We choose connections $\\blacktriangledown ^{E}$ and $\\blacktriangledown ^{F}$ , respectively, on ${E}$ and ${F}$ .",
"For simplicity, they are both denoted by $\\blacktriangledown $ .",
"Then $&\\quad (\\operatorname{id}\\otimes _{A}\\lambda ) \\circ \\operatorname{At}^{\\blacktriangledown }_{E}- (-1)^{\\vert \\lambda \\vert }\\operatorname{At}_{F}^{\\blacktriangledown } \\circ \\lambda \\\\&= (\\operatorname{id}_\\Omega \\otimes _{A}\\lambda ) \\circ (\\blacktriangledown \\circ \\partial _{A}- \\partial _{A}\\circ \\blacktriangledown ) - (-1)^{\\vert \\lambda \\vert }(\\blacktriangledown \\circ \\partial _{A}- \\partial _{A}\\circ \\blacktriangledown ) \\circ \\lambda \\\\&= ((\\operatorname{id}\\otimes _{A}\\lambda ) \\circ \\blacktriangledown - \\blacktriangledown \\circ \\lambda ) \\circ \\partial _{A}- (-1)^{\\vert \\lambda \\vert }\\partial _{A}\\circ ((\\operatorname{id}\\otimes _{A}\\lambda ) \\circ \\blacktriangledown - \\blacktriangledown \\circ \\lambda ).$ The map $(\\operatorname{id}\\otimes _{A}\\lambda ) \\circ \\blacktriangledown - \\blacktriangledown \\circ \\lambda :~{E}\\rightarrow \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{{A}} {F}$ is actually $ {{A}}$ -linear, by direct verifications.",
"Thus the two maps $(\\operatorname{id}\\otimes _{A}\\lambda ) \\circ \\operatorname{At}^{\\blacktriangledown }_{E}$ and $ (-1)^{\\vert \\lambda \\vert }\\operatorname{At}_{F}^{\\blacktriangledown } \\circ \\lambda $ are only differed by an exact term.",
"Composing with the dg ${A}$ -module morphism $\\bar{\\delta }:~\\Omega _{{A}\\mid \\mathbb {K}}^1 \\rightarrow \\Omega $ induced from the dg derivation $\\delta $ , we accomplish a commutative diagram in $\\operatorname{H}(\\operatorname{dg}{A})$ : ${{E}[r]^-{\\operatorname{At}_{E}} [d]_-{(-1)^{\\vert \\lambda \\vert }\\lambda } & \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{E}[d]^-{\\operatorname{id}\\otimes _{A}\\lambda } [r]^-{\\bar{\\delta }\\otimes _{A}\\operatorname{id}} & \\Omega \\otimes _{A}{E}[d]^-{\\operatorname{id}\\otimes _{A}\\lambda }\\\\{F}[r]^-{\\operatorname{At}_{F}} & \\Omega _{{A}\\mid \\mathbb {K}}^1 \\otimes _{A}{F}[r]^-{\\bar{\\delta }\\otimes _{A}\\operatorname{id}} &\\Omega \\otimes _{A}{F}.",
"}$ Now we study how Atiyah classes vary when twisted by different dg derivations.",
"So we need the category of dg derivations, denoted by $\\operatorname{dgDer}_{A}$ , whose objects are dg derivations ${A}\\xrightarrow{} \\Omega $ as in Definition REF , and whose morphisms are defined as follows: Definition 2.28 A morphism $\\phi $ from ${A}\\xrightarrow{} \\Omega $ to ${A}\\xrightarrow{} \\Omega ^\\prime $ is a morphism $\\phi : \\Omega \\rightarrow \\Omega ^\\prime $ of dg ${A}$ -modules such that $\\delta ^\\prime = \\phi \\circ \\delta : {A}\\rightarrow \\Omega ^\\prime .$ Now let us fix a dg ${A}$ -module ${E}$ .",
"We have the constant functor $(- \\mapsto {E})$ and the tensor functor $- \\otimes _{A}{E}$ , both from the category $\\operatorname{dgDer}_{A}$ of dg derivations to the homology category $\\operatorname{H}(\\operatorname{dg}{A})$ of dg ${A}$ -modules.",
"The Atiyah class is a functorial transformation from $(-\\mapsto {E})$ to $- \\otimes _{A}{E}$ : Proposition 2.29 Given a morphism $\\phi :~ ({A}\\xrightarrow{} \\Omega ) \\rightarrow ({A}\\xrightarrow{} \\widetilde{\\Omega })$ of dg derivations and a dg ${A}$ -module ${E}$ , let $\\operatorname{At}^\\delta _{E}$ and $\\operatorname{At}^{\\tilde{\\delta }}_{E}$ be the Atiyah classes of ${E}$ twisted, respectively, by ${A}\\xrightarrow{} \\Omega $ and ${A}\\xrightarrow{} \\widetilde{\\Omega }$ .",
"Then the following diagram commutes in $\\operatorname{H}(\\operatorname{dg}{A})$ : ${{E}[r]^-{\\operatorname{At}^\\delta _{E}} [d]_-{\\operatorname{id}_{E}} & \\Omega \\otimes _{A}{E}[d]^-{\\phi \\otimes _{A}\\operatorname{id}_{E}} \\\\{E}[r]^-{\\operatorname{At}^{\\tilde{\\delta }}_{E}} & \\widetilde{\\Omega } \\otimes _{A}{E}.",
"}$ The proof is easy and thus omitted.",
"Combining the previous two propositions, we have Theorem 2.30 With the same assumptions as in Propositions REF and REF , the following diagram in $\\operatorname{H}(\\operatorname{dg}{A})$ commutes: ${{E}[r]^-{\\operatorname{At}^{\\delta }_{E}} [d]_-{(-1)^{\\vert \\lambda \\vert }\\lambda } & \\Omega \\otimes _{A}{E}[d]^-{\\phi \\otimes _{A}\\lambda } \\\\{F}[r]^-{\\operatorname{At}^{\\tilde{\\delta }}_{F}} & \\widetilde{\\Omega } \\otimes _{A}{F}.",
"}$ The Kapranov functor In this section, we explore higher algebraic structures, called Kapranov Leibniz$_\\infty [1]$ algebras, induced from a dg derivation of a cdga ${A}$ .",
"Our main goal is to show that there exists a contravariant functor, called the Kapranov functor, from the category of dg derivations to the category of Leibniz$_\\infty [1]$ -algebras over ${A}$ .",
"Leibniz$_\\infty [1]$ algebras We recall some basic notions of homotopy Leibniz algebras (c.f.AP,CSX).",
"In what follows, all tensor products $\\otimes $ without adoration are assumed to be over $\\mathbb {K}$ .",
"Definition 3.1 A Leibniz$_\\infty [1]$ algebra (over $\\mathbb {K}$ ) is a graded $\\mathbb {K}$ -vector space $V = \\oplus _{n \\in \\mathbb {Z}}V^n$ , together with a sequence $\\lbrace \\lambda _k:~ \\otimes ^k V \\rightarrow V\\rbrace _{k \\ge 1}$ of degree 1, $\\mathbb {K}$ -multilinear maps satisfying $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (k-j)}\\vert } \\\\&\\lambda _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_n)=0,$ for all $n \\ge 1$ and all homogeneous elements $v_i \\in V$ , where $\\operatorname{sh}(p,q)$ denotes the set of $(p,q)$ -shuffles ($p,q \\ge 0$ ), and $\\epsilon (\\sigma )$ is the Koszul sign of $\\sigma $ .",
"Definition 3.2 A morphism of Leibniz$_\\infty [1]$ algebras from $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ to $(V^{\\prime },\\lbrace \\lambda ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a sequence $\\lbrace f_k:~ V^{\\otimes k} \\rightarrow V^{\\prime }\\rbrace _{k \\ge 1}$ of degree 0, $\\mathbb {K}$ -multilinear maps, satisfying the following compatibility condition: $&\\quad \\sum _{k+p \\le n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\dagger ^{\\sigma }_k}f_{n-p}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)}, \\lambda _{p+1}(b_{\\sigma (k+1)}, \\cdots ,b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_{n}) \\\\&= \\sum _{q \\ge 1}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^q) \\lambda _q^{\\prime }(f_{\\vert I^1\\vert }(b_{I^1}), \\cdots ,f_{\\vert I^q\\vert }(b_{I^q})),$ for all $n \\ge 1$ , where $\\dagger ^{\\sigma }_k = \\sum _{i=1}^k\\vert b_{\\sigma (i)}\\vert $ , $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n)} = \\lbrace 1,\\cdots ,n\\rbrace $ , and $(b_{I^j}) = (b_{i^j_1},\\cdots ,b_{i^j_{\\vert I^j\\vert }})$ for all $1\\le j \\le q$ .",
"In the above definition, the first component $f_1:~ (V,\\lambda _1) \\rightarrow (V^{\\prime },\\lambda _1^{\\prime })$ , called the tangent morphism, is a morphism of cochain complexes.",
"We call the Leibniz$_\\infty [1]$ morphism $f_\\bullet :~ (V,\\lambda _\\bullet ) \\rightarrow (V^{\\prime },\\lambda ^{\\prime }_\\bullet )$ a quasi-isomorphism (resp.",
"an isomorphism) if $f_1$ is a quasi-isomorphism (resp.",
"an isomorphism).",
"In fact, there is a standard way to find its quasi-inverse (resp.",
"inverse) $f^{-1}_\\bullet :~ (V^{\\prime },\\lambda ^{\\prime }_\\bullet ) \\rightarrow (V,\\lambda _\\bullet )$ (see ).",
"Definition 3.3 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ algebra.",
"A $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -module is a graded $\\mathbb {K}$ -vector space $W$ together with a sequence $\\lbrace \\mu _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W\\rbrace _{k \\ge 1}$ of degree 1, $\\mathbb {K}$ -multilinear maps satisfying the identities $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\dagger ^\\sigma _{k-j}} \\\\&\\qquad \\mu _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_{n-1},w) \\\\&\\qquad +\\sum _{1\\le j \\le n}\\sum _{\\sigma \\in \\operatorname{sh}(k,j)}\\epsilon (\\sigma )(-1)^{\\dagger ^\\sigma _{n-j}} \\mu _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (n-j)},\\mu _{j}(v_{\\sigma (n-j+1)}, \\cdots ,v_{\\sigma (n-1)}, w))=0,$ for all $n \\ge 1$ and all homogeneous vectors $v_1,\\cdots ,v_{n-1} \\in V, w \\in W$ , where $\\dagger ^\\sigma _j = \\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (j)}\\vert $ for all $j \\ge 0$ .",
"Definition 3.2 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ algebra.",
"A morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -modules from $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ to $(W^{\\prime },\\lbrace \\mu ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a sequence $\\lbrace \\psi _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W^{\\prime }\\rbrace _{k \\ge 1}$ of degree 0, $\\mathbb {K}$ -multilinear maps satisfying the identity $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (k-j)}\\vert } \\\\&\\quad \\psi _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_{n-1},w) \\\\&\\qquad +\\sum _{1\\le j \\le n}\\sum _{\\sigma \\in \\operatorname{sh}(k,j)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (n-j)}\\vert } \\psi _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (n-j)},\\mu _{j}(v_{\\sigma (n-j+1)},\\cdots , v_{\\sigma (n-1)}, w)) \\\\&= \\sum _{p \\ge 0}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^{p+1} = \\mathbb {N}^{(n-1)} \\\\ I^1,\\cdots ,I^{p+1} \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^{p+1}_{\\vert I^{p+1}\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^{p+1}) \\mu ^{\\prime }_{p+1}(\\lambda _{\\vert I^1\\vert }(v_{I^1}),\\cdots ,\\lambda _{\\vert I^p\\vert }(v_{I^p}), \\psi _{\\vert I^{p+1}\\vert +1}(v_{I^{p+1}},w)),$ for each $n \\ge 1$ and all homogeneous vectors $v_1,\\cdots ,v_{n-1} \\in V, w \\in W$ .",
"Here $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n-1)} = \\lbrace 1,\\cdots ,n-1\\rbrace $ , and $(v_{I^j}) = (v_{i^j_1},\\cdots ,v_{i^j_{\\vert I^j\\vert }})$ for all $1 \\le j \\le p+1$ .",
"In this note, we are particularly interested in Leibniz$_\\infty [1]$ algebras over a cdga ${A}$ (or Leibniz$_\\infty [1]$ ${A}$ -algebras).",
"Definition 3.0 A Leibniz$_\\infty [1]$ ${A}$ -algebra is a Leibniz$_\\infty [1]$ algebra $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ (in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ ) such that the cochain complex $(V,\\lambda _1)$ is a dg ${A}$ -module and all higher brackets $\\lambda _k: \\otimes ^k V \\rightarrow V$ ($k\\ge 2$ ) are ${A}$ -multilinear.",
"A morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ to $(V^{\\prime },\\lbrace \\lambda ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a morphism $\\lbrace f_k:V^{\\otimes k}\\rightarrow V^{\\prime }\\rbrace _{k \\ge 1}$ (in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ ) such that all structure maps $\\lbrace f_k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"In particular, its tangent morphism $f_1:~ (V,\\lambda _1) \\rightarrow (V^\\prime ,\\lambda ^\\prime _1)$ is a dg ${A}$ -module morphism.",
"Such a morphism $f_\\bullet :~ (V,\\lambda _\\bullet ) \\rightarrow (V^{\\prime },\\lambda ^{\\prime }_\\bullet )$ is called a quasi-isomorphism (resp.",
"an isomorphism) if its tangent morphism $f_1$ is a quasi-isomorphism (resp.",
"an isomorphism) of dg ${A}$ -modules.",
"Denote the category of Leibniz$_\\infty [1]$ ${A}$ -algebras by $\\operatorname{Leib}_\\infty ({A})$ .",
"It is a subcategory of the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ .",
"There are analogous notions of modules of a Leibniz$_\\infty [1]$ ${A}$ -algebra and their morphisms: Definition 3.1 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"A $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -module is a $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -module $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ (in the category of Leibniz$_\\infty [1]$ modules over $\\mathbb {K}$ ) such that $(W,\\mu _1)$ is a DG ${A}$ -module and all higher structure maps $\\lbrace \\mu _k\\rbrace _{k\\ge 2}$ are ${A}$ -multilinear.",
"A morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -modules from $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ to $(W^{\\prime },\\lbrace \\mu ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -modules $\\lbrace \\psi _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W^{\\prime }\\rbrace _{k \\ge 1}$ (in the category of Leibniz$_\\infty [1]$ modules over $\\mathbb {K}$ ) such that all maps $\\lbrace \\psi _k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"It follows that the collection of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -modules and their morphisms form a category.",
"The Kapranov functor In this section, we generalize Kapranov's construction of an $L_\\infty $ algebra structure and Chen-Stiénon-Xu's construction of a Leibniz$_\\infty [1]$ algebra structure in the setting of a dg derivation ${A}\\xrightarrow{} \\Omega $ of a cdga ${A}$ .",
"Kapranov Leibniz$_\\infty [1]$ algebras Let ${E}$ be a graded ${A}$ -module with a $\\delta $ -connection $\\nabla $ .",
"For each homogeneous $b \\in \\mathcal {B}$ , there is a degree $\\vert b\\vert $ derivation on the reduced tensor algebra $T({E})$ (over ${A}$ ) defined by $\\nabla _b(e_1 \\otimes \\cdots \\otimes e_n) = \\sum _{i=1}^{n}(-1)^{\\vert b\\vert \\ast _{i-1}} e_1 \\otimes \\cdots \\nabla _b e_i \\otimes \\cdots e_n,$ for all homogeneous $e_i \\in {E}$ , where $\\ast _{i} = \\sum _{j=1}^i\\vert e_j\\vert $ .",
"Let ${E}$ and ${F}$ be two graded ${A}$ -modules with $\\delta $ -connections $\\nabla ^{{E}}$ and $\\nabla ^{{F}}$ , respectively.",
"For $b \\in \\mathcal {B}$ and $\\lambda \\in \\operatorname{Hom}_{{A}}({E},{F})$ , there associates the derivation $\\nabla _b(\\lambda ) = [\\nabla _b,\\lambda ] = \\nabla ^{{F}}_b \\circ \\lambda - (-1)^{\\vert b\\vert \\vert \\lambda \\vert }\\lambda \\circ \\nabla ^{{E}}_b:~ {E}\\rightarrow {F}.$ It follows from a direct verification that $\\nabla _b(\\lambda ) \\in \\operatorname{Hom}_{{A}}({E},{F})$ .",
"Choose a $\\delta $ -connection on $\\mathcal {B}$ .",
"There associates a sequence of degree 1 maps $ \\mathcal {R}^\\nabla _k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}, k \\ge 1 $ defined as follows: $\\mathcal {R}^\\nabla _1 = \\partial _{A}:~ \\mathcal {B}\\rightarrow \\mathcal {B}$ ; $\\mathcal {R}^\\nabla _2$ is specified by the associated twisted Atiyah cocycles $\\operatorname{At}_\\mathcal {B}^\\nabla $ ; $\\lbrace \\mathcal {R}^\\nabla _{k+1}:~ \\mathcal {B}^{\\otimes (k+1)} \\rightarrow \\mathcal {B}\\rbrace _{k\\ge 2}$ are defined recursively by $\\mathcal {R}^\\nabla _{k+1}=\\nabla (\\mathcal {R}^\\nabla _{k})$ .",
"Explicitly, we have $&\\mathcal {R}^\\nabla _{k+1}(b_0,b_1,\\cdots ,b_{k}) = (-1)^{\\vert b_0\\vert }[\\nabla _{b_0},\\mathcal {R}^\\nabla _k] (b_1,\\cdots ,b_k), \\;\\;\\forall b_i \\in \\mathcal {B}.$ Proposition 3.3 The ${A}$ -module $\\mathcal {B}$ , together with the sequence of operators $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ , is a Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"The 2-bracket $\\mathcal {R}^\\nabla _2$ is the twisted Atiyah cocycles $\\operatorname{At}_\\mathcal {B}^\\nabla $ , which is certainly ${A}$ -bilinear.",
"By the recursive construction of higher brackets $\\mathcal {R}^\\nabla _{k+1}$ ($k\\ge 2$ ) in Equation (REF ), they are all ${A}$ -multilinear as well.",
"So it suffices to verify that $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ satisfies Equation (REF ).",
"We argue by induction.",
"The $n=1$ case follows from the fact that $\\mathcal {R}^\\nabla _1 = \\partial _{A}$ is a differential, and the $n=2$ case follows from the fact that the $\\delta $ -twisted Atiyah cocycle $\\operatorname{At}^\\nabla _\\mathcal {B}$ is a $\\partial _{A}$ -cocycle.",
"Now assume that the identity (REF ) holds for some $n \\ge 2$ , i.e., $&\\quad -\\partial _{A}(\\mathcal {R}^\\nabla _n)(b_1,\\cdots ,b_n) = -\\partial _{A}(\\mathcal {R}^\\nabla _n(b_1,\\cdots ,b_n)) + \\sum _{i=1}^n(-1)^{\\ast _{i-1}} \\mathcal {R}^\\nabla _n(b_1,\\cdots ,\\partial _{A}b_i,\\cdots ,b_{n}) \\\\&=\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_{\\sigma (1)}\\vert + \\cdots + \\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j} (b_{\\sigma (k-j+1)},\\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n).$ Consider the $(n+1)$ case: We first compute $-\\partial _{A}(\\mathcal {R}^\\nabla _{n+1})(b_0,b_1,\\cdots ,b_n) &= \\partial _{A}([\\mathcal {R}^\\nabla _{n},\\nabla _{b_0}])(b_1,\\cdots ,b_n) + [\\mathcal {R}^\\nabla _n,\\nabla _{\\partial _{A}b_0}](b_1,\\cdots ,b_n) \\\\&= [\\partial _{A}(\\mathcal {R}^\\nabla _n),\\nabla _{b_0}](b_1,\\cdots ,b_n) + [\\mathcal {R}^\\nabla _n,\\mathcal {R}^\\nabla _2(b_0,-)](b_1,\\cdots ,b_n).$ Here we have used the recursive definition (REF ) in the first equality and Equation (REF ) in the second one.",
"We introduce $b^{(i)}_k = {\\left\\lbrace \\begin{array}{ll}b_k, &\\;\\;\\;\\text{if}\\; k \\ne i \\\\\\nabla _{b_0}b_i, &\\;\\;\\;\\text{if}\\; k = i.\\end{array}\\right.",
"}$ Then the first summand in Equation (REF ) is, $&\\quad [\\partial _{A}(\\mathcal {R}^\\nabla _n),\\nabla _{b_0}](b_1,b_2,\\cdots ,b_{n}) \\\\&=\\sum _{i=1}^{n}(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\partial _{A}(\\mathcal {R}^\\nabla _n) (b_1,\\cdots ,\\nabla ^\\mathcal {B}_{b_0}b_i, \\cdots ,b_{n}) - \\nabla _{b_0}(\\partial _{A}(\\mathcal {R}^\\nabla _n)(b_1,\\cdots ,b_{n})) \\;\\\\&\\qquad \\qquad \\text{by assumption~(\\ref {casen})} \\\\&= -\\sum _{i=1}^n\\sum _{p,q \\ge 2, p+q=n+1}\\sum _{k=q}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-q,q-1)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert \\ast _{i-1}+\\vert b^{(i)}_{\\sigma (1)}\\vert + \\cdots +\\vert b^{(i)}_{\\sigma (k-q)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{p}(b^{(i)}_{\\sigma (1)},\\cdots , b^{(i)}_{\\sigma (k-q)}, \\mathcal {R}^\\nabla _{j}(b^{(i)}_{\\sigma (k-q+1)}, \\cdots ,b^{(i)}_{\\sigma (k-1)}, b^{(i)}_k), b^{(i)}_{k+1},\\cdots , b^{(i)}_n) \\\\&\\quad +\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_{\\sigma (1)}\\vert +\\cdots +\\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\nabla _{b_0}(\\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots , b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j}(b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n)) \\\\&=\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert + \\vert b_{\\sigma (1)}\\vert +\\cdots +\\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i+1}(b_0,b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j}(b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n)\\\\&\\quad + \\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{(\\vert b_{\\sigma (1)}\\vert +\\cdots + \\vert b_{\\sigma (k-j)}\\vert )(\\vert b_0\\vert +1)} \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j+1} (b_0,b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n).$ Here in the last step we used the recursive definition of $\\lbrace \\mathcal {R}^\\nabla _i\\rbrace $ .",
"Meanwhile, the second summand in Equation (REF ) is $&\\quad [\\mathcal {R}^\\nabla _n,\\mathcal {R}^\\nabla _2(b_0,-)](b_1,b_2,\\cdots ,b_{n}) \\\\&= \\sum _{i=1}^n(-1)^{(\\vert b_0\\vert +1)\\ast _{i-1}}\\mathcal {R}^\\nabla _n(b_1,\\cdots , \\mathcal {R}^\\nabla _2(b_0,b_i), \\cdots ,b_n) + (-1)^{\\vert b_0\\vert }\\mathcal {R}^\\nabla _2(b_0,\\mathcal {R}^\\nabla _n(b_1,\\cdots , b_{n})).$ Substituting Equations (REF ) and (REF ) into Equation (REF ), we see that Equation (REF ) holds for the case $(n+1)$ .",
"This proves that $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ satisfies Equation (REF ) for all $n \\ge 1$ .",
"The Leibniz$_\\infty [1] {A}$ -algebra $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ will be denoted by $\\operatorname{Kap}^c(\\delta )$ .",
"Here the superscript $c$ is to remind the reader that this Leibniz$_\\infty [1] {A}$ -algebra is defined via a particular $\\delta $ -connection on $\\mathcal {B}$ .",
"Remark 3.-6 This method is originated from Kapranov's construction of $L_\\infty $ algebra structure on the shifted tangent complex $\\Omega _X^{0,\\bullet -1}(T^{1,0}X)$ of a compact Kähler manifold $X$ .",
"For this reason, we call $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ the Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"Functoriality Next, we show that the assignment of a Leibniz$_\\infty [1]$ ${A}$ -algebra to each pair of dg derivation ${A}\\xrightarrow{} \\Omega $ and a $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ is functorial: Proposition 3.-5 Let $\\phi $ be a morphism from ${A}\\xrightarrow{} \\Omega ^\\prime $ to ${A}\\xrightarrow{} \\Omega $ in the category $\\operatorname{dgDer}_{A}$ of dg derivations (see Definition REF ).",
"Let $\\mathcal {B}=\\Omega ^\\vee $ and $\\mathcal {B}^{\\prime }=(\\Omega ^{\\prime })^\\vee $ be their dual dg ${A}$ -modules.",
"For a $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ and a $\\delta ^\\prime $ -connection $\\nabla ^\\prime $ on $\\mathcal {B}^\\prime $ , there exists a morphism $f_\\bullet = \\lbrace f_k\\rbrace _{k \\ge 1}$ of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ to $(\\mathcal {B}^\\prime , \\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1})$ , whose first map is $f_1=\\phi ^\\vee $ .",
"In other words, we have the following commutative diagram ${({A}\\xrightarrow{} \\Omega ^\\prime ) [d]_-{\\phi } [r]^-{\\operatorname{Kap}^{c}} & (\\mathcal {B}^\\prime ,\\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1}) \\\\({A}\\xrightarrow{} \\Omega ) [r]^-{\\operatorname{Kap}^c} & (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k \\ge 1}).",
"[u]_-{\\operatorname{Kap}^c(\\phi ) = f_\\bullet }}$ Define a sequence of ${A}$ -multilinear maps $f_k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}^\\prime $ recursively by setting $f_1(b_1) &= \\phi ^\\vee (b_1), & f_{k+1}(b_0,\\cdots ,b_k) &= \\nabla _{f_1(b_0)}^\\prime f_k(b_1,\\cdots ,b_k) - f_k(\\nabla _{b_0}(b_1,\\cdots ,b_k)), $ for all $k \\ge 1$ and $b_i \\in \\mathcal {B}$ .",
"It is easy to verify that all the maps $\\lbrace f_k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"Now we show that $\\lbrace f_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\mathcal {R}_k^{\\nabla })$ to $(\\mathcal {B}^\\prime , \\mathcal {R}_k^{\\nabla ^\\prime })$ .",
"We argue by induction: First of all, the $n=1$ case is obvious, since $f_{1} = \\phi ^\\vee : \\mathcal {B}\\rightarrow \\mathcal {B}^\\prime $ is a morphism of dg modules.",
"Now assume that Equation (REF ) holds for some $n \\ge 1$ , i.e., $&\\quad \\sum _{k+p \\le n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\dagger ^{\\sigma }_k}f_{n-p}(b_{\\sigma (1)}, \\cdots ,b_{\\sigma (k)}, \\mathcal {R}^\\nabla _{p+1}(b_{\\sigma (k+1)}, \\cdots ,b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_{n}) \\\\&= \\sum _{q \\ge 1}\\sum _{\\begin{array}{c}I^1 \\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^q) \\mathcal {R}_q^{\\nabla ^\\prime }(f_{\\vert I^1\\vert }(b_{I^1}), \\cdots ,f_{\\vert I^q\\vert }(b_{I^q})),$ where $\\dagger ^{\\sigma }_k = \\sum _{i=1}^k\\vert b_{\\sigma (i)}\\vert $ , $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n)} = \\lbrace 1,\\cdots ,n\\rbrace $ , and $(b_{I^j}) = (b_{i^j_1},\\cdots ,b_{i^j_{\\vert I^j\\vert }})$ for all $1\\le j \\le q$ .",
"We proceed to show that Equation (REF ) holds for $(n+1)$ homogeneous inputs $\\lbrace b_0,\\cdots ,b_{n+1}\\rbrace $ .",
"For simplicity, we denote the left-hand side and the right-hand side of Equation $(\\ref {casenprime})$ by $\\operatorname{LHS}(b_1,\\cdots ,b_n)$ and $\\operatorname{RHS}(b_1,\\cdots ,b_n)$ , respectively.",
"We write $\\operatorname{LHS}(b_0,b_1,\\cdots ,b_n) = I_1 + I_2 + I_3$ as the sum of three parts, where $I_1 &= f_{n+1}(\\partial _{{A}}b_0,b_1,\\cdots ,b_n) = \\nabla ^\\prime _{\\partial _{A}f_1(b_0)}\\circ f_n(b_1,\\cdots ,b_n) - \\sum _{i=1}^n(-1)^{\\ast _{i-1}(\\vert b_0\\vert +1)} f_n(b_1,\\cdots ,\\nabla _{\\partial _{A}b_0}(b_i),\\cdots ,b_n),\\\\I_2 &= \\sum _{k+p =0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert +\\dagger ^{\\sigma }_k} f_{n-p+1}(b_0,b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)},\\mathcal {R}^\\nabla _{p+1} (b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n) \\\\&= (\\nabla ^\\prime _{f_1(b_0)}\\circ f_{n-p} - f_{n-p}\\circ \\nabla _{b_0})(b_{\\sigma (1)}, \\cdots , \\mathcal {R}^\\nabla _{p+1}(b_{\\sigma (k+1)}, \\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n),$ by Equation (REF ), and $I_3 &= \\sum _{k+p=0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{(\\vert b_0\\vert +1)\\dagger ^{\\sigma }_k }f_{n-p}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)},\\mathcal {R}^\\nabla _{p+2}(b_0, b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n),\\\\&=\\sum _{k+p=0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)} \\epsilon (\\sigma ) (-1)^{(\\vert b_0\\vert +1)\\dagger ^{\\sigma }_k} f_{n-p}(b_{\\sigma (1)},\\cdots , b_{\\sigma (k)},-[\\mathcal {R}^\\nabla _{p+1}, \\nabla _{b_0}](b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n) \\\\&\\quad \\quad +\\sum _{k=0}^{n-1}(-1)^{\\ast _{k}(\\vert b_0\\vert +1)} f_n(b_1,\\cdots ,\\nabla _{\\partial _{A}b_0}(b_{k+1}),\\cdots ,b_n),$ by Equations (REF ) and (REF ).",
"Summing them up, we have $\\operatorname{LHS}(b_0,b_1,\\cdots ,b_n)&= \\nabla ^\\prime _{\\partial _{A}(f_1(b_0))} f_n(b_1,\\cdots ,b_n) -\\sum _{i=1}^n(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\operatorname{LHS}(b_1, \\cdots ,\\nabla _{b_0}b_i,\\cdots ,b_n) \\\\&\\quad + \\nabla ^\\prime _{f_1(b_0)} \\operatorname{LHS}(b_1,\\cdots ,b_n).$ Meanwhile, $&\\quad \\operatorname{RHS}(b_0,b_1,\\cdots ,b_n) \\\\ \\nonumber &= \\sum _{q=1}^{n}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}} \\epsilon (I^1,\\cdots ,I^q) (\\mathcal {R}^{\\nabla ^\\prime }_{q+1}(b_0, f_{\\vert I^1\\vert }(b_{I^1}),\\cdots ,f_{\\vert I^q\\vert }(b_{I^q})) \\\\\\nonumber &\\qquad + \\mathcal {R}^{\\nabla ^\\prime }_{q} (f_{\\vert I^1\\vert +1}(b_0,b_{I^1}), \\cdots , f_{\\vert I^q\\vert }(b_{I^q})) + \\cdots + (-1)^{\\vert b_0\\vert (\\vert b_{I^1}\\vert +\\cdots + \\vert b_{I^{q-1}}\\vert )} \\mathcal {R}^{\\nabla ^\\prime }_{q} (f_{\\vert I^1\\vert }(b_{I^1}),\\cdots , f_{\\vert I^q\\vert +1}(b_0,b_{I^q}))) \\\\&\\qquad \\qquad \\qquad \\text{by Equations~(\\ref {Atiyah cocycle}),(\\ref {Rnabla}),(\\ref {phik})} \\\\ &=\\nabla ^\\prime _{f_1(b_0)} \\operatorname{RHS}(b_1,\\cdots ,b_n) + \\nabla ^\\prime _{\\partial _{A}(f_1(b_0))}f_n(b_1,\\cdots ,b_n)-\\sum _{i=1}^n(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\operatorname{RHS}(b_1,\\cdots , \\nabla _{b_0}b_i, \\cdots , b_n).$ Applying the induction assumption to Equations (REF ) and (REF ), we see that Equation (REF ) holds for all $(n+1)$ entries.",
"Thus Equation (REF ) holds for all $n \\ge 1$ .",
"This proves that $f = \\lbrace f_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras.",
"As a consequence, the Kapranov's construction defines a contravariant functor $\\operatorname{Kap}^c: \\operatorname{dgDer}_{A}\\rightarrow \\operatorname{Leib}_\\infty ({A})$ from the category $\\operatorname{dgDer}_{A}$ of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ -algebras.",
"Remark 3.-5 The reason that we restrict to work in the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ ${A}$ -algebras is as follows: If we treat $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ merely as a Leibniz$_\\infty [1]$ algebra over $\\mathbb {K}$ , it is always isomorphic to the trivial one $(B,\\lbrace \\partial _{A},0,0,\\cdots \\rbrace )$ (all higher brackets are zero).",
"In fact, one can build a sequences of degree 0 maps $\\phi _k:~\\; &\\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B},\\quad k \\ge 1,$ where $\\phi _1 = \\operatorname{id}_\\mathcal {B}$ , and $\\lbrace \\phi _{k+1}\\rbrace _{k\\ge 1}$ are defined recursively by $\\phi _{k+1}(b_0,\\cdots ,b_k) &= {\\nabla }_{b_0} \\circ \\phi _k (b_1,\\cdots ,b_k), \\;\\;\\;\\forall b_i \\in \\mathcal {B}.$ The set $\\lbrace \\phi _k:~ \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}\\rbrace _{k\\ge 1}$ defines an isomorphism of Leibniz$_\\infty [1]$ algebras from $(\\mathcal {B},\\lbrace \\partial _{A},0,0,\\cdots \\rbrace )$ to $(\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ .",
"The proof is similar to that of Proposition REF .",
"However, the maps $\\lbrace \\phi _k\\rbrace _{k\\ge 2}$ are not ${A}$ -multilinear.",
"Next, we stress the independence from the choice of connections in the definition of Kapranov functors.",
"For a dg derivation ${A}\\xrightarrow{} \\Omega $ , suppose that we have another $\\delta $ -connection $\\widetilde{\\nabla }$ on $\\mathcal {B}= \\Omega ^\\vee $ .",
"Denote the corresponding Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra by $\\operatorname{Kap}^{\\tilde{c}}(\\delta ) = (\\mathcal {B},\\mathcal {R}_k^{\\widetilde{\\nabla }})$ .",
"By Proposition REF , there exists an isomorphism $g^{\\nabla ,\\widetilde{\\nabla }}_\\bullet : \\operatorname{Kap}^c(\\delta ) \\rightarrow \\operatorname{Kap}^{\\tilde{c}}(\\delta )$ of Leibniz$_\\infty [1]$ ${A}$ -algebras, where $g^{\\nabla ,\\widetilde{\\nabla }}_1 = \\operatorname{id}_\\mathcal {B}$ , and $\\lbrace g^{\\nabla ,\\widetilde{\\nabla }}_{k+1}\\rbrace _{k\\ge 1}$ are defined recursively as follows: $g^{\\nabla ,\\widetilde{\\nabla }}_{k+1}(b_0,\\cdots ,b_k) &= (\\widetilde{\\nabla }_{b_0} \\circ g^{\\nabla ,\\widetilde{\\nabla }}_k - g^{\\nabla ,\\widetilde{\\nabla }}_k \\circ \\nabla _{b_0})(b_1,\\cdots ,b_k), \\;\\;\\;\\forall b_i \\in \\mathcal {B}.$ Moreover, via a straightforward verification, we have Lemma 3.-4 There exists a natural equivalence between Kapranov functors $\\operatorname{Kap}^c$ and $\\operatorname{Kap}^{\\tilde{c}}$ with respect to different connections.",
"In other words, for any morphism $\\phi : ({A}\\xrightarrow{} \\Omega ^\\prime ) \\rightarrow ({A}\\xrightarrow{} \\Omega )$ of dg derivations of ${A}$ , we have the following commutative diagram ${\\operatorname{Kap}^c(\\delta ) [d]_-{\\operatorname{Kap}^c(\\phi )} [r]^-{g^{\\nabla ,\\widetilde{\\nabla }}_\\bullet } & \\operatorname{Kap}^{\\tilde{c}}(\\delta ) [d]^-{\\operatorname{Kap}^{\\tilde{c}}(\\phi )} \\\\\\operatorname{Kap}^c(\\delta ^\\prime ) [r]^-{g^{\\nabla ^\\prime ,\\widetilde{\\nabla ^\\prime }}} & \\operatorname{Kap}^{\\tilde{c}}(\\delta ^\\prime ).",
"}$ By this natural equivalence, we are allowed to drop the superscript $c$ to obtain the following Theorem 3.-3 The Kapranov's construction defines a contravariant functor $\\operatorname{Kap}: \\operatorname{dgDer}_{A}\\rightarrow \\operatorname{Leib}_\\infty ({A})$ from the category $\\operatorname{dgDer}_{A}$ of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ -algebras.",
"Remark 3.-2 By the universal property, the Kähler differential ${A}\\xrightarrow{} \\Omega _{{A}\\mid \\mathbb {K}}^1$ is the initial object in the category $\\operatorname{dgDer}_{A}$ of dg derivations.",
"Thus the corresponding Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra on the tangent complex $T_{{A}\\mid \\mathbb {K}} = (\\Omega _{{A}\\mid \\mathbb {K}}^1)^\\vee $ of ${A}$ is the final object of the subcategory in $\\operatorname{Leib}_\\infty ({A})$ consisting of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras arising from dg derivations of ${A}$ .",
"Let ${A}\\xrightarrow{} \\Omega $ be a dg derivation of ${A}$ and ${E}$ a dg ${A}$ -module.",
"By a similar argument, ${E}$ carries a Leibniz$_\\infty [1]$ ${A}$ -module structure over $\\operatorname{Kap}(\\delta )$ .",
"Moreover, we have Theorem 3.-1 Given a dg derivation ${A}\\xrightarrow{} \\Omega $ of ${A}$ , there exists a functor from the category $\\mathrm {dg}{A}$ of dg ${A}$ -modules to the category of Leibniz$_\\infty [1]$ ${A}$ -modules over $\\operatorname{Kap}(\\delta )$ .",
"Leibniz algebra structures Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ ${A}$ -algebra as in Definition REF .",
"Then $(V,\\lambda _1=\\partial _{A})$ is a dg ${A}$ -module.",
"Its cohomology $H^\\bullet (V)$ is called the tangent cohomology of the Leibniz$_\\infty [1]$ ${A}$ -algebra $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ .",
"According to *Proposition 3.10, the (degree $(-1)$ shifted) tangent cohomology $H^\\bullet (V[-1])$ is a Leibniz algebra (over $\\mathbb {K}$ ), when equipped with the bracket $\\check{\\lambda }_2:~H^\\bullet (V[-1]) \\times H^\\bullet (V[-1]) \\rightarrow H^\\bullet (V[-1])$ $\\check{\\lambda }_2([x],[y]):=(-1)^{\\vert x\\vert }[\\lambda _2(x,y)],$ where $x,y\\in V$ are $\\lambda _1$ -closed.",
"In a similar fashion, if $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ is a $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -module as in Definition REF , then $(W,\\mu _1=\\partial _{A})$ is also a dg ${A}$ -module.",
"The cohomology $H^\\bullet (W)$ is a Leibniz module over the aforesaid Leibniz algebra $ H^\\bullet (V[-1])$ (both over $\\mathbb {K}$ ), when equipped with the action $ \\check{\\mu }_2:~H^\\bullet (V[-1]) \\times H^\\bullet (W) \\rightarrow H^\\bullet (W)$ $\\check{\\mu }_2([x],[w]):=(-1)^{\\vert x\\vert }[\\mu _2(x,w)],$ where $x\\in V,~ w\\in W$ are, respectively, $\\lambda _1$ - and $\\mu _1$ -closed elements.",
"As a consequence of Theorem REF , Theorems REF and REF , we have the following Corollary 3.0 Let $\\phi $ be a morphism of dg derivations from ${A}\\xrightarrow{} \\Omega ^\\prime $ to ${A}\\xrightarrow{} \\Omega $ and let $\\mathcal {B}$ and $\\mathcal {B}^\\prime $ be the dual dg ${A}$ -modules of $\\Omega $ and $\\Omega ^\\prime $ , respectively.",
"The (degree $(-1)$ shifted) cohomology space $H^\\bullet ({A},\\mathcal {B}[-1])$ is a Leibniz algebra, whose bracket ${\\Bigl [-,-\\Bigr ]}_\\mathcal {B}$ is induced by the $\\delta $ -twisted Atiyah class of $\\mathcal {B}$ : $\\Bigl [[b_1 ],[b_2 ]\\Bigr ]_\\mathcal {B}= (-1)^{\\vert b_1\\vert }\\operatorname{At}^\\delta _\\mathcal {B}([b_1],[b_2]),$ where $b_1,b_2 \\in \\mathcal {B}$ are $\\partial _{A}$ -closed elements.",
"Moreover, $\\phi ^\\vee :~\\mathcal {B}\\rightarrow \\mathcal {B}^\\prime $ induces a morphism of Leibniz algebras, i.e., $\\Bigl [\\phi ^\\vee (b_1 ),\\phi ^\\vee [b_2]\\Bigr ]_{\\mathcal {B}^\\prime } = \\phi ^\\vee (\\Bigl [[b_1 ],[b_2 ]\\Bigr ]_\\mathcal {B}).$ For any dg ${A}$ -module ${E}$ , there exists a representation of $H^\\bullet ({A},\\mathcal {B}[-1])$ on the cohomology space $H^\\bullet ({A},{E})$ , with the action map $-\\triangleright -$ induced by the $\\delta $ -twisted Atiyah class of ${E}$ : $[b] \\triangleright [e]= (-1)^{\\vert b\\vert }\\operatorname{At}^\\delta _{E}([b],[e]),$ where $b\\in \\mathcal {B}$ , $e\\in {E}$ are both $\\partial _{A}$ -closed elements.",
"Moreover, this assignment is functorial, i.e., for each dg ${A}$ -module morphism $\\lambda : {E}\\rightarrow {F}$ (of degree 0), $[b] \\triangleright \\lambda (e) = \\lambda ([b] \\triangleright [e]).$ Remark 3.1 According to *Theorem 3.4, the Atiyah class of a Lie pair $(L,A)$ induces a Lie algebra structure on the cohomology $H^\\bullet _{\\mathrm {CE}}(A, L/A[-1])$ .",
"A similar result holds for $L_\\infty $ algebra pairs .",
"However, it is not the case in general (see an example below).",
"It is natural to ask when the Leibniz algebra structure in Corollary REF could be refined to a Lie algebra structure.",
"We will investigate this question somewhere else.",
"Example 3.2 Let $\\mathcal {LM}$ be the category of linear maps .",
"A Lie algebra object in $\\mathcal {LM}$ is a triple $E \\xrightarrow{} \\mathfrak {g}$ , where $\\mathfrak {g}$ is a Lie algebra, $E$ is a left $\\mathfrak {g}$ -module, and $\\psi $ is a $\\mathfrak {g}$ -equivariant linear map.",
"Consider the cdga ${A}=C^\\bullet (\\mathfrak {g}) = (\\wedge ^\\bullet \\mathfrak {g}^\\vee ,d_{\\mathrm {CE}})$ and dg $C^\\bullet (\\mathfrak {g})$ -module $\\Omega =C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]) = (\\wedge ^{\\bullet }\\mathfrak {g}^\\vee \\otimes E^\\vee [-1], d_{\\mathrm {CE}} )$ , i.e., the Chevalley-Eilenberg cochain complex of the dual $\\mathfrak {g}$ -module $E^\\vee [-1]$ .",
"The $\\mathfrak {g}$ -equivariant map $E \\xrightarrow{} \\mathfrak {g}$ gives rise to a dg derivation of $C^\\bullet (\\mathfrak {g}) $ : $C^\\bullet (\\mathfrak {g}) \\xrightarrow{} C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]).$ The dual module of $\\Omega =C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]) $ is $\\mathcal {B}=C^{\\bullet }(\\mathfrak {g},E[1])$ .",
"One can take the trivial $\\delta $ -connection on $\\mathcal {B}$ : $\\nabla : \\mathcal {B}=C^\\bullet (\\mathfrak {g},E[1]) \\rightarrow \\Omega \\otimes _{A}\\mathcal {B}=C^{\\bullet }(\\mathfrak {g},E^\\vee [-1] \\otimes E[1]),$ defined by $\\nabla (\\omega \\otimes e) = \\delta (\\omega ) \\otimes e,\\;\\;\\forall \\omega \\in \\wedge ^\\bullet \\mathfrak {g}^\\vee , e \\in E.$ By Equation (REF ), the associated Atiyah cocycle is a degree 1 element $\\operatorname{At}^\\nabla _\\mathcal {B}\\in E^\\vee [-1]\\otimes E^\\vee [-1] \\otimes E[1]$ specified by $\\mathcal {R}^\\nabla _2=\\operatorname{At}^\\nabla _\\mathcal {B}(e_1,e_2) = -\\psi (e_1)e_2 ,\\;\\;\\;\\forall e_1,e_2 \\in E .$ It can be easily seen that higher structures $\\mathcal {R}^\\nabla _j=0$ for all $j\\ge 3$ .",
"Hence, the Kapranov Leibniz$_\\infty [1]$ $C^\\bullet (\\mathfrak {g})$ -algebra $\\mathcal {B}=C^\\bullet (\\mathfrak {g},E[1])$ is simply a dg Leibniz$[1]$ algebra in this case, or equivalently, $\\mathcal {B}[-1]=C^\\bullet (\\mathfrak {g},E)$ is a dg Leibniz algebra.",
"In particular, the subspace $E$ is a Leibniz algebra, recovering the result in .",
"By Corollary REF , there is a Leibniz algebra structure on the graded vector space $H^\\bullet ({A},\\mathcal {B}[-1])=H_{\\mathrm {CE}}^\\bullet (\\mathfrak {g},E)$ , whose bracket is given by $\\Bigl [[e_1],[e_2]\\Bigr ] = (-1)^{\\vert e_1\\vert +1}[\\operatorname{At}^\\nabla _\\mathcal {B}(e_1 ,e_2 ) ] =\\pm [\\psi (e_1)e_2],$ for all $d_{\\mathrm {CE}}$ -closed elements $e_1,e_2 \\in C^\\bullet (\\mathfrak {g},E )$ .",
"Here the last term $\\psi (-)(-):~C^\\bullet (\\mathfrak {g},E)\\times C^\\bullet (\\mathfrak {g},E )\\rightarrow C^\\bullet (\\mathfrak {g},E )$ is a $(\\wedge ^\\bullet \\mathfrak {g}^\\vee )$ -bilinear map naturally extended from $\\psi (-)(-):~E\\times E\\rightarrow E$ .",
"In general, the Leibniz structure on $(H_{\\mathrm {CE}}^\\bullet (\\mathfrak {g},E),[-,-])$ is not skewsymmetric.",
"Homotopic invariance In this section, we prove that the isomorphism class of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras arising from dg derivations only depends on their homotopy classes.",
"Proposition 3.3 Let $\\delta \\sim \\delta ^\\prime $ be homotopic $\\Omega $ -valued dg derivations of ${A}$ .Then there exists an isomorphism $\\lbrace g_k\\rbrace _{k\\ge 1}$ sending the Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra $\\operatorname{Kap}(\\delta ^\\prime ) = (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1})$ (with respect to a $\\delta ^\\prime $ -connection $\\nabla ^\\prime $ ) to $\\operatorname{Kap}(\\delta ) = (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ (with respect to a $\\delta $ -connection $\\nabla $ ).",
"By assumption, there exists a degree $(-1)$ $\\Omega $ -valued derivation $h:{A}\\rightarrow \\Omega $ of ${A}$ such that $\\delta ^\\prime = \\delta + [\\partial _{A},h] = \\delta + \\partial _{A}\\circ h + h \\circ d_{A}.$ We choose an $h$ -connection on $\\mathcal {B}$ , i.e.",
"a degree $(-1)$ linear map $\\widehat{\\nabla }: \\mathcal {B}\\rightarrow \\Omega \\otimes _{A}\\mathcal {B}$ satisfying $\\widehat{\\nabla }(ab) = h(a)\\otimes b + (-1)^{\\vert a\\vert }a \\widehat{\\nabla }(b),\\;\\; \\forall a \\in {A}, b \\in \\mathcal {B}.$ For each $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ , it can be easily verified that $\\nabla ^{\\prime \\prime } := [\\partial _{A},\\widehat{\\nabla }]$ is a $[\\partial _{A},h]$ -connection on $\\mathcal {B}$ , and thus $\\nabla ^\\prime = \\nabla + \\nabla ^{\\prime \\prime } = \\nabla + [\\partial _{A},\\widehat{\\nabla }]: \\mathcal {B}\\rightarrow \\Omega \\otimes _{A}\\mathcal {B}$ is a $\\delta ^\\prime $ -connection on $\\mathcal {B}$ .",
"It follows that $\\mathcal {R}^{\\nabla ^\\prime }_2 = [\\nabla ^\\prime ,\\partial _{A}] = [\\nabla + [\\partial _{A}, \\widehat{\\nabla }],\\partial _{A}] = [\\nabla , \\partial _{A}] = \\mathcal {R}_2^{\\nabla }.$ Define a family of ${A}$ -multilinear maps $g_k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}$ inductively by setting $g_1 = \\operatorname{id}_\\mathcal {B}, g_2 = 0$ , and $g_{k+1}(b_0,\\cdots ,b_k) &= (-1)^{\\vert b_0\\vert }\\sum _{p=2}^k \\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^p = \\mathbb {N}^{(k)} \\\\ I_1,\\cdots ,I_p \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^p_{\\vert I^p\\vert }\\end{array}} \\epsilon (I^1,\\cdots ,I^p) [\\widehat{\\nabla }_{b_0},R^{\\nabla }_{p}](g_{\\vert I^1\\vert }(b_{I^1}),\\cdots , g_{\\vert I^p\\vert }(b_{I^p})) \\\\&\\quad + [\\nabla ^\\prime _{b_0}, g_k](b_1,\\cdots ,b_k),$ for all $k\\ge 2$ .",
"It follows from a straightforward inductive argument that $\\lbrace g_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\mathcal {R}_n^{\\nabla ^\\prime })$ to $(\\mathcal {B},\\mathcal {R}_n^\\nabla )$ .",
"Remark 3.4 Although the Kapranov functor $\\operatorname{Kap}$ maps homotopic derivations to isomorphic Leibniz$_\\infty [1]$ ${A}$ -algebra, it does not reduce to a functor from the category consisting of homology classes of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ ${A}$ -algebras.",
"Applications We first consider a Lie pair $(L,A)$ , and let $B=L/A$ be the Bott $A$ -module.",
"In the introduction, we explained that for each splitting $j: B \\rightarrow L$ of the short exact sequence (REF ) and for any $L$ -connection $\\nabla $ on $B$ extending the Bott $A$ -module structure, there associates a Leibniz$_\\infty [1]$ algebra structure $\\lbrace \\lambda _k\\rbrace _{k\\ge 1}$ on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B)$ .",
"As all $\\lbrace \\lambda _k\\rbrace _{k\\ge 2}$ are $\\Omega _A^\\bullet $ -multilinear, it is a Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebra.",
"Recall that we have a $\\Omega _A^\\bullet (B^\\vee )$ -valued dg derivation $\\delta _j$ of the cdga $\\Omega _A^\\bullet $ as in Equation (REF ).",
"By Proposition REF , the Atiyah cocycle $\\alpha _B^\\nabla $ of the Lie pair coincides with the Atiyah cocycle $\\operatorname{At}^{\\nabla ^{\\delta _j}}_\\mathcal {B}$ of the dg $\\Omega _A^\\bullet $ -module $\\mathcal {B}:= \\Omega _A^\\bullet (B)$ with respect to a $\\delta _j$ -connection $\\nabla ^{\\delta _j}$ as in Equation (REF ).",
"Comparing definitions of $\\lbrace \\lambda _k\\rbrace _{k\\ge 3}$ in the introduction and $\\lbrace \\mathcal {R}^{\\nabla ^{\\delta _j}}_k\\rbrace $ as in Equation (REF ), we see that the two Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebras $(\\mathcal {B},\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ and $(\\mathcal {B},\\mathcal {R}^{\\nabla ^{\\delta _j}}_k)$ are exactly the same.",
"Applying Proposition REF , Theorem REF , Theorem REF , and Proposition REF , we have the following Theorem 3.5 Let $(L,A)$ be a Lie pair over a smooth manifold $M$ .",
"Then the Leibniz$_\\infty [1]$ algebra structure constructed in *Theorem 3.13 on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ is unique up to isomorphisms in the category of Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebras.",
"Moreover, if $(E,\\partial _A^E)$ is an $A$ -module, then the representation of the above Leibniz$_\\infty [1]$ algebra on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes E)$ is also unique up to isomorphisms in the category of Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -modules.",
"Finally, we consider another interesting application: Let $X$ be a complex manifold and ${A}= (\\Omega _X^{0,\\bullet }, \\bar{\\partial })$ its Dolbeault dg algebra.",
"Let $\\Omega = (\\Omega _X^{0,\\bullet }(T^{1,0}X),\\bar{\\partial })$ be the dg ${A}$ -module generated by the smooth section space $\\Gamma (T^{1,0}X)$ of the holomorphic tangent bundle $T^{1,0}X$ .",
"Note that each holomorphic bivector field $\\pi \\in \\Gamma (\\wedge ^2 T^{1,0}X)$ determines an $\\Omega $ -valued dg derivation of ${A}$ , denoted by $\\delta _\\pi $ , which is the composition ${A}\\xrightarrow{} \\Omega _X^{1,\\bullet } = \\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee ) \\xrightarrow{} \\Omega .$ Here $\\pi ^\\sharp $ is the contraction along $\\pi $ from $(T^{1,0}X)^\\vee $ to $T^{1,0}X$ .",
"In fact, $\\pi ^\\sharp $ is a morphism of dg derivations of ${A}$ (from ${A}\\xrightarrow{} \\Omega _X^{1,\\bullet }$ to ${A}\\xrightarrow{} \\Omega $ ).",
"It sends the Atiyah class $\\alpha _E \\in H^1(X,(T^{1,0}X)^\\vee \\otimes \\operatorname{End}(E))$ of any holomorphic vector bundle $E$ to the $\\delta _\\pi $ -twisted Atiyah class $\\operatorname{At}^{\\delta _\\pi }_{E}\\in H^1(X,T^{1,0}X \\otimes \\operatorname{End}(E))$ of the associated dg ${A}$ -module ${E}= \\Omega _X^{0,\\bullet }(E)$ .",
"By Proposition REF , the $\\delta _\\pi $ -twisted Atiyah class $\\operatorname{At}^{\\delta _\\pi }_{E}$ measures the existence of holomorphic $\\delta _\\pi $ -connections on $E$ .",
"In particular, if $\\pi $ a holomorphic Poisson bivector field, then $(T^{1,0}X)^\\vee $ is a holomorphic Lie algebroid , and $\\operatorname{At}^{\\delta _\\pi }_{E}$ measures the existence of holomorphic $(T^{1,0}X)^\\vee $ -connections on $E$ .",
"Applying Theorem REF , we have the following Theorem 3.6 Let $X$ be a complex manifold, $\\pi $ a holomorphic bivector field.",
"Then, Both $\\Omega _X^{0,\\bullet }(T^{1,0}X)$ and $\\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee )$ carry canonical Kapranov Leibniz$_\\infty [1]$ $\\Omega _X^{0,\\bullet }$ -algebra structures; There is a morphism of Leibniz$_\\infty [1]$ $\\Omega _X^{0,\\bullet }$ -algebras $\\lbrace f_k\\rbrace _{k\\ge 1}: \\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee ) \\rightarrow \\Omega _X^{0,\\bullet }(T^{1,0}X)$ such that $f_1 = \\pi ^\\sharp $ .",
"Open questions and remarks In this note, we assume that each dg ${A}$ -module ${E}$ is projective in order that connections exist on ${E}$ .",
"In the non-projective case, one can follow Calaque-Van den Bergh's approach CV to define the Atiyah class of ${E}$ (which coincides with the Atiyah class of ${E}$ in Definition REF when ${E}$ admits connections)— The first step is to construct a short exact sequence, called the jet sequence, of dg ${A}$ -modules: $@C=0.5cm{0 [r] & \\Omega _{{A}\\mid \\mathbb {K}}^1\\otimes _{A}{E}[rr] && \\mathfrak {J}{E}[rr] && {E}[r] & 0 }.$ The Atiyah class of ${E}$ is then defined to be the extension class of the above jet sequence.",
"We would like to follow this approach to study twisted Atiyah classes of some cases when connections do not exist (singular foliations considered in for example).",
"Note that Kapranov's original construction on $\\Omega ^{0,\\bullet -1}_X(T_X)$ of a Kähler manifold $X$ is an $L_\\infty $ algebra, whereas Chen, Stiénon and Xu's construction of $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B)$ is a Leibniz$_\\infty [1]$ algebra.",
"In fact, this is due to the existence of Chern connection on $T_X$ which enjoys special properties (see *Section 3.4.4).",
"Meanwhile, when ${A}= C^\\infty (\\mathcal {M})$ is the cdga of functions of a smooth dg manifold $\\mathcal {M}$ .",
"According to , the tangent complex $T_{{A}\\mid \\mathbb {K}} = \\Gamma (T_\\mathcal {M})$ admits an $L_\\infty [1]$ algebra structure (by a construction different from the Kapranov's construction we discussed).",
"Moreover, Laurent-Gengoux, Stiénon and Xu have proved that for each Lie pair $(L,A)$ , there exists a canonical $L_\\infty [1]$ algebra structure on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ (which is different from the Chen-Stiénon-Xu's construction in ).",
"It is natural to ask how to tweak the Kapranov Leibniz$_\\infty [1]$ algebra of general dg derivations so as to produce an $L_\\infty [1]$ algebra rather than a mere Leibniz$_\\infty [1]$ algebra.",
"According to the perturbation lemmas proved by Huebschmann HueLie,HueshLie, many $L_\\infty $ algebras arise from dg Lie algebras or $L_\\infty $ algebras by homological perturbation theory.",
"It is interesting to investigate whether similar perturbation lemma holds for Leibniz$_\\infty [1]$ algebras.",
"Moreover, if this is the case, then it is natural to ask for which kind of dg derivations of a cdga ${A}$ , the associated Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra results from some perturbation.",
"These questions will be investigated somewhere else.",
"We would also like to mention other works that are related to the present paper: Batakidis and Voglaire showed how Atiyah classes of Lie pairs and of dg Lie algebroids give rises to Atiyah classes of dDG algebras .",
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"Moreover, given a perfect module $E$ over $X$ , there exists a representation of the aforesaid Lie algebra on $E$ induced by the Atiyah class of $E$ .",
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],
[
"The Kapranov functor",
"In this section, we explore higher algebraic structures, called Kapranov Leibniz$_\\infty [1]$ algebras, induced from a dg derivation of a cdga ${A}$ .",
"Our main goal is to show that there exists a contravariant functor, called the Kapranov functor, from the category of dg derivations to the category of Leibniz$_\\infty [1]$ -algebras over ${A}$ ."
],
[
"Leibniz$_\\infty [1]$ algebras",
"We recall some basic notions of homotopy Leibniz algebras (c.f.AP,CSX).",
"In what follows, all tensor products $\\otimes $ without adoration are assumed to be over $\\mathbb {K}$ .",
"Definition 3.1 A Leibniz$_\\infty [1]$ algebra (over $\\mathbb {K}$ ) is a graded $\\mathbb {K}$ -vector space $V = \\oplus _{n \\in \\mathbb {Z}}V^n$ , together with a sequence $\\lbrace \\lambda _k:~ \\otimes ^k V \\rightarrow V\\rbrace _{k \\ge 1}$ of degree 1, $\\mathbb {K}$ -multilinear maps satisfying $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (k-j)}\\vert } \\\\&\\lambda _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_n)=0,$ for all $n \\ge 1$ and all homogeneous elements $v_i \\in V$ , where $\\operatorname{sh}(p,q)$ denotes the set of $(p,q)$ -shuffles ($p,q \\ge 0$ ), and $\\epsilon (\\sigma )$ is the Koszul sign of $\\sigma $ .",
"Definition 3.2 A morphism of Leibniz$_\\infty [1]$ algebras from $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ to $(V^{\\prime },\\lbrace \\lambda ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a sequence $\\lbrace f_k:~ V^{\\otimes k} \\rightarrow V^{\\prime }\\rbrace _{k \\ge 1}$ of degree 0, $\\mathbb {K}$ -multilinear maps, satisfying the following compatibility condition: $&\\quad \\sum _{k+p \\le n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\dagger ^{\\sigma }_k}f_{n-p}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)}, \\lambda _{p+1}(b_{\\sigma (k+1)}, \\cdots ,b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_{n}) \\\\&= \\sum _{q \\ge 1}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^q) \\lambda _q^{\\prime }(f_{\\vert I^1\\vert }(b_{I^1}), \\cdots ,f_{\\vert I^q\\vert }(b_{I^q})),$ for all $n \\ge 1$ , where $\\dagger ^{\\sigma }_k = \\sum _{i=1}^k\\vert b_{\\sigma (i)}\\vert $ , $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n)} = \\lbrace 1,\\cdots ,n\\rbrace $ , and $(b_{I^j}) = (b_{i^j_1},\\cdots ,b_{i^j_{\\vert I^j\\vert }})$ for all $1\\le j \\le q$ .",
"In the above definition, the first component $f_1:~ (V,\\lambda _1) \\rightarrow (V^{\\prime },\\lambda _1^{\\prime })$ , called the tangent morphism, is a morphism of cochain complexes.",
"We call the Leibniz$_\\infty [1]$ morphism $f_\\bullet :~ (V,\\lambda _\\bullet ) \\rightarrow (V^{\\prime },\\lambda ^{\\prime }_\\bullet )$ a quasi-isomorphism (resp.",
"an isomorphism) if $f_1$ is a quasi-isomorphism (resp.",
"an isomorphism).",
"In fact, there is a standard way to find its quasi-inverse (resp.",
"inverse) $f^{-1}_\\bullet :~ (V^{\\prime },\\lambda ^{\\prime }_\\bullet ) \\rightarrow (V,\\lambda _\\bullet )$ (see ).",
"Definition 3.3 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ algebra.",
"A $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -module is a graded $\\mathbb {K}$ -vector space $W$ together with a sequence $\\lbrace \\mu _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W\\rbrace _{k \\ge 1}$ of degree 1, $\\mathbb {K}$ -multilinear maps satisfying the identities $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\dagger ^\\sigma _{k-j}} \\\\&\\qquad \\mu _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_{n-1},w) \\\\&\\qquad +\\sum _{1\\le j \\le n}\\sum _{\\sigma \\in \\operatorname{sh}(k,j)}\\epsilon (\\sigma )(-1)^{\\dagger ^\\sigma _{n-j}} \\mu _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (n-j)},\\mu _{j}(v_{\\sigma (n-j+1)}, \\cdots ,v_{\\sigma (n-1)}, w))=0,$ for all $n \\ge 1$ and all homogeneous vectors $v_1,\\cdots ,v_{n-1} \\in V, w \\in W$ , where $\\dagger ^\\sigma _j = \\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (j)}\\vert $ for all $j \\ge 0$ .",
"Definition 3.2 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ algebra.",
"A morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -modules from $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ to $(W^{\\prime },\\lbrace \\mu ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a sequence $\\lbrace \\psi _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W^{\\prime }\\rbrace _{k \\ge 1}$ of degree 0, $\\mathbb {K}$ -multilinear maps satisfying the identity $&\\sum _{i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (k-j)}\\vert } \\\\&\\quad \\psi _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (k-j)},\\lambda _{j}(v_{\\sigma (k-j+1)}, \\cdots ,v_{\\sigma (k-1)}, v_k), v_{k+1},\\cdots , v_{n-1},w) \\\\&\\qquad +\\sum _{1\\le j \\le n}\\sum _{\\sigma \\in \\operatorname{sh}(k,j)}\\epsilon (\\sigma )(-1)^{\\vert v_{\\sigma (1)}\\vert +\\cdots + \\vert v_{\\sigma (n-j)}\\vert } \\psi _{i}(v_{\\sigma (1)},\\cdots ,v_{\\sigma (n-j)},\\mu _{j}(v_{\\sigma (n-j+1)},\\cdots , v_{\\sigma (n-1)}, w)) \\\\&= \\sum _{p \\ge 0}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^{p+1} = \\mathbb {N}^{(n-1)} \\\\ I^1,\\cdots ,I^{p+1} \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^{p+1}_{\\vert I^{p+1}\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^{p+1}) \\mu ^{\\prime }_{p+1}(\\lambda _{\\vert I^1\\vert }(v_{I^1}),\\cdots ,\\lambda _{\\vert I^p\\vert }(v_{I^p}), \\psi _{\\vert I^{p+1}\\vert +1}(v_{I^{p+1}},w)),$ for each $n \\ge 1$ and all homogeneous vectors $v_1,\\cdots ,v_{n-1} \\in V, w \\in W$ .",
"Here $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n-1)} = \\lbrace 1,\\cdots ,n-1\\rbrace $ , and $(v_{I^j}) = (v_{i^j_1},\\cdots ,v_{i^j_{\\vert I^j\\vert }})$ for all $1 \\le j \\le p+1$ .",
"In this note, we are particularly interested in Leibniz$_\\infty [1]$ algebras over a cdga ${A}$ (or Leibniz$_\\infty [1]$ ${A}$ -algebras).",
"Definition 3.0 A Leibniz$_\\infty [1]$ ${A}$ -algebra is a Leibniz$_\\infty [1]$ algebra $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ (in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ ) such that the cochain complex $(V,\\lambda _1)$ is a dg ${A}$ -module and all higher brackets $\\lambda _k: \\otimes ^k V \\rightarrow V$ ($k\\ge 2$ ) are ${A}$ -multilinear.",
"A morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ to $(V^{\\prime },\\lbrace \\lambda ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a morphism $\\lbrace f_k:V^{\\otimes k}\\rightarrow V^{\\prime }\\rbrace _{k \\ge 1}$ (in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ ) such that all structure maps $\\lbrace f_k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"In particular, its tangent morphism $f_1:~ (V,\\lambda _1) \\rightarrow (V^\\prime ,\\lambda ^\\prime _1)$ is a dg ${A}$ -module morphism.",
"Such a morphism $f_\\bullet :~ (V,\\lambda _\\bullet ) \\rightarrow (V^{\\prime },\\lambda ^{\\prime }_\\bullet )$ is called a quasi-isomorphism (resp.",
"an isomorphism) if its tangent morphism $f_1$ is a quasi-isomorphism (resp.",
"an isomorphism) of dg ${A}$ -modules.",
"Denote the category of Leibniz$_\\infty [1]$ ${A}$ -algebras by $\\operatorname{Leib}_\\infty ({A})$ .",
"It is a subcategory of the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ .",
"There are analogous notions of modules of a Leibniz$_\\infty [1]$ ${A}$ -algebra and their morphisms: Definition 3.1 Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"A $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -module is a $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -module $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ (in the category of Leibniz$_\\infty [1]$ modules over $\\mathbb {K}$ ) such that $(W,\\mu _1)$ is a DG ${A}$ -module and all higher structure maps $\\lbrace \\mu _k\\rbrace _{k\\ge 2}$ are ${A}$ -multilinear.",
"A morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -modules from $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ to $(W^{\\prime },\\lbrace \\mu ^{\\prime }_k\\rbrace _{k\\ge 1})$ is a morphism of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ -modules $\\lbrace \\psi _k:~ V^{\\otimes (k-1)} \\otimes W \\rightarrow W^{\\prime }\\rbrace _{k \\ge 1}$ (in the category of Leibniz$_\\infty [1]$ modules over $\\mathbb {K}$ ) such that all maps $\\lbrace \\psi _k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"It follows that the collection of $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -modules and their morphisms form a category."
],
[
"The Kapranov functor",
"In this section, we generalize Kapranov's construction of an $L_\\infty $ algebra structure and Chen-Stiénon-Xu's construction of a Leibniz$_\\infty [1]$ algebra structure in the setting of a dg derivation ${A}\\xrightarrow{} \\Omega $ of a cdga ${A}$ ."
],
[
"Kapranov Leibniz$_\\infty [1]$ algebras",
"Let ${E}$ be a graded ${A}$ -module with a $\\delta $ -connection $\\nabla $ .",
"For each homogeneous $b \\in \\mathcal {B}$ , there is a degree $\\vert b\\vert $ derivation on the reduced tensor algebra $T({E})$ (over ${A}$ ) defined by $\\nabla _b(e_1 \\otimes \\cdots \\otimes e_n) = \\sum _{i=1}^{n}(-1)^{\\vert b\\vert \\ast _{i-1}} e_1 \\otimes \\cdots \\nabla _b e_i \\otimes \\cdots e_n,$ for all homogeneous $e_i \\in {E}$ , where $\\ast _{i} = \\sum _{j=1}^i\\vert e_j\\vert $ .",
"Let ${E}$ and ${F}$ be two graded ${A}$ -modules with $\\delta $ -connections $\\nabla ^{{E}}$ and $\\nabla ^{{F}}$ , respectively.",
"For $b \\in \\mathcal {B}$ and $\\lambda \\in \\operatorname{Hom}_{{A}}({E},{F})$ , there associates the derivation $\\nabla _b(\\lambda ) = [\\nabla _b,\\lambda ] = \\nabla ^{{F}}_b \\circ \\lambda - (-1)^{\\vert b\\vert \\vert \\lambda \\vert }\\lambda \\circ \\nabla ^{{E}}_b:~ {E}\\rightarrow {F}.$ It follows from a direct verification that $\\nabla _b(\\lambda ) \\in \\operatorname{Hom}_{{A}}({E},{F})$ .",
"Choose a $\\delta $ -connection on $\\mathcal {B}$ .",
"There associates a sequence of degree 1 maps $ \\mathcal {R}^\\nabla _k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}, k \\ge 1 $ defined as follows: $\\mathcal {R}^\\nabla _1 = \\partial _{A}:~ \\mathcal {B}\\rightarrow \\mathcal {B}$ ; $\\mathcal {R}^\\nabla _2$ is specified by the associated twisted Atiyah cocycles $\\operatorname{At}_\\mathcal {B}^\\nabla $ ; $\\lbrace \\mathcal {R}^\\nabla _{k+1}:~ \\mathcal {B}^{\\otimes (k+1)} \\rightarrow \\mathcal {B}\\rbrace _{k\\ge 2}$ are defined recursively by $\\mathcal {R}^\\nabla _{k+1}=\\nabla (\\mathcal {R}^\\nabla _{k})$ .",
"Explicitly, we have $&\\mathcal {R}^\\nabla _{k+1}(b_0,b_1,\\cdots ,b_{k}) = (-1)^{\\vert b_0\\vert }[\\nabla _{b_0},\\mathcal {R}^\\nabla _k] (b_1,\\cdots ,b_k), \\;\\;\\forall b_i \\in \\mathcal {B}.$ Proposition 3.3 The ${A}$ -module $\\mathcal {B}$ , together with the sequence of operators $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ , is a Leibniz$_\\infty [1]$ ${A}$ -algebra.",
"The 2-bracket $\\mathcal {R}^\\nabla _2$ is the twisted Atiyah cocycles $\\operatorname{At}_\\mathcal {B}^\\nabla $ , which is certainly ${A}$ -bilinear.",
"By the recursive construction of higher brackets $\\mathcal {R}^\\nabla _{k+1}$ ($k\\ge 2$ ) in Equation (REF ), they are all ${A}$ -multilinear as well.",
"So it suffices to verify that $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ satisfies Equation (REF ).",
"We argue by induction.",
"The $n=1$ case follows from the fact that $\\mathcal {R}^\\nabla _1 = \\partial _{A}$ is a differential, and the $n=2$ case follows from the fact that the $\\delta $ -twisted Atiyah cocycle $\\operatorname{At}^\\nabla _\\mathcal {B}$ is a $\\partial _{A}$ -cocycle.",
"Now assume that the identity (REF ) holds for some $n \\ge 2$ , i.e., $&\\quad -\\partial _{A}(\\mathcal {R}^\\nabla _n)(b_1,\\cdots ,b_n) = -\\partial _{A}(\\mathcal {R}^\\nabla _n(b_1,\\cdots ,b_n)) + \\sum _{i=1}^n(-1)^{\\ast _{i-1}} \\mathcal {R}^\\nabla _n(b_1,\\cdots ,\\partial _{A}b_i,\\cdots ,b_{n}) \\\\&=\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_{\\sigma (1)}\\vert + \\cdots + \\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j} (b_{\\sigma (k-j+1)},\\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n).$ Consider the $(n+1)$ case: We first compute $-\\partial _{A}(\\mathcal {R}^\\nabla _{n+1})(b_0,b_1,\\cdots ,b_n) &= \\partial _{A}([\\mathcal {R}^\\nabla _{n},\\nabla _{b_0}])(b_1,\\cdots ,b_n) + [\\mathcal {R}^\\nabla _n,\\nabla _{\\partial _{A}b_0}](b_1,\\cdots ,b_n) \\\\&= [\\partial _{A}(\\mathcal {R}^\\nabla _n),\\nabla _{b_0}](b_1,\\cdots ,b_n) + [\\mathcal {R}^\\nabla _n,\\mathcal {R}^\\nabla _2(b_0,-)](b_1,\\cdots ,b_n).$ Here we have used the recursive definition (REF ) in the first equality and Equation (REF ) in the second one.",
"We introduce $b^{(i)}_k = {\\left\\lbrace \\begin{array}{ll}b_k, &\\;\\;\\;\\text{if}\\; k \\ne i \\\\\\nabla _{b_0}b_i, &\\;\\;\\;\\text{if}\\; k = i.\\end{array}\\right.",
"}$ Then the first summand in Equation (REF ) is, $&\\quad [\\partial _{A}(\\mathcal {R}^\\nabla _n),\\nabla _{b_0}](b_1,b_2,\\cdots ,b_{n}) \\\\&=\\sum _{i=1}^{n}(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\partial _{A}(\\mathcal {R}^\\nabla _n) (b_1,\\cdots ,\\nabla ^\\mathcal {B}_{b_0}b_i, \\cdots ,b_{n}) - \\nabla _{b_0}(\\partial _{A}(\\mathcal {R}^\\nabla _n)(b_1,\\cdots ,b_{n})) \\;\\\\&\\qquad \\qquad \\text{by assumption~(\\ref {casen})} \\\\&= -\\sum _{i=1}^n\\sum _{p,q \\ge 2, p+q=n+1}\\sum _{k=q}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-q,q-1)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert \\ast _{i-1}+\\vert b^{(i)}_{\\sigma (1)}\\vert + \\cdots +\\vert b^{(i)}_{\\sigma (k-q)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{p}(b^{(i)}_{\\sigma (1)},\\cdots , b^{(i)}_{\\sigma (k-q)}, \\mathcal {R}^\\nabla _{j}(b^{(i)}_{\\sigma (k-q+1)}, \\cdots ,b^{(i)}_{\\sigma (k-1)}, b^{(i)}_k), b^{(i)}_{k+1},\\cdots , b^{(i)}_n) \\\\&\\quad +\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_{\\sigma (1)}\\vert +\\cdots +\\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\nabla _{b_0}(\\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots , b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j}(b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n)) \\\\&=\\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert + \\vert b_{\\sigma (1)}\\vert +\\cdots +\\vert b_{\\sigma (k-j)}\\vert } \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i+1}(b_0,b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j}(b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n)\\\\&\\quad + \\sum _{i,j \\ge 2, i+j=n+1}\\sum _{k=j}^{n}\\sum _{\\sigma \\in \\operatorname{sh}(k-j,j-1)}\\epsilon (\\sigma )(-1)^{(\\vert b_{\\sigma (1)}\\vert +\\cdots + \\vert b_{\\sigma (k-j)}\\vert )(\\vert b_0\\vert +1)} \\\\&\\qquad \\qquad \\mathcal {R}^\\nabla _{i}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k-j)}, \\mathcal {R}^\\nabla _{j+1} (b_0,b_{\\sigma (k-j+1)}, \\cdots ,b_{\\sigma (k-1)}, b_k), b_{k+1},\\cdots , b_n).$ Here in the last step we used the recursive definition of $\\lbrace \\mathcal {R}^\\nabla _i\\rbrace $ .",
"Meanwhile, the second summand in Equation (REF ) is $&\\quad [\\mathcal {R}^\\nabla _n,\\mathcal {R}^\\nabla _2(b_0,-)](b_1,b_2,\\cdots ,b_{n}) \\\\&= \\sum _{i=1}^n(-1)^{(\\vert b_0\\vert +1)\\ast _{i-1}}\\mathcal {R}^\\nabla _n(b_1,\\cdots , \\mathcal {R}^\\nabla _2(b_0,b_i), \\cdots ,b_n) + (-1)^{\\vert b_0\\vert }\\mathcal {R}^\\nabla _2(b_0,\\mathcal {R}^\\nabla _n(b_1,\\cdots , b_{n})).$ Substituting Equations (REF ) and (REF ) into Equation (REF ), we see that Equation (REF ) holds for the case $(n+1)$ .",
"This proves that $\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1}$ satisfies Equation (REF ) for all $n \\ge 1$ .",
"The Leibniz$_\\infty [1] {A}$ -algebra $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ will be denoted by $\\operatorname{Kap}^c(\\delta )$ .",
"Here the superscript $c$ is to remind the reader that this Leibniz$_\\infty [1] {A}$ -algebra is defined via a particular $\\delta $ -connection on $\\mathcal {B}$ .",
"Remark 3.-6 This method is originated from Kapranov's construction of $L_\\infty $ algebra structure on the shifted tangent complex $\\Omega _X^{0,\\bullet -1}(T^{1,0}X)$ of a compact Kähler manifold $X$ .",
"For this reason, we call $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ the Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra."
],
[
"Functoriality",
"Next, we show that the assignment of a Leibniz$_\\infty [1]$ ${A}$ -algebra to each pair of dg derivation ${A}\\xrightarrow{} \\Omega $ and a $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ is functorial: Proposition 3.-5 Let $\\phi $ be a morphism from ${A}\\xrightarrow{} \\Omega ^\\prime $ to ${A}\\xrightarrow{} \\Omega $ in the category $\\operatorname{dgDer}_{A}$ of dg derivations (see Definition REF ).",
"Let $\\mathcal {B}=\\Omega ^\\vee $ and $\\mathcal {B}^{\\prime }=(\\Omega ^{\\prime })^\\vee $ be their dual dg ${A}$ -modules.",
"For a $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ and a $\\delta ^\\prime $ -connection $\\nabla ^\\prime $ on $\\mathcal {B}^\\prime $ , there exists a morphism $f_\\bullet = \\lbrace f_k\\rbrace _{k \\ge 1}$ of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ to $(\\mathcal {B}^\\prime , \\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1})$ , whose first map is $f_1=\\phi ^\\vee $ .",
"In other words, we have the following commutative diagram ${({A}\\xrightarrow{} \\Omega ^\\prime ) [d]_-{\\phi } [r]^-{\\operatorname{Kap}^{c}} & (\\mathcal {B}^\\prime ,\\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1}) \\\\({A}\\xrightarrow{} \\Omega ) [r]^-{\\operatorname{Kap}^c} & (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k \\ge 1}).",
"[u]_-{\\operatorname{Kap}^c(\\phi ) = f_\\bullet }}$ Define a sequence of ${A}$ -multilinear maps $f_k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}^\\prime $ recursively by setting $f_1(b_1) &= \\phi ^\\vee (b_1), & f_{k+1}(b_0,\\cdots ,b_k) &= \\nabla _{f_1(b_0)}^\\prime f_k(b_1,\\cdots ,b_k) - f_k(\\nabla _{b_0}(b_1,\\cdots ,b_k)), $ for all $k \\ge 1$ and $b_i \\in \\mathcal {B}$ .",
"It is easy to verify that all the maps $\\lbrace f_k\\rbrace _{k\\ge 1}$ are ${A}$ -multilinear.",
"Now we show that $\\lbrace f_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\mathcal {R}_k^{\\nabla })$ to $(\\mathcal {B}^\\prime , \\mathcal {R}_k^{\\nabla ^\\prime })$ .",
"We argue by induction: First of all, the $n=1$ case is obvious, since $f_{1} = \\phi ^\\vee : \\mathcal {B}\\rightarrow \\mathcal {B}^\\prime $ is a morphism of dg modules.",
"Now assume that Equation (REF ) holds for some $n \\ge 1$ , i.e., $&\\quad \\sum _{k+p \\le n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\dagger ^{\\sigma }_k}f_{n-p}(b_{\\sigma (1)}, \\cdots ,b_{\\sigma (k)}, \\mathcal {R}^\\nabla _{p+1}(b_{\\sigma (k+1)}, \\cdots ,b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_{n}) \\\\&= \\sum _{q \\ge 1}\\sum _{\\begin{array}{c}I^1 \\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}}\\epsilon (I^1,\\cdots ,I^q) \\mathcal {R}_q^{\\nabla ^\\prime }(f_{\\vert I^1\\vert }(b_{I^1}), \\cdots ,f_{\\vert I^q\\vert }(b_{I^q})),$ where $\\dagger ^{\\sigma }_k = \\sum _{i=1}^k\\vert b_{\\sigma (i)}\\vert $ , $I^j = \\lbrace i^j_1 < \\cdots < i^j_{\\vert I^j\\vert }\\rbrace \\subset \\mathbb {N}^{(n)} = \\lbrace 1,\\cdots ,n\\rbrace $ , and $(b_{I^j}) = (b_{i^j_1},\\cdots ,b_{i^j_{\\vert I^j\\vert }})$ for all $1\\le j \\le q$ .",
"We proceed to show that Equation (REF ) holds for $(n+1)$ homogeneous inputs $\\lbrace b_0,\\cdots ,b_{n+1}\\rbrace $ .",
"For simplicity, we denote the left-hand side and the right-hand side of Equation $(\\ref {casenprime})$ by $\\operatorname{LHS}(b_1,\\cdots ,b_n)$ and $\\operatorname{RHS}(b_1,\\cdots ,b_n)$ , respectively.",
"We write $\\operatorname{LHS}(b_0,b_1,\\cdots ,b_n) = I_1 + I_2 + I_3$ as the sum of three parts, where $I_1 &= f_{n+1}(\\partial _{{A}}b_0,b_1,\\cdots ,b_n) = \\nabla ^\\prime _{\\partial _{A}f_1(b_0)}\\circ f_n(b_1,\\cdots ,b_n) - \\sum _{i=1}^n(-1)^{\\ast _{i-1}(\\vert b_0\\vert +1)} f_n(b_1,\\cdots ,\\nabla _{\\partial _{A}b_0}(b_i),\\cdots ,b_n),\\\\I_2 &= \\sum _{k+p =0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{\\vert b_0\\vert +\\dagger ^{\\sigma }_k} f_{n-p+1}(b_0,b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)},\\mathcal {R}^\\nabla _{p+1} (b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n) \\\\&= (\\nabla ^\\prime _{f_1(b_0)}\\circ f_{n-p} - f_{n-p}\\circ \\nabla _{b_0})(b_{\\sigma (1)}, \\cdots , \\mathcal {R}^\\nabla _{p+1}(b_{\\sigma (k+1)}, \\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n),$ by Equation (REF ), and $I_3 &= \\sum _{k+p=0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)}\\epsilon (\\sigma )(-1)^{(\\vert b_0\\vert +1)\\dagger ^{\\sigma }_k }f_{n-p}(b_{\\sigma (1)},\\cdots ,b_{\\sigma (k)},\\mathcal {R}^\\nabla _{p+2}(b_0, b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n),\\\\&=\\sum _{k+p=0}^{n-1}\\sum _{\\sigma \\in \\operatorname{sh}(k,p)} \\epsilon (\\sigma ) (-1)^{(\\vert b_0\\vert +1)\\dagger ^{\\sigma }_k} f_{n-p}(b_{\\sigma (1)},\\cdots , b_{\\sigma (k)},-[\\mathcal {R}^\\nabla _{p+1}, \\nabla _{b_0}](b_{\\sigma (k+1)},\\cdots , b_{\\sigma (k+p)},b_{k+p+1}),\\cdots ,b_n) \\\\&\\quad \\quad +\\sum _{k=0}^{n-1}(-1)^{\\ast _{k}(\\vert b_0\\vert +1)} f_n(b_1,\\cdots ,\\nabla _{\\partial _{A}b_0}(b_{k+1}),\\cdots ,b_n),$ by Equations (REF ) and (REF ).",
"Summing them up, we have $\\operatorname{LHS}(b_0,b_1,\\cdots ,b_n)&= \\nabla ^\\prime _{\\partial _{A}(f_1(b_0))} f_n(b_1,\\cdots ,b_n) -\\sum _{i=1}^n(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\operatorname{LHS}(b_1, \\cdots ,\\nabla _{b_0}b_i,\\cdots ,b_n) \\\\&\\quad + \\nabla ^\\prime _{f_1(b_0)} \\operatorname{LHS}(b_1,\\cdots ,b_n).$ Meanwhile, $&\\quad \\operatorname{RHS}(b_0,b_1,\\cdots ,b_n) \\\\ \\nonumber &= \\sum _{q=1}^{n}\\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^q = \\mathbb {N}^{(n)} \\\\ I_1,\\cdots ,I_q \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^q_{\\vert I^q\\vert }\\end{array}} \\epsilon (I^1,\\cdots ,I^q) (\\mathcal {R}^{\\nabla ^\\prime }_{q+1}(b_0, f_{\\vert I^1\\vert }(b_{I^1}),\\cdots ,f_{\\vert I^q\\vert }(b_{I^q})) \\\\\\nonumber &\\qquad + \\mathcal {R}^{\\nabla ^\\prime }_{q} (f_{\\vert I^1\\vert +1}(b_0,b_{I^1}), \\cdots , f_{\\vert I^q\\vert }(b_{I^q})) + \\cdots + (-1)^{\\vert b_0\\vert (\\vert b_{I^1}\\vert +\\cdots + \\vert b_{I^{q-1}}\\vert )} \\mathcal {R}^{\\nabla ^\\prime }_{q} (f_{\\vert I^1\\vert }(b_{I^1}),\\cdots , f_{\\vert I^q\\vert +1}(b_0,b_{I^q}))) \\\\&\\qquad \\qquad \\qquad \\text{by Equations~(\\ref {Atiyah cocycle}),(\\ref {Rnabla}),(\\ref {phik})} \\\\ &=\\nabla ^\\prime _{f_1(b_0)} \\operatorname{RHS}(b_1,\\cdots ,b_n) + \\nabla ^\\prime _{\\partial _{A}(f_1(b_0))}f_n(b_1,\\cdots ,b_n)-\\sum _{i=1}^n(-1)^{\\vert b_0\\vert \\ast _{i-1}}\\operatorname{RHS}(b_1,\\cdots , \\nabla _{b_0}b_i, \\cdots , b_n).$ Applying the induction assumption to Equations (REF ) and (REF ), we see that Equation (REF ) holds for all $(n+1)$ entries.",
"Thus Equation (REF ) holds for all $n \\ge 1$ .",
"This proves that $f = \\lbrace f_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras.",
"As a consequence, the Kapranov's construction defines a contravariant functor $\\operatorname{Kap}^c: \\operatorname{dgDer}_{A}\\rightarrow \\operatorname{Leib}_\\infty ({A})$ from the category $\\operatorname{dgDer}_{A}$ of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ -algebras.",
"Remark 3.-5 The reason that we restrict to work in the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ ${A}$ -algebras is as follows: If we treat $(\\mathcal {B},\\lbrace \\mathcal {R}^\\nabla _k\\rbrace _{k\\ge 1})$ merely as a Leibniz$_\\infty [1]$ algebra over $\\mathbb {K}$ , it is always isomorphic to the trivial one $(B,\\lbrace \\partial _{A},0,0,\\cdots \\rbrace )$ (all higher brackets are zero).",
"In fact, one can build a sequences of degree 0 maps $\\phi _k:~\\; &\\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B},\\quad k \\ge 1,$ where $\\phi _1 = \\operatorname{id}_\\mathcal {B}$ , and $\\lbrace \\phi _{k+1}\\rbrace _{k\\ge 1}$ are defined recursively by $\\phi _{k+1}(b_0,\\cdots ,b_k) &= {\\nabla }_{b_0} \\circ \\phi _k (b_1,\\cdots ,b_k), \\;\\;\\;\\forall b_i \\in \\mathcal {B}.$ The set $\\lbrace \\phi _k:~ \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}\\rbrace _{k\\ge 1}$ defines an isomorphism of Leibniz$_\\infty [1]$ algebras from $(\\mathcal {B},\\lbrace \\partial _{A},0,0,\\cdots \\rbrace )$ to $(\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ in the category of Leibniz$_\\infty [1]$ algebras over $\\mathbb {K}$ .",
"The proof is similar to that of Proposition REF .",
"However, the maps $\\lbrace \\phi _k\\rbrace _{k\\ge 2}$ are not ${A}$ -multilinear.",
"Next, we stress the independence from the choice of connections in the definition of Kapranov functors.",
"For a dg derivation ${A}\\xrightarrow{} \\Omega $ , suppose that we have another $\\delta $ -connection $\\widetilde{\\nabla }$ on $\\mathcal {B}= \\Omega ^\\vee $ .",
"Denote the corresponding Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra by $\\operatorname{Kap}^{\\tilde{c}}(\\delta ) = (\\mathcal {B},\\mathcal {R}_k^{\\widetilde{\\nabla }})$ .",
"By Proposition REF , there exists an isomorphism $g^{\\nabla ,\\widetilde{\\nabla }}_\\bullet : \\operatorname{Kap}^c(\\delta ) \\rightarrow \\operatorname{Kap}^{\\tilde{c}}(\\delta )$ of Leibniz$_\\infty [1]$ ${A}$ -algebras, where $g^{\\nabla ,\\widetilde{\\nabla }}_1 = \\operatorname{id}_\\mathcal {B}$ , and $\\lbrace g^{\\nabla ,\\widetilde{\\nabla }}_{k+1}\\rbrace _{k\\ge 1}$ are defined recursively as follows: $g^{\\nabla ,\\widetilde{\\nabla }}_{k+1}(b_0,\\cdots ,b_k) &= (\\widetilde{\\nabla }_{b_0} \\circ g^{\\nabla ,\\widetilde{\\nabla }}_k - g^{\\nabla ,\\widetilde{\\nabla }}_k \\circ \\nabla _{b_0})(b_1,\\cdots ,b_k), \\;\\;\\;\\forall b_i \\in \\mathcal {B}.$ Moreover, via a straightforward verification, we have Lemma 3.-4 There exists a natural equivalence between Kapranov functors $\\operatorname{Kap}^c$ and $\\operatorname{Kap}^{\\tilde{c}}$ with respect to different connections.",
"In other words, for any morphism $\\phi : ({A}\\xrightarrow{} \\Omega ^\\prime ) \\rightarrow ({A}\\xrightarrow{} \\Omega )$ of dg derivations of ${A}$ , we have the following commutative diagram ${\\operatorname{Kap}^c(\\delta ) [d]_-{\\operatorname{Kap}^c(\\phi )} [r]^-{g^{\\nabla ,\\widetilde{\\nabla }}_\\bullet } & \\operatorname{Kap}^{\\tilde{c}}(\\delta ) [d]^-{\\operatorname{Kap}^{\\tilde{c}}(\\phi )} \\\\\\operatorname{Kap}^c(\\delta ^\\prime ) [r]^-{g^{\\nabla ^\\prime ,\\widetilde{\\nabla ^\\prime }}} & \\operatorname{Kap}^{\\tilde{c}}(\\delta ^\\prime ).",
"}$ By this natural equivalence, we are allowed to drop the superscript $c$ to obtain the following Theorem 3.-3 The Kapranov's construction defines a contravariant functor $\\operatorname{Kap}: \\operatorname{dgDer}_{A}\\rightarrow \\operatorname{Leib}_\\infty ({A})$ from the category $\\operatorname{dgDer}_{A}$ of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ -algebras.",
"Remark 3.-2 By the universal property, the Kähler differential ${A}\\xrightarrow{} \\Omega _{{A}\\mid \\mathbb {K}}^1$ is the initial object in the category $\\operatorname{dgDer}_{A}$ of dg derivations.",
"Thus the corresponding Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra on the tangent complex $T_{{A}\\mid \\mathbb {K}} = (\\Omega _{{A}\\mid \\mathbb {K}}^1)^\\vee $ of ${A}$ is the final object of the subcategory in $\\operatorname{Leib}_\\infty ({A})$ consisting of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras arising from dg derivations of ${A}$ .",
"Let ${A}\\xrightarrow{} \\Omega $ be a dg derivation of ${A}$ and ${E}$ a dg ${A}$ -module.",
"By a similar argument, ${E}$ carries a Leibniz$_\\infty [1]$ ${A}$ -module structure over $\\operatorname{Kap}(\\delta )$ .",
"Moreover, we have Theorem 3.-1 Given a dg derivation ${A}\\xrightarrow{} \\Omega $ of ${A}$ , there exists a functor from the category $\\mathrm {dg}{A}$ of dg ${A}$ -modules to the category of Leibniz$_\\infty [1]$ ${A}$ -modules over $\\operatorname{Kap}(\\delta )$ ."
],
[
"Leibniz algebra structures",
"Let $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ be a Leibniz$_\\infty [1]$ ${A}$ -algebra as in Definition REF .",
"Then $(V,\\lambda _1=\\partial _{A})$ is a dg ${A}$ -module.",
"Its cohomology $H^\\bullet (V)$ is called the tangent cohomology of the Leibniz$_\\infty [1]$ ${A}$ -algebra $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ .",
"According to *Proposition 3.10, the (degree $(-1)$ shifted) tangent cohomology $H^\\bullet (V[-1])$ is a Leibniz algebra (over $\\mathbb {K}$ ), when equipped with the bracket $\\check{\\lambda }_2:~H^\\bullet (V[-1]) \\times H^\\bullet (V[-1]) \\rightarrow H^\\bullet (V[-1])$ $\\check{\\lambda }_2([x],[y]):=(-1)^{\\vert x\\vert }[\\lambda _2(x,y)],$ where $x,y\\in V$ are $\\lambda _1$ -closed.",
"In a similar fashion, if $(W,\\lbrace \\mu _k\\rbrace _{k\\ge 1})$ is a $(V,\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ ${A}$ -module as in Definition REF , then $(W,\\mu _1=\\partial _{A})$ is also a dg ${A}$ -module.",
"The cohomology $H^\\bullet (W)$ is a Leibniz module over the aforesaid Leibniz algebra $ H^\\bullet (V[-1])$ (both over $\\mathbb {K}$ ), when equipped with the action $ \\check{\\mu }_2:~H^\\bullet (V[-1]) \\times H^\\bullet (W) \\rightarrow H^\\bullet (W)$ $\\check{\\mu }_2([x],[w]):=(-1)^{\\vert x\\vert }[\\mu _2(x,w)],$ where $x\\in V,~ w\\in W$ are, respectively, $\\lambda _1$ - and $\\mu _1$ -closed elements.",
"As a consequence of Theorem REF , Theorems REF and REF , we have the following Corollary 3.0 Let $\\phi $ be a morphism of dg derivations from ${A}\\xrightarrow{} \\Omega ^\\prime $ to ${A}\\xrightarrow{} \\Omega $ and let $\\mathcal {B}$ and $\\mathcal {B}^\\prime $ be the dual dg ${A}$ -modules of $\\Omega $ and $\\Omega ^\\prime $ , respectively.",
"The (degree $(-1)$ shifted) cohomology space $H^\\bullet ({A},\\mathcal {B}[-1])$ is a Leibniz algebra, whose bracket ${\\Bigl [-,-\\Bigr ]}_\\mathcal {B}$ is induced by the $\\delta $ -twisted Atiyah class of $\\mathcal {B}$ : $\\Bigl [[b_1 ],[b_2 ]\\Bigr ]_\\mathcal {B}= (-1)^{\\vert b_1\\vert }\\operatorname{At}^\\delta _\\mathcal {B}([b_1],[b_2]),$ where $b_1,b_2 \\in \\mathcal {B}$ are $\\partial _{A}$ -closed elements.",
"Moreover, $\\phi ^\\vee :~\\mathcal {B}\\rightarrow \\mathcal {B}^\\prime $ induces a morphism of Leibniz algebras, i.e., $\\Bigl [\\phi ^\\vee (b_1 ),\\phi ^\\vee [b_2]\\Bigr ]_{\\mathcal {B}^\\prime } = \\phi ^\\vee (\\Bigl [[b_1 ],[b_2 ]\\Bigr ]_\\mathcal {B}).$ For any dg ${A}$ -module ${E}$ , there exists a representation of $H^\\bullet ({A},\\mathcal {B}[-1])$ on the cohomology space $H^\\bullet ({A},{E})$ , with the action map $-\\triangleright -$ induced by the $\\delta $ -twisted Atiyah class of ${E}$ : $[b] \\triangleright [e]= (-1)^{\\vert b\\vert }\\operatorname{At}^\\delta _{E}([b],[e]),$ where $b\\in \\mathcal {B}$ , $e\\in {E}$ are both $\\partial _{A}$ -closed elements.",
"Moreover, this assignment is functorial, i.e., for each dg ${A}$ -module morphism $\\lambda : {E}\\rightarrow {F}$ (of degree 0), $[b] \\triangleright \\lambda (e) = \\lambda ([b] \\triangleright [e]).$ Remark 3.1 According to *Theorem 3.4, the Atiyah class of a Lie pair $(L,A)$ induces a Lie algebra structure on the cohomology $H^\\bullet _{\\mathrm {CE}}(A, L/A[-1])$ .",
"A similar result holds for $L_\\infty $ algebra pairs .",
"However, it is not the case in general (see an example below).",
"It is natural to ask when the Leibniz algebra structure in Corollary REF could be refined to a Lie algebra structure.",
"We will investigate this question somewhere else.",
"Example 3.2 Let $\\mathcal {LM}$ be the category of linear maps .",
"A Lie algebra object in $\\mathcal {LM}$ is a triple $E \\xrightarrow{} \\mathfrak {g}$ , where $\\mathfrak {g}$ is a Lie algebra, $E$ is a left $\\mathfrak {g}$ -module, and $\\psi $ is a $\\mathfrak {g}$ -equivariant linear map.",
"Consider the cdga ${A}=C^\\bullet (\\mathfrak {g}) = (\\wedge ^\\bullet \\mathfrak {g}^\\vee ,d_{\\mathrm {CE}})$ and dg $C^\\bullet (\\mathfrak {g})$ -module $\\Omega =C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]) = (\\wedge ^{\\bullet }\\mathfrak {g}^\\vee \\otimes E^\\vee [-1], d_{\\mathrm {CE}} )$ , i.e., the Chevalley-Eilenberg cochain complex of the dual $\\mathfrak {g}$ -module $E^\\vee [-1]$ .",
"The $\\mathfrak {g}$ -equivariant map $E \\xrightarrow{} \\mathfrak {g}$ gives rise to a dg derivation of $C^\\bullet (\\mathfrak {g}) $ : $C^\\bullet (\\mathfrak {g}) \\xrightarrow{} C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]).$ The dual module of $\\Omega =C^{\\bullet }(\\mathfrak {g},E^\\vee [-1]) $ is $\\mathcal {B}=C^{\\bullet }(\\mathfrak {g},E[1])$ .",
"One can take the trivial $\\delta $ -connection on $\\mathcal {B}$ : $\\nabla : \\mathcal {B}=C^\\bullet (\\mathfrak {g},E[1]) \\rightarrow \\Omega \\otimes _{A}\\mathcal {B}=C^{\\bullet }(\\mathfrak {g},E^\\vee [-1] \\otimes E[1]),$ defined by $\\nabla (\\omega \\otimes e) = \\delta (\\omega ) \\otimes e,\\;\\;\\forall \\omega \\in \\wedge ^\\bullet \\mathfrak {g}^\\vee , e \\in E.$ By Equation (REF ), the associated Atiyah cocycle is a degree 1 element $\\operatorname{At}^\\nabla _\\mathcal {B}\\in E^\\vee [-1]\\otimes E^\\vee [-1] \\otimes E[1]$ specified by $\\mathcal {R}^\\nabla _2=\\operatorname{At}^\\nabla _\\mathcal {B}(e_1,e_2) = -\\psi (e_1)e_2 ,\\;\\;\\;\\forall e_1,e_2 \\in E .$ It can be easily seen that higher structures $\\mathcal {R}^\\nabla _j=0$ for all $j\\ge 3$ .",
"Hence, the Kapranov Leibniz$_\\infty [1]$ $C^\\bullet (\\mathfrak {g})$ -algebra $\\mathcal {B}=C^\\bullet (\\mathfrak {g},E[1])$ is simply a dg Leibniz$[1]$ algebra in this case, or equivalently, $\\mathcal {B}[-1]=C^\\bullet (\\mathfrak {g},E)$ is a dg Leibniz algebra.",
"In particular, the subspace $E$ is a Leibniz algebra, recovering the result in .",
"By Corollary REF , there is a Leibniz algebra structure on the graded vector space $H^\\bullet ({A},\\mathcal {B}[-1])=H_{\\mathrm {CE}}^\\bullet (\\mathfrak {g},E)$ , whose bracket is given by $\\Bigl [[e_1],[e_2]\\Bigr ] = (-1)^{\\vert e_1\\vert +1}[\\operatorname{At}^\\nabla _\\mathcal {B}(e_1 ,e_2 ) ] =\\pm [\\psi (e_1)e_2],$ for all $d_{\\mathrm {CE}}$ -closed elements $e_1,e_2 \\in C^\\bullet (\\mathfrak {g},E )$ .",
"Here the last term $\\psi (-)(-):~C^\\bullet (\\mathfrak {g},E)\\times C^\\bullet (\\mathfrak {g},E )\\rightarrow C^\\bullet (\\mathfrak {g},E )$ is a $(\\wedge ^\\bullet \\mathfrak {g}^\\vee )$ -bilinear map naturally extended from $\\psi (-)(-):~E\\times E\\rightarrow E$ .",
"In general, the Leibniz structure on $(H_{\\mathrm {CE}}^\\bullet (\\mathfrak {g},E),[-,-])$ is not skewsymmetric.",
"Homotopic invariance In this section, we prove that the isomorphism class of Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebras arising from dg derivations only depends on their homotopy classes.",
"Proposition 3.3 Let $\\delta \\sim \\delta ^\\prime $ be homotopic $\\Omega $ -valued dg derivations of ${A}$ .Then there exists an isomorphism $\\lbrace g_k\\rbrace _{k\\ge 1}$ sending the Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra $\\operatorname{Kap}(\\delta ^\\prime ) = (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla ^\\prime }_k\\rbrace _{k\\ge 1})$ (with respect to a $\\delta ^\\prime $ -connection $\\nabla ^\\prime $ ) to $\\operatorname{Kap}(\\delta ) = (\\mathcal {B},\\lbrace \\mathcal {R}^{\\nabla }_k\\rbrace _{k\\ge 1})$ (with respect to a $\\delta $ -connection $\\nabla $ ).",
"By assumption, there exists a degree $(-1)$ $\\Omega $ -valued derivation $h:{A}\\rightarrow \\Omega $ of ${A}$ such that $\\delta ^\\prime = \\delta + [\\partial _{A},h] = \\delta + \\partial _{A}\\circ h + h \\circ d_{A}.$ We choose an $h$ -connection on $\\mathcal {B}$ , i.e.",
"a degree $(-1)$ linear map $\\widehat{\\nabla }: \\mathcal {B}\\rightarrow \\Omega \\otimes _{A}\\mathcal {B}$ satisfying $\\widehat{\\nabla }(ab) = h(a)\\otimes b + (-1)^{\\vert a\\vert }a \\widehat{\\nabla }(b),\\;\\; \\forall a \\in {A}, b \\in \\mathcal {B}.$ For each $\\delta $ -connection $\\nabla $ on $\\mathcal {B}$ , it can be easily verified that $\\nabla ^{\\prime \\prime } := [\\partial _{A},\\widehat{\\nabla }]$ is a $[\\partial _{A},h]$ -connection on $\\mathcal {B}$ , and thus $\\nabla ^\\prime = \\nabla + \\nabla ^{\\prime \\prime } = \\nabla + [\\partial _{A},\\widehat{\\nabla }]: \\mathcal {B}\\rightarrow \\Omega \\otimes _{A}\\mathcal {B}$ is a $\\delta ^\\prime $ -connection on $\\mathcal {B}$ .",
"It follows that $\\mathcal {R}^{\\nabla ^\\prime }_2 = [\\nabla ^\\prime ,\\partial _{A}] = [\\nabla + [\\partial _{A}, \\widehat{\\nabla }],\\partial _{A}] = [\\nabla , \\partial _{A}] = \\mathcal {R}_2^{\\nabla }.$ Define a family of ${A}$ -multilinear maps $g_k: \\mathcal {B}^{\\otimes k} \\rightarrow \\mathcal {B}$ inductively by setting $g_1 = \\operatorname{id}_\\mathcal {B}, g_2 = 0$ , and $g_{k+1}(b_0,\\cdots ,b_k) &= (-1)^{\\vert b_0\\vert }\\sum _{p=2}^k \\sum _{\\begin{array}{c}I^1\\cup \\cdots \\cup I^p = \\mathbb {N}^{(k)} \\\\ I_1,\\cdots ,I_p \\ne \\emptyset \\\\ i_{\\vert I^1\\vert }^1 < \\cdots < i^p_{\\vert I^p\\vert }\\end{array}} \\epsilon (I^1,\\cdots ,I^p) [\\widehat{\\nabla }_{b_0},R^{\\nabla }_{p}](g_{\\vert I^1\\vert }(b_{I^1}),\\cdots , g_{\\vert I^p\\vert }(b_{I^p})) \\\\&\\quad + [\\nabla ^\\prime _{b_0}, g_k](b_1,\\cdots ,b_k),$ for all $k\\ge 2$ .",
"It follows from a straightforward inductive argument that $\\lbrace g_k\\rbrace _{k\\ge 1}$ is a morphism of Leibniz$_\\infty [1]$ ${A}$ -algebras from $(\\mathcal {B},\\mathcal {R}_n^{\\nabla ^\\prime })$ to $(\\mathcal {B},\\mathcal {R}_n^\\nabla )$ .",
"Remark 3.4 Although the Kapranov functor $\\operatorname{Kap}$ maps homotopic derivations to isomorphic Leibniz$_\\infty [1]$ ${A}$ -algebra, it does not reduce to a functor from the category consisting of homology classes of dg derivations of ${A}$ to the category $\\operatorname{Leib}_\\infty ({A})$ of Leibniz$_\\infty [1]$ ${A}$ -algebras.",
"Applications We first consider a Lie pair $(L,A)$ , and let $B=L/A$ be the Bott $A$ -module.",
"In the introduction, we explained that for each splitting $j: B \\rightarrow L$ of the short exact sequence (REF ) and for any $L$ -connection $\\nabla $ on $B$ extending the Bott $A$ -module structure, there associates a Leibniz$_\\infty [1]$ algebra structure $\\lbrace \\lambda _k\\rbrace _{k\\ge 1}$ on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B)$ .",
"As all $\\lbrace \\lambda _k\\rbrace _{k\\ge 2}$ are $\\Omega _A^\\bullet $ -multilinear, it is a Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebra.",
"Recall that we have a $\\Omega _A^\\bullet (B^\\vee )$ -valued dg derivation $\\delta _j$ of the cdga $\\Omega _A^\\bullet $ as in Equation (REF ).",
"By Proposition REF , the Atiyah cocycle $\\alpha _B^\\nabla $ of the Lie pair coincides with the Atiyah cocycle $\\operatorname{At}^{\\nabla ^{\\delta _j}}_\\mathcal {B}$ of the dg $\\Omega _A^\\bullet $ -module $\\mathcal {B}:= \\Omega _A^\\bullet (B)$ with respect to a $\\delta _j$ -connection $\\nabla ^{\\delta _j}$ as in Equation (REF ).",
"Comparing definitions of $\\lbrace \\lambda _k\\rbrace _{k\\ge 3}$ in the introduction and $\\lbrace \\mathcal {R}^{\\nabla ^{\\delta _j}}_k\\rbrace $ as in Equation (REF ), we see that the two Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebras $(\\mathcal {B},\\lbrace \\lambda _k\\rbrace _{k\\ge 1})$ and $(\\mathcal {B},\\mathcal {R}^{\\nabla ^{\\delta _j}}_k)$ are exactly the same.",
"Applying Proposition REF , Theorem REF , Theorem REF , and Proposition REF , we have the following Theorem 3.5 Let $(L,A)$ be a Lie pair over a smooth manifold $M$ .",
"Then the Leibniz$_\\infty [1]$ algebra structure constructed in *Theorem 3.13 on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ is unique up to isomorphisms in the category of Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -algebras.",
"Moreover, if $(E,\\partial _A^E)$ is an $A$ -module, then the representation of the above Leibniz$_\\infty [1]$ algebra on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes E)$ is also unique up to isomorphisms in the category of Leibniz$_\\infty [1]$ $\\Omega _A^\\bullet $ -modules.",
"Finally, we consider another interesting application: Let $X$ be a complex manifold and ${A}= (\\Omega _X^{0,\\bullet }, \\bar{\\partial })$ its Dolbeault dg algebra.",
"Let $\\Omega = (\\Omega _X^{0,\\bullet }(T^{1,0}X),\\bar{\\partial })$ be the dg ${A}$ -module generated by the smooth section space $\\Gamma (T^{1,0}X)$ of the holomorphic tangent bundle $T^{1,0}X$ .",
"Note that each holomorphic bivector field $\\pi \\in \\Gamma (\\wedge ^2 T^{1,0}X)$ determines an $\\Omega $ -valued dg derivation of ${A}$ , denoted by $\\delta _\\pi $ , which is the composition ${A}\\xrightarrow{} \\Omega _X^{1,\\bullet } = \\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee ) \\xrightarrow{} \\Omega .$ Here $\\pi ^\\sharp $ is the contraction along $\\pi $ from $(T^{1,0}X)^\\vee $ to $T^{1,0}X$ .",
"In fact, $\\pi ^\\sharp $ is a morphism of dg derivations of ${A}$ (from ${A}\\xrightarrow{} \\Omega _X^{1,\\bullet }$ to ${A}\\xrightarrow{} \\Omega $ ).",
"It sends the Atiyah class $\\alpha _E \\in H^1(X,(T^{1,0}X)^\\vee \\otimes \\operatorname{End}(E))$ of any holomorphic vector bundle $E$ to the $\\delta _\\pi $ -twisted Atiyah class $\\operatorname{At}^{\\delta _\\pi }_{E}\\in H^1(X,T^{1,0}X \\otimes \\operatorname{End}(E))$ of the associated dg ${A}$ -module ${E}= \\Omega _X^{0,\\bullet }(E)$ .",
"By Proposition REF , the $\\delta _\\pi $ -twisted Atiyah class $\\operatorname{At}^{\\delta _\\pi }_{E}$ measures the existence of holomorphic $\\delta _\\pi $ -connections on $E$ .",
"In particular, if $\\pi $ a holomorphic Poisson bivector field, then $(T^{1,0}X)^\\vee $ is a holomorphic Lie algebroid , and $\\operatorname{At}^{\\delta _\\pi }_{E}$ measures the existence of holomorphic $(T^{1,0}X)^\\vee $ -connections on $E$ .",
"Applying Theorem REF , we have the following Theorem 3.6 Let $X$ be a complex manifold, $\\pi $ a holomorphic bivector field.",
"Then, Both $\\Omega _X^{0,\\bullet }(T^{1,0}X)$ and $\\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee )$ carry canonical Kapranov Leibniz$_\\infty [1]$ $\\Omega _X^{0,\\bullet }$ -algebra structures; There is a morphism of Leibniz$_\\infty [1]$ $\\Omega _X^{0,\\bullet }$ -algebras $\\lbrace f_k\\rbrace _{k\\ge 1}: \\Omega _X^{0,\\bullet }((T^{1,0}X)^\\vee ) \\rightarrow \\Omega _X^{0,\\bullet }(T^{1,0}X)$ such that $f_1 = \\pi ^\\sharp $ .",
"Open questions and remarks In this note, we assume that each dg ${A}$ -module ${E}$ is projective in order that connections exist on ${E}$ .",
"In the non-projective case, one can follow Calaque-Van den Bergh's approach CV to define the Atiyah class of ${E}$ (which coincides with the Atiyah class of ${E}$ in Definition REF when ${E}$ admits connections)— The first step is to construct a short exact sequence, called the jet sequence, of dg ${A}$ -modules: $@C=0.5cm{0 [r] & \\Omega _{{A}\\mid \\mathbb {K}}^1\\otimes _{A}{E}[rr] && \\mathfrak {J}{E}[rr] && {E}[r] & 0 }.$ The Atiyah class of ${E}$ is then defined to be the extension class of the above jet sequence.",
"We would like to follow this approach to study twisted Atiyah classes of some cases when connections do not exist (singular foliations considered in for example).",
"Note that Kapranov's original construction on $\\Omega ^{0,\\bullet -1}_X(T_X)$ of a Kähler manifold $X$ is an $L_\\infty $ algebra, whereas Chen, Stiénon and Xu's construction of $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B)$ is a Leibniz$_\\infty [1]$ algebra.",
"In fact, this is due to the existence of Chern connection on $T_X$ which enjoys special properties (see *Section 3.4.4).",
"Meanwhile, when ${A}= C^\\infty (\\mathcal {M})$ is the cdga of functions of a smooth dg manifold $\\mathcal {M}$ .",
"According to , the tangent complex $T_{{A}\\mid \\mathbb {K}} = \\Gamma (T_\\mathcal {M})$ admits an $L_\\infty [1]$ algebra structure (by a construction different from the Kapranov's construction we discussed).",
"Moreover, Laurent-Gengoux, Stiénon and Xu have proved that for each Lie pair $(L,A)$ , there exists a canonical $L_\\infty [1]$ algebra structure on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ (which is different from the Chen-Stiénon-Xu's construction in ).",
"It is natural to ask how to tweak the Kapranov Leibniz$_\\infty [1]$ algebra of general dg derivations so as to produce an $L_\\infty [1]$ algebra rather than a mere Leibniz$_\\infty [1]$ algebra.",
"According to the perturbation lemmas proved by Huebschmann HueLie,HueshLie, many $L_\\infty $ algebras arise from dg Lie algebras or $L_\\infty $ algebras by homological perturbation theory.",
"It is interesting to investigate whether similar perturbation lemma holds for Leibniz$_\\infty [1]$ algebras.",
"Moreover, if this is the case, then it is natural to ask for which kind of dg derivations of a cdga ${A}$ , the associated Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra results from some perturbation.",
"These questions will be investigated somewhere else.",
"We would also like to mention other works that are related to the present paper: Batakidis and Voglaire showed how Atiyah classes of Lie pairs and of dg Lie algebroids give rises to Atiyah classes of dDG algebras .",
"Bordemann studied the Atiyah class as the obstruction to the existence of invariant connections on homogeneous spaces.",
"Hennion generalized Kapranov's construction to algebraic derived stack: There exists a Lie algebra structure on the shifted tangent complex $\\mathbb {T}_X[-1]$ of a derived Artin stack $X$ locally of finite presentation.",
"Moreover, given a perfect module $E$ over $X$ , there exists a representation of the aforesaid Lie algebra on $E$ induced by the Atiyah class of $E$ .",
"AParticle author=Ammar, Mourad, author=Poncin, Norbert, title=Coalgebraic approach to the Loday infinity category, stem differential for $2n$ -ary graded and homotopy algebras, language=English, with English and French summaries, journal=Ann.",
"Inst.",
"Fourier (Grenoble), volume=60, date=2010, number=1, pages=355–387, issn=0373-0956, Atiyaharticle author=Atiyah, Michael Francis, title=Complex analytic connections in fibre bundles, journal=Trans.",
"Amer.",
"Math.",
"Soc., volume=85, date=1957, pages=181–207, BVarticle author=Batakidis, Panagiotis, author=Voglaire, Yannick, title=Atiyah classes and dg-Lie algebroids for matched pairs, journal=J.",
"Geom.",
"Phys., volume=123, date=2018, pages=156–172, issn=0393-0440, Bordemannarticle author=Bordemann, Martin, title=Atiyah classes and equivariant connections on homogeneous spaces, conference= title=Travaux mathématiques.",
"Volume XX, , book= series=Trav.",
"Math., volume=20, publisher=Fac.",
"Sci.",
"Technol.",
"Commun.",
"Univ.",
"Luxemb., Luxembourg, , date=2012, pages=29–82, Bottacinarticle author=Bottacin, Francesco, title=Atiyah classes of Lie algebroids, conference= title=Current trends in analysis and its applications, , book= series=Trends Math., publisher=Birkhäuser/Springer, Cham, , date=2015, pages=375–393, CVarticle author=Calaque, Damien, author=Van den Bergh, Michel, title=Hochschild cohomology and Atiyah classes, journal=Adv.",
"Math., volume=224, date=2010, number=5, pages=1839–1889, issn=0001-8708, CLXarticle author=Chen, Zhuo, author=Lang, Honglei, author=Xiang, Maosong, title=Atiyah classes of Strongly homotopy Lie pairs, journal=Algebra Colloq., date=2019, volume=26, number=2, pages=195–230, CSXarticle author=Chen, Zhuo, author=Stiénon, Mathieu, author=Xu, Ping, title=From Atiyah classes to homotopy Leibniz algebras, journal=Comm.",
"Math.",
"Phys., volume=341, date=2016, number=1, pages=309–349, CXXarticle author=Chen, Zhuo, author=Xiang, Maosong, author=Xu, Ping, title=Atiyah and Todd classes arising from integrable distributions, journal=J.",
"Geom.",
"Phys., volume=136, date=2019, pages=52–67, issn=0393-0440, Costelloarticle author=Costello, Kevin, title=A geometric construction of the Witten genus II, eprint=1112.0816, Henarticle author=Hennion, Benjamin, title=Tangent Lie algebra of derived Artin stacks, journal=J.",
"Reine Angew.",
"Math., volume=741, date=2018, pages=1–45, issn=0075-4102, HueLiearticle author=Huebschmann, Johannes, title=The Lie algebra perturbation lemma, conference= title=Higher structures in geometry and physics, , book= series=Progr.",
"Math., volume=287, publisher=Birkhäuser/Springer, New York, , date=2011, pages=159–179, HueshLiearticle author=Huebschmann, Johannes, title=The sh-Lie algebra perturbation lemma, journal=Forum Math., volume=23, date=2011, number=4, pages=669–691, issn=0933-7741, Huesurveyarticle author=Huebschmann, Johannes, title=Origins and breadth of the theory of higher homotopies, conference= title=Higher structures in geometry and physics, , book= series=Progr.",
"Math., volume=287, publisher=Birkhäuser/Springer, New York, , date=2011, pages=25–38, Kaparticle author=Kapranov, Mikhail M., title=Rozansky-Witten invariants via Atiyah classes, journal=Compositio Math., volume=115, date=1999, number=1, pages=71–113, Kellerarticle author=Keller, Bernhard, title=On differential graded categories, conference= title=International Congress of Mathematicians.",
"Vol.",
"II, , book= publisher=Eur.",
"Math.",
"Soc., Zürich, , date=2006, pages=151–190, LGLSarticle author=Laurent-Gengoux, Camille, author=Lavau, Sylvain, author=Strobl, Thomas, title=The universal Lie $\\infty $ -algebroid of a singular foliation, eprint=1806.00475, LGSXarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Holomorphic Poisson manifolds and holomorphic Lie algebroids, journal=Int.",
"Math.",
"Res.",
"Not.",
"IMRN, date=2008, pages=Art.",
"ID rnn 088, 46, issn=1073-7928, LaurentSX-CRarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Exponential map and $L_\\infty $ algebra associated to a Lie pair, language=English, with English and French summaries, journal=C.",
"R. Math.",
"Acad.",
"Sci.",
"Paris, volume=350, date=2012, number=17-18, pages=817–821, issn=1631-073X, LaurentSXarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds, eprint=1408.2903, LSarticle author=Lada, Tom, author=Stasheff, Jim, title=Introduction to SH Lie algebras for physicists, journal=Internat.",
"J. Theoret.",
"Phys., volume=32, date=1993, number=7, pages=1087–1103, LParticle author=Loday, Jean-Louis, author=Pirashvili, Teimuraz, title=The tensor category of linear maps and Leibniz algebras, journal=Georgian Math.",
"J., volume=5, date=1998, number=3, pages=263–276, issn=1072-947X, MSXarticle author=Mehta, Rajan Amit, author=Stiénon, Mathieu, author=Xu, Ping, title=The Atiyah class of a dg-vector bundle, language=English, with English and French summaries, journal=C.",
"R. Math.",
"Acad.",
"Sci.",
"Paris, volume=353, date=2015, number=4, pages=357–362, issn=1631-073X, Molino1article author=Molino, Pierre, title=Classe d'Atiyah d'un feuilletage et connexions transverses projetables.",
", language=French, journal=C.",
"R. Acad.",
"Sci.",
"Paris Sér.",
"A-B, volume=272, date=1971, pages=A779–A781,"
],
[
"Open questions and remarks",
"In this note, we assume that each dg ${A}$ -module ${E}$ is projective in order that connections exist on ${E}$ .",
"In the non-projective case, one can follow Calaque-Van den Bergh's approach CV to define the Atiyah class of ${E}$ (which coincides with the Atiyah class of ${E}$ in Definition REF when ${E}$ admits connections)— The first step is to construct a short exact sequence, called the jet sequence, of dg ${A}$ -modules: $@C=0.5cm{0 [r] & \\Omega _{{A}\\mid \\mathbb {K}}^1\\otimes _{A}{E}[rr] && \\mathfrak {J}{E}[rr] && {E}[r] & 0 }.$ The Atiyah class of ${E}$ is then defined to be the extension class of the above jet sequence.",
"We would like to follow this approach to study twisted Atiyah classes of some cases when connections do not exist (singular foliations considered in for example).",
"Note that Kapranov's original construction on $\\Omega ^{0,\\bullet -1}_X(T_X)$ of a Kähler manifold $X$ is an $L_\\infty $ algebra, whereas Chen, Stiénon and Xu's construction of $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes B)$ is a Leibniz$_\\infty [1]$ algebra.",
"In fact, this is due to the existence of Chern connection on $T_X$ which enjoys special properties (see *Section 3.4.4).",
"Meanwhile, when ${A}= C^\\infty (\\mathcal {M})$ is the cdga of functions of a smooth dg manifold $\\mathcal {M}$ .",
"According to , the tangent complex $T_{{A}\\mid \\mathbb {K}} = \\Gamma (T_\\mathcal {M})$ admits an $L_\\infty [1]$ algebra structure (by a construction different from the Kapranov's construction we discussed).",
"Moreover, Laurent-Gengoux, Stiénon and Xu have proved that for each Lie pair $(L,A)$ , there exists a canonical $L_\\infty [1]$ algebra structure on the graded $\\mathbb {K}$ -vector space $\\Gamma (\\wedge ^\\bullet A^\\vee \\otimes L/A)$ (which is different from the Chen-Stiénon-Xu's construction in ).",
"It is natural to ask how to tweak the Kapranov Leibniz$_\\infty [1]$ algebra of general dg derivations so as to produce an $L_\\infty [1]$ algebra rather than a mere Leibniz$_\\infty [1]$ algebra.",
"According to the perturbation lemmas proved by Huebschmann HueLie,HueshLie, many $L_\\infty $ algebras arise from dg Lie algebras or $L_\\infty $ algebras by homological perturbation theory.",
"It is interesting to investigate whether similar perturbation lemma holds for Leibniz$_\\infty [1]$ algebras.",
"Moreover, if this is the case, then it is natural to ask for which kind of dg derivations of a cdga ${A}$ , the associated Kapranov Leibniz$_\\infty [1]$ ${A}$ -algebra results from some perturbation.",
"These questions will be investigated somewhere else.",
"We would also like to mention other works that are related to the present paper: Batakidis and Voglaire showed how Atiyah classes of Lie pairs and of dg Lie algebroids give rises to Atiyah classes of dDG algebras .",
"Bordemann studied the Atiyah class as the obstruction to the existence of invariant connections on homogeneous spaces.",
"Hennion generalized Kapranov's construction to algebraic derived stack: There exists a Lie algebra structure on the shifted tangent complex $\\mathbb {T}_X[-1]$ of a derived Artin stack $X$ locally of finite presentation.",
"Moreover, given a perfect module $E$ over $X$ , there exists a representation of the aforesaid Lie algebra on $E$ induced by the Atiyah class of $E$ .",
"AParticle author=Ammar, Mourad, author=Poncin, Norbert, title=Coalgebraic approach to the Loday infinity category, stem differential for $2n$ -ary graded and homotopy algebras, language=English, with English and French summaries, journal=Ann.",
"Inst.",
"Fourier (Grenoble), volume=60, date=2010, number=1, pages=355–387, issn=0373-0956, Atiyaharticle author=Atiyah, Michael Francis, title=Complex analytic connections in fibre bundles, journal=Trans.",
"Amer.",
"Math.",
"Soc., volume=85, date=1957, pages=181–207, BVarticle author=Batakidis, Panagiotis, author=Voglaire, Yannick, title=Atiyah classes and dg-Lie algebroids for matched pairs, journal=J.",
"Geom.",
"Phys., volume=123, date=2018, pages=156–172, issn=0393-0440, Bordemannarticle author=Bordemann, Martin, title=Atiyah classes and equivariant connections on homogeneous spaces, conference= title=Travaux mathématiques.",
"Volume XX, , book= series=Trav.",
"Math., volume=20, publisher=Fac.",
"Sci.",
"Technol.",
"Commun.",
"Univ.",
"Luxemb., Luxembourg, , date=2012, pages=29–82, Bottacinarticle author=Bottacin, Francesco, title=Atiyah classes of Lie algebroids, conference= title=Current trends in analysis and its applications, , book= series=Trends Math., publisher=Birkhäuser/Springer, Cham, , date=2015, pages=375–393, CVarticle author=Calaque, Damien, author=Van den Bergh, Michel, title=Hochschild cohomology and Atiyah classes, journal=Adv.",
"Math., volume=224, date=2010, number=5, pages=1839–1889, issn=0001-8708, CLXarticle author=Chen, Zhuo, author=Lang, Honglei, author=Xiang, Maosong, title=Atiyah classes of Strongly homotopy Lie pairs, journal=Algebra Colloq., date=2019, volume=26, number=2, pages=195–230, CSXarticle author=Chen, Zhuo, author=Stiénon, Mathieu, author=Xu, Ping, title=From Atiyah classes to homotopy Leibniz algebras, journal=Comm.",
"Math.",
"Phys., volume=341, date=2016, number=1, pages=309–349, CXXarticle author=Chen, Zhuo, author=Xiang, Maosong, author=Xu, Ping, title=Atiyah and Todd classes arising from integrable distributions, journal=J.",
"Geom.",
"Phys., volume=136, date=2019, pages=52–67, issn=0393-0440, Costelloarticle author=Costello, Kevin, title=A geometric construction of the Witten genus II, eprint=1112.0816, Henarticle author=Hennion, Benjamin, title=Tangent Lie algebra of derived Artin stacks, journal=J.",
"Reine Angew.",
"Math., volume=741, date=2018, pages=1–45, issn=0075-4102, HueLiearticle author=Huebschmann, Johannes, title=The Lie algebra perturbation lemma, conference= title=Higher structures in geometry and physics, , book= series=Progr.",
"Math., volume=287, publisher=Birkhäuser/Springer, New York, , date=2011, pages=159–179, HueshLiearticle author=Huebschmann, Johannes, title=The sh-Lie algebra perturbation lemma, journal=Forum Math., volume=23, date=2011, number=4, pages=669–691, issn=0933-7741, Huesurveyarticle author=Huebschmann, Johannes, title=Origins and breadth of the theory of higher homotopies, conference= title=Higher structures in geometry and physics, , book= series=Progr.",
"Math., volume=287, publisher=Birkhäuser/Springer, New York, , date=2011, pages=25–38, Kaparticle author=Kapranov, Mikhail M., title=Rozansky-Witten invariants via Atiyah classes, journal=Compositio Math., volume=115, date=1999, number=1, pages=71–113, Kellerarticle author=Keller, Bernhard, title=On differential graded categories, conference= title=International Congress of Mathematicians.",
"Vol.",
"II, , book= publisher=Eur.",
"Math.",
"Soc., Zürich, , date=2006, pages=151–190, LGLSarticle author=Laurent-Gengoux, Camille, author=Lavau, Sylvain, author=Strobl, Thomas, title=The universal Lie $\\infty $ -algebroid of a singular foliation, eprint=1806.00475, LGSXarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Holomorphic Poisson manifolds and holomorphic Lie algebroids, journal=Int.",
"Math.",
"Res.",
"Not.",
"IMRN, date=2008, pages=Art.",
"ID rnn 088, 46, issn=1073-7928, LaurentSX-CRarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Exponential map and $L_\\infty $ algebra associated to a Lie pair, language=English, with English and French summaries, journal=C.",
"R. Math.",
"Acad.",
"Sci.",
"Paris, volume=350, date=2012, number=17-18, pages=817–821, issn=1631-073X, LaurentSXarticle author=Laurent-Gengoux, Camille, author=Stiénon, Mathieu, author=Xu, Ping, title=Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds, eprint=1408.2903, LSarticle author=Lada, Tom, author=Stasheff, Jim, title=Introduction to SH Lie algebras for physicists, journal=Internat.",
"J. Theoret.",
"Phys., volume=32, date=1993, number=7, pages=1087–1103, LParticle author=Loday, Jean-Louis, author=Pirashvili, Teimuraz, title=The tensor category of linear maps and Leibniz algebras, journal=Georgian Math.",
"J., volume=5, date=1998, number=3, pages=263–276, issn=1072-947X, MSXarticle author=Mehta, Rajan Amit, author=Stiénon, Mathieu, author=Xu, Ping, title=The Atiyah class of a dg-vector bundle, language=English, with English and French summaries, journal=C.",
"R. Math.",
"Acad.",
"Sci.",
"Paris, volume=353, date=2015, number=4, pages=357–362, issn=1631-073X, Molino1article author=Molino, Pierre, title=Classe d'Atiyah d'un feuilletage et connexions transverses projetables.",
", language=French, journal=C.",
"R. Acad.",
"Sci.",
"Paris Sér.",
"A-B, volume=272, date=1971, pages=A779–A781,"
]
] | 1808.08576 |
[
[
"New Perspective On The Unruh Effect"
],
[
"Abstract In this work, based on the worldline path integral representation of the vacuum energy in spacetime with a Lorentzian metric, we provide a new but complementary interpretation of the Unruh effect.",
"We perform the quantization of the massless free scalar field in Rindler space specifying initial and final conditions.",
"After quantization, the final outcome for the vacuum energy is interpreted as world line path integrals.",
"In this picture we find that the Unruh radiation is made of real particles as well as real antiparticles.",
"The prediction regarding the presence of antiparticles in the radiation might open new lines for experimental detection of the effect.",
"We present a thought experiment which offers a clear picture and supports the new interpretation."
],
[
"Introduction",
"Since the seminal work [1], [2], the physics community has been struggling to figure out the origin of the Hawking radiation and how to solve the paradoxes that it generates.",
"Several interpretations have emerged over the last forty years [3], [4], [5], [6], [7], but the problem remains unsolved.",
"An easier but still conceptually rich setup is the so called Unruh effect [8], [9], [10].",
"A noninertial observer in flat space, having proper constant acceleration, i.e., a Rindler observer [11], measures a vacuum energy given by the Planck thermal distribution with a temperature proportional to the acceleration $T=\\frac{\\hbar \\text{a}}{2\\pi \\text{c} k}$ .",
"In what follows, we set $\\hbar =\\text{a}=\\text{c}=k=1$ , $E^{R}_{vac}\\propto \\int _{0}^{\\infty } d\\nu \\frac{\\nu }{\\text{e}^{\\frac{\\nu }{T}}-1}.",
"\\nonumber $ In contrast, an observer with a constant velocity, a Minkowski or inertial observer, measures a vanishing vacuum energy $E^{M}_{vac}=0$ .",
"From the canonical quantization point of view, it has been understood [8] that the Rindler observer experiences a different vacuum, in other words, different initial conditions compared to the Minkowski observer.",
"Thus she “sees” particles (radiation).",
"However, the controversy around where these particles are coming from keeps on, and it has not been fully understood yet, see, for instance [12] and references therein.",
"In this work, we provide a new interpretation to the Unruh effect which is complementary to the one originally proposed by Unruh [10].",
"Instead of performing the ordinary canonical quantization where the initial value of the field operator and its conjugate momentum have to be specified, we quantize the massless free scalar field in Rindler space imposing initial and final conditions at the points were the acceleration is turned on and off.",
"The advantage of this quantization scheme is that by means of the propagators, expressed as worldline path integrals, between the initial and final states in Rindler space, we can trace the particles running in loops in spacetime with a Lorentzian metric, which are the ones that contribute to the vacuum energy."
],
[
"Motivation",
"Let us start this section by collecting some features of quantum field theory (QFT).",
"Throughout the paper, we will use these ingredients to uncover a different facet of the Unruh effect.",
"In a noninteracting theory, the only processes that contribute to the vacuum energy are the loops in Fig.",
"REF .",
"These processes occur in spacetime, and they are events that are independent of any observer (coordinate system).",
"Figure: Pictorial representation of the infinitely many loops in spacetime contributing to the vacuum energy.",
"No particular coordinate system is needed to describe them.In the path integral formulation of QFT, the one loop partition function is indeed the vacuum energy of a given system.",
"A clearer picture comes out in the worldline path integral representation of the vacuum energy where we can interpret the quantum particles as moving over a collection of actual trajectories [13], [14], [15], [16], [17], [18].",
"For example, the propagator can be represented as $G(x_1,x_2)=\\int _0^{\\infty }ds\\int _{x(0)=x_1}^{x(1)=x_2}Dx^{\\mu }\\text{exp}\\Big [\\text{i}\\frac{1}{4s}\\int _{0}^{1}d\\tau \\ \\dot{{x}}^2(\\tau )\\Big ], \\nonumber $ where the functional integration is over all paths starting at $x_1$ and ending at $x_2$ in Minkowski space and $\\dot{{x}}^2(\\tau )=(\\dot{x}^{0}(\\tau ))^2-(\\dot{x}^{1}(\\tau ))^2-(\\dot{x}^{2}(\\tau ))^2-(\\dot{x}^{3}(\\tau ))^2$ .",
"The one loop partition function (vacuum energy) can also be written as a worldline functional integral as [15], [19], $E_{vac}\\propto \\frac{1}{2}\\int _0^{\\infty }\\frac{ds}{s}\\int _{PBC}Dx^{\\mu }\\text{exp}\\Big [\\text{i}\\frac{1}{4s}\\int _{0}^{1}d\\tau \\ \\dot{{x}}^2(\\tau )\\Big ],$ where PBC stands for periodic boundary conditions.",
"Of course, these are formal expressions, but they allow us to think about the quantum processes in geometrical terms in Minkowski space, see Appendix A of [13].",
"Although to make sense of them, a Wick rotation is needed.",
"A virtual particle in the Feynman approach of QFT is associated with an internal propagator between two different points in spacetime, or starting and ending at the same point (a loop).",
"The propagation of a real particle, on the other hand, is associated with a propagator (open paths), but, in addition, a definite on shell external momentum state is attached to it.",
"For the sake of completeness, we present a brief description of Rindler space [11].",
"Rindler space $M_R$ is defined as the region interior to the lines $x^0=x^1$ and $x^0=-x^1$ , with $x^1\\geqslant 0$ , in a two-dimensional Minkowski space.",
"It can be extended to higher dimensions.",
"In four dimensions, for instance, the metric of $M_R$ can be written in two equivalent forms $ds^2 & = & -\\rho ^2 d\\tau ^2+d\\rho ^2+(dx^2)^2+(dx^3)^2 \\\\{} & = & \\text{e}^{2 \\xi }(-d\\tau ^2+d\\xi ^2)+(dx^2)^2+(dx^3)^2.$ These two metrics are related to the Minkowski one, $ds^2 = -(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2$ , through the coordinate transformations $\\nonumber x^0 (\\rho ,\\tau ) & = & \\rho \\ \\text{sinh}(\\tau )\\\\ \\nonumber x^1(\\rho ,\\tau ) & = & \\rho \\ \\text{cosh}(\\tau ),$ and $\\nonumber x^0(\\xi ,\\tau ) & = & \\text{e}^{\\xi } \\ \\text{sinh}(\\tau )\\\\ \\nonumber x^1(\\xi ,\\tau ) & = & \\text{e}^{\\xi } \\ \\text{cosh}(\\tau ),$ respectively.",
"Now let us perform a thought experiment in order to visualize how each observer perceives the vacuum processes and how they would compute the vacuum energy.",
"In the calculation of the vacuum energy, a Minkowski observer regards all the loops in spacetime as in Fig.",
"REF .",
"Figure: Pictorial representation of the infinitely many loops in spacetime relevant to the vacuum energy calculation for an inertial observer.",
"The blue lines are initial and final times in the Minkowski slicing.",
"The arrows are representing the flow of time in the loop.For a Rindler observer, on the other hand, a naive calculation of the vacuum energy would proceed as follows.",
"This observer would consider all the loops in $M_R$ .",
"However, this calculation would be incomplete since there are contributions that are not being taken into account.",
"At this point, we stress that none of the observers actually see the virtual vacuum processes given by the particles running in loops.",
"Nevertheless, with the help of (REF ), they can build an intuitive and geometrical picture of these processes.",
"To see the missing contributions, let us add to Fig.",
"REF the Rindler horizons and an initial and final spacelike slice in $M_R$ , as indicated in Fig.",
"REF .",
"The times $\\tau _i$ and $\\tau _f$ can be placed anywhere in the Rindler wedge, even at the horizons, $\\tau _i\\rightarrow -\\infty $ , $\\tau _f\\rightarrow \\infty $ .",
"The portion of the spacetime the Rindler observer has access to is limited by the lines $\\tau _i$ and $\\tau _f$ , which mark the slices were the acceleration is turned on and off.",
"Figure: Pictorial representation of the infinitely many loops in spacetime as in Fig.",
", now in the context of the vacuum energy calculation for the Rindler observer.Now to make it easier to visualize the Rindler observer perspective, let us detach $M_R$ together with its content, from the rest of the space, as in Fig.",
"REF .",
"Figure: Pictorial representation of the infinitely many loops in spacetime.",
"The Rindler observer perspective of the vacuum processes contributing to the vacuum energy.",
"The arrows are representing the flow of time in the paths.From this we can see that the vacuum energy calculation from the Rindler observer point of view has to take into account, in addition to all loops in $M_R$ , the following contributions: All the paths starting at $\\tau _i$ and ending at $\\tau _i$ , including the case where the initial and final points on the slice $\\tau _i$ are the same, i.e., the path is a loop.",
"All the paths starting at $\\tau _f$ and ending at $\\tau _f$ , including the case where the initial and final points on the slice $\\tau _f$ are the same, i.e., the path is a loop.",
"All the paths starting at $\\tau _i$ and ending at $\\tau _f$ and vice versa.",
"Notice that from the Rindler perspective there will be paths running backward in time, representing real antiparticles.",
"The picture described above suggests that while a Minkowski observer experiences a vacuum energy full of virtual particles (loops), an observer in $M_R$ is able to measure a vacuum energy with real propagating particles.",
"From the non-inertial observer perspective, there are infinitely many paths that will never close into a loop; thus, according to QFT, these paths will look to him as real particles.",
"A more surprising consequence arising from this picture is that the Unruh radiation is made of real particles as well as real antiparticles.",
"This fact may be inferred just by looking at the arrows in Fig.",
"REF .",
"Whether these contributions, Fig.",
"REF , lead to a thermal radiation is something that we have to check.",
"In the next section we will give a mathematical proof of all the statements made in this section regarding the origin of the particles seen by the Rindler observer."
],
[
" Quantization and Vacuum Energy",
"In this section, we compute the vacuum energy of a free scalar field in Rindler space.",
"The usual canonical quantization in Minkoswki space is an initial value problem in which we solve the equation $\\Box \\varphi =0$ , with the conditions $\\varphi (t_i)=\\phi (t_i)$ , and $\\partial _ t \\varphi (t_i)=\\psi (t_i)$ .",
"It is a well-posed Cauchy problem, and this data determines the function $\\varphi $ for all $t$ in the region where we are solving the equation.",
"In addition, this initial condition is physical in the sense that we can always give the initial data at a given moment in time, i.e., prepare the initial state.",
"What would be non-physical is to give initial $\\varphi (t_i)=\\phi (t_i)$ , and final $\\varphi (t_f)=\\phi (t_f)$ , conditions.",
"While we can always prepare the initial state, the final state is impossible to prepare.",
"Nevertheless, the solution of $\\Box \\varphi =0$ , can be fully determined by the latter data too.",
"In the Rindler space, the situation could be different.",
"Since the Rindler patch does not cover the whole Minkowski space, the Rindler observer might have access to the final data.",
"Although the Rindler observer does not need to know explicitly the operator at the final time; she only needs to know the propagator in Rindler space, as we will see in the next section.",
"The final state boundary condition for the Rindler observer can be motivated as follows.",
"The Minkowski observer can either solve the equation of motion $\\Box \\varphi _M(x^0,x^1,\\bar{x})=0$ or evolve the initial field operator with his Hamiltonian, $\\varphi _M(x^0_f,x^1,\\bar{x})=\\text{e}^{\\text{i}(x^0_f-x^0_i)H} \\varphi _M(x^0_i,x^1,\\bar{x})\\text{e}^{-\\text{i}(x^0_f-x^0_i)H}\\ .",
"\\nonumber $ This data can be communicated to the Rindler observer.",
"Starting with the action in the Rindler patch $S=\\frac{1}{2}\\int _{\\tau _i}^{\\tau _f} d\\tau \\int _{0}^{\\infty }d\\rho \\int _{-\\infty }^{\\infty } d^2x\\Big [ \\\\\\rho ^{-1}(\\partial _{\\tau }\\varphi _{R})^2-\\rho \\big ((\\partial _{\\rho }\\varphi _{R})^2+(\\partial _{x^2}\\varphi _{R})^2+(\\partial _{x^3}\\varphi _{R})^2 \\big ) \\Big ],$ she can quantize her system in two different and equivalent ways.",
"The ordinary one, where the field and its time derivative are specified at some initial time, and she does not need the final data, $ \\nonumber \\Box \\varphi _R(\\tau ,\\rho ,\\bar{x}) & = & 0 \\\\ \\nonumber \\varphi _R(\\tau _i,\\rho ,\\bar{x}) & = & \\varphi _M(x^0(\\tau _i,\\rho ),x^1(\\tau _i,\\rho ),\\bar{x})\\\\ \\nonumber \\partial _{\\tau } \\varphi _R(\\tau _i,\\rho ,\\bar{x}) & = & \\partial _{\\tau }\\varphi _M(x^0(\\tau _i,\\rho ),x^1(\\tau _i,\\rho ),\\bar{x}),$ or $\\Box \\varphi _R(\\tau ,\\rho ,\\bar{x}) & = & 0 \\\\ \\nonumber \\varphi _R(\\tau _i,\\rho ,\\bar{x}) & = & \\varphi _M(x^0(\\tau _i,\\rho ),x^1(\\tau _i,\\rho ),\\bar{x})\\\\ \\nonumber \\varphi _R(\\tau _f,\\rho ,\\bar{x}) & = & \\varphi _M(x^0(\\tau _f,\\rho ),x^1(\\tau _f,\\rho ),\\bar{x}),$ where the field at the initial time and the field at the final time are specified.",
"A set of boundary conditions similar to (REF ) was previously considered in [20], but in a different context.",
"In both schemes, we can impose the usual equal time canonical commutation relation, $\\nonumber \\big [\\varphi _R(\\tau ,\\rho ,\\bar{x}), \\Pi _R(\\tau ,\\rho ,\\bar{x}^{\\prime })\\big ] =\\frac{\\delta (\\rho _i-\\rho _f)}{\\rho }\\delta ^2(\\bar{x}-\\bar{x}^{\\prime }),$ which holds for all times, and $\\Pi _R=\\rho ^{-1}\\partial _{\\tau }\\varphi _{R}$ .",
"Here, we choose to quantize the field using the second option.",
"In fact, the second option makes the variational problem of (REF ) inside the Rindler wedge a well-posed Lagrangian variational problem, where we fix the variations $\\delta \\varphi _{R}(\\tau _i,\\rho ,\\bar{x})=\\delta \\varphi _{R}(\\tau _f,\\rho ,\\bar{x})=0$ and as usual $\\delta \\varphi _{R}(\\tau ,\\rho \\rightarrow \\infty ,\\bar{x}\\rightarrow \\pm \\infty )=0$ .",
"The general solution of $\\Box \\varphi _M=0$ , in Minkowski space, is $\\varphi _M(x^0,x^1,\\bar{x})=\\int _{-\\infty }^{\\infty }\\frac{dk_1}{2\\pi }\\int _{-\\infty }^{\\infty }\\frac{d^2k}{(2\\pi )^2}\\frac{1}{(2k_0)^{\\frac{1}{2}}}\\\\\\Big (a_{(k_1,\\bar{k})}\\text{e}^{-\\text{i}(k_0x^0+k_1x^1+\\bar{k}\\cdot \\bar{x})}+a^{\\dagger }_{(k_1,\\bar{k})}\\text{e}^{\\text{i}(k_0x^0+k_1x^1+\\bar{k}\\cdot \\bar{x})}\\Big ),$ while in the Rindler patch with the metric (REF ), the general solution of $\\Box \\varphi _R=0$ can be expressed as $\\nonumber \\varphi _R(\\tau ,\\rho ,\\bar{x})=\\int _{0}^{\\infty }d\\nu \\int _{-\\infty }^{\\infty }\\frac{d^2k}{(2\\pi )^2}\\frac{1}{(2\\nu )^{\\frac{1}{2}}}\\\\ \\nonumber \\Big (b_{(\\nu ,-\\bar{k})}\\text{e}^{-\\text{i}\\nu \\tau }+b^{\\dagger }_{(\\nu ,\\bar{k})}\\text{e}^{\\text{i}\\nu \\tau }\\Big )\\psi _{\\nu ,\\bar{k}}(\\rho )\\text{e}^{\\text{i}\\bar{k}\\cdot \\bar{x}},$ where $k_0=(k_1^2+|\\bar{k}|^2)^{\\frac{1}{2}}$ , and $\\psi _{\\nu ,\\bar{k}}(\\rho )$ , are the normalized eigenfunctions of the equation $\\Big (\\rho ^2\\frac{d^2}{d\\rho ^2}+\\rho \\frac{d}{d\\rho }-|\\bar{k}|^2 \\rho ^2+\\nu ^2\\Big )\\psi _{\\nu ,\\bar{k}}(\\rho )=0,\\nonumber $ $\\psi _{\\nu ,\\bar{k}}(\\rho )=\\pi ^{-1}\\big (2\\nu \\text{sinh}(\\pi \\nu )\\big )^{\\frac{1}{2}}\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho ),$ with $\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho )$ , the modified Bessel function of the second kind [8].",
"By imposing (REF ) we get two equations of the form $\\nonumber \\text{e}^{-\\text{i}\\nu \\tau _{i,f}}b_{(\\nu ,-\\bar{k})}+\\text{e}^{\\text{i}\\nu \\tau _{i,f}}b^{\\dagger }_{(\\nu ,\\bar{k})}= \\\\ \\nonumber (2\\nu )^{\\frac{1}{2}}\\int _{0}^{\\infty }d\\rho \\int _{-\\infty }^{\\infty }d^2x\\frac{\\psi _{\\nu ,\\bar{k}}(\\rho )}{\\rho }\\text{e}^{-\\text{i}\\bar{k}\\cdot \\bar{x}}\\varphi _{M}(\\tau _{i,f},\\rho ,\\bar{x}),$ where $\\varphi _{M}(\\tau _{i,f},\\rho ,\\bar{x})$ , is the field operator as seen by the Minkowski observer but evaluated at the constant $\\tau =\\tau _i,\\tau _f$ slices.",
"Let us recall that the energy $E_{vac}^R$ seen by the Rindler observer is given by $E_{vac}^R=\\int _{-\\infty }^{\\infty } \\frac{d^2k}{(2\\pi )^2}\\int _{0}^{\\infty } d\\nu \\ \\nu \\ \\langle 0 | b^{\\dagger }_{(\\nu ,\\bar{k})} b_{(\\nu ,\\bar{k})}|0\\rangle \\ +E_{0}^R, \\nonumber $ where the term $E_{0}^R = \\frac{1}{2}\\delta (0)^3 \\int _{-\\infty }^{\\infty }d^2k \\int _{0}^{\\infty }d\\nu \\nu ,$ comes only from the loops inside the wedge.",
"As in the Minkowski calculation, for most of the purposes, the term $E_{0}^R$ can be discarded.",
"After solving for $b$ and $b^{\\dagger }$ and plugging the solution into the number of particles ${n}$ , we get ${n}=\\langle 0 | b^{\\dagger }_{(\\nu ,\\bar{k})} b_{(\\nu ,\\bar{k})}|0\\rangle =\\frac{\\nu }{2 \\big (\\text{sin}[\\nu (\\tau _f-\\tau _i)]\\big )^2}\\\\ \\nonumber \\times \\int _{0}^{\\infty }d\\rho \\int _{-\\infty }^{\\infty }d^2x\\frac{\\psi _{\\nu ,\\bar{k}}(\\rho )}{\\rho }\\text{e}^{-\\text{i}\\bar{k}\\cdot \\bar{x}}\\ \\ \\ \\\\ \\nonumber \\times \\int _{0}^{\\infty }d\\rho ^{\\prime }\\int _{-\\infty }^{\\infty }d^2x^{\\prime }\\frac{\\psi _{\\nu ,\\bar{k}}(\\rho ^{\\prime })}{\\rho ^{\\prime }}\\text{e}^{\\text{i}\\bar{k}\\cdot \\bar{x}^{\\prime }}\\Big [ \\\\ \\nonumber \\langle 0 |\\varphi _{M}(\\tau _i,\\rho ,\\bar{x})\\varphi _{M}(\\tau _i\\ ,\\rho ^{\\prime },\\bar{x}^{\\prime }) |0\\rangle \\\\ \\nonumber +\\langle 0 |\\varphi _{M}(\\tau _f,\\rho ,\\bar{x})\\varphi _{M}(\\tau _f,\\rho ^{\\prime },\\bar{x}^{\\prime }) |0\\rangle \\\\ \\nonumber -\\text{e}^{-\\text{i}\\nu (\\tau _f-\\tau _i)}\\langle 0 |\\varphi _{M}(\\tau _i,\\rho ,\\bar{x})\\varphi _{M}(\\tau _f,\\rho ^{\\prime },\\bar{x}^{\\prime }) |0\\rangle \\\\ \\nonumber -\\text{e}^{+\\text{i}\\nu (\\tau _f-\\tau _i)}\\langle 0 |\\varphi _{M}(\\tau _f,\\rho ,\\bar{x})\\varphi _{M}(\\tau _i,\\rho ^{\\prime },\\bar{x}^{\\prime }) |0\\rangle \\\\ \\nonumber \\Big ].$ Now, from (REF ) it is clear that the processes contributing to the vacuum energy are those listed in the previous section.",
"Notice that the propagators, which is the only information needed for knowing the energy, are given in terms of the field operator in Minkowski space, but we will regard them from the perspective of the Rindler observer.",
"The propagator in Rindler space can be represented as a functional integral over paths in $M_R$ only, as in Fig.",
"REF .",
"Notice in (REF ) there are definite momentum states attached to each propagator [15] .",
"These states could be consider as on shell states.",
"Expression (REF ) answers the question of where the particles in the radiation are coming from.",
"The particles have been there, in the spacetime, as vacuum fluctuations (loops) all the time.",
"However, by accelerating the Rindler observer breaks some loops in such a way that the particles trajectory looks to him as open path.",
"In addition, there are on shell definite momentum states attached to the propagators, thus real propagating particles and antiparticles [15] .",
"The appearance of $\\tau _i$ and $\\tau _f$ in (REF ) may be disturbing.",
"However, as we will show in the next section, the vacuum energy is independent of $\\tau _i$ and $\\tau _f$ and coincides with the Planck thermal distribution."
],
[
"Vacuum Energy Distribution",
"We shall explicitly compute the vacuum energy $E_{vac}^R$ .",
"To this end, we will first compute the integral $I=\\int _{0}^{\\infty }d\\rho \\int _{-\\infty }^{\\infty }d^2x\\frac{\\psi _{\\nu ,\\bar{k}}(\\rho )}{\\rho }\\text{e}^{-\\text{i}\\bar{k}\\cdot \\bar{x}}\\ \\ \\ \\\\ \\nonumber \\times \\int _{0}^{\\infty }d\\rho ^{\\prime }\\int _{-\\infty }^{\\infty }d^2x^{\\prime }\\frac{\\psi _{\\nu ,\\bar{k}}(\\rho ^{\\prime })}{\\rho ^{\\prime }}\\text{e}^{\\text{i}\\bar{k}\\cdot \\bar{x}^{\\prime }} \\\\ \\nonumber \\times \\langle 0 |\\varphi _{M}(\\tau _i,\\rho ,\\bar{x})\\varphi _{M}(\\tau _f\\ ,\\rho ^{\\prime },\\bar{x}^{\\prime }) |0\\rangle .",
"\\nonumber $ After plugging in, (REF ) and (REF ), and taking into account $a_{(p_1^{\\prime },\\bar{p}^{\\prime })}|0\\rangle =0$ , and $\\big [ a_{(p_1,\\bar{p})}\\ , a^{\\dagger }_{(p_1^{\\prime },\\bar{p}^{\\prime })}\\big ]=(2\\pi )^3\\delta (p_1-p_1^{\\prime })\\delta ^2(\\bar{p}-\\bar{p}^{\\prime }), \\nonumber $ we get $\\nonumber I=\\frac{2\\nu }{\\pi }\\text{sinh}(\\pi \\nu )\\delta ^2(0)\\int _{-\\infty }^{\\infty }dp\\frac{1}{\\sqrt{p^2+|\\bar{k}|^2}}\\\\ \\nonumber \\times \\int _0^{\\infty }d \\rho \\frac{\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho )}{\\rho }\\text{e}^{-\\text{i}\\rho \\big (\\sqrt{p^2+|\\bar{k}|^2}\\ \\text{sinh}(\\tau _i)+p \\ \\text{cosh}(\\tau _i)\\big )}\\\\ \\nonumber \\times \\int _0^{\\infty }d \\rho ^{\\prime }\\frac{\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho ^{\\prime })}{\\rho ^{\\prime }}\\text{e}^{\\text{i}\\rho ^{\\prime }\\big (\\sqrt{p^2+|\\bar{k}|^2}\\ \\text{sinh}(\\tau _f)+p \\ \\text{cosh}(\\tau _f)\\big )}.\\nonumber $ This integral can be further reduced by the change of variable $p=|\\bar{k}|\\text{sinh}(z)$ , to $\\nonumber I=\\frac{2\\nu }{\\pi }\\text{sinh}(\\pi \\nu )\\delta ^2(0)\\int _{-\\infty }^{\\infty }dz\\\\ \\nonumber \\times \\int _0^{\\infty }d \\rho \\frac{\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho )}{\\rho }\\text{e}^{-\\text{i}\\rho |\\bar{k}|\\text{sinh}(z+\\tau _i)}\\\\ \\nonumber \\times \\int _0^{\\infty }d \\rho ^{\\prime }\\frac{\\text{K}_{\\text{i}\\nu }(|\\bar{k}|\\rho ^{\\prime })}{\\rho ^{\\prime }}\\text{e}^{\\text{i}\\rho ^{\\prime }|\\bar{k}|\\text{sinh}(z+\\tau _f)}.",
"\\nonumber $ Interestingly enough $I$ does not depend on $|\\bar{k}|$ .",
"For $|\\bar{k}|=0$ , extra care is needed, but it is not difficult to check, using the asymptotic form of the modified Bessel function of the second kind, that the integration gives the same result and it is independent of $|\\bar{k}|$ when $|\\bar{k}|\\rightarrow 0$ .",
"For the massive scalar field, this precaution is not needed.",
"The Laplace transform like integrals in $\\rho $ and $\\rho ^{\\prime }$ can be easily computed.",
"After some algebra, it reduces to $\\nonumber I=\\frac{\\pi }{2\\nu \\text{sinh}(\\pi \\nu )}\\delta ^2(0)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\ \\nonumber \\times \\int _{-\\infty }^{\\infty }dz\\Big (\\text{e}^{\\text{i}\\nu (\\tau _f-\\tau _i)}(\\text{i})^{-2\\text{i}\\nu }+\\text{e}^{-\\text{i}\\nu (\\tau _f-\\tau _i)}(\\text{i})^{2\\text{i}\\nu }\\\\ \\nonumber +\\text{e}^{\\text{i}\\nu (2 z+\\tau _i+\\tau _f)}+\\text{e}^{-\\text{i}\\nu (2 z+\\tau _i+\\tau _f)} \\Big ).$ The last two term can be dropped since after $z$ -integration, they contribute with a Dirac delta function $\\delta (\\nu )$ , and the integral $I$ is defined for all $\\nu $ with $\\nu \\ne 0$ .",
"To make sense these expressions, we have to remove the zero mode $\\nu = 0$ .",
"Picking the branches $(\\text{i})^{-2\\text{i}\\nu }=(\\text{i})^{2\\text{i}\\nu }=\\text{e}^{-\\pi \\nu }$ , we obtain $I=\\frac{2\\pi }{\\nu }\\frac{1}{\\text{e}^{2\\pi \\nu }-1}\\text{cos}\\big [\\nu (\\tau _f-\\tau _i)\\big ]\\delta ^2(0)\\int _{-\\infty }^{\\infty }dz.",
"$ Plugging (REF ) into (REF ), the time dependence cancels out, giving the desired result ${n}=\\frac{1}{2\\pi }\\frac{1}{\\text{e}^{2\\pi \\nu }-1} V_3, \\nonumber $ where $V_3$ , is the volume of space and we have used $\\delta (0)=\\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty }dx$ ."
],
[
"Conclusions",
"The worldline path integral representation of the vacuum energy provides an intuitive and visual framework for describing quantum processes.",
"It makes it easier to translate these processes into geometrical terms.",
"As in the thought experiment of the ray light in Einstein's train or elevator, we have performed a similar thought experiment which allowed us to visualize the reality that a Rindler observer experiences.",
"The main difference with the classical experiment of Einstein is that we considered the quantum nature of the particles.",
"In other words, we have taken into account all the possible paths in the spacetime associated with the vacuum energy.",
"Putting all the ingredients together, the Unruh effect naturally emerges.",
"We have used the worldline path integral representation of the vacuum energy as motivation but the calculations were done within the operator formalism of QFT and then reinterpreted as worldline path integrals [15].",
"We have quantized the free massless scalar field in a rather arguable, but mathematically well justified, fashion.",
"Each step of the calculation admits a nice physical interpretation consistent with the thought experiment.",
"Furthermore, the final result is the one expected for the Unruh effect.",
"It would be worthwhile to conduct the same calculation entirely within the worldline formalism in spacetime with Lorentzian metric.",
"In this case, the vacuum energy would be given by a path integral where the integration is over two topologically different kinds of paths, i.e., those paths listed in section where the loops are restricted to the Rindler wedge.",
"If we believe in the premises of QFT, an irrefutable physical interpretation comes out from the picture presented above.",
"The particles that the Rindler observer experiences are those virtual particles moving in loops in the whole (Minkowski) space that from the Rindler observer's perspective never complete a loop, being seen as real particles and real antiparticles.",
"We would like to emphasize that this observation (the presence of antiparticles in the radiation) might open new possibilities for experimental confirmation of the effect.",
"We started this paper by citing Hawking's work on black holes, but we studied a flat space phenomenon called the Unruh effect that has many features in common.",
"The natural extension of this work should be including gravity into the game [21].",
"In particular the black hole geometry, in the same spirit of [4], [5], [6].",
"We believe that the physical interpretation presented here could shed some light onto the quantum aspects of the black hole geometry and gravity in general.",
"Let us remark that the interpretation of this picture in the black hole geometry would differ from that popular and conceptually inaccurate notion (see, for instance, the comments in page 31 of [22]) where the antiparticle falls into the black hole and the particle runs away to infinity, draining mass from the hole."
],
[
"Acknowledgments",
"I am grateful to the postdoc members of the “Fields, Gravity and Strings” group for discussions.",
"Especially to Pablo Diaz and Vladislav Vaganov for discussions and useful comments.",
"I would also to thank H. Casini for the reading of the manuscript and useful remarks."
],
[
"Vacuum Energy Distribution for $D=2$ ",
"In two dimensions, it is convenient to use the two dimensional version of the metric ().",
"Because of conformal symmetry, the solution in both patches takes a similar form $\\nonumber \\varphi _M(x^0,x^1)=\\int _{-\\infty }^{\\infty }\\frac{dk}{\\pi (8|k|)^{\\frac{1}{2}}}\\\\ \\nonumber \\Big (a_{k}\\text{e}^{-\\text{i}(|k|x^0+kx^1)}+a^{\\dagger }_{k}\\text{e}^{\\text{i}(|k|x^0+kx^1)}\\Big ),$ and $\\nonumber \\varphi _R(\\tau ,\\xi )=\\int _{-\\infty }^{\\infty }\\frac{dp}{\\pi (8|p|)^{\\frac{1}{2}}}\\nonumber \\Big (b_{-p}\\text{e}^{-\\text{i}|p|\\tau }+b^{\\dagger }_{p}\\text{e}^{\\text{i}|p|\\tau }\\Big )\\text{e}^{\\text{i}p\\xi }.$ In this case, the analogue of the integral (REF ) is I = 14-dk|k| -dexp(-ie(|k| sinh(i)+k cosh(i))-ip) -dexp(ie(|k| sinh(f)+k cosh(f))+ip).",
"It can be easily solved, giving $\\nonumber I & = & \\frac{1}{4\\pi }(-1)^{-\\text{i}p}\\Gamma (\\text{i}p)\\Gamma (-\\text{i}p)\\\\ \\nonumber {} & {} & \\int _{-\\infty }^{\\infty }\\frac{dk}{|k|}\\Big (\\frac{|k| \\text{sinh}(\\tau _i)+k \\ \\text{cosh}(\\tau _i)}{|k| \\text{sinh}(\\tau _f)+k \\ \\text{cosh}(\\tau _f)}\\Big )^{\\text{i}p}.$ Using $\\Gamma (\\text{i}p)\\Gamma (-\\text{i}p)=\\frac{\\pi }{|p|\\text{sinh}(\\pi p)}$ , after some algebra, it reduces to $I=\\frac{1}{|p|}\\frac{1}{\\text{e}^{2\\pi p}-1}\\text{cos}\\big [|p|(\\tau _f-\\tau _i)\\big ]\\int _{0}^{\\infty }\\frac{dk}{k}.$ Plugging (REF ) into the number of particles (REF ), in this case with $\\nu \\rightarrow |p|$ , we get $\\langle 0 | b^{\\dagger }_{p} b_{p}|0\\rangle =\\frac{1}{\\text{e}^{2\\pi |p|}-1}V_1,\\nonumber $ where $V_1=\\int _{0}^{\\infty }\\frac{dk}{k}=\\int _{-\\infty }^{\\infty }dx$ , is the volume of space.",
"It is exactly the result, computed within the ordinary canonical quantization framework, reported in the Appendix A of [23]."
]
] | 1808.08397 |
[
[
"Holographic control of information and dynamical topology change for\n composite open quantum systems"
],
[
"Abstract We investigate how the compositeness of a quantum system influences the characteristic time of equilibration.",
"We study the dynamics of open composite quantum systems strongly coupled to the environment after a quantum perturbation accompanied by non-equilibrium heating.",
"We use a holographic description of the evolution of entanglement entropy.",
"The non-smooth character of the evolution with holographic entanglement is a general feature of composite systems, which demonstrate a dynamical change of topology in the bulk space and a jump-like velocity change of entanglement entropy propagation.",
"Moreover, the number of jumps depends on the system configuration and especially on the number of composite parts.",
"The evolution of the mutual information of two composite systems inherits these jumps.",
"We present a detailed study of the mutual information for two subsystems with one of them being bipartite.",
"We have found 5 qualitatively different types of behavior of the mutual information dynamics and indicated the corresponding ranges of system parameters."
],
[
"Introduction",
"In recent years, there has been a growing interest in studying quantum entanglement entropy and quantum mutual information for various open quantum systems under conditions of environmental change and with control of it (see [1] and the references therein).",
"In [2], the holographic approach was applied to photosynthesis, which is an important example of non-trivial quantum phenomena relevant for life and is studied in the emerging field of quantum biology [1], [3].",
"Light-harvesting complexes of photosynthetic organisms are many-body quantum systems in which quantum coherence has recently been experimentally shown to survive for relatively long time scales at a physiological temperature despite the decoherence effects of the environment.",
"There are successful applications of the holographic AdS/CFT correspondence to high energy and condensed matter physics [4], [5], [6].",
"In [2], the holographic approach was used to evaluate the time dependence of entanglement entropy and quantum mutual information in the Fenna–Matthews–Olson complex during the transfer of an excitation from a chlorosome antenna to a reaction center.",
"It was demonstrated that the evolution of the quantum mutual information simulating the behavior of solutions of the GKSL master equation in some cases can be obtained using dual gravity describing black hole formation in the AdS space–time with the Vaidya metric (hereafter called AdS-Vaidya spacetime).",
"Estimates of the wake-up and scrambling times for various decompositions of the Fenna–Matthews–Olson complex were obtained.",
"Many photosynthetic light-harvesting complexes are conventionally modeled by a general three-part structure comprising an antenna, a transfer network, and a reaction center.",
"The antenna captures sunlight photons and excites pigment electrons from their ground state.",
"The excited electrons, which combine with holes to form excitons, travel from the antenna through the intermediate transfer complex to the reaction center, where they participate in chemical reactions.",
"Here, based on the results in [2] and [7], [8], we study whether the compositeness of the system increases the delocalization during equilibration.",
"For this, we consider the dynamics of open quantum systems consisting of separate parts strongly coupled to the environment after a quantum perturbation corresponding to nonequilibrium heating (also see [9]–[13]).",
"We use a holographic description of the time evolution of entanglement entropy during the nonequilibrium heating.",
"A nonsmooth character of the evolution is a general feature of the time evolution of the holographic entanglement of composite systems.",
"These systems exhibit a dynamical topology change in the bulk space and also a jumplike change of the velocity of the entanglement entropy propagation, and the number of jumps depends on the system configuration and especially on the number of composite parts.",
"The evolution of the mutual information of two composite systems inherits these discontinuities.",
"We present a detailed study of the mutual information for two subsystems, one of which is bipartite.",
"We show that there exist five qualitatively different types of dynamical behavior of the mutual information and indicate the corresponding regions of the system parameters."
],
[
"Holographic entanglement entropy in Vaidya-AdS$_3$",
"We use a holographic approach to study evolution of an open system after a quantum quench accompanied by non-equilibrium heating process.",
"As the dual model describing the time evolution of the entanglement during such a process, we consider a Vaidya shell collapsing on a black hole [2], [14].",
"The collapse of this shell leads to the formation of a heavier black hole, which corresponds to a temperature increase.",
"The initial thermal state is defined as the horizon position $z_H$ , and the final state as the horizon $z_h$ .",
"For simplicity, we consider the three-dimensional case.",
"The corresponding Vaidya metric defining the dual gravitational background consists of two parts and is given by $v<0:&\\,\\,\\,&ds^2=\\frac{1}{z^2}\\left( -f_{H}(z)d\\tau ^2+ \\frac{dz^2}{f_{H}(z)}+dx^2\\right),\\,\\,\\,\\,\\tau =v+z_H \\text{arctanh} \\frac{z}{z_H},\\\\v>0:&\\,\\,\\,&ds^2=\\frac{1}{z^2}\\left( -f_{h}(z)d\\tau ^2+ \\frac{dz^2}{f_{h}(z)}+dx^2\\right),\\,\\,\\,\\,\\,\\,\\tau =v+z_h \\text{arctanh} \\frac{z}{z_h},$ where the functions $f_{H}$ and $f_{h}$ are defined as $ f_H = 1-\\left(\\frac{z}{z_{H}}\\right)^2, \\,\\,\\,\\,f_{h} = 1-\\left(\\frac{z}{z_{h}}\\right)^2, \\,\\,\\,\\, 0<z_h<z_{H}.$ and are glued along the position of the shell $v=0$ .",
"This metric can also be written as $v<0: ~~~ ds^2 = \\frac{1}{z^2}\\left[ -\\left(1-\\frac{z^2}{z_H^2}\\right)dv^2 - 2\\,dv\\,dz + dx^2 \\right], \\\\v>0: ~~~ ds^2 = \\frac{1}{z^2}\\left[ -\\left(1-\\frac{z^2}{z_h^2}\\right)dv^2 - 2\\,dv\\,dz + dx^2 \\right].$ The initial and finite temperatures are $T_i=\\frac{1}{2\\pi z_H},~~~~ T_f=\\frac{1}{2\\pi z_h}.$ To define the holographic entanglement entropy [15], [16] corresponding to the simplest system, i.e., one segment, we must find a geodesic in metric (REF ), () anchored on this segment at a given instant.",
"The action for the geodesic connecting two points $(t,-\\ell /2)$ and $(t,\\ell /2)$ on the boundary is given by $ S(\\ell ,t)=\\int \\limits _{-\\ell /2}^{+\\ell /2}\\frac{1}{z}\\,\\sqrt{1-2v^{\\prime }z^{\\prime }-f(z,v)v^{\\prime 2}}\\,dx,$ where $t\\ge 0$ , $\\ell >0$ , $z=z(x)$ , $v=v(x)$ and $ z(\\pm \\ell /2)=0,\\,\\,\\,\\,v(\\pm \\ell /2)= t.$ To realize these symmetric boundary conditions, we also impose the conditions $z^{\\prime }(0)=v^{\\prime }(0)=0$ .",
"This problem was solved explicitly for a non-zero initial temperature in [7], [8] and the answer is given byThe entanglement entropy (REF ) has been obtained as a length of the geodesic with minimal divergence subtraction.",
"$S(\\ell ,t)= \\log \\left( \\frac{z_h}{\\ell \\,\\mathfrak {S}_\\kappa (\\rho ,s)}\\,\\sinh \\frac{t}{z_h}\\right), \\,\\,\\,\\,\\,0\\le t\\le \\ell ,$ where $\\mathfrak {S}_\\kappa $ is given by $\\mathfrak {S}_\\kappa (\\rho ,s)=\\frac{c \\rho + \\Delta }{\\Delta }\\cdot \\sqrt{\\frac{\\Delta ^2 -c^2 \\rho ^2}{\\rho \\left(c^2 \\rho +2 c\\Delta + \\rho \\right) - \\kappa ^2}}.$ The function $\\mathfrak {S}_\\kappa (\\rho ,s)$ depends on the new variables $(\\rho ,s)$ , which are related to the variables $(\\ell ,t)$ .",
"The new variables $\\rho ,s$ describe the geodesic in the bulk space and are expressed in terms of the position of the geodesic top $z_*$ and the point $z_c$ where the geodesic crosses the lightlike shell by the formulas $s=\\frac{z_c}{z_*}, \\;\\;\\;\\; \\rho =\\frac{z_h}{z_c},$ with the restrictions $z_*<z_H$ and $z_c<z_H$ satisfied.",
"The relations between $\\rho ,s$ and $\\ell ,t$ can be written in the explicit form $\\frac{t}{z_h}&=&{\\mbox{arccoth}}\\left(\\frac{-c \\kappa ^2+2 c \\rho ^2+c+2 \\Delta \\rho }{2 c \\rho +2 \\Delta }\\right),\\\\\\ell &=&\\frac{z_h}{2} \\log \\frac{c^2 \\gamma ^4-4 \\Delta \\left(c s \\left(\\kappa ^2-2 \\rho ^2+1\\right)+\\Delta +\\Delta \\left(\\rho ^2-2\\right) s^2\\right)}{c^2 \\gamma ^4-4 \\Delta ^2 (\\rho s-1)^2}+\\frac{z_h}{2 \\kappa } \\log \\frac{(c \\kappa +\\Delta s)^2}{\\rho ^2 s^2-\\kappa ^2},\\nonumber \\\\$ where we use the notations $\\kappa =\\frac{z_h}{z_H}<1,\\,\\,\\,\\,c=\\sqrt{1-s^2},\\,\\,\\,\\,\\gamma =1-\\kappa ^2,\\,\\,\\,\\,\\Delta =\\sqrt{\\rho ^2-\\kappa ^2}.$ For $t<0$ , the geodesic is entirely in the bulk space of the black hole with temperature $T_i$ .",
"The entanglement entropy is independent of $t$ and equal to $S_i= \\log \\left( \\frac{z_H}{\\ell }\\sinh \\frac{\\ell }{z_H}\\right)$ after the minimal renormalization.",
"The top of the corresponding geodesic $z_*<z_H$ , and $z_*\\rightarrow z_H$ as $\\ell $ increases in correspondence with [17].",
"We are interested in calculating $S=S(\\ell ,t)$ .",
"Hence, we have to find $\\rho =\\rho (\\ell ,t)$ and $s=s(\\ell ,t)$ first.",
"From the equation (REF ) it is possible to find $\\rho =\\rho (s,t)$ .",
"Inserting this expression for $\\rho $ into () we obtain $\\ell =\\ell ((s,\\rho (s,t))=\\ell (s,t)$ , from which we get numerically $s=s(\\ell ,t)$ and $\\rho =\\rho (s(\\ell ,t),t)=\\rho (\\ell ,t)$ .",
"And finally we can get $S=S(s(\\ell ,t),\\rho (\\ell ,t))=S(\\ell ,t)$ .",
"Let $\\ell $ be fixed and the time increases.",
"At very small $t>0$ , the geodesic intersects the null shell and for $t \\ll z_h$ the point of intersection is close to the boundary, $z_c\\ll z_h$ .",
"When $t$ is of the order of $z_h$ , the geodesic intersects the shell behind the horizon, i.e.",
"$z_c>z_h$ , and finally at some time the geodesic is entirely in the black hole (with the temperature $T_f$ ).",
"Figure: A. z H =4z_H=4, z h =1z_h=1, ℓ\\ell equals 2, 4, 6, 8, 10 and we vary tt.",
"B. z H =2z_H=2, z h =1z_h=1, tt equals 0, 1, 2, 3, 4 and we vary ℓ\\ell .The typical dependence of the holographic entanglement entropy on $t$ for given $\\ell $ and on $\\ell $ for given $t$ is presented in Fig.REF ."
],
[
"Mutual information",
"According to the general definition, holographic mutual information (HMI) is $ I(A;B)=S(A)+S(B)-S(A \\cup B),$ where $S(A)$ is the holographic entanglement entropy (HEE) of A, $S(A \\cup B)$ is the holographic entanglement entropy for the union of two subsystems.",
"This definition can be generalized for $m$ subsystems as $I(A_1;A_2\\cup ...\\cup A_m)=S(A_1)+S(A_2\\cup ...\\cup A_m)-S(A_1\\cup A_2\\cup ...\\cup A_m).$ We now consider the evolution of the mutual information of the system consisting of two subsystems, one of them (simple) is a segment $A$ and the second (composite) comprises two segments $B\\cup C$ .",
"In this case, formula (REF ) takes the form $I(A; B\\cup C)=S(A)+S(B\\cup C)-S(A\\cup B\\cup C).$"
],
[
"Phase diagrams for entanglement entropy of composite systems",
"We consider the system of 3 segments with the lengths $\\ell $ , $m$ , $n$ and the distances $x$ , $y$ between adjacent segments.",
"If we omit time dependence for brevity, then the formula of the holographic entanglement entropy is (minimum of 15 items): $S(\\ell ,x,m,y,n)=\\min \\Big \\lbrace S(\\ell )+S(m)+S(n),S(\\ell )+S(m+y)+S(y+n), \\\\S(\\ell )+S(m+y+n)+S(y),S(\\ell +x)+S(x+m)+S(n), \\\\S(\\ell +x)+S(x+m+y)+S(y+n),S(\\ell +x)+S(x+m+y+n)+S(y), \\\\S(\\ell +x+m)+S(x)+S(n),S(\\ell +x+m)+S(x+m+y)+S(m+y+n), \\\\S(\\ell +x+m)+S(x+m+y+n)+S(m+y),S(\\ell +x+m+y)+S(x)+S(y+n), \\\\S(\\ell +x+m+y)+S(x+m)+S(m+y+n),S(\\ell +x+m+y)+S(x+m+y+n)+S(m), \\\\S(\\ell +x+m+y+n)+S(x)+S(y),S(\\ell +x+m+y+n)+S(x+m)+S(m+y), \\\\S(\\ell +x+m+y+n)+S(x+m+y)+S(m) \\Big \\rbrace $ Among all configurations in this formula, there are so-called \"crossing configurations\" (see [2]).",
"Let's check whether crossing configurations contribute to the holographic entanglement entropy in static background.",
"For this purpose, we plot the phase diagrams (Fig.REF ) showing which configurations from (REF ) contribute to the holographic entanglement entropy.",
"Each region of the solid color corresponds to one configuration of (REF ).",
"Figure: Phase diagrams for HEE in static background.",
"A. z H =3z_H=3, x 1 =0.4x_1=0.4, x 2 =0.6x_2=0.6, l 3 =l 2 l_3=l_2 and we vary l 1 l_1, l 2 l_2.",
"B. z H =4z_H=4, l 1 =l 2 =l 3 =4l_1=l_2=l_3=4 and we vary x 1 x_1, x 2 x_2.",
"C. z H =2.5z_H=2.5, l 2 =5l_2=5, l 3 =3l_3=3 and we vary l 1 l_1, x 2 x_2.",
"D. z H =4z_H=4, l 1 =l 2 =l 3 =5l_1=l_2=l_3=5 and we vary l 2 l_2, x 2 x_2.",
"On all plots light purple color corresponds to (A)∥(B)∥(C)(A)\\Vert (B)\\Vert (C), light green – (A)∥(B,C)(A)\\Vert (B,C), light blue – (A,B)∥(C)(A,B)\\Vert (C), orange – (A,B,C)(A,B,C), darker green (in plot D) – engulfed configuration.From Fig.REF we can see that only the configurations without cross-section contribute to the holographic entanglement entropy.",
"These diagrams represented in Fig.REF .",
"This observation confirms the general statement that crossing configurations do not contribute [18], [19].",
"Figure: Five contributing configurations of the 3-segment system.",
"(A)(A) is a segment connected to itself, so (A)∥(B)∥(C)(A)\\Vert (B)\\Vert (C) is disjoint configuration.",
"(A,B)(A, B) means that segments AA and BB connected to each other without intersections.",
"The last one is so called \"engulfed\" configuration.Without crossing configurations, formula (REF ) is (minimum of 5 items) $S_{no\\:cr}(\\ell ,x,m,y,n)=\\min \\Big \\lbrace S(\\ell )+S(m)+S(n),~S(\\ell )+S(m+y+n)+S(y), \\\\S(\\ell +x+m)+S(x)+S(n),~S(\\ell +x+m+y+n)+S(x)+S(y), \\\\S(\\ell +x+m+y+n)+S(x+m+y)+S(m) \\Big \\rbrace $ Figure: Phase diagrams for HEE in Vaidya-AdS.",
"z H =3,z h =1,ℓ=2,m=3,n=4,x,y∈(0,2.5]z_H=3,~z_h=1,~\\ell =2,~m=3,~n=4,~x,y \\in (0, 2.5], t=0,1,2,3,4,5t=0, 1, 2, 3, 4, 5.",
"Light purple color corresponds to (A)∥(B)∥(C)(A)\\Vert (B)\\Vert (C), light green – (A)∥(B,C)(A)\\Vert (B,C), light blue – (A,B)∥(C)(A,B)\\Vert (C), orange – (A,B,C)(A,B,C).It is interesting to note that the same is also true for non-static case.",
"Phase diagrams for holographic entanglement entropy in the Vaidya–AdS background are presented in Fig.REF and Fig.REF .",
"Figure: Phase diagrams for HEE in Vaidya-AdS.",
"z H =4,z h =1,ℓ,m∈(0,4],n=3,x=y=1z_H=4,~z_h=1,~\\ell ,m \\in (0, 4],~n=3,~x=y=1, t=0,1,2,3,4,5t=0, 1, 2, 3, 4, 5.",
"Light purple color corresponds to (A)∥(B)∥(C)(A)\\Vert (B)\\Vert (C), light green – (A)∥(B,C)(A)\\Vert (B,C), light blue – (A,B)∥(C)(A,B)\\Vert (C), orange – (A,B,C)(A,B,C).",
"$ $"
],
[
"Discontinuity of the holographic entanglement entropy of composite systems",
"We study how the holographic entanglement entropy changes if we divide the system into parts.",
"We fix the total length of the system and divide it into simple or composite parts with distances between them.",
"Holographic entanglement entropy for the disconnected regions was previously studied in [17], [18], [19].",
"Here, we study specific features of the case where the total system size is fixed"
],
[
"2 segments",
"Let the system consist of two parts each of which is a segment.",
"On Fig.REF .A we plot the holographic entanglement entropy of this system, on Fig.REF .B we also plot connected and disjoint configurations.",
"From these plots one can see that the holographic entanglement entropy has one singular point where configurations change one another.",
"This well known fact has been noted in previous papers [18], [19].",
"Figure: The holographic entanglement entropy of two segments with one singular point (indicated by dotted vertical line).",
"Blue curve corresponds to connected configuration (A,B)(A,B), green curve corresponds to disjoint configuration (A)∥(B)(A)\\Vert (B).",
"In both plots z H =3.4z_H=3.4, z h =1z_h=1, l=m=3.3l=m=3.3, x=1x=1."
],
[
"3 segments",
"We now consider a system consisting of three segments On Fig.REF .A we plot the holographic entanglement entropy (thick curve) with two singular points (marked by vertical dashed lines) where configurations change.",
"To see this fact clearly on Fig.REF .B we plot the same HEE as on Fig.REF .A (thick gray curve) and four curves corresponding to the first four configurations represented on Fig.REF .",
"We can see that in different moments HEE coincides with one of those curves and at the two singular points (marked by vertical dashed lines) these curves change.",
"Figure: A.",
"The holographic entanglement entropy with two singular points.B.",
"The holographic entanglement entropy corresponding to the first four configurations represented in Fig.. Blue curve correspond to (A,B)∥(C)(A,B)\\Vert (C), red curve correspond to (A,B,C)(A,B,C), green curve correspond to (A)∥(B)∥(C)(A)\\Vert (B)\\Vert (C), purple curve correspond to (A)∥(B,C)(A)\\Vert (B,C).",
"On both plots z H =4.46z_H=4.46, l=1.77l=1.77, x=0.93x=0.93, m=3.33m=3.33, y=1.87y=1.87, n=3.76n = 3.76."
],
[
"n segments",
"It follows from the explicit form of the entanglement entropy evolution that if we divide a segment of length $\\ell $ into $n$ parts, calculate the entropy of each part, and multiply it by $n$ , then we obtain a value less than the entropy of the whole system: $nS(\\ell /n,t)<S(\\ell ,t)$ for $n>1$ .",
"Small distances between segments reduce the entanglement entropy, but there is a distance $x_{\\mathrm {cr}}$ such that the entanglement entropy does not change for at least some time $t<t_0$ , i.e., $S(\\ell ,t)\\approx S((\\ell -x_{\\mathrm {cr}})/2,x_{\\mathrm {cr}},(\\ell -x_{\\mathrm {cr}})/2,t), \\;\\;\\; t<t_0.$"
],
[
"Appearance of the bell-type mutual information",
"For the system consisting of two equal segments with lengths $\\ell $ and distance $x$ between them the formula for mutual information reads $I=2\\,S(\\ell ,t)-[S(2\\ell +x,t)+S(x,t)].$ From this formula, we can conclude that if $2\\,S(\\ell ,t)=S(2\\ell +x,t)+S(x,t)$ for two different values of $t$ , then the plot of the mutual information has a bell shape.",
"Figure: The bell-shape mutual information appears between two red curves.",
"On both plots z H =3.8,z h =1,ℓ 1 =ℓ 2 =5,x∈[1.7,2.3]z_H=3.8,~z_h=1,~\\ell _1=\\ell _2=5,~x\\in [1.7, 2.3].On Fig.REF .A we show the appearance of the bell-type time dependence of the holographic mutual information for two equal segments.",
"We can see that the curve representing $2S(\\ell ,t)$ (green curve) may cross the curves representing $S(2\\ell +x,t)+S(x,t)$ (dotted lines) for $x\\in [1, 3]$ .",
"For $x<1.7$ (the lower red curve) there is only one intersection, for $x\\in [1.7, 2.3]$ (between two red curves) there are two cross-sections and for $x>2.3$ (the upper red curve) there is no intersection.",
"The double intersection of these two lines corresponds to an appearance of the bell-shaped dependence.",
"The lower red curve corresponds to the minimal value of distance $x$ and the upper red curve – maximal, for which there is the bell-type time dependence of the holographic mutual information.",
"Corresponding mutual information is shown in Fig.REF .B.",
"Note that for these times and length scales $S(2\\ell +x,t)$ and $S(x,t)$ have linear time dependence and due to this the difference $2S(\\ell )-(S(2\\ell +x,t)+S(x,t))$ has a maximum."
],
[
"Case I",
"In this case, the first (simple) subsystem is a segment $A$ of length $\\ell $ , the second (composite) subsystem is a union of two segments $B$ , $C$ with lengths $m$ , $n$ respectively located on one side of segment $A$ ; in this system $x$ , $y$ are distances between segments $A$ , $B$ and segments $B$ , $C$ respectively (Fig.REF ).",
"Figure: 1–2-segment system, case I.Figure: Five types of holographic mutual information behavior for the three-segment system, case I.We have found that the behavior of the holographic mutual information has five types (Fig.REF ): 1) Wake-up and scrambling times are absent, and the holographic mutual information is always positive (blue curve); 2) Wake-up time is absent, but scrambling time is present (orange curve); 3) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a bell shape (green curve); 4) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a two-hump shape (red curve); 5) The holographic mutual information is identically equal to zero (purple curve).",
"Let $A$ and $B$ be two segments such that the corresponding holographic mutual information has a bell shape.",
"Fig.REF .A shows an appearance of the second hump as segment $C$ approaches segments $A$ and $B$ from the right (as $y$ decreases).",
"Fig.REF .B shows an appearance of the second hump as the length of segment $C$ increases with the distances fixed (as $n$ increases).",
"Figure: Appearance of a two-humped form of the HMI.",
"A. z H =4.5z_H=4.5, z h =1z_h=1, ℓ=1.77\\ell =1.77, m=3.3m=3.3, n=3.7n=3.7, x=0.93x=0.93, y=1.98,1.94,1.90,1.85,1.80y=1.98, 1.94, 1.90, 1.85, 1.80.",
"B. z H =4.5z_H=4.5, z h =1z_h=1, ℓ=1.77\\ell =1.77, m=3.3m=3.3, n=3.4,3.5,3.6,3.7,3.9n=3.4, 3.5, 3.6, 3.7, 3.9, x=0.93x=0.93, y=1.87y=1.87.Figure: Zones of different behavior of the HMI.",
"Blue color corresponds to HMI>0HMI>0, brown – presence of scrambling time, orange – bell shape, green – two-hump shape, yellow – HMI≡0HMI\\equiv 0.",
"A.",
"The system consists of two segments, horizontal axis corresponds to xx, vertical – to mm, z H =4.46z_H=4.46, z h =1z_h=1, ℓ=1.77\\ell =1.77 and x,mx, m are varied.",
"B.",
"The system consists of three segments, horizontal axis corresponds to xx, vertical – to yy, total length of the system is 12, z H =4.46z_H=4.46, z h =1z_h=1, ℓ=1.77\\ell =1.77, m=3.33m=3.33, n=12-1.77-3.33-x-yn=12-1.77-3.33-x-y and x,yx, y are varied.In Fig.REF .B, we show the zones of different behavior (see the five types above) of the holographic mutual information for the three-segment system depending on x and y.",
"It can be seen from this picture that the two-humped zone corresponds to rather narrow intervals of parameters."
],
[
"Case II",
"In this case, the first (simple) subsystem is a segment $A$ of length $\\ell $ , the second (composite) subsystem is a union of two segments $B$ , $C$ with lengths $m$ , $n$ respectively, located on opposite sides of segment $A$ ; in this system, $x$ , $y$ are distances between segments $A$ , $B$ and segments $A$ , $C$ respectively (Fig.REF ).",
"Figure: 1–2-segment system, case II.In this case, we have found generally the same typical holographic mutual information behavior as in the case I (Fig.REF ): 1) Wake-up and scrambling times are absent, and the holographic mutual information is always positive (purple curve); 2) Wake-up time is absent, but scrambling time is present (red curve); 3) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a bell shape (green curve); 4) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a two-hump shape (orange curve); 5) The holographic mutual information is identically equal to zero (blue curve).",
"Figure: Five types of holographic mutual information behavior for composite system of three segments, case II.Let $E$ and $F$ be two segments such that the corresponding holographic mutual information has a bell shape.",
"Fig.REF .A shows an appearance of the second hump as segment $D$ approaches segments $E$ and $F$ from the left (as $x$ decreases).",
"Fig.REF .B shows an appearance of the second hump as the length of segment $D$ increases with the distances fixed (as $m$ increases).",
"Figure: Appearance of a two-humped form of the HMI.",
"A. z H =3.5z_H=3.5, z h =1z_h=1, m=2.82m=2.82, ℓ=5.45\\ell =5.45, n=1.46n=1.46, x=1.65,1.63,1.60,1.57,1.53x=1.65, 1.63, 1.60, 1.57, 1.53, y=0.924y=0.924.",
"B. z H =3.5z_H=3.5, z h =1z_h=1, m=2.72,2.78,2.85,2.92,3.00m=2.72, 2.78, 2.85, 2.92, 3.00, ℓ=5.45\\ell =5.45, n=1.46n=1.46, x=1.6x=1.6, y=0.924y=0.924.In Fig.REF , we show the zones of different behavior (see the five types above) of the holographic mutual information for the three-segment system depending on x and y.",
"It can be seen from this picture that the two-humped zone corresponds to rather narrow intervals of parameters.",
"Figure: Zones of different behavior of HMI.",
"The system consists of three segments.",
"Horizontal axis corresponds to xx, vertical – to yy.",
"z H =3.5z_H=3.5, z h =1z_h=1, ℓ=5.48\\ell =5.48, m=2.9m=2.9, n=1.46n=1.46 and x,yx, y are varied.",
"Yellow zone corresponds to HMI≡0HMI\\equiv 0, orange zone – bell-type form, darker yellow – two-humped form, brown zone corresponds to the presence of scrambling time and local minimum, blue zone corresponds to the presence of scrambling time."
],
[
"Comparison of Case I and Case II",
"It is interesting to compare the holographic mutual information for the two cases.",
"We can do it in two ways.",
"First Way.",
"The geometry of the system does not change, but the partition of the system into two subsystems changes (Fig.",
"REF ) Figure: Left plot: I(A;B∪C)=S(A)+S(B∪C)-S(A∪B∪C)I(A;B \\cup C)=S(A)+S(B \\cup C)-S(A \\cup B \\cup C), right plot: I(B;A∪C)=S(B)+S(A∪C)-S(A∪B∪C)I(B;A \\cup C)=S(B)+S(A \\cup C)-S(A \\cup B \\cup C).In this case, all three situations are possible: $I(A;B\\cup C) < I(B;A\\cup C),~~ I(A;B\\cup C) > I(B;A\\cup C),~~ I(A;B\\cup C) = I(B;A\\cup C)$ for some $A, B, C$ and at different instants $t$ (Fig.REF ).",
"Figure: Blue curve corresponds to I(A;B∪C)I(A;B\\cup C), orange curve corresponds to I(B;A∪C)I(B;A\\cup C).",
"A. z H =3,z h =1,l=3,x=0.7,m=4,y=0.4,n=3z_H=3,~ z_h=1,~ l=3,~ x=0.7,~ m=4,~ y=0.4,~ n=3.",
"B. z H =4,z h =1,l=6,x=0.5,m=0.6,y=0.5,n=6z_H=4,~ z_h=1,~ l=6,~ x=0.5,~ m=0.6,~ y=0.5,~ n=6.",
"C. z H =4,z h =1,l=6,x=0.2,m=0.6,y=0.2,n=6z_H=4,~ z_h=1,~ l=6,~ x=0.2,~ m=0.6,~ y=0.2,~ n=6.Second Way.",
"The geometry of the system changes, but the partition of the system into two subsystems does not change.",
"(Fig.",
"REF ) Figure: Left plot: I(A;B∪C)=S(A)+S(B∪C)-S(A∪B∪C)I(A;B \\cup C)=S(A)+S(B \\cup C)-S(A \\cup B \\cup C), right plot: I(E;D∪F)=S(E)+S(D∪F)-S(D∪E∪F)I(E;D \\cup F)=S(E)+S(D \\cup F)-S(D \\cup E \\cup F).In this case, we numerically find that the following inequality is satisfied $I(A;B \\cup C) \\leqslant I(E;D \\cup F)$ for any $A, B, C$ and for any time $t$ (Fig.REF ).",
"Figure: Blue curve corresponds to I(A;B∪C)I(A;B\\cup C), orange curve corresponds to I(E;D∪F)I(E;D\\cup F).",
"A. z H =3,z h =1,l=2.7,x=0.5,m=2.7,y=0.4,n=4z_H=3,~ z_h=1,~ l=2.7,~ x=0.5,~ m=2.7,~ y=0.4,~ n=4.",
"B. z H =3,z h =1,l=5,x=0.8,m=1.3,y=0.5,n=2z_H=3,~ z_h=1,~ l=5,~ x=0.8,~ m=1.3,~ y=0.5,~ n=2.",
"C. z H =4,z h =1,l=2.9,x=1.4,m=2,y=1,n=3z_H=4,~ z_h=1,~ l=2.9,~ x=1.4,~ m=2,~ y=1,~ n=3."
],
[
"Conclusion",
"Using the holographic approach, we studied the time evolution of the mutual information of a composite system during the heating process and showed how the mutual information between two sub-systems can be controlled by changing the geometric system parameters.",
"We have presented a more detailed analysis for a three-segment system consisting of two subsystems, studying an asymmetric type of organization (case I) and a symmetric type of organization (case II).",
"In both cases, we have found 5 types of behavior of the holographic mutual information for a system containing one composite part (two segments) and one simple part (one segment): 1) Wake-up and scrambling times are absent, and the holographic mutual information is always positive; 2) Wake-up time is absent, but scrambling time is present; 3) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a bell shape; 4) Wake-up and scrambling times are present, and a plot of the holographic mutual information has a two-hump shape; 5) The holographic mutual information is identically equal to zero.",
"We have shown that the holographic mutual information in the symmetric case II is greater or equal to the holographic mutual information in the asymmetric case I (using the second way of comparison).",
"We paid particular attention to finding a special shape of the plot of the holographic mutual information: a double-hump.",
"We have found that this double-hump shape is realized only for quite small ranges of system parameters."
],
[
"Acknowledgments",
"This work was supported by the Russian Science Foundation (project №17-71-20154)."
]
] | 1808.08389 |
[
[
"Word Sense Induction with Neural biLM and Symmetric Patterns"
],
[
"Abstract An established method for Word Sense Induction (WSI) uses a language model to predict probable substitutes for target words, and induces senses by clustering these resulting substitute vectors.",
"We replace the ngram-based language model (LM) with a recurrent one.",
"Beyond being more accurate, the use of the recurrent LM allows us to effectively query it in a creative way, using what we call dynamic symmetric patterns.",
"The combination of the RNN-LM and the dynamic symmetric patterns results in strong substitute vectors for WSI, allowing to surpass the current state-of-the-art on the SemEval 2013 WSI shared task by a large margin."
],
[
"Introduction",
"We deal with the problem of word sense induction (WSI): given a target lemma and a collection of within-sentence usages it, cluster the usages (instances) according to the different senses of the target lemma.",
"For example, for the sentences: We spotted a large bass in the ocean.",
"The bass player did not receive the acknowledgment she deserves.",
"The black sea bass, is a member of the wreckfish family.",
"We would like to cluster (a) and (c) in one group and (b) in another.This example shows homonymy, a case where the same word form has two distinct meaning.",
"A more subtle case is polysemy, where the senses share some semantic similarity.",
"In “She played a low bass note”, the sense of bass is related to the sense in (b), but distinct from it.",
"The WSI task we tackle in this work deals with both cases.",
"Note that some mentions are ambiguous.",
"For example, (d) matches both the music and the fish senses: Bass scales are the worst.",
"This calls for a soft clustering, allowing to probabilistically associate a given mention to two senses.",
"The problem of WSI has been extensively studied with a series of shared tasks on the topic , , , the latest being SemEval 2013 Task 13 .",
"Recent state-of-the-art approaches to WSI rely on generative graphical models , , .",
"In these works, the sense is modeled as a latent variable that influences the context of the target word.",
"The later models explicitly differentiate between local (syntactic, close to the disambiguated word) and global (thematic, semantic) context features."
],
[
"Substitute Vectors",
" take a different approach to the problem, based on substitute vectors.",
"They represent each instance as a distribution of possible substitute words, as determined by a language model (LM).",
"The substitute vectors are then clustered to obtain senses.",
"derive their probabilities from a 4-gram language model.",
"Their system (AI-KU) was one of the best performing at the time of SemEval 2013 shared task.",
"Our method is inspired by the AI-KU use of substitution based sense induction, but deviate from it by moving to a recurrent language model.",
"Besides being more accurate, this allows us to further improve the quality of the derived substitutions by the incorporation of dynamic symmetric patterns."
],
[
"BiLM",
"Bidirectional RNNs were shown to be effective for word-sense disambiguation and lexical substitution tasks , , .",
"We adopt the ELMo biLM model of , which was shown to produce very competitive results for many NLP tasks.",
"We use the pre-trained ELMo biLM provided by .We thank the ELMo team for sharing the pre-trained models.",
"However, rather than using the LSTM state vectors as suggested in the ELMo paper, we opt instead to use the predicted word probabilities.",
"Moving from continuous and opaque state vectors to discrete and transparent word distributions allows far better control of the resulting representations (e.g.",
"by sampling, re-weighting and lemmatizing the words) as well as better debugging opportunities.",
"As expected, the move to the neural biLM already outperforms the AI-KU system, and matches the previous state-of-the-art.",
"However, we observe that the substitute vectors do not take into account the disambiguated word itself.",
"We find that this often results in noisy substitutions.",
"As a motivating example, consider the sentence “the doctor recommends oranges for your health”.",
"Here, running is a perfectly good substitution, as the “fruitness” of the target word itself isn't represented in the context.",
"We would like the substitutes word distribution representing the target word to take both kinds of information—the context as well as the target word—into account."
],
[
"Dynamic Symmetric Patterns",
"Our main proposal incorporates such information.",
"It is motivated by Hearst patterns , , , and made possible by neural LMs.",
"Neural LMs are better in capturing long-range dependencies, and can handle and predict unseen text by generalizing from similar contexts.",
"Conjunctions, and in particular the word and, are known to combine expressions of the same kind.",
"Recently, used conjunctive symmetric patterns to derive word embeddings that excel at capturing word similarity.",
"Similarly, search for doubly-anchored patterns including the word and in a large web-corpus to improve semantic-class induction.",
"The method of result in context-independent embeddings, while that of takes some context into account but is restricted to exact corpus matches and thus suffers a lot from sparsity.",
"We make use of the rich sequence representation capabilities of the neural biLM to derive context-dependent symmetric pattern substitutions.",
"Relying on the generalization properties of neural language models and the abundance of the “X and Y” pattern, we present the language model with a dynamically created incomplete pattern, and ask it to predict probable completion candidates.",
"Rather than predicting the word distribution following the doctor recommends , we instead predict the distribution following the doctor recommends oranges and .",
"This provides substantial improvement, resulting in state-of-the-art performance on the SemEval 2013 shared task.",
"The code for reproducing the experiments and our analyses is available at https://github.com/asafamr/SymPatternWSI."
],
[
"Method",
"Given a target word (lemma and its part-of-speech pair), together with several sentences in which the target word is used (instances), our goal is to cluster the word usages such that each cluster corresponds to a different sense of the target word.",
"Following the SemEval 2013 shared task and motivating example (d) from the introduction, we seek a soft (probabilistic) clustering, in which each word instance is assigned with a probability of belonging to each of the sense-clusters.",
"Our algorithm works in three stages: (1) We first associate each instance with a probability distribution over in-context word-substitutes.",
"This probability distribution is based on a neural biLM (section REF ).",
"(2) We associate each instance with $k$ representatives, each containing multiple samples from its associated word distributions (section REF ).",
"(3) Finally, we cluster the representatives and use the hard clustering to derive a soft-clustering over the instances (section REF ).",
"We use the pre-trained neural biLM as a black-box, but use linguistically motivated processing of both its input and its output: we rely on the generalization power of the biLM and query it using dynamic symmetric patterns (section REF ); and we lemmatize the resulting word distributions."
],
[
"Running example",
"In what follows, we demonstrate the algorithm using a running example of inducing senses from the word sound, focusing on the instance sentence: I liked the sound of the harpsichord."
],
[
"biLM Derived Substitutions",
"We follow the ELMo biLM approach and consider two separately trained language models, a forward model trained for predicting $p_{\\rightarrow }(w_i|w_1,...,w_{i-1})$ and a backward model $p_{\\leftarrow }(w_i|w_n,...,w_{i+1})$ .",
"Rather than combining the two models' predictions into a single distribution, we simply associate the target word with two distributions, one from $p_{\\rightarrow }$ and one from $p_{\\leftarrow }$ .",
"For convenience, we use $LM_{\\rightarrow }(w_1 w_2 ... w_{i-1} \\underline{\\hspace{10.0pt}})$ to denote the distribution $p_{\\rightarrow }(w_i|w_1,...,w_{i-1})$ and $LM_{\\leftarrow }(\\underline{\\hspace{10.0pt}} w_{i+1} w_{i+2} ... w_n)$ to denote $p_{\\leftarrow }(w_i|w_n,...,w_{i+1})$ .",
"Table: Predicted substitutes for two senses of sound, for context-only and the symmetric-pattern approaches."
],
[
"Context-based substitution",
"In the purely context-based setup (the one used in the AI-KU system) we represent the target word sounds by the two distributions:"
]
] | 1808.08518 |
[
[
"A Comparison of the Taguchi Method and Evolutionary Optimization in\n Multivariate Testing"
],
[
"Abstract Multivariate testing has recently emerged as a promising technique in web interface design.",
"In contrast to the standard A/B testing, multivariate approach aims at evaluating a large number of values in a few key variables systematically.",
"The Taguchi method is a practical implementation of this idea, focusing on orthogonal combinations of values.",
"This paper evaluates an alternative method: population-based search, i.e.",
"evolutionary optimization.",
"Its performance is compared to that of the Taguchi method in several simulated conditions, including an orthogonal one designed to favor the Taguchi method, and two realistic conditions with dependences between variables.",
"Evolutionary optimization is found to perform significantly better especially in the realistic conditions, suggesting that it forms a good approach for web interface design in the future."
],
[
"Introduction",
"In e-commerce, designing web interfaces (i.e.",
"web pages and interactions) that convert as many users as possible from casual browsers to paying customers is an important goal [2], [19].",
"While there are some well-known design principles, including simplicity and consistency, there are often also unexpected interactions between elements of the page that determine how well it converts.",
"The same element, such as a headline, image, or testimonial, may work well in one context but not in others—it is often hard to predict the result, and even harder to decide how to improve a given page.",
"An entire subfield of information technology has emerged in this area, called conversion rate optimization, or conversion science.",
"The standard method is A/B testing, i.e.",
"designing two different versions of the same page, showing them to different users, and collecting statistics on how well they each convert [11], [7], [21], [17], [12].",
"This process allows incorporating human knowledge about the domain and conversion optimization into the design, and then testing their effect.",
"After observing the results, new designs can be compared and gradually improved.",
"The A/B testing process is difficult and time-consuming: Only a very small fraction of page designs can be tested in this way, and subtle interactions in the design are likely to go unnoticed and unutilized.",
"An alternative to A/B is multivariate testing, where all value combinations of a few elements are tested at once [11], [17], [6].",
"While this process captures interactions between these elements, only a very small number of elements is usually included (e.g.",
"2-3); the rest of the design space remains unexplored.",
"The Taguchi method [18], [12] is a practical implementation of multivariate testing.",
"It avoids the computational complexity of full multivariate testing by evaluating only orthogonal combinations of element values.",
"Taguchi is the current state of the art in this area, included in commercial applications such as the Adobe Target [1].",
"However, it assumes that the effect of each element is independent of the others, which is unlikely to be true in web interface design.",
"It may therefore miss interactions that have a significant effect on conversion rate.",
"This paper evaluates an alternative approach based on evolutionary optimization.",
"Because such search is based on intelligent sampling of the entire space, instead of statistical modeling, it can potentially overcome those shortcomings.",
"Crossover and mutation can traverse massive solution spaces efficiently, discovering dependencies and using them as building blocks [9], [8], [5], [20].",
"It can therefore search the space in a more comprehensive manner, and effectively find solutions that the other methods miss.",
"This idea is implemented in Ascend by Evolv web-interface optimization system, deployed in numerous e-commerce websites of paying customers since September 2016 [15], [16].",
"Ascend uses a customer-designed search space as a starting point (Figure REF ).",
"It consists of a list of elements on the web page that can be changed, and their possible alternative values, such as a header text, font, and color, background image, testimonial text, and content order.",
"Ascend then automatically generates web-page candidates to be tested, and improves those candidates through evolutionary optimization.",
"This paper compares the evolutionary approach in Ascend to the state-of-the-art statistical method of Taguchi optimization in simulated experiments.",
"The results show that evolution is indeed more powerful optimizer, especially under realistic conditions where there are nonlinear dependencies between variables.",
"It is therefore a promising foundation for designing optimization applications in the future, and increases the potential for more powerful AI applications in related fields.",
"Figure: Web Interface Design as an Optimization Problem.",
"In thisexample, 13 elements of the page each have 2-4 possible values,resulting in 1.1M combinations.",
"The goal is to find combinationsthat make it as likely as possible for the user to click on theaction button in the middle.",
"The state-of-the-art multivariatetesting method, Taguchi, evaluates only orthogonal combinationsand therefore misses nonlinear interactions between elements.",
"Incontrast, population-based search in Ascend is sensitive to suchinteractions; it can therefore find good solutions that theTaguchi method misses."
],
[
"The Taguchi Method",
"Ideally, the best web interface design would be decided based on full factorial multivariate testing.",
"That is, each possible combination of $N$ variables with $K$ values would be implemented as a candidate.",
"For example, a variable might be the color or the position of a button; the values would then be the possible colors and positions.",
"A full factorial analysis would require testing all $K^N$ combinations, which is prohibitive in most cases.",
"Instead, the Taguchi method specifies a small subset of these combinations to test using orthogonal arrays.",
"An Taguchi orthogonal array is a matrix where each column corresponds to a variable and each row to a candidate to test.",
"Each value represents the setting for a given variable and experiment.",
"It has the following properties: The dot product between any two normalized column vectors is zero.",
"For every variable column, each value appears the same amount of times.",
"There are multiple ways of creating orthogonal arrays [4], [10].",
"Table REF shows an example of an orthogonal array of nine combinations, resulting from testing four variables of three values each.",
"Table: Example Taguchi array of four variables with three values eachTo compute the effect of a specific variable value, we average the performance scores of the candidates corresponding to combinations for that value setting.",
"Because in an orthogonal array, all values of the other variables are tested an equal amount of times, their effects cancel out, assuming each variable is independent[10].",
"For example, to compute the effect of value 2 of variable 3 in tableREF , we average the scores of candidates 2, 4 and 9.",
"Similarly, for value 1, we average the scores of candidates 3, 5 and 7.",
"In a Taguchi experiment, all the candidates (rows) in the orthogonal table are tested, and the scores for candidates that share the same value for each variable are averaged in this manner.",
"We can then predict a best performing combination by selecting, for each variable, the value with the best such average score.",
"The Taguchi method is a practical approximation of factorial testing.",
"However, the averaging steps assume that the effects of each variable are independent, which may or may not hold in real-world experiments.",
"In contrast, population-based search makes no such assumptions, as will be discussed next."
],
[
"Evolutionary Optimization",
"Evolutionary optimization is a broadly used method for combinatorial problems, building on population-based search.",
"It has certain advantages compared to diagnostic methods, including fewer restrictions on input variables, robustness to environmental changes, good scale-up to large and high-dimensional spaces, and robustness to deceptive search spaces and nonlinear interactions [3].",
"The main idea is that instead of constructing the winning combination though independence assumptions (as in Taguchi), the winner is searched for using crossover and mutation operators.",
"The evolution algorithm used in this paper is that of Ascend by Evolv, a conversion optimization product for web interfaces [15].",
"The basic unit of the method is the candidate's genome, which is a list representing the elements and values of the web interface.",
"For example, the genome $\\textbf {[2,4,5,3]}$ defines a web page with four changeable parts, i.e.",
"genes, with 2, 4, 5, and 3 different choices each.",
"The choices are represented as one-hot vectors, and are concatenated to form the genome.",
"The control candidate is the genome representing default web settings, consisting of the first dimension in each vector: $[[1, 0], [1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0]]$ The candidates in the first generation consist of all genomes that are one gene different from the control.",
"Thus, the number of candidates in this generation is the sum, over all genes, of the number of values minus one, i.e.",
"$1+3+4+2=10$ in this example.",
"In all future generations, the total number of candidates stays the same; a certain percentage of candidates (e.g.",
"20%) are chosen as elites, staying on to the next generation.",
"The remaining (e.g.",
"80%) candidates are formed by crossover from those elites [14].",
"The encoding of each candidate also has a chance to mutate (i.e.",
"specify a different choice for each part), according to probabilities specified in the parameter setting.",
"The evolution process ends after a prespecified number of generations or after a suitable candidate is found.",
"In this paper, the key value of evaluating the performance from different approaches is conversion rate, which is the ratio of people that convert to the total visitor of the web page.",
"After an evolutionary simulation, a prior estimate of the conversion rate is obtained as the average of all candidates tested.",
"A probability to beat control is computed for each candidate based on this prior and its individual estimate.",
"To compute the probability to beat control for a target candidate, a probability distribution of conversion rate is built for the control and the candidate, based on data collected from experiment, as demonstrated in Figure REF .",
"Then the proportion of area under the curve of candidate conversion rate distribution where it beats the one of control is computed as the probability to beat control for the candidate, and the one with the highest probability is selected as the winner.",
"Figure: Probability distribution of control and target candidate conversion rates.",
"The area under the candidate curve that is above the control curve stands for the probability to beat control.",
"Using that probability as the measure of performance, instead of the estimated conversion rate, leads to more reliable results."
],
[
"Evaluation in a Simulator",
"The performance of the Taguchi method and Evolutionary optimization was measured in simulated experiments of designing web interfaces for maximum conversion rate.",
"In the simulation, an evaluator is first constructed to calculate a candidate's true conversion rate based on the values it specifies for each variable.",
"Simulated traffic is then distributed to candidates and conversions are assigned probabilistically based on candidates' true conversion rate.",
"The observed conversion rates are then used as the scores of the candidates in Taguchi and evolution methods.",
"By setting the parameters in Table REF , different kinds of evaluators can be defined.",
"The conversion rate of simple linear evaluator is based on only bias and weight for each individual variable: $CR[c] = W^0+\\sum _{i=1}^nW_i^1(c).$ The bias represents the conversion rate of the control candidate; the different choices for each variable add or subtract from the control rate.",
"A non-linear evaluator is designed to include interactions between variables: $CR[c] = W^0+\\sum _{i=1}^nW_i^1(c)+\\sum _{j=1}^n\\sum _{k=j+1}^nW_{j,k}^2(c).$ In addition to bias and individual variable contributions, it includes contributions for each pair of variables.",
"Both the Taguchi candidates and the evolution candidates are represented in the same way, as concatenations of one-hot vectors representing the values for each variable in the Taguchi method, and actions for each gene in evolution.",
"The total traffic for the Taguchi method and evolution algorithm is set to be equal, distributed evenly to all Taguchi candidates, but differently for evolution candidates based on how many generations they survive.",
"Table REF specifies the parameter settings used in all experiments with evolution, and the evaluator bias rate."
],
[
"Experimental Results",
"Three experiments were run comparing the Taguchi method with evolutionary optimization: two experiments where the variables had independent effect, one with a uniform and the other with varied number of values; and one experiment with dependencies between pairs of variables.",
"In addition to comparing the ability of these methods to find good candidates as the final result of the experiment, their performance during the experiment was also compared.",
"The result curves demonstrate statistical means and 95% credible intervals of 20 repeated experiments, under the same example settings in each section.",
"Figure: True conversion rate performance with a more complex genome.",
"Both methods take longer to find good candidates; Taguchi is now comparable to evolution only with the highest amounts of traffic."
],
[
"Independent Variables with Uniform Values",
"The Taguchi method assumes that the variables are independent.",
"The first experiment was designed accordingly: It uses a linear evaluator that assumes all changes are independent, and a simple genome that results in few rows in the Taguchi array.",
"These are the ideal conditions for the Taguchi method, and it is expected to perform well.",
"The best settings for the Taguchi method are those with uniform numbers of values across all variables [1]: Setting 1: Three variables with two values each, i.e.",
"$\\textbf {[2, 2, 2]}$ , with $2^3=8$ combinations, resulting in four rows; Setting 2: Four variables with three values each, i.e.",
"$\\textbf {[3, 3, 3, 3]}$ , with $3^4=81$ combinations, resulting in nine rows; and Setting 3: Five variables with four values each, i.e.",
"$\\textbf {[4, 4, 4, 4, 4]}$ , with $4^5=1024$ combinations, resulting in 16 rows.",
"The Taguchi arrays for these settings can be found in orthogonal array libraries [13].",
"The learning curves under all three settings are similar, so Setting 2 will be used as an example.",
"Figure REF shows the true conversion rates of the best candidates under Setting 2 with increasing traffic.",
"The true conversion rate for the best evolution candidate is steady and high at all traffic values.",
"The best predicted Taguchi candidate's true conversion rate lags behind evolution with low traffic, but eventually catches up as traffic increases.",
"The best tested Taguchi candidate remains significantly below both curves, which shows that Taguchi method does achieve an improvement from its original tested candidates to the predicted candidates.",
"Thus, under ideal conditions for Taguchi, both methods find equally good solutions given enough traffic (i.e.",
"more than 500,000).",
"With low traffic, the best evolutionary approach performs significantly better.",
"Since the total traffic is equal with both approaches, and Taguchi method defines a set of candidates to distributed those traffic, it then has a fixed experiment plan that is indifferent to the allocation of traffic.",
"However, evolution algorithm consecutively generates better performed candidates to use the limited traffic through generations.",
"This way, it better exploits data to drive selection of good candidates in the end, because it allows testing to focus on good combinations adaptively, especially when total traffic is not sufficient."
],
[
"Independent Variables with Variable Values",
"In real world applications, such as optimization of commercial websites, the design space may be rather complex; in particular, the number of values for each variable is not likely to be the same.",
"In the second experiment, while still maintaining independence between variables, the genome structure is changed to: $\\textbf {[3, 6, 2, 3, 6, 2, 2, 6]},$ i.e.",
"three variables with two value each, two variables with three value each, and three variables with six value each.",
"In this setting with 15,552 combinations, the Taguchi array needs 36 rows [13].",
"Figure: Average true conversion rate of candidates with evolution and Taguchi methods during the experiment.",
"While Taguchi candidates do not change, evolution continuously comes up with better candidates, thus increasing performance during the experiment.",
"It therefore forms a good approach for campaigns with fixed duration as well.The result in Figure REF shows that with a more complex problem, both evolution and Taguchi require more traffic in order to find good solutions.",
"However, evolution produces significantly better candidates than Taguchi at almost all traffic values: The two methods are comparable only for very high traffic, i.e.",
"greater than 5,000,000.",
"The prediction process of Taguchi still provides a major advantage beyond its input set.",
"Similarly to the discussion above, evolution algorithms maintains better ability for choosing good candidates because its effort in distributing data to better candidates reactively.",
"In this more complex scenario, Taguchi method requires more candidates for testing, which greatly decrease the exposure to data for all its candidates, since they are equally tested and distributed with same amount of traffic.",
"This results in a much higher total traffic for Taguchi method to perform similarly as evolution algorithm, which still exploits data on good candidates."
],
[
"Interactions Between Variables",
"Another important challenge in real-world applications is that the variables are not likely to be independent.",
"For example, text color and background color may interact—for instance, blue text on a blue background would perform poorly compared to blue text on a white background.",
"The nonlinear evaluator is designed to test the ability of the two methods to handle this kind of interactions.",
"The example uses genome in Section REF .",
"Figure REF shows that when the independence assumption for Taguchi method is broken, the best predicted Taguchi candidate's true conversion rate is no longer comparable with evolution's.",
"Furthermore, its predicted best candidate does not even significantly outperform its best tested candidate.",
"Interestingly, the performance of the evolutionary algorithm is not significanly worse with interacting vs. independent variables, demonstrating its ability to adapt to complicated real-world circumstances."
],
[
"Performance During Experiment",
"The main goal in conversion optimization is to find good candidates that can be deployed after the experiment.",
"However, in many cases it is also important to not decrease the site's performance much during the experiment.",
"Evolution continuously creates improved candidates as it learns more about the system, whereas the Taguchi method generates a single set of candidates for the entire test—it therefore provides continual improvement on the site even during the experiment.",
"This principle is illustrated graphically in Figure REF , using the linear evaluator and genome from Section REF as the example setting.",
"The Taguchi's candidates' average performance stays the same throughout the increasing traffic, whereas evolution's candidates perform, on average, better with more traffic, i.e.",
"while the experiment progresses.",
"It therefore forms a good approach in domains where performance matters during the experiment, in particular in campaigns that run only for a limited duration."
],
[
"Discussion and Future Work",
"The two methods tested in the experiments of this paper, Taguchi and evolution, are both beneficial in web interface design.",
"Taguchi's appeal is its high reduction of rows compared to full factorial combinations, which works best when the genome structure is rather simple.",
"When the genome becomes larger and more complex, its performance falls behind that of evolution.",
"Most importantly, if there are nonlinear interactions between the variables, the method cannot keep up with them: the best-candidate construction does not improve upon its initial best candidates, and it performs much worse than evolution at all traffic values.",
"In contrast, the process of searching for good candidates in evolution is based on crossover and mutation, and therefore is not affected much by interactions.",
"Evolution discovers good combinations, and constructs future candidates using them as building blocks.",
"As long as the interactions occur within the building blocks, they will be included and utilized the same way as independent contributions.",
"Given how common interactions are in real-world problems, this ability should turn out important in applying optimization to web interface design in the future, enabling more powerful AI applications in related areas."
],
[
"Conclusion",
"This paper demonstrates that (1) even with ideal conditions, the Taguchi method does not exceed performance of evolution, and (2) with low traffic in ideal conditions, evolution performs significantly better.",
"(3) As the experiment configuration becomes more complex, Taguchi requires more traffic to match evolution's performance.",
"(4) With nonlinear interactions between variables, Taguchi's construction process breaks down, and it no longer improves upon best initial candidates.",
"(5) In contrast, evolution is able to find good candidates even with nonlinear interactions.",
"Furthermore, (6) evolution improves during the duration of the experiment, making it a good choice for campaigns as well.",
"Evolutionary optimization is thus a superior technique for improving conversion rates in web interface design."
]
] | 1808.08347 |
[
[
"Energy-preserving continuous-stage Runge-Kutta-Nystr\\\"om methods"
],
[
"Abstract Many practical problems can be described by second-order system $\\ddot{q}=-M\\nabla U(q)$, in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular momentum, etc.",
"However, conventional numerical integrators for such systems will fail to preserve any of these quantities which may lead to qualitatively incorrect numerical solutions.",
"This paper is concerned with the development of energy-preserving continuous-stage Runge-Kutta-Nystr\\\"om (csRKN) methods for solving second-order systems.",
"Sufficient conditions for csRKN methods to be energy-preserving are presented and it is proved that all the energy-preserving csRKN methods satisfying these sufficient conditions can be essentially induced by energy-preserving continuous-stage partitioned Runge-Kutta methods.",
"Some illustrative examples are given and relevant numerical results are reported."
],
[
"Introduction",
"In science and engineering fields, there are many problems that can be modelled by ordinary, or partial, differential equations, amongst which those special ones possessing geometric features have drawn much attention in numerical differential equations [3], [15], [19], [22], [29].",
"In this paper, we are concerned with the following Hamiltonian system of ordinary differential equations [2] $\\dot{p}=-\\nabla _q H(p,q), \\;\\;\\dot{q}=\\nabla _p H(p,q),\\quad p(t_0)=p_0\\in \\mathbb {R}^{d},\\;q(t_0)=q_0\\in \\mathbb {R}^{d},$ where $H(p,q)$ is called the Hamiltonian function (the total energy) of the system.",
"This system has two important geometric properties in phase space: symplecticity and energy conservation [2].",
"As is well known, for such system, a famous geometric integration approach called “symplectic integration\" has been placed on a central position in modern scientific computing since 1980s (see [3], [10], [13], [14], [15], [19], [22], [28], [29], [47] and references therein), while in more recent years there has been a rising interest in the subject of energy-preserving integration [4], [5], [9], [12], [20], [23], [24], [25], [26], [27], [31], [46].",
"Symplectic integrators are important and rather popular due to their global restriction of the numerical solutions in all directions by the symplectic structure in the phase space.",
"In contrast, as pointed out in [29], energy-preserving integrators may be more beneficial for numerical integration of low-dimensional Hamiltonian systems, by noticing the fact that the preservation of energy is a rather weak restriction for the numerical solutions when the dimension of the system is large.",
"However, compared to symplectic integrators, energy-preserving integrators can be more adaptable for variable time step computation and usually excellent for the integration of chaotic systems, molecular systems and stiff systems [1], [5], [17], [19], [30].",
"Unfortunately, in general it is impossible for us to construct a method preserving the symplecticity and energy at the same time for a general nonlinear Hamiltonian system [11], [16], hence we can not have the benefits of preserving both properties.",
"Nevertheless, symplectic methods are known to preserve a modified Hamiltonian [19] which implies a near-preservation of the energy, and there is another interesting result shown in [11] stating that a symplectic method is formally conjugate to a method that preserves the Hamitonian (the total energy) exactly.",
"Conversely, the existence of conjugate-symplectic (a symplectic-like conception in a weak sense) energy-preserving B-series integrators is affirmative — though it is still a great challenge to find a computational method of such type [21].",
"Recently, continuous-stage approaches are introduced and developed for solving initial value problems of ordinary differential equations (ODEs), following the pioneering work of Butcher [6], [7], [8] and Hairer [20].",
"Such approaches have led to many interesting applications in geometric integration.",
"Some typical applications can be found in literature, such as: symplectic integrators can be derived from Galerkin variational problems, and these integrators can be interpreted and analyzed by virtue of continuous-stage methods [31], [36], [43]; a number of newly-developed energy-preserving methods can be closely connected to continuous-stage methods [4], [9], [12], [20], [23], [24], [25], [26], [31], [46]; a wide variety of novel symplectic and symmetric methods can be constructed in use of continuous-stage approaches [32], [33], [34], [35], [37], [38], [39], [40], [41], [42], [44]; the conjugate-symplecticity of energy-preserving methods can be investigated in the context of continuous-stage methods [20], [21], [33].",
"Hopefully, other new applications of continuous-stage methods in geometric integration can be explored in the forthcoming future.",
"As is well known, second-order ordinary differential equations (ODEs) in the form $\\ddot{q}=-M\\nabla U(q)$ (with a constant symmetric matrix $M$ ) are frequently encountered in various fields such as celestial mechanics, molecular dynamics, plasma physics, biological chemistry and so on [15], [19], [29].",
"More recently, for solving such second-order ODEs, the present author et al.",
"[34], [37], [44], [45] have developed many new families of symplectic and symmetric integrators by using various weighted orthogonal polynomials in the context of continuous-stage Runge-Kutta-Nyström (csRKN) methods.",
"A highlighted advantage for adopting RKN-type methods in the numerical integration is that they can save about half of the storage and reduce the computational cost accordingly when compared to Runge-Kutta methods [18].",
"In this paper, we focus on the development of energy-preserving continuous-stage Runge-Kutta-Nyström (csRKN) methods.",
"For this sake, we shall first explore the sufficient conditions for csRKN methods to be energy-preserving, and then by virtue of the derived conditions we discuss the construction of new RKN-type energy-preserving integrators.",
"This paper will be organized as follows.",
"In Section 2, we first present the sufficient conditions for csRKN methods to be energy-preserving, and then it is shown that they can be closely related to continuous-stage partitioned Runge-Kutta (csPRK) methods.",
"This is followed by Section 3, where some illustrative examples for the construction of energy-preserving integrators will be included and some discussions on their numerical implementations will be given.",
"Section 4 is devoted to exhibit some numerical results.",
"At last, we end our paper in Section 5."
],
[
"Energy-preserving conditions",
"Consider the following initial value problem governed by a second-order system $\\ddot{q}=-M\\nabla U(q),\\;\\;q(t_0)=q_0\\in \\mathbb {R}^{d},\\;\\;\\dot{q}(t_0)=\\dot{q}_0\\in \\mathbb {R}^{d},$ where $M\\in \\mathbb {R}^{d\\times d}$ is a constant, symmetric and invertible matrix, and $U(q)$ (the potential energy) is a differentiable scalar function.",
"Such system can be transformed into a special separable Hamiltonian system with the Hamiltonian $H(p,q)=\\frac{1}{2}p^TMp+U(q)=\\frac{1}{2}\\dot{q}^TM^{-1}\\dot{q}+U(q)$ , which reads $\\dot{p}=-\\nabla U(q),\\;\\;\\dot{q}=Mp,$ and the corresponding initial value condition is given by $p(t_0)=M^{-1}\\dot{q}_0,\\,q(t_0)=q_0$ .",
"It is known that $H(p,q)$ (the total energy) is an invariant or a first integral of the system, say $H(p(t),q(t))=\\text{Const}$ along the solution curves of (REF ), which will to be considered in the energy-preserving time-discretization of the system later.",
"By using the notation $p_0=M^{-1}\\dot{q}_0$ , we introduce the following continuous-stage Runge-Kutta-Nyström (csRKN) method for solving (REF ) [34], [37] $&Q_\\tau =q_0 +hC_\\tau Mp_0-h^2M\\int _{0}^{1} \\bar{A}_{\\tau , \\sigma }\\nabla U(Q_\\sigma ) \\mathrm {d}\\sigma , \\;\\;\\tau \\in [0, 1], \\\\&q_{1}=q_0+ h Mp_0-h^2M\\int _{0}^{1} \\bar{B}_\\tau \\nabla U(Q_\\tau ) \\mathrm {d}\\tau , \\\\&p_1 = p_0 -h\\int _{0}^{1}B_\\tau \\nabla U(Q_\\tau ) \\mathrm {d}\\tau .$ where $\\bar{A}_{\\tau , \\sigma }$ is a smooth function of variables $\\tau , \\sigma \\in [0,1]$ and $\\bar{B}_\\tau ,\\;B_\\tau ,\\;C_\\tau $ are smooth functions of $\\tau \\in [0,1]$ .",
"The method () is said to have order $p$ , if for all sufficiently regular problem (REF ), as $h\\rightarrow 0$ , its local error satisfies [18] $q_1-q(t_0+h)=\\mathcal {O}(h^{p+1}),\\quad p_1-p(t_0+h)=\\mathcal {O}(h^{p+1}).$ By definition, to construct a energy-preserving csRKN method is to design suitable Butcher coefficient functions $\\bar{A}_{\\tau ,\\sigma },\\,\\bar{B}_\\tau ,\\,B_\\tau ,\\,C_\\tau $ so as to guarantee the preservation of energy, i.e., $H(p_{n+1},q_{n+1})=H(p_n,q_n),\\;\\;n=0,1,2,\\cdots .$ Without loss of generality, for a one-step method, it suffices to consider the case after one step computation.",
"In our case, we need to impose the following requirement $H(p_1,q_1)=H(p_0,q_0),$ on the one-step scheme ().",
"Theorem 2.1 If there exists a smooth binary function $A_{\\tau , \\sigma }$ , such that the coefficients of the csRKN method () satisfy $&\\;C_0=0\\;\\;\\text{and}\\;\\;C_1=1,\\\\[5pt]&\\,\\bar{A}_{0, \\sigma }=0,\\;\\bar{A}_{1, \\sigma }=\\bar{B}_\\sigma , \\;\\forall \\,\\sigma \\in [0, 1], \\\\[5pt]&\\,A_{0, \\sigma }=0,\\;A_{1, \\sigma }=B_\\sigma =C^{\\prime }_\\sigma , \\;\\forall \\,\\sigma \\in [0, 1],\\\\&\\int _{0}^{1} A^{\\prime }_{\\tau , \\eta }A_{\\tau , \\zeta }\\,\\mathrm {d}\\tau =\\bar{A}^{\\prime }_{\\eta , \\zeta },\\;\\forall \\,\\eta ,\\,\\zeta \\in [0,1],$ whereHereafter we always use primes for denoting the partial derivatives with respect to the first variable (subscript) of binary functions.",
"Moreover, the notation $A_{\\tau ,\\eta }\\Big |^{1}_{0}=A_{1, \\eta }-A_{0, \\eta }$ as well as other similar notations is also associated with the first variable.",
"$A^{\\prime }_{\\tau , \\sigma }=\\frac{\\partial }{\\partial \\tau }A_{\\tau ,\\sigma },\\;\\;\\bar{A}^{\\prime }_{\\tau , \\sigma }=\\frac{\\partial }{\\partial \\tau }\\bar{A}_{\\tau , \\sigma },$ then the method is energy-preserving for solving system (REF ).",
"Firstly, we define $P_\\tau $ as $p(t_0+\\tau h)\\approx P_\\tau =p_0-h\\int _{0}^{1}A_{\\tau , \\sigma }\\nabla U(Q_\\sigma )\\mathrm {d}\\sigma , \\;\\;\\tau \\in [0, 1],$ and here $A_{\\tau , \\sigma }$ is assumed to satisfy ().",
"From (REF )-(), it follows $P_0=p_0,\\;P_1=p_1,\\;Q_0=q_0,\\;Q_1=q_1,$ which means $P_\\tau $ and $Q_\\tau $ as continuous functions join the numerical solutions at the two ends of integration interval $[t_0,t_0+h]$ and they can be regarded as the approximations to the exact solutions $p(t)=p(t_0+\\tau h)$ and $q(t)=q(t_0+\\tau h),\\,\\tau \\in [0,1]$ .",
"Hence, by means of the fundamental theorem of calculus and using $M^T=M$ , we have $\\begin{split}&H(p_1,q_1)-H(p_0,q_0)\\\\&=\\int _{0}^{1}\\frac{\\mathrm {d}}{\\mathrm {d}\\tau }H(P_\\tau ,Q_\\tau )\\,\\mathrm {d}\\tau \\\\&=\\int _{0}^{1}P^T_\\tau MP^{\\prime }_\\tau + (Q^{\\prime }_\\tau )^T\\nabla U(Q_\\tau )\\mathrm {d}\\tau .\\end{split}$ By using (REF ), it gives $\\begin{split}&\\int _{0}^{1}P^T_\\tau MP^{\\prime }_\\tau \\,\\mathrm {d}\\tau \\\\&=\\int _{0}^{1}\\Big [\\Big (p^T_0-h\\int _{0}^{1}A_{\\tau , \\zeta }(\\nabla U(Q_\\zeta ))^T\\,\\mathrm {d}\\zeta \\Big )M\\Big (-h\\int _{0}^{1}A^{\\prime }_{\\tau ,\\eta }\\nabla U(Q_\\eta )\\mathrm {d}\\eta \\Big )\\Big ]\\,\\mathrm {d}\\tau \\\\&=-h\\int _{0}^{1}\\Big [\\underbrace{\\int _{0}^{1} A^{\\prime }_{\\tau , \\eta }\\mathrm {d}\\tau }_{=B_\\eta }p^T_0M\\nabla U(Q_\\eta )\\Big ]\\,\\mathrm {d}\\eta +h^2\\int _{0}^{1}\\int _{0}^{1}\\Big [\\big (\\int _{0}^{1}A_{\\tau ,\\zeta }A^{\\prime }_{\\tau , \\eta }\\mathrm {d}\\tau \\big )(\\nabla U(Q_\\zeta ))^TM\\nabla U(Q_\\eta )\\Big ]\\,\\mathrm {d}\\eta \\mathrm {d}\\zeta \\\\&=-h\\int _{0}^{1}B_\\eta p^T_0M\\nabla U(Q_\\eta )\\,\\mathrm {d}\\eta +h^2\\int _{0}^{1}\\int _{0}^{1}\\Big [\\Big (\\int _{0}^{1}A^{\\prime }_{\\tau ,\\eta }A_{\\tau , \\zeta }\\mathrm {d}\\tau \\Big )(\\nabla U(Q_\\zeta ))^TM\\nabla U(Q_\\eta )\\Big ]\\,\\mathrm {d}\\eta \\mathrm {d}\\zeta ,\\end{split}$ where we have used the following identity $\\int _{0}^{1}A^{\\prime }_{\\tau , \\eta }\\mathrm {d}\\tau =A_{\\tau ,\\eta }\\Big |^{1}_{0}=B_\\eta .$ Similarly, by using (REF ) we have $\\begin{split}&\\int _{0}^{1}(Q^{\\prime }_\\tau )^T\\nabla U(Q_\\tau )\\,\\mathrm {d}\\tau \\\\&=\\int _{0}^{1}(Q^{\\prime }_\\eta )^T\\nabla U(Q_\\eta )\\,\\mathrm {d}\\eta \\\\&=\\int _{0}^{1}\\Big [\\Big (hC^{\\prime }_\\eta p^T_0M-h^2\\int _{0}^{1}\\bar{A}^{\\prime }_{\\eta , \\zeta }(\\nabla U(Q_\\zeta ))^TM\\,\\mathrm {d}\\zeta \\Big )\\nabla U(Q_\\eta )\\Big ]\\,\\mathrm {d}\\eta \\\\&=h\\int _{0}^{1}C^{\\prime }_\\eta p^T_0M\\nabla U(Q_\\eta )\\,\\mathrm {d}\\eta -h^2\\int _{0}^{1}\\int _{0}^{1}\\Big [\\bar{A}^{\\prime }_{\\eta , \\zeta }(\\nabla U(Q_\\zeta ))^TM\\nabla U(Q_\\eta )\\Big ]\\,\\mathrm {d}\\eta \\mathrm {d}\\zeta .\\end{split}$ Substituting the two formulas above into (REF ) yields (REF ) which completes the proof.",
"Theorem 2.2 The conditions () and () imply $\\bar{A}^{\\prime }_{\\tau ,\\sigma }+\\bar{A}^{\\prime }_{\\sigma ,\\tau }\\equiv B_\\tau B_\\sigma ,\\;\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1].$ If we assume $B_\\tau \\equiv 1$ , then (REF ) becomes $\\bar{A}^{\\prime }_{\\tau ,\\sigma }+\\bar{A}^{\\prime }_{\\sigma ,\\tau }\\equiv 1,\\;\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1].$ By exchanging the variables $\\eta $ and $\\zeta $ in (), it yields $\\int _{0}^{1} A^{\\prime }_{\\tau , \\zeta }A_{\\tau , \\eta }\\,\\mathrm {d}\\tau =\\bar{A}^{\\prime }_{\\zeta , \\eta },\\;\\forall \\,\\eta ,\\,\\zeta \\in [0,1].$ Adding (REF ) with () and using () gives $\\bar{A}^{\\prime }_{\\eta , \\zeta }+\\bar{A}^{\\prime }_{\\zeta , \\eta }=\\int _{0}^{1}\\Big (A^{\\prime }_{\\tau , \\eta }A_{\\tau , \\zeta }+A^{\\prime }_{\\tau , \\zeta }A_{\\tau ,\\eta }\\Big )\\,\\mathrm {d}\\tau =A_{\\tau , \\eta }A_{\\tau ,\\zeta }\\Big |^{1}_{0}=B_\\eta B_\\zeta ,\\;\\forall \\,\\eta ,\\,\\zeta \\in [0,1],$ which leads to (REF ).",
"The formulae (REF ) is straightforward from (REF ) when $B_\\tau \\equiv 1$ .",
"Theorem 2.3 If we define $\\widehat{A}_{\\tau ,\\sigma }=\\int _0^\\tau A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\zeta ,\\;\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1],$ then it gives $\\widehat{A}_{0,\\sigma }=0,\\quad A^{\\prime }_{\\tau ,\\sigma }=\\widehat{A}^{\\prime }_{\\sigma ,\\tau },\\;\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1],$ and vice versa.",
"Moreover, under the condition $\\bar{A}_{0,\\sigma }=0$ (see also (REF )), the formula () implies $\\bar{A}_{\\tau ,\\sigma }=\\int _{0}^{1}\\widehat{A}_{\\tau ,\\rho }A_{\\rho ,\\sigma }\\,\\mathrm {d}\\rho ,\\;\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1].$ The equivalence between (REF ) and (REF ) is easy to verify by conducting some elementary calculations.",
"By taking integral of () with respect to $\\tau $ and using $\\bar{A}_{0, \\sigma }=0$ , it follows $\\int _{0}^{1} \\widehat{A}_{\\alpha ,\\rho } A_{\\rho , \\sigma }\\,\\mathrm {d}\\rho =\\int _{0}^{1} \\big (\\int _{0}^{\\alpha }A^{\\prime }_{\\rho , \\tau }\\,\\mathrm {d}\\tau \\big ) A_{\\rho , \\sigma }\\,\\mathrm {d}\\rho =\\int _{0}^{\\alpha }\\bar{A}^{\\prime }_{\\tau , \\sigma }\\,\\mathrm {d}\\tau =\\bar{A}_{\\tau , \\sigma }\\Big |^{\\alpha }_{0}=\\bar{A}_{\\alpha ,\\sigma },\\;\\forall \\,\\alpha ,\\,\\sigma \\in [0,1],$ which gives (REF ) by replacing the notation $\\alpha $ with $\\tau $ .",
"By virtue of Theorem REF , we derive a modified version of Theorem REF .",
"Theorem 2.4 If there exists a smooth binary function $A_{\\tau , \\sigma }$ , such that the coefficients of the csRKN method () satisfy $&C_0=0\\;\\;\\text{and}\\;\\;C_1=1,\\\\[5pt]&\\bar{A}_{0, \\sigma }=0,\\;\\bar{A}_{1, \\sigma }=\\bar{B}_\\sigma , \\;\\forall \\,\\sigma \\in [0, 1], \\\\[5pt]&A_{0, \\sigma }=0,\\;A_{1, \\sigma }=B_\\sigma =C^{\\prime }_\\sigma , \\;\\forall \\,\\sigma \\in [0, 1], \\\\&\\bar{A}_{\\tau , \\sigma }=\\int _{0}^{1} \\widehat{A}_{\\tau ,\\rho }A_{\\rho , \\sigma }\\,\\mathrm {d}\\rho ,\\;\\forall \\,\\tau ,\\,\\sigma \\in [0,1],$ where $\\widehat{A}_{\\tau , \\rho }$ is defined via (REF ), then the method is energy-preserving for solving system (REF ).",
"Particularly, if $B_\\tau =1,\\,C_\\tau =\\tau $ , then the first condition (REF ) can be removed and accordingly () should be replaced by $A_{0, \\sigma }=0,\\;A_{1, \\sigma }=1,\\;\\forall \\,\\sigma \\in [0, 1].$ This is a direct result of Theorem REF and Theorem REF .",
"In what follows, we show that all the energy-preserving csRKN methods determined by Theorem REF can be derived from energy-preserving continuous-stage partitioned Runge-Kutta (csPRK) methods.",
"To show this, we need the following two theorems.",
"Theorem 2.5 Suppose that $B_\\sigma $ satisfy (REF ) and (), and by means of (REF ) we define $\\widehat{B}_\\sigma $ as $\\widehat{B}_\\sigma =\\widehat{A}_{1,\\sigma }=\\int _0^1A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\zeta ,\\;\\;\\forall \\,\\sigma \\in [0,1],$ where $A_{\\sigma ,\\zeta }$ is assumed to satisfy (), then we have $\\int _0^1B_\\sigma \\mathrm {d}\\sigma =1,\\quad \\int _0^1\\widehat{B}_\\sigma \\mathrm {d}\\sigma =1.$ By using (REF ) and (), we have $\\int _0^1B_\\sigma \\mathrm {d}\\sigma =\\int _0^1C^{\\prime }_\\sigma \\mathrm {d}\\sigma =C_\\sigma \\Big |^1_0=1.$ Besides, by exchanging the order of integration and using (), it follows $\\int _0^1\\widehat{B}_\\sigma \\mathrm {d}\\sigma =\\int _0^1\\big (\\int _0^1A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\zeta \\big )\\mathrm {d}\\sigma =\\int _0^1\\big (\\int _0^1A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\sigma \\big )\\mathrm {d}\\zeta =\\int _0^1A_{\\sigma ,\\zeta }\\Big |^1_0\\mathrm {d}\\zeta =\\int _0^1B_\\zeta \\mathrm {d}\\zeta =1.$ Theorem 2.6 The conditions (REF ) and () implies that $C_\\tau =\\int _{0}^{1}\\widehat{A}_{\\tau ,\\sigma }\\mathrm {d}\\sigma ,\\;\\;\\forall \\,\\tau \\in [0,1],$ where $\\widehat{A}_{\\tau , \\rho }$ is defined via (REF ).",
"By using (REF ) and () we get $C_\\tau =C_\\tau -C_0=\\int _{0}^\\tau C^{\\prime }_\\sigma \\mathrm {d}\\sigma =\\int _{0}^{\\tau }A_{1,\\sigma }\\mathrm {d}\\sigma .$ On the other hand, for each fixed $\\tau $ , by exchanging the order of integration and using () it follows $\\int _{0}^{1}\\widehat{A}_{\\tau ,\\sigma }\\mathrm {d}\\sigma =\\int _{0}^{1}\\big (\\int _0^\\tau A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\zeta \\big )\\mathrm {d}\\sigma =\\int _0^\\tau \\big (\\int _{0}^{1}A^{\\prime }_{\\sigma ,\\zeta }\\,\\mathrm {d}\\sigma \\big )\\mathrm {d}\\zeta =\\int _0^\\tau A_{\\sigma ,\\zeta }\\Big |^1_0\\mathrm {d}\\zeta =\\int _{0}^{\\tau }A_{1,\\zeta }\\mathrm {d}\\zeta ,$ which gives rise to (REF ) by comparing the two formulae above.",
"In addition, let us review some existing results presented in [46].",
"For the numerical integration of a general Hamiltonian system (REF ), the so-called csPRK method (a kind of P-series integrators [19]) can be formulated as [46] $\\begin{split}P_\\tau &=p_0-h\\int _0^{1}A_{\\tau ,\\,\\sigma }\\nabla _qH(P_\\sigma ,Q_\\sigma )\\,\\mathrm {d}\\sigma ,\\quad \\tau \\in [0, 1],\\\\Q_\\tau &=q_0+h\\int _0^{1}\\widehat{A}_{\\tau ,\\,\\sigma }\\nabla _pH(P_\\sigma ,Q_\\sigma )\\,\\mathrm {d}\\sigma ,\\quad \\tau \\in [0, 1],\\\\p_{1}&=p_0-h\\int _0^{1}B_\\tau \\nabla _qH(P_\\tau ,Q_\\tau )\\,\\mathrm {d}\\tau ,\\\\q_{1}&=q_0+h\\int _0^{1}\\widehat{B}_\\tau \\nabla _pH(P_\\tau ,Q_\\tau )\\,\\mathrm {d}\\tau ,\\end{split}$ and the corresponding energy-preserving condition can be stated as follows.",
"Theorem 2.7 [46] If the coefficients of the csPRK method (REF ) satisfy $\\begin{split}& A_{0, \\sigma }=0,\\;A_{1, \\sigma }=B_\\sigma ,\\;\\,\\forall \\,\\sigma \\in [0,1], \\\\& \\widehat{A}_{0, \\sigma }=0,\\;\\widehat{A}_{1, \\sigma }=\\widehat{B}_\\sigma ,\\;\\,\\forall \\,\\sigma \\in [0, 1],\\\\& A^{\\prime }_{\\tau , \\sigma }=\\widehat{A}^{\\prime }_{\\sigma ,\\tau },\\;\\,\\forall \\,\\tau ,\\,\\sigma \\in [0,1],\\end{split}$ then the method is energy-preserving for solving Hamiltonian system (REF ).",
"Particularly, if we apply the csPRK method (REF ) with coefficients satisfying (REF ) to the Hamiltonian system (REF ), then it gives $&P_\\tau =p_0-h\\int _0^{1}A_{\\tau ,\\,\\sigma }\\nabla U(Q_\\sigma )\\,\\mathrm {d}\\sigma , \\quad \\tau \\in [0, 1], \\\\&Q_\\tau =q_0+h\\int _0^{1}\\widehat{A}_{\\tau ,\\,\\sigma }MP_\\sigma \\,\\mathrm {d}\\sigma ,\\quad \\tau \\in [0, 1], \\\\ &p_{1}=p_0-h\\int _0^{1}B_\\tau \\nabla U(Q_\\tau )\\,\\mathrm {d}\\tau ,\\\\ &q_{1}=q_0+h\\int _0^{1}\\widehat{B}_\\tau MP_\\tau \\,\\mathrm {d}\\tau ,$ and here we assumeThis assumption guarantees a P-series integrator to have order at least 1 [19].",
"$\\int _{0}^{1}B_\\tau \\mathrm {d}\\tau =1,\\quad \\int _{0}^{1}\\widehat{B}_\\tau \\mathrm {d}\\tau =1.$ It is observed that (REF ) is superfluous for obtaining the numerical solutions $p_1$ and $q_1$ , because we can substitute it into other formulae to get a simplified scheme and then it can be removed.",
"To be specific, by inserting (REF ) into (), it yields $Q_\\tau =q_0 +hC_\\tau Mp_0-h^2M\\int _{0}^{1} \\bar{A}_{\\tau , \\sigma }\\nabla U(Q_\\sigma ) \\mathrm {d}\\sigma ,$ where $C_\\tau =\\int _{0}^{1} \\widehat{A}_{\\tau ,\\sigma }\\mathrm {d}\\sigma ,\\quad \\bar{A}_{\\tau , \\sigma }=\\int _{0}^{1}\\widehat{A}_{\\tau ,\\rho }A_{\\rho ,\\sigma }\\mathrm {d}\\rho .$ Similarly, by inserting (REF ) into () and using (REF ), we have $q_1=q_0+ h Mp_0-h^2M\\int _{0}^{1} \\bar{B}_\\tau \\nabla U(Q_\\tau )\\mathrm {d}\\tau ,$ where $\\bar{B}_\\tau =\\int _{0}^{1}\\widehat{B}_\\rho A_{\\rho , \\tau }\\mathrm {d}\\rho .$ Moreover, by using (REF ), we get $\\bar{A}_{0, \\sigma }=\\int _{0}^{1}\\widehat{A}_{0,\\rho } A_{\\rho ,\\sigma }\\mathrm {d}\\rho =0,\\quad \\bar{A}_{1,\\sigma }=\\int _{0}^{1}\\widehat{A}_{1,\\rho } A_{\\rho ,\\sigma }\\mathrm {d}\\rho =\\int _{0}^{1}\\widehat{B}_\\rho A_{\\rho ,\\sigma }\\mathrm {d}\\rho =\\bar{B}_\\sigma .$ Consequently, (), (REF ) and (REF ) constitute a csRKN method in the form (), and the csRKN coefficients satisfy (REF ), (REF ) and (REF ).",
"On the basis of these analyses above, the following result is derived.",
"Theorem 2.8 An energy-preserving csRKN method acquired by Theorem REF is equivalent to an energy-preserving csPRK method originated from Theorem REF (with the condition (REF ) being satisfied).",
"Moreover, the csRKN method is at least of the same order of the associated csPRK method.",
"On account of the process from () to (REF ), the statement can be easily obtained by combining Theorem REF , Theorem REF , Theorem REF , Theorem REF and Theorem REF .",
"The only fact needs to be proved is the converse of Theorem REF .",
"From the first formula of (REF ) (see also (REF )), it is clear that $C_0=\\int _{0}^{1} \\widehat{A}_{0,\\sigma }\\mathrm {d}\\sigma =0,\\quad C_1=\\int _{0}^{1} \\widehat{A}_{1,\\sigma }\\mathrm {d}\\sigma =\\int _{0}^{1}\\widehat{B}_{\\sigma }\\mathrm {d}\\sigma =1,$ where we have used (REF ) and (REF ).",
"Besides, we have $C^{\\prime }_\\tau =\\int _{0}^{1} \\widehat{A}^{\\prime }_{\\tau ,\\sigma }\\mathrm {d}\\sigma .$ By Theorem REF , (REF ) implies (REF ).",
"Therefore, inserting (REF ) into (REF ) and using (REF ) yields $C^{\\prime }_\\tau =\\int _{0}^{1}A^{\\prime }_{\\sigma ,\\tau }\\mathrm {d}\\sigma =A_{\\sigma ,\\tau }\\Big |^1_0=A_{1,\\tau }=B_\\tau .$ Consequently, we get (REF ) and () from (REF ) and (REF ).",
"Remark 2.9 We must stress that, in general, a csRKN method (excluding the class of energy-preserving methods presented in this paper) is not necessarily equivalent to the method induced by a csPRK method.",
"The reason lies in the fact that the coefficients of a csRKN method do not necessarily satisfy (REF ) and (REF ).",
"This fact is similar to the classical case (see [18], P.284).",
"From Theorem REF , it is suggested that one might as well construct an energy-preserving csRKN method by virtue of an energy-preserving csPRK method, while the derivation of energy-preserving csPRK methods has been discussed in the previous study by the present author [46].",
"For convenience, in the following we mention two useful results which are based on the normalized shifted Legendre polynomial $L_j(x)$ : $L_0(x)=1,\\quad L_j(x)=\\frac{\\sqrt{2j+1}}{j!",
"}\\frac{{\\mathrm {d}}^j}{\\mathrm {d}x^j} \\Big (x^j(x-1)^j\\Big ),\\; \\;j=1,2,3,\\cdots .$ Theorem 2.10 [46] If the coefficients of the csPRK method (REF ) are of the following forms $\\begin{split}A_{\\tau ,\\sigma }&=\\sum \\limits _{i=0}^{s-1}\\sum \\limits _{j=0}^{r-1}\\alpha _{(i,j)}\\int _0^\\tau L_i(x)\\,\\mathrm {d}x L_j(\\sigma ),\\;\\;B_\\tau =\\sum \\limits _{j=0}^{r-1}\\alpha _{(0,j)}L_j(\\tau ),\\\\\\widehat{A}_{\\tau ,\\sigma }&=\\sum \\limits _{i=0}^{r-1}\\sum \\limits _{j=0}^{s-1}\\widehat{\\alpha }_{(i,j)}\\int _0^\\tau L_i(x)\\,\\mathrm {d}xL_j(\\sigma ),\\;\\;\\widehat{B}_\\tau =\\sum \\limits _{j=0}^{s-1}\\widehat{\\alpha }_{(0,j)}L_j(\\tau ),\\end{split}$ where the real coefficients $\\widehat{\\alpha }_{(i,j)}$ and $\\alpha _{(j,i)}$ satisfy $\\widehat{\\alpha }_{(i,j)}=\\alpha _{(j,i)}$ , then the method is energy-preserving for solving a general Hamiltonian system (REF ).",
"Moreover, the method has order at least 1 if and only if $\\widehat{\\alpha }_{(0,0)}=\\alpha _{(0,0)}=1$ .",
"Theorem 2.11 [46] The csPRK method (REF ) with coefficients given by ($s,r\\ge \\eta +1$ ) $\\begin{split}A_{\\tau , \\sigma }&=\\sum \\limits _{j=0}^{\\eta -1}\\int _0^\\tau L_j(x)\\,\\mathrm {d}xL_j(\\sigma )+\\sum \\limits ^{s-1}_{i=\\eta }\\sum \\limits ^{r-1}_{j=\\eta }\\alpha _{(i,j)}\\int _0^\\tau L_i(x)\\,\\mathrm {d}xL_j(\\sigma ),\\quad B_\\tau =1,\\\\\\widehat{A}_{\\tau , \\sigma }&=\\sum \\limits _{j=0}^{\\eta -1}\\int _0^\\tau L_j(x)\\,\\mathrm {d}xL_j(\\sigma )+\\sum \\limits ^{r-1}_{i=\\eta }\\sum \\limits ^{s-1}_{j=\\eta }\\widehat{\\alpha }_{(i,j)}\\int _0^\\tau L_i(x)\\,\\mathrm {d}xL_j(\\sigma ),\\quad \\widehat{B}_\\tau =1,\\end{split}$ is energy-preserving and at least of order $p=2\\eta $ ($\\eta \\ge 1$ ) for solving a general Hamiltonian system (REF ), where the real coefficients $\\widehat{\\alpha }_{(i,j)}$ and $\\alpha _{(j,i)}$ satisfy $\\widehat{\\alpha }_{(i,j)}=\\alpha _{(j,i)}$ ."
],
[
"Examples of energy-preserving methods and numerical implementations",
"In this section, we present some examples for illustrating the construction of energy-preserving RKN-type methods and give some comments about their numerical implementations.",
"We introduce two approaches to devise such methods.",
"The first one is a direct way by considering using the method of undetermined coefficients on the basis of Theorem REF and Theorem REF .",
"As an illustration, we present the following example.",
"Example 3.1 Assume $B_\\tau =1,\\,C_\\tau =\\tau $ and let $\\bar{A}_{\\tau ,\\sigma }=a\\tau ^2+b\\tau \\sigma +c\\tau +d,\\quad A_{\\tau ,\\sigma }=\\widehat{a}\\tau ^2\\sigma +\\widehat{b}\\tau \\sigma +\\widehat{c}\\tau +\\widehat{d},$ where $a,b,c,d$ are coefficients to be determined, noting that by Theorem REF it needs to verify the existence of $A_{\\tau ,\\sigma }$ by finding out the undetermined coefficients $\\widehat{a},\\widehat{b},\\widehat{c},\\widehat{d}$ .",
"By using (REF ), we get $b=-2a,c=\\frac{1}{2}$ .",
"Besides, from () and (), it gives $d=0,\\widehat{d}=0,\\widehat{c}=1,\\widehat{b}=-\\widehat{a}$ and $\\bar{B}_\\sigma =\\bar{A}_{1,\\sigma }=a-2a\\sigma +\\frac{1}{2}$ .",
"Finally, by inserting (REF ) into () it follows that $\\widehat{a}=12a,\\widehat{b}=-12a$ , which verifies the existence of $A_{\\tau ,\\sigma }$ .",
"As a consequence, we get a family of energy-preserving csRKN methods with coefficients given by $\\bar{A}_{\\tau ,\\sigma }=a\\tau ^2-2a\\tau \\sigma +\\frac{\\tau }{2},\\quad \\bar{B}_\\tau =a-2a\\tau +\\frac{1}{2},\\quad B_\\tau =1,\\quad C_\\tau =\\tau ,$ which is at least of orderThe order conditions for RKN-type methods can be expressed with SN-trees (see [18], page 291-292).",
"2.",
"Particularly, when $a=\\frac{1}{2}$ , the corresponding method is symmetricIt is easy to verify that the coefficients of the method satisfy the symmetric condition for csRKN methods [44].",
"and of order 4.",
"The second approach is not direct but very effective, the idea of which is based on Theorem REF .",
"To illustrate this approach, in what follows we make use of some available energy-preserving csPRK methods (derived by Theorem REF or Theorem REF , see [46] for more details) to get new energy-preserving csRKN methods.",
"Example 3.2 The $\\theta $ -parameter family of energy-preserving csPRK methods with coefficients given by [46] $A_{\\tau ,\\sigma }=\\theta \\tau ^2+(1-\\theta )\\tau ,\\;\\;B_\\tau =1;\\;\\;\\widehat{A}_{\\tau ,\\sigma }=(2\\theta \\sigma +1-\\theta )\\tau ,\\;\\;\\widehat{B}_\\tau =2\\theta \\tau +1-\\theta ,$ has order at least 1 (if and only if $\\theta =0$ the order becomes higher, say, 2).",
"Substituting (REF ) into (REF ) and (REF ), it gives $\\bar{A}_{\\tau ,\\sigma }=\\frac{\\tau }{2},\\quad \\bar{B}_\\tau =\\frac{1}{2},\\quad B_\\tau =1,\\quad C_\\tau =\\tau ,$ which corresponds to a special case of (REF ) when $a=0$ and the method is of order 2.",
"Moreover, if we interchange the role of $(A_{\\tau ,\\sigma },\\,B_\\tau )$ and $(\\widehat{A}_{\\tau ,\\sigma },\\,\\widehat{B}_\\tau )$ in (REF ), then it leads to $\\bar{A}_{\\tau ,\\sigma }=\\frac{(\\theta \\tau ^2+\\tau -\\theta \\tau )(2\\theta \\sigma +1-\\theta )}{2},\\;\\,\\bar{B}_\\tau =\\theta \\tau +\\frac{1-\\theta }{2},B_\\tau =2\\bar{B}_\\tau ,\\;\\, C_\\tau =\\theta \\tau ^2+(1-\\theta )\\tau ,$ which produces a family of 1-order energy-preserving csRKN methods.",
"Particularly, if we let $\\theta =0$ in (REF ), then we retrieve (REF ).",
"Example 3.3 A family of 4-order energy-preserving csPRK integrators is given by [46] $\\begin{split}A_{\\tau ,\\sigma }&=\\theta _2(30\\sigma ^2-30\\sigma +5)\\tau ^4+(2\\theta _1-10\\theta _2)(6\\sigma ^2-6\\sigma +1)\\tau ^3\\\\&\\;\\;\\;+\\big [(6\\theta _2-3\\theta _1)(6\\sigma ^2-6\\sigma +1)+6\\sigma -3\\big ]\\tau ^2\\\\&\\;\\;\\;+\\big [(\\theta _1-\\theta _2)(6\\sigma ^2-6\\sigma +1)-6\\sigma +4\\big ]\\tau ,\\quad B_\\tau =1,\\\\\\widehat{A}_{\\tau ,\\sigma }&=2\\big [\\theta _1(6\\sigma ^2-6\\sigma +1)+\\theta _2(20\\sigma ^3-30\\sigma ^2+12\\sigma -1)\\big ]\\tau ^3\\\\&\\;\\;\\;-3\\big [\\theta _1(6\\sigma ^2-6\\sigma +1)+\\theta _2(20\\sigma ^3-30\\sigma ^2+12\\sigma -1)-2\\sigma +1\\big ]\\tau ^2\\\\&\\;\\;\\;+\\big [\\theta _1(6\\sigma ^2-6\\sigma +1)+\\theta _2(20\\sigma ^3-30\\sigma ^2+12\\sigma -1)-6\\sigma +4\\big ]\\tau ,\\quad \\widehat{B}_\\tau =1.\\end{split}$ Substituting (REF ) into (REF ) and (REF ), it gives a family of 4-order energy-preserving csRKN methods with coefficients $\\begin{split}\\bar{A}_{\\tau ,\\sigma }&=\\frac{1}{10}\\big [(4\\theta _1\\sigma -2\\theta _1)\\tau ^3+(-6\\theta _1\\sigma ^2+2\\theta _1+5)\\tau ^2\\\\&\\;\\;\\;+(6\\theta _1\\sigma ^2-4\\theta _1\\sigma -10\\sigma +5)\\tau \\big ],\\\\\\bar{B}_\\tau &=1-\\tau ,\\quad B_\\tau =1,\\quad C_\\tau =\\tau .\\end{split}$ By exchanging the role of $(A_{\\tau ,\\sigma },\\,B_\\tau )$ and $(\\widehat{A}_{\\tau ,\\sigma },\\,\\widehat{B}_\\tau )$ in (REF ), it gives another family of 4-order energy-preserving csRKN methods with coefficients $\\begin{split}\\bar{A}_{\\tau ,\\sigma }&=\\frac{1}{10}\\big [(10\\theta _2\\sigma -5\\theta _2)\\tau ^4+(-20\\theta _2\\sigma +4\\theta _1\\sigma -2\\theta _1+10\\theta _2)\\tau ^3\\\\&\\;\\;\\;+(-20\\theta _2\\sigma ^3-6\\theta _1\\sigma ^2+30\\theta _2\\sigma ^2+2\\theta _1-5\\theta _2+5)\\tau ^2\\\\&\\;\\;\\;+(20\\theta _2\\sigma ^3+6\\theta _1\\sigma ^2-30\\theta _2\\sigma ^2-4\\theta _1\\sigma +10\\theta _2\\sigma -10\\sigma +5)\\tau \\big ],\\\\\\bar{B}_\\tau &=1-\\tau ,\\quad B_\\tau =1,\\quad C_\\tau =\\tau .\\end{split}$ It is observed that (REF ) contains (REF ) as a special case by considering taking $\\theta _2=0$ .",
"Besides, if we let $\\theta _1=\\theta _2=0$ in (REF ), then we retrieve the 4-order method given by (REF ) with $a=\\frac{1}{2}$ .",
"It is clear that the coefficients of csRKN methods are much simpler than those of the original csPRK methods.",
"As for the practical implementation, usually we have to approximate the integrals of () by numerical quadrature.",
"Let $b_i$ and $c_i$ be the weights and abscissae of the following $k$ -point interpolatory quadrature rule $\\int _0^1\\varphi (x)\\,\\mathrm {d}x\\approx \\sum \\limits _{i=1}^kb_i\\varphi (c_i),\\;\\;c_i\\in [0, 1],$ where $b_i=\\int _0^1\\ell _i(x)\\,\\mathrm {d}x,\\;\\;\\ell _i(x)=\\prod \\limits _{j=1,j\\ne i}^k\\frac{x-c_j}{c_i-c_j},\\;\\;i=1,\\cdots ,k.$ By applying the quadrature formula (REF ) of order $p$ to (), we derive a $k$ -stage classical RKN method $\\begin{split}&Q_i=q_0+hC_iM p_0 -h^2M\\sum \\limits _{j=1}^{k} b_j \\bar{A}_{ij} \\nabla U(Q_j),\\quad i=1,\\cdots , k, \\\\&q_1=q_0+hMp_0-h^2M\\sum \\limits _{i=1}^{k} b_i\\bar{B}_i \\nabla U(Q_i), \\\\&p_1= p_0-h\\sum \\limits _{i=1}^{k} b_iB_i \\nabla U(Q_i),\\end{split}$ where $\\bar{A}_{ij}=\\bar{A}_{c_i,c_j}, \\bar{B}_i=\\bar{B}_{c_i},B_i=B_{c_i}, C_i=C_{c_i}$ for $i,j=1,\\cdots , k$ .",
"Remark 3.1 Remark that usually the quadrature-based RKN scheme (REF ) possess the same order of the associated csRKN method when we use a quadrature formula with a high-enough degree of precision.",
"For the connection between the underlying csRKN method and its quadrature-based RKN method in terms of the order accuracy, we refer the readers to Theorem 3.7 of [34].",
"If the potential energy function $U(q)$ is a polynomial, then the integrands in () can be precisely computed by means of a suitable quadrature formula.",
"In such a case, the quadrature-based RKN scheme (REF ) produces an exact energy-preserving integration of (REF ) — as for the non-polynomial case, usually the RKN method (REF ) can also be able to preserve the nonlinear Hamiltonian $H(p,q)$ up to round-off error, given that we adopt a quadrature rule with high enough algebraic precision (some similar observations have been presented in [4], [5] for Hamiltonian boundary value methods).",
"Theorem 3.2 If the coefficients of the underlying energy-preserving csRKN method () acquired by Theorem REF are polynomial functions, then the RKN scheme (REF ) is exactly energy-preserving for the polynomial system (REF ) with a $\\nu $ -degree potential energy function $U(q)$ , provided that the quadrature formula (REF ) is of Gaussian typeThis means the quadrature formula is exact for all polynomial functions with degree $\\le 2k-1$ .",
"and the number of nodes, say $k$ , satisfies $k\\ge \\frac{\\max \\Big \\lbrace (\\nu -1)\\alpha +\\beta +1,\\;(\\nu -1)\\alpha +\\gamma +1\\Big \\rbrace }{2},$ where $\\bar{A}_{\\tau ,\\sigma }$ is assumed to be of degree $\\alpha $ in $\\tau $ and of degree $\\beta $ in $\\sigma $ , and $B_\\tau $ is assumed to be of degree $\\gamma $ .",
"The key of the proof lies in the fact that $k$ -point Gaussian-type quadrature formula can precisely compute the integrals of (), if the degrees of the integrands are no higher than the algebraic precision of the quadrature.",
"It is well to notice that the degree of $\\bar{B}_\\tau $ is $\\beta $ (since $\\bar{B}_\\sigma =\\bar{A}_{1,\\sigma }$ by Theorem REF ), the degree of $Q_\\tau $ is the same as that of $\\bar{A}_{\\tau ,\\sigma }$ in $\\tau $ , say $\\alpha $ , and then the degree of $\\nabla U(Q_\\tau )$ is $(\\nu -1)\\alpha $ ."
],
[
"Numerical tests",
"In this section, we report some numerical tests to verify our theoretical results.",
"The following eight methods are selected for comparisons in our experiments: (1) Method I: the 2-order energy-preserving csRKN method shown in (REF ) with $a=0.1$ ; (2) Method II: the 2-order energy-preserving csRKN method shown in (REF ) with $a=0.2$ ; (3) Method III: the 1-order energy-preserving csRKN method shown in (REF ) with $\\theta =0.1$ ; (4) Method IV: the 1-order energy-preserving csRKN method shown in (REF ) with $\\theta =0.2$ ; (5) Method V: the 4-order energy-preserving csRKN method shown in (REF ) with $\\theta _1=0.1$ ; (6) Method VI: the 4-order energy-preserving csRKN method shown in (REF ) with $\\theta _1=0.2$ ; (7) GLRK 2: the Gauss-Legendre Runge-Kutta method which is symplectic and of order 2 [18]; (8) GLRK 4: the Gauss-Legendre Runge-Kutta method which is symplectic and of order 4 [18].",
"Figure: Energy (Hamiltonian) errors by eight methods formathematical pendulum problem (), with step sizeh=0.1h=0.1."
],
[
"Test problem I",
"Consider the second-order system $\\ddot{q}=\\frac{1}{2}q^2-q,$ which can be transformed into a polynomial Hamiltonian system and the associated Hamiltonian function is $H=\\frac{1}{2}(p^2+q^2)-\\frac{1}{6}q^3,\\;\\;\\;\\text{with}\\;p=\\dot{q}.$ We take the initial value condition as $p_0=1,\\,q_0=0$ and use the time step size $h=0.1$ for numerical integration with $10,000$ steps.",
"Since the potential energy function $U(q)=-\\frac{1}{6}q^3$ is a cubic polynomial, by Theorem REF we can precisely compute the integrals of the associated csRKN methods.",
"For this problem, we use 3-point Gaussian quadrature for approximating the integrals of method I, II, III and IV, but 4-point Gaussian quadrature for method V and VI.",
"The numerical result is presented in Fig.",
"REF , which clearly shows the energy-preserving property of our new methods, while two symplectic methods only give a near-preservation of the energy."
],
[
"Test problem II",
"Consider the mathematical pendulum equation $\\ddot{q}=-\\sin q,$ which corresponds to a non-polynomial Hamiltonian system and the corresponding Hamiltonian function is $H=\\frac{1}{2}p^2-\\cos q,\\;\\;\\text{with}\\;p=\\dot{q}.$ In our experiments, we take $p_0=0.5,\\,q_0=0,\\,h=0.1$ for the numerical integration with $10,000$ steps and 4-point Gaussian quadrature is used for calculating the integrals of method I, II, V and VI, but for the method III and IV which possess the lowest order (order 1), the 6-point Gaussian quadrature is used.",
"Fig.",
"REF exhibits a very similar result as that shown in test problem I.",
"Figure: Angular momentum errors by eight methods for Kepler'sproblem (), with step size h=0.1h=0.1.Figure: Solution errors by eight methods for Kepler's problem(), with step size h=0.1h=0.1.Figure: Numerical orbits by eight methods for Kepler's problem(), with step size h=0.1h=0.1."
],
[
"Test problem III",
"Consider the well-known Kepler's problem described by the following second-order system [19] $\\ddot{q}_1=-\\frac{q_1}{(q_1^2+q_2^2)^{\\frac{3}{2}}}, \\quad \\ddot{q}_2=-\\frac{q_2}{(q_1^2+q_2^2)^{\\frac{3}{2}}}.$ By introducing $p_1=\\dot{q}_1, p_2=\\dot{q}_2$ , (REF ) can be recast as a nonlinear Hamiltonian system with the Hamiltonian (the total energy) $H=\\frac{1}{2}(p_1^2+p_2^2)-\\frac{1}{\\sqrt{q_1^2+q_2^2}}.$ It is known that such system possesses other two invariants: the quadratic angular momentum $I=q_1p_2-q_2p_1=q^T\\left(\\begin{array}{cc}0 & 1 \\\\ -1 & 0 \\\\\\end{array}\\right)\\dot{q},\\;\\;q=\\left(\\begin{array}{c}q_1 \\\\q_2 \\\\\\end{array}\\right),$ and the Runge-Lenz-Pauli-vector (RLP) invariant $L=\\left(\\begin{array}{c}p_1 \\\\p_2 \\\\0 \\\\\\end{array}\\right)\\times \\left(\\begin{array}{c}0 \\\\0 \\\\q_1p_2-q_2p_1 \\\\\\end{array}\\right)-\\frac{1}{\\sqrt{q_1^2+q_2^2}}\\left(\\begin{array}{c}q_1 \\\\q_2 \\\\0 \\\\\\end{array}\\right).$ We will take the initial values as $q_1(0)=1, \\;q_2(0)=0,\\;p_1(0)=0, \\;p_2(0)=1,$ and the corresponding exact solution is known as $q_1(t)=\\cos (t),\\;\\;q_2(t)=\\sin (t),\\;\\;p_1(t)=-\\sin (t),\\;\\;p_2(t)=\\cos (t).$ For such a non-polynomial system, we use 4-point Gaussian quadrature for approximating the integrals of method V and VI, and 5-point Gaussian quadrature for method I and II, while for method III and IV, 8-point Gaussian quadrature is applied.",
"In our numerical experiments, we compute and compare the accumulative errors of three invariants $H,\\,I$ and $L$ with $10,000$ -step integration.",
"These results are shown in Fig.",
"REF -REF , where the errors at each time step are carried out in the maximum norm $||x||_{\\infty }=\\max (|x_1|,\\cdots ,|x_n|)$ for $x=(x_1,\\cdots ,x_n)\\in \\mathbb {R}^n$ .",
"It indicates that our methods show a practical preservation of the energy but a near-preservation of other invariants, while two symplectic methods exhibit a practical preservation of the quadratic angular momentumIt is known that Gauss-Legendre Runge-Kutta methods can preserve all quadratic invariants of a general first-order system $\\dot{y}=f(y)$ [19]., but show a near-preservation of other invariants.",
"The global errors of numerical solutions are shown in Fig.",
"REF and from which linear error growths for all the methods are observed.",
"Moreover, the numerical solutions are plotted on the phase plane (see Fig.",
"REF ), showing that all the methods can mimic the phase orbits very well.",
"These numerical observations have well conformed with our theoretical results."
],
[
"Concluding remarks",
"The constructive theory of energy-preserving continuous-stage Runge-Kutta-Nyström methods is developed for solving a special class of second-order differential equations.",
"Sufficient conditions for a continuous-stage Runge-Kutta-Nyström method to be energy-preserving are presented.",
"With the presented conditions and relevant results, we can derive many new effective energy-preserving integrators.",
"Besides, the relationship between energy-preserving continuous-stage Runge-Kutta-Nyström methods and partitioned Runge-Kutta methods is examined.",
"Numerical experiments have verified our theoretical results very well."
],
[
"Acknowledgements",
"The author was supported by the National Natural Science Foundation of China (11401055), China Scholarship Council (No.201708430066) and Scientific Research Fund of Hunan Provincial Education Department (15C0028)."
]
] | 1808.08451 |
[
[
"A quasi-local characterisation of $L^p$-Roe algebras"
],
[
"Abstract Very recently, \\v{S}pakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity.",
"In this paper, we improve their method to deal with the $L^p$-version of (uniform) Roe algebras for any $p\\in [1,\\infty)$.",
"Due to the lack of reflexivity on $L^1$-spaces, some extra work is required for the case of $p=1$."
],
[
"Introduction",
"(Uniform) Roe algebras are $C^*$ -algebras associated to metric spaces, which reflect coarse properties of the underlying metric spaces.",
"These algebras have been well-studied and have fruitful applications, among which the most important ones would be the (uniform) coarse Baum-Connes conjecture and the Novikov conjecture (e.g., [32], [33], [40], [41], [42], [43]).",
"Meanwhile, they also provide a link between coarse geometry of metric spaces and the theory of $C^*$ -algebras (e.g., [1], [11], [15], [16], [19], [20], [23], [29], [30], [32], [37], [39]), and turn out to be useful in the study of topological phases of matter (e.g., [17], [10]) as well as the theory of limit operators in the study of Fredholmness of band-dominated operators (e.g., [14], [21], [31], [35]).",
"By definition, the (uniform) Roe algebra of a proper metric space $X$ is the norm closure of all bounded locally compact operators $T$ with finite propagation in the sense that there exists $R>0$ such that for any $f,g\\in C_b(X)$ acting on $L^2(X)$ by pointwise multiplication, we have $fTg=0$ provided their supports are $R$ -separated (i.e., the distance between the supports of $f$ and $g$ is at least $R$ ).",
"Since general elements in (uniform) Roe algebras may not have finite propagation, it is usually difficult to tell what operators exactly belong to them.",
"On the other hand, Roe [27] defined an asymptotic version of finite propagation as follows: An operator $T$ on $L^2(X)$ has finite $\\varepsilon $ -propagation for $\\varepsilon >0$ , if there is $R>0$ such that for any $f,g\\in C_b(X)$ , we have $\\Vert fTg\\Vert \\le \\varepsilon \\Vert f\\Vert \\cdot \\Vert g\\Vert $ provided their supports are $R$ -separated.",
"Operators with finite $\\varepsilon $ -propagation for all $\\varepsilon >0$ are called quasi-local in [26].",
"It is clear that limits of finite $\\varepsilon $ -propagation operators still have finite $\\varepsilon $ -propagation.",
"Consequently, all operators in (uniform) Roe algebras are quasi-local.",
"A natural question is that whether the converse holds as well, i.e., does every locally compact quasi-local operator belong to the (uniform) Roe algebra?",
"An affirmative answer to this question would provide a new approach to detect what operators belong to these algebras in a more practical way by estimating the norms of operator-blocks far from strips around the diagonal, and it has several immediate consequences including the followings.",
"The first one has its root in Engel's work [8], where he studied the index theory of pseudo-differential operators.",
"He showed that the indices of uniform pseudo-differential operators on Riemannian manifolds are quasi-local, while it is unclear to him whether they live in Roe algebras, which are well-understood.",
"Another application is in the work of White and Willett [38] on Cartan subalgebras of uniform Roe algebras.",
"They showed that if two uniform Roe algebras of bounded geometry metric spaces with Property A are $*$ -isomorphic, then the underlying metric spaces are bijectively coarsely equivalent provided that every quasi-local operator belongs to the uniform Roe algebras.",
"Historically, this question has been studied and partially addressed by many people including Lange and Rabinovich for $X=\\mathbb {Z}^n$ [18] (in fact they worked in a more general context, see the next paragraph), Engel for $X$ is a manifold of bounded geometry with polynomial volume growth [9], Špakula and Tikuisis [34] for $X$ has straight finite decomposition complexity in the sense of [7].",
"To our best knowledge, this question is still open for general metric spaces.",
"Based on the original definitions, various versions of Roe algebras are proposed and studied by different purposes.",
"In fact, in recent years there has been an uptick in interest in the $L^p$ -version of (uniform) Roe algebras for $p\\in [1,\\infty )$ , from the communities of both limit operator theory and coarse geometry (e.g.",
"[31], [21], [35], [14], [3], [44]).",
"And it is natural and important to study the same question in this context, i.e., does every locally compact and quasi-local operator belong to the $L^p$ -version of (uniform) Roe algebras for $p\\in [1,\\infty )$ ?",
"In this paper, we improve the method of Špakula and Tikuisis [34] in order to generalise their result from the case of $p=2$ to any $p\\in [1,\\infty )$ .",
"The main part of our result is the following (see Theorem REF for the complete version), which answers the $L^p$ -version of the question above under the condition that the underlying metric space has straight finite decomposition complexity.",
"Theorem A For a proper metric space with straight finite decomposition complexity and $p \\in [1,+\\infty )$ , quasi-locality is equivalent to being in the associated $L^p$ -Roe-like algebra.",
"Here the notion of $L^p$ -Roe-like algebra is the $L^p$ -analogue of Roe-like algebras Špakula and Tikuisis introduced for $p=2$ ([34]) and for which their main result is established.",
"However, we would like to point out that our definition of $L^p$ -Roe-like algebras are more general than Špakula and Tikuisis' definition even for $p=2$ , as we drop a commutant condition in [34], which is used in the proof of their main theorem.",
"However, we observe that this condition is redundant for the proof of the main theorem if we replace it with Lemma REF below.",
"The reason we drop this condition is inspired by the fact that it is not fulfilled for general $L^1$ -Roe-like algebras, and an obvious advantage of doing this is to allow more examples especially in the case of $p=1$ (see Remark REF and Example REF for more details).",
"The proof of our main theorem is closely modelled on their original one in [34] at least for $p\\in (1,\\infty )$ , except that the $L^p$ -Roe-like algebras need not possess a bounded involution and von Neumann algebra techniques are invalid.",
"Instead, we have to deal with asymmetric situation as in the proof of the implication “(iii) $\\Rightarrow $ (i)\" in Theorem REF and provide a direct and concrete proof of Lemma REF .",
"The case of $p=1$ is more complicated and in this case Proposition REF is established, which is the most technical part of the paper and is also a generalisation of [34].",
"The difficulty comes from the lack of reflexivity on $L^1$ -spaces, and the trick of the proof is to consider an artificial space $L^0(X)$ , which lies between $C_0(X)$ and $L^\\infty (X)$ .",
"It is worth pointing out that Proposition REF is based on a crucial intermediate result established in a more general setup of Banach spaces, and we hope that there might be some other applications in the future.After we finish this paper, Špakula and the third-named author informed us that the main theorem of this paper remains true if we only require Property A rather than straight finite decomposition complexity [36].",
"Their arguments include an essential application of Proposition REF .",
"The paper is organised as follows: we establish the settings of the paper by recalling some background in Banach algebra theory and coarse geometry in Section 2, where various examples of $L^p$ -Roe-like algebras are also provided.",
"In Section 3, we provide a complete version of our main result Theorem A, and prove the relatively easier part, where the assumption of straight finite decomposition complexity is not required.",
"In Section 4, we prove the technical tool, Proposition REF , and finish the remaining proof of the main theorem.",
"Conventions: Let $\\mathfrak {X}$ be a Banach space.",
"We denote the closed unit ball of $\\mathfrak {X}$ by $\\mathfrak {X}_1$ .",
"For any $a,b \\in \\mathfrak {X}$ and $\\varepsilon >0$ , we denote $\\Vert a-b\\Vert \\le \\varepsilon $ by $a\\approx _\\varepsilon b$ .",
"We also denote the bounded linear operators on $\\mathfrak {X}$ by $\\mathfrak {B}(\\mathfrak {X})$ , and the compact operators on $\\mathfrak {X}$ by $\\mathfrak {K}(\\mathfrak {X})$ .",
"Moreover, for a Banach algebra $A$ we define $A_\\infty := \\ell ^\\infty (\\mathbb {N}, A)\\big / \\big \\lbrace (a_n)_{n\\in \\mathbb {N}} \\in \\ell ^\\infty (\\mathbb {N}, A): \\lim _{n\\rightarrow \\infty }\\Vert a_n\\Vert =0\\big \\rbrace ,$ which is a Banach algebra with respect to the quotient norm.",
"Throughout the paper, we fix a proper metric space $(X,d)$ (i.e., every bounded subset is pre-compact).",
"Note that such a space is always locally compact and $\\sigma $ -compact.",
"We also fix a Radon measure $\\mu $ on $(X,d)$ with full support (i.e., $\\mu $ is a regular Borel measure on $X$ taking finite values on compact subsets, and for each $x \\in X$ , there exists a neighbourhood $U$ of $x$ such that $\\mu (U)>0$ )."
],
[
"Preliminaries",
"In this section, we provide the background settings of this paper by collecting several basic notions from Banach algebra theory and coarse geometry.",
"Throughout the section, let $E$ be a (complex) Banach space and $(X,d,\\mu )$ be a proper metric space with a Radon measure $\\mu $ on $X$ of full support.",
"Denote $C_b(X)$ the space of bounded continuous functions on $X$ , $C_0(X)$ the space of continuous functions on $X$ vanishing at infinity, and $C_c(X)$ the space of continuous functions on $X$ with compact supports."
],
[
"Banach space valued $L^p$ -spaces",
"In this subsection, we recall some basic notions and facts on Banach space valued $L^p$ -spaces.",
"Definition 2.1 Let $p \\in [1,\\infty ]$ .",
"For a Bochner measurable function (i.e., it equals $\\mu $ -almost everywhere to a pointwise limit of a sequence of simple functions)It follows from Pettis measurability theorem that Bochner measurability agrees with weak measurability when the Banach space $E$ is separable.",
"$\\xi : (X,\\mu ) \\rightarrow E$ , its $p$ -norm is defined by $\\Vert \\xi \\Vert _p:=\\big ( \\int _{X} \\Vert \\xi (x)\\Vert _E^p \\mathrm {d}\\mu (x) \\big )^{\\frac{1}{p}},$ and its infinity-norm is defined by $\\Vert \\xi \\Vert _\\infty :=\\mathrm {ess~sup}\\lbrace \\Vert \\xi (x)\\Vert _E: x\\in X\\rbrace .$ For $p \\in [1,\\infty ]$ , the space of $E$ -valued $L^p$ -functions on $(X,\\mu )$ is defined as follows: $L^p(X,\\mu ;E):=\\big \\lbrace \\xi :X \\rightarrow E ~\\big |~ \\xi \\mbox{~is~Bochner~measurable~and~} \\Vert \\xi \\Vert _p < \\infty \\big \\rbrace \\big / \\sim ,$ where $\\xi \\sim \\eta $ if and only if they are equal $\\mu $ -almost everywhere.",
"Equipped with the $p$ -norm, $L^p(X,\\mu ;E)$ becomes a Banach space, which is called the $L^p$ -Bochner space.",
"We also need the following closed linear subspace of $L^\\infty (X,\\mu ;E)$ : $L^0(X,\\mu ;E):=\\big \\lbrace [\\xi ] \\in L^\\infty (X,\\mu ;E)~\\big |~ \\forall \\varepsilon >0, \\exists \\mbox{~compact~}K \\subseteq X, \\mbox{~s.t.~} \\Vert \\xi |_{X\\setminus K}\\Vert _\\infty < \\varepsilon \\big \\rbrace ,$ equipped with the norm $\\Vert \\xi \\Vert _0:=\\Vert \\xi \\Vert _\\infty $ .",
"Clearly, $L^0(X,\\mu ;E)$ contains $C_0(X)$ but is more flexible, as it also contains all characteristic functions of bounded subsets of the proper metric space $(X,d)$ .",
"On the other hand, $L^0(X,\\mu ;E)$ inherits some nice behaviours of $C_0(X)$ , for example a representative can always be chosen for each element in $L^0(X,\\mu ;E)$ such that its norm goes to zero when the variable goes to infinity.",
"In order to simplify notations, we regard $\\xi $ as an element in $L^p(X,\\mu ;E)$ and write $L^p(X;E)$ instead if there is no ambiguity.",
"If $X$ is discrete and equipped with the counting measure, we simply write $\\ell ^p(X;E)$ .",
"If $p \\in (1,\\infty )$ , let $q$ be the conjugate exponent to $p$ (i.e., $\\frac{1}{p}+\\frac{1}{q}=1$ ) and if $p=1$ , we set $q=0$ instead of $q=\\infty $ .",
"It is worth noticing that the duality $L^p(X;E)^* \\cong L^q(X;E^*)$ does not hold in general (see e.g.",
"[2], [5], [6]), but we still have the following lemma.",
"Lemma 2.2 When $p \\in (1,\\infty )$ , set $q$ to be its conjugate exponent and when $p=1$ , set $q=0$ .",
"Then there is an isometric embedding $L^q(X;E^*)\\rightarrow L^p(X;E)^*$ defined by $\\eta (\\xi ):=\\int _X \\eta (x)(\\xi (x)) \\mathrm {d}\\mu (x)$ where $\\eta \\in L^q(X;E^*)$ and $\\xi \\in L^p(X;E)$ .",
"On the other hand, there is another isometric embedding $L^p(X;E)\\rightarrow L^q(X;E^*)^*$ defined by $\\xi (\\zeta ):=\\int _X \\zeta (x)(\\xi (x)) \\mathrm {d}\\mu (x)$ where $\\xi \\in L^p(X;E)$ and $\\zeta \\in L^q(X;E^*)$ .",
"For the first statement, when $p>1$ it follows from the same argument showing the classical result that $L^q(X;\\mathbb {C})$ embeds isometrically into $L^p(X;\\mathbb {C})^*$ (which are indeed isomorphic).",
"And for $p=1$ , we have the following maps $L^0(X;E^*) \\subseteq L^\\infty (X;E^*) \\hookrightarrow L^1(X,E)^*$ where the second isometric embedding follows from the same argument showing the classical result that $L^\\infty (X;\\mathbb {C})$ embeds isometrically into $L^1(X;\\mathbb {C})^*$ .",
"For the second statement, it suffices to show that for any $\\xi \\in L^p(X;E)$ , we have $\\Vert \\xi \\Vert _p = \\sup \\lbrace |\\xi (\\zeta )|: \\zeta \\in L^q(X;E^*)\\mbox{~and~} \\Vert \\zeta \\Vert _q \\le 1\\rbrace .$ It is clear that the right hand side does not exceed the left.",
"Conversely we may assume, by the inner regularity of $\\mu $ , that $\\xi $ is non-zero and $\\xi =\\sum _{i=1}^{n} y_i \\chi _{\\Omega _i}$ for some $y_i\\in E$ and mutually disjoint compact subsets $\\Omega _i$ in $X$ .",
"Note that $\\Vert \\xi \\Vert _p^p=\\sum _{i=1}^n \\Vert y_i\\Vert _E^p\\mu (\\Omega _i)$ .",
"For each $y_i$ , choose a $y_i^* \\in (E^*)_1$ such that $y_i^*(y_i)=\\Vert y_i\\Vert _E$ .",
"Define $\\zeta :=\\sum _{i=1}^{n} \\frac{\\Vert y_i\\Vert _E^{p-1}}{\\Vert \\xi \\Vert _p^{p-1}}y_i^* \\chi _{\\Omega _i}.$ Note that when $p=1$ , $\\zeta $ can be written simply as $\\sum _{i=1}^{n} y_i^* \\chi _{\\Omega _i}$ .",
"It is straightforward to check that $\\zeta \\in L^q(X;E^*)$ with $\\Vert \\zeta \\Vert _q=1$ and $\\xi (\\zeta )=\\Vert \\xi \\Vert _p$ (note that when $p=1$ , we set $q=0$ ).",
"Hence, we finish the proof.",
"Finally we recall $L^p$ -tensor products (more details can be found in [4], [24] and [22]), which will be used in Section REF without further reference.",
"For $p\\in [1,\\infty )$ , there is a tensor product of $L^p$ -spaces with $\\sigma $ -finite measures such that there is a canonical isometric isomorphism $L^p(X,\\mu )\\otimes L^p(Y,\\nu )\\cong L^p(X\\times Y,\\mu \\times \\nu )$ , which identifies the element $\\xi \\otimes \\eta $ with the function $(x,y)\\mapsto \\xi (x)\\eta (y)$ on $X\\times Y$ for every $\\xi \\in L^p(X,\\mu )$ and $\\eta \\in L^p(Y,\\nu )$ .",
"Moreover, the following properties hold: Under the identification above, the linear spans of all $\\xi \\otimes \\eta $ are dense in $L^p(X\\times Y,\\mu \\times \\nu )$ .",
"$||\\xi \\otimes \\eta ||_p=||\\xi ||_p||\\eta ||_p$ for all $\\xi \\in L^p(X,\\mu )$ and $\\eta \\in L^p(Y,\\nu )$ .",
"The tensor product is commutative and associative.",
"If $a\\in \\mathfrak {B}(L^p(X_1,\\mu _1),L^p(X_2,\\mu _2))$ and $b\\in \\mathfrak {B}(L^p(Y_1,\\nu _1),L^p(Y_2,\\nu _2))$ , then there exists a unique element $c\\in \\mathfrak {B}(L^p(X_1\\times Y_1,\\mu _1\\times \\nu _1),L^p(X_2\\times Y_2,\\mu _2\\times \\nu _2))$ such that under the identification above, $c(\\xi \\otimes \\eta )=a(\\xi )\\otimes b(\\eta )$ for all $\\xi \\in L^p(X_1,\\mu _1)$ and $\\eta \\in L^p(Y_1,\\nu _1)$ .",
"We will denote this operator by $a \\otimes b$ .",
"Moreover, $||a\\otimes b||=||a|| \\cdot ||b||$ .",
"The above tensor product of operators is associative, bilinear, and satisfies $(a_1\\otimes b_1)(a_2\\otimes b_2)=a_1a_2\\otimes b_1b_2$ .",
"If $A\\subseteq \\mathfrak {B}(L^p(X,\\mu ))$ and $B\\subseteq \\mathfrak {B}(L^p(Y,\\nu ))$ are closed subalgebras, we define $A\\otimes B\\subseteq \\mathfrak {B}(L^p(X\\times Y,\\mu \\times \\nu ))$ to be the closed linear span of all $a\\otimes b$ with $a\\in A$ and $b\\in B$ ."
],
[
"Block cutdown maps",
"Now we introduce block cutdown maps, providing an approach to cut an operator into the form of block diagonals.",
"First, let us recall some more notions.",
"For $p \\in \\lbrace 0\\rbrace \\cup [1,\\infty ]$ , the multiplication representation $\\rho :C_b(X) \\rightarrow \\mathfrak {B}(L^p(X;E))$ is defined by pointwise multiplications: $(\\rho (f)\\xi )(x)=f(x)\\xi (x)$ , where $f \\in C_b(X)$ , $\\xi \\in L^p(X;E)$ and $x \\in X$ .",
"Without ambiguity, we write $fT$ and $Tf$ instead of $\\rho (f)T$ and $T\\rho (f)$ , for $f\\in C_b(X)$ and $T \\in \\mathfrak {B}(L^p(X;E))$ , respectively.",
"It is worth noticing that $\\mu $ has full support if and only if $\\rho $ is injective.",
"We also recall that a net $\\lbrace T_\\alpha \\rbrace $ converges in strong operator topology (SOT) to $T$ in $\\mathfrak {B}(L^p(X;E))$ if and only if $\\Vert T_\\alpha (\\xi )-T(\\xi )\\Vert _p \\rightarrow 0$ for any $\\xi \\in L^p(X;E)$ .",
"Definition 2.3 Given an equicontinuous family $(e_j)_{j \\in J}$ of positive contractions in $C_b(X)$ with pairwise disjoint supports, define the block cutdown map $\\theta _{(e_j)_{j \\in J}}: \\mathfrak {B}(L^p(X;E))\\rightarrow \\mathfrak {B}(L^p(X;E))$ by $\\theta _{(e_j)_{j \\in J}}(a):=\\sum _{j \\in J}e_jae_j,$ where the sum converges in (SOT) by Lemma REF below.",
"We say that a closed subalgebra $B \\subseteq \\mathfrak {B}(L^p(X;E))$ is closed under block cutdowns, if $\\theta _{(e_j)_{j \\in J}}(B)\\subseteq B$ for every equicontinuous family $(e_j)_{j \\in J}$ of positive contractions in $C_b(X)$ with pairwise disjoint supports.",
"Lemma 2.4 Let $(e_j)_{j \\in J}$ and $(f_j)_{j \\in J}$ be two equicontinuous families of positive contractions in $C_b(X)$ with pairwise disjoint supports, and $a \\in \\mathfrak {B}(L^p(X;E))$ .",
"Then the sum $\\sum _{j \\in J}f_jae_j$ converges in (SOT) to an operator in $\\mathfrak {B}(L^p(X;E))$ .",
"Furthermore, we have: $\\big \\Vert \\sum _{j \\in J}f_jae_j\\big \\Vert =\\sup _{j \\in J}\\Vert f_jae_j\\Vert .$ First of all, we prove in the case of $p\\in (1,\\infty )$ and let $q$ be the conjugate exponent to $p$ .",
"Let $Y_j:=\\mathrm {supp}(e_j)$ and $Z_j:=\\mathrm {supp}(f_j)$ .",
"For any $\\xi \\in L^p(X;E)$ , any finite subset $F \\subseteq J$ and any $\\eta \\in L^q(X;E^*)$ with $\\Vert \\eta \\Vert _q \\le 1$ , we have that $\\big |\\eta \\big (\\sum _{j \\in F}f_jae_j\\xi \\big )\\big | &=& \\big |\\sum _{j \\in F} (\\eta \\chi _{Z_j})(f_jae_j\\chi _{Y_j}\\xi )\\big | \\\\&\\le &\\sum _{j \\in F} \\Vert \\eta |_{Z_j}\\Vert _q\\cdot \\Vert f_jae_j(\\xi |_{Y_j})\\Vert _p\\\\&\\le &\\big (\\sum _{j \\in F} \\Vert \\eta |_{Z_j}\\Vert _q^q\\big )^{\\frac{1}{q}}\\cdot \\big (\\sum _{j \\in F} \\Vert f_jae_j(\\xi |_{Y_j})\\Vert _p^p\\big )^{\\frac{1}{p}}\\\\& \\le & \\sup _{j \\in J}\\Vert f_jae_j\\Vert \\cdot \\Vert \\xi |_{\\sqcup _{j\\in F}Y_j}\\Vert _p.$ Hence, it follows from Lemma REF that $\\big \\Vert \\big (\\sum _{j \\in F}f_jae_j\\big )\\xi \\big \\Vert _p \\le \\sup _{j \\in J}\\Vert f_jae_j\\Vert \\cdot \\Vert \\xi |_{\\sqcup _{j\\in F}Y_j}\\Vert _p.$ Since $\\Vert \\xi |_{\\sqcup _{j\\in J}Y_j} \\Vert _p\\le \\Vert \\xi \\Vert _p<\\infty $ , we know $\\big \\lbrace \\xi |_{\\sqcup _{j\\in F}Y_j}\\big \\rbrace _F$ is a Cauchy net.",
"Hence, $\\sum _{j \\in J}f_jae_j$ converges in (SOT) and $\\Vert \\sum _{j \\in J}f_jae_j\\Vert \\le \\sup _{j \\in J}\\Vert f_jae_j\\Vert $ .",
"On the other hand, it is clear that $\\Vert \\sum _{j \\in J}f_jae_j\\Vert $ $\\ge \\sup _{j \\in J}\\Vert f_jae_j\\Vert $ .",
"Hence we finish the proof for $p>1$ .",
"Since the proof for the case of $p=1$ is more direct, we leave the details to the reader.",
"Remark 2.5 Note that the multiplication by $C_b(X)$ commutes with the block cutdowns, i.e., for any $a \\in \\mathfrak {B}(L^p(X;E))$ and $f \\in C_b(X)$ , we have $f\\theta _{(e_j)_{j \\in J}}(a)=\\theta _{(e_j)_{j \\in J}}(fa) \\quad \\mbox{~and~}\\quad \\theta _{(e_j)_{j \\in J}}(a)f=\\theta _{(e_j)_{j \\in J}}(af).$ Definition 2.6 Suppose $\\mathcal {X}$ is a metric family of subsets in $X$ (recall that a metric family is a set of metric spaces), each of the subset is equipped with the induced metric and $a \\in \\mathfrak {B}(L^p(X;E))$ .",
"We say that $a$ is block diagonal with respect to $\\mathcal {X}$, if there exist an equicontinuous family $(e_j)_{j\\in J}$ of positive contractions in $C_b(X)$ with pairwise disjoint supports and $\\lbrace Y_j\\rbrace _{j\\in J} \\subseteq \\mathcal {X}$ , such that $a=\\theta _{(e_j)_{j \\in J}}(a),$ and $\\mathrm {supp}(e_j) \\subseteq Y_j$ .",
"In this case, we shall denote $a_{Y_j}:=e_jae_j$ , which is called the $Y_j$ -block of $a$."
],
[
"$L^p$ -Roe-like algebras",
"Now we introduce $L^p$ -Roe-like algebras, which are our main objects in this paper.",
"Definition 2.7 Let $R \\ge 0$ and $a \\in \\mathfrak {B}(L^p(X;E))$ .",
"We say that $a$ has propagation at most $R$, if for any $f,f^{\\prime } \\in C_b(X)$ with $d(\\mathrm {supp}(f),\\mathrm {supp}(f^{\\prime })) > R$ , then $faf^{\\prime }=0$ .",
"$a$ has $\\varepsilon $ -propagation at most $R$ for some $\\varepsilon >0$ , if for any $f,f^{\\prime } \\in C_b(X)_1$ with $d(\\mathrm {supp}(f),\\mathrm {supp}(f^{\\prime })) > R$ , then $\\Vert faf^{\\prime }\\Vert < \\varepsilon $ .",
"$a$ is quasi-local, if it has finite $\\varepsilon $ -propagation for every $\\varepsilon >0$ .",
"Definition 2.8 Let $(X,d)$ be a proper metric space equipped with a Radon measure $\\mu $ whose support is $X$ , and $p \\in [1,+\\infty )$ .",
"Suppose $E$ is a Banach space and $B \\subseteq \\mathfrak {B}(L^p(X;E))$ is a Banach subalgebra such that $C_b(X)BC_b(X)=B$ and is closed under block cutdowns.",
"Define: $\\mathrm {Roe}(X,B)$ to be the norm-closure of all the operators in $B$ with finite propagations.",
"$\\mathrm {Roe}(X,B)$ is called the $L^p$ -Roe-like algebra of $(X,d,\\mu )$; $\\mathcal {K}(X,B)$ to be the norm-closure of $C_0(X)BC_0(X)$ in $\\mathfrak {B}(L^p(X;E))$ .",
"Remark 2.9 The definition of $L^2$ -Roe-like algebras come from [34], in which the following extra condition is also imposed: $[C_0(X),B] \\subseteq \\mathcal {K}(X,B).$ This condition is used in the proof of their main theorem, [34].",
"However, it turns out to be redundant if we apply our Lemma REF below.",
"On the other hand, this condition is fulfilled by most of the well-known $L^p$ -Roe-like algebras for $p\\in (1,\\infty )$ (as we will see in the following examples), but not for $p=1$ (see the explanation in Example REF ).",
"This is exactly our starting point to explore whether condition (REF ) is necessary, and it turns out that we may omit it in Definition REF without affecting the main theorem.",
"In this way, our main result (Theorem REF ) is a slight generalisation of [34].",
"We notice that in the case of $p=2$ , it has been pointed out in [34] that $\\mathcal {K}(X,B)$ is an ideal in $\\mathrm {Roe}(X,B)$ under the additional condition (REF ).",
"Now we show that it still holds in our settings.",
"Lemma 2.10 For any $p \\in [1,+\\infty )$ , $\\mathcal {K}(X,B)$ is a closed two-sided ideal in $\\mathrm {Roe}(X,B)$ .",
"It suffices to show that for any $b=f_1b_1g_1 \\in C_c(X)BC_c(X)$ and $a\\in B$ with finite propagation at most $R$ , $ba \\in \\mathcal {K}(X,B)$ .",
"Take a function $g_2 \\in C_c(X)$ such that $g_2$ is 1 on the compact subset $\\overline{\\mathcal {N}_R(\\mathrm {supp}(g_1))}$ .",
"It follows that $g_1a(1-g_2)=0$ , which implies that $g_1a=g_1ag_2$ .",
"Hence, we have $ba=f_1b_1g_1a=f_1(b_1g_1a)g_2.$ Recall that $C_b(X)BC_b(X)=B$ , so we have $b_1g_1\\in B$ and $a \\in B$ , which implies that $ba \\in C_c(X)BC_c(X)$ .",
"Similarly, $ab \\in C_c(X)BC_c(X)$ as well.",
"So we finish the proof.",
"Before we illustrate several examples of $L^p$ -Roe-like algebras, let us recall the following notion related to matrix algebras.",
"Definition 2.11 Let $(X,d)$ be a discrete proper metric space and $p \\in [1,+\\infty )$ .",
"Denote $\\overline{M}^p_X:=\\overline{C_c(X)\\mathfrak {B}(\\ell ^p(X))C_c(X)}^{\\mathfrak {B}(\\ell ^p(X))},$ i.e., for any fixed point $x_0 \\in X$ $\\overline{M}^p_X=\\overline{\\bigcup _{n\\in \\mathbb {N}}M^p_{B_n(x_0)}}$ where $M^p_{B_n(x_0)}=\\mathfrak {B}(\\ell ^p(B_n(x_0))) \\subseteq \\mathfrak {B}(\\ell ^p(X))$ , which is the matrix algebra over the closed ball of radius $n$ and centered in $x_0$ .",
"In other words, operators in $\\overline{M}^p_X$ are exactly those that can be approximated by finite matrices.",
"Phillips studied the relation between $\\overline{M}^p_X$ and compact operators $\\mathfrak {K}(\\ell ^p(X))$ in [25].",
"He showed that when $p>1$ , $\\overline{M}^p_X=\\mathfrak {K}(\\ell ^p(X))$ ([25]); and when $p=1$ , $\\overline{M}^1_X \\subsetneq \\mathfrak {K}(\\ell ^1(X))$ in general as illustrated in [25] (see also Example REF ).",
"Now we are ready to provide various of examples of $L^p$ -Roe-like algebras, which include $\\ell ^p$ -uniform Roe algebras, band-dominated operator algebras, $L^p$ -Roe algebras, $\\ell ^p$ -uniform algebras and stable $\\ell ^p$ -uniform Roe algebras.",
"Example 2.12 ($\\ell ^p$ -Uniform Roe Algebra) Let $(X,d)$ be a discrete proper metric space and $p \\in [1,+\\infty )$ .",
"Take $E=\\mathbb {C}$ to be the complex number and $B=\\mathfrak {B}(\\ell ^p(X))$ , which is clearly closed under block cutdowns, and satisfies $C_b(X)BC_b(X)=B$ .",
"In this case, $\\mathrm {Roe}(X,B)$ is called the $\\ell ^p$ -uniform Roe algebra of $X$ , which is defined in [3] and denoted by $B^p_u(X)$ , and $\\mathcal {K}(X,B)$ is $\\overline{M}^p_X$ introduced above.",
"It may be worth noting that $\\overline{M}^1_X$ is structurally different from $\\overline{M}^p_X$ for $p>1$ .",
"$\\bullet ~ p>1$ : As pointed out above, $\\overline{M}^p_X=\\mathfrak {K}(\\ell ^p(X))$ .",
"And condition (REF ) follows from the fact that $C_0(X)B \\subseteq \\mathcal {K}(X,B)$ and $BC_0(X) \\subseteq \\mathcal {K}(X,B)$ .",
"$\\bullet ~ p=1$ : The algebra $\\mathcal {K}(X,B)$ is in general properly contained in $\\mathfrak {K} (\\ell ^1(X))$ (see Example 1.10 in [25]).",
"For example, taking $X$ to be the natural number $\\mathbb {N}$ , consider the operator $T: \\ell ^1(\\mathbb {N}) \\rightarrow \\ell ^1(\\mathbb {N})$ defined by $T(\\xi ):=\\big (\\sum _{n\\in \\mathbb {N}} \\xi (n)\\big ) \\delta _0,$ where $\\xi \\in \\ell ^1(\\mathbb {N})$ and $\\delta _0 \\in \\ell ^1(\\mathbb {N})$ is the function taking value 1 at the origin point 0, and 0 elsewhere.",
"Since $T$ has rank 1, it belongs to $\\mathfrak {K} (\\ell ^1(\\mathbb {N}))$ .",
"However, it is not hard to see that $T\\notin \\mathcal {K}(\\mathbb {N}, \\mathfrak {B}(\\ell ^1(\\mathbb {N})))=\\overline{M}^1_{\\mathbb {N}}$ .",
"Furthermore, the operator $T$ also illuminates that condition (REF ) does not hold in general, since $[\\delta _0,T] \\notin \\mathcal {K}(\\mathbb {N}, \\mathfrak {B}(\\ell ^1(\\mathbb {N})))$ .",
"Example 2.13 (Band-Dominated Operator Algebra) Let $(X,d)$ be a uniformly discrete metric space of bounded geometry (in the sense that for a given $R>0$ , all closed balls $B(x,R)$ have a uniform bound on cardinalities for all $x\\in X$ ), $p \\in (1,+\\infty )$ and $E$ be a Banach space.",
"Take $B=\\mathfrak {B}(\\ell ^p(X;E))$ , which is clearly closed under block cutdowns and satisfies $C_b(X)BC_b(X)=B$ .",
"Elements in $B$ can be represented in the matrix form $b=(b_{x,y})_{x,y\\in X} \\in \\mathfrak {B}(\\ell ^p(X;E)), \\quad \\mbox{where} \\quad b_{x,y} \\in \\mathfrak {B}(E).$ In this case, $\\mathrm {Roe}(X,B)=\\mathcal {A}^p_E(X)$ , which is the algebra of band-dominated operators (see [35]) and it is clear that $\\mathcal {K}(X,B)=\\mathcal {K}^p_E(X)$ , which is the set of all $\\mathcal {P}$ -compact operators on $\\ell ^p(X;E)$ , defined in [35].",
"Example 2.14 ($L^p$ -Roe Algebra) Let $(X,d)$ be a proper metric space equipped with a Radon measure $\\mu $ with support $X$ , and $p \\in [1,+\\infty )$ .",
"We say that an operator $b$ in $\\mathfrak {B}(L^p(X; \\ell ^p(\\mathbb {N}))) \\cong \\mathfrak {B}(L^p(X \\times \\mathbb {N}))$ is locally compact if for any $f \\in C_0(X)$ , $fb$ and $bf$ belong to $\\mathfrak {K}(L^p(X \\times \\mathbb {N}))$ .",
"Now take $E=\\ell ^p(\\mathbb {N})$ and $B$ to be the set of all locally compact operators in $\\mathfrak {B}(L^p(X; \\ell ^p(\\mathbb {N})))$ , which is clearly closed under block cutdowns and satisfies $C_b(X)BC_b(X)=B$ .",
"The corresponding $L^p$ -Roe-like algebra $\\mathrm {Roe}(X,B)$ is called the $L^p$ -Roe algebra of $X$, denoted by $B^p(X)$ .",
"It is, by definition, the norm closure of all locally compact and finite propagation operators in $\\mathfrak {B}(L^p(X; \\ell ^p(\\mathbb {N})))$ .",
"Analogous to the arguments in Example REF , one can check that when $p>1$ , $\\mathcal {K}(X,B)=\\mathfrak {K}(L^p(X \\times \\mathbb {N}))$ and it does not hold in general when $p=1$ .",
"When $X$ is discrete, the $L^p$ -Roe algebra $B^p(X)$ coincides with the $\\ell ^p$ -Roe algebra defined in [3] and in the case of $p=2$ , the $L^2$ -Roe algebra is the classical Roe algebra in the literature.",
"Remark 2.15 As explained in [3], there is another version of locally compactness: we say that an operator $b$ in $\\mathfrak {B}(L^p(X; \\ell ^p(\\mathbb {N})))$ is locally compact if for any $f \\in C_0(X)$ , $fb$ and $bf$ belong to $\\mathfrak {K}(L^p(X)) \\otimes \\overline{M}^p_{\\mathbb {N}} \\subseteq \\mathfrak {B}(L^p(X \\times \\mathbb {N}))$ .",
"Note that the subalgebra $\\mathfrak {K}(L^p(X)) \\otimes \\overline{M}^p_{\\mathbb {N}}$ is isomorphic to the norm closure of $\\bigcup _{n\\in \\mathbb {N}} M^p_n(\\mathfrak {K}(L^p(X)))$ .",
"Therefore, we can alternatively define another version of the $L^p$ -Roe algebra of $X$ to be the norm closure of all locally compact (in this new sense) and finite propagation operators in $\\mathfrak {B}(L^p(X; \\ell ^p(\\mathbb {N})))$ .",
"When $p>1$ , it coincides with $B^p(X)$ defined in Example REF as $\\mathfrak {K}(L^p(X)) \\otimes \\overline{M}^p_{\\mathbb {N}} \\cong \\mathfrak {K}(L^p(X \\times \\mathbb {N}))$ .",
"However, it is strictly contained in $B^1(X)$ when $p=1$ .",
"Example 2.16 ($\\ell ^p$ -Uniform Algebra) Let $(X,d)$ be a discrete metric space with bounded geometry and $p \\in [1,+\\infty )$ .",
"Set $E=\\ell ^p(\\mathbb {N})$ , and $B$ to be the closure of the set of all $b=(b_{x,y})_{x,y \\in X} \\in \\mathfrak {B}(\\ell ^p(X; \\ell ^p(\\mathbb {N})))$ for which the rank of $b_{x,y} \\in \\mathfrak {B}(\\ell ^p(\\mathbb {N}))$ is uniformly bounded.",
"Clearly, $B$ is closed under block cutdowns, and satisfies $C_b(X)BC_b(X)=B$ .",
"In this case, $\\mathrm {Roe}(X,B)=UB^p(X)$ , the $\\ell ^p$ -uniform algebra of $X$, introduced in [3].",
"When $p>1$ , we have that $\\mathcal {K}(X,B)=\\mathfrak {K}(l^p(X \\times \\mathbb {N}))$ .",
"But it does not hold in general when $p=1$ .",
"Example 2.17 (Stable $\\ell ^p$ -Uniform Roe Algebra) Let $(X,d)$ be a discrete metric space with bounded geometry and $p \\in [1,+\\infty )$ .",
"Set $E=\\ell ^p(\\mathbb {N})$ , and $B$ to be the closure of the set of all $b=(b_{x,y})_{x,y \\in X} \\in \\mathfrak {B}(\\ell ^p(X; \\ell ^p(\\mathbb {N})))$ for which there exists a finite-dimensional subspace $E_b \\subseteq \\ell ^p(\\mathbb {N})$ such that $b_{x,y} \\in \\mathfrak {B}(E_b) \\subseteq \\mathfrak {B}(\\ell ^p(\\mathbb {N}))$ .",
"Clearly, $B$ is closed under block cutdowns and satisfies $C_b(X)BC_b(X)=B$ .",
"In this case, $\\mathrm {Roe}(X,B)=B^p_s(X)$ , the stable $\\ell ^p$ -uniform Roe algebra of $X$, introduced in [3].",
"Moreover, $B^p_s(X)\\cong B^p_u(X) \\otimes \\mathfrak {K}(\\ell ^p(\\mathbb {N}))$ , which explains the terminology.",
"Analogous to the arguments in Example REF , one can check that when $p>1$ , $\\mathcal {K}(X,B)=\\mathfrak {K}(l^p(X \\times \\mathbb {N}))$ and it does not hold in general when $p=1$ .",
"Remark 2.18 As explained in [3], there is another version of the stable $\\ell ^p$ -uniform Roe algebra of $X$ , defined to be the norm closure of finite propagation operators $b=(b_{x,y})_{x,y \\in X} \\in \\mathfrak {B}(\\ell ^p(X; \\ell ^p(\\mathbb {N})))$ for which there exists some $k\\in \\mathbb {N}$ such that $b_{x,y} \\in M_k(\\mathbb {C}) \\subseteq \\mathfrak {B}(\\ell ^p(\\mathbb {N}))$ (here $M_k(\\mathbb {C})$ is embedded as a subalgebra of $\\mathfrak {B}(\\ell ^p(\\mathbb {N}))$ in a fixed way, independent of the points in $X$ ).",
"It is clear that this algebra is isomorphic to $B^p_u(X) \\otimes \\overline{M}^p_\\mathbb {N}$ for all $p\\in [1,\\infty )$ .",
"As before for $p>1$ , it coincides with $B^p_s(X)$ defined in Example REF .",
"Remark 2.19 In general, we have that for discrete space $X$ , $B^p_u(X) \\subseteq B^p_s(X) \\subseteq UB^p(X) \\subseteq B^p(X).$ It is worth noticing that $UB^1(X)$ is not contained in the weak version of the $L^1$ -Roe algebra defined in Remark REF .",
"Indeed, Example REF provides a rank one operator $T \\in \\mathfrak {B}(\\ell ^1(\\mathbb {N}))$ which does not sit in $\\overline{M}^1_\\mathbb {N}$ .",
"Define the diagonal operator $b \\in \\mathfrak {B}(\\ell ^1(X; \\ell ^1(\\mathbb {N})))$ by $b_{x,x}:=T$ for $x\\in X$ , and $b_{x,y}=0$ for $x \\ne y$ .",
"Clearly, $b$ is such an example as desired."
],
[
"Straight finite decomposition complexity",
"In this subsection, we explain the notion of straight finite decomposition complexity, which will be used in the sequel.",
"Straight finite decomposition complexity (sFDC) was introduced in [7] as a weak version of the original notion of finite decomposition complexity (FDC), which was introduced and studied by Guentner, Tessera and Yu in their study of topological rigidity in [12].",
"In general, finite asymptotic dimension implies finite decomposition complexity [13], which consequently implies straight finite decomposition complexity [7].",
"Moreover, it was also shown in [7] that straight finite decomposition complexity does imply Yu's Property A.",
"However, it is still unknown whether (FDC), (sFDC) and Yu's Property A are all equivalent or not.",
"Definition 2.20 Let $(X,d)$ be a proper metric space and $Z, Z^{\\prime } \\subseteq X$ .",
"Let $\\mathcal {X},\\mathcal {Y}$ be metric families of subsets in $X$ , and $R\\ge 0$ .",
"$\\mathcal {X}$ is uniformly bounded, if $\\sup _{X\\in \\mathcal {X}}\\mathrm {diam}(X)<\\infty .$ Denote the $R$ -neighbourhood of $Z$ by $\\mathcal {N}_R(Z):=\\lbrace z\\in X: d(z,Z)\\le R\\rbrace $ .",
"Set $\\mathcal {N}_R(\\mathcal {X}):=\\lbrace \\mathcal {N}_R(X):X\\in \\mathcal {X}\\rbrace .$ A metric family $(Y_j)_{j\\in J}$ of subsets of $X$ is $R$ -disjoint, if $d(Y_j,Y_j^{\\prime })>R$ for all $j\\ne j^{\\prime }$ .",
"Write $\\bigsqcup _{R-disjoint}Y_j$ for their union to indicate that the family is $R$ -disjoint.",
"$Z$ can $R$ -decompose over $\\mathcal {Y}$, if $Z$ can be decomposed into $Z=X_0\\cup X_1$ and $X_i=\\bigsqcup _{R-\\text{disjoint}}X_{ij}, \\quad i=0,1,$ such that $X_{ij}\\in \\mathcal {Y}$ for all $i,j.$ $\\mathcal {X}$ can $R$ -decompose over $\\mathcal {Y}$, denoted by $\\mathcal {X}\\xrightarrow{}\\mathcal {Y}$ , if every $Y\\in \\mathcal {X}$ can $R$ -decompose over $\\mathcal {Y}$ .",
"$X$ has straight finite decomposition complexity, if for any sequence $0\\le R_1<R_2< \\cdots ,$ there exists $m\\in \\mathbb {N}$ and metric families $\\lbrace X\\rbrace =\\mathcal {X}_0,\\mathcal {X}_1,\\ldots ,\\mathcal {X}_m$ , such that $\\mathcal {X}_{i-1} \\xrightarrow{} \\mathcal {X}_i$ for $i=1,\\ldots , m$ , and the family $\\mathcal {X}_m$ is uniformly bounded.",
"We remark here briefly that one way to define finite decomposition complexity is to use a “decomposition game\", which means a priori that the choices of $R_i$ might depend on the previous families $\\mathcal {X}_0,\\mathcal {X}_1,\\ldots ,\\mathcal {X}_{i-1}$ .",
"Consequently, sFDC can be obviously implied from FDC."
],
[
"The main theorem",
"In this section, we present our main result (Theorem REF ), which gives several different pictures of how elements in $L^p$ -Roe-like algebras may look like.",
"We also prove the relatively easier part where straight finite decomposition complexity is not required, while leaving the rest of the proof to the next section after more technical tools are developed.",
"To state our main theorem, we need to introduce some notions as follows.",
"Definition 3.1 ([28]) Let $(X,d)$ be a proper metric space.",
"A function $g \\in C_b(X)$ is called a Higson function (also called a slowly oscillating function), if for every $R>0$ and $\\varepsilon >0$ , there exists a compact set $A \\subseteq X$ such that for any $x,y \\in X\\backslash A$ with $d(x,y) < R$ , then $|g(x)-g(y)| < \\varepsilon $ .",
"The set of all Higson functions on $X$ is denoted by $C_h(X)$ .",
"Definition 3.2 [34], Let $(X,d)$ be a metric space.",
"A bounded sequence $(f_n)_{n \\in \\mathbb {N}}$ in $C_b(X)$ is called very Lipschitz, if for every $L>0$ , there exists $n_0 \\in \\mathbb {N}$ such that $f_n$ is $L$ -Lipschitz for any $n \\ge n_0$ .",
"Let $\\mathrm {VL}(X)$ denote the set of all very Lipschitz bounded sequences in $C_b(X)$ .",
"Define $\\mathrm {VL}_\\infty (X):=\\mathrm {VL}(X)\\big / \\big \\lbrace (f_n)_{n\\in \\mathbb {N}} \\in \\mathrm {VL}(X): \\lim _{n \\rightarrow \\infty } \\Vert f_n\\Vert =0\\big \\rbrace .$ It is known from [34] that $\\mathrm {VL}(X)$ is a $C^*$ -subalgebra of $\\ell ^\\infty (\\mathbb {N},C_b(X))$ and $\\mathrm {VL}_\\infty (X)$ is a $C^*$ -subalgebra of $(C_b(X))_\\infty $ .",
"In the following, we will view both $\\mathrm {VL}_\\infty (X)$ and $B \\subseteq \\mathfrak {B}(L^p(X;E))$ as Banach subalgebras of $\\mathfrak {B}(L^p(X;E))_\\infty $ , and consider the relative commutant: $B \\cap \\mathrm {VL}_\\infty (X)^{\\prime }=\\lbrace b\\in B: b \\mbox{~commutes~with~elements~in~}\\mathrm {VL}_\\infty (X)\\rbrace .$ It is clear that any operator in $\\mathfrak {B}(L^p(X;E))$ with finite propagation commutes with $\\mathrm {VL}_\\infty (X)$ .",
"Hence, by taking limits it follows that $\\mathrm {Roe}(X,B)\\subseteq B \\cap \\mathrm {VL}_\\infty (X)^{\\prime }.$ The converse inclusion is also true provided the space $X$ has straight finite decomposition complexity and this is included in our main theorem as follows (this is the complete version of Theorem A in Section 1): Theorem 3.3 Let $(X,d)$ be a proper metric space equipped with a Radon measure $\\mu $ whose support is $X$ , and $p \\in [1,+\\infty )$ .",
"Suppose $E$ is a Banach space and $B \\subseteq \\mathfrak {B}(L^p(X;E))$ is a Banach subalgebra such that $C_b(X)BC_b(X)=B$ and $B$ is closed under block cutdowns.",
"Then for $b \\in B$ , the following are equivalent: $[b,f]=0$ for all $f \\in \\mathrm {VL}_\\infty (X)$ ; $b$ is quasi-local; $[b,g] \\in \\mathcal {K}(X,B)$ for any $g \\in C_h(X)$ .",
"If $X$ has straight finite decomposition complexity, then these are also equivalent to: $b \\in \\mathrm {Roe}(X,B)$ .",
"Recall that we have already explained in Remark REF that Theorem REF is a slight generalisation of [34] as condition (REF ) is not required here.",
"Also notice that (REF ) implies that “(iv) $\\Rightarrow $ (i)\" holds generally and the converse implication is also true under the extra condition of straight finite decomposition complexity.",
"In the remaining of this section, we prove that (i), (ii) and (iii) in Theorem REF are all equivalent, and leave the implication “(i) $\\Rightarrow $ (iv)\" to the next section, after we develop some technical tools such as Proposition REF ."
],
[
"“(i) $\\Leftrightarrow $ (ii)\"",
"We start with the proof of Theorem REF , “(i) $\\Leftrightarrow $ (ii)\".",
"The implication “(i) $\\Rightarrow $ (ii)\" follows exactly from the same arguments in [34], while the proof of “(ii) $\\Rightarrow $ (i)\" is slightly different from that one given in [34] due to the absence of inner products.",
"Fortunately, since both proofs are relativity short, we include the details for the convenience of the reader.",
"Let us begin with the following characterisation of the condition (i) in Theorem REF , which is proved in [34] when $p=2$ and actually holds for general $p$ .",
"Lemma 3.4 [34] Let $p\\in [1,\\infty )$ , $b \\in \\mathfrak {B}(L^p(X;E))$ and $\\varepsilon >0$ .",
"Then $\\Vert [b,f]\\Vert < \\varepsilon $ for every $f \\in \\mathrm {VL}_\\infty (X)_1$ if and only if there exists some $L>0$ such that $b \\in \\mathrm {Commut}(L,\\varepsilon )$ , where $\\mathrm {Commut}(L,\\varepsilon ):=\\big \\lbrace a \\in \\mathfrak {B}(L^p(X;E)): \\Vert [a,f]\\Vert < \\varepsilon , \\mbox{~for~any~}L\\mbox{-Lipschitz~} f\\in C_b(X)_1\\big \\rbrace .$ Assume $b\\in \\mathfrak {B}(L^p(X;E))$ such that $[b,\\mathrm {VL}_\\infty (X)_1]=0$ and let $\\varepsilon >0$ .",
"By Lemma REF , there exists some $L>0$ such that $b \\in \\mathrm {Commut}(L,\\varepsilon )$ .",
"For any $f,g \\in C_b(X)_1$ with $L^{-1}$ -disjoint supports, we may choose an $L$ -Lipschitz $h \\in C_b(X)_1$ such that $h|_{\\mathrm {supp}f} \\equiv 1$ and $h|_{\\mathrm {supp}g} \\equiv 0$ .",
"In particular, $\\Vert [b,h]\\Vert < \\varepsilon $ .",
"Therefore, $\\Vert fbg\\Vert =\\Vert fhbg\\Vert \\le \\Vert [h,b]\\Vert +\\Vert fbhg\\Vert <\\varepsilon +0 = \\varepsilon .$ Hence, $b$ is quasi-local as desired.",
"On the other hand, we assume that for any $\\varepsilon >0$ , $b$ has finite $\\varepsilon $ -propagation.",
"Without loss of generality, we may assume that $b$ is a contraction.",
"Given $\\varepsilon >0$ , pick $N$ such that $6/N<\\varepsilon /2$ .",
"By the hypothesis, $b$ has $\\varepsilon /(2N^2)$ -propagation at most $R>0$ .",
"For any $(2RN)^{-1}$ -Lipschitz $f\\in C_b(X)_1$ , we claim that $\\Vert [b,f]\\Vert <\\varepsilon $ .",
"In fact, take $A_1:=f^{-1}([0,\\frac{1}{N}]), \\quad \\mbox{~and~}\\quad A_i:=f^{-1}((\\frac{i-1}{N},\\frac{i}{N}]), \\quad i=2,\\ldots ,N.$ These sets partition $X$ , and $A_i$ is $2R$ -disjoint from $A_j$ for $|i-j|>1$ .",
"Now choose a partition of unity $e_1,\\ldots ,e_N \\in C_b(X)$ such that $e_i$ is supported in $\\mathcal {N}_{R/2}(A_i)$ .",
"Thus, $\\Vert e_ibe_j\\Vert < \\varepsilon /(2N^2)$ for $|i-j|>1$ .",
"Meanwhile, we have $f \\approx _{1/N} \\sum _{i=1}^{N}\\frac{i}{N}e_i.$ Hence, it follows that $\\Vert [f,b]\\Vert &\\le & \\frac{2}{N} + \\big \\Vert \\sum _{i=1}^N [\\frac{i}{N}e_i,b] \\big \\Vert \\\\&=& \\frac{2}{N} + \\big \\Vert \\big (\\sum _{i=1}^N \\frac{i}{N}e_ib\\big )\\big ( \\sum _{j=1}^N e_j \\big ) - \\big ( \\sum _{i=1}^N e_i \\big ) \\big (\\sum _{j=1}^N \\frac{j}{N}be_j\\big ) \\big \\Vert \\\\&\\le & \\frac{2}{N} + \\sum _{|i-j|>1} \\Vert e_ibe_j\\Vert + \\left\\Vert \\sum _{|i-j|\\le 1}(\\frac{i}{N}-\\frac{j}{N})e_ibe_j \\right\\Vert .$ Each term in the first sum is dominated by $\\frac{\\varepsilon }{2N^2}$ , hence $\\sum _{|i-j|>1} \\Vert e_ibe_j\\Vert < \\varepsilon /2$ .",
"The second sum can be broken into four sums: note that the terms vanish when $i=j$ ; what remain are $j=i+1$ and $j=i-1$ , and we break each of these further into even and odd parts.",
"By Lemma REF , each of these terms has norm at most $\\frac{1}{N}$ .",
"Hence, we have that $\\Vert [f,b]\\Vert < \\frac{2}{N}+\\frac{\\varepsilon }{2}+\\frac{4}{N}< \\varepsilon .$ So we complete the proof by Lemma REF ."
],
[
"“(i) $\\Leftrightarrow $ (iii)\"",
"Now we move on to Theorem REF , “(i) $\\Leftrightarrow $ (iii)\".",
"Here our major work is focused on omitting condition (REF ), as well as providing a “non-symmetric\" version of the argument given in [34] for $p=2$ .",
"However, the main body of the proof is still very similar to that of the original $p=2$ case [34], so we just outline the proof and highlight the differences we make here.",
"First of all, we recall that the proof of “(i) $\\Rightarrow $ (iii)\" given in [34] requires condition (REF ): $[C_0(X),B] \\subseteq \\mathcal {K}(X,B).$ After a careful reading of the proof, we realise that it is unnecessary to assume the entire $B$ essentially commuting with $C_0(X)$ but only a closed subalgebra of $B$ as shown in the following lemma: Lemma 3.5 Let $p\\in [1,\\infty )$ and $B$ be a Banach subalgebra of $\\mathfrak {B}(L^p(X;E))$ such that $C_b(X)BC_b(X)=B$ .",
"If $b \\in B$ satisfies $[b,\\mathrm {VL}_\\infty (X)]=0$ , then $[b,C_0(X)] \\subseteq \\mathcal {K}(X,B)$ .",
"Let $b\\in B$ such that $[b,\\mathrm {VL}_\\infty (X)]=0$ .",
"Since $\\mathcal {K}(X,B)$ is closed, we only need to prove $[b,g] \\in \\mathcal {K}(X,B)$ for any $g \\in C_c(X)$ .",
"Fix a base point $x_0 \\in X$ .",
"For each $k \\in \\mathbb {N}$ , we may choose a $(k^{-1})$ -Lipschitz function $f_k\\in C_b(X)_1$ such that $f_k$ vanishes on $\\mathrm {supp}(g)$ and $f_k|_{B_{R_k}(x_0)^c}=1$ for some sufficiently large $R_k>0$ .",
"Hence, the sequence $(f_k)_{k \\in \\mathbb {N}}\\in \\mathrm {VL}_\\infty (X)$ and $\\Vert [b,f_k]\\Vert \\rightarrow 0$ for $k \\rightarrow \\infty $ by assumption.",
"Since $gf_k=0$ for any $k\\in \\mathbb {N}$ , it follows that $\\Vert [b,g]f_k\\Vert = \\Vert bgf_k-gbf_k\\Vert = \\Vert gbf_k\\Vert =\\Vert g[b,f_k]\\Vert \\le \\Vert g\\Vert \\cdot \\Vert [b,f_k]\\Vert \\rightarrow 0,$ as $k \\rightarrow \\infty $ .",
"Similarly, we have that $\\Vert f_k[b,g]\\Vert \\rightarrow 0$ and $\\Vert f_k[b,g]f_k\\Vert \\rightarrow 0$ as $k \\rightarrow \\infty $ .",
"Moreover, we have that $\\Vert [b,g]-(1-f_k)[b,g](1-f_k)\\Vert \\le \\Vert [b,g]f_k\\Vert +\\Vert f_k[b,g]\\Vert +\\Vert f_k[b,g]f_k\\Vert \\rightarrow 0,$ as $k \\rightarrow \\infty $ .",
"Since $\\mathrm {supp}(1-f_k) \\subseteq B_{R_k}(x_0)$ and $C_b(X)BC_b(X)=B$ , it follows that $(1-f_k)[b,g](1-f_k) \\in C_c(X)BC_c(X)$ .",
"Hence, $[b,g] \\in \\mathcal {K}(X,B)$ .",
"Replacing condition (REF ) by Lemma REF in the original proof for $p=2$ [34], we obtain a proof of Theorem REF “(i) $\\Rightarrow $ (iii)\" without any further changes.",
"Hence we omit the details.",
"Now we outline the proof for the other direction, “(iii) $\\Rightarrow $ (i)\".",
"Since $L^p$ -Roe-like algebras may not possess a bounded involution in general, the proof becomes slightly different.",
"Fix a base point $x_0 \\in X$ .",
"For $R>0$ , we define $e_R \\in C_0(X)$ by $e_R(x):=\\max \\lbrace 0,1-d(x,B_R(x_0))/R \\rbrace .$ Lemma 3.6 ([34], Lemma 5.4) For $(f_k)_{k=1}^\\infty \\in \\mathrm {VL}(X)$ , a subsequence $(f_{k_i})_{i=1}^\\infty $ , and a sequence of positive numbers $(R_i)_{i=0}^\\infty $ with $R_{i+1} \\ge 6R_i$ for each $i$ , define $g_{(f_{k_i}),(R_i)} := \\sum _{i=1}^{\\infty } f_{k_i}(e_{R_i}-e_{3R_{i-1}}).$ Then $g_{(f_{k_i}),(R_i)} \\in C_h(X)$ .",
"Fix a point $x_0 \\in X$ , and set $B_R:=B_R(x_0)$ .",
"Recall $b$ satisfies the condition that $[b,g] \\in \\mathcal {K}(X,B)$ for any $g \\in C_h(X)$ .",
"Now assume $b$ is a contraction, and that there exists some $f=(f_k)_{k=1}^\\infty \\in \\mathrm {VL}_\\infty (X)$ such that $[b,f] \\ne 0$ .",
"Take an $\\varepsilon : 0< \\varepsilon < \\Vert [b,f]\\Vert $ , and consider two cases.",
"Case I.",
"There exists $R_0>0$ such that for any $S>0$ , there exists infinitely many $k$ : $\\mbox{either}\\quad \\Vert \\chi _{B_{R_0}}[b,f_k](1-\\chi _{B_S})\\Vert > \\frac{\\varepsilon }{5}, \\quad \\mbox{or} \\quad \\Vert (1-\\chi _{B_S})[b,f_k]\\chi _{B_{R_0}}\\Vert > \\frac{\\varepsilon }{5}.$ In other words, there exists $R_0>0$ with the property that either: there exists a sequence $S_1<S_2<\\ldots $ tending to $\\infty $ , such that for any $n \\in \\mathbb {N}$ , there exist infinitely many $k$ such that $\\Vert \\chi _{B_{R_0}}[b,f_k](1-\\chi _{B_{S_n}})\\Vert > \\frac{\\varepsilon }{5}$ ; or: there exists a sequence $S_1<S_2<\\ldots $ tending to $\\infty $ , such that for any $n \\in \\mathbb {N}$ , there exist infinitely many $k$ such that $\\Vert (1-\\chi _{B_{S_n}})[b,f_k]\\chi _{B_{R_0}}\\Vert > \\frac{\\varepsilon }{5}$ .",
"We only prove in the first situation, while the second is similar.",
"Since $(f_k)$ is very Lipschitz, $f_k|_{B_{R_0}}$ tends towards constant as $k \\rightarrow \\infty $ .",
"So without loss of generality, we can assume that $f_k|_{B_{R_0}}\\equiv \\gamma _k$ , for all $k$ .",
"Setting $\\hat{f_k}:=f_k-\\gamma _k$ gives us another very Lipschitz sequence $(\\hat{f_k})_{k=1}^\\infty $ satisfying the same condition and $\\hat{f_k}|_{B_{R_0}}\\equiv 0$ .",
"Additionally, for all $k$ , we have that $\\chi _{B_{R_0}}[b,\\hat{f_k}]=\\chi _{B_{R_0}}b\\hat{f_k}.$ By assumption, there exists a sequence $S_1<S_2<\\ldots $ tending to $\\infty $ , such that for any $n \\in \\mathbb {N}$ , we can find infinitely many $k$ such that $\\Vert \\chi _{B_{R_0}}b\\hat{f_k}(1-\\chi _{B_{S_n}})\\Vert > \\frac{\\varepsilon }{5}$ .",
"As the original proof for $p=2$ [34], we may choose sequences: $k_1 < k_2 < \\ldots $ and $R_1,R_2,\\ldots $ satisfying $\\Vert \\chi _{B_{R_0}}b\\hat{f_{k_i}}(e_{R_i}-e_{3R_{i-1}})\\Vert > \\frac{\\varepsilon }{10}.$ Applying Lemma REF to $(\\hat{f_{k_i}})$ and $(R_i)$ , we obtain $g \\in C_h(X)$ defined by: $g:=g_{(\\hat{f_{k_i}}),(R_i)} = \\sum _{i=1}^{\\infty } \\hat{f_{k_i}}(e_{R_i}-e_{3R_{i-1}}).$ Hence for any $S>R_0$ , choose an $i$ such that $3R_{i-1}>S$ , and we have $\\Vert [b,g](1-\\chi _{B_S})\\Vert & \\ge & \\Vert \\chi _{B_{R_0}}[b,g](1-\\chi _{B_S})(\\chi _{B_{2R_i}}-\\chi _{B_{3R_{i-1}}})\\Vert \\\\&= & \\Vert \\chi _{B_{R_0}}b\\hat{f_{k_i}}(e_{R_i}-e_{3R_{i-1}})\\Vert \\\\&> & \\frac{\\varepsilon }{10},$ which contradicts with the hypothesis that $[b,g] \\in \\mathcal {K}(X,B)$ .",
"Case II.",
"For every $R>0$ , there exists $S>0$ such that, for all but finitely many $k \\in \\mathbb {N}$ , we have $\\Vert \\chi _{B_R}[b,f_k](1-\\chi _{B_S})\\Vert \\le \\frac{\\varepsilon }{5} \\quad \\mbox{~and~}\\quad \\Vert (1-\\chi _{B_S})[b,f_k]\\chi _{B_R}\\Vert \\le \\frac{\\varepsilon }{5}.$ Without loss of generality, we may assume that $S>R$ .",
"Suppose we are given $R>0$ and $K \\in \\mathbb {N}$ , and let $S$ be given as above.",
"Then there exists $k \\ge K$ such that $\\Vert \\chi _{B_R}[b,f_k](1-\\chi _{B_S})\\Vert \\le \\frac{\\varepsilon }{5}, \\quad \\mbox{~and~}\\quad \\Vert (1-\\chi _{B_S})[b,f_k]\\chi _{B_R}\\Vert \\le \\frac{\\varepsilon }{5}.$ By assumption, we have $\\Vert [b,f_k]\\Vert > \\varepsilon $ ; and since $(f_k)$ is very Lipschitz, we can also assume that $f_k|_{B_S} \\approx _{\\varepsilon /10} \\gamma $ for some constant $\\gamma $ .",
"Hence, we have: $\\Vert \\chi _{B_S}[b,f_k]\\chi _{B_S}\\Vert \\le 2 \\cdot \\frac{\\varepsilon }{10}\\cdot \\Vert b\\Vert \\le \\frac{\\varepsilon }{5}.$ Now cutting the space by $B_R, B_R^c$ and $B_S, B_S^c$ , we have the following decomposition for the operator $T=[b,f_k]$ (recall we assume that $S>R$ ): $T=(1-\\chi _{B_R})T(1-\\chi _{B_R})+\\chi _{B_R}T\\chi _{B_S}+\\chi _{B_R}T(1-\\chi _{B_S})+(\\chi _{B_S}-\\chi _{B_R})T\\chi _{B_R}+(1-\\chi _{B_S})T\\chi _{B_R}.$ From Inequalities (REF ), the norms of the third and fifth items are less than or equal to $\\frac{\\varepsilon }{5}$ ; and from Equality (REF ), the norms of the second and fourth items are less than or equal to $\\frac{\\varepsilon }{5}$ as well.",
"Hence by triangle inequality, we have: $\\Vert (1-\\chi _{B_R})[b,f_k](1-\\chi _{B_R})\\Vert & \\ge & \\Vert [b,f_k]\\Vert -(\\Vert \\chi _{B_R}[b,f_k]\\chi _{B_S}\\Vert + \\Vert (\\chi _{B_S}-\\chi _{B_R})[b,f_k]\\chi _{B_R}\\Vert ) \\\\& & - (\\Vert \\chi _{B_R}[b,f_k](1-\\chi _{B_S})\\Vert + \\Vert (1-\\chi _{B_S})[b,f_k]\\chi _{B_R}\\Vert )\\\\&>& \\varepsilon - \\frac{2\\varepsilon }{5} - \\frac{2\\varepsilon }{5}\\\\&=& \\frac{\\varepsilon }{5}.$ In conclusion, for every $R>0$ and $K \\in \\mathbb {N}$ , there exists $k \\ge K$ such that: $\\Vert (1-\\chi _{B_R})[b,f_k](1-\\chi _{B_R})\\Vert > \\frac{\\varepsilon }{5}.$ As the original proof for $p=2$ [34], we may choose sequences: $k_1 < k_2 < \\ldots $ and $R_1,R_2,\\ldots $ satisfying $R_i \\ge 6R_{i-1}$ and $\\Vert (\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})[b,f_{k_i}](\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})\\Vert > \\frac{\\varepsilon }{5}.$ Applying Lemma REF to $(f_{k_i})$ and $(R_i)$ , we obtain $g \\in C_h(X)$ defined by: $g:=g_{(f_{k_i}),(R_i)} = \\sum _{i=1}^{\\infty } f_{k_i}(e_{R_i}-e_{3R_{i-1}}).$ By the choice of $(R_i)$ above, for any $i$ , we have $(\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})[b,g](\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})=(\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})[b,f_{k_i}](\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}}).$ Hence for any $S>0$ , choose an $i$ such that $6R_{i-1}>S$ , and we have $\\Vert [b,g](1-\\chi _{B_S})\\Vert & \\ge & \\Vert (\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})[b,g](\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})\\Vert \\\\&= & \\Vert (\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})[b,f_{k_i}](\\chi _{B_{R_i}}-\\chi _{B_{6R_{i-1}}})\\Vert \\\\&> & \\frac{\\varepsilon }{5},$ which contradicts with the hypothesis that $[b,g] \\in \\mathcal {K}(X,B)$ ."
],
[
"Proof of “(i) $\\Leftrightarrow $ (iv)”",
"In this section, we will prove the remaining case of “(i) $\\Leftrightarrow $ (iv)\" in Theorem REF .",
"Recall that as explained in Section 3, “(iv) $\\Rightarrow $ (i)\" holds in general.",
"So we will only focus on the opposite implication “(i) $\\Rightarrow $ (iv)\".",
"A key ingredient to prove “(i) $\\Rightarrow $ (iv)” is to approximate a bounded operator via its block cutdowns as indicated in [34] for the case of $p=2$ .",
"Unfortunately, their proof uses the technique of von Nuemann algebra, which is no longer available for $p \\ne 2$ .",
"Hence we need to search for a substitution of [34], and we figure out the following crucial result, which might be of independent interest to experts in Banach space theory.",
"Proposition 4.1 Let $(X,d)$ be a proper metric space equipped with a Radon measure $\\mu $ whose support is $X$ and $p \\in [1,+\\infty )$ .",
"Suppose $E$ is a Banach space, $a \\in \\mathfrak {B}(L^p(X;E))$ and $a\\in \\mathrm {Commut}(L,\\varepsilon )$ for some $L,\\varepsilon >0$ .",
"Let $(e_j)_{j\\in J}$ be an equicontinuous family of positive contractions in $C_b(X)$ with $2/L$ -disjoint supports, and define $e:=\\sum _{j\\in J} e_j$ .",
"Then, we have $\\big \\Vert eae-\\sum _{j\\in J}e_jae_j\\big \\Vert \\le \\varepsilon .$ The proof of the above proposition is technical and relatively long, so we decide to postpone it to Section REF for the convenience of the reader, and first show how to use the proposition to prove “(i) $\\Rightarrow $ (iv)\".",
"Let us start with the following lemma, which is a consequence of Proposition REF by the same proof of [34].",
"It may be worth reminding the reader that for any $L,\\varepsilon >0$ , we denote $\\mathrm {Commut}(L,\\varepsilon )=\\big \\lbrace a \\in \\mathfrak {B}(L^p(X;E)): \\Vert [a,f]\\Vert < \\varepsilon , \\mbox{~for~any~}L\\mbox{-Lipschitz~} f\\in C_b(X)_1\\big \\rbrace .$ Lemma 4.2 Let $\\mathcal {X}$ and $\\mathcal {Y}$ be metric families of $X$ such that $\\mathcal {X}{4L^{-1}+4} \\mathcal {Y}$ for some $L>0$ , and $a\\in \\mathfrak {B}(L^p(X;E))$ be block diagonal with respect to $\\mathcal {X}$ for $p\\in [1,\\infty )$ .",
"Let $\\varepsilon >0$ be such that $a\\in \\mathrm {Commut}(L,\\varepsilon )$ .",
"Then we can write: $a\\approx _{8\\varepsilon }a_{00}+a_{01}+a_{10}+a_{11},$ where each $a_{ii^{\\prime }}$ is of the form $\\theta _{(f_k)_{k\\in K}}(gag^{\\prime })$ (see (REF ))for some $g,g^{\\prime }\\in C_b(X)_1$ and some equicontinuous positive family $(f_k)_{k\\in K}$ in $C_b(X)_1$ with disjoint supports, such that each $\\mathrm {supp}(f_k)$ is contained in some set in $\\mathcal {N}_{L^{-1}+1}(\\mathcal {Y})$ .",
"In particular: (i) each $a_{ii^{\\prime }}$ is block diagonal with respect to $\\mathcal {N}_{L^{-1}+1}(\\mathcal {Y})$ , (ii) if $a\\in \\mathrm {Commut}(L^{\\prime },\\varepsilon ^{\\prime })$ for some $L^{\\prime },\\varepsilon ^{\\prime }>0$ , then each $a_{ii^{\\prime }}$ is in $\\mathrm {Commut}(L^{\\prime },\\varepsilon ^{\\prime })$ as well, and (iii) if $B\\subseteq \\mathfrak {B}(L^p(X;E))$ is a Banach subalgebra such that $C_b(X)BC_b(X)= B$ and $B$ is closed under block cutdowns, and if $a$ is in $B$ , then each $a_{ii^{\\prime }}$ is in $B$ as well.",
"Although the proof is exactly the same as the one given in [34], we decide to include it here for the completeness and show the reader how straight finite decomposition complexity is used in the proof.",
"Take $b\\in B$ such that it commutes with all $f\\in \\mathrm {VL}_\\infty (X)$ .",
"Given $\\varepsilon > 0$ , we aim to construct a finite propagation operator in $B$ , which is $\\varepsilon $ -close to $b$ .",
"It follows from Lemma REF that for every $\\varepsilon _n:=\\varepsilon /(2\\cdot 8^n),$ there exists some $L_n>0$ such that $b\\in Commut(L_n,\\varepsilon _n)$ .",
"Set $R_n:= 4(L_n^{-1}+1)+2(L_{n-1}^{-1}+1)+\\cdots +2(L_1^{-1}+1).$ Since $X$ has straight finite decomposition complexity, there exist metric families $\\mathcal {X}_0=\\lbrace X\\rbrace ,\\mathcal {X}_1,\\ldots ,\\mathcal {X}_m$ such that $\\mathcal {X}_{n-1}{R_n}\\mathcal {X}_n$ for $n\\in \\lbrace 1,\\ldots , m\\rbrace $ and $\\mathcal {X}_m$ is uniformly bounded.",
"An elementary observation shows that $\\mathcal {N}_{(L_{n-1}^{-1}+1)+\\cdots +(L_1^{-1}+1)}(\\mathcal {X}_{n-1}){4(L_n^{-1}+1)}\\mathcal {N}_{(L_{n-1}^{-1}+1)+\\cdots +(L_1^{-1}+1)}(\\mathcal {X}_{n}).$ Thus, we can apply Lemma REF inductively with $L_n$ , $\\varepsilon _n$ , the operators obtained in the previous iteration, and metric families in (REF ).",
"After $m$ steps, we approximate $b$ by an operator $b^{\\prime }$ which is a sum of $4^m$ operators in $B$ , each of which is block diagonal with respect to the bounded family $\\mathcal {N}_{(L_{m}^{-1}+1)+\\cdots +(L_1^{-1}+1)}(\\mathcal {X}_m)$ .",
"Hence, operators which are block diagonal with respect to it clearly have finite propagation.",
"Consequently, $b^{\\prime }\\in \\mathrm {Roe}(X,B)$ .",
"Finally, the distance between $b$ and $b^{\\prime }$ is at most $8\\varepsilon _{1}+4(8\\varepsilon _{2}+4(8\\varepsilon _{3}+4(\\ldots )))=\\varepsilon (\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\ldots )=\\varepsilon $ by Lemma REF .",
"So we finish the proof."
],
[
"Approximation via block cutdowns",
"Finally, we complete the proof of Proposition REF as promised before.",
"The main difficulty is the lack of reflexivity of the $L^p$ -Bochner space $L^p(X;E)$ for general $p$ and general Banach space $E$ (see e.g.",
"[2], [5], [6]), which impedes us from applying the original proof in [34] directly.",
"Instead, we establish some substituting results in functional analysis and state them in the context of general Banach spaces, which conceivably would be of independent interests.",
"In the rest of this subsection, suppose $\\mathfrak {X}$ is a Banach space and $\\hat{\\mathfrak {X}}$ is a closed subspace of the dual space $\\mathfrak {X}^*$ , which separates points in $\\mathfrak {X}$ (i.e., for any nonzero $\\xi \\in \\mathfrak {X}$ , there exists some $\\eta \\in \\hat{\\mathfrak {X}}$ such that $\\eta (\\xi ) \\ne 0$ ).",
"The inclusion $i: \\hat{\\mathfrak {X}}\\hookrightarrow \\mathfrak {X}^*$ induces a surjective adjoint map $i^*: \\mathfrak {X}^{**} \\rightarrow \\hat{\\mathfrak {X}}^*$ .",
"Composing it with the canonical map from $\\mathfrak {X}$ into its double dual $\\mathfrak {X}^{**}$ , we obtain the following map $\\tau : \\mathfrak {X}\\rightarrow \\hat{\\mathfrak {X}}^*.$ It is clear that $\\tau $ is injective, as $\\hat{\\mathfrak {X}}$ separates points in $\\mathfrak {X}$ .",
"For any $\\theta \\in \\hat{\\mathfrak {X}}^*$ and $\\eta \\in \\hat{\\mathfrak {X}}$ , we use the notation $\\langle \\theta ,\\eta \\rangle $ for $\\theta (\\eta )$ .",
"Consider the Banach space $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ of all bounded operators from $\\mathfrak {X}$ to $\\hat{\\mathfrak {X}}^*$ , equipped with the weak* operator topology (W*OT)In [34] Špakula and Tikuisis considered the weak operator topology (WOT) instead.",
"However, (WOT) and (W*OT) agree when $\\mathfrak {X}^* \\cong \\mathfrak {X}$ and taking $\\hat{\\mathfrak {X}}:=\\mathfrak {X}^*$ .",
"defined as follows: a net $\\lbrace T_\\alpha \\rbrace $ converges to $T$ in $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ if and only if for any $\\xi \\in \\mathfrak {X}$ and any $\\eta \\in \\hat{\\mathfrak {X}}$ , we have $\\langle T_\\alpha (\\xi ),\\eta \\rangle \\rightarrow \\langle T(\\xi ),\\eta \\rangle .$ The strong* topology with respect to $\\hat{\\mathfrak {X}}$ on $\\mathfrak {B}(\\mathfrak {X})$ is defined as follows: a net $\\lbrace T_\\alpha \\rbrace $ converges to $T$ in $\\mathfrak {B}(\\mathfrak {X})$ if and only if for any $\\xi \\in \\mathfrak {X}$ and any $\\eta \\in \\hat{\\mathfrak {X}}$ , we have $\\Vert T_\\alpha (\\xi )-T(\\xi )\\Vert \\rightarrow 0 \\quad \\mbox{~and~}\\quad \\Vert T_\\alpha ^*(\\eta )-T^*(\\eta )\\Vert \\rightarrow 0.$ We say that $\\hat{\\mathfrak {X}}$ is $a^*$ -invariant for $a\\in \\mathfrak {B}(\\mathfrak {X})$ if $a^*(\\hat{\\mathfrak {X}}) \\subseteq \\hat{\\mathfrak {X}}$ .",
"In this case, the restriction $a^*|_{\\hat{\\mathfrak {X}}}$ belongs to $\\mathfrak {B}(\\hat{\\mathfrak {X}})$ .",
"Hence, its adjoint $(a^*|_{\\hat{\\mathfrak {X}}})^*$ belongs to $\\mathfrak {B}(\\hat{\\mathfrak {X}}^*)$ as well.",
"In order to simplify notations, we write $a^{(**)}$ instead of $(a^*|_{\\hat{\\mathfrak {X}}})^*$ .",
"Clearly, for any $\\zeta \\in \\hat{\\mathfrak {X}}^*$ and $\\eta \\in \\hat{\\mathfrak {X}}$ we have: $\\langle a^{(**)}\\zeta , \\eta \\rangle =\\langle \\zeta , a^*\\eta \\rangle .$ Moreover, it is easy to check that if $\\hat{\\mathfrak {X}}$ is $a^*$ -invariant for some $a\\in \\mathfrak {B}(\\mathfrak {X})$ , then $a^{(**)}\\tau =\\tau a.$ In other words, the following diagram commutes: $@=1.3cm{\\mathfrak {X}@{^{(}->}[r]^-{\\textstyle \\tau } [d]_{\\textstyle a}& \\hat{\\mathfrak {X}}^* [d]^{\\textstyle a^{(**)}} \\\\\\mathfrak {X}@{^{(}->}[r]^-{\\textstyle \\tau } & \\hat{\\mathfrak {X}}^*.",
"}$ We say that $\\hat{\\mathfrak {X}}$ is $\\mathcal {A}^*$ -invariant for a subset $\\mathcal {A}\\subseteq \\mathfrak {B}(\\mathfrak {X})$ if $\\hat{\\mathfrak {X}}$ is $a^*$ -invariant for all $a\\in \\mathcal {A}$ .",
"If $G$ is a subgroup of invertible isometries in $\\mathfrak {B}(\\mathfrak {X})$ and $\\hat{\\mathfrak {X}}$ is $G^*$ -invariant, then $u^*(\\hat{\\mathfrak {X}})=\\hat{\\mathfrak {X}}$ for all $u\\in G$ .",
"It is clear that if $u$ is an invertible isometry, then so are $u^{*}$ and $u^{(**)}$ .",
"Moreover, $(u^{*})^{-1}=(u^{-1})^{*}$ and $(u^{(**)})^{-1}=(u^{-1})^{(**)}$ , which are denoted by $u^{-*}$ and $u^{-(**)}$ , respectively.",
"Now suppose $(u_\\alpha )$ and $u$ are invertible isometries in $G$ and $u_\\alpha \\rightarrow u$ in the strong* topology with respect to $\\hat{\\mathfrak {X}}$ , then $\\Vert u_\\alpha ^{-*}\\eta - u^{-*}\\eta \\Vert \\rightarrow 0$ for all $\\eta \\in \\hat{\\mathfrak {X}}$ .",
"Indeed, we have $\\Vert u_\\alpha ^{-*}\\eta - u^{-*}\\eta \\Vert =\\Vert \\eta - u_\\alpha ^*u^{-*}\\eta \\Vert =\\Vert u^*(u^{-*}\\eta )- u_\\alpha ^*(u^{-*}\\eta )\\Vert \\rightarrow 0.$ We have the following technical lemma, which generalises [34].",
"Lemma 4.3 Suppose $G$ is an abelian subgroup of the group of invertible isometries in $\\mathfrak {B}(\\mathfrak {X})$ , which is compact in the strong* topology with respect to $\\hat{\\mathfrak {X}}$ .",
"Suppose $\\hat{\\mathfrak {X}}$ is $G^*$ -invariant, and define $\\mathfrak {G}^{\\prime }:= \\lbrace a\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*): au=u^{(**)}a, \\forall u \\in G \\rbrace $ .",
"Then there exists a unique idempotent linear contraction $\\mathcal {E}_G: \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)\\rightarrow \\mathfrak {G}^{\\prime }$ with the following properties: For any $b_1,b_2 \\in G$ and $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , $\\mathcal {E}_G(b_1^{(**)}ab_2)=b_1^{(**)}\\mathcal {E}_G(a)b_2$ .",
"The restriction of $\\mathcal {E}_G$ to the unit ball of $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ is (W*OT)-continuous.",
"In this case, for any $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , we have that $\\Vert \\mathcal {E}_G(a)-a\\Vert \\le \\sup _{u \\in G} \\Vert au-u^{(**)}a\\Vert .$ Since $G$ is compact with respective to the strong* topology, we consider the normalised Haar measure $\\mu _G$ on $G$ .",
"Fix $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , the map $(G,\\mbox{strong* topology}) \\rightarrow (\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*), \\mathrm {W^*OT})$ defined by $u \\mapsto u^{-(**)}au$ is clearly continuous.",
"For each $\\xi \\in \\mathfrak {X}$ and each $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , we may consider the following functional on $\\hat{\\mathfrak {X}}$ : $\\phi _{\\xi ,a}: \\eta \\mapsto \\int _G\\langle u^{-(**)}au\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u),$ whose norm is bounded by $\\Vert a\\Vert \\cdot \\Vert \\xi \\Vert $ .",
"Therefore, we obtain a linear contraction $\\mathcal {E}_G: \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)\\rightarrow \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ given by $\\mathcal {E}_G(a)(\\xi )=\\phi _{\\xi ,a}$ , where $\\xi \\in \\mathfrak {X}$ and $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ .",
"It remains to check that $\\mathcal {E}_G$ satisfies the required properties.",
"First of all, we show that $\\mathcal {E}_G$ has image in $\\mathfrak {G}^{\\prime }$ .",
"More precisely, $\\mathcal {E}_G(a)v=v^{(**)}\\mathcal {E}_G(a)$ for any $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ and any $v \\in G$ .",
"Given $\\xi \\in \\mathfrak {X}$ and $\\eta \\in \\hat{\\mathfrak {X}}$ , it follows from the right-invariance of the Haar measure $\\mu _G$ that $\\langle \\mathcal {E}_G(a)v\\xi , \\eta \\rangle &=& \\int _G\\langle u^{-(**)}auv\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) \\\\&=& \\int _G\\langle v^{(**)}u^{-(**)}au\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) \\\\&=& \\int _G\\langle u^{-(**)}au\\xi , v^*\\eta \\rangle \\mathrm {d}\\mu _G(u)\\\\&=& \\langle \\mathcal {E}_G(a)\\xi , v^*\\eta \\rangle \\\\&=& \\langle v^{(**)}\\mathcal {E}_G(a)\\xi , \\eta \\rangle .$ Hence, it follows that $\\mathcal {E}_G(a)v=v^{(**)}\\mathcal {E}_G(a)$ .",
"Given $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , $\\xi \\in \\mathfrak {X}$ and $\\eta \\in \\hat{\\mathfrak {X}}$ , we have $|\\langle (\\mathcal {E}_G(a)-a)\\xi ,\\eta \\rangle | & \\le & \\int _G \\Vert u^{-(**)}au-a\\Vert \\cdot \\Vert \\xi \\Vert \\cdot \\Vert \\eta \\Vert \\mathrm {d}\\mu _G(u)\\\\& = & \\int _G \\Vert au-u^{(**)}a\\Vert \\cdot \\Vert \\xi \\Vert \\cdot \\Vert \\eta \\Vert \\mathrm {d}\\mu _G(u)\\\\& \\le & \\left(\\sup _{u \\in G} \\Vert au-u^{(**)}a\\Vert \\right) \\cdot \\Vert \\xi \\Vert \\cdot \\Vert \\eta \\Vert .$ Hence, (REF ) holds.",
"In particular, $\\mathcal {E}_G(a)=a$ for any $a \\in \\mathfrak {G}^{\\prime }$ , which implies that $\\mathcal {E}_G: \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)\\rightarrow \\mathfrak {G}^{\\prime }$ is an idempotent.",
"Now let us check that $\\mathcal {E}_G(b_1^{(**)}ab_2)=b_1^{(**)}\\mathcal {E}_G(a)b_2$ for any $b_1,b_2 \\in G$ and $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ .",
"Since $G$ is abelian, we have $\\langle \\mathcal {E}_G(b_1^{(**)}ab_2)\\xi ,\\eta \\rangle &=& \\int _G \\langle u^{-(**)}b_1^{(**)}ab_2u\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) \\\\&=& \\int _G \\langle b_1^{(**)}u^{-(**)}aub_2\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) \\\\&=& \\int _G \\langle u^{-(**)}aub_2\\xi , b_1^{*}\\eta \\rangle \\mathrm {d}\\mu _G(u)\\\\&=& \\langle \\mathcal {E}_G(a)b_2\\xi ,b_1^{*}\\eta \\rangle \\\\&=& \\langle b_1^{(**)}\\mathcal {E}_G(a)b_2\\xi ,\\eta \\rangle ,$ for any $\\xi \\in \\mathfrak {X}$ and any $\\eta \\in \\hat{\\mathfrak {X}}$ .",
"Hence, $\\mathcal {E}_G(b_1^{(**)}ab_2)=b_1^{(**)}\\mathcal {E}_G(a)b_2$ .",
"In order to prove the (W*OT)-continuity of the restriction of $\\mathcal {E}_G$ to the unit ball of $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , we have to approximate the integration by finite Riemann sums uniformly in the weak* operator topology: Indeed, fix $\\xi \\in \\mathfrak {X}$ , $\\eta \\in \\hat{\\mathfrak {X}}$ and $u \\in G$ and for any $\\varepsilon >0$ , from (REF ) there exists an open neighbourhood $V_u$ of $u$ in the strong* topology such that for all $v \\in V_u$ and all $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)_1$ , we have $|\\langle v^{-(**)}av\\xi , \\eta \\rangle - \\langle u^{-(**)}au\\xi , \\eta \\rangle | <\\varepsilon .$ Since $\\lbrace V_u: u \\in G\\rbrace $ forms an open cover of $G$ and $G$ is compact in the strong* topology, there exists a finite subcover $\\lbrace V_{u_1},\\ldots , V_{u_n}\\rbrace $ of $G$ .",
"Let $W_1=V_{u_1}$ and we put $W_k=V_{u_k}\\setminus \\bigcup _{i=1}^{k-1} W_i$ for $1<k\\le n$ .",
"Without loss of generality, we may assume that $\\lbrace W_k\\rbrace _{k=1}^n$ forms a non-empty Borel partition of $G$ .",
"Take an arbitrary point $w_k$ in each $W_k$ for $k=1,\\ldots ,n$ .",
"Then for any $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)_1$ and $u \\in W_k$ , we have that $|\\langle u^{-(**)}au\\xi , \\eta \\rangle - \\langle w_k^{-(**)}aw_k\\xi , \\eta \\rangle | <2\\varepsilon .$ In particular, we have that $& &\\big | \\langle \\mathcal {E}_G(a) \\xi , \\eta \\rangle - \\sum _{k=1}^n \\langle w_k^{-(**)} a w_k \\xi , \\eta \\rangle \\mu _G(W_k)\\big | \\\\& = & \\big | \\sum _{k=1}^n \\int _{W_k} \\langle u^{-(**)}a u\\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) - \\sum _{k=1}^n \\int _{W_k} \\langle w_k^{-(**)}a w_k \\xi , \\eta \\rangle \\mathrm {d}\\mu _G(u) \\big | \\\\& \\le & \\sum _{k=1}^n \\int _{W_k}\\big | \\langle u^{-(**)}a u\\xi , \\eta \\rangle - \\langle w_k^{-(**)}a w_k \\xi , \\eta \\rangle \\big | \\mathrm {d}\\mu _G(u)\\\\& \\le & \\sum _{k=1}^n \\int _{W_k} 2\\varepsilon \\mathrm {d}\\mu _G(u) = 2\\varepsilon ,$ for all $a\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)_1$ .",
"Since the map $a\\mapsto \\sum _{k=1}^n\\mu _G(W_k)w_k^{-(**)}aw_k$ is continuous in the weak* operator topology, it is not hard to see that the restriction of $\\mathcal {E}_G$ to the unit ball of $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ is (W*OT)-continuous as well.",
"Finally, we check the uniqueness of $\\mathcal {E}_G$ .",
"If we have another $\\mathcal {E}: \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)\\rightarrow \\mathfrak {G}^{\\prime }$ satisfying all the conditions in the lemma, then: $\\mathcal {E}_G(a) &= \\mathcal {E}(\\mathcal {E}_G(a)) && (\\mathcal {E}\\mbox{~fixes~} \\mathfrak {G}^{\\prime } ) \\\\&= \\mathcal {E}\\big ({\\scriptstyle \\mathrm {W^*OT}-}\\int _G u^{-(**)}au \\mathrm {d}\\mu _G(u)\\big ) && \\\\&= {\\scriptstyle \\mathrm {W^*OT}}-\\int _G \\mathcal {E}(u^{-(**)}au) \\mathrm {d}\\mu _G(u) &&(\\mathrm {W^*OT}\\mbox{-continuity on the unit ball} ) \\\\&= {\\scriptstyle \\mathrm {W^*OT}-}\\int _G u^{-(**)}\\mathcal {E}(a)u \\mathrm {d}\\mu _G(u) &&(\\mbox{Property~1)}) \\\\&= \\mathcal {E}_G(\\mathcal {E}(a)) && \\\\&= \\mathcal {E}(a) && (\\mathcal {E}_G \\mbox{~fixes~} \\mathfrak {G}^{\\prime } )$ for all $a\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)_1$ .",
"Thus, $\\mathcal {E}_G=\\mathcal {E}$ and we complete the proof.",
"Now let us return to the setting of Proposition REF .",
"Let $(X,d)$ be a proper metric space equipped with a Radon measure $\\mu $ whose support is $X$ .",
"Let $q$ be the conjugate exponent to $p$ when $p\\in (1,+\\infty )$ , and $q=0$ when $p=1$ .",
"Suppose $E$ is a Banach space and $(e_j)_{j\\in J}$ is an equicontinuous family of positive contractions in $C_b(X)$ with uniformly disjoint supports.",
"In order to apply Lemma REF , we put $\\mathfrak {X}=L^p(X;E)$ and $\\hat{\\mathfrak {X}}=L^q(X;E^*)$ .",
"Clearly, $\\hat{\\mathfrak {X}}$ is a closed subspace of the dual space $\\mathfrak {X}^*$ , and separates points in $\\mathfrak {X}$ by Lemma REF .",
"For each $j\\in J$ , set $A_j= \\mathrm {supp}(e_j)$ and $B=X\\setminus \\big (\\bigsqcup _{j\\in J} A_j\\big )$ .",
"We consider $p_j$ and $q_c$ in $\\mathfrak {B}(L^p(X;E))$ given by $p_j(\\xi )=\\chi _{A_j}\\xi $ and $q_c(\\xi )=\\chi _{B}\\xi $ for $\\xi \\in L^p(X;E)$ .",
"We define that $G=\\left\\lbrace \\sum _{j\\in J} (-1)^{\\alpha _j} p_j + (-1)^\\beta q_c: (\\alpha _j)_{j\\in J} \\subseteq (\\mathbb {Z}/2)^J, \\beta \\in \\mathbb {Z}/2 \\right\\rbrace ,$ where the sum converges in (SOT) and each element in $G$ can be presented by a function of the form $\\sum _{j\\in J} (-1)^{\\alpha _j} \\chi _{A_j} + (-1)^\\beta \\chi _{B}$ (in the pointwise convergence) via the faithful multiplication representation $\\rho :L^\\infty (X) \\rightarrow \\mathfrak {B}(L^p(X;E))$ .",
"Since $g^2=\\text{id}$ for all $g\\in G$ , $G$ becomes a subgroup of the invertible isometry group in $\\mathfrak {B}(L^p(X;E))$ , and clearly $G$ is abelian.",
"Also notice that $\\hat{\\mathfrak {X}}$ is $L^\\infty (X)^*$ -invariant as for any $f\\in L^\\infty (X)\\subseteq \\mathfrak {B}(L^p(X;E))$ and $\\eta \\in \\hat{\\mathfrak {X}}$ , we have that $f^*(\\eta )=f\\cdot \\eta $ by pointwise multiplications as functions on $X$ .It is worth noting that $C_0(X,E^*)$ is not $L^\\infty (X)^*$ -invariant and this is the reason why we use $L^0(X;E^*)$ instead of $C_0(X,E^*)$ when $p=1$ .",
"Consequently, $\\hat{\\mathfrak {X}}$ is $G^*$ -invariant since $G\\subseteq \\rho (L^\\infty (X))$ .",
"Moreover, the strong* topology on $G$ with respect to $\\hat{\\mathfrak {X}}$ is compact, as it is homeomorphic to the product topology on $(\\mathbb {Z}/2)^{J\\cup \\lbrace \\beta \\rbrace }$ .However, it is false for $L^\\infty (X;E^*)$ and this is the reason why we use $L^0(X;E^*)$ instead of $L^\\infty (X;E^*)$ when $p=1$ .",
"The next lemma is a replacement of [34], where Špakula and Tikuisis work within the setting of von Neumann algebras.",
"Instead, we provide a direct and concrete proof here as follows: Lemma 4.4 As above, the group $G$ is defined as in (REF ) and $q$ is the conjugate exponent to $p$ when $p\\in (1,\\infty )$ , and $q=0$ when $p=1$ .",
"Let $\\mathfrak {X}=L^p(X;E)$ and $\\hat{\\mathfrak {X}}=L^q(X;E^*)$ .",
"If $\\mathfrak {G}^{\\prime }=\\lbrace a\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*): au=u^{(**)}a, \\forall u \\in G \\rbrace $ , then there exists a (W*OT)-continuous idempotent linear contraction $\\mathcal {E}: \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)\\rightarrow \\mathfrak {G}^{\\prime }$ given by the formula $\\mathcal {E}(x)=\\sum _{j\\in J} p_j^{(**)}xp_j + q_c^{(**)}xq_c,$ where the sum converges in (SOT).",
"Moreover, $\\mathcal {E}(b_1^{(**)}ab_2)=b_1^{(**)}\\mathcal {E}(a)b_2$ for any $b_1,b_2 \\in G$ and $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ .",
"Consequently, we have that $\\Vert \\mathcal {E}(a)-a\\Vert \\le \\sup _{u \\in G} \\Vert au-u^{(**)}a\\Vert , \\quad \\text{for any $a \\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$.",
"}$ It is clear that $\\mathcal {E}$ is a (W*OT)-continuous linear map on $\\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ and the sum defining $\\mathcal {E}$ converges in (SOT), so we leave the details to the readers.",
"Let us first verify that $\\mathcal {E}$ is a contraction.",
"When $p=1$ , we have $\\Vert \\mathcal {E}(x)\\xi \\Vert \\le \\sum _{j\\in J} \\Vert xp_j\\xi \\Vert + \\Vert xq_c\\xi \\Vert \\nonumber \\le \\Vert x\\Vert \\cdot \\big (\\sum _{j\\in J} \\Vert \\chi _{A_j}\\xi \\Vert _1 + \\Vert \\chi _{B}\\xi \\Vert _1 \\big ) = \\Vert x\\Vert \\cdot \\Vert \\xi \\Vert _1,$ for any $\\xi \\in L^1(X;E)$ by Lemma REF .",
"It implies that $\\mathcal {E}$ is a contraction in this case.",
"When $p>1$ , it follows from Hölder's inequality that $\\big | \\big \\langle \\sum _{j\\in J} p_j^{(**)}xp_j\\xi + q_c^{(**)}xq_c\\xi , \\eta \\big \\rangle \\big |&\\le & \\sum _{j\\in J} |\\langle xp_j\\xi , p_j^*\\eta \\rangle | + |\\langle xq_c\\xi , q_c^*\\eta \\rangle | \\\\& \\le & \\Vert x\\Vert \\cdot \\big (\\sum _{j\\in J} \\Vert p_j\\xi \\Vert _p \\cdot \\Vert p_j^*\\eta \\Vert _q + \\Vert q_c\\xi \\Vert _p \\cdot \\Vert q_c^*\\eta \\Vert _q \\big ) \\\\& \\le & \\Vert x\\Vert \\cdot \\big (\\sum _{j\\in J} \\Vert p_j\\xi \\Vert _p^p + \\Vert q_c\\xi \\Vert _p^p\\big )^{\\frac{1}{p}} \\cdot \\big (\\sum _{j\\in J} \\Vert p_j^*\\eta \\Vert _q^q + \\Vert q_c^*\\eta \\Vert _q^q\\big )^{\\frac{1}{q}} \\\\& = & \\Vert x\\Vert \\cdot \\Vert \\xi \\Vert _p \\cdot \\Vert \\eta \\Vert _q,$ for any $\\xi \\in L^p(X;E)$ and $\\eta \\in L^q(X;E^*)$ .",
"This implies that $\\Vert \\mathcal {E}(x)\\xi \\Vert =\\big \\Vert \\sum _{j\\in J} p_j^{(**)}xp_j\\xi + q_c^{(**)}xq_c\\xi \\big \\Vert \\le \\Vert x\\Vert \\cdot \\Vert \\xi \\Vert _p$ by Lemma REF .",
"Hence, $\\mathcal {E}$ is a contraction in this case as well.",
"Now we show that the image of $\\mathcal {E}$ sits inside $\\mathfrak {G}^{\\prime }$ .",
"Indeed, given any $x\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ , any $u=\\sum _{j\\in J} (-1)^{\\alpha _j} p_j + (-1)^\\beta q_c \\in G$ , $\\xi \\in \\mathfrak {X}$ and $\\eta \\in \\hat{\\mathfrak {X}}$ , we have that $\\langle \\mathcal {E}(x)u\\xi , \\eta \\rangle &=& \\big \\langle \\big ( \\sum _{j\\in J} p_j^{(**)}xp_j + q_c^{(**)} x q_c\\big ) \\big (\\sum _{j\\in J}(-1)^{\\alpha _j} p_j +(-1)^\\beta q_c \\big )\\xi , \\eta \\big \\rangle \\\\&=& \\sum _{j\\in J} (-1)^{\\alpha _j} \\langle p_j^{(**)}x p_j\\xi ,\\eta \\rangle + (-1)^\\beta \\langle q_c^{(**)}xq_c\\xi , \\eta \\rangle \\\\&=& \\sum _{j\\in J} (-1)^{\\alpha _j} \\langle x p_j\\xi , p_j^*\\eta \\rangle + (-1)^\\beta \\langle xq_c\\xi , q_c^*\\eta \\rangle .$ On the other hand, $\\langle u^{(**)}\\mathcal {E}(x)\\xi , \\eta \\rangle &=& \\big \\langle \\big ( \\sum _{j\\in J} p_j^{(**)}xp_j + q_c^{(**)} x q_c\\big )\\xi , u^*\\eta \\big \\rangle \\\\&=& \\sum _{j\\in J} \\langle x p_j\\xi , p_j^*u^*\\eta \\rangle + \\langle xq_c\\xi , q_c^*u^*\\eta \\rangle \\\\&=& \\sum _{j\\in J} (-1)^{\\alpha _j} \\langle x p_j\\xi , p_j^*\\eta \\rangle + (-1)^\\beta \\langle xq_c\\xi , q_c^*\\eta \\rangle .$ Hence, $\\mathcal {E}(x)u=u^{(**)}\\mathcal {E}(x)$ for all $u\\in G$ .",
"Next, we show that $\\mathcal {E}(x)=x$ for all $x\\in \\mathfrak {G}^{\\prime }$ .",
"In other words, $\\mathcal {E}$ is an idempotent onto $\\mathfrak {G}^{\\prime }$ .",
"Fix an $x \\in \\mathfrak {G}^{\\prime }$ and for any $u=\\sum _{j\\in J} (-1)^{\\alpha _j} p_j + (-1)^\\beta q_c$ in $G$ , we have that $p_j^{(**)}xp_i=(-1)^{\\alpha _i}p_j^{(**)}x(up_i)=(-1)^{\\alpha _i}(p_j^{(**)}u^{(**)})xp_i=(-1)^{\\alpha _i+\\alpha _j}p_j^{(**)}xp_i.$ It follows that $p_{j}^{(**)}xp_i=0$ for any $i\\ne j$ .",
"Similarly, $p_j^{(**)}xq_c=q_c^{(**)}xp_{j}=0$ for any $j \\in J$ .",
"Therefore, we have that $x = \\big ( \\sum _{i\\in J}p_i +q_c \\big )^{(**)}~x~\\big ( \\sum _{j\\in J}p_j +q_c \\big )=\\sum _{j\\in J} p_j^{(**)}xp_j + q_c^{(**)}xq_c = \\mathcal {E}(x)\\ \\text{for all $x\\in \\mathfrak {G}^{\\prime }$.",
"}$ Moreover, for any $b_1,b_2 \\in G$ and any $x\\in \\mathfrak {B}(\\mathfrak {X},\\hat{\\mathfrak {X}}^*)$ we have that $\\mathcal {E}(b_1^{(**)}xb_2) &=& \\sum _{j\\in J} p_j^{(**)}b_1^{(**)}xb_2p_j + q_c^{(**)}b_1^{(**)}xb_2q_c\\\\&=& \\sum _{j\\in J} b_1^{(**)}p_j^{(**)}xp_jb_2 + b_1^{(**)}q_c^{(**)}xq_cb_2\\\\ &=&b_1^{(**)}\\mathcal {E}(x)b_2,$ where we use the fact that $b_kp_j=p_jb_k$ and $b_kq_c=q_cb_k$ for any $j\\in J$ and $k\\in \\lbrace 1,2\\rbrace $ .",
"The final conclusion follows from the uniqueness of $\\mathcal {E}$ in Lemma REF and (REF ) therein.",
"So we finish the proof.",
"Let the group $G$ be defined as in (REF ), and the map $\\mathcal {E}: \\mathfrak {B}(L^p(X;E),L^q(X;E^*)^*)\\rightarrow \\mathfrak {G}^{\\prime }$ be the idempotent defined in Lemma REF .",
"Recall that by Lemma REF , the map $\\tau : L^p(X;E)\\rightarrow L^q(X;E^*)^*$ defined in (REF ) is an isometric embedding, hence it induces the following isometric embedding $\\iota : \\mathfrak {B}(L^p(X;E))=\\mathfrak {B}(L^p(X;E),L^p(X;E)) \\hookrightarrow \\mathfrak {B}(L^p(X;E),L^q(X;E^*)^*).$ In other words, $\\iota (a)=\\tau \\circ a$ for any $a \\in \\mathfrak {B}(L^p(X;E))$ .",
"Now we define another map $\\mathcal {E}^{\\prime }: \\mathfrak {B}(L^p(X;E))\\rightarrow \\mathfrak {B}(L^p(X;E))$ by the formula $\\mathcal {E}^{\\prime }(z)=\\sum _{j\\in J} p_jzp_j + q_czq_c$ for $z \\in \\mathfrak {B}(L^p(X;E))$ and the sum converges in (SOT) by Lemma REF .",
"It follows easily from Equation (REF ) that the following diagram commutes $@=1.5cm{\\mathfrak {B}(L^p(X;E))@{^{(}->}[r]^-{\\textstyle \\iota } & \\mathfrak {B}(L^p(X;E),L^q(X;E^*)^*)\\\\\\mathfrak {B}(L^p(X;E))[u]^{\\textstyle \\mathcal {E}^{\\prime }} @{^{(}->}[r]^-{\\textstyle \\iota } & \\mathfrak {B}(L^p(X;E),L^q(X;E^*)^*).",
"[u]_{\\textstyle \\mathcal {E}} }$ Furthermore, we have that $\\Vert \\mathcal {E}^{\\prime }(z)-z\\Vert =\\Vert \\iota (\\mathcal {E}^{\\prime }(z))-\\iota (z)\\Vert =\\Vert \\mathcal {E}(\\iota (z))-\\iota (z)\\Vert \\le \\sup _{u \\in G}\\lbrace \\Vert \\iota (z)u-u^{(**)}\\iota (z)\\Vert \\rbrace ,$ for any $z \\in \\mathfrak {B}(L^p(X;E))$ .",
"While for $u \\in G$ , it follows from Equation (REF ) that $\\iota (z)u-u^{(**)}\\iota (z)=\\tau z u-u^{(**)} \\tau z=\\tau z u-\\tau u z.$ Combining the above facts together, we obtain that $\\Vert \\mathcal {E}^{\\prime }(z)-z\\Vert \\le \\sup _{u \\in G}\\lbrace \\Vert zu-uz\\Vert \\rbrace .$ Let $e:=\\sum _{j\\in J} e_j$ .",
"Since $p_je=e_j=ep_j$ and $q_ce=eq_c=0$ , we have that $\\mathcal {E}^{\\prime }(eae) &=& \\sum _{j\\in J} p_jeaep_j + q_ceaeq_c= \\sum _{j\\in J} e_jae_j.$ Also notice that for any $u=\\sum _{j\\in J} (-1)^{\\alpha _j} p_j + (-1)^\\beta q_c$ in $G$ , we have that $eu=ue= \\sum _{j \\in J} (-1)^{\\alpha _j} e_j.$ Since $\\lbrace A_j\\rbrace _{j \\in J}$ are pairwise $2/L$ -disjoint, there exists an $L$ -Lipschitz map $f \\in C_b(X)_1$ such that $f|_{A_j}\\equiv (-1)^{\\alpha _j}\\chi _{A_j}$ for all $j$ .",
"Hence, $\\Vert [a,f]\\Vert \\le \\varepsilon $ since $a\\in \\mathrm {Commut}(L,\\varepsilon )$ , and we clearly have $e_jf=fe_j=(-1)^{\\alpha _j}e_j$ .",
"Therefore, we obtain that $ueae &= \\big ( \\sum _{j \\in J} (-1)^{\\alpha _j} e_j \\big ) ae= efae\\approx _{\\varepsilon } eafe= ea\\big ( \\sum _{j \\in J} (-1)^{\\alpha _j} e_j \\big )= eaeu.$ Finally, we complete the proof by the following computation: $ \\Vert eae-\\sum _{j\\in J}e_jae_j\\Vert = \\Vert \\mathcal {E}^{\\prime }(eae)-eae\\Vert \\le \\sup _{u \\in G} \\Vert eaeu-ueae\\Vert \\le \\varepsilon ,$ for any $a\\in \\mathrm {Commut}(L,\\varepsilon )$ .",
"Acknowledgments.",
"The first-named author would like to thank Tomasz Kania for helpful discussions on Banach space valued $L^p$ -spaces."
]
] | 1808.08593 |
[
[
"Dynamics of a spatially developing liquid jet with slower coaxial gas\n flow"
],
[
"Abstract A three-dimensional round liquid jet within a low-speed coaxial gas flow is numerically simulated and explained via vortex dynamics ($\\lambda_2$ method).",
"The instabilities on the liquid-gas interface reflect well the vortex interactions around the interface.",
"Certain key features are identified for the first time.",
"Two types of surface deformations are distinguished, which are separated by a large indentation on the jet stem: First, those near the jet start-up cap are encapsulated inside the recirculation zone behind the cap.",
"These deformations are directly related to the dynamics of the growing cap and well explained by the vortices generated there.",
"Second, deformations occurring farther upstream of the cap are mainly driven by the Kelvin-Helmholtz (KH) instability at the interface.",
"Three-dimensional deformations occur in the vortex structures first, and the initially axisymmetric KH vortices deform and lead to several liquid lobes, which stretch first as thinning sheets and then either continue stretching directly into elongated ligaments - at lower relative velocity - or perforate to create liquid bridges and holes - at higher relative velocity.",
"The different scenarios depend on Weber and Reynolds numbers based on the relative gas-liquid velocity as was found in the temporal studies.",
"The deformations in the upstream region are well portrayed in a frame moving with the convective velocity of the liquid jet.",
"The usefulness of the temporal analyses are now established."
],
[
"Introduction",
"When a liquid jet discharges into a gaseous medium, it becomes unstable and breaks into droplets due to the growth of instabilities.",
"In combustion and jet propulsion applications, the common purpose of breaking a liquid stream into spray is to increase the liquid surface area so that subsequent heat and mass transfer can be increased.",
"Even though the liquid jet breakup has been studied theoretically, experimentally and numerically for more than half a century, the liquid surface deformation mechanisms and its causes are still not satisfactorily understood and categorized at different flow conditions.",
"In this study, a spatially developing round liquid jet with slower coaxial gas flow is analyzed numerically.",
"The main objective here is to examine the interaction of the vortices near the liquid-gas interface, and to see how those interactions vary with gas-to-liquid velocity ratio, and their consequent effects on the surface deformation and growth of instabilities.",
"Lasheras and Hopfinger [1] in their review of the liquid jet atomization in a coaxial gas stream, categorized the regimes of liquid jet breakup and showed the effects of gas-to-liquid momentum ratio on those regimes.",
"However, they did not relate those regimes to the dynamics of vortices generated prior to atomization.",
"Shinjo and Umemura [2] briefly touched upon the axial and radial vortices generated in a round liquid jet atomization process (without coaxial flow) and showed that the orientation of vortices determines the orientation of ligaments created during the primary breakup; however, they mainly focused on the vortices near the jet tip and claimed that the primary breakup is mainly affected by the vortices that are convected upstream from the jet tip, without detailing the vortex interactions.",
"More recently, Jarrahbashi and Sirignano [3] and Jarrahbashi et al.",
"[4] studied the details of vortex dynamics in a temporal study of a round liquid jet segment, and showed that the vortices can also form far upstream of the jet cap, independent of the vortices shed behind the cap region.",
"They were able to relate the vortex interactions to the surface deformation, lobe formation and perforation.",
"Later, Zandian et al.",
"[5], [6] extended the vortex dynamics analysis to the atomization of planar liquid sheets.",
"They identified three main atomization regimes with different characteristic length and time scales and unique breakup mechanisms based only on the liquid Reynolds number ($Re_l$ ) and gas Weber number ($We_g$ ).",
"They showed that one can understand each breakup mechanism by following the vortex interactions near the gas-liquid interface.",
"Ling et al.",
"[7] also observed the hairpin vortex structures emphasized by Zandian et al.",
"[6] at the surface of a spatial liquid jet, but failed to explain the details of those vortex interactions.",
"Here, we perform an analysis similar to Zandian et al.",
"[6], but with inclusion of a slow coaxial gas flow and for a spatially developing jet.",
"This study shows the validity of the prior temporal studies and their relevance to a real atomization application."
],
[
"Numerical Methods",
"The three-dimensional Navier-Stokes (NS) with volume-of-fluid (VoF) interface-capturing method yield computational results for the round liquid jet which captures the liquid-gas interface deformations after injection.",
"The incompressible continuity and Navier-Stokes equations follow $\\nabla \\cdot \\textbf {u}=0 , \\hspace{15.0pt} \\frac{\\partial (\\rho \\textbf {u})}{\\partial t} + \\nabla \\cdot (\\rho \\textbf {u} \\textbf {u})= - \\nabla p+\\nabla \\cdot (2\\mu \\textbf {D}) - \\sigma \\kappa \\delta (d) \\textbf {n},$ where $\\textbf {D}$ is the rate of deformation tensor, and $\\textbf {u}$ is the velocity field; $p$ , $\\rho $ , and $\\mu $ are the pressure, density and dynamic viscosity of the fluid, respectively.",
"The last term in the NS equation is the surface tension force per unit volume, where $\\sigma $ is the surface tension coefficient, $\\kappa $ is the surface curvature, $\\delta (d)$ is the Dirac delta function and $\\textbf {n}$ is the unit vector normal to the liquid/gas interface pointing away from the liquid.",
"Direct numerical simulation is done by using an unsteady three-dimensional finite-volume solver for the NS equations for the round incompressible liquid jet and its coaxial gas stream.",
"A uniform staggered grid is used with the mesh size of $2~\\mu m$ and a time step of $5~ns$ .",
"A third-order accurate QUICK scheme is used for spatial discretization and the Crank-Nicolson scheme for time marching.",
"The continuity and momentum equations are coupled through the SIMPLE algorithm.",
"The VoF method developed by Hirt and Nichols [8] captures the liquid-gas interface through time and space.",
"The volume fraction $f$ , which represents the volume of liquid phase fraction at each cell, is advected by the velocity field [8]: $\\frac{\\partial f}{\\partial t} + \\textbf {u} \\cdot \\nabla f = 0.$ The 3D computational domain forms a rectangular box, which is discretized into uniform-sized cells.",
"The domain is initially filled with quiescent gas.",
"The liquid jet of diameter $D=200~\\mu m$ is injected from the left boundary at time zero with a constant velocity of $U_l=50~m/s$ .",
"The domain size is $15D\\times 6D\\times 6D$ , in the $x$ (axial), $y$ and $z$ (radial) directions, respectively.",
"The coaxial gas stream fills the rest of the inlet boundary with a constant velocity of $U_g=5$ , 10, and $25~m/s$ resulting in velocity ratios of $\\hat{U}=0.1$ , $0.2$ and $0.5$ , respectively.",
"The other sides are outlet boundaries, where the Lagrangian derivatives of the velocity components are set to zero.",
"Similar axisymmetric (2D) domain is also considered and solved for comparison, where necessary.",
"The most important dimensionless groupings in this study are the liquid Reynolds number ($Re_l$ ), the gas Weber number ($We_g$ ), and gas-to-liquid density ratio ($\\hat{\\rho }$ ), viscosity ratio ($\\hat{\\mu }$ ), and velocity ratio ($\\hat{U}$ ), as defined below.",
"$Re_l = \\frac{\\rho _l UR}{\\mu _l} , \\hspace{10.0pt}We_g = \\frac{\\rho _g U^2 R}{\\sigma } , \\hspace{10.0pt}\\hat{\\rho } = \\frac{\\rho _g}{\\rho _l} , \\hspace{10.0pt}\\hat{\\mu } = \\frac{\\mu _g}{\\mu _l} , \\hspace{10.0pt}\\hat{U} = \\frac{U_g}{U_l}.$ The jet radius $R$ is the characteristic length, and for the characteristic velocity the liquid jet velocity $U_l$ is mainly used in the literature.",
"However, our results show that the more relevant characteristic velocity in this problem is the relative velocity of the liquid with respect to gas; i.e.",
"$U_{r}=U_l-U_g$ .",
"The subscripts $l$ and $g$ refer to the liquid and gas, respectively.",
"In this study we mainly focus on $\\hat{U}$ effects and for this purpose the three values indicated above are considered.",
"For the other four parameters, the chosen values are kept constant: $Re_l=2000$ , $We_g=420$ , $\\hat{\\rho }=0.05$ , and $\\hat{\\mu }=0.01$ .",
"Our goal is to study the vortex dynamics and its influence on the liquid surface dynamics in order to understand breakup mechanisms at different coaxial flow conditions.",
"To this end, the $\\lambda _2$ criterion introduced by Jeong and Hussain [9] is used to define a vortex."
],
[
"Results and Discussion",
"Two types of surface perturbations are distinguished when a liquid jet is injected in a gaseous medium.",
"These two surface deformations and their regions of occurrence are shown schematically in Fig.",
"REF and the 3D simulation result shown in Fig.",
"REF .",
"The detached droplets and ligaments have been removed in Fig.",
"REF to better display the surface waves on the liquid jet core.",
"The first region is right behind the jet start-up cap, to the right of the broken red line in Fig.",
"REF , and is called the Behind the Cap Region (BCR).",
"In this region, the gas phase velocity is faster compared to the liquid jet and thus, the relative local velocity of the gas stream points downstream.",
"The gas shear creates negative azimuthal vorticity ($\\omega _z$ ) near the interface, which generates KH vortices and consequently downstream facing KH waves, as shown in Fig.",
"REF .",
"The vortices in this region are encapsulated inside the recirculation zone behind the cap – the region inside the box in Fig.",
"REF – and the surface deformations are directly related to the dynamics of the growing cap and can be explained by the vortex interactions in that region.",
"Shinjo and Umemura [2] briefly discussed this region in their 3D simulation of a round liquid jet with no coaxial flow, and while describing that the vortex dynamics in this region are very complex, they showed that the vortex orientation determines the orientation of the ligaments that are broken from the cap.",
"The surface pattern is not periodic in BCR.",
"Even though this region is not the main focus of our study, kinematics of the cap and the BCR waves are analyzed later in this section.",
"Figure: Azimuthal vorticity contour (ω z \\omega _z) and the various vortices generated near the interface in the axisymmetric jet; U ^=0.1\\hat{U}=0.1.The second kind of surface deformations occurs farther upstream of the cap, to the left of the broken red line in Fig.",
"REF , and is called the Upstream Region (UR).",
"In this region, the liquid-phase velocity is faster than the gas, and the relative gas velocity points upstream.",
"The gas shear creates positive azimuthal vorticity which creates KH vortices that roll-up the liquid surface and generate upstream facing KH wave pattern.",
"As also mentioned by Shinjo and Umemura [2], relatively periodic wave patterns can be observed in this region.",
"They claim that the UR dynamics are highly affected by the BCR dynamics since the shed vortices and the broken droplets are transmitted upstream from the jet cap.",
"However, our analysis shows that the vortices shed from the cap rim mainly affect the droplet propagation in the upstream region and are far from the interface and have minor interactions with the UR KH waves (see Fig.",
"REF ); thus, the surface dynamics in UR can be studied separately from the BCR in a temporal analysis with periodic boundary conditions similar to studies conducted by Jarrahbashi et al.",
"[3], [4] and Zandian et al.",
"[5], [6].",
"As shown in Figs.",
"REF –REF , a large indentation exists between the UR and BCR regions.",
"This indentation is caused by the radially inward gas flow which impinges on the jet stem upstream of the cap.",
"The gas stream then branches into two opposite streams, one flowing downstream and the other upstream in the frame of reference of the jet.",
"The same indentation was also observed in results of Shinjo and Umemura [2] (Fig.",
"15a), although not emphasized by them.",
"We draw the line between the two regions at the center of this indentation.",
"Shinjo and Umemura [2] however, did not identify an exact criterion for the borderline between these two regions, and their BCR and UR regions overlapped at some places and also stretched beyond our proposed segmentation line.",
"Figure: Definition of the tip length (L t L_t), the smooth region length (L s L_s), and the first UR-KH-wave length (L w L_w).Figure REF shows the time and length of the first KH perturbation occurrence at different velocity ratios.",
"Time has been non-dimensionalized by the injection velocity and jet diameter ($t^*=U_lt/D$ ), and the length is normalized by the jet diameter.",
"Both jet smooth length and time increase substantially by increasing the coaxial gas velocity.",
"The increase in the length seems to be almost linear, which means that the perturbations are transmitted downstream according to the gas stream velocity.",
"However, the time at which the first perturbation occurs gets delayed exponentially by increasing the velocity ratio.",
"Figure REF compares the average length of surface waves in the axial direction for different $\\hat{U}$ .",
"By increasing $\\hat{U}$ from 0.1 to 0.5, the average wavelength increases from 80 to 100 $\\mu m$ .",
"The effect of coaxial gas velocity on axial wavelength becomes less significant as $\\hat{U}$ increases.",
"Results of Figs.",
"REF and REF clearly indicate that the most relevant characteristic velocity in coaxial injection problems is the relative velocity between the liquid and gas streams; i.e.",
"$U_r$ .",
"Thus, since obviously $Re_l$ is the same for all three cases, the most pertinent Reynolds number should also be based on $U_r$ and not the injection velocity as is used in many studies in the literature.",
"As $\\hat{U}$ decreases, hence $Re_{l,r}$ increases, axial wavelength decreases, as intuitively expected.",
"Figure REF schematically defines the length of the jet tip ($L_t$ ), the jet smooth length ($L_s$ ), and the length of the first surface wave in the UR region ($L_w$ ), which are measured for three $\\hat{U}$ cases in time in Figs.",
"REF –REF .",
"All of the lengths are measured from the injection plane and all are normalized by the jet diameter.",
"The average velocity of each length (in terms of the fraction of $U_l$ ), measured from the slope of each trend, is also computed and presented on the plots.",
"The solid lines indicate the simple convection of the first perturbation with the Dimotakis velocity $U_d=(U_l+\\sqrt{\\hat{\\rho }}U_g)/(1+\\sqrt{\\hat{\\rho }})$ .",
"Figure: Temporal plot of L s /DL_s/D, L t /DL_t/D, and L w /DL_w/D for U ^=0.5\\hat{U}=0.5.",
"The solid line indicates simple convection with Dimotakis velocity.In all cases, the wave speed follows the Dimotakis speed at early times after its appearance, but it diverges and becomes slightly larger at later times.",
"This divergence is more apparent at lower $\\hat{U}$ .",
"The smooth length also grows in time for all cases, but it is always below $L_w$ .",
"Shinjo and Umemura [2] also observed a similar difference in the rate of smooth region growth and the injection speed.",
"They concluded that this difference means that the size of the region of influence of the tip is spreading toward upstream as time passes.",
"However, we do not see a direct connection between these perturbations and the vortices generated in the BCR, and the only conclusion that can be drawn here is that new instabilities keep forming upstream of the initial perturbation, resulting in growth of the UR.",
"We showed earlier that UR is not much affected by the BCR and thus, the only reason for the upstream spreading of the instabilities is concluded to be the increase in the strain rate on the surface after the growth of the former KH waves.",
"This triggers new KH vortices, which result in new waves upstream.",
"The rate of growth of the smooth length increases as $\\hat{U}$ increases.",
"Figure: Change in roll-up direction of the KH wave as it moves from UR to BCR region; U ^=0.2\\hat{U}=0.2.The tip velocity is always smaller than the wave speed.",
"This means that the KH waves that form in UR, finally enter BCR and catch up with the tip.",
"The rate at which the KH waves reach the tip is directly proportional to the difference between the tip speed and the wave speed.",
"This velocity difference becomes smaller at higher $\\hat{U}$ and thus, it takes more time for the UR waves to merge with the cap and disappear.",
"This happens at $t^*=15$ for $\\hat{U}=0.1$ , and at $t^*=17.5$ for $\\hat{U}=0.2$ .",
"For $\\hat{U}=0.5$ , this catchup is so slow that the cap moves out of the domain before the KH wave has enough time to reach it.",
"When the UR waves get into BCR, their roll-up direction changes since the relative gas stream direction changes.",
"This phenomenon is clearly shown in Fig.",
"REF , where the KH wave indicated by the black arrow faces upstream at $t^*=12.5$ while it is in UR, becomes neutral as it enters BCR boundary at $t^*=13$ , and faces downstream while it is in BCR at $t^*=13.5$ .",
"Notice the formation of a negative KH vortex just as the wave enters the BCR region.",
"Figure: Liquid jet surface at t * =6t^*=6; U ^=0.2\\hat{U}=0.2.The liquid jet at a few time steps before the start of perturbations is shown in Fig.",
"REF .",
"The vortex structures indicated by the $\\lambda _2$ isosurface are also shown in the same figure at the same time step.",
"The vortices that are attached to the jet tip include the vortex structure that covers the front of the jet cap (Tip Vortex), which is caused by the gas flow that goes around the mushroom-shaped cap, and a vortex ring that is shed from the rim of the cap (Rim Vortex), due to flapping of the rim.",
"As discussed earlier, on the stem of the jet, there are two sets of oppositely oriented KH vortex rings which are formed due to the shear caused by the entrained gas in the downstream and upstream directions.",
"As seen in Fig.",
"REF , the KH vortices deform and take a hairpin structure before the surface of the jet is deformed.",
"Figure REF , shows that the first liquid lobes are formed at the exact same location where the KH vortices are turned streamwise and create a hairpin structure, at a later time.",
"This conveys that the vortex dynamics drives the surface dynamics, as was first identified by Jarrahbashi et al.",
"[4] and Zandian et al. [6].",
"Even though both those studies were temporal with periodic conditions on a liquid segment, our spatial results show that the temporal study of vortex dynamics can capture well the mechanisms in the UR region, where fairly similar physical behaviors occur.",
"Figure: Liquid-jet surface (a), and vortex structures indicated by λ 2 =-10 11 s -2 \\lambda _2=-10^{11}~s^{-2} isosurface (b) at t * =11.5t^*=11.5; U ^=0.1\\hat{U}=0.1.Figure REF shows the liquid jet and its vortices at $t^*=11.5$ for $\\hat{U}=0.1$ .",
"The KH vortex rings start from an axisymmetric form and grow and deform as they move downstream.",
"Following this change in the KH vortex structure, we can see that the initially axisymmetric KH waves (closer to the nozzle) also become more corrugated as they move downstream.",
"The mode number (azimuthal wavelength) of the lobes is directly related to the number of counter-rotating streamwise vortex pairs that form as the KH vortices become streamwise oriented.",
"Thus, a detailed analysis of the causes of streamwise vortex generation and its growth rate (similar to analyses of Jarrahbashi and Sirignano [3] for round jets and Zandian et al.",
"[6] for planar sheets) can explain a lot regarding the future behavior of the lobes and their breakup mechanism and size of ligaments and droplets.",
"This analysis will not be quantified here, but the effects of velocity ratio on the mode number will be discussed qualitatively here.",
"Figure: Liquid-jet surface and the axial vorticity (ω x \\omega _x) contours on the plane intersecting the jet at x/D=8.75x/D=8.75 at t * =11.5t^*=11.5; U ^=0.1\\hat{U}=0.1.The axial vorticity ($\\omega _x$ ) contours on a spanwise plane intersecting with the liquid jet stem at $x/D=8.5$ and $x/D=8.75$ at $t^*=11.5$ (same time as Fig.",
"REF ) are shown in Figs.",
"REF and REF , respectively.",
"The plane in Fig.",
"REF cuts the jet at the braid of one of the newer KH waves slightly upstream of the former wave that is located at $x/D=8.75$ and is the subject of Fig.",
"REF .",
"Two layers of counter-rotating streamwise vorticity are seen in Fig.",
"REF .",
"The inner layer closer to the liquid surface is the hairpin vortex ring with several counter-rotating axial vortices, a pair of which is indicated by the simple arrows.",
"This hairpin stretches upstream and over the next consecutive KH wave upstream of the current wave.",
"Right on the outer side of this vorticity layer is another layer of counter-rotating vorticity, shown by double-lined arrows.",
"Since these counter-rotating vortex pairs are $180^\\circ $ out of phase with respect to the inner layer, it is concluded that this layer belongs to another hairpin vortex layer with opposite direction.",
"This layer is called the outer hairpin layer and is stretched downstream and underneath the next downstream KH wave, shown in Fig.",
"REF .",
"The reason why much axial deflection is still not seen in this vortex ring in Fig.",
"REF is that the $\\omega _x$ magnitude is an order of magnitude smaller than the azimuthal vorticity ($\\omega _\\theta $ ) magnitude at this time.",
"Even though the vorticity layers are not very neat and organized at all azimuthal locations of this picture, seven counter-rotating vortex pairs are distinguished in Fig.",
"REF , which indicates that seven lobes are expected to form on this KH wave later.",
"In the downstream cross-sectional plane shown in Fig.",
"REF , three counter-rotating axial vortices are observed.",
"The outer layer – indicated by thick arrows – is the outer hairpin vortex layer for this new wave which stretches downstream.",
"This hairpin layer is right on the outer surface of the KH wave.",
"Right on the inner side of this wave, there is another layer of counter-rotating hairpin vortex, indicated as inner hairpin layer.",
"This layer is the same outer hairpin layer seen in Fig.",
"REF and is stretched underneath the wave shown in Fig.",
"REF .",
"A comparison between the counter-rotating vortex pairs of these two layers shows that they both belong to the same hairpin structure that wraps over the upstream wave and under the next downstream wave.",
"There is another vortex ring on the inner side of this hairpin, which is less organized and more chaotic, but with smaller axial vorticity component.",
"This layer is part of the KH vortex located underneath the wave and slightly deflected.",
"The effects of the counter-rotating axial vortex pairs shown in Figs.",
"REF and REF are more clear at a later time ($t^*=12$ ) shown in Fig.",
"REF .",
"When $\\omega _x$ grows enough to become comparable to $\\omega _\\theta $ , three-dimensional instabilities occur and the vortices lose their axisymmetry.",
"This phenomenon creates corrugations in the KH vortex ring and also larger axial stretch on the hairpin vortices that are also stretched by the KH vortex.",
"The corrugated KH vortex and the hairpin vortex that stretches over it are shown in Fig.",
"REF .",
"The inner hairpin vortex is not clearly seen in this figure since the KH vortex and the liquid lobes on the outer side of this hairpin block those hairpins from the view.",
"As shown by Zandian et al.",
"[6], overlapping of these oppositely oriented counter-rotating hairpins that are on the outer and inner sides of the lobe, thins the lobe at its center and creates holes on the lobes.",
"Thinning of the lobes (wave) can be clearly seen in the cross-sectional view of the plane illustrated in Fig.",
"REF .",
"Figure: Liquid-jet surface at t * =12.5t^*=12.5; U ^=0.1\\hat{U}=0.1.Following the same wave at a later time ($t^*=12.5$ ), shown in Fig.",
"REF , it is seen that holes form on the lobes and they expand as the lobes get stretched in the axial direction.",
"The liquid bridges that are formed around the lobe rim finally break and create the first ligaments, which then break into droplets or detach from the jet core.",
"Figure REF also shows that there are in fact seven liquid lobes on each KH wave, as was inferred earlier by the number of counter-rotating axial vorticity pairs.",
"Four of these lobes are already seen in this figure, and the other three are on the hidden side of the jet stem, which are blocked in this view.",
"As we follow the jet structure at much later time ($t^*=15.5$ shown in Fig.",
"REF ), we see that the same hole formation and breakup mechanism repeats for other waves as well.",
"This confirms that the breakup mechanism on the jet stem in UR is periodic and occurs for all the waves formed in that range until they reach the BCR region and break into droplets and/or coalesce with the cap.",
"This validates the temporal studies of Jarrahbashi et al.",
"[3], [4] and Zandian et al.",
"[5], [6].",
"The formation of holes on the rim of the jet cap (see Fig.",
"REF ) is also conjectured to follow the same vortex overlapping mechanism, where the overlapping vortices in that case are the tip vortex and the downstream KH vortex (see Fig.",
"REF ) as it runs along the inner side of the mushroom-shaped cap; however, these vortex structures are much harder to follow and are not analyzed in this study.",
"Figure: Liquid-jet surface at t * =15.5t^*=15.5; U ^=0.1\\hat{U}=0.1.Based on $Re_l$ and $We_g$ values of the current study, this jet (without coaxial gas flow) should belong to Domain I as indicated by Zandian et al. [5].",
"In Domain I, lobes stretch directly into ligaments without formation of holes.",
"The hole formation mechanism occurs in Domain II, at higher ranges of $Re_l$ and $We_g$ .",
"Thus, we can conclude from this study that addition of coaxial gas flow shifts the breakup mechanism, as this jet now belongs to Domain II, which is consistent with the use of relative velocity for defining $We_g$ and $Re_l$ .",
"The same jet with a higher velocity ratio is simulated and the results are depicted in Fig.",
"REF for $t^*=15$ and 17.",
"A few main differences are observed at a first glance between these results and the lower $\\hat{U}$ cases.",
"First, the azimuthal mode number has significantly decreased from seven (for $\\hat{U}=0.1$ ) to four.",
"Four liquid lobes are seen in this figure – one in the front view, one on top, one on bottom, and one on the hind view which cannot be seen here, following the four axial pairs of KH vortices shown in Fig.",
"REF (b,d).",
"The next main change is that the lobes, ligaments and cap rim seem much thicker compared to lower $\\hat{U}$ .",
"Because of this thickening in the lobe structures, the lobes do not thin easily and they are stretched directly into thick ligaments.",
"This means that the breakup mechanism has moved from Domain II towards Domain I by increasing $\\hat{U}$ .",
"This clearly shows that the Reynolds and Weber numbers in such coaxial flow should be based on the relative gas-liquid velocity rather than just liquid jet velocity.",
"By increasing $\\hat{U}$ , the relative velocity $U_r$ decreases, and thus, $Re_{l,r}$ and $We_{g,r}$ decrease too.",
"This decrease in the Reynolds and Weber numbers is consistent with shifting from the hole formation breakup (Domain II) to lobe stretching mechanism (Domain I), as predicted by Zandian et al. [5].",
"Therefore, a more thorough analysis of the effects of velocity ratio is required to generalize the breakup mechanisms formerly developed for non-coaxial jet flows.",
"This is left for a later study.",
"Figure: Liquid jet surface (a,c) and vortex structures (b,d) at t * =15t^*=15 (a,b) and t * =17t^*=17 (c,d); U ^=0.5\\hat{U}=0.5.",
"The surface of liquid jet is colored by the axial velocity contours."
],
[
"Summary and Conclusions",
"A three-dimensional round liquid jet with coaxial, outer gas flow is numerically analyzed.",
"The evolution of instabilities on the liquid-gas interface were observed to be correlated with the vortex interactions around the liquid-gas interface using a $\\lambda _2$ analysis.",
"Two main regions were defined on the liquid jet separated by a large indentation on the jet stem with distinguished surface deformations.",
"The Behind the Cap Region (BCR) is encapsulated inside the recirculation zone behind the mushroom-shaped cap.",
"The KH waves formed on the jet core roll downstream in BCR and flow downstream until they coalesce with the cap.",
"The vortices and surface deformations in BCR are not periodic and are controlled by the dynamics of vortices in the recirculation zone.",
"The second region (Upstream Region, UR) is farther upstream of the cap.",
"The gas speed is lower than the liquid jet in UR, and the shear caused by this upstream flowing gas stream relative to the liquid triggers a KH instability.",
"The 3D deformation of the initially axisymmetric KH vortices leads to several liquid lobes.",
"The lobes either thin and form holes at lower velocity ratios or stretch directly into elongated ligaments at higher velocity ratios, which could be explained by the vortex interactions in the UR region.",
"The deformations developed in UR can be portrayed better in a frame moving with the convective velocity of the liquid jet with periodic conditions.",
"The azimuthal and axial wavelengths of the instabilities and the breakup mechanism in UR can be well defined using a Reynolds and Weber number based on the relative gas-liquid velocity."
],
[
"Acknowledgements",
"Access to the XSEDE supercomputer under Allocation CTS170036 and to the UCI HPC cluster were very valuable in performing our high resolution computations."
]
] | 1808.08322 |
[
[
"Phase factors of periodically driven two-level systems"
],
[
"Abstract Using a perturbative solution for a periodically driven two-level quantum system, we show how to obtain phase factors for both a two-level quantum system and two two-level quantum systems non-interacting and interacting.",
"The method is easily implemented by numerical routines and presents the advantage of being stable for long-time periods.",
"We furthermore explore the possibility of implementing a quantum phase gate using the perturbative solution."
],
[
"introduction",
"The study of geometric phases has attracted significant interest since it was shown that they could be used to process quantum information [18] and, due to its geometric properties, they present an inherent resilience to fluctuation errors in the control parameters.",
"The experimental implementations of geometric phase in the context of quantum computation, sometimes refered to as geometric quantum computation (GQC), has been fruitful [13], [1].",
"Nevertheless, to obtain the expression for the geometric phase acquired by a two-level quantum system, many works implement the rotating wave approximation (RWA) [10], [15], [13].",
"As every approximation, the RWA has its realm of validity and applicability that has been extensively studied [7], [17], [8], [11], [16].",
"In this work, we consider the evolution of a two-level quantum system driven by periodic fields.",
"Instead of the RWA, we use the solution of the Schrödinger equation obtained in [3], [4] (see also [6], [2], [5], [12]) to compute the total, dynamical and geometric phase for a two-level quantum system and two two-level quantum systems.",
"Since the solution used is uniformly convergent in time, the expressions for the phases present a robustness when long-time periods are considered.",
"We first make a brief overview of the perturbative method developed in [3], [4].",
"The phases of a two-level quantum system are then obtained using the perturbative expansion.",
"The discussion is extended to two two-level quantum systems, non-interacting and interacting.",
"In each case, we present the expressions for calculating each phase factor.",
"Finally, we obtain the phase factors for the composite two two-level quantum system with a delta interaction.",
"We show that for a specific choice of parameters, it is possible to build a phase shift gate."
],
[
"Description of the model and methods",
"Let us start by considering a system with the following Hamiltonian: $H_1(t) = \\epsilon \\sigma _3 - f(t)\\sigma _1,$ where $\\epsilon $ is a real constant and $f(t)$ is a periodic function of time with frequency $\\omega >0$ .",
"Let us consider a rotation of $\\pi /2$ around the $y$ -axis, denoted by $R_y(\\pi /2)$ , and the Schrödinger equation on this new rotated frame is given by $i\\frac{d}{dt}\\psi _2(t) = H_2(t)\\psi _2(t),$ where $\\psi _2(t) = R_y(\\pi /2)\\psi (t) = \\exp (-i\\pi \\sigma _2/4)\\psi (t) $ and $H_2(t) = \\epsilon \\sigma _1 + f(t)\\sigma _3.$ The Hamiltonian (REF ) can be interpreted as describing a system with a Hamiltonian independent of time $\\epsilon \\sigma _3$ subjected to a time-dependent perturbation $-f(t)\\sigma _1$ .",
"The later is responsible for transitions between the two states of the system.",
"The method developed in [3] and [4] is valid for small $\\epsilon $ and periodic $f$ .",
"The quasi-periodic case was analysed in [12].",
"It consists in writing a perturbative expansion in $\\epsilon $ for the time evolution operator.",
"This method has proven to have the following advantages: the series expansion are uniformly convergent in time, the expression obtained for the time evolution operator is given in terms of series and so are easily implementable in numerical calculations and they can be employed for any periodic function.",
"The uniform convergence is of great importance, since it means the results lead to stable numerical calculations and therefore allows the study of long-time behaviour of the observable quantities of the system.",
"It was shown in [3] that the time evolution operator $U(t)$ for the system described by (REF ) can be written as $U(t) = \\left(\\begin{array}{cc}R(t)(1+ig_0S(t)) & -i\\epsilon R(t)S(t) \\\\-i\\epsilon \\overline{R(t)} \\overline{S(t)} &\\overline{R(t)}(1-i\\overline{g_0}\\overline{S(t)})\\end{array}\\right).$ where $R(t)$ and $S(t)$ are given by $R(t) = e^{-i\\Omega t} \\sum _{m\\in \\mathbb {Z}} R_m e^{im\\omega t}$ and $S(t) = \\sigma _0 + e^{2i\\Omega t} \\sum _{m\\in \\mathbb {Z}} S_m e^{im\\omega t}.$ $R_m$ and $S_m$ are coefficients of the Fourier expansion of $R(t)$ and $S(t)$ , respectively.",
"Together with the Rabi frequency $\\Omega $ and the constants $g_0$ and $\\sigma _0$ , they can all be obtained from rather complex but convergent power series expansions in $\\epsilon $ , involving the the Fourier coefficients of $f$ and its frequency $\\omega $ .",
"See [4] as well as [3], [6], [2], [5] for explicit formulas and examples.",
"Sometimes we will refer to the matrix elements of $U(t)$ , for example, $U_{11}(t)=R(t)(1+ig_0S(t))$ and $U_{12}(t)=-i\\epsilon R(t)S(t)$ .",
"As done in [4], we implemented numerically the method developed there for a perturbation of the form $f(t) = F_0 + A\\cos (\\omega t),$ where $F_0$ is a real number, $A$ and $\\omega $ is the amplitude and the frequency of the periodic perturbation, respectively.",
"Following the directions of the original paper, the method was applied to several values of $\\omega $ and $\\epsilon $ , the former ranging from $1.0$ to $10.0$ and the later from $0.01$ to $0.40$ .",
"For all these values, the unitarity test was sufficiently satisfactory, since the error is bounded by $3\\times 10^{-3}$ in one specific case (for $\\omega =1.0$ and $\\epsilon =0.40$ ), but for most cases, is bounded by $10^{-5}$ or even $10^{-10}$ ."
],
[
"Total, dynamical and geometric phases",
"We now show the calculations of the total, dynamical and geometric phases for the two-level system considered.",
"The total phase of the system is simply given by $\\phi _{tot}(t) = \\arg \\langle \\psi (0),\\psi (t)\\rangle ,$ and the dynamical phase $\\alpha _{dyn}$ is given by $\\alpha _{dyn}(t) = i\\int _0^t \\langle \\psi (t^{\\prime }),\\dot{\\psi }(t^{\\prime })\\rangle \\text{d}t^{\\prime } ,$ where $\\psi (0)$ and $\\psi (t)$ are the state vectors of the system at the initial instant of time and for an instant of time $t$ , respectively.",
"The dot indicates derivation relative to time.",
"The geometric phase $\\gamma _{geo}$ is simply the difference between the total and dynamical phases: $\\gamma _{geo}(t) = \\phi _{tot}(t)-\\alpha _{dyn}(t).$ We note that the phase factors are functions of time, since they are defined by the evolution of the state vector $\\psi (t)$ .",
"When performing the following calculations, we shall consider the state vector correspondent to the rotated Hamiltonian given by (REF ).",
"The resulting expressions become $\\phi _{tot}(t) &= \\arg \\lbrace \\operatorname{Re}U_{11}(t) +i(-2\\operatorname{Re}(\\overline{\\alpha }\\beta )\\operatorname{Im}U_{11}(t) \\nonumber \\\\&\\quad + 2\\operatorname{Im}(\\overline{\\alpha }\\beta )\\operatorname{Re}U_{12}(t) + (2|\\alpha |^2-1)\\operatorname{Im}U_{12}(t))\\rbrace $ and $&\\alpha _{dyn}(t) = |\\alpha |^2\\left(-\\operatorname{Im}\\int _0^t a_{11}(t^{\\prime })dt^{\\prime } +i\\operatorname{Re}\\int _0^t a_{12}(t^{\\prime })dt^{\\prime }\\right) \\nonumber \\\\&\\quad - 2i\\operatorname{Re}(\\overline{\\alpha }\\beta ) \\operatorname{Re}\\int _0^t a_{11}(t^{\\prime })dt^{\\prime } -2i\\operatorname{Im}(\\overline{\\alpha }\\beta ) \\operatorname{Im}\\int _0^t a_{12}(t^{\\prime })dt^{\\prime } \\nonumber \\\\&\\quad + |\\beta |^2 \\left(-\\operatorname{Im}\\int _0^t a_{11}(t^{\\prime })dt^{\\prime } -i\\operatorname{Re}\\int _0^t a_{12}(t^{\\prime })dt^{\\prime }\\right),$ where $a_{11}(t)$ and $a_{12}(t)$ are matrix elements of the product of $U^*(t)$ and $\\dot{U}(t)$ : $U^*(t)\\dot{U}(t) = \\left(\\begin{array}{cc}a_{11}(t) & a_{12}(t) \\\\-\\overline{a}_{12}(t) & \\overline{a}_{11}(t)\\end{array}\\right).$ The expression for the dynamical phase involves integrations over time of the expansions.",
"Although there are lots of integration routines, using them in the highly oscillatory functions that constitute the expansions often results in a large error due to the routine.",
"Thus, the integrations were carried out analytically term by term in the Fourier expansions and then implemented numerically.",
"The previous expressions determine the total and dynamical phase for the system for any instant of time.",
"Next, it is necessary to define the instant of time that is physically meaningful to the calculations of the phase acquired by the system.",
"One could argue that the appropriate instant of time would be the “natural” frequency of the system, characterised by the Rabi frequency $\\Omega $ .",
"But we must recollect the nature of the geometric phase, that is, the phase acquired over the course of the evolution of the system resulted from the geometrical properties of the parameter space of the Hamiltonian.",
"In our case, the parameter space is two-dimensional, with each dimension associated to the parameters $A$ and $\\omega $ in (REF ).",
"So, if we consider a cyclic evolution on the parameter space and a fixed amplitude $A$ of the external field, the relevant instant of time is precisely $t_\\omega = \\frac{2\\pi }{\\omega }.$ Therefore, the expressions (REF ), (REF ) and (REF ) for the respective total phase, dynamical phase and geometric phase of the system are taken at $t_\\omega $ .",
"Next, we present some results of our calculations for the phase factors of the system as graphical representations.",
"Without loss of generality, we considered the initial state vector to be $\\psi (0)=|0\\rangle $ , that is, the state vector is initially aligned with the $z$ -axis.",
"The calculations were performed for values of $\\epsilon $ ranging from $0.01$ to $0.40$ with steps of $0.01$ ; and values of $\\omega $ ranging from $1.0$ to $10.0$ with steps of $0.5$ .",
"As previously stated, the numerical implementation of the total phase was easily accomplished.",
"We note that since the total phase is defined as an argument, there was no need to test if the numerical function had relevant imaginary parts due to built-in machine errors.",
"Figure REF shows the relation between the values of the total phase and the parameter $\\epsilon $ and Figure REF presents a three-dimensional representation of the total phase as a function of $\\omega $ and $\\epsilon $ .",
"We can see that the absolute value of the total phase is proportional to the value of $\\epsilon $ .",
"According to the interpretation of (REF ) in which $\\epsilon $ is the energy gap between the two eigenstates of $\\sigma _3$ , we can say that the total phase is proportional to this gap.",
"Moreover, we note that as the value of $\\omega $ increases, the rate in which the total phase increases with $\\epsilon $ decreases.",
"In other words, the value of $\\omega $ modulates the curve $\\phi _{tot} \\times \\epsilon $ .",
"Figure REF shows graphs of the total phase as a function of $\\omega $ with fixed values of $\\epsilon $ .",
"The same behaviour observed in Figure REF is present in Figure REF , but in this case, the value of $\\epsilon $ modulates the curve $\\phi _{tot}\\times \\omega $ in the following way: as $\\epsilon $ increases, the curve gets more accentuated.",
"It is also notable that for $\\omega $ around $2.0$ , the absolute value of the total phase is maximised.",
"Figure: Total phase plotted as a function of ϵ\\epsilon andω\\omega .Figure: Graphical representation of the total phaseφ tot \\phi _{tot} as a function of ω\\omega and ϵ\\epsilon .The numerical implementation of the dynamical phase is not as straightforward as that of the total phase, since it involves several integrations over time (equation (REF )).",
"These integrations, as we said before, were done analytically and then numerically implemented.",
"The dynamical phase is expected to be real, but the expansions in our implementation are truncated, so we tested if the imaginary part of the dynamical phase had relevant contributions.",
"The imaginary parts equal zero within the machine accuracy.",
"The relation between the dynamical phase and the values of $\\omega $ has a particular behaviour: for $\\omega =1.0,1.5,2.0,2.5$ the curve $\\alpha _{dyn}\\times \\epsilon $ resembles a parabola and for higher values the curve resembles a linear function.",
"Figure REF shows the dynamical phase as a function of $\\epsilon $ for some fixed values of $\\omega $ .",
"Figure REF shows the curve $\\alpha _{dyn}\\times \\omega $ for some values of $\\epsilon $ .",
"We can see that, similar to Figure REF , $\\epsilon $ seems to modulate the curve and there is a value of $\\omega $ that maximises $\\alpha _{dyn}$ , but this value shifts according to the value of $\\epsilon $ .",
"Figure REF shows a three-dimensional representation of the dynamical phase as a function of $\\omega $ and $\\epsilon $ .",
"Figure: Dynamical phase plotted as a function of ϵ\\epsilon andω\\omega .Figure: Graphical representation of the dynamical phaseα dyn \\alpha _{dyn} as a function of ω\\omega and ϵ\\epsilon .A similar behaviour of the total phase is observed for the geometric phase in Figures REF and (REF ): the absolute value of the geometric phase increases as $\\epsilon $ increases, the curve $\\gamma _{geo}\\times \\omega $ is modulated by $\\epsilon $ and it presents a value of $\\omega $ that maximises the absolute value of the geometric phase.",
"Figure REF shows the graphical representation of the geometric phase as a function of $\\omega $ and $\\epsilon $ .",
"Figure: Geometric phase plotted as a function of ϵ\\epsilon andω\\omega .Figure: Graphical representation of the geometric phaseγ geo \\gamma _{geo} as a function of ω\\omega and ϵ\\epsilon .We next consider two two-level quantum systems with individual Hamiltonians given by (REF ).",
"When considering that the two systems do not interact with each other, the phase factors obtained for the composite system are simply the algebraic sum of the individual phase factors.",
"In order to explore how the phase factors of the composite system change when interactions are taken into account, we considered an interaction given by $H^{\\prime }(t) = \\kappa v(t) \\ \\sigma _3^{(a)}\\otimes \\sigma _3^{(b)},$ where $\\kappa $ is a real constant and $v(t)$ is a real function of time.",
"The corresponding Hamiltonian in the rotated frame is given by $H^{\\prime }_2(t) &= R_y(\\pi /2) \\kappa v(t) \\ \\sigma _3^{(a)}\\otimes \\sigma _3^{(b)} R_y^*(\\pi /2) \\nonumber \\\\&= \\kappa v(t) \\ \\sigma _1^{(a)}\\otimes \\sigma _1^{(b)}, $ where $\\epsilon _a$ and $\\epsilon _b$ are the respective constants of the individual systems and $f_a(t)$ and $f_b(t)$ are the external fields applied to each subsystem.",
"The Hamiltonian of the composite system is $H_2(t) = \\left( {\\begin{matrix}f_a(t)+f_b(t) & \\epsilon _b & \\epsilon _a & \\kappa v(t) \\\\\\epsilon _b & f_a(t)-f_b(t) & \\kappa v(t) & \\epsilon _a \\\\\\epsilon _a & \\kappa v(t) & -f_a(t)+f_b(t) & \\epsilon _b \\\\\\kappa v(t) & \\epsilon _a(t) & \\epsilon _b & -f_a(t)-f_b(t)\\end{matrix}} \\right)$ In order to obtain the phase factors for the composite system, we consider the interaction picture.",
"We will denote the state vector in this picture by $\\psi _I(t)$ and it relates to the state vector in the Schrödinger picture by the unitary transformation $\\psi _I(t) = U^*(t)\\psi (t),$ where $U(t)$ is the time evolution operator.",
"In the interaction picture, the time evolution operator $U_I(t)$ is given by the Dyson series $U_I(t) = {1}+ \\sum _{n=1}^\\infty (-i)^n \\int _0^t V_I(t_1) \\,\\text{d}t_1 \\ldots \\int _0^{t_{n-1}} V_I(t_n)\\,\\text{d}t_n.$ where $V_I(t)$ is the interaction Hamiltonian in the interaction picture given by $V_I(t) &= \\kappa v(t) \\begin{pmatrix}V_{11}(t) & V_{12}(t) \\\\\\overline{V}_{12}(t) & -V_{11}(t)\\end{pmatrix}^{(a)} \\otimes \\begin{pmatrix}V_{11}(t) & V_{12}(t) \\\\\\overline{V}_{12}(t) & -V_{11}(t)\\end{pmatrix}^{(b)}, $ with $V_{11}(t) &= -\\overline{U}_{11}(t)\\overline{U}_{12}(t) - U_{11}(t)U_{12}(t), \\\\V_{12}(t) &= \\overline{U}_{11}(t)^2 - U_{12}(t)^2.",
"$ The time evolution operator in the interaction picture given by the Dyson expansion in (REF ), considering the expression for the operator $V_I(t)$ in (REF ), is $U_I(t) &= {1}-i \\kappa \\int _0^t(U^*_a(t^{\\prime })\\sigma _1^{(a)}U_a(t^{\\prime })) \\\\&\\otimes ((U^*_b(t^{\\prime })\\sigma _1^{(b)}U_b(t^{\\prime }))\\,\\text{d}t^{\\prime } + \\mathcal {O}(\\kappa ^2).$ We shall consider the Dyson expansion up to first order.",
"The matrix form of the time evolution operator in the interaction picture, in first order, is given by $U_I(t) = {1}_4 -i\\kappa V^{(1)}(t) ,$ where ${1}_4$ is the identity operator acting on a four-dimensional Hilbert space and $V^{(1)}(t) = \\int _0^t v(t^{\\prime })\\left({\\begin{matrix}V_{11}^{(a)} V_{11}^{(b)} & V_{11}^{(a)} V_{12}^{(b)} &V_{12}^{(a)} V_{11}^{(b)} & V_{12}^{(a)} V_{12}^{(b)} \\\\V_{11}^{(a)} \\overline{V}_{12}^{(b)} & -V_{11}^{(a)} V_{11}^{(b)} &V_{12}^{(a)} \\overline{V}_{12}^{(b)} & -V_{12}^{(a)} V_{11}^{(b)} \\\\\\overline{V}_{12}^{(a)} V_{11}^{(b)} & \\overline{V}_{12}^{(a)} V_{12}^{(b)} &-V_{11}^{(a)} V_{11}^{(b)} & -V_{11}^{(a)} V_{12}^{(b)} \\\\\\overline{V}_{12}^{(a)} \\overline{V}_{12}^{(b)} & -\\overline{V}_{12}^{(a)}V_{11}^{(b)} &-V_{11}^{(a)} \\overline{V}_{12}^{(b)} & V_{11}^{(a)} V_{11}^{(b)}\\end{matrix}} \\right) \\,\\text{d}t^{\\prime }.",
"$ We omitted the time-dependency of the expressions for $V_{11}(t)$ and $V_{12}(t)$ given by equations (REF ) and (), respectively.",
"The operator $V^{(1)}(t)$ will be useful for evaluating the expressions for the phase factors of the composite system.",
"Also, we must note that $V^{(1)}(t)$ is a self-adjoint operator, since $v(t)$ is a real function of $t$ and the matrix operator in the integrand on the right hand side of (REF ) is self-adjoint.",
"The total phase factor for the composite system is $\\phi _{tot}(t) = \\arg \\langle \\psi _2(0),\\psi _2(t)\\rangle \\\\= \\arg \\Big \\lbrace \\langle \\psi _2^{(a)}(0),U_a(t)\\psi _2^{(a)}(0)\\rangle \\langle \\psi _2^{(b)}(0),U_b(t)\\psi _2^{(b)}(0)\\rangle \\Big .",
"\\\\-i\\kappa \\langle \\psi _2(0),U(t)V^{(1)}(t) \\psi _2(0)\\rangle \\Big \\rbrace .$ Note that for $\\kappa =0$ the expression above reduces itself to the total phase of two non-interacting systems.",
"Using (REF ) for the dynamical phase and the expansion in $\\kappa $ for the time evolution operator in the interaction picture, we have $\\alpha _{dyn}(t) &= i\\int _0^t \\langle \\psi _2(t^{\\prime }),\\dot{\\psi _2}(t^{\\prime })\\rangle \\,\\text{d}t^{\\prime }\\nonumber \\\\&= i\\int _0^t \\langle \\psi _2(0),U^*(t^{\\prime }) \\dot{U}(t^{\\prime }) \\psi _2(0)\\rangle \\,\\text{d}t^{\\prime }\\nonumber \\\\&+ \\kappa \\int _0^t \\langle \\psi _2(0),U^*(t^{\\prime }) U(t^{\\prime }) V^{(1)}(t^{\\prime })\\psi _2(0)\\rangle \\,\\text{d}t^{\\prime } \\nonumber \\\\&+ \\kappa \\int _0^t \\langle \\psi _2(0),U^*(t^{\\prime }) \\dot{U}(t^{\\prime })\\dot{V}^{(1)}(t^{\\prime }) \\psi _2(0)\\rangle \\,\\text{d}t^{\\prime } \\nonumber \\\\&- \\kappa \\int _0^t \\langle \\psi _2(0),V^{(1)}(t^{\\prime })^*U^*(t^{\\prime }) \\dot{U}(t^{\\prime })\\rangle \\,\\text{d}t^{\\prime } + \\mathcal {O}(\\kappa ^2), \\nonumber $ since $U(t)$ is unitary, the identity $U^*(t)\\dot{U}(t) =-\\dot{U^*}(t)U(t)$ holds and the third term on the right hand side of the expression above can be rewritten as the complex conjugate of the second term.",
"Hence, the dynamical phase up to first order in $\\kappa $ is given by $\\alpha _{dyn}(t) &= \\alpha _{dyn}^{(0)}(t) \\nonumber \\\\& + 2\\kappa \\operatorname{Re}\\int _0^t \\langle \\psi _2(0),U^*(t^{\\prime }) \\dot{U}(t^{\\prime })V^{(1)}(t^{\\prime }) \\psi _2(0)\\rangle \\,\\text{d}t^{\\prime } \\nonumber \\\\&\\quad \\quad + \\kappa \\int _0^t \\langle \\psi _2(0),\\dot{V}^{(1)}(t^{\\prime }) \\psi _2(0)\\rangle \\,\\text{d}t^{\\prime } + \\mathcal {O}(\\kappa ^2),$ where the $\\alpha _{dyn}^{(0)}(t)$ is exactly the expression for the dynamical phase for two non-interacting two-level systems.",
"Also, the third term on the right hand side of (REF ) is the integral over time of the expectation value of the self-adjoint operator $V^{(1)}(t)$ .",
"Therefore, this term is also real and so is the expression for the dynamical phase.",
"The geometric phase for the composite system is still given by the difference between the total phase and the dynamical phase.",
"Now, let us consider the case in which the interaction is given by $v(t) = \\delta (t-t_0),$ where $t_0$ is any instant of time.",
"The time evolution operator in the interaction picture, according to (REF ) and (REF ), is $U_I(t) = {1}_4 -i\\kappa \\left({\\begin{matrix}V_{11}^{(a)} V_{11}^{(b)} & V_{11}^{(a)} V_{12}^{(b)} &V_{12}^{(a)} V_{11}^{(b)} & V_{12}^{(a)} V_{12}^{(b)} \\\\V_{11}^{(a)} \\overline{V}_{12}^{(b)} & -V_{11}^{(a)} V_{11}^{(b)} &V_{12}^{(a)} \\overline{V}_{12}^{(b)} & -V_{12}^{(a)} V_{11}^{(b)} \\\\\\overline{V}_{12}^{(a)} V_{11}^{(b)} & \\overline{V}_{12}^{(a)} V_{12}^{(b)} &-V_{11}^{(a)} V_{11}^{(b)} & -V_{11}^{(a)} V_{12}^{(b)} \\\\\\overline{V}_{12}^{(a)} \\overline{V}_{12}^{(b)} & -\\overline{V}_{12}^{(a)}V_{11}^{(b)} &-V_{11}^{(a)} \\overline{V}_{12}^{(b)} & V_{11}^{(a)} V_{11}^{(b)}\\end{matrix}} \\right)_{t=t_0}, $ where the time dependency of $V_{11}(t)$ and $V_{12}(t)$ are respectively given by (REF ) and ().",
"The time dependency in the second term on the right hand side was omitted, but we assume that $0<t_0<t$ and so, both $V_{11}(t)$ and $V_{12}(t)$ are calculated for $t_0$ , as is indicated by the subscript on the matrix on the right hand side of (REF ).",
"Up to first order in $\\kappa $ , the time evolution operator in (REF ) is constant in time.",
"Thus, the third term of the expression for the dynamical phase in (REF ), that involves the time derivative of $V^{(1)}(t)$ , is null.",
"We implemented in our code routines that calculate the phase factors for the interaction given by (REF ).",
"To investigate the relation between the phase factors and the constant $\\kappa $ , we considered a system composed of two commensurable subsystems with fixed $\\omega _a$ , $\\omega _b$ , $\\epsilon _a$ and $\\epsilon _b$ , a fixed $t_0$ that characterises the delta interaction and we varied $\\kappa $ from 0 to $0.2$ , with steps of $0.01$ .",
"Considering this set of parameters, the code calculates the phase factors for each of the computational basis states ($|00\\rangle $ , $|01\\rangle $ , $|10\\rangle $ and $|11\\rangle $ ).",
"Figure REF shows the results for the initial state $|00\\rangle $ and $\\omega _q=1.0$ , $\\omega _b=2.0$ , $\\epsilon _a=\\epsilon _b=0.01$ and $t_0=0.5$ .",
"The results are similar for others sets of parameters.",
"We note that since our approximation of the Dyson expansion (equation (REF )) is only up to first order, the dependency of the phase factors on $\\kappa $ is linear.",
"Figure: Plots of the phase factors for the initial state|00〉|00\\rangle as functions of the parameter κ\\kappa .",
"The thick linerepresents the value of the phase factors for a system withnon-interacting subsystems.",
"The dashed line represents theinteraction given by ().",
"We considered subsystems withω a =1.0\\omega _a=1.0, ω b =2.0\\omega _b=2.0, ϵ a =ϵ b =0.01\\epsilon _a=\\epsilon _b=0.01 andt 0 =0.5t_0=0.5.The parameter $\\kappa $ is not, as one could imagine, a parameter of the control space of the system.",
"It simply modulates the interaction between the subsystems and can be thought of as an structural constant.",
"Figure REF shows the dependency of the phase factors on the instant of time $t_0$ of the interaction for the initial state $|00\\rangle $ .",
"The presented relation between the phase factors and $t_0$ is similar for the others states of the computational basis and for different sets of parameters.",
"We note that there is a value of $t_0$ that maximises the absolute value of the geometric phase, but we cannot state that this is a global maximum.",
"Figure: Plots of the phase factors for the initial state|00〉|00\\rangle as functions of the instant of time of the interactiont 0 t_0.",
"The thick line represents the value of the phase factorsfor a system with non-interacting subsystems.",
"The dashed linerepresents the interaction given by ().",
"We consideredsubsystems with ω a =1.0\\omega _a=1.0, ω b =2.0\\omega _b=2.0,ϵ a =ϵ b =0.01\\epsilon _a=\\epsilon _b=0.01 and κ=0.1\\kappa =0.1.",
"The time is measuredin unites of 2π/ω2\\pi /\\omega ."
],
[
"Further results",
"Using the results obtained so far for two two-level quantum systems, we may investigate once again the appropriate instant of time to calculate the phase factors.",
"Following the same prerogative, that the instant to be considered corresponds to the time interval in which the system undergoes a cyclic evolution, we consider the probability of transition for the composite system: $P(t) = |\\langle \\psi (0),U(t)\\psi (0)\\rangle |^2.$ Figure REF shows $P(t)$ as a function of time.",
"We observe that the system returns to its initial state after a time $T_\\Omega \\cong 456 t_\\omega $ , where $t_\\omega =2\\pi /\\omega $ .",
"$T_\\Omega $ is also obtained through $T_\\Omega =2\\pi /\\Omega $ , where $\\Omega $ is the Rabi frequency and is calculated numerically.",
"We considered a system with $\\omega _a=1.0$ , $\\omega _b=2.0$ , $\\epsilon _a=\\epsilon _b=0.01$ .",
"The constants that determine the interaction are $\\kappa =0.1$ and $t_0=0.5=0.16\\,t_\\omega $ .",
"For this values, the correspondent Rabi frequency is $\\Omega =0.0022$ , resulting in $T_\\Omega \\cong 456\\,t_\\omega $ , as observed in Figure REF .",
"Figure: Probability of the system remaining in its initial state.Starting from the graph in the left column and first row, in clockwiseorder the graphs correspond to the initial states |00〉|00\\rangle , |01〉|01\\rangle ,|10〉|10\\rangle and |11〉|11\\rangle .",
"The full line corresponds to non-interactingsubsystems and the dashed line corresponds to an interaction of the form().",
"The time is measured in units of t ω =2π/ωt_\\omega =2\\pi /\\omega .The relevant constants of the systems are ω a =1.0\\omega _a=1.0, ω b =2.0\\omega _b=2.0,ϵ a =ϵ b =0.01\\epsilon _a=\\epsilon _b=0.01, κ=0.1\\kappa =0.1 and t 0 =.16t ω t_0=.16\\,t_\\omega .Once we determined the period that the system takes to return to its initial state ($T_\\Omega $ ), we can calculate the total phase factor of the composite system.",
"Table: Values of the total phase φ tot \\phi _{tot} for each stateof the computational basis, considering differentvalues of ω b \\omega _b and fixed ω a =1.0\\omega _a=1.0.",
"The subscriptφ tot (0) \\phi _{tot}^{(0)} and φ tot (δ) \\phi _{tot}^{(\\delta )} indicate systemswith no interaction and interaction given by a delta function,respectively.Table REF shows values of the total phase for a set of $\\omega _a$ and $\\omega _b$ values.",
"We note that when $\\omega _a=\\omega _b$ , we can write the following transformation: $B(\\phi ) = \\left( \\begin{array}{cccc} e^{i\\phi } & 0 &0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1& 0 \\\\ 0 & 0 & 0 & e^{-i\\phi } \\end{array} \\right),$ where $\\phi $ is the total phase associated with the basis state $|00\\rangle $ .",
"This transformation implements a conditional evolution of the basis states, we can say that (REF ) is a conditional phase gate in the sense that the state of one system influences the state of the other, although it does present the usual symmetric form of controlled phase shift gates.",
"This gate is not purely geometrical, since the total phase factor involves both the dynamical and geometric phases.",
"When $\\omega _a\\ne \\omega _b$ , the transformation on the basis state can no longer be represented by (REF ), as can be seen in Table REF ."
],
[
"Conclusions",
"The main contribution of this work is the implementation of the method developed in [3] and [4] to obtain phase factors for a two-level quantum system and two two-level quantum systems interacting and non-interacting.",
"Since this method presents a solution stable for long-time periods, the resulting phase factors also present this property.",
"The implementation of a quantum gate, when RWA is considered [10], [15] is valid for an adiabatic evolution and, in the context of two two-level systems interacting, only one is subjected to an external time-dependent field.",
"In our case, both systems are subjected to an external periodic field and neither the adiabatic approximation nor the rotating wave approximation are necessary.",
"Using the results for phase factors we were able to implement a controlled phase shift gate.",
"The resulting gate is not purely geometrical and removing the dynamical contribution to the overall phase is not a straightforward task.",
"One possibility is finding a Hamiltonian that cancels the dynamical phase of the system along a cyclic trajectory.",
"Nevertheless, our work can be extended in many ways.",
"For example, the time evolution operator obtained for a two-level quantum system could be used in the calculation of geometric phases in open quantum systems under the Quantum Jump Approach [9].",
"Or, for non-unitary evolutions in the context of interferometry, it is even possible to combine the method developed in [14] with our work to obtain a time evolution operator for a system subjected to a time-dependent perturbation and derivate the corresponding phase factors."
]
] | 1808.08324 |
[
[
"Highly localized kernels on the sphere induced by Newtonian kernels"
],
[
"Abstract The purpose of this article is to construct highly localized summability kernels on the unit sphere in ${\\mathbb R}^d$ that are restrictions to the sphere of linear combinations of a small number of shifts of the fundamental solution of the Laplace equation (Newtonian kernel) with poles outside the unit ball in ${\\mathbb R}^d$.",
"The same problem is also solved for the subspace ${\\mathbb R}^{d-1}$ in ${\\mathbb R}^d$."
],
[
"Introduction",
"The shifts of the fundamental solution of the Laplace equation $\\frac{1}{|x|^{d-2}}$ in dimensions $d>2$ or $\\ln \\frac{1}{|x|}$ if $d=2$ with $|x|$ being the Euclidean norm of $x\\in {\\mathbb {R}}^d$ are basic building blocks in Potential theory.",
"As is customary, we shall term the harmonic function $\\frac{1}{|x|^{d-2}}$ or $\\ln \\frac{1}{|x|}$ “Newtonian kernel”.",
"We are interested in the problem for approximation of harmonic functions on the unit ball $B^d$ in ${\\mathbb {R}}^d$ from finite linear combinations of shifts of the Newtonian kernel.",
"More explicitly, the problem is for a given harmonic function $U$ on $B^d$ and $n\\ge 1$ to find $n$ locations $\\lbrace y_j\\rbrace $ in ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and coefficients $\\lbrace c_j\\rbrace $ in ${\\mathbb {C}}$ such that $c_0+\\sum _{j=1}^n \\frac{c_j}{|x-y_j|^{d-2}}\\quad \\hbox{if} \\;\\; d>2\\quad \\hbox{or}\\quad c_0+\\sum _{j=1}^n c_j\\ln \\frac{1}{|x-y_j|}\\quad \\hbox{if} \\;\\; d=2$ approximates $U$ well (near best) in the harmonic Hardy space $\\mathcal {H}^p(B^d)$ , $0<p\\le \\infty $ .",
"This problem is also important in the case when $U$ is harmonic on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ and the poles $\\lbrace y_j\\rbrace $ are in $B^d$ or $U$ is harmonic on ${\\mathbb {R}}^d_+$ and the poles $\\lbrace y_j\\rbrace $ are in ${\\mathbb {R}}^d_-$ .",
"An alternative formulation of the problem is to approximate a given potential $U$ by the potential of $n$ point masses (using terminology from Geodesy) or by the potential of $n$ point charges (in terms of Electrostatics) or by the potential of $n$ magnetic poles (in Magnetism).",
"It should be pointed out that there is a great deal of work done on the Method of Fundamental Solutions for the Dirichlet problem of the Laplace equation in Numerical Analysis.",
"This theme is directly related to the problems we consider here.",
"We refer the reader to [2], [6], [8] for the basics of Potential theory.",
"The poor localization of the Newtonian kernel makes the above approximation problem unamenable and challenging.",
"An important step in solving this problem (see [7]) is to construct highly localized summability kernels on the unit sphere ${{\\mathbb {S}}^{d-1}}$ in ${\\mathbb {R}}^d$ that are restrictions to the sphere of linear combinations of finitely many (fixed number) shifts of the Newtonian kernel.",
"This is the main goal of this article.",
"The simple fact that $|x-a\\eta |^2=a^2+1-2a(x\\cdot \\eta ),\\quad x,\\eta \\in {{\\mathbb {S}}^{d-1}},$ implies that the restriction of any shift of the Newtonian kernel to ${{\\mathbb {S}}^{d-1}}$ is a zonal function, i.e.",
"it is the composition $F(x\\cdot \\eta )$ of an appropriate univariate function $F:[-1,1]\\rightarrow {\\mathbb {R}}$ and the dot product $x\\cdot \\eta $ , $x,\\eta \\in {{\\mathbb {S}}^{d-1}}$ .",
"This leads us to the following explicit formulation of the problem at hand: Problem 1.",
"Let $M>d-1$ .",
"For given ${\\varepsilon }\\in (0,1]$ find $2m+1$ constants $b_\\nu \\in {\\mathbb {R}}$ , $a_\\nu >1$ so that the restriction $F_{\\varepsilon }(x\\cdot \\eta )$ of the function $f_{{\\varepsilon },\\eta }(x) = b_0 + \\sum _{\\nu =1}^{m} \\frac{b_\\nu }{|x-a_\\nu \\eta |^{d-2}},\\quad \\eta \\in {{\\mathbb {S}}^{d-1}},\\; x\\in {\\mathbb {R}}^d\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu \\eta \\rbrace ,\\quad \\hbox{if $d>2$},$ or $f_{{\\varepsilon },\\eta }(x) = b_0 + \\sum _{\\nu =1}^{m} b_\\nu \\ln \\frac{1}{|x-a_\\nu \\eta |},\\quad \\eta \\in {\\mathbb {S}}^1,\\; x\\in {\\mathbb {R}}^2\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu \\eta \\rbrace ,\\quad \\hbox{if $d=2$,}$ to the unit sphere ${{\\mathbb {S}}^{d-1}}\\subset {\\mathbb {R}}^d$ satisfies the following conditions: $|F_{\\varepsilon }(x\\cdot \\eta )| \\le \\frac{c{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^M},\\quad \\forall x,\\eta \\in {{\\mathbb {S}}^{d-1}};$ $\\int _{{{\\mathbb {S}}^{d-1}}}F_{\\varepsilon }(x\\cdot \\eta )d\\sigma (x)=1, \\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}}$ with constants $m\\in {\\mathbb {N}}$ and $c>0$ depend only on $M$ and $d$ .",
"Here $\\rho (x, \\eta ):=\\arccos \\,(x\\cdot \\eta )$ is the geodesic distance between $x,\\eta \\in {{\\mathbb {S}}^{d-1}}$ and $\\sigma $ denotes the Lebesgue measure on ${{\\mathbb {S}}^{d-1}}$ .",
"It should be pointed out that the localization required in (REF )–(REF ) is only on the boundary ${{\\mathbb {S}}^{d-1}}$ of the unit ball.",
"As far as every such $f_{{\\varepsilon },\\eta }$ is a harmonic function on $B^d$ it cannot be well localized in the interior of the ball.",
"We shall present two solutions (even three in dimension $d=2$ ) of Problem 1.",
"To solve this problem it suffices to solve either of the following two problems: Problem 2.",
"Let $M>d-1$ .",
"For given ${\\varepsilon }\\in (0,1]$ find constants $a_j>1$ and $b_j, c_j\\in {\\mathbb {R}}$ so that the restriction $F_{\\varepsilon }(x\\cdot \\eta )$ of the function $f_{{\\varepsilon },\\eta }(x) = \\sum _{j=1}^{m} \\frac{b_j}{|x-a_j\\eta |^{d-2}}+ \\sum _{j=1}^{m} c_j(\\eta \\cdot \\nabla )\\Big (\\frac{1}{|x-a_j\\eta |^{d-2}}\\Big ),$ $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , $x\\in {\\mathbb {R}}^d\\setminus \\lbrace a_1\\eta , \\dots , a_m\\eta \\rbrace $ , if $d>2$ or $f_{{\\varepsilon },\\eta }(x) = b_0 + \\sum _{j=1}^{m} c_j(\\eta \\cdot \\nabla )\\ln \\frac{1}{|x-a_j\\eta |},\\;\\; \\eta \\in {\\mathbb {S}}^1,\\; x\\in {\\mathbb {R}}^d\\setminus \\lbrace a_1\\eta , \\dots , a_m\\eta \\rbrace ,\\;\\;\\hbox{if $d=2$,}$ to ${{\\mathbb {S}}^{d-1}}$ satisfies conditions (REF )–(REF ), where as above the constants $m\\in {\\mathbb {N}}$ and $c>0$ depend only on $M$ and $d$ .",
"Problem 3.",
"Let $M>d-1$ .",
"For given ${\\varepsilon }\\in (0,1]$ find $m+1$ constants $b_\\ell \\in {\\mathbb {R}}$ and $a>1$ so that the restriction $F_{\\varepsilon }(x\\cdot \\eta )$ of the function $f_{{\\varepsilon },\\eta }(x) = \\sum _{\\ell =0}^{m} b_\\ell (\\eta \\cdot \\nabla )^\\ell \\Big (\\frac{1}{|x-a\\eta |^{d-2}}\\Big ),\\quad \\eta \\in {{\\mathbb {S}}^{d-1}}, \\; x\\in {\\mathbb {R}}^d\\setminus \\lbrace a\\eta \\rbrace ,\\quad \\hbox{if $d>2$};$ or $f_{{\\varepsilon },\\eta }(x) = b_0 + \\sum _{\\ell =1}^{m} b_\\ell (\\eta \\cdot \\nabla )^\\ell \\ln \\frac{1}{|x-a\\eta |},\\quad \\eta \\in {\\mathbb {S}}^1,\\; x\\in {\\mathbb {R}}^2\\setminus \\lbrace a\\eta \\rbrace ,\\quad \\hbox{if $d=2$,}$ to ${{\\mathbb {S}}^{d-1}}$ satisfies conditions (REF )–(REF ), where as above the constants $m\\in {\\mathbb {N}}$ and $c>0$ depend only on $M$ and $d$ .",
"As is well known the $\\ell $ th directional derivative operator $(\\eta \\cdot \\nabla )^\\ell $ , where $\\nabla $ stands for the gradient operator, is approximated well by the finite difference operator ${\\mathfrak {D}}^\\ell _t(\\eta ):=t^{-\\ell }\\sum _{k=0}^\\ell (-1)^{\\ell -k} \\binom{\\ell }{k} T(\\eta ,kt)$ , where $T(\\eta ,t)f(x):=f(x+t\\eta )$ , $x\\in {\\mathbb {R}}^d$ .",
"More precisely, if $d>2$ , $\\ell \\ge 1$ , $a>1$ , and $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , then $\\big \\Vert (\\eta \\cdot \\nabla )^\\ell |x-a\\eta |^{2-d}- {\\mathfrak {D}}^\\ell _t(\\eta )|x-a\\eta |^{2-d}\\big \\Vert _{L^\\infty (\\overline{B^d})} \\rightarrow 0\\quad \\hbox{as}\\quad t \\rightarrow 0,$ and a similar claim is valid when $d=2$ .",
"Having in mind that ${\\mathfrak {D}}^\\ell _t(\\eta )|x-a\\eta |^{2-d}$ is a linear combination of Newtonian kernels with poles at $(a-kt)\\eta $ , $k=0,\\dots ,\\ell $ , we see that, a solution of Problem 2 or Problem 3 leads immediately to a solution of Problem 1.",
"It is easy to see that a properly dilated and normalized version of the Poisson kernel provides a solution of Problem 2 and Problem 3 in the case $M=d$ .",
"Indeed, the Poisson kernel for a ball of radius $a>1$ in ${\\mathbb {R}}^d$ takes the form $P(y,x)=\\frac{1}{a\\omega _d}\\frac{a^2-|x|^2}{|x-y|^d}, \\quad |y|=a, \\quad |x|<a,$ where $\\omega _d:=2\\pi ^{d/2}/\\Gamma (d/2)$ is the Lebesgue measure of ${{\\mathbb {S}}^{d-1}}$ .",
"Restricting $P(y,x)$ to ${{\\mathbb {S}}^{d-1}}$ as a function of $x$ and setting $y=a\\eta $ with $\\eta \\in {{\\mathbb {S}}^{d-1}}$ and $a:=1+{\\varepsilon }$ we get $P(a\\eta ,x)=\\frac{1}{a\\omega _d}\\frac{a^2-1}{|x-a\\eta |^d}$ .",
"A straightforward derivation shows that $(\\eta \\cdot \\nabla )|x-a\\eta |^{2-d} = (d-2)(2a)^{-1}|x-a\\eta |^{2-d}+2^{-1}(d-2)\\omega _d P(a\\eta ,x),\\;\\;\\hbox{if}\\; d>2.$ Hence, the kernel $F_{\\varepsilon }(x\\cdot \\eta ):=P(a\\eta ,x)$ is of the forms (REF ) and (REF ) with $m=1$ .",
"It is also easy to see that in dimension $d=2$ $(\\eta \\cdot \\nabla )\\ln \\frac{1}{|x-a\\eta |} = \\frac{1}{2a}+\\pi P(a\\eta ,x).$ and hence the kernel $F_{\\varepsilon }(x\\cdot \\eta ):=P(a\\eta ,x)$ is of the forms (REF ) and (REF ) with $m=1$ .",
"Furthermore, it is easy to show that (see (REF )) $5^{-1}({\\varepsilon }+\\rho (x, \\eta )) \\le |x-a\\eta | \\le 2({\\varepsilon }+\\rho (x, \\eta )),\\;\\;x\\in {{\\mathbb {S}}^{d-1}},\\quad \\hbox{if $\\;0<{\\varepsilon }\\le 1$.",
"}$ Therefore, $0< F_{\\varepsilon }(x\\cdot \\eta )\\le c{\\varepsilon }^{-d+1}(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^{-d}$ and hence $F_{\\varepsilon }(x\\cdot \\eta ):=P(a\\eta ,x)$ solves Problem 2 and Problem 3 for $M=d$ .",
"To solve Problem 2 or Problem 3 for an arbitrary $M>d$ is not so easy.",
"When trying to solve Problem 3 in the general case the first question that occurs is whether the $m$ th directional derivative $(\\eta \\cdot \\nabla )^m|x-a\\eta |^{2-d}$ if $d>2$ or $(\\eta \\cdot \\nabla )^m\\ln 1/|x-a\\eta |$ if $d=2$ for sufficiently large $m$ , depending on $M$ , can solve the problem.",
"The well known Maxwell formula (see e.g.",
"[1]) asserts that if $d\\ge 1$ , $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , $\\lambda >0$ , $m\\in {\\mathbb {N}}$ , then $\\left(\\eta \\cdot \\nabla \\right)^m \\frac{1}{|x|^\\lambda }= (-1)^m m!C_m^{(\\lambda /2)}\\left(\\frac{x\\cdot \\eta }{|x|}\\right) \\frac{1}{|x|^{\\lambda +m}},\\quad x\\in {\\mathbb {R}}^d\\backslash \\lbrace 0\\rbrace ,$ where $C_m^{(\\mu )}$ is the $m$ th degree ultraspherical polynomial normalized by the identity $C_m^{(\\mu )}(1)=\\binom{m+2\\mu -1}{m}$ .",
"Now, using that $\\lim _{\\mu \\rightarrow 0+}\\mu ^{-1}(|t|^{-\\mu }-1)=\\ln \\frac{1}{|t|}$ and $\\lim _{\\mu \\rightarrow 0+}\\mu ^{-1}C_m^{(\\mu )}(t)=2m^{-1}T_m(t)$ one obtains by letting $\\mu \\rightarrow 0$ in (REF ) $\\left(\\eta \\cdot \\nabla \\right)^m \\ln \\frac{1}{|x|}=(-1)^{m}(m-1)!",
"T_m\\left(\\frac{x\\cdot \\eta }{|x|}\\right) \\frac{1}{|x|^m},\\quad x\\in {\\mathbb {R}}^2\\backslash \\lbrace 0\\rbrace ,$ where $T_m$ is the $m$ th degree Chebyshev polynomial of the first kind normalized by $T_m(1)=1$ .",
"Let $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , $a:=1+{\\varepsilon }$ , ${\\varepsilon }>0$ , and $m\\in {\\mathbb {N}}$ .",
"Then (REF )-(REF ) yield $\\left(\\eta \\cdot \\nabla \\right)^m \\frac{1}{|x-a\\eta |^{d-2}}&= (-1)^m m!C_{m}^{(d/2-1)}\\left(\\frac{(x-a\\eta )\\cdot \\eta }{|x-a\\eta |}\\right)\\frac{1}{|x-a\\eta |^{d-2+m}},\\;\\; d>2,\\\\\\left(\\eta \\cdot \\nabla \\right)^m \\ln \\frac{1}{|x-a\\eta |}&= (-1)^{m}(m-1)!T_{m}\\left(\\frac{(x-a\\eta )\\cdot \\eta }{|x-a\\eta |}\\right)\\frac{1}{|x-a\\eta |^m}, \\;\\;d=2.$ Now, using (REF ) and (REF ) we obtain the sharp estimate $\\left|\\left(\\eta \\cdot \\nabla \\right)^m \\frac{1}{|x-a\\eta |^{d-2}}\\right|\\le c(m, d)\\frac{{\\varepsilon }^{-m+1}{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^{m+d-2}},\\quad x\\in {{\\mathbb {S}}^{d-1}}.$ On the other hand, since $\\left(\\eta \\cdot \\nabla \\right)^m |x-a\\eta |^{2-d}$ is a harmonic function we have $\\int _{{\\mathbb {S}}^{d-1}}\\left(\\eta \\cdot \\nabla \\right)^m \\frac{1}{|x-a\\eta |^{d-2}} d\\sigma (x)&= \\omega _d\\left(\\eta \\cdot \\nabla \\right)^m \\frac{1}{|x-a\\eta |^{d-2}}\\Big |_{x=0}\\\\= \\omega _d m!C_{m}^{(d/2-1)}(1)a^{-d+2-m}&= \\omega _d m!\\binom{m+d-3}{m}a^{-d+2-m}=\\frac{c(m, d)}{(1+{\\varepsilon })^{m+d-2}}.$ Therefore, if we set $F_{\\varepsilon }(x\\cdot \\eta ):=c^*\\left(\\eta \\cdot \\nabla \\right)^m |x-a\\eta |^{2-d}$ with a normalization constant $c^*$ so that $F_{\\varepsilon }(x\\cdot \\eta )$ obeys (REF ) then in light of the additional multiplier ${\\varepsilon }^{-m+1}$ in (REF ) $|F_{\\varepsilon }(x\\cdot \\eta )|$ with $m\\ge 2$ cannot have the decay from (REF ) for any $M>d-1$ .",
"The same argument applies if $d=2$ .",
"The conclusion is that Problem 2 cannot be solved by using a single $m$ th directional derivative of the Newtonian kernel.",
"In this article we present two main results.",
"First, modifying Lemma 2.5 in L. Colzani [3] we show that the function $F_{\\varepsilon }(x\\cdot \\eta ) = \\sum _{j=1}^m (-1)^{j+1} \\binom{m}{j} (1+j{\\varepsilon })^{d-1}P((1+j{\\varepsilon })\\eta , x),$ where $P$ is the Poisson kernel (REF ) and $m \\ge M-d$ , solves Problem 2.",
"Secondly, we show that Problem 3 is solved by the simpler kernel $F_{\\varepsilon }(x\\cdot \\eta ) := \\frac{c^\\star {\\varepsilon }^{2m-1}}{|x-a\\eta |^{2m+d-2}}, \\;\\;x\\in {{\\mathbb {S}}^{d-1}},\\quad \\hbox{with}\\quad \\int _{{\\mathbb {S}}^{d-1}}F_{\\varepsilon }(x\\cdot \\eta )d\\sigma (x)=1,$ where $m\\ge (M-d+2)/2$ , $a=1+{\\varepsilon }$ , and $c^\\star >0$ is a normalization constant.",
"While the proof of the first result is straightforward, the proof of the second (more surprising) result is quite involved and this is the main novelty in this paper.",
"Our solution of Problem 3 (and hence of Problem 1) has an obvious advantage over Colzani's solution of Problem 2 - it is amenable to generalizations.",
"Our scheme can be used for the solution of the analog of Problem 3 and consequently Problem 1 for domains with much more complicated geometry than the ball, while Colzani's solution of Problem 2 relying on the Poisson kernel is limited to domains for which the Poisson kernel is available in a convenient concrete form.",
"The rest of this article is organized as follows.",
"In Section we presents a solution of Problem 2 based on an idea of L. Colzani from [3].",
"In Section we present the solution of Problem 3 mentioned above.",
"In Section we present a second solution of Problem 3 in dimension $d=2$ .",
"Section treats in brief the localization on ${{\\mathbb {S}}^{d-1}}$ of harmonic functions on ${\\mathbb {R}}^d\\setminus \\overline{B^d}$ .",
"As a natural progression of our development, in Section we also solve the analogues of Problems 2 and 3 and as consequence the analogue of Problem 1 with ${{\\mathbb {S}}^{d-1}}$ replaced by ${\\mathbb {R}}^{d-1}$ ."
],
[
"Localized kernels on ${{\\mathbb {S}}^{d-1}}$ in terms of Newtonian kernels: Solution of Problem 2",
"In this section we present a solution of Problem 2 from § based on the idea from [3].",
"Theorem 2.1 Let $m\\in {\\mathbb {N}}$ , $d\\ge 2$ , $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , and $0<{\\varepsilon }\\le 1$ .",
"Consider the function $f_{{\\varepsilon }, \\eta }(x) := \\sum _{j=1}^m (-1)^{j+1} \\binom{m}{j} (1+j{\\varepsilon })^{d-1}P((1+j{\\varepsilon })\\eta , x),\\quad x\\in {\\mathbb {R}}^d\\setminus \\cup _{j=1}^m\\lbrace (1+j{\\varepsilon })\\eta \\rbrace ,$ where $P$ is the Poisson kernel (REF ).",
"Then the restriction $F_{{\\varepsilon }}(x\\cdot \\eta )$ of the function $f_{{\\varepsilon }, \\eta }(x)$ on ${{\\mathbb {S}}^{d-1}}$ has these properties: $|F_{{\\varepsilon }}(x\\cdot \\eta )| \\le \\frac{c{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^{m+d-1}},\\quad \\forall x, \\eta \\in {{\\mathbb {S}}^{d-1}},$ and $\\int _{{{\\mathbb {S}}^{d-1}}} F_{{\\varepsilon }}(x\\cdot \\eta )d\\sigma (x) =1,\\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}},$ where $c>0$ is a constant depending only on $m$ and $d$ .",
"Furthermore, $f_{{\\varepsilon }, \\eta }(x)$ can be represented in the form $f_{{\\varepsilon },\\eta }(x) = \\sum _{j=1}^{m} \\frac{b_j}{|x-a_j\\eta |^{d-2}}+ \\sum _{j=1}^{m} c_j(\\eta \\cdot \\nabla )\\Big (\\frac{1}{|x-a_j\\eta |^{d-2}}\\Big ),\\quad \\hbox{if}\\quad d>2,$ or $f_{{\\varepsilon },\\eta }(x) = b_0 + \\sum _{j=1}^{m} c_j(\\eta \\cdot \\nabla )\\ln \\frac{1}{|x-a_j\\eta |},\\quad \\hbox{if} \\quad d=2,$ where $a_j:=1+j{\\varepsilon }$ .",
"From the definition of $f_{{\\varepsilon },\\eta }(x)$ and (REF )–(REF ) it readily follows $f_{{\\varepsilon },\\eta }(x)$ can be represented in the form (REF ) or (REF ).",
"From the harmonicity of the Poisson kernel we know that $\\int _{{\\mathbb {S}}^{d-1}}P(a\\eta , x)d\\sigma (x)=\\omega _d P(a\\eta , 0)=a^{-d+1}$ , $a>1$ , implying $\\int _{{\\mathbb {S}}^{d-1}}f_{{\\varepsilon }, \\eta }(x)d\\sigma (x)= \\sum _{j=1}^m (-1)^{j+1} \\binom{m}{j} =1,$ which confirms (REF ).",
"To prove (REF ) we first observe that for $x, \\eta \\in {{\\mathbb {S}}^{d-1}}$ and $a>1$ (see (REF )) $|x-a\\eta |^2= (1-a)^2+a\\sin ^2(\\beta /2)\\quad \\hbox{with}\\;\\; \\beta :=\\rho (x,\\eta ),$ and hence, using (REF ), $P(a\\eta , x)=\\frac{1}{a\\omega _d}\\frac{a^2-1}{[(a-1)^2+a\\sin ^2(\\beta /2)]^{d/2}}.$ If $\\rho (x, \\eta )\\le {\\varepsilon }$ , then from above it readily follows that $|P((1+j{\\varepsilon })\\eta , x)|\\le c{\\varepsilon }^{-d+1}$ .",
"This and the definition of $f_{{\\varepsilon }, \\eta }(x)$ yield (REF ).",
"Let $\\rho (x, \\eta )> {\\varepsilon }$ .",
"Clearly, $P(\\eta , x)=0$ since $x, \\eta \\in {{\\mathbb {S}}^{d-1}}$ , $x\\ne \\eta $ .",
"Hence, $f_{{\\varepsilon }, \\eta }(x) = \\sum _{j=0}^m (-1)^{j+1} \\binom{m}{j} (1+j{\\varepsilon })^{d-1}P((1+j{\\varepsilon })\\eta , x).$ Denote $g(u):= (1+u)^{d-1}P((1+u)\\eta , x)$ with $x, \\eta \\in {{\\mathbb {S}}^{d-1}}$ fixed.",
"Then $f_{{\\varepsilon }, \\eta }(x)= (-1)^{m+1}\\Delta _{\\varepsilon }^m g(0)= (-1)^{m+1}\\int _0^{\\varepsilon }\\cdots \\int _0^{\\varepsilon }g^{(m)}(u_1+\\dots +u_m) du_1\\dots du_m.$ We claim that $|g^{(m)}(u)| \\le \\frac{c}{|x-(1+u)\\eta |^{m+d-1}},\\quad 0<u<m,$ where $c$ is a constant depending only on $m$ and $d$ .",
"Indeed, from (REF ) $g(u) &= \\frac{(2+u)(1+u)^{d-2}}{\\omega _d}\\frac{u}{(u^2+(1+u)\\sin ^2(\\beta /2))^{d/2}}\\\\&=:\\phi (u)u(u^2+(1+u)\\sin ^2(\\beta /2))^{-d/2}.$ Using this representation of $g(u)$ it easily follows that (REF ) holds.",
"Finally, (REF ) coupled with (REF ) yields (REF )."
],
[
"Localized kernels on ${{\\mathbb {S}}^{d-1}}$ in terms of Newtonian kernels: Solution of Problem 3",
"The solution of Problem 3 from the introduction is essentially contained in the following Theorem 3.1 Let $m\\in {\\mathbb {N}}$ , $d\\ge 2$ , $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , and $0<{\\varepsilon }\\le 1$ .",
"Set $a:=1+{\\varepsilon }$ and $\\delta :=1-a^{-2}$ .",
"Consider the function $\\mathcal {F}_{{\\varepsilon }}(t) := \\frac{(d/2)_{m-1}}{2m!",
"}a^{2m}\\delta ^{2m-1} (a^2+1-2at)^{-d/2+1-m},\\quad t\\in [-1,1].$ The function $\\mathcal {F}_{{\\varepsilon }}$ has these properties: $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta ) = \\frac{(d/2)_{m-1}}{2m!",
"}a^{2m}\\delta ^{2m-1} |x-a\\eta |^{-d+2-2m},\\quad x\\in {{\\mathbb {S}}^{d-1}},$ $0< \\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta ) \\le \\frac{c_1{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}\\rho (x, \\eta ))^{2m+d-2}},\\quad \\forall x, \\eta \\in {{\\mathbb {S}}^{d-1}},$ and $\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )d\\sigma (x) \\ge c_2>0,\\quad \\forall \\eta \\in {{\\mathbb {S}}^{d-1}},$ where $c_1, c_2>0$ are constants depending only on $m$ and $d$ .",
"Furthermore, $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )$ is the restriction on ${{\\mathbb {S}}^{d-1}}$ of the harmonic function, defined on ${\\mathbb {R}}^d\\setminus \\lbrace a\\eta \\rbrace $ , $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x):=q_0|x-a\\eta |^{2-d}+\\sum _{\\ell =1}^m\\frac{q_{\\ell }\\delta ^{\\ell -1} a^\\ell }{\\ell !",
"(d-2)} (\\eta \\cdot \\nabla )^\\ell |x-a\\eta |^{2-d}\\quad \\mbox{if}~d\\ge 3,$ or $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x):=q_0+\\sum _{\\ell =1}^m\\frac{q_{\\ell }\\delta ^{\\ell -1} a^\\ell }{\\ell !}",
"(\\eta \\cdot \\nabla )^\\ell \\ln \\frac{1}{|x-a\\eta |} \\quad \\mbox{if}~d=2,$ where the coefficients $q_0,\\dots , q_m$ are determined as the solution of the linear system of $m+1$ equations: $q_0+\\sum _{\\ell =1}^m (d/2)_{\\ell -1}\\frac{\\delta ^{\\ell -1}}{2\\ell !}",
"q_{\\ell }&=0,\\nonumber \\\\\\sum _{\\ell =0}^{m-\\nu }\\left[\\sum _{k=(\\ell -\\nu )_+}^{\\ell }(-1)^{\\ell -k} \\binom{\\nu }{\\ell -k}(d/2+\\nu -1)_k \\frac{\\delta ^k}{k!",
"}\\right] q_{\\nu +\\ell } &=0,\\\\\\nu &=1,\\dots ,m-1,\\nonumber \\\\q_m&=1.\\nonumber $ Here $(u)_0:=1$ , $(u)_k:=u(u+1)\\cdots (u+k-1)$ denotes the Pochhammer's symbol and $(u)_+:=\\max \\lbrace 0,u\\rbrace $ .",
"Remark 3.2 In fact, the function $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)$ from (REF ) or (REF ) for $x\\in B^d$ is the harmonic extension of $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )$ to $B^d$ .",
"Note also that unlike (REF ) identity (REF ) contains the constant term $q_0$ instead of a Newtonian kernel term like $q_0\\ln \\frac{1}{|x-a\\eta |}$ .",
"Theorem REF immediately implies Corollary 3.3 Let $d \\ge 2$ , $M>d-1$ .",
"Under the hypotheses of Theorem REF define $f_{{\\varepsilon },\\eta }(x):=\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)\\Big (\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,y) d\\sigma (y)\\Big )^{-1},\\quad x\\in {\\mathbb {R}}^d\\setminus \\lbrace a\\eta \\rbrace ,$ where $\\mathcal {F}_{{\\varepsilon },m}$ is from (REF ) or (REF ) and $m=\\left\\lceil (M-d+2)/2\\right\\rceil $ .",
"Then the function $f_{{\\varepsilon },\\eta }$ solves Problem 3 from the introduction.",
"We shall carry out the proof of Theorem REF in three steps."
],
[
"Proof of (",
"Representation (REF ) is immediate from the definition of $\\mathcal {F}_{{\\varepsilon }}$ and (REF ).",
"We claim that $5^{-1}({\\varepsilon }+\\rho (x, \\eta )) \\le |x-a\\eta | \\le 2({\\varepsilon }+\\rho (x, \\eta )), \\quad x, \\eta \\in {{\\mathbb {S}}^{d-1}}.$ Indeed, let $x, \\eta \\in {{\\mathbb {S}}^{d-1}}$ and denote by $\\beta $ ($0\\le \\beta \\le \\pi $ ) the angle between $x$ and $\\eta $ .",
"Using $\\eta \\cdot x=\\cos \\rho (x,\\eta )=\\cos \\beta $ in (REF ) we get $|x-a\\eta |^2 = \\sin ^2\\beta + (a-\\cos \\beta )^2= \\sin ^2\\beta + ({\\varepsilon }+2\\sin ^2(\\beta /2))^2.$ Assume $0\\le \\beta \\le \\pi /2$ .",
"Using the obvious inequalities $(2/\\pi )\\beta \\le \\sin \\beta \\le \\beta $ we obtain $(2/\\pi )^2\\beta ^2+{\\varepsilon }^2\\le |x-a\\eta |^2 \\le \\beta ^2+({\\varepsilon }+\\beta ^2/2)^2$ , which implies (REF ).",
"In the case $\\pi /2< \\beta \\le \\pi $ inequalities (REF ) are trivial.",
"Now estimate (REF ) readily follows by (REF ) and (REF ).",
"Also, from (REF ) we derive $\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )\\,d\\sigma (x)=\\omega _{d-1}\\int _{-1}^1 \\mathcal {F}_{{\\varepsilon }}(u) (1-u^2)^{(d-3)/2}du\\\\=\\frac{(d/2)_{m-1}}{2m!",
"}a^{2m}\\delta ^{2m-1}\\omega _{d-1}\\int _{-1}^1 (a^2+1-2au)^{(-2m-d+2)/2} (1-u^2)^{(d-3)/2}du.$ Restricting the interval of integration to $[1-{\\varepsilon }^2,1]$ and using that $a^2+1-2au\\le 5{\\varepsilon }^2$ for $u$ in this range we get $\\int _{{{\\mathbb {S}}^{d-1}}}\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )\\,d\\sigma (x)&\\ge c{\\varepsilon }^{2m-1} \\int _{1-{\\varepsilon }^2}^1 {\\varepsilon }^{-2m-d+2} (1-u)^{(d-3)/2}du\\\\&\\ge c{\\varepsilon }^{-d+1}{\\varepsilon }^{2((d-3)/2+1)}=c^{\\prime }$ with $c^{\\prime }>0$ depending only on $d$ and $m$ .",
"This proves (REF )."
],
[
"Solution of linear system (",
"Clearly, system (REF ) has an upper triangular matrix with 1's on the main diagonal.",
"Hence $q_0,\\dots , q_m$ are uniquely determined by (REF ).",
"Also from (REF ) we get by induction on $\\nu =m-1,m-2,\\dots ,0$ that the $q_\\ell $ 's satisfy $q_{\\ell }=q_{\\ell }(d,m,\\delta )=\\sum _{k=0}^{m-\\ell }\\alpha _{\\ell ,k}\\delta ^k,\\quad \\ell =0,1,\\dots ,m,$ with some coefficients $\\alpha _{\\ell ,k}=\\alpha _{\\ell ,k}(d,m)$ depending only on $d$ and $m$ , where $q_m=\\alpha _{m,0}=1$ .",
"Moreover, $\\alpha _{\\ell ,k}(d,m)$ is a polynomial of $d$ of degree $k$ and, hence, $\\alpha _{\\ell ,0}$ does not depend on $d$ .",
"Observe also that $\\alpha _{0,m}=0$ , i.e.",
"$q_0$ is a polynomial of degree $m-1$ .",
"The $q_\\ell $ 's for $m=1,2,3,4$ are given in Remark REF .",
"Lemma 3.4 For $m\\in {\\mathbb {N}}$ the numbers $\\alpha _{\\ell }(m):=\\alpha _{\\ell ,0}(d,m)$ , $\\ell =1,\\dots ,m$ , satisfy $\\alpha _\\ell (m)=\\frac{\\ell (2m-\\ell -1)!}{m!",
"(m-\\ell )!",
"},\\quad \\ell =1,2,\\dots ,m,$ and $\\alpha _0(m)=-\\alpha _1(m)/2$ .",
"The numbers $\\alpha _\\ell (m)$ satisfy the limit case of (REF ) when $\\delta =0$ , i.e.",
"$\\alpha _0(m)+\\alpha _1(m)/2=0$ and $\\sum _{\\ell =0}^{\\min \\lbrace \\nu ,m-\\nu \\rbrace } (-1)^\\ell \\binom{\\nu }{\\ell } \\alpha _{\\nu +\\ell }(m)=0,\\quad \\nu =1,\\dots ,m-1;\\qquad \\alpha _m(m)=1.$ Note that (REF ) has coefficients independent of $d$ , which also justifies that $\\alpha _\\ell (m)$ does not depends on $d$ .",
"In order to remove the dependence of the upper bound of the sum in (REF ) on $m-\\nu $ we set $\\alpha _\\ell (m):=0$ for $\\ell >m$ .",
"Then (REF ) becomes ${\\mathfrak {D}}^\\nu \\alpha _\\nu (m)=(-1)^\\nu \\delta _{\\nu ,m},\\quad \\nu =1,2,\\dots ,m,$ where $\\delta _{\\nu ,m}$ is the Kroneker $\\delta $ and ${\\mathfrak {D}}^\\nu $ denotes the $\\nu $ th forward finite difference operator, i.e.",
"${\\mathfrak {D}}^\\nu z_j:=\\sum _{k=0}^\\nu (-1)^{\\nu +k}\\binom{\\nu }{k}z_{j+k}$ .",
"We shall show that the solutions $\\alpha _\\nu (m)$ of (REF ) for all $m\\in {\\mathbb {N}}$ are uniquely determined by the following recursive procedure: $\\alpha _k(m):=\\delta _{k,m},\\quad k\\ge m,\\quad m\\in {\\mathbb {N}};$ $\\alpha _k(m):=\\alpha _{k+1}(m)+\\alpha _{k-1}(m-1), \\quad k=m-1,m-2,\\dots ,2,\\quad m\\ge 3;$ $\\alpha _1(m):=\\alpha _2(m),\\quad m\\ge 2,$ where (REF ) is applied inductively on $m$ and for given $m$ inductively on $k$ .",
"In order to establish this we prove by induction on $m\\in {\\mathbb {N}}$ that $\\alpha _k(m)$ , $k\\in {\\mathbb {N}}$ , from (REF )–(REF ) satisfy (REF ).",
"Observe that (REF ) trivially follows from (REF ) for $\\nu =1$ , $m\\ge 2$ , and from (REF ) for $\\nu =m$ , $m\\ge 1$ .",
"Hence (REF ) is true for $m=1$ and $m=2$ .",
"For $m\\ge 3$ assume (REF ) is true for for $m-1$ .",
"Using (REF ) we get for $\\nu =2,\\dots ,m-1$ ${\\mathfrak {D}}^\\nu \\alpha _\\nu (m)={\\mathfrak {D}}^{\\nu -1}\\alpha _{\\nu +1}(m)-{\\mathfrak {D}}^{\\nu -1}\\alpha _\\nu (m)=-{\\mathfrak {D}}^{\\nu -1}(\\alpha _\\nu (m)-\\alpha _{\\nu +1}(m))\\\\=-{\\mathfrak {D}}^{\\nu -1}\\alpha _{\\nu -1}(m-1)=0.$ This verifies (REF ) by induction.",
"Now, one establishes directly that the non-zero entries in (REF )–(REF ) are given by (REF ) and hence (REF ) solves (REF ).",
"This completes the proof of Lemma REF .",
"Remark 3.5 The numbers $\\alpha _\\ell (m)$ from (REF ) are known as ballot numbers, see [5].",
"The numbers $C_n=\\alpha _1(n+1)$ , $n=0,1,\\dots $ , are known as Catalan numbers, see [5].",
"Several values of $\\alpha _\\ell (m)$ are given in the following table.",
"Table: α ℓ (m)\\alpha _\\ell (m) for 1≤ℓ,m≤101\\le \\ell , m \\le 10."
],
[
"Completion of the proof of Theorem ",
"Using the fact that $C_\\ell ^{(\\mu )}$ and $T_\\ell $ are even functions for even $\\ell $ and odd functions for odd $\\ell $ we rewrite the derivatives of the Newtonian kernel (REF )–() as $\\left(\\eta \\cdot \\nabla \\right)^\\ell |x-a\\eta |^{2-d}&= \\ell !C_{\\ell }^{(d/2-1)}\\left(\\frac{(a\\eta -x)\\cdot \\eta }{|a\\eta -x|}\\right)|a\\eta -x|^{-d+2-\\ell },\\\\\\left(\\eta \\cdot \\nabla \\right)^\\ell \\ln 1/|x-a\\eta |&= (\\ell -1)!T_{\\ell }\\left(\\frac{(a\\eta -x)\\cdot \\eta }{|a\\eta -x|}\\right)|a\\eta -x|^{-\\ell }.$ By [9] for $\\ell \\ge 1$ we have $C_{\\ell }^{(d/2-1)}(t)&=(d/2-1)\\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor }\\frac{(-1)^s(d/2)_{\\ell -s-1}}{s!",
"(\\ell -2s)!",
"}(2t)^{\\ell -2s},\\\\T_{\\ell }(t)&=\\frac{\\ell }{2}\\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor } \\frac{(-1)^s(\\ell -s-1)!}{s!",
"(\\ell -2s)!",
"}(2t)^{\\ell -2s}.$ Now, by (REF ) and (REF ) substituted in the right-hand side of (REF ) or by () and () substituted in the right-hand side of (REF ) we get for $d\\ge 2$ $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)=q_0|a\\eta -x|^{2-d}\\\\+\\sum _{\\ell =1}^mq_{\\ell }\\delta ^{\\ell -1} a^\\ell \\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor }\\frac{(-1)^s(d/2)_{\\ell -s-1}}{2s!",
"(\\ell -2s)!",
"}\\left(2\\frac{(a\\eta -x)\\cdot \\eta }{|a\\eta -x|}\\right)^{\\ell -2s}|a\\eta -x|^{-d+2-\\ell }.$ To find a convenient representation of the values of $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)$ for $|x|=1$ we denote by $\\theta $ the angle between the vectors $a\\eta -x$ and $\\eta $ , $|a\\eta -x|\\cos \\theta =(a\\eta -x)\\cdot \\eta $ .",
"By the Law of Cosines we have $|a\\eta -x|^2+a^2-2|a\\eta -x|a\\cos \\theta =|x|^2,$ which, with the notation $r:=|a\\eta -x|/a,\\quad |x|=1,$ can be written as (recall $\\delta =(a^2-1)/a^2$ ) $2\\cos \\theta =r+\\delta r^{-1},\\quad |x|=1.$ Using (REF ), (REF ) and (REF ) in (REF ) we get $\\left.|a\\eta -x|^{d-2}\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)\\right|_{|x|=1}=q_0+\\sum _{\\ell =1}^m q_{\\ell }\\delta ^{\\ell -1} \\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor }b_{\\ell ,s}(r+\\delta r^{-1})^{\\ell -2s} r^{-\\ell }\\\\=q_0+\\sum _{\\ell =1}^m q_{\\ell }\\delta ^{\\ell -1} \\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor }b_{\\ell ,s}(1+\\delta r^{-2})^{\\ell -2s} r^{-2s}=:A,$ where $b_{\\ell ,s}=b_{\\ell ,s}(d):=\\frac{(-1)^s(d/2)_{\\ell -s-1}}{2s!",
"(\\ell -2s)!",
"}.$ We rewrite (REF ) as follows.",
"In the expression $A=q_0+\\sum _{\\ell =1}^m \\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor } \\sum _{k=0}^{\\ell -2s}q_{\\ell } b_{\\ell ,s} \\binom{\\ell -2s}{k} \\delta ^{\\ell +k-1} r^{-2s-2k}$ we set $k=\\nu -s$ and get $A=q_0+\\sum _{\\ell =1}^m \\sum _{s=0}^{\\left\\lfloor \\ell /2\\right\\rfloor } \\sum _{\\nu =s}^{\\ell -s}q_{\\ell } b_{\\ell ,s} \\binom{\\ell -2s}{\\nu -s} \\delta ^{\\ell +\\nu -s-1} r^{-2\\nu }\\\\=q_0+\\sum _{\\ell =1}^m \\sum _{\\nu =0}^{\\ell } \\sum _{s=0}^{\\min \\lbrace \\nu ,\\ell -\\nu \\rbrace }q_{\\ell } b_{\\ell ,s} \\binom{\\ell -2s}{\\nu -s} \\delta ^{\\ell +\\nu -s-1} r^{-2\\nu }.$ Separating the terms for $\\nu =0$ (which implies $s=0$ ) and shifting the order of summation in $\\ell $ and $\\nu $ in the triple sum above we get $A=q_0+\\sum _{\\ell =1}^m q_{\\ell } b_{\\ell ,0} \\delta ^{\\ell -1}+\\sum _{\\nu =1}^{m} \\sum _{\\ell =\\nu }^m \\sum _{s=0}^{\\min \\lbrace \\nu ,\\ell -\\nu \\rbrace } \\!\\!q_{\\ell } b_{\\ell ,s} \\binom{\\ell -2s}{\\nu -s} \\delta ^{\\ell +\\nu -s-1} r^{-2\\nu }.$ In the triple sum we set $s=\\ell -\\nu -k$ with $(\\ell -2\\nu )_+\\le k \\le \\ell -\\nu $ and get $A=q_0+\\sum _{\\ell =1}^m q_{\\ell } b_{\\ell ,0} \\delta ^{\\ell -1}\\\\+\\sum _{\\nu =1}^{m} \\sum _{\\ell =\\nu }^m q_{\\ell } \\left[\\sum _{k=(\\ell -2\\nu )_+}^{\\ell -\\nu }b_{\\ell ,\\ell -\\nu -k} \\binom{2\\nu -\\ell +2k}{k} \\delta ^k\\right]\\delta ^{2\\nu -1} r^{-2\\nu }\\\\=q_m b_{m,0}\\delta ^{2m-1}r^{-2m},$ where we used (REF ) for the last equality.",
"Indeed, if the summation index in the $\\nu +1$ -st row of (REF ) is changed from $\\ell $ to $\\ell -\\nu $ and this equation is multiplied by $b_{\\nu ,0}$ then $\\sum _{\\ell =\\nu }^m q_{\\ell } \\left[\\sum _{k=(\\ell -2\\nu )_+}^{\\ell -\\nu }b_{\\ell ,\\ell -\\nu -k} \\binom{2\\nu -\\ell +2k}{k} \\delta ^k\\right]=0.$ Now, (REF ), $q_m=1$ , (REF ) and (REF ) yield $\\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )=\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)$ for $x\\in {{\\mathbb {S}}^{d-1}}$ , which completes the proof of Theorem REF .",
"$\\Box $ The asymptotic of $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)$ as ${\\varepsilon }\\rightarrow 0$ (and of $\\int _{{{\\mathbb {S}}^{d-1}}} \\mathcal {F}_{{\\varepsilon }}(x\\cdot \\eta )\\,d\\sigma (x)$ as well) is given by Proposition 3.6 Under the assumptions of Theorem REF we have $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)=a^{2-d}\\left(q_0+\\sum _{\\ell =1}^m\\frac{(\\ell +d-3)!",
"}{\\ell !",
"(d-2)!",
"}q_{\\ell }\\delta ^{\\ell -1}\\right)\\\\=\\frac{1}{2}\\alpha _{1,0}(m)+ O({\\varepsilon })=\\frac{1}{2}\\frac{(2m-2)!}{m!(m-1)!",
"}+ O({\\varepsilon }).$ In order to evaluate $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)$ in the case $d\\ge 3$ we substitute (REF ) in (REF ) and use that $C_{\\ell }^{(d/2-1)}(1)=\\binom{\\ell +d-3}{\\ell }$ to get $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)=a^{2-d}\\left(q_0+\\sum _{\\ell =1}^m\\frac{(\\ell +d-3)!",
"}{\\ell !",
"(d-2)!",
"}q_{\\ell }\\delta ^{\\ell -1}\\right)=q_0+q_1+O({\\varepsilon }).$ The validity of (REF ) in the case $d=2$ is obtain by substituting () in (REF ) and the use of $T_\\ell (1)=1$ .",
"Note that (REF ) is the first equality in (REF ).",
"From the first equation of (REF ) we get $q_0=-q_1/2+O({\\varepsilon })$ , which together with (REF ) and (REF ) gives $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)=q_1/2+O({\\varepsilon })=\\alpha _{1,0}/2+O({\\varepsilon }).$ Finally, (REF ) and (REF ) with $\\ell =1$ prove (REF ).",
"Remark 3.7 The values of the $q_\\ell $ 's and $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,0)$ for $m=1,2,3,4$ are as follows: If $m=1$ , then $q_0=-\\frac{1}{2}$ , $q_1=1$ , $a^{d-2}\\mathcal {F}_{{\\varepsilon },1}(a\\eta ,0)=\\frac{1}{2}.",
"$ Note that $\\mathcal {F}_{{\\varepsilon },1}(a\\eta ,x)=\\frac{1}{2}(a-|x|^2)/|a\\eta -x|^{d}$ , i.e.",
"$\\lim _{{\\varepsilon }\\rightarrow 0}\\mathcal {F}_{{\\varepsilon },1}(a\\eta ,x)$ is a constant multiple of the Poisson kernel.",
"If $m=2$ , then $q_0=-\\frac{1}{2}+\\frac{d}{8}\\delta $ , $q_1=1-\\frac{d}{2}\\delta $ , $q_2=1$ , $a^{d-2}\\mathcal {F}_{{\\varepsilon },2}(a\\eta ,0)=\\frac{1}{2}+\\frac{d-4}{8}\\delta .",
"$ If $m=3$ , then $q_0=-1+\\frac{d+2}{4}\\delta -\\frac{d(d+2)}{48}\\delta ^2$ , $q_1=2-(d+1)\\delta +\\frac{d(d+2)}{8}\\delta ^2$ , $q_2=2-\\frac{d+2}{2}\\delta $ , $q_3=1$ , $a^{d-2}\\mathcal {F}_{{\\varepsilon },3}(a\\eta ,0)=1+\\frac{d-6}{4}\\delta +\\frac{(d-4)(d-6)}{48}\\delta ^2.",
"$ If $m=4$ , then $q_0=-\\frac{5}{2}+\\frac{5(d+4)}{8}\\delta -\\frac{(d+2)(d+4)}{16}\\delta ^2+\\frac{d(d+2)(d+4)}{384}\\delta ^3$ , $q_1=5-\\frac{5(d+2)}{2}\\delta +\\frac{(3d+2)(d+4)}{8}\\delta ^2-\\frac{d(d+2)(d+4)}{48}\\delta ^3$ , $q_2=5-\\frac{3d+10}{2}\\delta +\\frac{(d+2)(d+4)}{8}\\delta ^2$ , $q_3=3-\\frac{d+4}{2}\\delta $ , $q_4=1$ , $a^{d-2}\\mathcal {F}_{{\\varepsilon },4}(a\\eta ,0)=\\frac{5}{2}+\\frac{5(d\\!-\\!8)}{8}\\delta +\\frac{(d\\!-\\!6)(d\\!-\\!8)}{16}\\delta ^2+\\frac{(d\\!-\\!4)(d\\!-\\!6)(d\\!-\\!8)}{384}\\delta ^3.",
"$"
],
[
"Localized kernels on ${\\mathbb {S}}^1$ : Second solution",
"In dimension $d=2$ we next identify another linear combination of a single shift of the Newtonian kernel directional derivatives with excellent localization on the unit sphere ${\\mathbb {S}}^1$ .",
"Theorem 4.1 Let $0<{\\varepsilon }\\le 1$ , $a=e^{\\varepsilon }$ , $m\\in {\\mathbb {N}}$ , and $\\eta \\in {\\mathbb {S}}^1$ .",
"The function $G_{{\\varepsilon }}(x\\cdot \\eta ):=\\frac{2^{2m-2}}{m}\\sum _{n\\in {\\mathbb {Z}}} \\frac{{\\varepsilon }^{-1}}{(1+{\\varepsilon }^{-2}(\\rho (x,\\eta )+2\\pi n)^2)^{m}},\\quad x\\in {\\mathbb {S}}^1,$ has the following properties: $0<G_{{\\varepsilon }}(x\\cdot \\eta )\\le c \\frac{{\\varepsilon }^{-1}}{(1+{\\varepsilon }^{-1}\\rho (x,\\eta ))^{2m}}, \\quad x\\in {\\mathbb {S}}^1,$ with a constant $c>0$ depending only on $m$ , and $\\int _{{\\mathbb {S}}^1} G_{{\\varepsilon }}(x\\cdot \\eta )\\,d\\sigma (x)=\\frac{\\pi (2m-2)!}{(m-1)!m!",
"}.$ Moreover, $G_{{\\varepsilon }}(x\\cdot \\eta )$ is the restriction to ${\\mathbb {S}}^1$ of the following harmonic function, defined on ${\\mathbb {R}}^2\\backslash \\lbrace a\\eta \\rbrace $ , $\\mathcal {G}_{{\\varepsilon },m}(a\\eta ,x):=-\\frac{1}{2}\\frac{(2m-2)!}{m!(m-1)!",
"}+\\sum _{\\ell =1}^m Q_\\ell (2{\\varepsilon })\\frac{(2{\\varepsilon })^{\\ell -1}a^{\\ell }}{\\ell !",
"}(\\eta \\cdot \\nabla )^\\ell \\ln \\frac{1}{|x-a\\eta |},$ where $Q_\\ell (u)=\\sum _{k=\\ell }^{m}\\frac{\\ell (2m-k-1)!}{m!(m-k)!",
"}\\frac{A_{k-1,\\ell -1}}{(k-1)!",
"}u^{k-\\ell },\\quad 1\\le \\ell \\le m,$ $A_{k,\\ell }=\\sum _{\\nu =\\ell }^k (-1)^{\\nu -\\ell }\\binom{\\nu }{\\ell }\\nu !S_{k,\\nu },\\quad 0\\le \\ell \\le k,$ and $S_{k,\\nu }$ denote the Stirling numbers of the second kind, defined by $u^k=\\sum _{\\nu =0}^k S_{k,\\nu }u(u-1)\\cdots (u-\\nu +1),\\quad k=0,1,\\dots .$ Note that $S_{k,0}=\\delta _{k,0}$ , $S_{k,k}=1$ .",
"The proof of Theorem REF is based on several auxiliary statements.",
"Lemma 4.2 Let $m\\in {\\mathbb {N}}$ , ${\\varepsilon }>0$ and $g_m(u):=2\\pi {\\varepsilon }^{-1}(1+(2\\pi {\\varepsilon }^{-1}u)^2)^{-m}$ for $u\\in {\\mathbb {R}}$ .",
"Then the Fourier transform of $g_m$ has the representation $\\hat{g}_m(v) :=\\int _{{\\mathbb {R}}} g_m(u)e^{-iuv}\\,du= e^{-|v|{\\varepsilon }/(2\\pi )} \\sum _{k=1}^{m} \\beta _{m-1,k-1} \\left(\\frac{|v|{\\varepsilon }}{2\\pi }\\right)^{k-1},$ where $\\beta _{m,k}:=\\frac{\\pi (2m-k)!2^k}{k!",
"(m-k)!m!2^{2m}},\\quad 0\\le k\\le m.$ We have $g_m(u)=b^{2m-1}(b^2+u^2)^{-m}$ with $b:={\\varepsilon }/(2\\pi )$ .",
"Clearly, the function $g_m$ is even.",
"Hence, it suffices to prove (REF ) only for $v\\ge 0$ .",
"From identity 1.3.7 in [4] (which gives the Fourier cosine transform) with $\\nu =m-\\frac{1}{2}$ we get $\\hat{g}_m(v) = 2b^{2m-1}\\pi ^{1/2}\\left(\\frac{v}{2b}\\right)^{m-1/2} \\Gamma (m)^{-1} K_{m-1/2}(bv),$ where $K_{m-1/2}$ is the modified Bessel function of the second kind for half an odd integer index.",
"According to identity 10.47.9 in [9] $K_{m-1/2}$ is related to the modified spherical Bessel function $\\mathbf {\\textsf {k}}_{m-1}$ by $K_{m-1/2}(z) = \\sqrt{\\frac{2z}{\\pi }}\\mathbf {\\textsf {k}}_{m-1}(z)$ and $\\mathbf {\\textsf {k}}_{m-1}$ has the explicit form (identities 10.49.12 and 10.49.1 in [9]) $\\mathbf {\\textsf {k}}_{m-1}(z) = \\frac{\\pi }{2}e^{-z}\\sum _{\\nu =0}^{m-1} \\frac{(m-1+\\nu )!",
"}{(m-1-\\nu )!\\nu !2^\\nu }z^{-\\nu -1}.$ Now (REF ) follows from (REF ), (REF ) and (REF ).",
"Lemma 4.3 Let $k\\in {\\mathbb {N}}$ and $t\\in {\\mathbb {C}}$ , $|t|<1$ .",
"Then $\\sum _{n=0}^\\infty n^{k-1} t^n=\\sum _{\\ell =1}^k A_{k-1,\\ell -1} (1-t)^{-\\ell },$ where $A_{k,\\ell }$ are defined in (REF ).",
"Identity (REF ) for $k=1$ reduces to the geometric series $\\sum _{n=0}^\\infty t^n=(1-t)^{-1}$ .",
"Let $k\\ge 2$ .",
"We differentiate the previous identity $\\nu $ times, then multiply by $t^\\nu $ and finaly apply the binomial formula to obtain $\\sum _{n=0}^\\infty n(n-1)\\dots (n-\\nu +1)t^n=\\nu !",
"t^\\nu (1-t)^{-\\nu -1}\\\\=\\nu !",
"\\sum _{\\ell =0}^\\nu \\binom{\\nu }{\\ell } (t-1)^\\ell (1-t)^{-\\nu -1}=\\nu !",
"\\sum _{\\ell =0}^\\nu (-1)^{\\nu -\\ell }\\binom{\\nu }{\\ell } (1-t)^{-\\ell -1}.$ This coupled with (REF ), where $k$ replaced by $k-1$ , leads to $\\sum _{n=0}^\\infty n^{k-1} t^n=\\sum _{\\nu =0}^{k-1} S_{k-1,\\nu }\\sum _{n=0}^\\infty n(n-1)\\dots (n-\\nu +1)t^n\\\\=\\sum _{\\nu =0}^{k-1} S_{k-1,\\nu }\\nu !",
"\\sum _{\\ell =0}^\\nu (-1)^{\\nu -\\ell }\\binom{\\nu }{\\ell } (1-t)^{-\\ell -1},$ which proves the lemma.",
"Theorem 4.4 Let $m\\in {\\mathbb {N}}$ , ${\\varepsilon }>0$ , $a=e^{\\varepsilon }$ , $g_m(u):=2\\pi {\\varepsilon }^{-1}(1+(2\\pi {\\varepsilon }^{-1}u)^2)^{-m}$ for $u\\in {\\mathbb {R}}$ , and $z=e^{-2\\pi i u}$ .",
"Then $\\sum _{\\nu \\in {\\mathbb {Z}}} g_m(\\nu +u)\\\\= -\\beta _{m-1,0} +2\\sum _{\\ell =1}^{m} \\sum _{k=\\ell }^{m} \\beta _{m-1,k-1}A_{k-1,\\ell -1} {\\varepsilon }^{k-1}a^\\ell Re\\left\\lbrace (a-z)^{-\\ell }\\right\\rbrace .$ Applying Lemma REF and the Poisson summation formula: $\\sum _{\\nu =-\\infty }^\\infty g_m(\\nu +u)= \\sum _{n=-\\infty }^\\infty \\hat{g}_m(2\\pi n) e^{-2\\pi i n u}$ we get $\\sum _{\\nu =-\\infty }^\\infty g_m(\\nu +u)= \\sum _{k=1}^{m} \\beta _{m-1,k-1}{\\varepsilon }^{k-1} \\sum _{n=-\\infty }^\\infty |n|^{k-1} e^{-|n|{\\varepsilon }} e^{-2\\pi i n u}.$ For the evaluation of the inner sum in the right-hand side of (REF ) we use Lemma REF with $t=a^{-1}z$ and with $t=a^{-1}\\bar{z}$ to get $\\sum _{n=-\\infty }^\\infty |n|^{k-1} e^{-|n|{\\varepsilon }} e^{-2\\pi i n u}\\\\=\\sum _{\\ell =1}^k A_{k-1,\\ell -1} \\left[(1-a^{-1}z)^{-\\ell }+(1-a^{-1}\\bar{z})^{-\\ell }-\\delta _{k,1}\\right]\\\\=-\\delta _{k,1}+\\sum _{\\ell =1}^k A_{k-1,\\ell -1} 2a^\\ell Re\\left\\lbrace (a-z)^{-\\ell }\\right\\rbrace .$ Substituting (REF ) in (REF ) we arrive at (REF ).",
"Due to the rotational invariance we may assume that the vector $\\eta =(1,0)$ in (REF ).",
"For any $x=(x_1,x_2)\\in {\\mathbb {S}}^1$ we apply Theorem REF with $z=x_1+ix_2=e^{-2\\pi i u}$ , $|u|\\le 1/2$ .",
"Thus $\\rho (x,\\eta )=2\\pi |u|$ and $a-z=|x-a\\eta |e^{i\\varphi }$ , where $\\cos \\varphi =-\\frac{(x-a\\eta )\\cdot \\eta }{|x-a\\eta |}$ .",
"Using the Maxwell formula () we get $Re\\left\\lbrace (a-z)^{-\\ell }\\right\\rbrace =Re\\left\\lbrace (a-\\bar{z})^{-\\ell }\\right\\rbrace \\\\= (-1)^{\\ell } T_{\\ell }\\left(\\frac{(x-a\\eta )\\cdot \\eta }{|x-a\\eta |}\\right)\\frac{1}{|x-a\\eta |^\\ell }=\\frac{1}{(\\ell -1)!",
"}\\left(\\eta \\cdot \\nabla \\right)^\\ell \\ln \\frac{1}{|x-a\\eta |}.$ Now, combining (REF ) with (REF ) we get $\\sum _{\\nu =-\\infty }^\\infty 2\\pi {\\varepsilon }^{-1}(1+{\\varepsilon }^{-2}(\\rho (x,\\eta )+2\\pi \\nu )^2)^{-m}\\\\= -\\beta _{m-1,0} +2\\sum _{\\ell =1}^{m} \\sum _{k=\\ell }^{m} \\beta _{m-1,k-1}A_{k-1,\\ell -1} {\\varepsilon }^{k-1}a^\\ell \\frac{1}{(\\ell -1)!",
"}\\left(\\eta \\cdot \\nabla \\right)^\\ell \\ln \\frac{1}{|x-a\\eta |}$ whenever $u\\ge 0$ .",
"Identity (REF ) is also valid for $u< 0$ because the left-hand side of (REF ) is an even function of $u$ .",
"Now, multiplying both sides of (REF ) by $\\frac{2^{2m-3}}{\\pi m}$ we obtain (REF ).",
"Inequalities (REF ) follow readily by (REF ).",
"From (REF ) and (REF ) we get $\\int _{{\\mathbb {S}}^1}\\mathcal {G}_{{\\varepsilon },m}(a\\eta ,x)\\,d\\sigma (x)=\\frac{2^{2m-2}}{m}\\int _{{\\mathbb {R}}} (1+u^2)^{-m}du=\\frac{2^{2m-2}}{m}\\hat{g}_m(0)=\\frac{\\pi (2m-2)!}{(m-1)!m!",
"},$ which confirms (REF ).",
"Remark 4.5 Some similarities and differences between the functions $\\mathcal {F}_{{\\varepsilon },m}(a\\eta ,x)$ defined in (REF ) and $\\mathcal {G}_{{\\varepsilon },m}(a\\eta ,x)$ defined by (REF ) are: $\\mathcal {F}_{{\\varepsilon },m}$ is defined for every $d\\ge 2$ , while $\\mathcal {G}_{{\\varepsilon },m}$ is defined only for $d=2$ .",
"In (REF ) $a=1+{\\varepsilon }$ , while $a=e^{\\varepsilon }$ in (REF ).",
"$q_\\ell $ and $Q_\\ell $ are polynomials of the same degree and $q_\\ell (\\delta )-Q_\\ell (2{\\varepsilon })= O({\\varepsilon })$ , $\\ell =1,\\dots ,m$ .",
"The polynomials $Q_\\ell $ are given explicitly, while the $q_\\ell $ 's are only known recursively."
],
[
"Localization on ${{\\mathbb {S}}^{d-1}}$ of harmonic functions on {{formula:2319e1d3-bfa1-4b10-ad38-c52ddd463a33}}",
"Having solved Problem 1 one can easily solved the analogous problem for localization on ${{\\mathbb {S}}^{d-1}}$ of linear combinations of shifts of the Newtonian kernel with poles inside the unit ball.",
"The answer is given by the simple Proposition 5.1 For $d>2$ , $\\eta \\in {{\\mathbb {S}}^{d-1}}$ and $a_\\nu >1$ the harmonic functions on ${\\mathbb {R}}^d\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu \\eta \\rbrace $ $b_0 + \\sum _{\\nu =1}^{m} \\frac{b_\\nu }{|x-a_\\nu \\eta |^{d-2}}$ and the harmonic functions on ${\\mathbb {R}}^d\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu ^{-1}\\eta \\rbrace $ $b_0 + \\sum _{\\nu =1}^{m} \\frac{b_\\nu |a_\\nu |^{2-d}}{|x-a_\\nu ^{-1}\\eta |^{d-2}}$ coincide on ${{\\mathbb {S}}^{d-1}}$ .",
"For $d=2$ , $\\eta \\in {\\mathbb {S}}^1$ and $a_\\nu >1$ the harmonic functions on ${\\mathbb {R}}^2\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu \\eta \\rbrace $ $b_0 + \\sum _{\\nu =1}^{m} b_\\nu \\ln \\frac{1}{|x-a_\\nu \\eta |}$ and the harmonic functions on ${\\mathbb {R}}^2\\setminus \\cup _{\\nu =1}^m\\lbrace a_\\nu ^{-1}\\eta \\rbrace $ $b_0 + \\sum _{\\nu =1}^{m}b_\\nu \\ln \\frac{1}{|a_\\nu |} + \\sum _{\\nu =1}^{m} b_\\nu \\ln \\frac{1}{|x-a_\\nu ^{-1}\\eta |}$ coincide on ${\\mathbb {S}}^1$ .",
"The proof follows immediately by the symmetry lemma: $a|x-a^{-1}\\eta |=|x-a\\eta |,\\quad x,\\eta \\in {{\\mathbb {S}}^{d-1}},~a>0.$"
],
[
"Localized kernels on ${\\mathbb {R}}^{d-1}$ in terms of Newtonian kernels",
"In this section we construct highly localised kernels on the subspace ${\\mathbb {R}}^{d-1}:=\\lbrace x\\in {\\mathbb {R}}^d: x=(x_1, \\dots , x_{d-1},0)\\rbrace \\quad \\hbox{of $\\;{\\mathbb {R}}^d$.",
"}$ In this case the problem is less involved compared to the case on ${{\\mathbb {S}}^{d-1}}$ and the solution is simpler.",
"Theorem 6.1 Let $m\\in {\\mathbb {N}}$ , $d\\ge 2$ , ${\\varepsilon }>0$ , and $\\eta =(0,\\dots ,0,-1)$ .",
"Denote $\\mathcal {F}_{{\\varepsilon },m}^*(x) := \\frac{2^{2m-2}(d/2)_{m-1}}{m!",
"}\\frac{{\\varepsilon }^{2m-1}}{|x-{\\varepsilon }\\eta |^{2m+d-2}},\\quad x\\in {\\mathbb {R}}^{d-1}.$ The function $\\mathcal {F}_{{\\varepsilon },m}^*$ has the following properties: $0< \\mathcal {F}_{{\\varepsilon },m}^*(x) \\le \\frac{c_1{\\varepsilon }^{-d+1}}{(1+{\\varepsilon }^{-1}|x|)^{2m+d-2}},\\quad \\forall x\\in {\\mathbb {R}}^{d-1},$ and $\\int _{{\\mathbb {R}}^{d-1}}\\mathcal {F}_{{\\varepsilon },m}^*(x)\\,dx_1\\dots dx_{d-1} \\ge c_2>0,$ where $c_1, c_2>0$ are constants depending only on $m$ and $d$ .",
"Furthermore, $\\mathcal {F}_{{\\varepsilon },m}^*$ is the restriction to ${\\mathbb {R}}^{d-1}$ of the harmonic function $\\mathcal {F}_{{\\varepsilon },m}^*$ , defined on ${\\mathbb {R}}^{d}\\setminus \\lbrace {\\varepsilon }\\eta \\rbrace $ , $\\mathcal {F}_{{\\varepsilon },m}^*(x)=\\sum _{\\ell =1}^m \\frac{(-1)^\\ell 2^{\\ell -1}(2m-\\ell -1)!",
"}{(\\ell -1)!m!",
"(m-\\ell )!",
"(d-2)}{\\varepsilon }^{\\ell -1} \\partial _d^\\ell \\frac{1}{|x-{\\varepsilon }\\eta |^{d-2}}\\quad \\mbox{if}~d\\ge 3,$ $\\mathcal {F}_{{\\varepsilon },m}^*(x)= \\sum _{\\ell =1}^m \\frac{(-1)^{\\ell } 2^{\\ell -1}(2m-\\ell -1)!",
"}{(\\ell -1)!m!",
"(m-\\ell )!",
"}{\\varepsilon }^{\\ell -1} \\partial _d^\\ell \\ln \\frac{1}{|x-{\\varepsilon }\\eta |}\\quad \\mbox{if}~d=2,$ where $\\partial _d$ stands for the partial derivative with respect to $x_d$ .",
"From Theorem REF we immediately get Corollary 6.2 Under the hypotheses of Theorem REF define $F_{{\\varepsilon },m}^*(x):=\\mathcal {F}_{{\\varepsilon },m}^*(x)\\Big (\\int _{{\\mathbb {R}}^{d-1}}\\mathcal {F}_{{\\varepsilon },m}^*(y)dy\\Big )^{-1}, \\quad x\\in {\\mathbb {R}}^{d-1}.$ Then $F_{{\\varepsilon },m}^*(x)$ is a summability kernel with decay just as in $(\\ref {main-22})$ that can be represented as a linear combination of $\\partial _d^\\ell |x-{\\varepsilon }\\eta |^{2-d}$ if $d >2$ or $\\partial _d^\\ell \\ln 1/|x-{\\varepsilon }\\eta |$ if $d >2$ for $\\ell =1,\\dots , m$ .",
"We shall derive this result from Theorem REF by a limiting process.",
"Our first step is to obtain a version of Theorem REF for an arbitrary sphere of radius $R$ in ${\\mathbb {R}}^d$ .",
"Let $m\\in {\\mathbb {N}}$ , $d\\ge 2$ , ${\\varepsilon }>0$ , $\\eta \\in {{\\mathbb {S}}^{d-1}}$ , $\\bar{x}\\in {\\mathbb {R}}^d$ , and $R>{\\varepsilon }$ .",
"Set $\\bar{y}=\\bar{x}+(R+{\\varepsilon })\\eta $ .",
"Denote by ${\\mathbb {S}}(\\bar{x}, R)$ the sphere in ${\\mathbb {R}}^d$ centered at $\\bar{x}$ of radius $R$ , i.e.",
"${\\mathbb {S}}(\\bar{x}, R):= \\lbrace \\bar{x}\\rbrace +R{{\\mathbb {S}}^{d-1}}$ .",
"Scaling by a factor of $1/R$ the sphere ${\\mathbb {S}}(\\bar{x}, R)$ and the pole location $\\bar{y}$ we arrive at the sphere ${\\mathbb {S}}(\\bar{x}/R, 1)$ and pole location at $\\bar{y}/R=\\bar{x}/R+(1+{\\varepsilon }/R)\\eta $ .",
"By (REF ) with ${\\varepsilon }/R$ and $\\bar{x}/R$ in the place of ${\\varepsilon }$ and $\\bar{x}$ we get for $x/R\\in {\\mathbb {S}}(\\bar{x}/R, 1)$ $\\mathcal {F}_{{\\varepsilon }/R,m}\\left(\\frac{x-\\bar{x}}{R}\\cdot \\eta \\right)= \\frac{(d/2)_{m-1}}{2m!",
"}\\frac{({\\varepsilon }/R)^{2m-1}(2+{\\varepsilon }/R)^{2m-1}}{(1+{\\varepsilon }/R)^{2m-2}}\\left|\\frac{x}{R}-\\frac{\\bar{y}}{R}\\right|^{-d+2-2m}.$ We multiply both sides above by $R^{1-d}$ and factor $1/R$ out of the norm to obtain $R^{1-d}\\mathcal {F}_{{\\varepsilon }/R,m}\\left(\\frac{x-\\bar{x}}{R}\\cdot \\eta \\right)= \\frac{(d/2)_{m-1}}{2m!",
"}\\frac{(2+{\\varepsilon }/R)^{2m-1}}{(1+{\\varepsilon }/R)^{2m-2}}{\\varepsilon }^{2m-1}|x-\\bar{y}|^{-d+2-2m}.$ Now, using Theorem REF and (REF ) we obtain the follow representations of the functions $R^{1-d}\\mathcal {F}_{{\\varepsilon }/R,m}((x-\\bar{x})R^{-1}\\cdot \\eta )$ for $x\\in {\\mathbb {S}}(\\bar{x},R)$ : In the case $d\\ge 3$ we have $&\\frac{(d/2)_{m-1}}{2m!",
"}\\frac{(2+{\\varepsilon }/R)^{2m-1}}{(1+{\\varepsilon }/R)^{2m-2}}{\\varepsilon }^{2m-1}|x-\\bar{y}|^{-d+2-2m} \\\\&=\\sum _{k=0}^{m-1}\\alpha _{0,k}\\frac{({\\varepsilon }/R)^{k}(2+{\\varepsilon }/R)^{k}}{(1+{\\varepsilon }/R)^{2k}}R^{-1}|x-\\bar{y}|^{2-d}\\\\& +\\sum _{\\ell =1}^m \\frac{(2+{\\varepsilon }/R)^{\\ell -1}}{(1+{\\varepsilon }/R)^{\\ell -2}}\\frac{{\\varepsilon }^{\\ell -1}}{\\ell !",
"(d-2)}\\sum _{k=0}^{m-\\ell }\\alpha _{\\ell ,k}\\frac{({\\varepsilon }/R)^{k}(2+{\\varepsilon }/R)^{k}}{(1+{\\varepsilon }/R)^{2k}} (\\eta \\cdot \\nabla )^\\ell |x-\\bar{y}|^{2-d} $ and in the case $d=2$ $\\frac{1}{2m}\\frac{(2+{\\varepsilon }/R)^{2m-1}}{(1+{\\varepsilon }/R)^{2m-2}}\\frac{{\\varepsilon }^{2m-1}}{|x-\\bar{y}|^{2m}}=\\sum _{k=0}^{m-1}\\alpha _{0,k}\\frac{({\\varepsilon }/R)^{k}(2+{\\varepsilon }/R)^{k}}{(1+{\\varepsilon }/R)^{2k}}R^{-1}\\\\+\\sum _{\\ell =1}^m \\frac{(2+{\\varepsilon }/R)^{\\ell -1}}{(1+{\\varepsilon }/R)^{\\ell -2}}\\frac{{\\varepsilon }^{\\ell -1}}{\\ell !",
"}\\sum _{k=0}^{m-\\ell }\\alpha _{\\ell ,k}\\frac{({\\varepsilon }/R)^{k}(2+{\\varepsilon }/R)^{k}}{(1+{\\varepsilon }/R)^{2k}} (\\eta \\cdot \\nabla )^\\ell \\ln \\frac{1}{|x-\\bar{y}|}.$ We are prepared to prove identities (REF )–(REF ).",
"Let ${\\varepsilon }>0$ and $\\eta =(0,\\dots ,0,-1)$ .",
"Fix $x^\\star =(x_1^\\star ,\\dots , x_{d-1}^\\star , 0)\\in {\\mathbb {R}}^{d-1}$ (see (REF )) and let $R>\\max \\lbrace |x^\\star |,{\\varepsilon }\\rbrace $ .",
"We choose $\\bar{x}:=-R\\eta $ , $\\bar{y}:={\\varepsilon }\\eta =(0,\\dots ,0,-{\\varepsilon })$ , and consider the point $x\\in {\\mathbb {S}}(\\bar{x},R)$ defined by $x:= (x_1^\\star ,\\dots , x_{d-1}^\\star , x_d), \\quad \\hbox{where}\\quad x_d:=|x^\\star |^2/(R+\\sqrt{R^2-|x^\\star |^2}).$ It is easy to verify that $x-\\bar{x}\\in R{{\\mathbb {S}}^{d-1}}$ .",
"Then (REF ) and (REF ) hold.",
"Letting $R\\rightarrow \\infty $ in (REF ) or (REF ), using Lemma REF and observing that $x\\rightarrow x^*$ we conclude that the restriction of $\\mathcal {F}_{{\\varepsilon },m}^*$ from (REF )–(REF ) coincides with $\\mathcal {F}_{{\\varepsilon },m}^*$ from (REF ) at every point $x^\\star \\in {\\mathbb {R}}^{d-1}$ .",
"Inequalities (REF )–(REF ) follow trivially from (REF )."
]
] | 1808.08637 |
[
[
"Online Human Activity Recognition using Low-Power Wearable Devices"
],
[
"Abstract Human activity recognition~(HAR) has attracted significant research interest due to its applications in health monitoring and patient rehabilitation.",
"Recent research on HAR focuses on using smartphones due to their widespread use.",
"However, this leads to inconvenient use, limited choice of sensors and inefficient use of resources, since smartphones are not designed for HAR.",
"This paper presents the first HAR framework that can perform both online training and inference.",
"The proposed framework starts with a novel technique that generates features using the fast Fourier and discrete wavelet transforms of a textile-based stretch sensor and accelerometer.",
"Using these features, we design an artificial neural network classifier which is trained online using the policy gradient algorithm.",
"Experiments on a low power IoT device (TI-CC2650 MCU) with nine users show 97.7% accuracy in identifying six activities and their transitions with less than 12.5 mW power consumption."
],
[
"Introduction",
"Advances in wearable electronics has potential to disrupt a wide range of health applications [11], .",
"For example, diagnosis and follow-up for many health problems, such as motion disorders, depend currently on the behavior observed in a clinical environment.",
"Specialists analyze gait and motor functions of patients in a clinic, and prescribe a therapy accordingly.",
"As soon as the person leaves the clinic, there is no way to continuously monitor the patient and report potential problems , .",
"Another high-impact application area is obesity related diseases, which claim about 2.8 million lives every year [4], [2].",
"Automated tracking of physical activities of overweight patients, such as walking, offers tremendous value to health specialists, since self recording is inconvenient and unreliable.",
"As a result, human activity recognition (HAR) using low-power wearable devices can revolutionize health and activity monitoring applications.",
"There has been growing interest in human activity recognition with the prevalence of low cost motion sensors and smartphones.",
"For example, accelerometers in smartphones are used to recognize activities such as stand, sit, lay down, walking, and jogging , , [3].",
"This information is used for rehabilitation instruction, fall detection of elderly, and reminding users to be active , .",
"Furthermore, activity tracking also facilitates physical activity, which improves the wellness and health of its users [8], , [9].",
"HAR techniques can be broadly classified based on when training and inference take place.",
"Early work collects the sensor data before processing.",
"Then, both classifier design and inference are performed offline [5].",
"Hence, they have limited applicability.",
"More recent work trains a classifier offline, but processes the sensor data online to infer the activity [3], .",
"However, to date, there is no technique that can perform both online training and inference.",
"Online training is crucial, since it needs to adapt to new, and potentially large number of, users who are not involved in the training process.",
"To this end, this paper presents the first HAR technique that continues to train online to adapt to its user.",
"The vast majority, if not all, of recent HAR techniques employ smartphones.",
"Major motivations behind this choice are their widespread use and easy access to integrated accelerometer and gyroscope sensors .",
"We argue that smartphones are not suitable for HAR for three reasons.",
"First, patients cannot always carry a phone as prescribed by the doctor.",
"Even when they have the phone, it is not always in the same position (e.g., at hand or in pocket), which is typically required in these studies , [10].",
"Second, mobile operating systems are not designed for meeting real-time constraints.",
"For example, the Parkinson's Disease Dream Challenge [1] organizers shared raw motion data collected using iPhones in more than 30K experiments.",
"According to the official spec, the sampling frequency is 100 Hz.",
"However, the actual sampling rate varies from 89 Hz to 100 Hz, since the phones continue to perform many unintended tasks during the experiments.",
"Due to the same reason, the power consumption is in the order of watts (more than 100$\\times $ of our result).",
"Finally, researchers are limited to sensors integrated in the phones, which are not specifically designed for human activity recognition.",
"Figure: Wearable system setup, sensors and the low-power IoT device .We knitted the textile-based stretch sensor to a knee sleeve to accurately capture the leg movements.This paper presents an online human activity recognition framework using the wearable system setup shown in Figure REF .",
"The proposed solution is the first to perform online training and leverage textile-based stretch sensors in addition to commonly used accelerometers.",
"Using the stretch sensor is notable, since it provides low-noise motion data that enables us to segment the raw data in non-uniform windows ranging from one to three seconds.",
"In contrast, prior studies are forced to divide the sensor data into fixed windows , [4] or smoothen noisy accelerometer data over long durations [10] (detailed in Section ).",
"After segmenting the stretch and accelerometer data, we generate features that enable classifying the user activity into walking, sitting, standing, driving, lying down, jumping, as well as transitions between them.",
"Since the stretch sensor accurately captures the periodicity in the motion, its fast Fourier transform (FFT) reveals invaluable information about the human activity in different frequency bands.",
"Therefore, we judiciously use the leading coefficients as features in our classification algorithm.",
"Unlike the stretch sensor, the accelerometer data is notoriously known to be noisy.",
"Hence, we employ the approximation coefficients of its discrete wavelet transform (DWT) to capture the behavior as a function of time.",
"We evaluate the performance of these features for HAR using commonly used classifiers including artificial neural network, random forest, and k-nearest neighbor (k-NN).",
"Among these, we focus on artificial neural network, since it enables online reinforcement learning using policy gradient with low implementation cost.",
"Finally, this work is the first to provide a detailed power consumption and performance break-down of sensing, processing and communication tasks.",
"We implement the proposed framework on the TI-CC2650 MCU , and present an extensive experimental evaluation using data from nine users and a total of 2614 activity windows.",
"Our approach provides 97.7% overall recognition accuracy with 27.60 ms processing time, 1.13 mW sensing and 11.24 mW computation power consumption.",
"The major contributions of this work are as follows: A novel technique to segment the sensor data non-uniformly as a function of the user motion, Online inference and training using an NN, and reinforcement learning based on policy gradient, A low power implementation on a wearable device and extensive experimental evaluation of accuracy, performance and power consumption using nine users.",
"The rest of the paper is organized as follows.",
"We review the related work in Section .",
"Then, we present the feature generation and classifier design techniques in Section .",
"Online learning using policy gradient algorithm is detailed in Section .",
"Finally, the experimental results are presented in Section , and our conclusions are summarized in Section ."
],
[
"Related Work and Novelty",
"Human activity recognition has been an active area of research due to its applications in health monitoring, patient rehabilitation and in promoting physical activity among the general population [4], [8], [7].",
"Advances in sensor technology have enabled activity recognition to be performed using body mounted sensors .",
"Typical steps for activity recognition using sensors include data collection, segmentation, feature extraction and classification.",
"HAR studies typically use a fixed window length to infer the activity of a person , [4].",
"For instance, the studies in , [4] use 10 second windows to perform activity recognition.",
"Increasing the window duration improves accuracy [7], since it provides richer data about the underlying activity.",
"However, transitions between different activities cannot be captured with long windows.",
"Moreover, fixed window lengths rarely capture the beginning and end of an activity.",
"This leads to inaccurate classification as the window can have features of two different activities [7].",
"A recent work proposes action segmentation using step detection algorithm on the accelerometer data [10].",
"Since the accelerometer data is noisy, they need to smoothen the data using a one-second sliding window with 0.5 second overlap.",
"Hence, this approach is not practical for low-cost devices with limited memory capacity.",
"Furthermore, the authors state that there is a strong need for better segmentation techniques to improve the accuracy of HAR [10].",
"To this end, we present a robust segmentation technique which produces windows whose sizes vary as a function of the underlying activity.",
"Most existing studies employ statistical features such as mean, median, minimum, maximum, and kurtosis to perform HAR , [4], .",
"These features provide useful insight, but there is no guarantee that they are representative of all activities.",
"Therefore, a number of studies use all the features or choose a subset of them through feature selection .",
"Fast Fourier transform and more recently discrete wavelet transform have been employed on accelerometer data.",
"For example, the work in [10] computes the 5th order DWT of the accelerometer data.",
"Eventually, it uses only a few of the coefficients to calculate the wavelet energy in the 0.625 - 2.5 Hz band.",
"In contrast, we use only the approximation coefficients of a single level DWT with $O(N/2)$ complexity.",
"Unlike prior work, we do not use the FFT of the accelerometer data, since it entails significant high frequency components without clear implications.",
"In contrast, we employ leading FFT coefficients of the stretch sensor data, since it gives a very good indication of the underlying activity.",
"Early work on HAR used wearable sensors to perform data collection while performing various activities [5].",
"This data is then processed offline to design the classifier and perform the inference.",
"However, offline inference has limited applicability since users do not get any real time feedback.",
"Therefore, recent work on HAR has focused on implementation on smartphones , [3], [8], .",
"Compared to wearable HAR devices, smartphones have limited choice of sensors and high power consumption.",
"In addition, results on smartphones are harder to reproduce due to the variability in different phones, operating systems and usage patterns [9], .",
"Finally, existing studies on HAR approaches employ commonly used classifiers, such as k-NN , support vector machines , decision trees , and random forest , which are trained offline.",
"In strong contrast to these methods, the proposed framework is the first to enable online training.",
"We first train an artificial neural network offline to generate an initial implementation of the HAR system.",
"Then, we use reinforcement learning at runtime to improve the accuracy of the system.",
"This enables our approach to adapt to new users in the field."
],
[
"Goals and Problem Statement",
"The goal of the proposed HAR framework is to recognize the six common daily activities listed in Table REF and the transitions between them in real-time with more than 90% accuracy under mW power range.",
"These goals are set to make the proposed system practical for daily use.",
"The power consumption target enables day-long operation using ultrathin lithium polymer cells [12].",
"Figure: Overview of the proposed human activity recognition framework.Table: List of activities used in the HAR frameworkThe stretch sensor is knitted to a knee sleeve, and the IoT device with a built-in accelerometer is attached to it, as shown in Figure REF .",
"All the processing outlined in Figure REF is performed locally on the IoT device.",
"More specifically, the streaming stretch sensor data is processed to generate segments ranging from one to three seconds (Section REF ).",
"Then, the raw accelerometer and stretch data in each window are processed to produce the features used by the classifier (Section REF ).",
"Finally, these features are used both for online inference (Section REF ) and reinforcement learning using policy gradient (Section ).",
"Since communication energy is significant, only the recognized activity and time stamps are transmitted to a gateway, such as a phone or PC, using Bluetooth whenever they are nearby (within 10m).",
"The following sections provide a theoretical description of the proposed framework without tying them to specific parameters values.",
"These parameters are chosen to enable a low-overhead implementation using streaming data.",
"The actual values used in our experiments are summarized in Section REF while describing the experimental setup."
],
[
"Sensor Data Segmentation",
"Activity windows should be sufficiently short to catch transitions and fast movements, such as fall and jump.",
"However, short windows can also waste computation time and power for idle periods, such as sitting.",
"Furthermore, a fixed window may contain portions of two different activities, since perfect alignment is not possible.",
"Hence, activity-based segmentation is necessary to maintain a high accuracy with minimum processing time and power consumption.",
"To illustrate the proposed segmentation algorithm, we start with the snapshot in Figure REF from our user studies.",
"Both the 3-axis accelerometer and stretch sensor data are preprocessed using a moving average filter similar to prior studies.",
"The unit of acceleration is already normalized to gravitational acceleration.",
"The stretch sensor outputs a capacitance value which changes as a function of its state.",
"This value ranges from around 390 pF (neutral) to close to 500 pF when it is stretched .",
"Therefore, we normalize the stretch sensor output by subtracting its neutral value and scaling by a constant: $s(t) = [s_{raw}(t) - min(s_{raw})] / S_{const}$ .",
"We adopted $S_{const} = 8$ to obtain a comparable range to accelerometer.",
"First, we note that the 3-axis accelerometer data exhibits significantly larger variations compared to the normalized stretch capacitance.",
"Therefore, decisions based on accelerations are prone to false hits [10].",
"In contrast, we propose a robust solution which generates the segments specified with red $\\ast $ markers in Figure REF .",
"Figure: Illustration of the segmentation algorithm.The boundaries between different activities can be identified by detecting the deviation of the stretch sensor from its neutral value.",
"For example, the first segment in Figure REF corresponds to a step during walk.",
"The sensor value starts increasing from a local minima to a peak in the beginning of the step.",
"The beginning of the second segment ($t \\approx 21$ s) exhibits similar behavior, since it is another step.",
"Although the second step is followed by a longer neutral period (the user stops and sits to a chair at $t \\approx 23$ s), the beginning of the next segment is still marked by a rise from a local minima.",
"In general, we can observe a distinct minima (fall followed by rise as in walk) or a flat period followed by rise (as in walk to sit) at the boundaries of different activity windows.",
"Therefore, the proposed segmentation algorithm monitors the derivative of the stretch sensor to detect the activity boundaries.",
"We employ the 5-point derivative formula given below to track the trend of the sensor value: $s^{\\prime }(t) = \\frac{s(t-2) - 8s(t-1) + 8s(t+1) - s(t+2)}{12}$ where $s(t)$ and $s^{\\prime }(t)$ are the stretch sensor value and its derivative time step $t$ , respectively.",
"When the derivative is positive, we know that the stretch value is increasing.",
"Similarly, a negative value means a decrease, and $s^{\\prime }(t)=0$ implies a flat region.",
"Looking at a single data point can catch sudden peaks and lead to false alarms.",
"To improve the robustness, one can look at multiple consecutive data points before determining the trend.",
"In our implementation, we conclude that the trend changes only if the last three derivatives consistently signal the new trend.",
"For example, if the current trend is flat, we require that the derivative is positive for three consecutive data points to filter glitches in the data point.",
"Whenever we detect that the trend changes from flat or decreasing to positive, we produce a new segment.",
"Finally, we bound the window size from below and above to prevent excessively short or long windows.",
"We start looking for a new segment, only if a minimum duration (one second in this work) passes after starting a new window.",
"Besides preventing unnecessarily small segments, this approach saves computation time.",
"Similarly, a new segment is generated automatically after exceeding an upper threshold.",
"This choice improves robustness in case a local minima is missed.",
"We use $t_{max} = 3$ s as the upper bound, since it is long enough to cover all transitions.",
"Figure REF shows the segmented data for the complete duration of the illustrative example given in Figure REF .",
"The proposed approach is able to clearly segment each step of walk.",
"Moreover, it is able to capture the transitions from walking to sitting and sitting to standing very well.",
"This segmentation allows us to extract meaningful features from the sensor data, as described in the next section.",
"Figure: Illustration of the sensor data segmentation."
],
[
"Feature Generation",
"To achieve a high classification accuracy, we need to choose representative features that capture the underlying movements.",
"We note that human movements typically do not exceed 10-Hz.",
"Since statistical features, such as mean and variance, are not necessarily representative, we focus on FFT and DWT coefficients, which have clear frequency interpretations.",
"Prior studies typically choose the largest transform coefficients to preserve the maximum signal power as in compression algorithms.",
"However, sorting loses the frequency connotation, besides using valuable computational resources.",
"Instead, we focus on the coefficients in the frequency bins of interest by preserving the number of data samples in each segment, as described next.",
"Stretch sensor features: The stretch sensor shows a periodic pattern for walking, and remains mostly constant during sitting and standing, as shown in Figure REF .",
"As the level of activity changes, the segment duration varies in the (1,3] second interval.",
"We can preserve 10 Hz sampling rate for the longest duration (3 s during low activity), if we maintain $2^5=32$ data samples per segment.",
"As the level of activity intensifies, the sampling rate grows to 32 Hz, which is sufficient to capture human movements.",
"We choose a power of 2, since it enables efficient FFT computation in real-time.",
"When the segment has more than 32 samples due to larger sensor sampling rate, we first sub-sample and smooth the input data as follows: $ s_s[k] = \\frac{1}{2S_R}\\sum _{i = -S_R}^{S_R} s(tS_R+i), \\hspace{19.91692pt} 0 \\le k <32$ where $S_R = \\lfloor N/32 \\rfloor $ is the subsampling rate, and $s_s[k]$ is the sub-sampled and smoothed data point.",
"When there are less than 32 samples, we simply pad the segment with zeros.",
"After standardizing the size, we take the FFT of the current window and the previous window.",
"We use two windows as it allows us to capture any repetitive patterns in the data.",
"With 32 Hz sampling rate during high activity regions, we cover $F_s/2=$ 16 Hz activity per Nyquist theorem.",
"We observe that the leading 16 FFT coefficients, which cover the [0-8] Hz frequency range, carry most of the signal power in our experimental data.",
"Therefore, they are used as features in our classifiers.",
"The level of the stretch sensor also gives useful information.",
"For instance, it can reliably differentiate sit from stand.",
"Hence, we also add the minimum and maximum value of the stretch sensor to the feature set.",
"Accelerometer features: Acceleration data contains faster changes compared to the stretch data, even though the underlying human motion is slow.",
"Therefore, we sub-sample and smoothen the acceleration to $2^6=64$ points following the same procedure given in Equation REF .",
"Three axis accelerometers provide acceleration $a_x$ , $a_y$ and $a_z$ along $x-$ , $y-$ and $z-$ axes, respectively.",
"In addition, we compute the body acceleration excluding the effect of gravity $g$ as $b_{acc} = \\sqrt{a_x^2 + a_y^2 + a_z^2} - g$ , since it carries useful information.",
"Discrete wavelet transform is an effective method to recursively divide the input signal to approximation $A_i$ and detail $D_i$ coefficients.",
"One can decompose the input signal to $\\log _2 N$ samples where $N$ is the number of data points.",
"After one level of decomposition, $A_1$ coefficients in our data correspond to 0-32 Hz, while and $D_1$ coefficients cover 32-64 Hz band.",
"Since the former is more than sufficient to capture acceleration due to human activity, we only compute and preserve $A_1$ coefficients with $O(N/2)$ complexity.",
"The number of features could be further reduced by computing the lower level coefficients and preserving largest ones.",
"As shown in the performance break-down in Table REF , using the features in the NN computations takes less time than computing the DWT coefficients.",
"Moreover, keeping more coefficients and preserving the order maintains the shape of the underlying data.",
"Feature Overview: In summary, we use the following features: Stretch sensor: We use 16 FFT coefficients, the minimum and maximum values in each segment.",
"This results in 18 features.",
"Accelerometer: We use 32 DWT coefficients for $a_x$ , $a_z$ and $b_{acc}$ .",
"In our experiments, we use only the mean value of $a_y$ , since no activity is expected in the lateral direction, and $b_{acc}$ already captures its effect given the other two directions.",
"This results in 97 features.",
"General features: The length of the segment also carries important information, since the number of data points in each segment is normalized.",
"Similarly, the activity in the previous window is useful to detect transitions.",
"Therefore, we also add these two features to obtain a total of 117 features."
],
[
"Supervised Learning for State Classification",
"In the offline phase of our framework, the feature set is assigned a label corresponding to the user activity.",
"Then, a supervised learning technique takes the labeled data to train a classifier which is used at runtime.",
"Since one of our major goals is online training using reinforcement learning, we employ a cost-optimized neural network (NN).",
"We also compare our solution to most commonly used classifiers by prior work, and provide brief explanations.",
"Support Vector Machine (SVM): SVM finds a hyperplane that can separate the feature vectors of two output classes.",
"If a separating hyperplane does not exist, SVM maps the data into higher dimensions until a separating hyperplane is found.",
"Since SVM is a two class classifier, multiple classifiers need to be trained for recognizing more than two output classes.",
"Due to this, SVM is not suitable for reinforcement learning with multiple classes , which is the case in our HAR framework.",
"Random Forests and Decision Trees: Random forests use an ensemble of tree-structured classifiers, where each tree independently predicts the output class as a function of the feature vector.",
"Then, the class which is predicted most often is selected as the final output class.",
"C4.5 decision tree is another commonly used classifier for HAR.",
"Instead of using multiple trees, C4.5 uses a single tree.",
"Random forests typically shows a higher accuracy than decision trees, since it evaluates multiple decision trees.",
"Reinforcement learning using random forests has been recently investigated in .",
"As part of the reinforcement learning process, additional trees are constructed and then a subset of trees is chosen to form the new random forest.",
"This adds additional processing and memory requirements on the system, making it unsuitable for implementation on a wearable system with limited memory.",
"k-Nearest Neighbors (k-NN): k-Nearest Neighbors is one of the most popular techniques used by many previous HAR studies.",
"k-NN evaluates the output class by first calculating k nearest neighbors in the training dataset.",
"Then, it chooses the class that is most common among the k neighbors and assigns it as the output class.",
"This requires storing all the training data locally.",
"Since storing the training data on a wearable device with limited memory is not feasible, k-NN is not suitable for online training.",
"Proposed NN Classifier: We use the artificial neural network shown in Figure REF as our classifier.",
"The input layer processes the features denoted by $\\mathbf {X}$ , and relay to the hidden layer with the ReLU activation.",
"It is important to choose an appropriate number of neurons ($N_h$ ) in the hidden layer to have a good accuracy, while keeping the computational complexity low.",
"To obtain the best trade-off, we evaluate the recognition accuracy and memory requirements as a function of neurons, as detailed in Section REF .",
"The output layer includes a neuron for each activity $a_i \\in \\mathbf {A} = \\lbrace D, J, L, S, Sd, W, T\\rbrace , 1 \\le i \\le N_A$ , where $N_A$ is the number of activities in set $\\mathbf {A}$ , which are listed in Table REF .",
"Output neuron for activity $a_i$ computes $O_{a_i}(\\mathbf {X},\\mathbf {\\theta }_{in}, \\mathbf {\\theta })$ as a function of the input features $\\mathbf {X}$ and the weights of the NN.",
"To facilitate the policy gradient approach described in Section , we express the output $O_{a_i}$ in terms of the hidden layer outputs as: $\\vspace{-2.84526pt}O_{a_i}(\\mathbf {X},\\mathbf {\\theta }_{in}, \\mathbf {\\theta }) =O_{a_i}(\\mathbf {h},\\mathbf {\\theta }) = \\sum _{j=1}^{N_{h} + 1} h_{j}\\theta _{j,i}, ~~~~~~~1\\le i \\le N_A\\vspace{-2.84526pt}$ where $h_{j}$ is the output of the $j^{\\mathrm {th}}$ neuron in the hidden layer, and $\\theta _{j,i}$ is the weight from $j^{\\mathrm {th}}$ neuron to output activity $a_i$ .",
"Note that $h_j$ is a function of $\\mathbf {X}$ and $\\mathbf {\\theta }_{in}$ .",
"The summation goes to $N_{h} + 1$ , since there are $N_h$ neurons and one bias term in the hidden layer.",
"After computing the output functions, we use the softmax activation function to obtain the probability of each activity: $ \\vspace{-2.84526pt}\\pi (a_i|\\hspace{1.42262pt}\\mathbf {h}, \\mathbf {\\theta }) =\\frac{e^{O_{a_i}(\\mathbf {h},\\mathbf {\\theta })}}{\\sum _{j=1}^{N_A}{e^{O_{a_j}(\\mathbf {h},\\mathbf {\\theta })}}}, ~~~~~~~1\\le i\\le N_A$ We express $\\pi (a_i|\\hspace{1.42262pt}\\mathbf {h}, \\mathbf {\\theta })$ as a function of the hidden layer outputs $\\mathbf {h}$ instead of the input features, since our reinforcement learning algorithm will leverage it.",
"Finally, the activity which has the maximum probability is chosen as the output.",
"Implementation cost: Our optimized classifier requires 264 multiplications for the FFT of stretch data, $118N_h + (N_h+1)N_A$ multiplications for the NN and uses only 2 kB memory.",
"Figure: The NN used for activity classifier and reinforcement learning.The trained ANN classifier is implemented on the IoT device to recognize the human activities in real-time.",
"In addition to online activity recognition, we employ the policy gradient based reinforcement learning (RL) to continue training the classifier in the field.",
"Online training improves the recognition accuracy for new users by as much as 33%, as demonstrated in our user studies.",
"We use the following definitions for the state, action, policy, and the reward.",
"State: Stretch sensor and accelerometer readings within a segment are used as the continuous state space.",
"We process them as described in Section REF to generate the input feature vector $\\mathbf {X}$ (Figure REF ).",
"Policy: The ANN processes input features as shown in Figure REF to generate the hidden layer outputs $\\mathbf {h} = \\lbrace h_j, 1 \\le j \\le N_h+1\\rbrace $ and the activity probabilities $\\pi (a_i|\\mathbf {h}, \\mathbf {\\theta })$ , i.e., the policy given in Equation REF .",
"Action: The activity performed in each sensor data segment is interpreted as the action in our RL framework.",
"It is given by $argmax~\\pi (a_i|\\mathbf {h}, \\mathbf {\\theta })$ , i.e., the activity with maximum probability.",
"Reward: Online training requires user feedback, which is defined as the reward function.",
"When no feedback is provided by the user, the weights of the network remain the same.",
"The user can give feedback upon completion of an activity, such as walking, which contains multiple segments (i.e., non-uniform action windows).",
"If the classification in this period is correct, a positive reward (in our implementation $+1$ ) is given.",
"Otherwise, the reward is negative ($-1$ ).",
"We define the sequence of segments for which a reward is given as an epoch.",
"The set of epochs in a given training session is called an episode following the RL terminology .",
"Objective: The value function for a state is defined as the total reward that can be earned starting from that state and following the given policy until the end of an episode.",
"Our objective is to maximize the total reward $J(\\mathbf {\\theta })$ as a function of the classifier weights.",
"Proposed Policy Gradient Update: In general, all the weights in the policy network can be updated after an epoch .",
"This is useful when we start with an untrained network with random weights.",
"When a policy network is trained offline as in our example, its first few layers generate broadly applicable intermediate features .",
"Consequently, we can update only the weights of the output layer to take advantage of offline training and minimize the computation cost.",
"More precisely, we update the weights denoted by $\\mathbf {\\theta }$ in Figure REF to tune our optimized ANN to individual users.",
"Since we use the value function as the objective, the gradient of $J(\\theta )$ is proportional to the gradient of the policy .",
"Using this result, the update equation for $\\mathbf {\\theta }$ is given as: $ \\vspace{2.84526pt}\\mathbf {\\theta }_{t+1} \\doteq \\mathbf {\\theta }_t + \\alpha r_t\\frac{\\nabla _\\mathbf {\\theta } \\pi (a_t|\\hspace{1.42262pt}\\mathbf {h},\\mathbf {\\theta }_t)}{\\pi (a_t|\\hspace{1.42262pt}\\mathbf {h}, \\mathbf {\\theta }_t)},\\hspace{8.53581pt} \\alpha : \\mathrm {Learning~rate}$ where $\\mathbf {\\theta }_{t}$ and $\\mathbf {\\theta }_{t+1}$ are the current and updated weight matrices, respectively.",
"Similarly, $a_t$ is the current action at time $t$ , $r_t$ is the corresponding reward, and $\\mathbf {h}$ denotes the hidden layer outputs.",
"Hence, we need to compute the gradient of the policy to update the weights.",
"To facilitate this computation and partial update, we partition the weights into two disjoint sets as $\\mathcal {S}_t$ and $\\overline{\\mathcal {S}_t}$ .",
"The weights that connect to the output $O_{a_t}$ corresponding to the current action are in $\\mathcal {S}_t$ .",
"The rest of the weights belong to the complementary set $\\overline{\\mathcal {S}_t}$ .",
"With this definition, we summarize the weight update rule in a theorem in order not to disrupt the flow of the paper with derivations.",
"Interested readers can go through the proof.",
"Weight Update Theorem: Given the current policy, reward and the learning rate $\\alpha $ , the weights in the output layer of the ANN given in Figure REF are updated online as follows: $\\theta _{t+1,j,i} \\doteq {\\left\\lbrace \\begin{array}{ll}\\mathbf {\\theta }_{t,j,i} + \\alpha r_t (1 - \\pi (a_t|\\hspace{1.42262pt}\\mathbf {h},\\theta _t)) \\cdot h_j &\\mathbf {\\theta }_{t,j,i} \\in \\mathcal {S}_t \\\\\\mathbf {\\theta }_{t,j,i} - \\alpha r_t \\pi (a_i|\\hspace{1.42262pt}\\mathbf {h},\\theta _t)) \\cdot h_j &\\mathbf {\\theta }_{t,j,i} \\in \\overline{\\mathcal {S}_t}\\end{array}\\right.",
"}$ Proof: The partial derivative of the policy $\\pi (a_t|\\hspace{1.42262pt}\\mathbf {h}, \\mathbf {\\theta })$ with respect to the weights $\\theta _{j,i}$ can be expressed using the chain rule as: $\\frac{\\partial \\pi (a_t|\\hspace{1.42262pt}\\mathbf {h},\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }_{j,i}} =\\frac{\\partial \\pi (a_t|\\hspace{1.42262pt}\\mathbf {h},\\mathbf {\\theta })}{\\partial O_{a_i}(\\mathbf {h},\\mathbf {\\theta })}\\frac{\\partial O_{a_i}(\\mathbf {h},\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }_{j,i}}$ where $1 \\le j \\le N_h+1$ and $1 \\le i \\le N_A$ .",
"When $\\mathbf {\\theta }_{t,j,i} \\in \\mathcal {S}_t$ , action $a_t$ corresponds to output $O_{a_t}(\\mathbf {h},\\mathbf {\\theta })$ .",
"Hence, we can express the first partial derivative using Equation REF as follows: (at|h,)Oat(h,) = eOat(h,) j=1NaeOaj(h,) - ( eOat(h,))2 ( j=1NaeOaj(h,) )2 = (at|h,)(1 - (at| h,) ) Otherwise, i.e., $\\mathbf {\\theta }_{t,j,i} \\in \\overline{\\mathcal {S}_t}$ , the derivative is taken with respect to another output.",
"Hence, we can find the partial derivative as: (at|h,)Oai(h,) = - eOat(h,) eOai(h,) ( j=1NAeOaj(h,) )2 = -(at|h,) (ai|h,) The second partial derivative in Equation REF , $\\partial O_{a_i}(\\mathbf {h},\\mathbf {\\theta })/\\partial \\mathbf {\\theta }_{j,i}$ , can be easily computed as $h_j$ using Equation REF .",
"The weight update is the product of learning rate $\\alpha $ , reward $r_t$ , $h_j$ and the partial derivative of the policy with respect to the output functions.",
"For the weights $\\mathbf {\\theta }_{t,j,i} \\in \\mathcal {S}_t$ , we use the partial derivative in Equation .",
"For the remaining weights, we use Equation .",
"Hence, we obtain the first and second lines in Equation REF , respectively.",
"Q.E.D $\\square $ In summary, the weights of the output layer are updated online using Equation REF after a user feedback.",
"Detailed results for the improvement in accuracy using RL are presented in Section REF ."
],
[
"Experimental Setup",
"Wearable System Setup: The proposed HAR framework is implemented on the TI-CC2650 IoT device, which includes a motion processing unit.",
"It also integrates a radio that runs Bluetooth Low Energy (BLE) protocol.",
"This device is placed on the ankle, since this allows for a maximum swing in the accelerometer We plan to integrate the stretch sensor and the TI-CC2650 into single flexible hybrid electronics device , as shown in Figure REF , in our future work..",
"The users wear the flexible stretch sensor on the right knee to capture the knee movements of the user.",
"In our current implementation, the stretch sensor transmits its output to the IoT device over BLE to provide flexibility in placement.",
"To synchronize the sensors, we record the wall clock time of each sensor at the beginning of the experiment.",
"Then, we compute the offset between the sensors, and use this offset to align the sensor readings, as proposed in .",
"After completing the processing on the IoT device, the recognized activities and their time durations are transmitted to a host, such as a smartphone, for debugging and offline analysis.",
"Parameter Selection: We use the default sampling frequencies: 100 Hz for the stretch sensor and 250 Hz for the accelerometer.",
"Lower sampling frequencies did not produce any significant power savings.",
"The raw sensor readings are preprocessed using a moving average filter with a window of nine samples.",
"User Studies: We evaluate the accuracy of the proposed approach using data from nine users, as summarized in Table REF .",
"The users consist of eight males and one female, with ages 20–40 years and heights 160–180 cm.",
"Data from only five of them are employed during the training phase.",
"This data is divided into 80% training/validation and 20% test following the common practice.",
"The rest of the user data is saved for evaluating only the online reinforcement learning framework.",
"Each user performs the activities listed in Table REF while wearing the sensors.",
"For example, the illustration in Figure REF is from an experiment where the user jumps, takes 10 steps, sits on a chair, and finally stands up.",
"The experiments vary from 21 seconds to 6 minutes in length, and have different composition of activities.",
"We report results from 58 different experiments with a 100 minutes total duration, as summarized in Table REF .",
"After each experiment, the segmentation algorithm presented in Section REF is used to identify non-uniform activity windows.",
"This results in 2614 unique different segments in our experimental data.",
"Then, each window is labeled manually through visual inspection by four human experts.",
"Finally, the labeled data is used for offline training.",
"Comparing specific HAR approaches is challenging, since data is collected using different platforms, sensors and settings.",
"Therefore, we compare our results with all commonly used classifiers in the next section.",
"We also release the labeled experimental data to the public on the eLab web pagehttp://elab.engineering.asu.edu/public-release/ to enable other researchers to make comparisons using a common data set.",
"Table: Summary of user studies"
],
[
"Training by Supervised Learning",
"We use an artificial neural network to perform online activity recognition and training.",
"The NN has to be implemented on the wearable device with a limited memory (in our case 20kB).",
"Therefore, it should have small memory footprint, i.e., number of weights, while giving a high recognition accuracy.",
"To achieve robust online weight updates during reinforcement learning, we first fix the number of hidden layers to one.",
"Then, we vary the number of neurons in the hidden layer to study the effect on the accuracy and memory requirements.",
"Specifically, we vary the number of hidden layer neurons from one to seven.",
"Note that the number of neurons in the output layer remains constant as we do not change the number of activities being recognized.",
"Figure REF shows the recognition accuracy (left axis) and memory requirements (right axis) of the network as a function of number of neurons in the hidden layer.",
"We observe that the accuracy is only about 80%, when a single neuron is used in the hidden layer.",
"As we increase the number of neurons, both the memory requirements and accuracy increase.",
"The accuracy starts saturating after the third neuron, while the number of weights and memory requirements increase.",
"In fact, the increase in memory requirement is linear, with an increase of around 500 bytes with every additional neuron in the hidden layer.",
"Thus, there is a trade-off between the memory requirements and accuracy.",
"In our HAR framework, we choose an NN with four neurons in the hidden layer as it gives an overall accuracy of about 97.7% and has a memory requirement of 2 kB, leaving the rest of the memory for operating system and other tasks.",
"Figure: Comparison of accuracy with number of neurons5.2.1 Confusion Matrix We analyze the accuracy of recognizing each activity in our experiment in Table REF .",
"There is one column and one row corresponding to the activities of interest.",
"The numbers on the diagonal show the recognition accuracy for each activity.",
"For example, the first row in the first column shows that driving is recognized with 99.4% accuracy.",
"According to the first row, only 0.6% of the driving activity windows are classified falsely as “Transition”.",
"To provide also the absolute numbers, the number in parenthesis at the end of each row shows the total number of activity windows with the corresponding label.",
"For instance, a total of 155 windows were labeled “Drive” according to row 1.",
"We achieve an accuracy greater than 97% for five of the seven activities.",
"The accuracy is slightly lower for jump because it is more dynamic than all the other activities.",
"Moreover, there is a higher variability in the jump patterns for each user, leading to slightly lower accuracy.",
"It is also harder to recognize transitions due to the fact that each transition segment contains features of two activities.",
"This can lead to a higher confusion for the NN, but we still achieve more than 90% accuracy.",
"We also note that the loss in accuracy is acceptable for transitions, since we can indirectly infer a transition by looking at the segments before and after the transition.",
"Table: Confusion matrix for 5 training users5.2.2 Comparison with other classifiers It is not possible to do a one to one comparison with existing approaches because they use different devices, data sets and activities.",
"Therefore, we use our data set with the commonly used classifiers described in Section REF .",
"The results are summarized in Table REF .",
"Although we use only a single hidden layer and minimize the number of neurons, our implementation achieves competitive test and overall accuracy compared to the other classifiers.",
"We also emphasize that our NN is used for both online classification and training on the IoT device.",
"Table: Comparison of accuracy for different classifiers"
],
[
"Reinforcement Learning with new users",
"The NN obtained in the offline training stage is used to recognize the activities of four new users that are previously unseen by the network.",
"This capability provides a real world evaluation of the approach, since a device cannot be trained for all possible users.",
"Due to variations in usage patterns, it is possible that the initial accuracy for a new user is low.",
"Indeed, the initial accuracy for users 6 and 9 is only about 60 – 70 %.",
"Therefore, we use reinforcement learning using policy gradients to continuously adapt the HAR system to each user.",
"Figure REF shows the improvement achieved using reinforcement learning for four users.",
"Each episode in the x-axis corresponds to an iteration of RL using the data set for new users.",
"The weights of the NN are updated after each segment as a function of the user feedback for a total of 100 episodes.",
"Moreover, we run 5 independent runs, each consisting of 100 epochs, to obtain an average accuracy of the NN at each episode.",
"We observe consistent improvement in accuracy for all four users.",
"The accuracy for users 6 and 9 starts low and increases to about 93% after about 20 episodes.",
"User 8 starts with a higher accuracy of about 85%.",
"The accuracy increases quickly to about 98% after 10 episodes.",
"In summary, reinforcement improves the accuracy for users not previously seen by the network.",
"This ensures that the device can adapt to new users very easily.",
"Figure: Reinforcement learning results for four new users."
],
[
"Power, Performance and Energy Evaluation",
"To fully assess the cost of the proposed HAR framework, we present a detailed breakdown of execution time, power consumption and energy consumption for each step.",
"The first part of HAR involves data acquisition from the sensors and segmentation.",
"The segmentation algorithm is continuously running while the data is being acquired.",
"Therefore, we include its energy consumption in the sensing block.",
"Table REF shows the power and energy consumption for a typical segment of 1.5 s. The average power consumption for the data acquisition is 1.13 mW, leading to a total energy consumption of 1695 $\\mu $ J.",
"If the segments are of a longer duration, the energy consumption for data sensing increases linearly.",
"Following the data segmentation, we extract the features and run the classifier.",
"The execution time, power and energy for these blocks are shown in the \"Compute\" rows in Table REF .",
"As expected, the FFT block has the largest execution time and energy consumption.",
"However, it is still two orders of magnitude lower than the duration of a typical segment.",
"Finally, the energy consumption of the BLE communication block is given in the last row of Table REF .",
"Since we transmit the inferred activity, the energy consumed by the BLE communication is only about 43 $\\mu $ J.",
"In summary, with less than 12.5 mW average power consumption, our approach enables close to 60-hour uninterrupted operation using a 200mAh @ 3.7V battery [12].",
"Hence, it can enable self-powered wearable devices [6] that can harvest their own energy .",
"Table: Execution time, power and energy consumption"
],
[
"Conclusions",
"We presented a HAR framework on a wearable IoT device using stretch accelerometer sensors.",
"The first step of our solution is a novel technique to segment the sensor data non-uniformly as a function of the user motion.",
"Then, we generate FFT and DWT features using the segmented data.",
"Finally, these features are used for online inference and training using an ANN.",
"Our solution is the first to perform online training.",
"Experiments on TI-CC2650 MCU with nine users show 97.7% accuracy in identifying six activities and their transitions with less than 12.5 mW power consumption."
]
] | 1808.08615 |
[
[
"Driven tabu search: a quantum inherent optimisation"
],
[
"Abstract Quantum computers are different from binary digital electronic computers based on transistors.",
"Common digital computing encodes the data into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits).",
"A circuit-based qubit quantum computer exists and is available for experiments via cloud, the IBM quantum experience project.",
"We implemented a Quantum Tabu Search in order to obtain a quantum combinatorial optimisation, suggesting that an entanglement-metaheuristic can display optimal solutions and accelerate the optimisation process by using entangled states.",
"We show by building optimal coupling maps that the distribution of our results gave similar shape as shown previous results in an existing teleport circuit.",
"Our research aims to find which graph of coupling better matches a quantum circuit."
],
[
"Introduction",
"In Quantum Mechanics, a qubit is a quantum system in which the Boolean states 0 and 1 are represented by a pair of normalised and mutually orthogonal quantum states.",
"The two states form a computational basis and any other state of the qubit can be written as a linear combination of $|0\\rangle $ and $|1\\rangle $ : $|\\psi \\rangle = \\alpha |0\\rangle + \\beta |1\\rangle $ , where $\\alpha $ and $\\beta $ are probability amplitudes and can be complex numbers.",
"A qubit can be in a superposition of both states at the same time.",
"Multiple qubits can exhibit quantum entanglement - pairs are generated or interact in ways that the quantum state of each particle cannot be described independently of the state of the other(s) [9].",
"The IBM (International Business Machines Corporation) Quantum Experience (QX) allows us the possibility to connect to an IBM quantum processor via the IBM Cloud.",
"We developed a driven quantum version of the Tabu search algorithm [4], [5], which has been well understood for solving combinatorial or nonlinear problems [6].",
"The experiments were implemented in the Python programming language using the Quantum Information Software Kit (QISKit) - a software development kit (SDK) for working with the Open Quantum Assembly Language (OpenQASM) and the IBM QX.",
"We use as backend IBM Q 16 Rueschlikon (16-qubits) and IBM Q 5 Yorktown (5-qubits) simulators.",
"In a quantum-metaheuristic procedure, we challenge the quantum search space to be maximized, derived from the advantage of interacting quantum technologies with classical implementations.",
"In a quantum combinatorial optimisation, an entanglement-metaheurisc can uncover optimal solutions and accelerate the optimisation process by using entangled states.",
"Therefore, we conduct simulation-based experiments with two types of quantum initial populations.",
"The sample solutions are composed by 16 qubits of combined pairs - with non replacement, and with replacement.",
"Until we obtain the best solution, we build the neighborhood, evaluate the system and detect whether the algorithm falls in a local optimum.",
"In those cases, we perform the entanglement between qubits if they are unequal; otherwise, we use superposition in the redundant qubit.",
"This particular feature of enhanced-entanglement showed different best solutions, and different system evaluation according to different combinatorial inputs.",
"The quantum inherent optimisation allows us to highlight the best solution and algorithm performance according to the input combined set.",
"We show the application of the proposed research by defining coupling maps for quantum devices with a driven Tabu search approach."
],
[
"Quantum computing",
"Quantum programming is like composing, where qubits in superposition interfere and where the quantum programmer ensures useful interference of qubit states.",
"Experimental measurements disturbs, or can destroy, the wave-like quantum states.",
"In order to better understand such states through experiments, it is possible to gather detailed properties of the quantum system by averaging many weak nondisturbing quantum measurements.",
"However, the frontier between what is quantum and what is classical becomes small when averaging over a large number of weak measurements.",
"Quantum mechanics allows the qubit to be in a superposition of both states at the same time.",
"In quantum computing, a qbit or qubit or quantum bits a unit of quantum information.",
"Therefore, a qubit is a two-state quantum-mechanical system, similar to the polarisation of a single photon, where the two states are vertical polarisation and horizontal polarisation, since it can be described as a polarised photon.",
"A quantum register of size n is defined by n qubits.",
"A qubit is a quantum system in which the Boolean states 0 and 1 are represented by a pair of normalised and mutually orthogonal quantum states.",
"The two states form a computational basis and any other (pure) state of the qubit can be written as a linear combination of $|0\\rangle $ and $|1\\rangle $ : $|\\psi \\rangle = \\alpha |0\\rangle + \\beta |1\\rangle $ , where $\\alpha $ and $\\beta $ are probability amplitudes and can be complex numbers.",
"For example, $|\\psi \\rangle = \\frac{1}{\\@root \\of {3}}|0\\rangle + \\@root \\of {\\frac{2}{3}}|1\\rangle $ , with probabilities $|\\alpha _0|^2 = \\frac{1}{3}$ and $|\\alpha _1|^2 = \\frac{2}{3}$ , and measurement result 0 and 1, respectively.",
"In quantum mechanics, bra-ket notation is a standard notation for representing quantum states.",
"In order to calculate the scalar product of vectors, the notation uses angle brackets $\\langle $ $\\rangle $ , and a vertical bar |.",
"The scalar product is then $\\langle \\phi |\\psi \\rangle $ where the right part is the \"psi ket\" (a column vector) and the left part is the bra - the Hermitian transpose of the ket (a row vector).",
"Table: Exponential state space.",
"Where |α i | 2 |\\alpha _i|^2 is the probability of finding the qubit in state |i〉|i\\rangle when we measure it (in the computational basis)."
],
[
"Superposition",
"Since a pure qubit state is a linear superposition of the basis states when we measure the qubit, according to the Born rule, the probability of outcome $|0\\rangle $ is $|\\@root 2 \\of {\\alpha |}$ and the probability of outcome is $|1\\rangle $ is $|\\beta |^2$ , $\\alpha $ and $\\beta $ are constrained by the equation $|\\alpha |^2 + |\\beta |^2$ .",
"Superposition is then similar to waves in classical physics, any two (or more) quantum states can be added, becoming superposed resulting in a new quantum state - i.e.",
"in 0 and 1 simultaneously, a linear combination of the two classical states - for example, the states $|+\\rangle ={\\frac{1}{{\\sqrt{2}}}}(|0\\rangle +|1\\rangle )$ or $|-\\rangle ={\\frac{1}{{\\sqrt{2}}}}(|0\\rangle -|1\\rangle )$ ."
],
[
"Entanglement",
"An important difference between a qubit and a classical bit is that multiple qubits can exhibit quantum entanglement - a physical phenomenon that occurs when pairs or more than two particles are generated or interact in forms that the quantum state of each particle can't be described independently of the state of the other(s).",
"In the case of two entangled qubits in the Bell state $\\frac{1}{\\@root \\of {2}}(|00\\rangle + |11\\rangle )$ , an equal superposition, there are equal probabilities of measuring either $|00\\rangle $ or $|11\\rangle $ , as $|\\frac{1}{\\@root \\of {2}}|^2 = \\frac{1}{2}$ .",
"Considering the two qubits are entangled qubits separately, Alice's qubit and Bob's qubit.",
"Alice measure her qubit, obtaining equal probabilities in either $|0\\rangle $ or $|1\\rangle $ .",
"Since the qubits are entangled, Bob must now get exactly the same measurement as Alice."
],
[
"Quantum Tabu Search",
"Some perspectives of Quantum Tabu Search (QTS) have been presented [1], [8].",
"The classical Tabu Search (TS) is a metaheuristic that explores search spaces and conduct a local heuristic search procedure to explore the solution space beyond local optimum using a Tabu list with forbidden moves.",
"The algorithm REF main steps can be briefly described by, a) generating the neighbors, b) evaluating each neighbor and c) getting the neighbor with maximum evaluation.",
"The algorithm stops at any iteration where there are no feasible moves into the local neighborhood of the current solution.",
"The Quantum Tabu Search is proposed for solving combinatorial or nonlinear problems with a Knapsack problem approach in a quantum procedure: qubit ($i$ ), weight ($w_i$ ) and profit ($b_i$ ), $f(s)=\\sum _{i=1}^{n}b_is_i*\\left(1-max\\left(0,\\sum _{i=1}^{n}w_is_i-max capacity\\right)\\right)$ where $s$ is the best solution and the goal is to achieve a global optimum by maximizing the quantum parameters.",
"Coupling maps for quantum devices, can be an application of the QTS, where qubit–qubit couplings are explored based on combinations of qubits taking into account the maximum number in a given quantum architecture.",
"A coupling map in quantum architectures is a directed graph representing superconducting bus connections between qubits [2] as shown in Figure REF .",
"Figure: IBMQX5 connections, 16-qubits.To obtain the neighbor with maximum evaluation we get the tabu qubit, find the best neighbor position, afterwards we check if the neighbor is a result of a tabu move, if it is, then we get the position of the qubit and check if it is in a tabu list.",
"The tabu list is built by storing the qubit considering if the neighbors evaluation is greater that best evaluation.",
"The best iteration is an incremented value when the condition occurs and the best solution are the neighbors given by the position of best neighborhood evaluation.",
"The qubit is given by the position that occurs in the range of the length of the best solution when the best solution differs from the best neighbor.",
"The input and output states will be in a superposition, so that the qubits can be also entangled.",
"Figure: General QTS algorithm"
],
[
"Results: find the best coupling map",
"In Figure REF and Figure REF we present two combinatorial configurations of pairs in 100 runs each.",
"The figures show the scores for each solution and the respectively number of iterations.",
"A maximum number of iterations do not particularly mean a high score solution.",
"Figure: 100 runs to find best coupling map.To validate our results we choose the quantum teleport circuithttps://github.com/Qiskit/qiskit-terra with parameters initialization and quantum gates performing operations, such us, Pauli-$\\it {X}$ (NOT Gate) to obtain a $\\pi $ -rotation around the $\\it {X}$ axis where $\\it {X}\\rightarrow \\it {X}$ and $\\it {Z}\\rightarrow -\\it {Z}$ (bit-flip), Pauli-$\\it {Z}$ ($\\it {R}_\\pi gate)$ ) to obtain $\\pi $ -rotation around the $\\it {Z}$ axis where $\\it {X}\\rightarrow -\\it {X}$ and $\\it {Z}\\rightarrow \\it {Z}$ (phase-flip), Hadamard $\\it {H}$ to map $\\it {X}\\rightarrow \\it {Z}$ , and Controlled-NOT, a two-qubit gate that flips the target qubit (applies Pauli-$\\it {X}$ ).",
"Figure: Quantum teleport circuit.",
"Coupling map = None Coupling map = [[0, 1], [0, 2], [1, 2], [3, 2], [3, 4], [4, 2]] Coupling map = [[0, 1], [0, 4], [1, 2], [1, 3], [1, 4], [3, 4]] (Tabu) Figure: 5-qubits coupling map for quantum teleportation.Our results show that an quantum coupling map algorithm based on tabu search can produce the same behaviour as not using any mapping or by setting a predefined map.",
"However, by restricting multi-qubit operations to coupling maps, we decrease the state decoherenceA process that separates states so that they can no longer interfere.",
"in qubits."
],
[
"Conclusion and future work",
"Quantum computers calculate multiple functions at once, exponentially increasing processing speed with each added qubit.",
"[3], [7] A quantum computer uses quantum states and dynamics of particles to store and process information.",
"Transistors have continually decreasing the size, and doubled the power of computers.",
"When technologies reaches the scale of atoms, quantum effects can disrupt its operations due to effects as tunneling and entanglement.",
"QTS allows us to discover solutions efficiently, where the entangled state can solve high-dependency problems, by using fewer evaluation to determine the global optimum, it also increases the search speed and probes to be a quantum procedure to escape from local optima.",
"In the future we intend to validate our results by comparing different circuits, and by displaying all the results in a descending order instead of just the best one.",
"Also, we intend to implement the algorithm in a D-Wave machine and compare with the current approach.",
"D-Wave Systems has been developing their own version of a quantum computer that uses annealing, which is different from the gate model based approaches from IBM.",
"In the D-Wave’s regular (forward) quantum annealing computing approach, we begin with a massive amount of data that must be mapped to an energy space.",
"This mapping process is actually a mathematical strategy, like simulated annealingA metaheuristic to approximate global optimization in a large search space.",
"or parallel temperingA super simulated annealing where a system at high temperature can supply new local optimizers to a system at low temperature, allowing tunneling and improving convergence to a global optimum..",
"This procedure allows to map a highly quantum mechanical state as a superposition of all the potential solutions.",
"Then the D-Wave machine slowly fades the quantum state and quantum tunnelingEffect where a particle crosses through a classically forbidden potential energy barrier., superposition occurs, and entanglement and coherence manages interactions.",
"As the quantum mechanical wave function laid across possible solutions, it shows the solution sets that are most accurate.",
"Nowadays, D-Wave makes quantum leap with reverse annealing.",
"It is now possible to give results from classical algorithms to the quantum annealer and work backwards.",
"The system starts with the classical state, then falls back in the annealing process introducing quantum dynamics, rather than coming from a superposition of all possible states (candidate states) with equal weights, which is a more similar approach to our implementation."
]
] | 1808.08429 |
[
[
"A Simplified Weak Galerkin Finite Element Method: Algorithm and Error\n Estimates"
],
[
"Abstract In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations.",
"The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method.",
"A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions.",
"Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions."
],
[
"Introduction",
"This paper is concerned with the development of a simplified formulation for the weak Galerkin finite element method for second order elliptic equations.",
"For simplicity, consider the model problem that seeks an unknown function $u=u(x)$ satisfying $-\\nabla \\cdot (\\alpha \\nabla u) + {\\beta }\\cdot \\nabla u + cu&=&f\\quad {\\rm in}\\ \\Omega \\\\u&=&g\\quad {\\rm on}\\ \\partial \\Omega $ where $\\Omega $ is a bounded polytopal domain in $\\mathbb {R}^d \\;(d\\ge 2)$ with boundary $\\partial \\Omega $ , $\\alpha =\\alpha (x)$ is the diffusion coefficient, ${\\beta }={\\beta }(x)$ is the convection, and $c=c(x)$ is the reaction coefficient in relevant applications.",
"We assume that $\\alpha $ is sufficient smooth, ${\\beta }\\in [W^{1,\\infty }(\\Omega )]^d$ , and $c$ is piecewise smooth with respect to a partition of the domain.",
"For well-posedness of the problem (REF )-(), we assume $f=f(x)\\in L^2(\\Omega )$ , $g=g(x)\\in H^{\\frac{1}{2}}(\\partial \\Omega )$ , and $c-\\frac{1}{2}\\nabla \\cdot {\\beta }\\ge 0,\\qquad \\alpha (x) \\ge \\alpha _0 \\qquad \\forall x \\in \\Omega $ for a constant $\\alpha _0>0$ .",
"The model problem (REF )-() arises from many scientific applications such as fluid flow in porous media.",
"Mostly importantly, this model problem has served, and still serves, the scientific computing community as a testbed in the search and design of new and efficient computational algorithms for partial differential equations.",
"The classical Galerkin finite element method (see, e.g., [10], [28], [16]) is particularly a numerical technique originated from the study of elliptic problems closed related to (REF )-() or its variations.",
"In the last three decades, various finite element methods using discontinuous trial and test functions, including discontinuous Galerkin (DG) methods and weak Galerkin (WG) methods, have been developed for numerical solutions of partial differential equations.",
"These developments were often tested over testbed problems such as (REF )-() before they were generalized or applied to more complex problems in science and engineering.",
"The DG method, also known as the interior penalty method in different contexts, was originated in early 70s of the last century for a numerical study of model problems such as (REF )-(); see, e.g., [3], [14], [25], [38] for early incubations and [1], [13], [17], [27] for a detailed discussion and recent developments.",
"The weak Galerkin finite element method is a recently developed discretization framework for partial differential equations [36], [37], [24], [34].",
"With new concepts referred to as weak differential operators (e.g., weak gradient, weak curl, weak Laplacian etc.)",
"and weak continuity through the use of various stabilizers, the method allows the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent or free of locking [33].",
"For the convection-diffusion-reaction equation (REF )-(), the recent work in the context of weak Galerkin includes the algorithm developed and analyzed in [9], the one in [20] for singularly perturbed problems, and an earlier one in [39].",
"The WG finite element method has been rapidly developed and applied to several different types of problems, including second order elliptic problems, the Stokes and Navier-Stokes equations, the biharmonic and elasticity equations, div-curl systems and the Maxwell's equations, etc.",
"The latest development of the WG methods is the prime-dual formulation for problems that are either nonsymmetric or do not have variational forms friendly for numerical use.",
"Details on the new developments can be found in [30] for second order elliptic equations in nondivergence form, [31] for the Fokker-Planck equation, and [32] for elliptic Cauchy problems.",
"The typical WG method for the model problem (REF )-() seeks weak finite element approximations $u_h=\\lbrace u_0, u_b\\rbrace $ satisfying $u_b|_{\\partial \\Omega } = Q_b g$ and $S(u_h, v)+ (\\alpha \\nabla _w u_h, \\nabla _w v) + ({\\beta }\\cdot \\nabla _w u_h, v_0) + (c u_0, v_0) = (f,v_0)$ for all test functions $v=\\lbrace v_0, v_b\\rbrace $ satisfying $v_b|_{\\partial \\Omega }=0$ , where $Q_bg$ is an interpolation of the Dirichlet boundary data, $\\nabla _w$ is the discrete weak gradient operator, and $S(\\cdot ,\\cdot )$ is a properly selected stabilizer that gives weak continuities for the numerical solutions.",
"The numerical solution $u_h$ consists of two components: the approximation $u_0$ on each element and the approximation $u_b$ on the boundary of each element.",
"To reduce the computational complexity, some hybridized formulations have been introduced in [22], [29] for the method when applied to the diffusion equation and the biharmonic equation through the elimination of the degrees of freedom associated with the unknown function $u_0$ locally on each element.",
"In the superconvergence study for WG [18] on rectangular elements, this hybridized formulation was further simplified in the description of the numerical algorithm, yielding a simplified weak Galerkin (SWG) finite element scheme for the diffusion equation.",
"In our further investigation of the SWG to the convection-diffusion-reaction equation (REF ), we came to the conclusion that SWG represents a new discretization scheme that is different from the usual WG through a simple elimination of the unknown $u_0$ .",
"As a result, we believe that a systematic study of the SWG for the convection-diffusion-reaction problem (REF )-() should be conducted for its stability and convergence.",
"This paper is in response to this observation and shall provide a mathematical theory for the stability and the convergence of the simplified weak Galerkin finite element method for the model problem (REF )-().",
"We believe that the result of this paper can be extended to other types of modeling equations.",
"The paper is organized as follows: In Section , we shall describe the simplified weak Galerkin finite element method for (REF )-() on general polygonal partitions.",
"In Section , we shall present a computational formula for the element stiffness matrices and the element load vectors from SWG.",
"In Section , we provide a mathematical theory for the stability and well-posedness of the SWG scheme.",
"Sections and are devoted to a discussion of the error estimates in a discrete $H^1$ and the $L^2$ norm for the numerical solutions.",
"Finally, in Section , we present some numerical results to demonstrate the efficiency and accuracy of the SWG method.",
"Throughout the rest of the paper, we assume $d=2$ and shall use the standard notations for Sobolev spaces and norms [10], [16].",
"For any open set $D\\subset \\mathbb {R}^{2}$ , $\\Vert \\cdot \\Vert _{s,D}$ and $(\\cdot ,\\cdot )_{s,D}$ denote the norm and inner-product in the Sobolev space $H^s(D)$ consisting of square integrable partial derivatives up to order $s$ .",
"When $s=0$ or $D=\\Omega $ , we shall drop the corresponding subscripts in the norm and inner-product notation."
],
[
"Algorithm on Polymesh",
"Assume that the domain is of polygonal type and is partitioned into non-overlap polygons ${\\mathcal {T}}_h=\\lbrace T\\rbrace $ that are shape regular.",
"For each $T\\in {\\mathcal {T}}_h$ , denote by $h_T$ its diameter and by $N$ the number of edges.",
"For each edge $e_i, \\ i=1,\\ldots , N$ , denote by $M_i$ the midpoints and ${\\bf n}_i$ the outward normal direction of $e_i$ (see Fig.",
"REF for an illuatration).",
"The meshsize of ${\\mathcal {T}}_h$ is defined as $h=\\max _{T\\in {\\mathcal {T}}_h} h_T$ .",
"Let $v_b$ be a piecewise constant function defined on the boundary of $T$ , i.e., $v_b|_{e_i} =v_{b,i},$ with $v_{b,i}$ being a constant.",
"We define the weak gradient of $v_b$ on $T$ by: $\\nabla _w v_b:=\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\bf {n_i},$ where $|e_i|$ is the length of the edge $e_i$ and $|T|$ is the area of the element $T$ .",
"It is not hard to see that the weak gradient $\\nabla _w v_b$ satisfies the following equation: $(\\nabla _w v_b, \\mathbf {\\phi })_T=\\langle v_b, \\mathbf {\\phi }\\cdot {\\bf n}\\rangle _{\\partial T}$ for all constant vector $\\mathbf {\\phi }$ .",
"Here and in what follows of the paper, $\\langle \\cdot ,\\cdot \\rangle _{\\partial T}$ stands for the usual inner product in $L^2({\\partial T})$ .",
"Denote by $W(T)$ the space of piecewise constant functions on ${\\partial T}$ .",
"The global finite element space $W({\\mathcal {T}}_h)$ is constructed by patching together all the local elements $W(T)$ through single values on interior edges.",
"The subspace of $W_h({\\mathcal {T}}_h)$ consisting of functions with vanishing boundary value is denoted as $W_h^0({\\mathcal {T}}_h)$ .",
"We use the conventional notation of ${P}_j(T)$ for the space of polynomials of degree $j\\ge 0$ on $T$ .",
"For each $v_b\\in W(T)$ , we associate it with a linear extension in $T$ , denoted as ${\\mathfrak {s}}(v_b)\\in {P}_1 (T)$ , satisfying $\\sum _{i=1}^{N}({\\mathfrak {s}}(v_b)(M_i) -v_{b,i})\\phi (M_i)|e_i|=0,\\quad \\forall \\; \\phi \\in {P}_1(T).$ It is easy to see that ${\\mathfrak {s}}(u_b)$ is well defined by (REF ), and its computation is local and straightforward.",
"In fact, ${\\mathfrak {s}}(u_b)$ can be viewed as an extension of $u_b$ from $\\partial T$ to $T$ through a least-squares fitting.",
"Figure: An illustrative polygonal element.On each element $T \\in {\\mathcal {T}}_h $ , we introduce the following bilinear forms: $a_T(u_b,v_b)&:= & (\\alpha \\nabla _w u_b, \\nabla _w v_b)_T, \\\\b_T(u_b,v_b)&:= & ({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T,\\\\c_T(u_b,v_b)&:= & (c{\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ For simplicity, we set $\\mathcal {B}_T(u_b,v_b):=a_T(u_b,v_b) + b_T(u_b,v_b) + c_T(u_b,v_b)$ for $u_b, v_b \\in W(T)$ .",
"We further introduce the stabilizer $\\begin{split}S_T(u_b,v_b):= & h^{-1}\\sum _{i=1}^N ({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\= & h^{-1}\\langle Q_b{\\mathfrak {s}}(u_b)-u_{b},Q_b{\\mathfrak {s}}(v_b)-v_{b}\\rangle _{\\partial T},\\end{split}$ where $Q_b$ is the $L^2$ projection operator onto $W(T)$ ; namely $Q_b u$ is the average of $u$ on each edge.",
"In particular, $Q_b(g)$ is well-defined and takes the average of the Dirichlet data on each boundary edge.",
"SWG Algorithm 2.1 The simplified weak Galerkin (SWG) scheme for the elliptic equation (REF )-() seeks $u_b \\in W_h({\\mathcal {T}}_h)$ satisfying $u_b = Q_b(g)$ on $\\partial \\Omega $ and ${\\mathcal {A}}(u_b, v_b) =(f, {\\mathfrak {s}}(v_b))\\qquad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where ${\\mathcal {A}}(u_b,v_b):=\\kappa S(u_b,v_b)+ {\\mathcal {B}}(u_b, v_b)$ , $S(u_b,v_b) &=& \\sum _{T\\in {\\mathcal {T}}_h} S_T(u_b,v_b),\\\\{\\mathcal {B}}(u_b,v_b)&=& \\sum _{T\\in {\\mathcal {T}}_h} \\mathcal {B}_T(u_b, v_b)$ are bilinear forms in $W_h({\\mathcal {T}}_h)$ and $(f,{\\mathfrak {s}}(v_b)):=\\sum _{T\\in {\\mathcal {T}}_h} (f, {\\mathfrak {s}}(v_b))_T$ is a linear form in $W_h({\\mathcal {T}}_h)$ ."
],
[
"Element Stiffness Matrices",
"The simplified weak Galerkin finite element method (REF ) is user-friendly in computer implementation.",
"In this section, we present a formula for the computation of the element stiffness matrices and the element load vector on general polygonal elements.",
"Let $T\\in {\\mathcal {T}}_h$ be a polygonal element of $N$ sides.",
"Denote by $X_{u_b}$ the vector representation of $u_b$ given by $(u_{b,1},u_{b,2},\\ldots ,u_{b,N})^T$ .",
"Then, the element stiffness matrix and the element load vector for the SWG scheme (REF ) are given in a block matrix form as follows: $(\\kappa h^{-1}A^T + B + R + C)X_{u_b} \\cong F,$ where the block components in (REF ) are given by: (1) $A:=\\lbrace a_{i,j}\\rbrace _{i,j=1}^N = E- EM(M^TEM)^{-1}M^TE$ , (2) $B:=\\lbrace b_{i,j}\\rbrace _{i,j=1}^N$ , with $b_{i,j} = (\\alpha \\mathbf {n}_i,\\mathbf {n}_j)_{T}\\displaystyle \\frac{|e_i||e_j|}{|T|^2}$ , (3) $R:=\\lbrace r_{ij}\\rbrace _{i,j=1}^N$ , with $r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT$ , (4) $C:=\\lbrace c_{ij}\\rbrace _{i,j=1}^N$ , with $c_{ij}= \\int _T c\\zeta _j\\zeta _i dT$ , (5) $F:=\\lbrace f_{i}\\rbrace _{i=1}^N$ , with $f_{i} = \\int _{T}f(x,y) \\zeta _i(x,y) dT$ , (6) $D:=\\lbrace d_{j,i}\\rbrace _{3\\times N}=(M^TEM)^{-1}M^TE$ and $\\zeta _i = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T)$ , (7) $M$ and $E$ are given by $M=\\begin{bmatrix}1 & x_{1} - x_T & y_{1} - y_T\\\\1 & x_{2} - x_T & y_{2} - y_T\\\\\\vdots & \\vdots & \\vdots \\\\1 & x_{N} - x_T & y_{N} - y_T\\\\\\end{bmatrix}_{N\\times 3},\\;E=\\begin{bmatrix}|e_1| & & & \\\\& |e_2| & & \\\\& & \\ddots & \\\\& & & |e_N| \\\\\\end{bmatrix}_{N\\times N}.$ Here $M_T=(x_T,y_T)$ is any point on the plane (e.g., the center of $T$ as a specific case), $(x_i, y_i)$ is the midpoint of $e_i$ , $|e_i|$ is the length of edge $e_i$ , $\\mathbf {n}_i$ is the unit outward normal vector on $e_i$ , and $|T|$ is the area of the element $T$ .",
"From (REF ), the element stiffness matrix on $T\\in {\\mathcal {T}}_h$ consists of two sub-matrices corresponding to the following forms: $S_T(u_b,v_b)\\ \\mbox{and } {\\mathcal {B}}_T(u_b, v_b).$ The bilinear form ${\\mathcal {B}}_T(\\cdot ,\\cdot )$ is composed of three bilinear forms given by (REF ).",
"The rest of this section is devoted to a computation of the element stiffness matrices for each of the bilinear forms involved."
],
[
"The stiffness matrix for $S_T(\\cdot ,\\cdot )$",
"For the element stiffness matrix corresponding to $S_T(u_b,v_b)$ , the key is to compute ${\\mathfrak {s}}(u_b)$ and ${\\mathfrak {s}}(v_b)$ which can be accomplished through its definition (REF ); readers are referred to [21] for a detailed derivation.",
"Specifically, let $M_T=(x_T,y_T)$ be the center of T (or any point on the plane), the extension ${\\mathfrak {s}}(u_b)$ can be represented as follows: ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ From ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T)$ and (REF ), we have $\\begin{bmatrix}{\\mathfrak {s}}(u_{b})(M_1) \\\\{\\mathfrak {s}}(u_{b})(M_2) \\\\\\vdots \\\\{\\mathfrak {s}}(u_{b})(M_N) \\\\\\end{bmatrix}=M\\begin{bmatrix}\\gamma _0\\\\\\gamma _1\\\\\\gamma _2\\\\\\end{bmatrix}=M(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ Let $v_b\\in W(T)$ be the basis function corresponding to the edge $e_j$ of $T$ : $v_b=\\left\\lbrace \\begin{array}{lllll}1, \\qquad \\text{ on } e_j,\\\\0, \\qquad \\text{ otherwise}.\\\\\\end{array}\\right.$ Then the coefficient $(\\tilde{\\gamma }_0,\\tilde{\\gamma }_1,\\tilde{\\gamma }_2)^T$ for ${\\mathfrak {s}}(v_b)$ is given by $\\begin{bmatrix}\\tilde{\\gamma }_0 \\\\\\tilde{\\gamma }_1 \\\\\\tilde{\\gamma }_2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1}\\\\\\vdots \\\\v_{b,j}\\\\\\vdots \\\\v_{b,N}\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}0\\\\\\vdots \\\\1\\\\\\vdots \\\\0\\\\\\end{bmatrix}\\triangleq \\begin{bmatrix}d_{1,j}\\\\d_{2,j}\\\\d_{3,j}\\\\\\end{bmatrix}.$ It follows that $\\begin{split}S_T(u_b,v_b)=&h^{-1}\\sum _{i=1}^N({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\=&h^{-1}\\sum _{i=1}^N (u_{b,i}-{\\mathfrak {s}}(u_b)(M_i))v_{b,i}|e_i|\\\\=&h^{-1}\\left((I_N-M(M^TEM)^{-1}M^TE)\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}\\right)_j |e_j| \\\\=&h^{-1}\\sum _{i=1}^N a_{j,i}u_{b,i},\\end{split}$ where $I_N$ is the identity matrix of size $N\\times N$ ."
],
[
"The stiffness matrix for $a_T(\\cdot ,\\cdot )$",
"For a computation of the element stiffness matrix corresponding to the bilinear form $a_T(u_b,v_b)=(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T$ , we have from the weak gradient formula (REF ) that $(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T&=&(\\alpha \\displaystyle \\frac{1}{|T|}\\sum _{j=1}^N u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}\\mathbf {n}_i|e_i|)_{T}\\\\&=&\\sum _{i,j=1}^N(\\alpha \\displaystyle \\frac{1}{|T|} u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|} v_{b,i}\\mathbf {n}_i|e_i|)_{T} \\nonumber \\\\&=&\\sum _{i,j=1}^N \\frac{|e_j||e_i|}{|T|^2}(\\alpha \\mathbf {n}_j,\\mathbf {n}_i)_{T}u_{b,j}v_{b,i},\\nonumber \\\\&=&\\sum _{i,j=1}^N b_{i,j}u_{b,j}v_{b,i},\\nonumber $ which leads to the block matrix $B$ in the element stiffness matrix."
],
[
"The stiffness matrix for $b_T(\\cdot ,\\cdot )$",
"Recall that the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $b_T(u_b, v_b)=({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T.$ Note that the extension ${\\mathfrak {s}}(v_b)$ has the following representation: ${\\mathfrak {s}}(v_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1} \\\\v_{b,2} \\\\\\vdots \\\\v_{b,N} \\\\\\end{bmatrix}.$ Thus, with $D=(M^TEM)^{-1}M^TE$ , we have from the weak gradient formula (REF ) that $\\begin{split}& ({\\beta }\\cdot \\nabla _w u_b,{\\mathfrak {s}}(v_b))_T \\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N ({\\beta }\\cdot \\mathbf {n}_j, d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))_T |e_j| u_{b,j} v_{b,i}\\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N \\int _T {\\beta }\\cdot \\mathbf {n}_j (d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))dT |e_j| u_{b,j} v_{b,i}.\\end{split}$ For simplicity, we introduce the following functions: $\\zeta _i(x,y) = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T),\\qquad i=1,\\ldots , N.$ Then, the equation (REF ) indicates that the element stiffness matrix corresponding to the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $R=\\lbrace r_{ij}\\rbrace _{N\\times N},\\ \\ r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT.$"
],
[
"The stiffness matrix for $c_T(\\cdot ,\\cdot )$",
"Recall that the bilinear form $c_T(\\cdot ,\\cdot )$ is given by $c_T(u_b, v_b)=(c {\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ Thus, the element stiffness matrix corresponding to $c_T(\\cdot ,\\cdot )$ has the following formula: $C=\\lbrace c_{ij}\\rbrace _{N\\times N},\\quad c_{ij} = \\int _T c(x,y) \\zeta _j\\zeta _i dT,$ where $\\zeta _i$ is the function defined in (REF )."
],
[
"The element load vector",
"Finally, the element load vector can be obtained from $(f, {\\mathfrak {s}}(v_b))_T& =& \\int _{T}f {\\mathfrak {s}}(v_b)dT \\\\& =& \\int _{T}f(x,y) (d_{1,i} + d_{2,i}(x-x_T) + d_{3,i}(y-y_T)) dT\\\\& =& \\int _{T}f(x,y) \\zeta _i(x,y) dT$ for $i=1,\\ldots , N$ ."
],
[
"Stability and Well-Posedness",
"The SWG scheme (REF ) can be derived from the classical weak Galerkin finite element method [36], [24], [37] by eliminating the degrees of freedom associated with the interior of each element when ${\\beta }=0$ and $c=0$ .",
"But for the general case of ${\\beta }$ and $c$ , the SWG finite element method (REF ) is different from the weak Galerkin schemes in existing literature.",
"It is thus necessary to provide a mathematical theory for the stability and well-posedness of the numerical scheme (REF ).",
"Let ${\\mathcal {T}}_h$ be a shape-regular polygonal partition of the domain $\\Omega $ .",
"There exists a constant $C$ such that $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),\\\\\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}^2&\\le & C h \\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1} \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).$ Moreover, the following Poincaré-type estimate holds true: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right).$ From the formula (REF ) for the weak gradient, we have for any constant vector $\\mathbf {\\phi }$ that $\\begin{split}(\\nabla _w v_b, \\mathbf {\\phi })_T=&\\langle v_b, \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ (\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T,\\end{split}$ which gives $(\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T = (\\nabla _w v_b, \\mathbf {\\phi })_T - \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}.$ Hence, by letting $\\mathbf {\\phi } = \\nabla {\\mathfrak {s}}(v_b)$ we arrive at $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 \\le C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),$ which verifies (REF ).",
"Next, from the usual error estimate for the $L^2$ projection operator $Q_b$ and the estimate (REF ), we have $\\begin{split}\\Vert {\\mathfrak {s}}(v_b) - Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}} \\le & C h^2 \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2 \\\\\\le & C h \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert ^2_T \\\\\\le & C\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).\\end{split}$ It follows that $\\begin{split}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\le & \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}} + \\Vert {\\mathfrak {s}}(v_b)-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\\\\\le & C \\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{1/2},\\end{split}$ which verifies the estimate ().",
"To derive the inequality (REF ), we note the following discrete Poincaré inequality: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 \\le C\\sum _{T\\in {\\mathcal {T}}_h} \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2 + h_T^{-1}\\Vert {\\mathfrak {s}}(v_b) - v_b\\Vert _{\\partial T}^2\\right).$ Combining the above estimate with (REF ) and (REF ) gives rise to the desired inequality (REF ).",
"This completes the proof of the lemma.",
"On each element $T\\in {\\mathcal {T}}_h$ , the following identity holds true: $\\begin{split}b_T(v_b,v_b) =&\\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where $\\overline{{\\mathfrak {s}}(v_b){\\beta }}$ is the average of ${\\mathfrak {s}}(v_b){\\beta }$ on the element $T$ .",
"From the formula (REF ), we have $\\begin{split}b_T(v_b,v_b) =& ({\\beta }\\cdot \\nabla _w v_b, {\\mathfrak {s}}(v_b))_T \\\\=& (\\nabla _w v_b, {{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& (\\nabla _w v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& \\langle v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Note that $\\begin{split}\\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}&= (\\nabla {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\&= (\\nabla {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b){\\beta })_T\\\\&= \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T.\\end{split}$ Substituting the above identity into (REF ) yields $\\begin{split}b_T(v_b,v_b) =& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle v_b,{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ + \\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T,\\end{split}$ which leads to the identify (REF ).",
"In the finite element space $W_h({\\mathcal {T}}_h)$ , we introduce the following semi-norm: ${|\\!|\\!|} v_b{|\\!|\\!|}^2: = \\sum _{T\\in {\\mathcal {T}}_h} \\left( \\kappa S_T(v_b, v_b) + a_T(v_b, v_b) \\right)$ We claim that ${|\\!|\\!|}\\cdot {|\\!|\\!|}$ defines a norm in the closed subspace $W_h^0({\\mathcal {T}}_h)$ .",
"It suffices to show that $v_b \\equiv 0$ for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ satisfying ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ .",
"In fact, if ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ , then from (REF ) we have $\\kappa \\sum _{T}S_T(v_b,v_b)+\\sum _{T}(\\alpha \\nabla _w v_b,\\nabla _w v_b)_T =0.$ It follows that on each element $T\\in {\\mathcal {T}}_h$ $\\nabla _w v_b=0,\\quad (v_b - {\\mathfrak {s}}(v_b))(M_i)=0$ for $i=1,\\ldots , N$ .",
"Thus, $\\displaystyle \\nabla {\\mathfrak {s}}(v_b)=\\frac{1}{|T|}\\sum _{i=1}^N {\\mathfrak {s}}(v_b)(M_i)|e_i|\\mathbf {n}_i=\\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\mathbf {n}_i=\\nabla _w v_b=0,$ so that ${\\mathfrak {s}}(v_b)$ has constant value on each element $T\\in {\\mathcal {T}}_h$ .",
"By using (REF ) we see that $v_b={\\mathfrak {s}}(v_b)=const$ on each edge, which, together with the fact that $v_b=0$ on $\\partial \\Omega $ , leads to $v_b\\equiv 0$ in $\\Omega $ .",
"For the model problem (REF ), assume that ${\\beta }\\in W^{1,\\infty }(\\Omega )$ and the condition (REF ) is satisfied.",
"Then, the bilinear form $\\kappa S(\\cdot ,\\cdot )+{\\mathcal {B}}(\\cdot ,\\cdot )$ is bounded and coercive in the finite element space $W_h^0({\\mathcal {T}}_h)$ ; i.e., there exist constants $M$ and $\\Lambda >0$ such that $|\\kappa S(v_b,w_b)+{\\mathcal {B}}(v_b, w_b)| & \\le & M {|\\!|\\!|} v_b{|\\!|\\!|} {|\\!|\\!|} w_b{|\\!|\\!|} \\qquad \\forall v_b, w_b \\in W_h^0({\\mathcal {T}}_h),\\\\\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) & \\ge & \\Lambda {|\\!|\\!|} v_b{|\\!|\\!|}^2\\qquad \\forall v_b\\in W_h^0({\\mathcal {T}}_h),$ provided that the meshsize $h$ of ${\\mathcal {T}}_h$ is sufficiently small.",
"Recall that for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ we have $\\begin{split}{\\mathcal {B}}(v_b, w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} \\left(a_T(v_b,w_b)+ b_T(v_b, w_b) + c_T(v_b,w_b)\\right),\\\\S(v_b,w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} S_T(v_b,w_b).\\end{split}$ The boundedness estimate (REF ) is then straightforward from the usual Cauchy-Schwarz and the inequality (REF ).",
"We shall focus on the derivation of the coercivity inequality () in the rest of the proof.",
"In comparison with (REF ), the key to the coercivity inequality () is to derive an estimate of the following type: $\\sum _{T\\in {\\mathcal {T}}_h} \\left(b_T(v_b, v_b) + c_T(v_b,v_b)\\right)\\ge \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2,$ where $\\eta \\ge 0$ and $\\varepsilon (h)$ is a parameter satisfying $\\varepsilon (h) \\rightarrow 0$ as $h\\rightarrow 0$ .",
"If (REF ) indeed holds true, then we have from (REF ) that $\\begin{split}\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) \\ge & {|\\!|\\!|} v_b {|\\!|\\!|}^2 + \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2\\\\\\ge & (1- \\varepsilon (h)) {|\\!|\\!|} v_b{|\\!|\\!|}^2,\\end{split}$ which implies the coercivity () for sufficiently small $h$ .",
"It remains to derive the estimate (REF ).",
"To this end, we sum up the identify in Lemma to obtain $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} b_T(v_b,v_b)= & -\\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h} ( \\nabla \\cdot {\\beta }{\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where we have used the fact that $\\sum _{T\\in {\\mathcal {T}}_h} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}=0$ .",
"Thus, $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} (b_T(v_b,v_b) + c_T(v_b,v_b))= & \\sum _{T\\in {\\mathcal {T}}_h} ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Next, from () we have $\\begin{split}\\left|\\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| & \\le C \\left( h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b - Q_b {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right) \\\\& \\le C h \\sum _{T\\in {\\mathcal {T}}_h} \\left( a_T(v_b,v_b) + S_T(v_b,v_b)\\right)\\\\&\\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.\\end{split}$ As to the last term in (REF ), we have $\\begin{split}& \\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\\\\\le & \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\Vert \\overline{{\\mathfrak {s}}(v_b){\\beta }} - {{\\mathfrak {s}}(v_b){\\beta }}\\Vert _{\\partial T}\\\\\\le & C h^{\\frac{1}{2}} \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T\\right)\\\\\\le & C h \\left(\\sum _{T\\in {\\mathcal {T}}_h} h^{-1}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right)^{\\frac{1}{2}} \\left(\\sum _{T\\in {\\mathcal {T}}_h} \\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T^2 + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2\\right)\\right)^{\\frac{1}{2}}\\end{split}$ Combining the estimates (REF ), (), and (REF ) with (REF ) yields $\\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.$ Now by substituting (REF ) and (REF ) into (REF ) we obtain the inequality (REF ) with $\\eta = ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))\\ge 0$ and $\\varepsilon (h) = Ch$ .",
"This completes the proof of the lemma.",
"The following is a direct application of Lemma .",
"Under the assumptions of Lemma , there exists a small, but fixed number $h_0>0$ , such that the numerical scheme (REF ) has one and only one solution $u_b\\in W_h({\\mathcal {T}}_h)$ for sufficiently fine finite element partitions ${\\mathcal {T}}_h$ satisfying $h\\le h_0$ .",
"It suffices to show that the homogeneous problem has only the trivial solution.",
"To this end, let $u_b \\in W_h^0({\\mathcal {T}}_h)$ , be the solution of scheme (REF ) with homogeneous data $f=0$ and $g=0$ .",
"By taking $v_b = u_b$ in (REF ) we obtain $\\kappa S(u_b,u_b)+{\\mathcal {B}}(u_b,u_b)=0,$ which, from the coercivity inequality (), gives $\\Lambda {|\\!|\\!|} u_b{|\\!|\\!|}^2 \\le \\kappa S(u_b,u_b) + {\\mathcal {B}}(u_b,u_b)=0$ , and hence $u_b\\equiv 0$ for sufficiently small $h$ ."
],
[
"Error Estimates in $H^1$",
"Let $u$ be the exact solution of the model problem (REF )-() and $u_b \\in W_h^0({\\mathcal {T}}_h)$ be the numerical approximation arising from the SWG scheme (REF ).",
"Let $Q_b u$ be the $L^2$ projection of $u$ in the space $W_h^0({\\mathcal {T}}_h)$ .",
"The error function refers to the difference between the $L^2$ projection and the SWG approximation: $e_b := Q_b u - u_b,$ The goal of this section is to establish an estimate for the error function $e_b$ in a discrete Sobolev norm.",
"Let us first state an error equation which plays an important role in the convergence analysis of the SWG scheme.",
"Assume that the coefficient $\\alpha $ of the model problem (REF )-() has piecewise constant values with respect to the finite element partition ${\\mathcal {T}}_h$ .",
"Then the following equation holds true $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b,v_b)= \\ell _u(v_b) \\quad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where $\\ell _u(\\cdot )$ is a linear functional given by $\\begin{split}\\ell _u(v_b):=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}+\\kappa S(Q_bu,v_b)\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ where $Q_0(\\nabla u)$ is the $L^2$ projection of $\\nabla u$ in the space $[P_0({\\mathcal {T}}_h)]^2$ , and $\\mathbf {n}$ is the outward normal vector on ${\\partial T}$ .",
"We first consider the weak gradient of $Q_b u$ , for any constant vector $\\mathbf {\\phi }$ , we have $(\\nabla _w Q_b u,\\mathbf {\\phi })_{T}&=& \\langle Q_b u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}= \\langle u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}\\\\&=& (\\nabla u, \\mathbf {\\phi })_{T}= (Q_0(\\nabla u), \\mathbf {\\phi })_{T},$ which implies $\\nabla _w Q_b u \\equiv Q_0(\\nabla u)$ .",
"Thus, for any $v_b \\in W_h^0(T_h)$ , we have $\\begin{split}&(\\alpha \\nabla _w Q_b u,\\nabla _w v_b)= \\sum _{T}(\\alpha Q_0(\\nabla u),\\nabla _w v_b)_T \\\\= & \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}-\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} + \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(\\alpha \\nabla u,\\nabla {\\mathfrak {s}}(v_b))_{T} \\\\=& \\sum _{T}\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))_{T} + \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& (-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))+ \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}.\\end{split}$ Next, from $\\nabla _w (Q_bu) = Q_0(\\nabla u)$ , we have $\\begin{split}\\sum _T ({\\beta }\\cdot \\nabla _w (Q_bu), {\\mathfrak {s}}(v_b))_T = & \\sum _T ({\\beta }\\cdot (Q_0 \\nabla u), {\\mathfrak {s}}(v_b))_T \\\\= & \\sum _T ({\\beta }\\cdot \\nabla u, {\\mathfrak {s}}(v_b))_T + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T,\\end{split}$ and $\\sum _T (c{\\mathfrak {s}}(Q_bu), {\\mathfrak {s}}(v_b))_T = (cu, {\\mathfrak {s}}(v_b)) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).$ The sum of (REF ), (REF ), and (REF ) gives rise to $\\begin{split}{\\mathcal {B}}(Q_b u, v_b) = &\\ (f, {\\mathfrak {s}}(v_b)) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ which, combined with $ (f, {\\mathfrak {s}}(v_b) = \\kappa S(u_b, v_b) + {\\mathcal {B}}(u_b,v_b)$ , leads to $\\begin{split}{\\mathcal {B}}(Q_b u - u_b, v_b) = &\\ \\kappa S(u_b,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ and $\\begin{split}&\\kappa S(Q_bu-u_b,v_b)+{\\mathcal {B}}(Q_b u - u_b, v_b) \\\\= &\\ \\kappa S(Q_bu,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).\\end{split}$ This completes the proof of the lemma.",
"Remark 5.1 It should be pointed out that Lemma can be extended to the case when $\\alpha $ is in $L^\\infty (\\Omega )$ and piecewise smooth with respect to the finite element partition ${\\mathcal {T}}_h$ .",
"Detailed analysis can be established by following the approach presented in [35].",
"The following result is concerned with the error estimate for the SWG numerical solutions in a discrete $H^1$ norm.",
"Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).",
"Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and that (REF ) is satisfied.",
"Then, the following error estimate holds true $\\kappa S(e_b ,e_b) +(\\alpha \\nabla _w e_b, \\nabla _w e_b )\\le Ch^2\\Vert u\\Vert _2^2,$ provided that the meshsize $h$ is sufficiently small.",
"Consequently, we have $\\Vert \\nabla _w u_b- \\nabla u\\Vert _0 \\le Ch \\Vert u\\Vert _2, $ The proof is based on the error equation (REF ) through a thorough analysis for the linear functional $\\ell _u(\\cdot )$ given in (REF ).",
"For the first term on the righ-hand side of (REF ), from the usual Cauchy-Schwarz inequality we have $\\begin{split}&\\ |\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & \\ \\Vert \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n}\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}\\\\\\le & \\ \\Vert \\alpha \\Vert _{\\infty } \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}.\\end{split}$ Now using the estimate () in the above inequality and then summing over all the element $T\\in {\\mathcal {T}}_h$ we arrive at the following: $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h}|\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & C \\Vert \\alpha \\Vert _{\\infty } \\sum _{T\\in {\\mathcal {T}}_h} \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{\\partial T}\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T +\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{\\frac{1}{2}}\\\\\\le & C\\Vert \\alpha \\Vert _{\\infty }\\left(\\Vert \\nabla u -Q_0(\\nabla u)\\Vert ^2_{0} + h^2\\Vert \\nabla ^2 u \\Vert _{0}^2\\right)^{\\frac{1}{2}}\\left( \\Vert \\nabla _w v_b\\Vert ^2+ \\kappa S(v_b, v_b)\\right)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b {|\\!|\\!|}.\\end{split}$ As to the second term on the right hand side of (REF ), we have $\\begin{split}|S(Q_bu,v_b)| =&\\sum _{T}h^{-1}\\langle Q_bu-Q_b{\\mathfrak {s}}(Q_bu),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu -Q_b(Q_1 u),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=&\\sum _{T}h^{-1}\\langle u -Q_1 u,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\\\le &\\left(\\sum _{T} h^{-1} \\int _{\\partial T}|u-Q_1u|^2ds \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C\\left(h^{-2}\\Vert u-Q_1 u\\Vert ^2 + \\Vert u-Q_1 u\\Vert _1^2 \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ The third term on the right hand side of (REF ) can be bounded by using the usual error estimate for $L^2$ projections as follows: $\\begin{split}|((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })| = & |((Q_0-I) \\nabla u, (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta }))| \\\\\\le & \\Vert (Q_0-I) \\nabla u\\Vert \\ \\Vert (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta })\\Vert \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert + \\Vert {\\mathfrak {s}}(v_b)\\Vert \\right) \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert {|\\!|\\!|} v_b{|\\!|\\!|},\\end{split}$ where we have used the estimates (REF ) and (REF ) in the last line.",
"The last term on the right hand side of (REF ) can be estimated as follows: $\\begin{split}|(c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b))|\\le & \\Vert c\\Vert _\\infty \\Vert {\\mathfrak {s}}(Q_bu)-u\\Vert \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C (\\Vert {\\mathfrak {s}}(Q_bu)-Q_1 u\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu)-{\\mathfrak {s}}(Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu-Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C h^2 \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ Substituting the estimates (REF )-(REF ) into the error equation (REF ) yields $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b, v_b) \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|},$ which, together with the coercivity (), leads to $\\Lambda {|\\!|\\!|} e_b{|\\!|\\!|}^2 \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}.$ The last inequality implies the error estimate (REF ).",
"Finally, from the triangle inequality and the error estimate (REF ), we obtain $\\begin{split}\\Vert \\nabla _w u_b - \\nabla u\\Vert \\le & \\Vert \\nabla _w (u_b - Q_b u)\\Vert + \\Vert \\nabla _w (Q_b u) - \\nabla u\\Vert \\\\= & \\Vert \\nabla _w e_b\\Vert + \\Vert Q_0 (\\nabla u) - \\nabla u\\Vert \\\\\\le & C h \\Vert u\\Vert _2,\\end{split}$ which gives rise to (REF ).",
"This completes the proof of the theorem."
],
[
"Error Estimates in $L^2$",
"We use the usual duality argument to derive an error estimate in $L^2$ for the numerical solutions arising from (REF ).",
"The analysis to be presented is a modified version of those developed in [36], [24], [35].",
"Consider the following auxiliary problem that seeks $\\Phi \\in H_0^1(\\Omega )$ such that $-\\nabla \\cdot (\\alpha \\nabla \\Phi ) - \\nabla \\cdot ({\\beta }\\Phi ) + c\\Phi &=&\\chi \\quad {\\rm in}\\ \\Omega \\\\\\Phi &=&0\\quad {\\rm on}\\ \\partial \\Omega , $ where $\\chi \\in L^2(\\Omega )$ .",
"Assume that the solution of the problem (REF )-() exists and has the $H^2$ -regularity: $\\Vert \\Phi \\Vert _2 \\le C \\Vert \\chi \\Vert ,$ where $C$ is a constant depending only on the domain and the coefficients $\\alpha , {\\beta }$ , and $c$ .",
"Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).",
"Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and the conditions (REF ) and (REF ) are satisfied.",
"Then, the following $L^2$ error estimate holds true $\\Vert u - {\\mathfrak {s}}(u_b)\\Vert \\le Ch^2\\Vert u\\Vert _2,$ provided that the meshsize $h$ is sufficiently small.",
"On each element $T\\in {\\mathcal {T}}_h$ , we test (REF ) against the linear function ${\\mathfrak {s}}(e_b)$ to obtain $(\\chi , {\\mathfrak {s}}(e_b))_T &=& (\\alpha \\nabla \\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + ({\\beta }\\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (Q_0({\\beta }\\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla _w e_b)_T + (Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&& - \\langle \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}- \\langle Q_0({\\beta }\\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}$ By using $Q_0(\\nabla \\Phi ) = \\nabla _w (Q_b\\Phi )$ and $(Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T = ({\\beta }\\cdot \\nabla _w e_b, \\Phi )_T$ in the above equation, we have from summing over all $T\\in {\\mathcal {T}}_h$ that $\\begin{split}(\\chi , {\\mathfrak {s}}(e_b))=& (\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi ) \\\\& - \\sum _T \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}- \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}\\\\& -\\langle {\\beta }\\Phi \\cdot {\\bf n}- Q_0({\\beta }\\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}.\\end{split}$ The last two terms on the right-hand side of (REF ) can be bounded by $C h \\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}$ through the Cauchy-Schwarz inequality.",
"Thus, we have $\\begin{split}|(\\chi , {\\mathfrak {s}}(e_b))|\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi )| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}\\\\\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|},\\end{split}$ where have also used $\\Vert \\Phi -{\\mathfrak {s}}(Q_b\\Phi )\\Vert \\le C h^2\\Vert \\Phi \\Vert _2$ .",
"Now, recall that $(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))= {\\mathcal {B}}(e_b, Q_b\\Phi ),$ and from the error equation (REF ), we have $\\begin{split}{\\mathcal {B}}(e_b, Q_b\\Phi ) = & \\ell _u(Q_b\\Phi ) - \\kappa S(e_b, Q_b\\Phi )\\\\=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}+\\kappa S(u_b,Q_b\\Phi )\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi )),\\end{split}$ The last two terms on the right-hand side of (REF ) have the following estimate: $\\left|((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi ))\\right| \\le C h^2 \\Vert u\\Vert _2 \\Vert \\Phi \\Vert _1.$ The second term, $\\kappa S(u_b,Q_b\\Phi )$ , can be dealt with as follows: $\\begin{split}\\kappa S(u_b,Q_b\\Phi ) = & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), Q_b\\Phi - Q_b{\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\= & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\\\le & \\kappa h^{-1} \\sum _T \\Vert u_b-Q_b{\\mathfrak {s}}(u_b)\\Vert _{\\partial T}\\Vert \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\Vert _{\\partial T}\\\\\\le & C h ({|\\!|\\!|} e_b{|\\!|\\!|} + h\\Vert u\\Vert _2)\\Vert \\Phi \\Vert _2.\\end{split}$ As to the first term, we note from the definition of $Q_b$ and $\\Phi |_{\\partial \\Omega }=0$ that $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= 0.\\end{split}$ Thus, we have $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}\\\\= & \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-\\Phi \\rangle _{\\partial T}\\\\\\le & C h^2 \\Vert u\\Vert _2\\Vert \\Phi \\Vert _2.\\end{split}$ Substituting (REF ), (REF ), and (REF ) into (REF ) yields the following estimate: $|{\\mathcal {B}}(e_b, Q_b\\Phi )| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2,$ which, together with (REF ), leads to $|(\\chi , {\\mathfrak {s}}(e_b))| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2 \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\chi \\Vert ,$ where the regularity assumption (REF ) has been employed in the last inequality.",
"Next, from (REF ) and the $H^1$ error estimate (REF ) in Theorem , we have $|(\\chi , {\\mathfrak {s}}(e_b))| \\le Ch^2 \\Vert u\\Vert _2 \\Vert \\chi \\Vert ,$ which leads to $\\Vert {\\mathfrak {s}}(e_b\\Vert \\le C h^2 \\Vert u\\Vert _2.$ Finally, we arrive at $\\Vert u-{\\mathfrak {s}}(u_b)\\Vert \\le \\Vert u-{\\mathfrak {s}}(Q_bu)\\Vert + \\Vert {\\mathfrak {s}}(e_b)\\Vert \\le C h^2\\Vert u\\Vert _2,$ which completes the proof of the theorem."
],
[
"Numerical Experiments",
"The goal of this section is to numerically verify the error estimates developed in the previous sections for the numerical scheme (REF ).",
"The following metrics are employed to measure the magnitude of the error function: $&&\\text{Discrete $L^2$-norm: }\\\\&&\\Vert u_b-u \\Vert _{0}=h\\left(\\sum _{i=1}^{n+1}\\sum _{j=1}^{n}|u_{i-\\frac{1}{2},j}-u(x_{i-\\frac{1}{2}},y_j)|^2 + \\sum _{i=1}^{n}\\sum _{j=1}^{n+1}|u_{i,j-\\frac{1}{2}}-u(x_{i},y_{j-\\frac{1}{2}})|^2\\right)^{1/2},\\\\&&\\text{Discrete $H^1$-norm: }\\\\&&\\Vert u_b-u\\Vert _1= h \\left(\\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i+\\frac{1}{2},j}-u_{i-\\frac{1}{2},j}}{h} - \\frac{\\partial u}{\\partial x} (x_{i},y_{j})\\right|^2 \\right.",
"\\\\&&\\qquad \\qquad \\qquad \\left.",
"+ \\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i,j+\\frac{1}{2}}-u_{i,j-\\frac{1}{2}}}{h} - \\frac{\\partial u}{\\partial y} (x_{i},y_{j})\\right|^2 \\right)^{1/2} ,\\\\$ Our numerical experiments are conducted for the model problem (REF )-() on polygonal domains.",
"The following set of test cases are considered: $\\left\\lbrace \\begin{split}&u=xy, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u= 3x^2+2xy, \\\\&\\alpha =\\begin{bmatrix}2 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y) + x^2-y^2, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\2\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y), \\\\&\\alpha =\\begin{bmatrix}xy+1 &0\\\\0 & 3xy\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}x^3y+xy+1\\\\3x^2y+xy+2\\end{bmatrix},\\quad c=x^4y^2+xy+1;\\end{split}\\right.$ The right-hand side function $f$ and the Dirichlet boundary data $g$ are chosen to match the exact solution $u=u(x,y)$ for each test case.",
"Table: Error and convergence performance of the SWG scheme () with κ=4.0\\kappa =4.0 and uniform square partitions on the unit square domain Ω=(0,1) 2 \\Omega =(0,1)^2.Table REF shows the performance of the SWG scheme for each of the above test problems with the stabilizer parameter $\\kappa =4$ on uniform square partitions.",
"The results indicate that the numerical approximation is in the machine accuracy for the test problem (REF ) where the exact solution is a bilinear function.",
"For the other three test problems, the numerical solutions have the optimal rate of convergence $r=2$ in the discrete $L^2$ norm and a superconvergence of order ${\\mathcal {O}}(h^2)$ in the discrete $H^1$ norm.",
"The numerical results are consistent with the theoretical prediction in the discrete $L^2$ norm, but they outperform the theory in the discrete $H^1$ norm.",
"It should be pointed out that the superconvergence theory in [18] was developed for the diffusion equation only; but a slight modification of the analysis there will yield a superconvergence of order ${\\mathcal {O}}(h^2)$ for the SWG solutions of the full convection-diffusion equation (REF )-().",
"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."
],
[
"On the influence of the stabilizer parameter",
"The goal of this subsection is to test the influence of the stabilizer parameter $\\kappa $ on the numerical solutions.",
"This part of the numerical experiment considers only the test cases (REF ) and (REF ) with the following six values of $\\kappa =0.01, \\ 0.1, \\ 1.0, \\ 4.0, \\ 6.0, \\ 20.0$ .",
"The case of $\\kappa =0$ is not a viable choice , as it was not covered in the convergence theory.",
"In fact, our computation does not suggest any convergence of the scheme when $\\kappa =0$ .",
"Tables REF -REF illustrate the numerical performance of the SWG scheme with different values of the stabilizer parameter $\\kappa $ .",
"Note that, for both test cases, the rate of convergence deteriorates as $\\kappa $ gets small (e.g.",
"$\\kappa =0.01$ ), particulary on coarse finite element partitions, but the rate of convergence begins to improve when the meshsize $h$ gets small.",
"Optimal rate of convergence and the supercovergence of order ${\\mathcal {O}}(h^2)$ are clearly shown in the tables when $\\kappa $ is away from 0 (e.g., $\\kappa \\ge 0.1$ ).",
"The stability and accuracy of the SWG scheme is insensitive to the value of $\\kappa $ as long as it stays away from 0.",
"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."
],
[
"SWG with general polygonal partitions",
"The SWG scheme was applied to the test problem (REF ) with general polygonal partitions.",
"Table REF shows the error and convergence performance of the scheme on four types of polygonal partitions.",
"The stabilization parameter was set as $\\kappa =4$ in all these tests.",
"Optimal order of convergence in the discrete $L^2$ norm can be observed for each polygonal partition, but the superconvergence in the discrete $H^1$ norm was only seen for rectangular partitions.",
"The table shows a numerical rate of convergence of $r=1$ in the discrete $H^1$ norm for three other type of partitions.",
"The result is clearly in consistency with the error estimate developed in Section .",
"Fig.",
"REF illustrates the contour plots of the numerical solutions on different type of polygonal partitions.",
"It also shows the shape of the polygonal elements in our computation.",
"Table: Error and convergence performance of the SWG scheme () for the test problem () on general polygonal partitions for Ω=(0,1) 2 \\Omega =(0,1)^2, with κ=4\\kappa = 4.Figure: Comparison of numerical solutions obtained from SWG and the exact solution for the test problem () on various polygonal partitions of h=1/8h=1/8 and κ=1\\kappa =1."
],
[
"Numerical results on a non-convex domain",
"The SWG scheme with the stabilization parameter $\\kappa =4$ was applied to the test problem (REF ) on the L-shaped domain $\\Omega :=(-1,1)\\times (-1,1)/ (0,1)\\times (-1,0)$ partitioned into triangles or rectangles.",
"The corresponding numerical results are summarized in Table REF , which shows a convergence of order ${\\mathcal {O}}(h^2)$ in the $L^2$ norm for both the triangular and rectangular partitions.",
"A superconvergence of order ${\\mathcal {O}}(h^2)$ was observed in the discrete $H^1$ norm on rectangular partitions, while the optimal order of convergence with $r=1$ is confirmed numerically on triangular partitions.",
"It should be pointed out that the $H^2$ -regularity assumption (REF ) is not valid for non-convex polygonal domains so that the optimal order of error estimate (REF ) is not known theoretically on the L-shaped domain.",
"The numerical results therefore outperform the theory in the usual $L^2$ norm.",
"Table: Error and convergence performance of the SWG scheme () for test case () on Lshape domain, κ=4\\kappa = 4.Figure: Comparison of numerical solution obtained from SWG and the exact solution for test case () on L-shaped domain with h=1/8h=1/8."
]
] | 1808.08667 |
[
[
"Symmetries and Mass Degeneracies in the Scalar Sector"
],
[
"Abstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars.",
"In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum.",
"In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars.",
"We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars.",
"Here, CP4 is a generalized CP symmetry with the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.",
"The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields.",
"The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists).",
"A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices."
],
[
"Introduction",
"After the initial discovery of the Higgs boson in 2012[1], [2], certain anomalies in the Higgs data (which have since disappeared) motivated the exploration of the possibility that the 125 GeV Higgs signal was comprised of two nearly mass-degenerate scalar states[3], [4], [5], [6], [7], [8], [9], [10], [11].",
"Although the present Higgs data is consistent with the Standard Model[12], [13], [14], one cannot yet rule out the presence of a mass degenerate scalar state at 125 GeV[15].",
"In this work, we consider the implications of a mass degeneracy among two (or more) scalar states of an extended Higgs sector.",
"Such a mass degeneracy can be either accidental or the result of a symmetry.",
"A trivial example of such a phenomenon arises in any doublet extended Higgs model.",
"All such models possess a mass degenerate state, namely the charged Higgs boson, $H^\\pm $ .",
"Indeed, $H^+$ and $H^-$ are mass-degenerate due to the U(1)$_{\\rm EM}$ gauge symmetry.",
"Moreover, the $H^+$ and $H^-$ are distinguishable by their electric charge, which can be experimentally probed using photons.",
"Suppose that this probe were unavailable (or equivalently, suppose one could turn off electromagnetism).",
"In this case, would it be possible for an experiment to reveal the existence of a mass-degenerate scalar?",
"In this very simple example, one could not physically distinguish (on an event by event basis) between the two degenerate states that comprise the charged Higgs scalar.",
"Nevertheless, there would in principle be observables that are sensitive to the number of mass-degenerate scalar states present.",
"For example, in the CP-conserving two Higgs doublet model, the decay rate for the decay of a heavy CP-even neutral scalar, $H\\rightarrow H^+ H^-$ (if kinematically allowed) is proportional to the number of degrees of freedom in the final state.",
"If we express the charged Higgs field as a linear combination of real scalar fields, $H^\\pm =(\\phi _1\\pm i\\phi _2)/\\sqrt{2}$ , then the decay rate for $H\\rightarrow H^+ H^-$ is the (incoherent) sum of the decay rates for $H\\rightarrow \\phi _1\\phi _1$ and $\\phi _2\\phi _2$ .",
"These two rates are identical, and the sum yields a multiplicity factor of 2.",
"This multiplicity factor provides the experimental signal for mass-degenerate scalars.Note that the decay rate for $Z\\rightarrow H^+ H^-$ is equal to the decay rate for $Z\\rightarrow \\phi _1\\phi _2$ .",
"In this case, the off-diagonal nature of the $Z\\phi _1\\phi _2$ coupling implies that no multiplicity factor is present.",
"Nevertheless, one can still infer the existence of mass-degenerate states, since the decays $Z\\rightarrow \\phi _1\\phi _1, \\phi _2\\phi _2$ are forbidden by Bose statistics.",
"Apart from the trivial mass degeneracy of $H^\\pm $ , we would like to explore in this paper the possibility of exactly mass-degenerate neutral scalars and/or mass-degenerate charged Higgs pairs in extended Higgs sectors.",
"In each case, the critical questions to ask are: (i) is the origin of the exact mass degeneracy natural?",
"and (ii) how can the mass degenerate scalars be distinguished experimentally?",
"Exact mass degeneracies are natural if they are a consequence of an unbroken symmetry.",
"In particular, accidental mass degeneracies require an artificial fine-tuning of the scalar potential parameters, and in this sense we shall call them unnatural (and in our view not especially interesting).",
"If mass-degenerate states are present, it is of interest to determine how to probe them experimentally.",
"In some cases, one can identify the presence of mass degenerate states on an event by event basis.",
"In other cases, the only signal of the mass degeneracy is a measurable multiplicity factor that can be determined when averaging over initial state degeneracies and summing over final state degeneracies.For example, a quark of a given flavor is a mass degenerate state due to its three possible colors.",
"Although the color of a quark cannot be identified experimentally, the presence of the color degree of freedom can be experimentally verified by the color multiplicity factor (most famously exhibited in the observed cross section for $e^+e^-$ annihilation into quark-antiquark pairs.)",
"Our focus in this paper is extended multi-doublet Higgs sectors with mass-degenerate scalar states.",
"It is particularly instructive to discuss mass degeneracy in the scalar sector starting from the so-called Higgs basis.",
"This corresponds to a subset of all possible scalar field parameterizations in which only one Higgs doublet, denoted by $H_1$ , acquires a non-zero positive vacuum expectation value (vev), while all the other scalar fields of the Higgs basis ($H_2$ , $H_3,\\ldots H_n$ ) have zero vev [16], [17], [18], [19], [20].",
"The neutral and charged Goldstone bosons reside entirely in $H_1$ , as this is the only doublet that possesses a non-zero vev, together with a neutral Higgs field that, in the absence of mixing with the neutral fields of the other Higgs doublets, behaves like the Standard Model (SM) Higgs boson.",
"In this sense, this subset of scalar bases may be viewed as Standard Model aligned bases.",
"That is, the Higgs basis is actually a family of basis choices, since one is always free to perform an arbitrary U($n-1$ ) transformation among $H_2$ , $H_3,\\ldots H_n$ while preserving the vev of $H_1$ [21].",
"There is no loss of generality in choosing any particular scalar basis as a starting point.",
"In order to identify the physical neutral scalars one must diagonalize the corresponding scalar squared-mass matrix.",
"As expected, the end result is independent of the initial choice of basis for the scalar fields.",
"In section , we study under what circumstances there is an exact mass degeneracy in the familiar two-Higgs-doublet model (2HDM) [22], [23].",
"In this model there are three physical neutral fields and one charged field, so we only consider potential mass degeneracies among the neutral fields.One can also examine mass degeneracies between the charged Higgs boson and one of the neutral Higgs bosons.",
"For example, in a custodial symmetric 2HDM, the CP-odd Higgs scalar is degenerate with the charged Higgs boson [24], [25], [26], [27].",
"However, custodial symmetry is not an exact symmetry of the full electroweak Lagrangian.",
"Thus, given a custodial symmetric 2HDM scalar potential, any potential mass degeneracies between neutral and charged scalars is at best approximate.",
"We do not consider such mass degeneracies further in this work.",
"We begin our 2HDM analysis by studying possible mass degeneracies among the neutral scalar states of the inert doublet model (IDM) [28], [29].",
"The scalar potential of this model possesses a discrete $\\mathbb {Z}_2$ symmetry that is unbroken by the vacuum.",
"In this case, the CP-even neutral component of $H_1$ in the Higgs basis is a mass eigenstate whose tree-level couplings are precisely those of the SM Higgs boson.",
"The real and imaginary parts of the neutral component of $H_2$ are odd under the discrete $\\mathbb {Z}_2$ symmetry, and have opposite signs under CP.",
"We will denote these two neutral states by $H$ and $A$ , although there is no way to identify which of these two states is CP-even and which is CP-odd.Equivalently, one can propose two different definitions of CP (called, say, CPa and CPb), such that $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.",
"Either definition can be consistently used to define the CP symmetry of the bosonic sector of the IDM.",
"It is possible that $h$ is degenerate in mass with either $H$ or $A$ , but such mass degeneracies are accidental in nature since neither case can arise due to a symmetry.",
"Moreover, these mass-degenerate states are physically distinguishable, since $h$ is even whereas $H$ and $A$ are odd under the $\\mathbb {Z}_2$ symmetry.",
"In contrast, an exact mass degeneracy of $H$ and $A$ can arise if the $\\mathbb {Z}_2$ symmetry of the scalar potential is promoted to a continuous U(1) symmetry.",
"In our terminology, this mass degeneracy of $H$ and $A$ is natural.",
"Nevertheless, the two mass-degenerate states can still be physically distinguished due to the coupling of these states to $W^\\pm H^\\mp $ .",
"One can now extend the above analysis to an arbitrary 2HDM.",
"One can show that with one exception, all 2HDM mass degeneracies are accidental.",
"The one exceptional case of a natural mass degeneracy is precisely the case of $m_H=m_A$ in the IDM.",
"This conclusion can also be obtained by considering all possible symmetries of the 2HDM scalar potential.",
"Among these symmetries, we can identify those that can potentially guarantee the mass degeneracy of scalar states.",
"By examining the consequences of these symmetries, we again confirm that the only possible neutral scalar mass degeneracy in the 2HDM arises in the IDM as previously noted.",
"In section we consider possible mass-degeneracies in the three Higgs doublet model.",
"Using the previous 2HDM analysis of mass degeneracies of the IDM, we construct a three Higgs doublet model (3HDM) generalization of the IDM, which we call the replicated inert doublet model (RIDM).",
"In this model, two of the three Higgs doublets are inert, and four mass-degenerate scalar pairs exist (two involving the charged scalar states from the inert doublets and two involving the neutral scalar states from the inert doublets).",
"We can explicitly identify the symmetries that are responsible for these mass degeneracies.",
"We then investigate the possibility of adding new terms to the scalar potential that partially break these symmetries while preserving the mass degeneracies.",
"In this way, we arrive at a model first proposed by Ivanov and Silva [30].",
"The Ivanov and Silva scalar potential possesses a discrete subgroup of the continuous symmetries that govern the RIDM, that maintains the mass degeneracies of the RIDM.",
"This discrete subgroup is the generalized CP symmetry, CP4, which has the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.",
"The CP4 symmetry is distinguished from the conventional CP symmetry (denoted henceforth by CP2), which has the property that $(\\rm {CP}2)^2$ is the identity operator.",
"Some properties of specialized 3HDMs have also been analyzed recently in Ref.[31].",
"One of the most notable properties of the Ivanov-Silva (IS) model is that one can write down the most general CP4-invariant scalar potential with three Higgs doublets, which has the feature that at least one of the coefficients of the quartic terms of the scalar potential must be complex (with a nonvanishing imaginary part).",
"Indeed, as demonstrated explicitly in Appendix A, one cannot redefine the scalar fields within the family of Higgs bases such that all the coefficients of the scalar potential are real.",
"In this case, we say that no real Higgs basis exists.",
"This means that CP2 is not a symmetry of the IS scalar potential and vacuum.",
"In section , we identify the existence of a physical observable that is present if no CP2 symmetry exists that commutes with the CP4 symmetry of the IS model.However, this leaves open the possibility of the existence of a CP2 symmetry that does not commute with the CP4 symmetry [32]; in this case, a real Higgs basis exists and a conventional CP symmetry can be defined.",
"As an example, we focus on $Z$ decay into four inert neutral scalars (with some details relegated to Appendix B).",
"Nevertheless, the CP4 invariance guarantees that all CP-violating observables involving the Higgs/gauge boson sector of the theory must be absent.",
"For example, we provide an instructive analysis in section that shows how the CP4 symmetry of the IS model with no real Higgs basis ensures the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZW^+ W^-$ vertices up to three-loop order.",
"Finally, we state our conclusions in section ."
],
[
"2HDM mass degeneracies",
"Consider the 2HDM, consisting of two hypercharge-one, doublet scalar fields, $\\Phi _1$ and $\\Phi _2$ .",
"The most general gauge-invariant renormalizable scalar potential is $\\mathcal {V}&=& m_{11}^2 \\Phi _1^\\dagger \\Phi _1+ m_{22}^2 \\Phi _2^\\dagger \\Phi _2 -[m_{12}^2\\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}]+\\tfrac{1}{2}\\lambda _1(\\Phi _1^\\dagger \\Phi _1)^2+\\tfrac{1}{2}\\lambda _2(\\Phi _2^\\dagger \\Phi _2)^2+\\lambda _3(\\Phi _1^\\dagger \\Phi _1)(\\Phi _2^\\dagger \\Phi _2)\\nonumber \\\\&&\\quad +\\lambda _4( \\Phi _1^\\dagger \\Phi _2)(\\Phi _2^\\dagger \\Phi _1)+\\left\\lbrace \\tfrac{1}{2}\\lambda _5 (\\Phi _1^\\dagger \\Phi _2)^2 +\\big [\\lambda _6 (\\Phi _1^\\dagger \\Phi _1) +\\lambda _7 (\\Phi _2^\\dagger \\Phi _2)\\big ] \\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}\\right\\rbrace \\,.$ We shall assume that the minimum of the scalar potential is electric charge conserving, in which case only the neutral scalar fields possess a nonzero vacuum expectation value (vev), $\\langle \\Phi _i^0 \\rangle =v_i/\\sqrt{2}$ , where the $v_i$ are potentially complex.",
"The Fermi constant, $G_F$ fixes the value of $v^2\\equiv |v_1|^2+|v_2|^2=(\\sqrt{2}G_F)^{-1}\\simeq (246~{\\rm GeV})^2\\,.$ Employing a new scalar field basis consisting of two orthonormal linear combinations of $\\Phi _1$ and $\\Phi _2$ does not modify the physical predictions of the model.",
"One convenient choice is the Higgs basis, in which the redefined doublet fields (denoted below by $H_1$ and $H_2$ ) have the property that $H_1$ has a non-zero vev whereas $H_2$ has a zero vev [16], [17], [18].",
"In particular, we define new Higgs doublet fields: $H_1=\\begin{pmatrix}H_1^+\\\\ H_1^0\\end{pmatrix}\\equiv \\frac{1}{v}(v_1^* \\Phi _1+v_2^*\\Phi _2)\\,,\\qquad \\quad H_2=\\begin{pmatrix} H_2^+\\\\ H_2^0\\end{pmatrix}\\equiv \\frac{1}{v}(-v_2 \\Phi _1+v_1\\Phi _2)\\,.$ It follows that $\\langle H_1^0 \\rangle =v$ and $\\langle H_2^0 \\rangle =0$ .",
"The Higgs basis is uniquely defined up to an overall rephasing, $H_2\\rightarrow e^{i\\chi } H_2$ (which does not alter the fact that $\\langle H_2^0 \\rangle =0$ ).",
"In the Higgs basis, the scalar potential of Eq.",
"(REF ) is denoted as [19], [20]: $\\mathcal {V}&=& Y_1 H_1^\\dagger H_1+ Y_2 H_2^\\dagger H_2 +[Y_3H_1^\\dagger H_2+{\\rm h.c.}]+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2)^2+Z_3(H_1^\\dagger H_1)(H_2^\\dagger H_2)\\nonumber \\\\&&\\quad +Z_4( H_1^\\dagger H_2)(H_2^\\dagger H_1)+\\left\\lbrace \\tfrac{1}{2}Z_5 (H_1^\\dagger H_2)^2 +\\big [Z_6 (H_1^\\dagger H_1) +Z_7 (H_2^\\dagger H_2)\\big ] H_1^\\dagger H_2+{\\rm h.c.}\\right\\rbrace \\,,$ where $Y_1$ , $Y_2$ and $Z_1,\\ldots ,Z_4$ are real parameters, whereas $Y_3$ , $Z_5$ , $Z_6$ and $Z_7$ are potentially complex parameters.",
"Imposing the scalar potential minimum conditions yields, $Y_1=-\\tfrac{1}{2}Z_1 v^2\\,,\\qquad \\quad Y_3=-\\tfrac{1}{2}Z_6 v^2\\,.$"
],
[
"Mass degeneracies of the inert doublet model (IDM)",
"We wish to study the consequences of a 2HDM in which two or three of the neutral Higgs scalars are degenerate in mass.",
"For simplicity, we shall first specialize to the inert 2HDM (the so-called IDM)[28], [29] in which there is an exact discrete $\\mathbb {Z}_2$ symmetry that is preserved by the vacuum, under which all particles of the SM and one of the two Higgs doublet fields (which contains the observed Higgs boson) are even and the second Higgs doublet field is odd under the multiplicative discrete symmetry.",
"In particular, the discrete symmetry of the IDM is manifest in the Higgs basis, where we identify $H_1$ as even and $H_2$ as odd under the $\\mathbb {Z}_2$ symmetry.",
"It then follows that $Y_3=Z_6=Z_7=0$ .",
"The IDM scalar potential is CP-conserving since one can eliminate the phase of $Z_5$ (the only remaining potentially complex scalar potential parameter) by appropriately rephasing the Higgs basis field $H_2$ .",
"The Higgs basis fields are $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h+iG^0\\bigr ] \\end{pmatrix}\\,,\\qquad \\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix}\\,,$ where $G^\\pm $ and $G^0$ are the Goldstone bosons that provide the longitudinal degrees of freedom of the massive $W^\\pm $ and $Z^0$ gauge bosons.",
"The physical mass spectrum of the IDM is given by, $&&m_h^2=Z_1 v^2\\,,\\qquad \\qquad \\qquad \\qquad \\quad m^2_{H^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\\\&&m_{A}^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4- Z_5)v^2\\,,\\qquad \\,\\, m_H^2=m_A^2+Z_5 v^2\\,.$ For completeness, we exhibit the Higgs couplings of the IDM in the unitary gauge below (where the Goldstone fields are set to zero).",
"First, the interactions of the Higgs bosons and the gauge bosons are governed by,The photon field $A_\\mu $ should not be confused with the scalar field $A$ .",
"${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h^2+H^2+A^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right]H^+H^-\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)H^-(H+iA) +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\[8pt]{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H -12g[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA) +h.c.]",
"+[ieA+igcW(12-sW2) Z]H+$\\leftrightarrow $$\\partial $ H-, where $s_W\\equiv \\sin \\theta _W$ , $c_W\\equiv \\cos \\theta _W$ .",
"The trilinear and quadrilinear Higgs self-interactions are governed by ${L}_{3h}&=&-\\tfrac{1}{2}v\\bigl [Z_1 h^3+(Z_3+Z_4)h(H^2+A^2)+Z_5h(H^2-A^2)\\bigr ]-vZ_3hH^+H^-\\,, \\\\[6pt]{L}_{4h}&=&-\\tfrac{1}{8}\\bigl [Z_1 h^4+Z_2(H^2+A^2)^2+2(Z_3+Z_4)h^2(H^2+A^2)+2Z_5 h^2(H^2-A^2)\\bigr ]\\nonumber \\\\[6pt]&&-\\tfrac{1}{2}H^+ H^-\\bigl [Z_2(H^2+A^2+H^+ H^-)+Z_3 h^2\\bigr ].",
"$ The tree-level couplings of $h$ to SM particles obtained above are precisely those of the SM Higgs boson, corresponding to the exact Higgs alignment limit [33], [34], [35], [36], [37], [38], [39], [40], [41] (as expected in light of $Z_6=0$ ).",
"Moreover, an examination of the above couplings implies that $h$ is CP-even (to be identified with the SM Higgs boson) and $H$ and $A$ have opposite CP-quantum numbers (one is odd and the other is even) based on the $ZAH$ coupling.Under the rephasing, $H_2\\rightarrow iH_2$ , we note that $Z_5\\rightarrow -Z_5$ , $H\\rightarrow -A$ and $A\\rightarrow H$ .",
"One can check that the masses of $H$ and $A$ and their couplings are invariant under this rephasing.",
"Note that the CP is not uniquely defined by the IDM interactions, since two candidate definitions of CP exist (called CPa and CPb in footnote REF ), where $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.",
"Either definition of CP can be used consistently in exploring the phenomenology of the IDM.",
"Finally, we note that under the $\\mathbb {Z}_2$ symmetry of the IDM, the quarks and leptons can be chosen to be even.",
"Consequently, the tree-level couplings of $h$ to fermion pairs are identical to those of the SM Higgs boson, whereas $H$ , $A$ and $H^\\pm $ do not couple to the SM fermions.",
"We now examine the possibility of mass degeneracies in the IDM.",
"First, consider the case of $m_h=m_H$ or $m_h=m_A$ .",
"In this case, it is possible to physically distinguish between $h$ and its mass-degenerate partner due to their opposite $\\mathbb {Z}_2$ quantum numbers.",
"For example, since all SM bosons and fermions are even under the $\\mathbb {Z}_2$ symmetry, it follows that the gluon-gluon (via a top quark loop), $WW$ and $ZZ$ fusion processes can only produce $h$ whereas Drell-Yan production (via virtual $s$ -channel $Z$ exchange) can only produce $H$ in association with $A$ .",
"Hence, despite the mass degeneracy, the two mass-degenerate scalars are physically distinguishable.",
"Note that the mass degeneracy of $h$ and its scalar partner is not radiatively stable.",
"For example, if $h$ and $H$ are mass degenerate states, then the one-loop contributions to the $hh$ two-point function (such as $ZZ$ and $WW$ intermediate states) differ from the corresponding contributions to the $HH$ two-point function (e.g.",
"the $AZ$ intermediate state).",
"Indeed, the tree-level condition for the mass degeneracy of $h$ and $H$ , $Z_1v^2=Y_2+\\tfrac{1}{2}Z_{345}v^2,$ where $Z_{345}\\equiv Z_3+Z_4+Z_5$ , is unnatural; i.e., Eq.",
"(REF ) is not the result of some symmetry.The scenario where $m_h=m_H=m_A$ is a special case of the $h$ –$H$ mass degeneracy.",
"In the triply mass-degenerate scenario, $h$ is also distinguished from $\\phi ^\\pm $ by its U(1) charge, which is zero.",
"For example, there is no coupling of $ZW^\\pm H^\\mp h$ in contrast to the $Z W^\\pm H^\\mp \\phi ^\\pm $ couplings exhibited in Eq.",
"(REF ).",
"Second, consider the case of $m_H=m_A$ , which corresponds to $Z_5=0$ .",
"In this case, the IDM scalar potential possesses a continuous U(1) symmetry, which is not spontaneously broken by the vacuum.This case has also been noted in Ref. [41].",
"It is this symmetry that is responsible for the mass degenerate states $H$ and $A$ .",
"One can now define eigenstates of U(1) charge, $\\phi ^\\pm =\\frac{1}{\\sqrt{2}}\\bigl [H\\pm iA\\bigr ]\\,.$ The relevant interaction terms of $\\phi ^{\\pm }$ are ${L}_{\\rm int}&=&\\left[\\tfrac{1}{2}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{4c_W^2}Z_\\mu Z^\\mu \\right]\\phi ^+\\phi ^- +\\frac{ig}{2c_W}Z^\\mu \\phi ^-\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ + -g2[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ + +h.c.]",
"+eg2 (AW+H-+ + AW-H+-) -g2sW22cW(ZW+H-+ + ZW-H+-) - v(Z3+Z4)h+–12[Z2(+-)2+(Z3+Z4)h2+-]-Z2 H+ H-+- .",
"Although $\\phi ^\\pm $ are mass degenerate states, they can be distinguished.",
"For example, Drell-Yan production via a virtual $s$ -channel $W^+$ exchange can produce $H^+$ in association with $\\phi ^-$ , whereas virtual $s$ -channel $W^-$ exchange can produce $H^-$ in association with $\\phi ^+$ .",
"Thus, the sign of the charged Higgs boson reveals the U(1)-charge of the produced neutral scalar.",
"The origin of this correlation lies in the fact that, by construction, $H^+$ and $\\phi ^+$ both reside in $H_2$ , whereas $H^-$ and $\\phi ^-$ both reside in $H_2^\\dagger $ ."
],
[
"2HDM mass degeneracies beyond the IDM",
"Although the IDM is a rather special case among all possible 2HDMs, the conclusions concerning mass degeneracies are robust.",
"Allowing for the most general 2HDM scalar potential, the Higgs sector is CP-violating if the relative phases among $Z_5$ , $Z_6$ and $Z_7$ cannot be removed by rephasing the Higgs basis field $H_2$ .",
"It is convenient to introduce three invariant quantities[42], [18], [20], whose imaginary parts are given by,Basis-invariant expressions for the $J_i$ are given in Ref.[20].",
"$\\operatorname{Im}J_1=\\operatorname{Im}(Z_6^* Z_7)\\,,\\qquad \\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)\\,,\\qquad \\operatorname{Im}J_3=\\operatorname{Im}\\bigl [Z_5^*(Z_6+Z_7)^2\\bigr ]\\,.$ The Higgs sector of the 2HDM is CP-violating unless $\\operatorname{Im}J_1=\\operatorname{Im}J_2=\\operatorname{Im}J_3=0$ .",
"The origin of the CP-violation can either be explicit or spontaneous [43], [44].To test for explicit CP-violation, one must employ invariant quantities that are independent of the scalar field vacuum expectation values [45], [46], [47].",
"Note that the neutral scalar squared-mass matrix does not involve the Higgs basis parameter $Z_7$ .",
"In particular, $Z_7$ only enters in the Higgs boson cubic and quartic self-couplings.",
"Hence, if $\\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)=0$ , then the neutral Higgs mass eigenstates behave like eigenstates of CP in their tree-level interactions with the gauge bosons and fermions (independently of the values of $\\operatorname{Im}J_1$ and $\\operatorname{Im}J_3$ ).",
"Moreover, the neutral scalar squared-mass matrix breaks up into a block diagonal form consisting of a $2\\times 2$ block (whose diagonalization yields the two CP-even neutral scalars) and a $1\\times 1$ block (which yields the CP-odd neutral scalar).",
"Consider the possibility of mass degeneracies among the neutral scalars of the most general 2HDM.",
"We now recall a remarkable tree-level relation of the CP-violating 2HDM[48], [42], [49], $ \\operatorname{Im}J_2&={\\rm Im}(Z_5^* Z_6^2)=\\frac{2 s_{13}c_{13}^2 s_{12}c_{12}}{v^6}(m_2^2-m_1^2)(m_3^2-m_1^2)(m_3^2-m_2^2)\\,,$ where the $m_i$ ($i=1,2,3$ ) are the masses of the three neutral Higgs bosons of the 2HDM, $s_{12}\\equiv \\sin \\theta _{12}$ , $c_{12}\\equiv \\cos \\theta _{12}$ , etc., and $\\theta _{12}$ and $\\theta _{13}$ are invariant mixing angles that are associated with the diagonalization of the neutral Higgs squared-mass matrix in the Higgs basis.Details on the definition of the mixing angles and their relations to the Higgs basis scalar potential parameters can be found in Ref.[49].",
"However, we will not need any of these details for the present argument.",
"In Ref.",
"[50], the three CP-odd invariants $\\operatorname{Im}J_i$ have been expressed in terms of the neutral scalar masses and the couplings of the neutral Higgs mass eigenstates to charged pairs, $e_i$ $(H_iW^+W^-)$ and $q_i$ $(H_iH^+H^-)$ .",
"In particular,In Ref.",
"[42], the invariant defined in Eq.",
"(REF ) is called $J_1$ .",
"$ \\operatorname{Im}J_2=\\frac{2e_1 e_2 e_3}{v^9}(m_1^2-m_2^2)(m_2^2-m_3^2)(m_3^2-m_1^2).$ If any two of the three neutral Higgs bosons are mass-degenerate, then either Eq.",
"(REF ) or (REF ) implies that $\\operatorname{Im}J_2=0$ , and the corresponding neutral scalar mass-eigenstates will behave as states of definite CP in their interactions with gauge bosons and fermions.",
"Nevertheless, if $\\operatorname{Im}J_1\\ne 0$ and/or $\\operatorname{Im}J_3\\ne 0$ (which would imply that $\\operatorname{Im}Z_7\\ne 0$ in the Higgs basis where $Z_5$ and $Z_6$ are simultaneously real), then CP-violating Higgs self-couplings must be present.",
"Moreover, radiative corrections will generate a non-zero $\\operatorname{Im}J_2$ and yield neutral Higgs states of indefinite CP.",
"That is, if CP-violation in the scalar sector is present, the tree-level relation $\\operatorname{Im}J_2=0$ can only be realized via an artificial fine-tuning of the parameters.",
"Nevertheless, one can consider the implications of a tree-level mass degeneracy among the neutral Higgs scalars of the 2HDM.",
"The above discussion illustrates the power of using scalar basis invariant conditions to analyze the CP properties of multi-Higgs models [45], [46], [51], [52], [53].",
"In light of Eq.",
"(REF ), if a tree-level mass degeneracy among the neutral Higgs scalars is present, then it is possible to rephase the Higgs basis field $H_2$ such that $Z_5$ and $Z_6$ are simultaneously real.",
"Thus, in the analysis that follows, we shall analyze the most general 2HDM scalar potential assuming that $Z_5$ and $Z_6$ are real parameters.",
"The squared-masses of the charged Higgs boson, $H^\\pm $ , and the CP-odd Higgs boson, $A$ are given by,In the notation employed in Eqs.",
"(REF )–(), $h$ and $H$ [$A$ ] refer to the neutral scalars that behave as CP-even [odd] mass eigenstates in their tree-level interactions with the gauge bosons and fermions.",
"Indeed, CP-violating interactions are present in Eqs.",
"(REF ) and () if $\\operatorname{Im}Z_7\\ne 0$ .",
"$m_{H^\\pm }^2=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad \\quad m_A^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4-Z_5)v^2\\,.$ The squared-masses of the CP-even Higgs bosons, $h$ and $H$ are the eigenvalues of the $2\\times 2$ matrix, $\\mathcal {M}_H^2=\\begin{pmatrix} Z_1 v^2 & \\quad Z_6 v^2 \\\\ Z_6 v^2 & \\quad m_A^2+Z_5 v^2\\end{pmatrix}.$ That is, $m^2_{H,h}=\\frac{1}{2}\\biggl \\lbrace m_A^2+(Z_1+Z_5)v^2\\pm \\sqrt{[m_A^2-(Z_1-Z_5)v^2]^2+4Z_6^2 v^4\\,}\\biggr \\rbrace \\,.$ Mass degenerate states arise if one of the following two quantities is zero, $Z_5(m_A^2-Z_1 v^2)+Z_6^2 v^2=0\\quad \\text{or}\\quad \\bigl [m_A^2-(Z_1-Z_5)v^2\\bigr ]^2+4Z_6^2 v^4=0\\,,$ where $m_A^2$ is given by Eq.",
"(REF ).",
"The case of $m_h=m_H$ arises when the second condition given in Eq.",
"(REF ) is satisfied.",
"It then follows that $m_A^2=(Z_1-Z_5)v^2$ and $Z_6=0$ , and the latter then yields the IDM mass spectrum.",
"As in the case of the IDM, the mass degeneracy of $h$ and $H$ requires a fine tuning of the parameters shown in Eq.",
"(REF ).",
"In principle, it is possible that $Z_7\\ne 0$ , but in this case, $Z_6=0$ is not a natural condition since the $\\mathbb {Z}_2$ symmetry of the IDM is not present.",
"Nevertheless, even if one accepts the two fine tuned conditions needed in this scenario, the arguments presented above Eq.",
"(REF ) still apply.",
"Namely, $Z_6=0$ corresponds to the exact alignment limit (at tree-level), in which case the tree-level interactions of the Higgs bosons and gauge bosons are still the same as those of the IDM [cf. Eqs.",
"(REF )–(REF )], whereas the tree-level trilinear and quadrilinear Higgs self-interactions given in Eqs.",
"(REF ) and () are modified by the addition of the following terms, $\\delta {L}_{3h}&=&-\\tfrac{1}{4} v \\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)\\,, \\\\[5pt]\\delta {L}_{4h}&=&-\\tfrac{1}{4}\\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)h\\,,$ after rephasing the Higgs basis field $H_2$ such that $Z_5$ is real.",
"The cases $m_h=m_A$ or $m_H=m_A$ arise when the first condition given in Eq.",
"(REF ) is satisfied.",
"This condition also requires a fine-tuning of the parameters.",
"Moreover, approximate Higgs alignment (as suggested by the LHC Higgs data) is not achieved unless $m_A^2\\gg Z_1 v^2$ or $|Z_6|\\ll 1$ .",
"Nevertheless, the physical distinction of the mass degenerate states is due to the CP quantum numbers of the neutral scalar states (which are preserved in the tree-level Higgs interactions with gauge bosons and with fermions).",
"One can therefore distinguish between the corresponding production mechanisms of the degenerate scalars that are mediated by gauge boson fusion or Drell-Yan production via $s$ -channel gauge boson exchange.",
"Finally, we consider the triply mass-degenerate case of $m_h=m_H=m_A$ .",
"In this case, both conditions given in Eq.",
"(REF ) must be satisfied, which yields $Z_5=Z_6=0$ and $m_A^2=Z_1 v^2$ .",
"This leaves $Z_7$ as the only potentially complex parameter of the scalar potential in the Higgs basis.",
"Thus, one is free to rephase the Higgs basis field $H_2$ such that $Z_7$ is real, and we conclude that the Higgs scalar potential and vacuum must be CP-conserving.",
"However, as long as $Z_7\\ne 0$ , the triply mass-degenerate case is unnatural, since the $\\mathbb {Z}_2$ symmetry of the IDM is not present."
],
[
"Natural 2HDM mass degeneracies: a symmetry based approach",
"In sections REF and REF , we derived the conditions that yield mass degeneracies among the neutral scalars of the 2HDM by brute force.",
"Namely, we obtained explicit expressions for the neutral scalar masses and then derived the corresponding relations among Higgs basis parameters for which mass degeneracies were present.",
"We then checked whether any of these relations were a consequence of a symmetry, and if yes we concluded that the corresponding mass degeneracy was natural.",
"In this section, we will obtain the same result by considering all possible symmetries of the 2HDM scalar potential.",
"Since the complete list of such symmetries is known [54], [55], [56], [57], [58], [59], we can be sure that our catalog of natural mass degeneracies of the 2HDM is complete.",
"We shall make use of the classification of symmetries presented in Ref.",
"[56], which identifies three possible Higgs family symmetries, $\\mathbb {Z}_2$ , U(1) and SO(3), and three classes of generalized CP-symmetries, denoted by GCP1, GCP2 and GCP3, respectively, as summarized in Table .In Ref.",
"[56] the three classes of generalized CP transformations are denoted by CP1, CP2 and CP3 respectively.",
"This nomenclature for the generalized CP-symmetries is awkward, in light of the notation that will be employed in section .",
"To avoid confusion, we have appended the letter G (for “general”) in denoting the three classes of generalized CP transformations of the 2HDM.",
"In the GCP transformation laws of Table , we have introduced the conjugation symbol ${}^{\\scriptscriptstyle {\\bigstar }}$ , which when applied to an SU(2) multiplet of scalar fields is defined by $\\Phi ^{\\scriptscriptstyle {\\bigstar }}\\equiv \\bigl [\\Phi ^\\dagger \\bigr ]^{\\scriptstyle T}$$,where the dagger refers both to hermitian conjugation of the quantum field operator when acting on the Hilbert space, and to complex conjugate transpose when acting on an SU(2) multiplet of fields.\\vspace{-0.72229pt}\\begin{table}[hb!",
"]\\begin{tabular}{|cccc|}\\hline \\rule {0pt}{}symmetry & \\hspace{36.135pt} transformation law & \\hspace{50.58878pt} \\phantom{transformation law} & \\\\[2pt]\\hline \\rule {0pt}{}\\mathbb {Z}_2 & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow -\\Phi _2 &\\\\U(1) & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow e^{2i\\theta }\\Phi _2 &\\\\SO(3) & \\Phi _a\\rightarrow U_{ab}\\Phi _b & U\\in {\\rm U}(2)/{\\rm U}(1)_{\\rm Y}& (\\text{for $a$, $b=1,2$} )\\\\GCP1 & \\Phi _1\\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP2 & \\Phi _1\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP3 & \\Phi _1 \\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} \\cos \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }} \\sin \\theta &\\Phi _2 \\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }}\\sin \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }}\\cos \\theta & \\text{(for $0<\\theta <\\tfrac{1}{2}\\pi $)} \\\\[2pt]\\hline \\hline \\rule {0pt}{}\\Pi _2& \\Phi _1\\rightarrow \\Phi _2 & \\Phi _2\\rightarrow \\Phi _1 & \\\\[2pt]\\hline \\end{tabular}\\caption {\\small Possible symmetries of the 2HDM scalar potential that are respected by the SU(2)\\times U(1)_{\\rm Y} gauge kinetic terms of the scalar fields.",
"The corresponding symmetry transformation laws are given in a basis where the symmetry is manifest.",
"Note that a scalar potential that is invariant under the mirror discretesymmetry, \\Pi _2, is also invariant under the \\mathbb {Z}_2 in another scalar field basis~\\cite {Davidson:2005cw}.",
"}\\end{table}$ Table: Impact of the symmetries defined in Table on the coefficients of the 2HDM scalar potential [cf. Eq.",
"()] in abasis where the symmetry is manifest.",
"A short dash indicates the absence of a constraint.",
"A scalar potential that is simultaneously invariant under ℤ 2 \\mathbb {Z}_2 and Π 2 \\Pi _2 isalso invariant under GCP2 in another scalar field basis , .",
"Likewise, a scalar potential that is simultaneously invariant under U(1) and Π 2 \\Pi _2 is also invariant under GCP3 in another scalar field basis .",
"The symbol ⊕\\oplus is being used above to indicate that two symmetries are enforced simultaneously within the same scalar field basis.We shall not consider the seven additional accidental symmetries of the 2HDM scalar potential identified in Refs.",
"[58], [59], that utilized mixed Higgs family and generalized CP transformations that leave the SU(2) gauge kinetic terms of the scalar fields invariant.",
"An example of such a symmetry is the well known custodial symmetry that is respected by the 2HDM scalar potential when $m_{H^\\pm }=m_A$ [24], [25], [26], [27].",
"However, this class of symmetries is violated by the U(1)$_{\\rm Y}$ gauge kinetic term of the scalar potential (as well as by the Yukawa couplings that are responsible for mass differences between up and down-type fermions).",
"Hence, any exact mass degeneracies arising from these seven accidental symmetries will be spoiled, in the absence of an artificial fine tuning of the Higgs scalar potential parameters.In cases of accidental symmetries, i.e.",
"symmetries of the scalar potential that are not respected by the full theory, the would-be mass degeneracies are only approximate, with calculable mass splittings.",
"The possibility of such approximate mass degeneracies, although technically natural, is not the subject of this paper.",
"Possible natural mass degeneracy of the 2HDM must be the consequence of one of the symmetries listed in Table .",
"Starting from a generic scalar potential given by Eq.",
"(REF ), if the scalar potential respects one of the symmetries listed in Table , then a scalar basis is picked out in which the symmetry is manifest.",
"In this basis, the coefficients of the scalar potential are constrained according to Table REF .It can be shown that for each of the symmetries listed in Table REF , a scalar field basis exists in which all scalar potential parameters and the neutral scalar field vacuum expectation values are simultaneously real, in which case CP (as defined by GCP1 in Table ) is conserved by the scalar sector Lagrangian and vacuum.",
"It is straightforward to check that the possible discrete symmetries of the 2HDM, namely $\\mathbb {Z}_2$ , GCP1, GCP2 (or equivalently, $\\mathbb {Z}_2\\oplus \\Pi _2$ ), do not yield scalar potentials that lead to scalar mass degeneracies.",
"Thus, we henceforth focus on U(1), SO(3) and GCP3 (and the related U(1)$\\oplus \\Pi _2$ symmetry).",
"Given a 2HDM scalar potential with a Peccei-Quinn [U(1)$_{\\rm PQ}$ ] symmetry [60] (or equivalently the U(1) transformation specified in Table In Ref.",
"[60], a U(1)$_{\\rm PQ}$ transformation of the 2HDM scalar fields is given by $\\Phi _1\\rightarrow e^{-i\\theta }\\Phi _1$ and $\\Phi _2\\rightarrow e^{i\\theta }\\Phi _2$ .",
"The U(1) transformation specified in Table corresponds to combining the U(1)$_{\\rm PQ}$ transformation with a hypercharge U(1)$_{\\rm Y}$ transformation, $\\Phi _i\\rightarrow e^{i\\theta }\\Phi _i$ (for $i=1,2$ ).)",
"that is spontaneously broken by the vacuum, the scalar sector will contain a massless CP-odd (Goldstone) scalar [61], [62].",
"In such cases, no mass degeneracy is present (without further constraints on the scalar potential parameters).",
"However, if the U(1) symmetry is manifestly realized in the Higgs basis, then the U(1) symmetry is unbroken by the vacuum, resulting in a mass degeneracy between the two neutral scalars residing in the Higgs basis field $H_2$ .",
"Indeed, this has already been shown in Section REF [see text above Eq.",
"(REF )], in the case of the mass degeneracy, $m_H=m_A$ , of the IDM with $Z_5=0$ .",
"In the case of a 2HDM scalar potential with an SO(3) symmetry, the form of the scalar potential is invariant with respect to all possible changes of the scalar basis.",
"Hence, it follows that the scalar potential parameters in the Higgs basis satisfy $Y_1=Y_2$ , $Z_1=Z_2=Z_3+Z_4$ and $Y_3=Z_5=Z_6=Z_7=0$ .",
"Using Eqs.",
"(REF ), (REF ) and (), it follows that $m_H=m_A=0$ .",
"The presence of two massless (Goldstone) scalars is a consequence of the spontaneous breaking of the SO(3) global symmetry by the vacuum.",
"This scalar mass degeneracy is a special case of the mass degeneracy in the case of the IDM with $Z_5=0$ , in which additional constraints among the Higgs basis parameters result in the pair of massless scalar states.",
"Finally, let us consider the case of a 2HDM scalar potential with a GCP3 symmetry.",
"Suppose that the GCP3 symmetry is manifestly realized in a basis where $\\langle \\Phi _1^0 \\rangle =\\frac{v_1}{\\sqrt{2}}\\,,\\qquad \\quad \\langle \\Phi _2^0 \\rangle =\\frac{v_2}{\\sqrt{2}}e^{i\\xi }\\,,$ where $v_1$ and $v_2$ are positive.",
"We define $\\tan \\beta \\equiv v_2/v_1$ (so that $0<\\beta <\\tfrac{1}{2}\\pi $ ).",
"Then, in light of the constraints on the GCP3 scalar potential parameters given in Table REF , the scalar potential minimum conditions yield (e.g., see eqs.",
"(3)–(5) of Ref.",
"[63]), $m_{11}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\sin ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{22}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\cos ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{12}^2\\sin \\xi &=& v^2\\lambda _5 \\sin \\beta \\cos \\beta \\sin \\xi \\cos \\xi \\,.$ We can assume that $\\lambda _5\\ne 0$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.",
"Setting $m_{11}^2=m_{22}^2$ and $m_{12}^2=0$ then yields two conditions, $\\sin ^2 \\xi \\cos 2\\beta =0\\,,\\qquad \\quad \\sin \\xi \\cos \\xi \\sin 2\\beta =0\\,.$ Hence, there are two classes of vacua, $\\sin \\xi =0$ and $\\beta $ arbitrary ($0<\\beta <\\tfrac{1}{2}\\pi $ ) , $\\cos \\xi =0$ and $\\cos 2\\beta =0$ .",
"We now can calculate the parameters of the GCP3 scalar potential in the Higgs basis (e.g., see eqs.",
"(11)–(20) of Ref.",
"[63]) in the two Cases A and B defined above, $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1, \\qquad \\quad Z_3+Z_4=\\lambda _1-\\lambda _5,\\quad Z_5=\\lambda _5$ , $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1-\\lambda _5, \\quad Z_3+Z_4=\\lambda _1+\\lambda _5,\\quad Z_5=0$ .",
"In particular, $Z_1=Z_2=Z_3+Z_4+Z_5$ in Case A, and $Z_1=Z_2\\ne Z_3+Z_4$ (and $Z_5=0$ ) in Case B.",
"We can now make use of Eqs.",
"(REF ) and () [along with Eq.",
"(REF ) to eliminate $Y_2$ by virtue of $Y_1=Y_2$ ] to compute the neutral scalar mass spectrum in the two cases,Note that the positivity of the squared masses are consistent with the conditions on the 2HDM scalar potential parameters first obtained in Ref. [54].",
"$m_h^2=Z_1 v^2,\\quad m_H^2=0,\\quad m_A^2=(Z_3+Z_4-Z_1)v^2,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ , $m_h^2=Z_1 v^2,\\quad m^2_H=m^2_A=\\tfrac{1}{2}(Z_3+Z_4-Z_1) v^2,\\qquad \\,\\,\\, m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ .",
"Note that Cases A and B correspond to degenerate vacua, since in both cases the value of the scalar potential (in the Higgs basis) at its minimum is $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .",
"In light of Table REF , we can identify case A as corresponding to realizing a GCP3 symmetry in the Higgs basis,This means that we can relax the restrictions of $\\beta \\ne 0, \\tfrac{1}{2}\\pi $ in defining Case A.",
"Note that for $\\beta =\\tfrac{1}{2}\\pi $ one can simply interchange the definitions of $\\Phi _1$ and $\\Phi _2$ to recover the Higgs basis result (corresponding to $\\beta =0$ ).",
"and case B corresponding to realizing a U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.This result is not surprising given that U(1)$\\oplus \\Pi _2$ is equivalent to GCP3 in another scalar field basis.",
"In particular, case A exhibits a massless Goldstone boson corresponding to the spontaneous breaking of GCP3 by the vacuum.",
"In contrast, in case B, the GCP3 symmetry possesses a continuous U(1) subgroup, which is unbroken by the vacuum, that protects the mass degeneracy, $m_H=m_A$ .",
"Indeed, this case is again a special case of the IDM with $Z_5=0$ , where the additional constraints, $Y_1=Y_2$ and $Z_1=Z_2$ are imposed.",
"As a check, it is instructive to evaluate the consequences of a 2HDM scalar potential with a U(1)$\\oplus \\Pi _2$ symmetry that is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.",
"In the following, we employ primed coefficients, $\\lambda _i^\\prime $ to distinguish this case from the one above where the GCP3 symmetry is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.",
"Using the results of Table REF , the scalar potential minimum conditions yield, $m_{11}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\cos ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\sin ^2\\beta \\bigr ]\\,,\\\\m_{22}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\sin ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\cos ^2\\beta \\bigr ]\\,,$ under the assumption that $0<\\beta <\\tfrac{1}{2}\\pi $ .",
"We can assume that $\\lambda ^\\prime _1\\ne \\lambda ^\\prime _3+\\lambda ^\\prime _4$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.",
"Setting $m_{11}^2=m_{22}^2$ then yields $\\cos 2\\beta =0$ , with $\\xi $ arbitrary.",
"Without loss of generality, one can set $\\xi =0$ , since the scalar potential is unchanged under a rephasing of $\\Phi _2$ .",
"As before, we can now compute the scalar potential parameters in the Higgs basis, $&&Y_1=Y_2,\\qquad Y_3=Z_6=Z_7=0,\\qquad Z_1=Z_2=\\tfrac{1}{2}(\\lambda ^\\prime _1+\\lambda ^\\prime _3+\\lambda ^\\prime _4),\\nonumber \\\\&& Z_3+Z_4=\\lambda ^\\prime _1,\\qquad Z_5=\\tfrac{1}{2}(\\lambda ^\\prime _1-\\lambda ^\\prime _3-\\lambda ^\\prime _4)\\,.",
"$ In particular, note that $Z_1=Z_2=Z_3+Z_4-Z_5$ .",
"Using Eqs.",
"(REF ), (REF ) and (), we obtain $m_h^2=Z_1 v^2\\,,\\quad m_H^2=(Z_3+Z_4-Z_1)v^2\\,,\\quad m_A^2=0\\,,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2\\,,$ which is the same mass spectrum as Case A of the GCP3-symmetric scalar potential with $H$ and $A$ interchanged.",
"This result can be understood by noting that Eq.",
"(REF ) takes the standard form of the GCP3-symmetric scalar potential in the Higgs basis after rephasing the Higgs basis field $H_2\\rightarrow iH_2$ , which interchanges $H$ and $A$ and transforms $Z_5\\rightarrow -Z_5$ .",
"Finally, the case of $\\beta =0$ or $\\beta =\\tfrac{1}{2}\\pi $ must be treated separately and corresponds to a manifest realization of the U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.",
"This vacuum is degenerate with the one considered above, since in both cases, $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .",
"Indeed, this latter case corresponds to Case B of the GCP3-symmetric scalar potential treated above, where the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .",
"In summary, massless scalar (Goldstone boson) states $A$ , $H$ or ($A$ , $H$ ) exist in the 2HDM with a scalar potential that exhibits, respectively, a U(1), GCP3 or SO(3) symmetry manifestly realized in a generic $\\Phi _1$ –$\\Phi _2$ basis, which agrees with the results of Table 2 of Ref. [59].",
"Nevertheless, in the special cases where U(1) or U(1)$\\oplus \\Pi _2$ are manifestly realized in the Higgs basis (the latter corresponding to the Case B solution of the GCP3-symmetric scalar potential), the corresponding U(1) subgroups of theses symmetries are not spontaneously broken by the vacuum, and the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .",
"In the case of the SO(3)-symmetric scalar potential, this mass degeneracy is realized by a pair of massless Goldstone boson states.",
"Thus, we conclude that mass-degenerate neutral scalars can arise naturally in the 2HDM only in the case of the IDM with $Z_5=0$ .",
"All other cases of mass-degenerate scalars require an artificial fine-tuning of the scalar potential parameters, in agreement with the analysis of section REF .",
"Furthermore, this conclusion is unaffected by the interactions of the scalars with the vector bosons.",
"Indeed, the Higgs boson–gauge boson interactions, ${L}_{\\rm int}$ , given by Eq.",
"(REF ) show that the global U(1) symmetry responsible for the mass degeneracy of $H$ and $A$ is an exact symmetry of ${L}_{\\rm int}$ .",
"Finally, as previously noted, the Higgs basis field $H_2$ of the IDM is odd whereas all other scalar, fermion and vector fields are even under the discrete $\\mathbb {Z}_2$ symmetry.",
"This can be achieved by employing Type-I Yukawa couplings [64] where fermions couple only to the Higgs basis field $H_1$ .",
"In this case, the global U(1) symmetry of the IDM scalar potential with $Z_5=0$ will also be respected by the Yukawa interactions.",
"However, a GCP3 [or equivalently U(1)$\\oplus \\Pi _2$ ] or SO(3) symmetry of the IDM scalar potential will be explicitly broken by the Yukawa interactions.",
"Hence, the U(1)-symmetric IDM is the only 2HDM for which an exact mass degeneracy of $H$ and $A$ can be preserved."
],
[
"3HDM mass degeneracies and the Ivanov Silva model",
"In extended Higgs sectors with more than two scalar doublets, it is now possible to have mass-degenerate charged Higgs pairs as well as mass-degenerate neutral scalars [65].",
"In this section, we explore new phenomena associated with mass degenerate scalars that arises for the first time in the three-Higgs doublet model (3HDM).",
"As a warmup exercise, we return to the IDM and add a second inert doublet and consider possible mass degeneracies among the scalar fields of the two inert doublets.",
"We then perturb the resulting model to obtain a version of the 3HDM that is equivalent to a model first introduced by Ivanov and Silva[30]."
],
[
"The replicated inert doublet model (RIDM)",
"The IDM introduced in section REF can be generalized by introducing additional inert scalar doublets.",
"In this section, we consider a 3HDM that consists of two inert hypercharge-one electroweak doublets, in which the inert doublets contain mass-degenerate scalar states.",
"The resulting models shall be called the replicated inert doublet model (RIDM).",
"As in the case of the IDM, we work in the Higgs basis in which the first Higgs doublet field $H_1$ contains the SM Higgs boson.",
"The RIDM consists of $H_1$ , with $\\langle H_1^0 \\rangle =v/\\sqrt{2},$ and two inert doublet fields $H_2$ and $H_3$ , with $\\langle H_2 \\rangle =\\langle H_3 \\rangle =0$ , and a scalar potential given by, $\\mathcal {V}_{\\rm RIDM}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\left(H_2^\\dagger H_2 +H_3^\\dagger H_3\\right)+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2+H_3^\\dagger H_3)^2\\nonumber \\\\&&+Z_3(H_1^\\dagger H_1)\\left(H_2^\\dagger H_2+H_3^\\dagger H_3\\right)+Z_4\\left[( H_1^\\dagger H_2)(H_2^\\dagger H_1)+( H_1^\\dagger H_3)(H_3^\\dagger H_1)\\right]\\nonumber \\\\&&+\\tfrac{1}{2}Z_5\\left\\lbrace (H_1^\\dagger H_2)^2 +(H_2^\\dagger H_1)^2+(H_1^\\dagger H_3)^2 +(H_3^\\dagger H_1)^2\\right\\rbrace \\,.$ Without loss of generality, we have chosen $Z_5$ real and non-negative in Eq.",
"(REF ), which is always possible by an appropriate rephasing of the scalar fields $H_2$ and $H_3$ .",
"Hence, if follows that the bosonic sector of the RIDM is CP-conserving.",
"The charged and neutral components of the Higgs basis doublet fields of the RIDM are also mass eigenstate fields, $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h_{\\rm SM}+iG^0\\bigr ] \\end{pmatrix},\\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix},\\quad H_3=\\begin{pmatrix} h^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [h+ia\\bigr ]\\end{pmatrix},$ with a minor change of notation from the IDM.",
"The corresponding squared masses of the neutral and charged scalars are given by, $m^2_{H^\\pm }&=&m^2_{h^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad m^2_H=m^2_h=Y_2+\\tfrac{1}{2}(Z_3+Z_4+Z_5)v^2\\,,\\nonumber \\\\m^2_A&=&m^2_a=Y_2+\\tfrac{1}{2}(Z_3+Z_4-Z_5)v^2\\,.$ By assumption, $Z_5\\ge 0$ , in which case $m_H=m_h\\ge m_A=m_a$ .In particular, note that if $Z_5=0$ then there is an enhanced mass degeneracy in which $m_H=m_h=m_A=m_a$ .",
"Thus, the RIDM possesses four mass-degenerate scalar pairs: $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ .",
"These mass degeneracies can be understood as a consequence of a continuous global Higgs flavor symmetry (where Higgs flavor corresponds to the multiplicity of Higgs doublets).",
"In order to explicitly exhibit the relevant symmetries, it is convenient to focus on the neutral scalar states of the doublet fields $H_2$ and $H_3$ , denoted henceforth by the complex fields, $H^0\\equiv \\frac{H+iA}{\\sqrt{2}}\\,,\\qquad \\quad h^0\\equiv \\frac{h+ia}{\\sqrt{2}}\\,,$ respectively.",
"Let us first focus on the kinetic energy terms and the terms in Eq.",
"(REF ) in the absence of the term proportional to $Z_5$ .",
"Then, one can check that the neutral complex scalar fields $H^0$ and $h^0$ appear only in the combination $H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0=\\tfrac{1}{2}(H^2+h^2+A^2+a^2)$ .",
"Thus, excluding $Z_5$ , the scalar Lagrangian possesses an O(4) global symmetry, that is responsible for four mass-degenerate neutral scalar states.",
"It is instructive to see how this symmetry arises when employing the complex basis $\\varphi _i=\\lbrace H^0,h^0\\rbrace $ (for $i=1,2$ ).",
"Noting that $\\varphi ^{\\dagger \\,i}\\varphi _i=H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0$ (the sum over the repeated index $i$ is implicit), it is clear that the scalar Lagrangian (in the absence of $Z_5$ ) is invariant under a U(2) global symmetry, $\\varphi _i\\rightarrow U_i{}^j\\varphi _j$ , with $U\\in {\\rm U(2)}$ .",
"However, the corresponding symmetry group is in fact larger than U(2).",
"Working in the complex basis, it is straightforward to verify that the quantity $\\varphi ^{\\dagger \\,i}\\varphi _i$ is invariant with respect to $\\varphi _i\\rightarrow U_i{}^j\\varphi _j+(V^*)^i{}_j\\varphi ^{\\dagger \\,j}\\,,$ where $U$ and $V$ are complex $2\\times 2$ matrices (which are not in general unitary), provided that the following two conditions are satisfied: $&&(i)~~~~(U^\\dagger U+V^\\dagger V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii)~~~V^T U~\\hbox{\\rm is an antisymmetric matrix}\\,.$ One can now check that Eq.",
"(REF ) corresponds to an O(4) symmetry transformation.",
"More explicitly, the $4\\times 4$ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}\\operatorname{Re}(U+V) &\\,\\, -\\operatorname{Im}(U+V) \\\\\\operatorname{Im}(U-V) & \\phantom{-}\\,\\, \\operatorname{Re}(U-V) \\end{array}\\right)$ is an orthogonal matrix if and only if $U$ and $V$ satisfy Eqs.",
"(REF ) and ().",
"Indeed, one can check that in light of Eqs.",
"(REF ) and (), the global symmetry specified by Eq.",
"(REF ) is governed by 6 continuous parameters as expected for an O(4) transformation.",
"Two special cases of Eq.",
"(REF ) are noteworthy.",
"First, if $V=0$ , then $U$ is unitary and we recover the U(2) global symmetry mentioned previously.",
"Second, if $U=0$ then Eq.",
"(REF ) corresponds to a generalized CP transformation [cf. Eq.",
"(REF )].The unified treatment of Higgs family transformations and generalized CP transformations has been advocated previously in Ref. [58].",
"A related discussion emphasizing the promotion of the U(2) basis transformation to an enlarged group of O(4) transformations appears in Ref. [66].",
"Both symmetries are present in the scalar Lagrangian if the $Z_5$ coupling is neglected, and either one would be sufficient to guarantee the mass degeneracy of $H$ , $h$ , $A$ and $a$ .",
"In the absence of the $Z_5$ coupling, the full O(4) global symmetry is respected by the pure scalar Lagrangian.",
"However, when we include the coupling of the scalar doublets to the gauge bosons, one must replace the ordinary derivative, $\\partial _\\mu $ , with the SU(2)$\\times $ U(1) gauge covariant derivative, $D_\\mu $ , in the scalar kinetic energy term.",
"The resulting coupling of the scalars to the vector bosons partially breaks the O(4) symmetry.",
"Employing the complex basis, it is easy to check that the symmetry transformation specified by Eq.",
"(REF ) is unbroken if and only if either $U=0$ or $V=0$ , namely the two special cases just highlighted above.This result is not surprising given that Eq.",
"(REF ) transforms the scalar field into a linear combination of two fields of opposite hypercharge unless either $U=0$ or $V=0$ .",
"That is, the kinetic energy term $(D^\\mu \\varphi )^{i\\,\\dagger } (D_\\mu \\varphi )_i$ is invariant under a U(2) symmetry (corresponding to $V=0$ ) and under the generalized CP symmetry (corresponding to $U=0$ ).",
"Mathematically, the unbroken global symmetry that remains is the semi-direct product U(2)$\\rtimes \\mathbb {Z}_2$ .This symmetry is a generalization of the U(1) symmetry (and the associated CP symmetry) of the IDM with $Z_5=0$ treated in section REF , and provides the motivation for our choice of the RIDM scalar potential given in Eq.",
"(REF ).",
"We now examine the consequence of including the term of Eq.",
"(REF ) proportional to $Z_5$ .",
"Focusing again on the neutral complex scalar fields $H^0$ and $h^0$ [cf. Eq.",
"(REF )], we see that a new combination of fields arises, $\\varphi _i\\varphi _i+{\\rm h.c.}=(H^0)^2+(H^{0\\,\\dagger })^2+(h^0)^2+(h^{0\\,\\dagger })^2$ .",
"This term is invariant with respect to Eq.",
"(REF ) provided that the conditions specified in Eqs.",
"(REF ) and () are replaced by 1.25 $&&(i^\\prime )~~~~(U^T U+V^T V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii^\\prime )~~~V^\\dagger U~\\hbox{\\rm is an antihermitian matrix}\\,.$ The conditions specified by Eqs.",
"(REF ) and () are compatible with those of Eqs.",
"(REF ) and () if $U$ and $V$ are real matrices.",
"Consequently, $\\mathcal {Q}$ specified in Eq.",
"(REF ) is now a block diagonal orthogonal $4\\times 4 $ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}U+V &\\,\\, 0 \\\\0 & \\phantom{-}\\,\\, U-V \\end{array}\\right)\\,,$ where $(U\\pm V)^T(U\\pm V)={1}_{2\\times 2}$ as a consequence of Eqs.",
"() and (REF ).",
"That is, the scalar Lagrangian is invariant under a global O(2)$\\times $ O(2) symmetry, which explains the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.",
"The breaking of the four-fold mass degeneracy to the two mass-degenerate pairs is due to the scalar potential term proportional to $Z_5$ , as is evident from Eq.",
"(REF ).",
"Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is O(2)$\\rtimes \\mathbb {Z}_2$ .",
"For completeness, we note that the degeneracy of the charged Higgs scalars $(H^\\pm ,h^\\pm )$ is governed by the full O(4) symmetry, which is broken down to U(2)$\\rtimes \\mathbb {Z}_2$ after promoting the derivatives of the scalar kinetic energy term to gauge covariant derivatives.",
"This is easily seen by noting that in the unitary gauge (in which the Goldstone fields do not explicitly appear), the physical charged scalar fields do not appear in the scalar potential term proportional to $Z_5$ .",
"Finally, if $Z_4=Z_5=0$ , we can make use of the vertical SU(2) global symmetry (which when gauged corresponds to the SU(2) electroweak gauge group) to conclude that all eight charged and neutral inert scalars are mass-degenerate.",
"Next, we examine all the bosonic couplings of the RIDM in the unitary gauge (where the Goldstone fields are set to zero).",
"The Higgs boson interactions with the gauge bosons and the Higgs boson self couplings of the RIDM are listed below.",
"${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h_{\\rm SM}\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h_{\\rm SM}^2+H^2+h^2+A^2+a^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](H^+H^- +h^+ h^-)\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [H^-(H+iA)+h^-(h+ia)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu ( A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H+a $\\leftrightarrow $$\\partial $ h) -12g{iW+[ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA)+h-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (h+ia)] +h.c.}",
"+[ieA+igcW(12-sW2) Z](H+$\\leftrightarrow $$\\partial $ H-+h+$\\leftrightarrow $$\\partial $ h-), In the RIDM, there is no experimental measurement that can physically distinguish the degenerate scalars, $(H^\\pm ,h^\\pm )$ , $(H,h)$ and $(A,a)$ .",
"However, a multiplicity factor will appear after summing over final mass-degenerate states, e.g., $Z\\rightarrow HA$ , $ha$ doubles the rate into a pair of neutral scalars."
],
[
"An alternative basis choice for the RIDM",
"So far, our discussion has employed the $\\lbrace H_1,H_2,H_3\\rbrace $ basis of doublet scalar fields.",
"This is one choice among a family of Higgs bases defined such that $\\langle H_1^0 \\rangle =v/\\sqrt{2}$ and $\\langle H_2^0 \\rangle =\\langle H_3^0 \\rangle =0$ .",
"Indeed, the Higgs basis is unique only up to an arbitrary U(2) transformation of the doublet fields $H_2$ and $H_3$ .",
"In the following, we shall denote the $\\lbrace H_1,H_2,H_3\\rbrace $ basis as the $H23$ -basis, since the scalar potential of Eq.",
"(REF ) provides a simple 3HDM extension of the inert 2HDM.",
"It will prove useful to consider another choice of scalar field basis that is related to the $H23$ -basis as follows,Further details are provided in Appendix REF .",
"$&&\\mathcal {R}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2+iH_3\\bigr )=\\begin{pmatrix} R^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P+iQ^\\dagger \\bigr )\\end{pmatrix}\\,,\\nonumber \\\\[15pt]&&\\mathcal {S}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2-iH_3\\bigr )=\\begin{pmatrix} S^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P^\\dagger +iQ\\bigr )\\end{pmatrix}\\,.$ This defines the $\\lbrace H_1,\\mathcal {R},\\mathcal {S}\\rbrace $ basis of doublet scalar field, henceforth denoted as the $RS$ -basis.",
"Note that since the real neutral fields $(H,h)$ and $(A,a)$ are mass-degenerate pairs, respectively, one can combine the mass-degenerate real fields into complex fields, $P\\equiv \\frac{H+ih}{\\sqrt{2}}\\,,\\qquad Q\\equiv \\frac{A-ia}{\\sqrt{2}}\\,,$ where $M_P\\ge M_Q$ (in our convention where $Z_5\\ge 0$ ).The relative minus sign in the definition of the imaginary parts of $P$ and $Q$ has been introduced for later convenience.",
"The corresponding conjugate fields are $P^\\dagger \\equiv \\frac{H-ih}{\\sqrt{2}}\\,,\\qquad Q^\\dagger \\equiv \\frac{A+ia}{\\sqrt{2}}\\,,$ Likewise, since $H^\\pm $ and $h^\\pm $ are mass-degenerate charged fields, one is free to define, $R&=\\frac{H^- - ih^-}{\\sqrt{2}}, &\\quad S&=\\frac{H^- + ih^-}{\\sqrt{2}}, \\\\R^\\dagger &=\\frac{H^+ + ih^+}{\\sqrt{2}}, &\\quad S^\\dagger &=\\frac{H^+ - ih^+}{\\sqrt{2}},$ where $R$ and $S$ are negatively charged mass-degenerate scalars and the corresponding conjugate fields, $R^\\dagger $ and $S^\\dagger $ , are positively charged mass-degenerate scalars.",
"In the $RS$ -basis, the scalar potential is given by $\\!\\!\\!\\!\\!\\!\\mathcal {V}_{\\rm RIDM-RS}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\bigl (\\mathcal {R}^\\dagger \\mathcal {R} +\\mathcal {S}^\\dagger \\mathcal {S}\\bigr )+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}{\\bar{Z}}_2(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S})^2+Z_3(H_1^\\dagger H_1)(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S}) \\nonumber \\\\&& +Z_4\\bigl [( H_1^\\dagger \\mathcal {R})(\\mathcal {R}^\\dagger H_1)+( H_1^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger H_1)\\bigr ]+ \\bar{Z}^\\prime _5\\bigl [(H_1^\\dagger \\mathcal {R})(H_1^\\dagger \\mathcal {S}) +(\\mathcal {R}^\\dagger H_1)(\\mathcal {S}^\\dagger H_1)\\bigr ]\\,,$ where $\\bar{Z}_2=Z_2$ and $\\bar{Z}_5^\\prime =Z_5$ .The reason for introducing the notation $\\bar{Z}_2$ and $\\bar{Z}^\\prime _5$ in Eq.",
"(REF ) is clarified in section REF .",
"One can then rewrite the RIDM couplings given in Eqs.",
"(REF )–() in terms of the neutral scalar fields $P$ and $Q$ and the charged scalar fields $R$ and $S$ (and the corresponding conjugated fields), $&&\\hspace{-14.45377pt}{L}_{VVHH}=\\left[\\tfrac{1}{4} g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right]\\bigl (h^2_{\\rm SM}+2|P|^2+2|Q|^2\\bigr ) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](R^\\dagger R+S^\\dagger S)\\nonumber \\\\&&+\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [R(P+iQ^\\dagger )+S(P^\\dagger +iQ)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace ,\\\\&&\\hspace{-14.45377pt}{L}_{VHH}=\\frac{g}{2c_W}\\,Z^\\mu ( Q\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ P+Q $\\leftrightarrow $$\\partial $ P) +[ieA+igcW(12-sW2) Z] (R$\\leftrightarrow $$\\partial $ R+S$\\leftrightarrow $$\\partial $ S) -12g{iW+[R $\\leftrightarrow $$\\partial $ (P+iQ) +S $\\leftrightarrow $$\\partial $ (P+iQ)] +h.c.}",
"L3h=-12vZ1 hSM3-v[(Z3+Z4)hSM(|P|2+|Q|2)+Z5hSM(|P|2-|Q|2)] -vZ3hSM(RR+SS) , line .",
"L4h=-18 Z1 hSM4 -12Z2(|P|2+|Q|2)(|P|2+|Q|2+2RR + 2SS)-12(Z3+Z4) hSM2(|P|2+|Q|2) -12Z5 hSM2 (|P|2-|Q|2)-12[Z2(RR+SS) +Z3 h2SM] (RR+SS) ."
],
[
"Mass degeneracies beyond the RIDM",
"In this section, we add additional terms to the RIDM scalar potential while preserving the mass degeneracies of the model.",
"Naively, one can add to the RIDM scalar potential any gauge invariant quartic term involving the doublet fields $H_2$ and $H_3$ without upsetting the mass degeneracies of Eq.",
"(REF ).",
"However, the resulting tree-level mass degeneracies will be unnatural unless they are a consequence of a symmetry.",
"The simplest possible modification of the RIDM is to remove the $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ term entirely from the scalar potential.",
"That is, we can define a RIDM$^\\prime $ scalar potential as, $\\mathcal {V}_{\\rm RIDM^{\\prime }}= \\mathcal {V}_{\\rm RIDM}-Z_2(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)\\,.$ Note that the term in $\\mathcal {V}_{\\rm RIDM^{\\prime }}$ that is proportional to $Z_2$ is now given by $\\tfrac{1}{2}\\bigl [(H_2^\\dagger H_2)^2+(H_3^\\dagger H_3)^2\\bigr ]$ .",
"Indeed, one can argue that Eq.",
"(REF ) provides the simplest 3HDM generalization of the IDM.",
"In the case of the RIDM$^\\prime $ , the tree-level mass degeneracies are no longer a consequence of a continuous symmetry, which is now explicitly broken by the presence of the explicit term in Eq.",
"(REF ) that is proportional to $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ .",
"Indeed, this term is invariant only under a discrete subgroup of O(2)$\\times $ O(2) [which is the symmetry group of the RIDM scalar Lagrangian as discussed in section REF ].",
"In the notation of Eqs.",
"(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=g\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=g\\,,$ where $g$ is a $2\\times 2$ matrix that acts on the Higgs basis fields $H_2$ and $H_3$ regarded as a two dimensional vector.",
"Then, the term $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ is invariant under the two discrete subgroups above if $g\\in D_4\\mathchoice{\\cong }{\\cong }{\\vbox {{ \\scriptstyle \\hfil # \\hfil \\crcr \\sim \\crcr = \\crcr }}}{\\cong }\\lbrace {1},-{1}, R,-R,S,-S,Z,-Z\\rbrace $ , where 1 is the $2\\times 2$ identity matrix and $R=\\begin{pmatrix} 1 &\\,\\,\\,\\phantom{-}0 \\\\ 0 &\\,\\,\\, \\phantom{-}1\\end{pmatrix}\\,,\\qquad \\quad S=\\begin{pmatrix} 1 &\\,\\,\\, \\phantom{-}0 \\\\ 0 &\\,\\,\\,-1\\end{pmatrix}\\,,\\qquad \\quad Z=\\begin{pmatrix} 0 &\\,\\,\\, -1 \\\\ 1 &\\,\\,\\, \\phantom{-}0\\end{pmatrix}\\,.$ We recognize $D_4$ as the dihedral group of order eight, which is the symmetry group of the square [67].",
"Both discrete subgroups [Eqs.",
"(REF ) and ()] are isomorphic to $D_4$ .",
"Following the discussion below Eq.",
"(REF ), we conclude that the RIDM$^\\prime $ scalar Lagrangian is invariant under a discrete $D_4\\times D_4$ symmetry, which is responsible for the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.",
"Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is $D_4\\rtimes \\mathbb {Z}_2$ .",
"A comprehensive treatment of natural scalar mass degeneracies in the 3HDM would require a complete classification of 3HDM scalar potential symmetries, along the lines of the 2HDM analysis given in section REF .A complete catalog of all possible finite symmetry groups of the 3HDM is known (as well as some additional partial results); however the complete classification of all possible symmetries of the 3HDM remains an open problem [68], [69].",
"In this paper, we shall ask a less ambitious question: can one break the discrete symmetry identified above further while still naturally maintaining the mass-degenerate states of the RIDM.",
"The answer turns out to be affirmative.",
"This investigation led us to a particular 3HDM originally introduced by Ivanov and Silva [30] for other reasons that will be reviewed below.",
"The Ivanov-Silva (IS) model was constructed to exhibit a number of curious properties [30], [66], which appear to rely on the existence of degenerate states in the scalar spectrum.",
"In particular, the IS scalar potential does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ , where the latter satisfies $({\\rm CP})^2={1}$ , but instead respects a generalized CP symmetry of the form $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ for some unitary matrix $X$ .",
"In particular, the generalized CP symmetry of the IS scalar potential, denoted by CP4, is of order 4, signifying that $({\\rm CP4})^4={1}$ and $({\\rm CP4})^2\\ne {1}$ .",
"Moreover, no Higgs basis of scalar fields exists in which all the parameters of the IS scalar potential are simultaneously real.",
"As noted in section REF , this property is in stark contrast with the 2HDM in which the existence of any generalized CP symmetry implies that the 2HDM scalar potential automatically respects the conventional CP symmetry, i.e.",
"a basis of scalar fields exists such that the corresponding 2HDM scalar potential parameters are real [56], [57].",
"In Appendix A, we demonstrate that starting from the IS scalar potential, one can perform a basis change in order to obtain a more convenient form of the scalar potential.",
"By making an appropriate U(2) transformation to define the Higgs basis fields, $H_2$ and $H_3$ , we find that the IS scalar potential takes on the following form in the $H23$ -basis, $\\mathcal {V}_{\\rm IS}= \\mathcal {V}_{\\rm RIDM}+Z_3^\\prime (H_2^\\dagger H_2)(H_3^\\dagger H_3)+Z_4^\\prime (H_2^\\dagger H_3)(H_3^\\dagger H_2) +\\bigl [Z_8 (H_2^\\dagger H_3)^2+Z_9(H_2^\\dagger H_3)(H_2^\\dagger H_2-H_3^\\dagger H_3)+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM}$ is given in Eq.",
"(REF ).",
"In general, $Z_8$ and $Z_9$ are complex parameters.As shown in Appendix REF , one can perform an SO(2) rotation to redefine the fields $H_2$ and $H_3$ to remove the complex phase from either $Z_8$ or $Z_9$ .",
"We shall continue to use Eq.",
"(REF ) to express the Higgs basis fields in terms of mass-eigenstate fields.",
"Since none of the extra terms in Eq.",
"(REF ) involve the Higgs basis field $H_1$ , the tree-level mass relations of Eq.",
"(REF ) are not modified.",
"We now argue that the mass-degeneracies of $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ are stable due to the presence of a symmetry.",
"The O(2)$\\times $ O(2) symmetry of the RIDM (prior to gauging the scalar kinetic energy terms) that is responsible for the mass degeneracies among the neutral Higgs mass eigenstates is broken by the new terms beyond $\\mathcal {V}_{\\rm RIDM}$ contained in Eq.",
"(REF ).",
"Indeed, after the extra terms are included, no unbroken continuous subgroup of O(2)$\\times $ O(2) remains.",
"In the notation of Eqs.",
"(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=Z\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=Z\\,,$ where $Z$ is given by Eq.",
"(REF ).",
"The $2\\times 2$ matrix $Z$ acts on the Higgs basis fields $H_2$ and $H_3$ .",
"Both discrete subgroups [Eqs.",
"(REF ) and ()] are isomorphic to $\\mathbb {Z}_4=\\bigl \\lbrace {1},-{1},Z,-Z\\bigr \\rbrace $ .",
"Note that $Z^2=-{1}$ , where 1 is the $2\\times 2$ identity matrix.In this case, gauging the scalar kinetic energy terms does not reduce the symmetry group further.",
"Consider first the discrete symmetry defined in Eq.",
"(REF ).",
"The fields $H_2$ and $H_3$ are odd under $-{1}$ , which simply identifies the two inert doublets.",
"The elements $Z$ (and $-Z$ ) act non-trivially on the inert doublets.",
"However, Eq.",
"(REF ) is invariant with respect to $\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\rightarrow \\begin{pmatrix} 0 & -1\\\\ 1 & \\phantom{-}0\\end{pmatrix}\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\,,$ if and only if $Z_8$ and $Z_9$ are both real.",
"In the model of IS where there is an unremovable complex phase in the scalar potential, only the subgroup $\\mathbb {Z}_2=\\lbrace {1},-{1}\\rbrace $ of $\\mathbb {Z}_4$ survives.",
"In particular, the residual symmetry in this case is not sufficient to explain the mass degeneracies of the IS model.",
"The discrete symmetry defined in Eq.",
"() is a generalized CP symmetry.",
"In particular, the IS scalar potential is invariant under $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}\\,, \\qquad \\quad \\text{where $X=\\begin{pmatrix} 1 &\\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1 \\\\ 0 & \\phantom{-}1 & \\phantom{-}0\\end{pmatrix}$,}$ This symmetry, which is also isomorphic to $\\mathbb {Z}_4$ , is the CP4 symmetry advertised above.",
"Moreover, this discrete symmetry is sufficient to explain the mass degeneracies of the IS model (in the case of an unremovable complex phase in the IS scalar potential).",
"It is instructive to consider the Higgs couplings of the IS model.",
"Only the quartic Higgs couplings of the RIDM are modified as follows, $\\delta {L}_{4h}&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime )\\bigl [(H^2+A^2)(h^2+a^2)+4H^+ H^- h^+ h^-\\bigr ]-\\tfrac{1}{2}Z_3^\\prime \\bigl [(H^2+A^2)h^+ h^-+(h^2+a^2)H^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{2}Z^{\\prime }_4\\bigl [\\bigl (Hh+Aa+i(Ha-hA)\\bigr )H^+ h^- + \\bigl (Hh+Aa-i(Ha-hA)\\bigr )h^+ H^-\\bigr ] \\nonumber \\\\&& -\\tfrac{1}{4} Z_8\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]^2-\\tfrac{1}{4} Z^*_8\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]^2 \\nonumber \\\\&& -\\tfrac{1}{4} Z_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{4} Z^*_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]\\,.$ It is convenient to re-express the neutral scalar fields appearing in Eq.",
"(REF ) in terms of the complex neutral fields $P$ and $Q$ and their conjugates introduced in Eqs.",
"(REF ) and (REF ), and the charged fields $R$ and $S$ and their conjugates defined in Eqs.",
"(REF ) and ().",
"Note that the fields $P$ , $Q$ and the corresponding conjugate fields $P^\\dagger $ and $Q^\\dagger $ are each eigenstates of CP4.This means that each of the four states, $P$ , $Q$ , $P^\\dagger $ and $Q^\\dagger $ , are CP4-self conjugate (they are their own antiparticles).",
"Moreover, $P$ and the corresponding conjugate state $P^\\dagger $ are mass-degenerate, but are otherwise unrelated fields (and similarly for $Q$ and $Q^\\dagger $ ).",
"In particular, under a CP4 transformation, $P\\rightarrow iP$ , $Q\\rightarrow iQ$ , $P^\\dagger \\rightarrow -iP^\\dagger $ , and $Q^\\dagger \\rightarrow -iQ^\\dagger $ .",
"Likewise, under a CP4 transformation, $R\\rightarrow -iS^\\dagger $ , $R^\\dagger \\rightarrow iS$ , $S\\rightarrow iR^\\dagger $ and $S^\\dagger \\rightarrow -iR$ .",
"Note that these transformation properties are consistent with the requirement that $({\\rm CP}4)^4={1}$ .",
"We can evaluate the four-scalar interaction Lagrangian directly in the $RS$ -basis.",
"We first must rewrite Eq.",
"(REF ) in the RS-basis, $\\mathcal {V}_{\\rm IS-RS}= \\mathcal {V}_{\\rm RIDM-RS}+\\bar{Z}_3^\\prime (\\mathcal {R}^\\dagger \\mathcal {R})(\\mathcal {S}^\\dagger \\mathcal {S})+\\bar{Z}_4^\\prime (\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger \\mathcal {R}) +\\bigl [\\bar{Z}_8 (\\mathcal {R}^\\dagger \\mathcal {S})^2+\\bar{Z}_9(\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {R}^\\dagger \\mathcal {R}-\\mathcal {S}^\\dagger \\mathcal {S})+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM-RS}$ is given by Eq.",
"(REF ).",
"The relations between the unbarred and barred parameters are derived in Appendix REF , $\\bar{Z}_2&=&Z_2+\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\quad \\,\\,\\,\\,\\bar{Z}_3^\\prime =-Z_4^\\prime +2\\operatorname{Re}Z_8\\,,\\\\\\bar{Z}_4^\\prime &=&\\tfrac{1}{2}(Z_4^\\prime -Z_3^\\prime +2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\qquad \\qquad \\bar{Z}_5^\\prime =Z_5\\,,\\\\\\bar{Z}_8&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime +2\\operatorname{Re}Z_8)+i\\operatorname{Re}Z_9\\,,\\qquad \\quad \\bar{Z}_9= \\operatorname{Im}Z_9+i\\operatorname{Im}Z_8\\,.",
"$ The quartic interactions given in Eq.",
"(REF ) are then modified by employing the new definition of $\\bar{Z_2}$ given in Eq.",
"(REF ) and adding the following terms, We now consider the possible effects of the Yukawa interactions.",
"It is remarkable that it is possible to construct a CP4-invariant Yukawa interaction Lagrangian where the fermions transform nontrivially under a CP4 transformation [66], [70], [71].",
"In such a model, the mass degeneracies identified above that are a consequence of the CP4 symmetry are of course maintained.",
"Alternatively, if the fermions couple exclusively to the Higgs basis field $H_1$ (as in the case of the IDM), then the Yukawa interactions are invariant with respect to the $\\mathbb {Z}_2$ discrete symmetry defined below Eq.",
"(REF ),Note that this $\\mathbb {Z}_2$ symmetry is isomorphic to $(\\rm {CP}4)^2$ , which remains an exact symmetry of the model.",
"under which the inert doublet fields, $H_2$ and $H_3$ , are odd and all other fields of the model ($H_1$ , gauge bosons and fermions) are even.",
"However, the CP4 symmetry is no longer a symmetry of the complete model.",
"That is, if we define the CP4 transformation to be the conventional CP transformation when acting on the fermions and gauge fields, then the CP4 symmetry of the model will be violated by the presence of the unremovable CP-violating phase in the CKM mixing matrix.",
"Nevertheless, it is not clear whether this violation is sufficient to remove the scalar mass degeneracies of the IS model that were protected by the (now accidental) CP4 symmetry of the scalar potential.",
"This is an open question that we hope to revisit in a future work.",
"Finally, it is instructive to note that the scalar mass degeneracies of the CP4-invariant 3HDM is just the simplest example of a larger class of multi-Higgs models with degenerate scalars that are a consequence of a generalized CP symmetry.",
"In Ref.",
"[72], Ivanov and Laletin demonstrate how to construct $N$ Higgs doublet models with a generalized CP symmetry of order $2k$ (denoted by CP$2k$ ) with positive integer $k$ .",
"Nontrivial cases arise only for $2k=2^p$ with integer $p\\ge 1$ .",
"The simplest nontrivial models of this type (CP8 and CP16) require at least $N=5$ Higgs doublets.",
"Such models necessarily have mass-degenerate neutral scalars and mass-degenerate charged Higgs pairs.",
"A further exploration of models of this type is beyond the scope of this work."
],
[
"An observable distinction between CP2 and CP4",
"The distinction between the IS scalar potential in the $H23$ -basis with $Z_8$ and $Z_9$ real or complex is physical.In making this assertion, we have implicitly assumed that $Z_5\\ne 0$ .",
"The case of $Z_5=0$ , which is special due to the enhanced mass degeneracy noted in footnote REF , will be treated at the end of this section.",
"To demonstrate this assertion, we focus on the neutral scalar self-interactions in $\\delta {L}_{4h}$ that are linear in the fields $P$ or $Q$ (or their complex conjugates), $\\delta {L}_{4h}\\ni \\tfrac{1}{2}\\operatorname{Im}Z_8\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2-Q^2-P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]+\\tfrac{1}{2}i\\operatorname{Im}Z_9\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2+Q^2+P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]\\,,$ where we have used Eq.",
"() to re-express $\\bar{Z}_9$ [which appears in Eq.",
"(REF )] in terms of the $H23$ -basis parameters, $\\operatorname{Im}Z_8$ and $\\operatorname{Im}Z_9$ .",
"Self-interaction terms of this type are absent if $Z_8$ and $Z_9$ are both real.",
"Hence, the presence of these terms signals a CP4-symmetric IS scalar potential that does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ .",
"Here we provide two specific examples.",
"First, Eq.",
"(REF ) shows the existence of a $ZPQ$ interaction, which would permit the decay $Z\\rightarrow PQ, P^*Q^*$ , if kinematically available.",
"Since $M_Q\\le M_P$ , let us further suppose that $M_Q<\\tfrac{1}{4} m_Z<M_P$ .",
"In this case, the $P$ and $P^*$ would be virtual.",
"One possible decay of the virtual $P$ or $P^*$ makes use of the existence of the four-scalar interaction given in Eq.",
"(REF ).",
"If this interaction is present, the decay $Z\\rightarrow QQQQ^*$ , $Q^* Q^* Q^* Q$ is allowed and provides unambiguous evidence that either $Z_8$ and/or $Z_9$ possesses a nonzero imaginary part.",
"A second example makes use of the $W^+H^-P$ , $W^+h^-P$ , $W^+H^-Q$ , and $W^+h^-Q$ interactions of Eq.",
"(REF ).",
"In this case, we can consider the decay of a charged $W$ into a charged Higgs boson and $P$ (or $P^*$ ).",
"We can now make use of Eq.",
"(REF ) to decay the virtual $P$ or $P^*$ into $QQQ$ , $QQQ^*$ , $QQ^*Q^*$ , $Q^* Q^* Q^*$ .",
"Note that in each of the two cases above, there are multiple four-scalar final states involving mass-degenerate scalars.",
"In computing the experimentally observed rates, one must compute the squared amplitude for each of the possible final states, and then multiply the final result by a multiplicity factor that counts the number of possible final states.",
"In contrast, suppose that Eq.",
"(REF ) were a symmetry of the IS scalar potential.",
"In this case, the corresponding transformation properties of the scalar fields are, $P\\rightarrow iP$ , $Q\\rightarrow -iQ$ , $P^\\dagger \\rightarrow -iP^\\dagger $ , $Q^\\dagger \\rightarrow iQ^\\dagger $ , $H^{\\pm }\\rightarrow -h^\\pm $ , and $h^\\pm \\rightarrow H^\\pm $ .",
"One would then immediately conclude that $Z_8=Z_8^*$ and $Z_9=Z_9^*$ , as expected.",
"In particular, Eq.",
"(REF ) is not invariant under Eq.",
"(REF ), and thus the four scalar decay modes listed above would necessarily be absent.",
"As an exercise, we have evaluated the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , in an approximation where $M_Q=0$ and $M_P\\gg m_Z$ .",
"The computation is presented in Appendix .",
"The end result is $\\frac{\\Gamma (Z\\rightarrow QQQQ^*,QQ^* Q^* Q^*)}{\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })}=\\frac{(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2}{3\\cdot 5\\cdot 2^{8}\\, \\pi ^4 }\\left(\\frac{m_Z}{M_P}\\right)^4\\,.$ This result implies that the quantity $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2$ must be a physical quantity, and hence invariant with respect to scalar basis changes that are consistent with the form of the IS scalar potential given by Eq.",
"(REF ) in the $H23$ -basis.",
"However, the family of Higgs bases is larger than the set of scalar field bases in which the IS scalar potential has the form of Eq.",
"(REF ), as discussed in Appendix REF .",
"In special cases, it is possible that there exists a real Higgs basis even if $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\ne 0$ .",
"In such cases, one can transform the fields ($H_2, H_3)\\rightarrow (\\bar{H}_2,\\bar{H}_3)$ , where $\\bar{H}_i\\rightarrow \\bar{H_i}^{\\scriptscriptstyle {\\bigstar }}$ is a symmetry of the Lagrangian; i.e., the model exhibits a CP2 symmetry.The notation CP2 derives from the property, $(\\text{CP2})^2={1}$ .",
"In the IS model, the existence of a nonzero decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ implies that no CP2 symmetry that commutes with the CP4 symmetry is present.We say that the CP2 symmetry commutes with CP4 if the application of these two symmetry transformations on the scalar fields does not depend on the order in which the transformations are applied.",
"For further details see Appendix B of Ref. [32].",
"However, this leaves open the possibility of a CP2 symmetry that does not commute with CP4.",
"In Appendix REF , we provide two examples in which $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\ne 0$ in the $H23$ basis, but nevertheless a real Higgs basis exists: (i) $\\operatorname{Im}Z_8\\ne 0$ and $Z_9=0$ and (ii) $\\operatorname{Im}Z_8=0$ , $\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .",
"In both these examples, the CP2 symmetry that exists does not commute with the CP4 symmetry, even though the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is nonzero.",
"In light of the results of Appendix REF , this is a generic feature of a noncommuting CP2 symmetry in the IS model.",
"Equivalently, the nonexistence or existence of the decay $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is a physical distinction between the 3HDM with a CP4 symmetric IS scalar potential that either preserves or does not preserve a commuting CP2 symmetry.",
"Finally, we return to the special case of $Z_5=0$ (cf.",
"footnote REF ).",
"In Appendix REF , we have demonstrated explicitly that if $Z_5\\ne 0$ , then there exists a ratio of two basis-invariant quantities, which when evaluated in the $H23$ -basis yields $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2$ .",
"Moreover, if $Z_5=0$ , then it is possible to change the basis of scalar fields of the IS model, in which the form of the IS potential is still given by Eq.",
"(REF ) but $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ .",
"This result appears to be in contradiction to the result of Eq.",
"(REF ).",
"The resolution of this apparent paradox can be obtained by noting that if $Z_5=0$ , then $M_P=M_Q$ .",
"Since Eq.",
"(REF ) was derived under the assumption that $M_Q=0$ and $M_P\\gg m_Z$ , Eq.",
"(REF ) no longer applies if $Z_5=0$ .",
"But, more importantly, if $Z_5=0$ (so that $M_P=M_Q$ ), then the decay $Z\\rightarrow QQQQ^*,QQ^* Q^* Q^*$ is no longer an experimental observable, since one must also include four scalar decays involving $P$ and $P^*$ .",
"The possible four-body final states involve all possible combinations of $P$ and $Q$ scalars, such that either one or three of the final state scalars are complex-conjugated.",
"Some of the vertices that contribute to these final states are present even if $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ .",
"For example, there is a four-scalar $|P|^2 |Q|^2$ interaction that contributes to $Z\\rightarrow QQPP^*$ .",
"One must compute the squared amplitude for each possible final state and then add the amplitudes incoherently to obtain the final experimentally observable decay rate.",
"This decay rate will involve a complicated combination of the IS potential coefficients, which will correspond to the appropriate invariant quantity in the case of $Z_5=0$ .",
"Thus, the possibility of finding a new basis for the IS potential in which $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ when $Z_5=0$ is no longer paradoxical."
],
[
"The $ZZZ$ and {{formula:a1ad6645-e9b6-4fb4-b6f1-a07eb57f0b31}} vertices",
"In the CP-violating 2HDM, CP violation may manifest itself at loop level in the effective $ZZZ$ and $ZWW$ vertices.",
"In that model, one finds that CP-violating form factors can be described in terms of the three invariants introduced in Eq.",
"(REF ) [73].",
"In section , we noted the existence of a physical observable that could distinguish between the CP4-conserving IS models in which a CP2 symmetry that commutes with CP4 is either present or absent.",
"However, from a spacetime viewpoint, this physical observable was CP-even.",
"This raises the question as to whether any observable can exist in a CP4-invariant theory that is CP2-odd.",
"The answer to this question is no.",
"For example, there is no way to distinguish between CP2 and CP4 on the level of the form factors themselves.",
"Thus, if the theory respects at least one generalized CP symmetry, then all CP-violating form factors must be absent.",
"It is instructive to check the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices in a CP4-conserving, CP2-violating theory (neglecting any effects from the Higgs-fermion Yukawa interactions).",
"The general $ZZZ$ vertex function (with all $Z$ bosons off-shell) can be expressed in terms of 14 different Lorentz structures [74], [75], [76], [77], [78], all preserving parity.",
"Some of these vanish when one or more $Z$ are on-shell.",
"Let us characterize them by momenta and Lorentz indices ($p_1,\\mu $ ), ($p_2,\\alpha $ ) and ($p_3,\\beta $ ), and let $Z_1$ be off-shell while $Z_2$ and $Z_3$ are on-shell.",
"Furthermore, we assume that $Z_1$ couples to a pair of leptons such as $e^+e^-$ , and terms proportional to the lepton mass will be neglected.",
"Denoting $\\ell \\equiv p_2-p_3 \\equiv 2p_2-p_1$ , the $ZZZ$ vertex structure reduces to the form [76] $-i\\Gamma _{ZZZ}^{\\alpha \\beta \\mu }=\\frac{p_1^2-M_Z^2}{M_Z^2}\\left[f_4^Z(p_1^{\\alpha }g^{\\mu \\beta }+p_1^{\\beta }g^{\\mu \\alpha })+f_5^Z\\epsilon ^{\\mu \\alpha \\beta \\rho }\\ell _\\rho \\right]\\,.$ The dimensionless form factor $f_4^Z$ violates CP while $f_5^Z$ conserves CP.",
"Figure: A typical pair of Feynman diagrams for Z→ZZZ\\rightarrow ZZ at two-loop order.For example, consider the case of the 2HDM.",
"At the one-loop level, CP violating effects yield a non-zero contribution to the $ZZZ$ vertex function, $f_4$ , that is proportional to $\\operatorname{Im}J_2$ of Eq.",
"(REF ) [73].",
"Thus, only one of the three invariants of Eq.",
"(REF ) contributes.",
"Indeed, in light of Eq.",
"(REF ), it follows that a non-zero $\\operatorname{Im}J_2$ requires all three neutral Higgs bosons to be non-degenerate in mass, and the $Z$ boson couples to all three non-diagonal neutral Higgs pairs.",
"In order to understand how the IS model conserves CP (while not respecting CP2), it is instructive to see how the CP-violating effects cancel at loop level in the effective $ZZZ$ (and $ZWW$ ) vertices.",
"In order to do this we have employed the software package FeynArts[79] and written a FeynArts model file containing all the bosonic couplings of the IS-model.",
"We have automated the construction of the diagrams contributing to the effective $ZZZ$ -vertex and evaluated their amplitude (the loop integrals are kept unevaluated in symbolic form).",
"We are only interested in those contributions to each diagram that contain $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ , since such contributions could be a signal of CP violation.",
"At the one-loop level there are no diagrams containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ .",
"Such contributions can only arise from a four-point scalar vertex.",
"This means that this four-point vertex must be “internal\"; i.e., none of the external $Z$ -fields can be part of this vertex.",
"None of the $ZZZ$ one-loop topologies can accommodate this.",
"Diagrams containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ first appear at two-loop order.",
"But even if there are individual diagrams with this type of contribution, the sum of the contributions is zero when we add the amplitudes for all the individual diagrams within each topology.",
"A pair of cancelling diagrams are shown in Fig.",
"REF .",
"The same happens for diagrams at three-loop order.",
"Repeating this exercise for the $ZWW$ vertex we find the same result.",
"Hence, there are no contributions at one-, two- or three-loop order containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ after adding the amplitudes for all the individual diagrams within each topology.",
"The arguments presented at the beginning of this section imply that this cancellation persists to all orders in perturbation theory."
],
[
"Conclusions",
"In this work we discussed the interplay between symmetries and natural mass degeneracies in the scalar sector.",
"Some cases of scalar mass degeneracy are accidental, i.e.",
"they are not the result of an exact symmetry and therefore can be implemented only by an artificial fine tuning of the scalar potential parameters.",
"The Higgs basis [16], [17], [18], in which the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields, is especially suitable for our study.",
"We began by examining the two Higgs doublet model (2HDM), with particular attention given to the special case of the inert doublet model (IDM), which possesses an unbroken $\\mathbb {Z}_2$ symmetry under which one “inert” scalar doublet is odd, and all other fields of the model are even.",
"In all cases in which the 2HDM exhibited scalar mass degeneracies (whether natural or accidental), the mass degenerate states can be experimentally distinguished from each other.",
"Moreover, with one exception, we found that all 2HDM mass degeneracies are accidental.",
"The one exceptional case of 2HDM scalars that can be naturally degenerate in mass are the two neutral scalar states of the inert doublet of the IDM.",
"This result was also confirmed by examining all possible symmetries of the 2HDM scalar potential and analyzing which of these symmetries can guarantee the presence of mass degenerate scalar states.",
"For models with three Higgs doublets, the analysis of the general case becomes significantly more elaborate.",
"We focused first on a 3HDM generalization of the IDM with mass degenerate scalars, which we denoted as the replicated IDM (RIDM), where the two doublets $H_2$ and $H_3$ are invariant under two separate unbroken $\\mathbb {Z}_2$ symmetries and the model is CP conserving.",
"In this framework $H_2$ and $H_3$ are composed of mass eigenstate fields, that do not mix with the SM like Higgs boson, forming four mass degenerate pairs.",
"Furthermore, each mass degenerate pair picks one field from each one of these doublets.",
"We also identified the symmetry obeyed by the neutral mass eigenstates themselves, which is responsible for the twofold mass degeneracies.",
"In the absence of $Z_5$ (which appears in the RIDM scalar potential) there are four mass degenerate neutral scalars and the symmetry of the scalar potential consists of an O$(4)$ global symmetry.",
"Introducing in the potential the term proportional to $Z_5$ , partially breaks the O$(4)$ symmetry down to an O(2)$\\times $ O(2) symmetry and the fourfold mass degeneracy is lifted, leaving a pairwise mass degeneracy.",
"The mass degeneracy of the two charged physical fields is governed by the full O(4) symmetry.",
"In the case of $Z_4 = Z_5 =0 $ there is further enhancement of the symmetry and all eight physical scalars contained in $H_2$ and $H_3$ are mass degenerate.",
"It is instructive to examine the Higgs boson interactions with the gauge bosons as well as the Higgs self couplings of the RIDM, since in the RIDM the components of $H_2$ and $H_3$ are already states with well defined masses.",
"We are then led to the conclusion that there is no experimental measurement that can physically distinguish the mass degenerate scalars of the RIDM on an event by event basis.",
"Nevertheless, multiplicity factors due to the production of different scalar states of the same mass do appear in physical observables and signal the existence of the mass degeneracy.",
"Starting with the RIDM, one can consider perturbations in which the mass degeneracies persist and yet remain natural.",
"By reducing the RIDM symmetries responsible for the mass degeneracies to the smallest discrete subgroup that maintains the mass degenerate scalar states, we are led to a model that is equivalent to a particular 3HDM that was originally proposed by Ivanov and Silva (IS) [30].",
"The IS model exhibits very special properties.",
"The original form of the IS scalar potential is given in Appendix REF and is the most general potential respecting the symmetry given by Eq.",
"(REF ).",
"We have rewritten the IS potential in the notation of Eq.",
"(REF ) where the symmetry is now given by Eq.",
"(REF ).",
"In particular, the scalar mass terms (and the corresponding mass degeneracies) are the same as in the RIDM; only the quartic couplings of the physical scalar states differ.",
"One must apply the symmetry given by Eq.",
"(REF ) [denoted by CP4] four times in order to obtain the identity transformation.",
"This is to be contrasted with the conventional CP symmetry transformation (denoted by CP2) whose square is the identity.",
"On the other hand, if we apply the CP4 transformation while at the same time transforming the space coordinates from $x$ into $-x$ , the end result can be identified as a generalized CP transformation.",
"This is a very unusual type of CP transformation since applying it twice does not yield the identity transformation.",
"However, identifying CP4 with a CP transformation is possible because from the spacetime point of view the transformation remains of order two, as it should.",
"Likewise, one can define a generalized time reversal operator with properties analogous to CP4 while transforming the time coordinate from $t$ to $-t$ .",
"Consequently, there is no contradiction with the CPT theorem, which remains intact.",
"A very interesting feature of the IS scalar potential is that the symmetry requires some of its coefficients to be complex (in a particular Higgs basis).",
"Moreover, for generic choices of the scalar potential parameters, there is no scalar basis transformation within the family of Higgs bases, of the form given by Eqs.",
"(REF )–(REF ), that can transform the scalar potential into a new potential with only real coefficients.",
"This is a surprising result in light of the statement that the IS potential is CP-conserving.",
"The IS model conserves CP independently of the existence or nonexistence of a real Higgs basis, although in the case where no real Higgs basis exists, the IS model is only invariant with respect to the generalized CP symmetry, CP4 (whereas CP2 is not a symmetry of the IS scalar potential).",
"Nevertheless, any CP-violating observable of the IS model must vanish.",
"For example, the contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices generated in the IS model must exactly cancel.",
"As a check of this statement, we confirmed this cancellation up to three-loop order in the IS model with no real Higgs basis.",
"We identified a physical quartic scalar interaction made up of an odd number of mass-degenerate neutral scalar states (e.g., $P^3 Q$ and $Q^3 P$ ) that is consistent with the CP4 symmetry, but would vanish if the IS scalar potential exhibits a CP2 symmetry that commutes with CP4.",
"This leaves open the possibility of the existence of a CP2 symmetry that does not commute with CP4.",
"However, we were unable to find an observable quantity of the IS model that can distinguish between the presence or absence of a noncommuting CP2 symmetry.",
"Finally, we stress that the possibility of a scalar potential and vacuum that is invariant with respect to a generalized CP symmetry without the existence of a real basis appears to be inexorably connected with the existence of mass-degenerate scalar states.",
"We strongly suspect that this connection, which has been demonstrated in this paper for the IS model, is applicable more generally to any multi-Higgs doublet model.",
"If true, then the existence of a generalized CP symmetry in the absence of mass degenerate scalars necessarily implies the presence of a conventional CP symmetry; i.e., the existence of a real basis of scalar fields in which the CP symmetry corresponds simply to conjugation of the scalar fields."
],
[
"Acknowledgments",
"We are very grateful to Igor Ivanov for many fruitful and enlightening conversations.",
"We also thank Gustavo C. Branco, Nick Mavromatos, Palash B. Pal, Apostolos Pilaftsis, and Graham Ross with whom several aspects of this work were discussed.",
"H.E.H.",
"and P.O.",
"acknowledge the Galileo Galilei Institute for Theoretical Physics, where this work was initiated.",
"H.E.H., P.O.",
"and M.N.R.",
"also appreciate the hospitality of the CERN Theory group where some of this work was carried out, and M.N.R also acknowledges partial support from CERN.",
"H.E.H.",
"is supported in part by the U.S. Department of Energy grant number DE-SC0010107, and in part by the grant H2020-MSCA-RISE-2014 No.",
"645722 (NonMinimalHiggs).",
"P.O.",
"is supported by the Research Council of Norway.",
"The work of M.N.R.",
"was partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CFTP-FCT Unit 777 (UID/FIS/00777/2013), CERN/FIS-PAR/0004/2017 and PTDC/FIS-PAR/29436/2017 which are partially funded through POCTI (FEDER), COMPETE, QREN and EU.",
"H.E.H.",
"and M.N.R.",
"benefited from discussions that took place at the University of Warsaw during visits supported by the HARMONIA project of the National Science Centre, Poland, under contract UMO-2015/18/M/ST2/00518 (2016–2019).",
"M.N.R.",
"and P.O.",
"also thank the University of Bergen and CFTP/IST/University of Lisbon, where collaboration visits took place."
],
[
"The Ivanov Silva model revisited",
"Consider the most general 2HDM with a scalar potential as specified in Eq.",
"(REF ).",
"Including the kinetic energy terms with SU(2)$\\times $ U(1) gauge covariant derivatives, the 2HDM [after electroweak symmetry breaking under the assumption that the vacuum preserves U(1)$_{\\rm EM}$ ] consists of a model of two scalar doublets coupled to the gauge bosons, $W^\\pm $ , $Z$ and $\\gamma $ .",
"We shall ignore the couplings of the bosonic sector of the 2HDM to the fermions of the SM in the following discussion.",
"We now ask the following question.",
"Does the bosonic Lagrangian conserve CP?",
"For CP to be conserved, two conditions must be verified.",
"First, the scalar potential must exhibit explicit CP conservation.",
"Second, the vacuum must conserve CP.",
"If the former is true but the latter is false, we say that CP is spontaneously broken.",
"However, in this discussion, we are interested in whether both explicit and spontaneous CP violation are absent.",
"In the 2HDM, the answer to this question is simple.",
"We first transform to the Higgs basis and examine the scalar potential given in Eq.",
"(REF ).",
"The Higgs basis is unique up to a possible rephasing of the Higgs basis field, $H_2\\rightarrow e^{i\\chi }H_2$ .",
"Then, CP is conserved if and only if there exists a choice of $\\chi $ such that all Higgs basis scalar potential parameters are real.",
"In the discussion above, we have not specified in detail how the scalar fields transform under a CP transformation.",
"Starting from the generic $\\Phi _1$ –$\\Phi _2$ basis employed in writing Eq.",
"(REF ), the conventional CP transformation corresponds to conjugation, $\\Phi _i^{\\rm CP}= \\Phi _i^{\\scriptscriptstyle {\\bigstar }}$ .",
"However, this is a basis-dependent statement.",
"Indeed, one is always free to change the basis, $\\Phi _i^\\prime =U_{ij}\\Phi _j$ , where $U\\in $ U(2).",
"In the new basis, $\\Phi ^{\\prime \\,CP}_i=X_{ij}\\Phi ^{\\prime \\,CP}_j$ , where $X=UU^T$ is a symmetric unitary matrix.",
"More generally, we can consider the generalized CP transformation, $\\Phi _i^{\\rm CP}(x,t)= X_{ij}\\Phi _j^{\\scriptscriptstyle {\\bigstar }}(-x,t)\\,,$ where $X\\in $ U(2).Note that it is not consistent to simply define the CP transformation of a multi-Higgs doublet model without including the matrix $X$ in Eq.",
"(REF ), since the form of the CP transformation depends on the choice of the scalar basis, as noted above.",
"Consequently, some authors prefer to call this transformation a general CP transformation rather than generalized CP transformation.",
"If $X$ is both unitary and symmetric, then one can find a basis in which the CP transformation is simply conjugation.As shown in Appendix D of Ref.",
"[80] [see the Lemma below eq.",
"(D.3.1)], for any symmetric unitary matrix $X$ , there exists a unitary matrix $U$ such that $X=UU^T$ .",
"In Ref.",
"[56], it is shown that in the 2HDM there are three possible classes of generalized CP transformations (GCPs): (i) $X$ is unitary and symmetric; (ii) $X$ is unitary and antisymmetric; and (iii) $X$ is unitary but is neither symmetric nor antisymmetric.",
"Clearly, no basis change can convert a GCP transformation of types (ii) or (iii) into the transformation of the field into its conjugate.",
"Nevertheless, as shown in Ref.",
"[56], any 2HDM scalar potential that is invariant under GCP transformations of types (ii) or (iii) is also separately invariant under a GCP transformation of type (i).",
"Do the above results generalize to arbitrary Higgs sectors?",
"In particular, consider an extended Higgs sector with $N$ hypercharge-one, complex doublets (denoted henceforth as the NHDM).",
"To address the question of CP invariance, we transform to the so-called charged Higgs basis defined in Ref. [81].",
"If the scalar fields of the charged Higgs basis are denoted by $H_i$ ($i=1,\\ldots ,n$ ), then $\\langle H_1^0 \\rangle =v/\\sqrt{2}$ , $\\langle H_j^0 \\rangle =0$ for $j=2,3,\\ldots ,n$ , and the fields $H_j^\\pm $ (for $j=2,3,\\ldots ,n$ ) are the physical, mass-eigenstate charged Higgs fields.",
"Note that for $N=2$ , the Higgs basis and the charged Higgs basis coincide.",
"For $N\\ge 3$ , consider first the case in which the physical charged Higgs bosons are mass non-degenerate.",
"In this case, the charged Higgs basis is uniquely defined up to a possible rephasing, $H_j\\rightarrow e^{i\\chi _j}H_j$ .",
"In this case, CP is conserved if and only if there exist a choice of the $\\chi _j$ such that all charged Higgs basis scalar potential parameters are real.",
"This generalizes the result of the 2HDM quoted above.",
"If there exist mass degeneracies among the physical charged Higgs fields, then one must re-evaluate the conditions for CP invariance.",
"To simplify the discussion, we focus on the case of $N=3$ , in which the two physical charged Higgs bosons are mass degenerate.",
"In this case, the charged Higgs basis is unique up to a U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ .",
"Ivanov and Silva[30] constructed a 3HDM whose scalar potential and vacuum are invariant under a generalized CP transformation such that $({\\rm GCP})^2\\ne {1}$ , where 1 is the identity operator.",
"Moreover, some of the scalar potential parameters of the charged Higgs basis of the Ivanov–Silva (IS) scalar potential are complex, and no U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ can be performed to remove all the complex phases.",
"Hence, the IS scalar potential is not invariant under a separate GCP transformation that is equivalent to conjugation in another basis, in contrast to the corresponding 2HDM result.",
"Ivanov and Silva denote the GCP transformation of the IS scalar potential by CP4, since it has the property that $({\\rm CP}4)^4={1}$ and $({\\rm CP}4)^2\\ne {1}$ .",
"Indeed, one consequence of the CP4 symmetry of the scalar potential and the vacuum is the mass degeneracy of the physical charged Higgs bosons, as well as two additional mass degeneracies among pairs of neutral Higgs bosons.",
"In this Appendix, we consider the 3HDM scalar potential of the Ivanov and Silva model and examine some of its properties."
],
[
"The IS scalar potential",
"Consider the 3HDM consisting of three hypercharge-one, complex doublet fields, $\\phi _i$ ($i=1,2,3$ ).",
"In the Higgs basis, the form of the scalar potential proposed initially by Ivanov and Silva (IS) in Ref.",
"[30] is fixed by imposing the following generalized CP symmetry, $ \\phi _i \\rightarrow W_{ij}\\phi _j^{\\scriptscriptstyle {\\bigstar }}, \\quad W=\\begin{pmatrix}1 & \\phantom{-}0 & \\phantom{-}0 \\\\0 & \\phantom{-}0 & \\phantom{-}i \\\\0 & -i & \\phantom{-}0\\end{pmatrix},$ which has the property that applying it four times yields the identity operator.",
"This is the CP4 symmetry transformation noted above.",
"The resulting IS scalar potential is given by $V=V_0+V_1\\,,$ with $V_0&=-m_{11}^2(\\phi _1^\\dagger \\phi _1)-m_{22}^2(\\phi _2^\\dagger \\phi _2+\\phi _3^\\dagger \\phi _3)+\\lambda _1(\\phi _1^\\dagger \\phi _1)^2+\\lambda _2[(\\phi _2^\\dagger \\phi _2)^2+(\\phi _3^\\dagger \\phi _3)^2]+\\lambda _3^\\prime (\\phi _2^\\dagger \\phi _2)(\\phi _3^\\dagger \\phi _3)\\nonumber \\\\& \\qquad +\\lambda _3(\\phi _1^\\dagger \\phi _1)[(\\phi _2^\\dagger \\phi _2)+(\\phi _3^\\dagger \\phi _3)]+\\lambda _4^\\prime (\\phi _2^\\dagger \\phi _3)(\\phi _3^\\dagger \\phi _2)+\\lambda _4[(\\phi _1^\\dagger \\phi _2)(\\phi _2^\\dagger \\phi _1)+(\\phi _1^\\dagger \\phi _3)(\\phi _3^\\dagger \\phi _1)],\\\\[6pt]V_1&=\\lambda _5(\\phi _3^\\dagger \\phi _1)(\\phi _2^\\dagger \\phi _1)+\\tfrac{1}{2}\\lambda _6[(\\phi _2^\\dagger \\phi _1)^2-(\\phi _1^\\dagger \\phi _3)^2]+\\lambda _8(\\phi _2^\\dagger \\phi _3)^2+\\lambda _9(\\phi _2^\\dagger \\phi _3)[(\\phi _2^\\dagger \\phi _2)-(\\phi _3^\\dagger \\phi _3)] +{\\rm h.c.}$ The hermiticity of the scalar potential implies that the coefficients of $V_0$ are real.",
"In contrast, the coefficients of $V_1$ are potentially complex.",
"However, having imposed the CP4 symmetry given by Eq.",
"(REF ), we see that $\\lambda _5$ is real.",
"Under the CP4 symmetry specified in Eq.",
"(REF ), the gauge-invariant bilinear quantities, $B_{ij}\\equiv \\phi _i^\\dagger \\phi _j$ , transform as follows: $B_{11} &\\rightarrow B_{11}, \\\\B_{22}&\\rightarrow B_{33} &\\quad B_{33}&\\rightarrow B_{22}, \\\\B_{12}&\\rightarrow i B_{31}, &\\quad B_{21}&\\rightarrow -i B_{13}, \\\\B_{13}&\\rightarrow -i B_{21}, &\\quad B_{31}&\\rightarrow i B_{12}, \\\\B_{23}&\\rightarrow -B_{23}, &\\quad B_{32}&\\rightarrow -B_{32}.$ It follows that $V$ given by Eqs.",
"(REF )–(REF ), with $\\lambda _6$ , $\\lambda _8$ and $\\lambda _9$ complex and all other scalar potential parameters real, is the most general 3HDM potential that is invariant under the CP4 transformation given in (REF ).",
"Without loss of generality, one can furthermore assume that $\\lambda _6$ is real after an appropriate rephasing of the scalar fields $\\phi _2$ and $\\phi _3$ .",
"At this stage, we have not yet found the minimum of the scalar potential and determined whether the CP4 symmetry is respected by the vacuum.",
"There exist a range of scalar potential parameters in which the vacuum preserves U(1)$_{\\rm EM}$ , in which case one can decompose the scalar doublets as, $ \\phi _i=\\left(\\begin{array}{c}\\varphi _i^+\\\\ (v_i+\\eta _i+i \\chi _i)/\\sqrt{2}\\end{array}\\right), \\quad i=1,2,3.$ In particular, the vacuum conserves CP4 if the minimum of the scalar potential corresponds to $(v_1,v_2,v_3)=(v,0,0)$[30].",
"Indeed, there exists a range of scalar potential parameters for which this corresponds to the global minimum, in which case the value of $m_{11}^2$ is fixed by the scalar potential minimum condition to be $m_{11}^2=\\lambda _1 v^2.$ In this case, the scalar field basis employed in Eqs.",
"(REF ) and (REF ) is the Higgs basis, with the freedom to perform U(2) transformations on $\\lbrace \\phi _2,\\phi _3\\rbrace $ .",
"We shall take advantage of this freedom in the next two subsections.",
"It is now straightforward to determine the scalar mass spectrum of the IS model.",
"Since we are in the Higgs basis, we can immediately identify the Goldstone bosons, $\\varphi _1^\\pm =G^\\pm $ and $\\chi _1=G^0$ .",
"Moreover, $\\eta _1$ is a neutral mass-eigenstate with mass $m^2_{\\eta _1}=2\\lambda _1 v^2$ , whose tree-level couplings to the gauge bosons and to itself are precisely those of the SM Higgs boson (corresponding to the exact alignment limit).",
"Indeed, this is analogous to the IDM in which $\\phi _1$ is equivalent to the hypercharge-one, complex scalar doublet of the SM and $\\phi _2$ and $\\phi _3$ are inert doublets.",
"The two physical charged Higgs fields, $\\varphi _2^\\pm $ and $\\varphi _3^\\pm $ , are mass-degenerate, $ m_{\\varphi _2^\\pm ,\\varphi _3^\\pm }^2=\\tfrac{1}{2}\\lambda _3 v^2 -m_{22}^2.$ The neutral scalar spectrum consist of the SM-like Higgs boson $\\eta _1$ and a pair of mass degenerate neutral scalars made up of linear combinations of the $\\eta _{2,3}$ and $\\chi _{2,3}$ , with masses given by[30], $M^2= a+\\sqrt{b^2+c^2}\\,,\\qquad \\quad m^2= a-\\sqrt{b^2+c^2}\\,,$ where $a=\\tfrac{1}{2}(\\lambda _3+\\lambda _4)v^2-m_{22}^2, \\qquad b=\\tfrac{1}{2}\\lambda _6 v^2, \\qquad c=\\tfrac{1}{2}\\lambda _5 v^2.$"
],
[
"A simpler form for the IS scalar potential",
"Given the IS scalar potential in the Higgs basis, we still have the freedom to perform a U(2) transformation on $\\lbrace \\phi _2,\\phi _3\\rbrace $ .",
"It is possible to remove the $\\lambda _5$ term in Eq.",
"(REF ) by the following basis transformation, $\\bar{\\phi }_i=U_{ij}\\phi _j,$ where $U=\\begin{pmatrix}1 & 0 & 0 \\\\0 & \\phantom{-}\\cos \\theta & -\\sin \\theta \\\\0 & \\phantom{-}\\sin \\theta & \\phantom{-}\\cos \\theta \\end{pmatrix}\\,,$ with $0\\le \\theta \\le \\pi $ .",
"With respect to the new basis, the CP4 transformation specified in Eq.",
"(REF ) is given by, $\\bar{\\phi }_i\\rightarrow V_{ij}\\bar{\\phi }^{\\scriptscriptstyle {\\bigstar }}_j\\,, \\qquad \\text{where $V=UWU^T$}.$ Using the form for $U$ given in Eq.",
"(REF ), it follows that $V=W$ .",
"Thus, in this new basis, the IS symmetry takes the same form as in the original basis.",
"When the scalar potential is expressed in terms of the fields $\\bar{\\phi }_i$ , the resulting scalar potential parameters will be denoted by $\\bar{m}^2_{ii}$ and $\\bar{\\lambda }_i$ .",
"It is straightforward to obtain expressions for $\\bar{m}_{11}^2$ , $\\bar{m}_{22}^2$ and the $\\bar{\\lambda }_i$ in terms of the scalar potential parameters defined in Eq.",
"(REF ).",
"In particular, $\\bar{m}_{11}^2=m_{11}^2$ , $\\bar{m}_{22}^2=m_{22}^2$ , and $\\bar{\\lambda }_i=\\lambda _i$ for $i=1,3$ and 4.",
"Next, we note that the CP4 symmetry does not mandate that $\\bar{\\lambda }_6$ is real.",
"However, it is straightforward to check that $\\operatorname{Im}\\bar{\\lambda }_6=\\operatorname{Im}\\lambda _6$ .",
"Having previously chosen $\\lambda _6$ real (after an appropriate rephasing of $\\phi _2$ and $\\phi _3$ ), it follows that $\\bar{\\lambda }_6$ is also real.",
"The remaining transformed coefficients are given by, $\\bar{\\lambda }_2&= \\lambda _2-\\tfrac{1}{2}\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]-\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }^\\prime _3&=\\lambda ^\\prime _3+\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+2\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }^\\prime _4&=\\lambda ^\\prime _4+\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+2\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }_5&=\\lambda _5\\cos 2\\theta +\\lambda _6\\sin 2\\theta \\,,\\\\\\bar{\\lambda }_6&=\\lambda _6\\cos 2\\theta -\\lambda _5\\sin 2\\theta \\,,\\\\\\operatorname{Re}\\bar{\\lambda }_8&=\\operatorname{Re}\\lambda _8+\\tfrac{1}{2}\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,,\\\\\\operatorname{Re}\\bar{\\lambda }_9&=(1-2\\sin ^2 2\\theta )\\operatorname{Re}\\lambda _9+\\sin 2\\theta \\cos 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]\\,,\\\\\\operatorname{Im}\\bar{\\lambda }_8&=\\cos 2\\theta \\operatorname{Im}\\lambda _8+\\sin 2\\theta \\operatorname{Im}\\lambda _9\\,,\\\\\\operatorname{Im}\\bar{\\lambda }_9&=\\cos 2\\theta \\operatorname{Im}\\lambda _9-\\sin 2\\theta \\operatorname{Im}\\lambda _8\\,.$ One can now choose the angle $\\theta $ such that $\\bar{\\lambda }_5=0$ .Note that a different choice of $\\tan 2\\theta $ could have been made to set either $\\bar{\\lambda }_6=0$ , $\\operatorname{Im}\\bar{\\lambda }_8=0$ or $\\operatorname{Im}\\bar{\\lambda }_9=0$ .",
"That is, one can always perform a change of Higgs basis to remove one degree of freedom from the coefficients of the IS scalar potential.",
"This yields $\\tan 2\\theta =-\\lambda _5/\\lambda _6$ .",
"Then, $\\sin 2\\theta $ and $\\cos 2\\theta $ are determined up to an overall sign.",
"Introducing the following notation, $\\lambda _{56}\\equiv \\sqrt{\\lambda _5^2+\\lambda _6^2}\\,,$ we choose the angle $\\theta $ such that, $\\sin 2\\theta =\\frac{\\lambda _5}{\\lambda _{56}}\\,,\\qquad \\quad \\cos 2\\theta =-\\,\\frac{\\lambda _6}{\\lambda _{56}}\\,.$ Thus, the $\\lambda _5$ -term in Eq.",
"(REF ) is actually redundant.A similar simplification was presented recently in Ref. [70].",
"Inserting the results of Eq.",
"(REF ) back into Eq.",
"(REF ) yields $\\bar{\\lambda }_5=0$ and, $\\bar{\\lambda }_2&=\\frac{\\lambda _5 \\left[\\lambda _5 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4+2 \\operatorname{Re}\\lambda _8\\right)+4 \\lambda _6 \\operatorname{Re}\\lambda _9\\right]+2 \\lambda _2 \\left(\\lambda _5^2+2 \\lambda _6^2\\right)}{4 \\lambda ^2_{56}},\\\\\\bar{\\lambda }^\\prime _3&=\\frac{\\lambda _5^2 \\left(\\lambda ^\\prime _3-\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)+2 \\lambda _6^2 \\lambda ^\\prime _3-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+2 \\lambda _2 \\lambda _5^2}{2\\lambda ^2_{56}},\\\\\\bar{\\lambda }^\\prime _4&=\\frac{\\lambda _5^2 \\left(-\\lambda ^\\prime _3+\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)+2 \\lambda _6^2 \\lambda ^\\prime _4-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+2 \\lambda _2 \\lambda _5^2}{2 \\lambda ^2_{56}},\\\\\\bar{\\lambda }_6&=-\\lambda _{56},\\\\\\operatorname{Re}\\bar{\\lambda }_8&=\\frac{-\\lambda _5^2 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+4 \\lambda _6^2 \\operatorname{Re}\\lambda _8+2 \\lambda _2 \\lambda _5^2}{4 \\lambda ^2_{56}},\\\\\\operatorname{Re}\\bar{\\lambda }_9&=\\frac{\\lambda _5\\lambda _6 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4+2 \\operatorname{Re}\\lambda _8\\right)-2 \\lambda _5^2 \\operatorname{Re}\\lambda _9+2 \\lambda _6^2 \\operatorname{Re}\\lambda _9-2 \\lambda _2 \\lambda _5 \\lambda _6}{2 \\lambda ^2_{56}},\\\\\\operatorname{Im}\\bar{\\lambda }_8&=\\frac{\\lambda _5 \\operatorname{Im}\\lambda _9-\\lambda _6 \\operatorname{Im}\\lambda _8}{\\lambda _{56}},\\\\\\operatorname{Im}\\bar{\\lambda }_9&=\\frac{-\\lambda _5 \\operatorname{Im}\\lambda _8-\\lambda _6 \\operatorname{Im}\\lambda _9}{\\lambda _{56}}.$ An additional feature of the IS scalar potential with $\\bar{\\lambda }_5=0$ is that the real and imaginary parts of the neutral fields $\\bar{\\phi }_2^0$ and $\\bar{\\phi }_3^0$ are mass eigenstates.",
"That is, the neutral squared-mass matrices are already diagonal in the $\\lbrace \\bar{\\phi }_1,\\bar{\\phi }_2,\\bar{\\phi }_3\\rbrace $ basis.",
"In particular, the lightest of the two mass-degenerate states lives in the imaginary part of $\\bar{\\phi }_2$ and in the real part of $\\bar{\\phi }_3$ .",
"The heaviest of the two mass-degenerate neutral states lives in the real part of $\\bar{\\phi }_2$ and in the imaginary part of $\\bar{\\phi }_3$ .",
"It is convenient to make an additional field redefinition, $\\bar{\\phi }_3\\rightarrow i\\bar{\\phi }_3$ .",
"The effect of this modification is to modify $\\bar{V}_1$ by flipping the sign of $(\\bar{\\phi }^\\dagger _1\\bar{\\phi }_3)^2$ in the term proportional to $\\bar{\\lambda }_6$ and to transform $\\bar{\\lambda }_8\\rightarrow -\\bar{\\lambda }_8$ and $\\bar{\\lambda }_9\\rightarrow -i\\bar{\\lambda }_9$ .",
"To make contact with the $H23$ -basis employed in Eq.",
"(REF ), we define, $H_1=\\bar{\\phi }_1\\,,\\qquad H_2=\\bar{\\phi }_2\\,,\\qquad H_3=i\\bar{\\phi }_3\\,,$ corresponding to a basis change, $H_i\\rightarrow \\widetilde{U}_{ij}\\bar{\\phi }_j$ , with $\\widetilde{U}={\\rm diag}(1\\,,\\,1\\,,\\,i)$ .",
"Note that the heaviest mass degenerate neutral fields now reside in the real part of the neutral components of $H_2$ and $H_3$ , and the lightest mass degenerate neutral fields reside in the imaginary part of the neutral components of $H_2$ and $H_3$ .",
"When expressed in the $H23$ -basis, the IS scalar potential is given by, $\\mathcal {V}_{\\rm IS}= \\mathcal {V}_{\\rm RIDM}+Z_3^\\prime (H_2^\\dagger H_2)(H_3^\\dagger H_3)+Z_4^\\prime (H_2^\\dagger H_3)(H_3^\\dagger H_2) +\\bigl [Z_8 (H_2^\\dagger H_3)^2+Z_9(H_2^\\dagger H_3)(H_2^\\dagger H_2-H_3^\\dagger H_3)+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM}$ is given by Eq.",
"(REF ), with $Z_8$ and $Z_9$ potentially complex and all other scalar potential parameters real.",
"Eq.",
"(REF ) is the version of the IS scalar potential employed in section REF .",
"To make contact with the previous notation used above, we note that $Y_1&=& -m_{11}^2\\,,\\qquad Y_2=-m_{22}^2\\,,\\qquad Z_1=2\\lambda _1\\,,\\qquad Z_2=2\\bar{\\lambda }_2\\,,\\qquad Z_3=\\lambda _3\\,,\\qquad Z_4=\\lambda _4 \\nonumber \\\\Z_3^\\prime &=&\\bar{\\lambda }_3^\\prime -2\\bar{\\lambda }_2\\,,\\qquad Z_4^\\prime =\\bar{\\lambda }_4^\\prime \\,,\\qquad Z_5=\\bar{\\lambda }_6\\,,\\qquad Z_8=-\\bar{\\lambda }_8\\,,\\qquad Z_9=-i\\bar{\\lambda }_9\\,.$ The corresponding CP4 symmetry transformation now takes the form $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}\\,,\\qquad \\text{where $X=\\widetilde{U}W\\widetilde{U}^T=\\begin{pmatrix} 1 & \\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1 \\\\ 0 & \\phantom{-}1 & \\phantom{-}0\\end{pmatrix}$}\\,,$ as indicated in Eq.",
"(REF )."
],
[
"Non-existence of a real Higgs basis",
"Consider the IS scalar potential [cf. Eq.",
"(REF ) with $\\mathcal {V}_{\\rm RIDM}$ given by Eq.",
"(REF )] expressed in terms of the Higgs basis of scalar doublet fields, $\\lbrace H_1,H_2,H_3\\rbrace $ , where $\\langle H_1^0 \\rangle \\ne 0$ and the vevs of the other two doublet fields vanish.",
"The coefficients $Z_8$ and $Z_9$ are potentially complex and all other scalar potential parameters are real.",
"Recall that the Higgs basis is unique only up to an arbitrary U(2) transformation of $\\lbrace H_2, H_3\\rbrace $ .",
"Is it possible to transform to a new Higgs basis in which all the IS scalar potential parameters are real?",
"Such a Higgs basis, if it exists, is called a real Higgs basis.",
"The most general basis transformation that preserves the general class of Higgs bases is given (in block diagonal form) by, $\\begin{pmatrix} \\bar{H}_1 \\\\ \\bar{H}_{23}\\end{pmatrix}=\\begin{pmatrix} 1 & \\,\\,0 \\\\ 0 &\\,\\, \\widetilde{V}\\end{pmatrix} \\begin{pmatrix} H_1 \\\\ H_{23}\\end{pmatrix}\\,,$ where $H_{23}\\equiv \\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\,,\\qquad \\quad \\bar{H}_{23}\\equiv \\begin{pmatrix} \\bar{H}_2\\\\ \\bar{H}_3\\end{pmatrix}\\,,$ and $\\widetilde{V}$ is the most general U(2) matrix, $\\widetilde{V}=e^{i\\psi /2}\\begin{pmatrix} e^{i\\alpha }\\cos \\phi & -e^{-i\\beta }\\sin \\phi \\\\ e^{i\\beta }\\sin \\phi & \\phantom{-}e^{-i\\alpha }\\cos \\phi \\end{pmatrix}\\,,$ where $0\\le \\phi <\\pi $ , $-\\pi <\\psi \\le \\pi $ , $0\\le \\alpha \\le \\pi $ and $0\\le \\beta \\le \\pi $ .",
"Applying Eq.",
"(REF ) to the IS scalar potential given in Eq.",
"(REF ) yields $\\mathcal {V}_{\\rm IS}&=&Y_1 \\bar{H}_1^\\dagger \\bar{H}_1+ Y_2 \\left( \\bar{H}_2^\\dagger \\bar{H}_2 + \\bar{H}_3^\\dagger \\bar{H}_3\\right)+\\tfrac{1}{2}Z_1( \\bar{H}_1^\\dagger \\bar{H}_1)^2+\\tfrac{1}{2}\\bar{Z}_2( \\bar{H}_2^\\dagger \\bar{H}_2+ \\bar{H}_3^\\dagger \\bar{H}_3)^2 \\\\&&+Z_3(\\bar{H}_1^\\dagger \\bar{H}_1)( \\bar{H}_2^\\dagger \\bar{H}_2+ \\bar{H}_3^\\dagger \\bar{H}_3)+Z_4\\bigl [( \\bar{H}_1^\\dagger \\bar{H}_2)( \\bar{H}_2^\\dagger \\bar{H}_1)+( \\bar{H}_1^\\dagger \\bar{H}_3)( \\bar{H}_3^\\dagger \\bar{H}_1)\\bigr ]\\nonumber \\\\&&+\\bar{Z}_3^\\prime ( \\bar{H}_2^\\dagger \\bar{H}_2)( \\bar{H}_3^\\dagger \\bar{H}_3)+\\bar{Z}_4^\\prime ( \\bar{H}_2^\\dagger \\bar{H}_3)( \\bar{H}_3^\\dagger \\bar{H}_2)+i\\bar{Z}_5^\\prime \\bigl [e^{i\\psi } (\\bar{H}_3^\\dagger \\bar{H}_1)( \\bar{H}_2^\\dagger \\bar{H}_1)-e^{-i\\psi }( \\bar{H}_1^\\dagger \\bar{H}_2)( \\bar{H}_1^\\dagger \\bar{H}_3)\\bigr ]\\,,\\nonumber \\\\&&+\\bigl \\lbrace \\tfrac{1}{2}\\bar{Z}_5\\bigl [e^{i\\psi } (\\bar{H}_2^\\dagger \\bar{H}_1)^2 +e^{-i\\psi }( \\bar{H}_1^\\dagger \\bar{H}_3)^2\\bigr ]+\\bar{Z}_8 ( \\bar{H}_2^\\dagger \\bar{H}_3)^2+\\bar{Z}_9( \\bar{H}_2^\\dagger \\bar{H}_3)(\\bar{H}_2^\\dagger \\bar{H}_2- \\bar{H}_3^\\dagger \\bar{H}_3)+{\\rm h.c.} \\bigr \\rbrace \\,.\\nonumber $ The coefficients $Y_1$ , $Y_2$ , $Z_1$ , $Z_3$ and $Z_4$ are unmodified, whereas, $&& \\bar{Z}_2=Z_2+\\tfrac{1}{2}\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )-\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_3^\\prime =Z_3^\\prime -\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )+2\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_4^\\prime =Z_4^\\prime -\\tfrac{1}{2}\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )+\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_5^\\prime =Z_5\\sin 2\\phi \\sin \\xi \\,, \\\\&&\\bar{Z}_5=e^{i\\chi }Z_5\\bigl (e^{i\\xi }\\cos ^2\\phi +e^{-i\\xi }\\sin ^2\\phi \\bigr )\\,, \\\\&&\\bar{Z}_8= e^{2i\\chi }\\bigl \\lbrace -\\tfrac{1}{4} \\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime \\bigr )+e^{2i\\xi }\\cos ^4\\phi \\,Z_8+e^{-2i\\xi }\\sin ^4\\phi \\, Z_8^* \\nonumber \\\\&& \\qquad \\qquad \\qquad +\\sin 2\\phi \\bigl [e^{i\\xi }\\cos ^2\\phi \\,Z_9-e^{-i\\xi }\\sin ^2\\phi \\, Z_9^*\\bigr ]\\bigr \\rbrace \\,, \\\\&&\\bar{Z}_9=e^{i\\chi }\\bigl \\lbrace -\\tfrac{1}{2}\\sin 2\\phi \\cos 2\\phi (Z_3^\\prime +Z_4^\\prime ) -\\sin 2\\phi \\bigl [e^{2i\\xi }\\cos ^2\\phi \\,Z_8-e^{-2i\\xi } \\sin ^2\\phi \\,Z_8^*\\bigr ] \\nonumber \\\\&& \\qquad \\qquad \\qquad +\\tfrac{1}{2}e^{i\\xi }(\\cos 4\\phi +\\cos 2\\phi )Z_9+\\tfrac{1}{2}e^{-i\\xi }(\\cos 4\\phi -\\cos 2\\phi )Z_9^*\\bigr \\rbrace \\,,$ where $\\xi \\equiv \\alpha +\\beta \\,,\\qquad \\quad \\chi \\equiv \\alpha -\\beta \\,.$ By definition of the $H23$ -basis, $Z_5$ is real and $Z_5^\\prime =0$ [the latter is a consequence of the absence of a term in Eq.",
"(REF ) that involves $(H_3^\\dagger H_1)(H_2^\\dagger H_1)$ and its hermitian conjugate].",
"After employing a generic U(2) basis change [Eq.",
"(REF )], a nonzero $\\bar{Z}_5^\\prime $ and a complex $\\bar{Z}_5$ are generated [cf. Eqs.",
"() and ()], such that $Z_5^2=|\\bar{Z}_5|^2+\\bar{Z}_5^{\\prime \\,2}\\,.$ It is instructive to examine the form of the CP4 transformation in the $\\lbrace \\bar{H}_1$ , $\\bar{H}_2$ , $\\bar{H_3}\\rbrace $ basis.",
"Starting from Eq.",
"(REF ) and transforming $H_i\\rightarrow \\bar{H}_i= \\widetilde{V}_{ij}H_j$ ($i,j=2,3$ ), it follows that the CP4 transformation of the barred fields is given by, $\\bar{H}_i\\rightarrow \\bar{X}_{ij}\\bar{H}_j^{\\scriptscriptstyle {\\bigstar }}\\,,\\qquad \\text{where $\\bar{X}=VW{V}^T$}\\,,$ where the $3\\times 3$ matrices $\\bar{X}$ , ${V}$ and $W$ in block form are given by $\\bar{X}=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\widetilde{X}\\end{pmatrix}\\,,\\qquad {V}=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\widetilde{V}\\end{pmatrix}\\,,\\qquad W=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\epsilon \\end{pmatrix}\\,,$ and $\\epsilon \\equiv \\left({\\begin{matrix} 0 & -1 \\\\ 1 & \\phantom{-}0\\end{matrix}}\\right)$ .",
"For any $\\widetilde{V}\\in $ U(2), we have $\\widetilde{X}=\\widetilde{V}\\epsilon \\widetilde{V}^T=e^{i\\psi }\\epsilon \\,,$ after taking the determinant of Eq.",
"(REF ) and noting that $\\det \\widetilde{V}=e^{i\\psi }$ .",
"Indeed, if we impose invariance of the scalar potential under CP4 in the $\\lbrace \\bar{H}_1$ , $\\bar{H}_2$ , $\\bar{H_3}\\rbrace $ basis, then the IS scalar potential must have the form given by Eq.",
"(REF ), with $\\bar{Z}_8$ and $\\bar{Z}_9$ potentially complex and all other scalar potential coefficients [excluding factors of $i$ or $e^{\\pm i\\psi }$ that explicitly appear in Eq.",
"(REF )] real.",
"It is possible to choose a basis in which all but one of the scalar potential parameters are real.",
"This can be achieved by choosing $\\psi =\\alpha =\\beta =0$ in Eq.",
"(REF ).",
"In this case, Eqs.",
"() and () yield $\\bar{Z}_5^\\prime =0$ , $\\bar{Z}_5=Z_5$ and $\\operatorname{Im}\\bar{Z}_8&=&\\cos 2\\phi \\operatorname{Im}Z_8+\\sin 2\\phi \\operatorname{Im}Z_9\\,,\\nonumber \\\\\\operatorname{Im}\\bar{Z}_9&=&\\cos 2\\phi \\operatorname{Im}Z_9-\\sin 2\\phi \\operatorname{Im}Z_8\\,.$ Indeed, there is a choice of $\\phi $ in Eq.",
"(REF ) such that $\\operatorname{Im}\\bar{Z}_9=0$ (and another choice of $\\phi $ such that $\\operatorname{Im}\\bar{Z}_8=0$ ).",
"Thus, by a series of basis changes, we have reduced the number of independent parameters in the IS scalar potential from 14 to 12."
],
[
"Transforming to a Higgs basis where $Z_8$ and {{formula:67350f71-efd2-4657-aecf-2e42710893ea}} are real",
"We now examine whether a choice of $\\psi $ , $\\chi $ , $\\xi $ and $\\phi $ exists such that $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ , $\\bar{Z}_5 e^{\\pm i\\psi }$ , $\\bar{Z}_8$ and $\\bar{Z}_9$ are all real.",
"To begin, we first show that a Higgs basis exists in which $\\bar{Z}_8$ and $\\bar{Z}_9$ are both real.",
"Here, we follow the analysis given in Appendix C of Ref. [46].",
"By assumption, $Z_9$ is real and $Z_8=|Z_8|e^{i\\theta _8}$ (where $\\theta _8$ is not an integer multiple of $\\pi $ so that $\\operatorname{Im}Z_8\\ne 0$ ).",
"Setting $\\operatorname{Im}\\bar{Z_8}=\\operatorname{Im}\\bar{Z_9}=0$ in Eqs.",
"() and () yields, $\\operatorname{Im}\\bar{Z}_8=f_a\\cos 2\\chi -f_b\\sin 2\\chi =0\\,,\\qquad \\quad \\operatorname{Im}\\bar{Z}_9=f_c\\cos \\chi -f_d\\sin \\chi =0\\,,\\ $ where $f_a&=& |Z_8|\\cos 2\\phi \\sin (2\\xi +\\theta _8)+Z_9\\sin 2\\phi \\sin \\xi \\,, \\\\f_b&=&\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime )\\sin ^2 2\\phi -|Z_8|(1-\\tfrac{1}{2}\\sin ^2 2\\phi )\\cos (2\\xi +\\theta _8)-Z_9\\sin 2\\phi \\cos 2\\phi \\cos \\xi \\,,\\\\f_c&=&-|Z_8|\\sin 2\\phi \\sin (2\\xi +\\theta _8)+Z_9\\cos 2\\phi \\sin \\xi \\,,\\\\f_d&=&\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime ) \\sin 2\\phi \\cos 2\\phi +|Z_8|\\sin 2\\phi \\cos 2\\phi \\cos (2\\xi +\\theta _8)-Z_9\\cos 4\\phi \\cos \\xi \\,.$ Assuming that $f_a\\ne 0$ and $f_c\\ne 0$ , Eq.",
"(REF ) implies that $\\cot \\chi =\\frac{f_d}{f_c}\\,,\\qquad \\quad \\cot 2\\chi =\\frac{f_b}{f_a}\\,.$ Employing the trigonometric identity, $\\cot 2\\chi =(\\cot ^2\\chi -1)/(2\\cot \\chi )$ , we end up with, $G(\\phi ,\\xi )\\equiv f_a(f_d^2-f_c^2)-2f_b f_c f_d=0\\,.$ Note that the above condition is independent of the angle $\\psi $ .",
"Inserting the results of Eqs.",
"(REF )–() into Eq.",
"(REF ) leads to a very complicated expression.",
"However, it is quite easy to check that $G(0,\\xi )=-G(\\tfrac{1}{2}\\pi ,\\xi )=Z_9^2\\operatorname{Im}Z_8\\,.$ As a consequence of Eq.",
"(REF ), for any choice of $\\xi $ , there must exist a value of $\\phi $ between 0 and $\\tfrac{1}{2}\\pi $ such that $G(\\phi ,\\xi )=0$ .",
"Plugging these values of $\\phi $ and $\\xi $ back into Eqs.",
"(REF )–(), we can then use Eq.",
"(REF ) to determine $\\chi $ .",
"Thus, we have shown that for any choice of $\\xi $ and $\\psi $ , there must exist a corresponding $\\phi $ and $\\chi $ (whose values depend on the choice of $\\xi $ ) such that $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ .Although we have reached this conclusion under the assumption that $f_c$ and $f_a$ are nonzero, it is straightforward to modify the analysis if either $f_a=0$ and/or $f_c=0$ .",
"If $f_a=f_b=f_c=f_d=0$ , then Eq.",
"(REF ) immediately yields $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ .",
"If at least one of the quantities $f_a$ , $f_b$ , $f_c$ and $f_d$ is nonzero, then $\\chi $ can be determined from one of the two expressions in Eq.",
"(REF )."
],
[
"Does a Higgs basis exist were all scalar potential parameters are real?",
"Having found a Higgs basis with real $\\bar{Z}_8$ and $\\bar{Z}_9$ for an arbitrary choice of $\\xi $ and $\\psi $ (where the parameters $\\phi $ and $\\chi $ have been determined), we now examine whether it is also possible to choose particular values of $\\xi $ and $\\psi $ such that $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ and $\\bar{Z}_5 e^{\\pm i\\psi }$ are both real.",
"If this were possible, then one would have succeeded in finding a U(2) transformation, $\\bar{H}_i = \\widetilde{V}_{ij} H_j$ ($i,j=2,3$ ) such that all the coefficients of the IS scalar potential are real.",
"For example, if $Z_5=0$ , then it follows from Eqs.",
"() and () that $\\bar{Z}_5^\\prime =\\bar{Z}_5=0$ , in which case all the coefficients of the IS scalar potential, when expressed in terms of the barred scalar doublet fields, are real.",
"Thus a real Higgs basis exists when $Z_5=0$ .",
"It therefore follows that if $Z_5=0$ , then the IS scalar potential must possess a CP2 symmetry of the form, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , where $Y$ is a symmetric unitary matrix, which in block diagonal form [cf. Eq.",
"(REF )] is given by, $Y=\\begin{pmatrix} 1 & \\,\\,\\,0 \\\\ 0 &\\,\\,\\, \\widetilde{Y}\\end{pmatrix}\\,,\\qquad \\quad \\text{with $\\widetilde{Y}\\equiv (\\tilde{V}^T\\tilde{V})^*$},$ and $\\tilde{V}$ [given by Eq.",
"(REF )] is the unitary matrix that transforms the $H23$ basis into a real Higgs basis.",
"Suppose one performs a CP4 transformation [Eq.",
"(REF )] followed by a CP2 transformation, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , and compares the result obtained by performing these two transformations in the opposite order.",
"The results of applying CP4 followed by CP2 as compared to CP2 followed by CP4 are equivalent to the Higgs family transformations, $YX^*$ and $XY^*$ , respectively [32].",
"Using Eqs.",
"(REF ), (REF ) and (REF ), it follows that $XY^*=e^{2i\\psi }YX^*\\,.$ That is, the CP2 and CP4 transformations commute if and only if $\\det Y= e^{2i\\psi }=1$ .",
"For example, the $H23$ basis is a real Higgs basis in the trivial case where $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ , independently of the value of $Z_5$ .",
"In this case, the corresponding CP2 symmetry, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y={1}$ , commutes with the CP4 symmetry of the IS potential.",
"If $Z_5=0$ , the real Higgs basis obtained above is independent of $\\psi $ .",
"In this case, it is convenient to choose $\\psi =0$ in defining the CP2 transformation.",
"It then follows from Eq.",
"(REF ) that the CP2 and CP4 transformations commute if $Z_5=0$ and either $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ is nonzero.",
"Two other special cases, first pointed out in Appendix B of Ref.",
"[32], are noteworthy.",
"First, suppose that $Z_9=0$ .",
"In this case, the choice of $\\psi =\\chi =\\tfrac{1}{2}\\pi $ , $\\xi =0$ and $\\phi =\\tfrac{1}{4}\\pi $ inserted into Eqs.",
"()–() will yield a real Higgs basis, with $\\bar{Z}_5^\\prime =0$ , $e^{\\pm i\\psi }\\bar{Z}_5=\\mp Z_5$ , $\\bar{Z_8}=\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)$ and $\\bar{Z}_9=\\operatorname{Im}Z_8$ .",
"In light of Eqs.",
"(REF ) and (REF ), the barred and unbarred scalar fields are related by $\\bar{H}_2=\\frac{i}{\\sqrt{2}}(H_2-H_3)\\,,\\qquad \\quad \\bar{H}_3=\\frac{1}{\\sqrt{2}}(H_2+H_3)\\,.$ In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y=\\begin{pmatrix} 1 & \\,\\,\\, 0 &\\,\\,\\, 0 \\\\ 0 &\\,\\,\\, 0 &\\,\\,\\, 1 \\\\ 0 &\\,\\,\\, 1 &\\,\\,\\, 0\\end{pmatrix}\\,.$ Since $XY^*\\ne YX^*$ , it follows that the CP2 and CP4 transformations do not commute.",
"Second, suppose that $\\operatorname{Im}Z_8=0$ , $\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .",
"In this case, we simply choose $\\bar{H}_2=H_2$ and $\\bar{H_3}=iH_3$ , corresponding to $\\psi =\\tfrac{1}{2}\\pi $ , $\\chi =\\xi =-\\tfrac{1}{4}\\pi $ and $\\phi =0$ in Eq.",
"(REF ).",
"A real Higgs basis is then achieved with $\\bar{Z}_5^\\prime =0$ , $e^{\\pm i\\psi }\\bar{Z}_5=\\pm Z_5$ , $\\bar{Z}_8=-\\operatorname{Re}Z_8$ , and $\\bar{Z}_9=\\operatorname{Im}Z_9$ .",
"In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y=\\begin{pmatrix} 1 & \\phantom{-}0 &\\phantom{-}0 \\\\ 0 & \\phantom{-}1 &\\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1\\end{pmatrix}\\,.$ Once again, the CP2 and CP4 transformations do not commute.",
"It should be noted that the last two cases are related by a simple basis transformation.",
"Namely, starting from an $H23$ basis with $\\operatorname{Im}Z_8=0$ and $\\operatorname{Re}Z_9=0$ and employing $\\chi =\\psi =\\xi =0$ and $\\phi =\\tfrac{1}{4}\\pi $ in Eqs.",
"()–() yields $\\bar{Z}_5^\\prime =0$ , $\\bar{Z}_5=Z_5$ , $\\operatorname{Im}\\bar{Z}_8\\ne 0$ and $\\bar{Z}_9=0$ , thereby reducing to the previous case above.",
"We now consider the IS scalar potential with generic parameters (excluding the special cases considered above) and investigate whether a real Higgs basis exists.",
"In particular, consider the $H23$ basis with $Z_5$ , $Z_9\\ne 0$ .",
"As noted above, we can assume without loss of generality that $Z_5$ and $Z_9$ are real.The case of $\\operatorname{Im}Z_8=0$ and $\\operatorname{Re}Z_9=0$ is thus eliminated from consideration, since it is related by a scalar basis transformation to the case of $Z_9=0$ as noted above.",
"We examine two different cases: Case 1: $\\psi \\ne \\pm \\tfrac{1}{2}\\pi $ Case 2: $\\psi =\\pm \\tfrac{1}{2}\\pi $ In case 1, a real Higgs basis would require $\\bar{Z}_5^\\prime =0$ .",
"Since $\\sin 2\\phi \\ne 0$ [in light of Eq.",
"(REF )], it follows that $\\sin \\xi =0$ , in which case $e^{\\pm i\\psi }\\bar{Z}_5=\\pm e^{i(\\chi \\pm \\psi )}Z_5$ is real if and only if $\\sin (\\chi \\pm \\psi )=0$ .",
"This equation must be satisfied for both sign choices, which yields $\\sin \\chi \\cos \\psi =\\cos \\chi \\sin \\psi =0$ .",
"For generic values of the parameters, Eqs.",
"(REF )–(REF ) imply that $\\sin \\chi \\ne 0$ and $\\cos \\chi \\ne 0$ .",
"Thus, in general no value of $\\psi $ exists such that $\\sin (\\chi \\pm \\psi )=0$ holds for both sign choices.",
"That is, case 1 cannot yield a real Higgs basis for a generic choice of the IS scalar potential parameters.",
"In case 2, $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ is real for all choices of $\\xi $ and one must check whether there exists a $\\xi $ that yields a real value of $i\\bar{Z}_5=ie^{i\\chi }Z_5(e^{i\\xi }\\cos ^2\\phi +e^{-i\\xi }\\sin ^2\\phi $ ).",
"The condition that $i\\bar{Z}_5$ is real is equivalent to $\\operatorname{Re}\\bigl [e^{i(\\chi +\\xi )}\\cos ^2\\phi +e^{i(\\chi -\\xi )}\\sin ^2\\phi \\bigr ]=0\\,,$ which can be simplified to the condition, $\\cot \\chi =\\cos 2\\phi \\tan \\xi \\,.$ It follows that either $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ or $\\cos 2\\phi =\\cot \\chi \\cot \\xi $ .",
"If $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ , then either $\\chi $ is a half odd integer of $\\tfrac{1}{2}\\pi $ and $\\xi $ is an integer multiple of $\\pi $ or vice versa.",
"If $\\chi $ is a half odd integer multiple of $\\tfrac{1}{2}\\pi $ and $\\xi $ is an integer multiple of $\\pi $ , then Eq.",
"(REF ) yields $f_a=f_d=0$ .",
"However, these latter two equations cannot be simultaneously satisfied if $Z_9\\ne 0$ .",
"Similarly, if $\\chi $ is an integer multiple of $\\pi $ and $\\xi $ is a half odd integer multiple of $\\tfrac{1}{2}\\pi $ , then Eq.",
"(REF ) yields $f_a=f_c=0$ which cannot be simultaneously satisfied if $\\operatorname{Im}Z_8$ and $Z_9$ are nonzero.",
"Thus, if $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ , then no real Higgs basis exists for generic values of the IS scalar potential parameters.",
"Finally, we examine the possibility that $i\\bar{Z}_5$ is real due to $\\cos 2\\phi =\\cot \\chi \\cot \\xi $ .",
"We can also assume that $\\xi $ is not an integer multiple of $\\tfrac{1}{2}\\pi $ , as this case was already treated above.",
"In order that $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ , one must satisfy $\\cot \\chi =f_d/f_c$ and $G(\\phi ,\\xi )=0$ , under the assumption of $f_c\\ne 0$ .",
"In this case we can satisfy $\\operatorname{Im}\\bar{Z}_9=0$ if $\\phi =\\phi _\\xi $ , where $\\cos 2\\phi _\\xi =\\left(\\frac{f_d}{f_c}\\right)_{\\phi =\\phi _\\xi }\\cot \\xi \\,.$ Using Eqs.",
"() and (), one can employ Eq.",
"(REF ) to obtain a quadratic equation for $\\cot 2\\phi _\\xi $ , whose solution is given by $\\cot 2\\phi _\\xi &=&\\frac{1}{2Z_9}\\biggl [\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|\\cos (\\xi +\\theta _8) \\nonumber \\\\&&\\qquad \\qquad \\quad \\pm \\sqrt{[\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|\\cos (\\xi +\\theta _8)]^2+4Z_9^2\\cos ^2\\xi }\\biggr ]\\,.$ Eq.",
"(REF ) determines $\\sin 2\\phi _\\xi $ up to an overall sign.",
"It is convenient to choose this sign to be positive.",
"It is sufficient to demonstrate one example of the IS scalar potential parameters in which no real Higgs basis exists.",
"Thus, consider an example where $Z_8=-\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )+2iZ_9\\,.$ Then, $\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|(\\cos (\\xi +\\theta _8) =-2Z_9\\sin \\xi $ .",
"It follows that $\\cot 2\\phi _\\xi =\\pm 1-\\sin \\xi \\,.$ We now investigate whether a value of $\\xi \\ne \\tfrac{1}{2}n\\pi $ (where $n$ is an integer) exists such that $G(\\phi _\\xi ,\\xi )=0$ .",
"We introduce the notation, $\\tilde{f}\\equiv f(\\phi _\\xi ,\\xi )$ , where $\\phi _\\xi $ has been determined from Eq.",
"(REF ).",
"Then, Eq.",
"(REF ) implies that $\\tilde{f}_d=\\tilde{f}_c\\cos 2\\phi _\\xi \\tan \\xi $ .",
"Inserting this result into Eq.",
"(REF ) yields, $G(\\phi _\\xi ,\\xi )= \\tilde{f}_c^2\\bigl [\\tilde{f}_a(\\cos ^2 2\\phi _\\xi \\tan ^2\\xi -1)-2\\tilde{f}_b \\cos 2\\phi _\\xi \\tan \\xi \\bigr ]\\,.$ An explicit calculation yields $\\tilde{f}_c=\\sin 2\\phi _\\xi \\bigl [-\\operatorname{Re}Z_8\\sin 2\\xi +(3\\sin \\xi \\mp 2)(\\sin \\xi \\pm 1)Z_9\\bigr ]\\,,$ and $\\tilde{f}_a(\\cos ^2 2\\phi _\\xi \\tan ^2\\xi -1)-2\\tilde{f}_b \\cos 2\\phi _\\xi \\tan \\xi =2Z_9\\frac{\\sin 2\\phi _\\xi }{\\cos ^2\\xi }(\\sin \\xi \\mp 1)\\,.$ Hence, we end up withWe have made use of the identity, $1+\\cot ^2 2\\phi _\\xi =1/\\sin ^2 2\\phi _\\xi $ .",
"As noted below Eq.",
"(REF ) , we have assumed that $\\sin 2\\phi _\\xi $ is positive.",
"$G(\\phi _\\xi ,\\xi )=\\frac{2Z_9}{\\cos ^2\\xi }\\bigl [1+(1\\mp \\sin \\xi )^2\\bigr ]^{-3/2}(\\sin \\xi \\mp 1)\\bigl [\\operatorname{Re}Z_8\\sin 2\\xi -(3\\sin \\xi \\mp 2)(\\sin \\xi \\pm 1)Z_9\\bigr ]^2\\,.$ Since the above analysis has assumed that $\\tilde{f}_c\\ne 0$ and $\\xi \\ne \\tfrac{1}{2}n\\pi $ (for integer $n$ ) it follows that $G(\\phi _\\xi ,\\xi )$ is strictly nonzero, which implies that $\\operatorname{Im}\\bar{Z}_8\\ne 0$ .By expanding to squared expression in Eq.",
"(REF ), one sees that the factor of $\\cos ^2\\xi $ in the denominator is canceled by terms in the numerator.",
"Hence, there is no singularity in the limit of $\\cos \\xi \\rightarrow 0$ ."
],
[
"Special cases",
"Cases where $f_c=0$ need to be treated separately.",
"First, we examine the case of $f_c=0$ and $f_d\\ne 0$ .",
"Inserting Eq.",
"(REF ) into Eq.",
"(), one can solve for $\\cot 2\\phi _\\xi $ , $\\cot 2\\phi _\\xi =\\frac{\\operatorname{Re}Z_8\\sin 2\\xi +2Z_9\\cos 2\\xi }{Z_9\\sin \\xi }\\,.$ We next impose $\\operatorname{Im}\\bar{Z}_9=0$ .",
"Then Eq.",
"(REF ) implies that $\\sin \\chi =0$ , in which case Eq.",
"(REF ) yields $\\cos \\xi =0$ .",
"Inserting the latter result back into Eq.",
"(REF ) yields $\\cot 2\\phi _\\xi =\\pm 2$ .",
"If one now attempts to impose $\\operatorname{Im}\\bar{Z}_8=0$ using Eq.",
"(REF ) with $\\sin \\chi =0$ , then one would conclude that $f_a=0$ .",
"However, one can explicitly show that $f_a\\ne 0$ by inserting Eq.",
"(REF ) into Eq.",
"() and employing $\\cot 2\\phi _\\xi =\\pm 2$ and $\\cos \\xi =0$ .",
"Finally, we briefly consider the case of $f_c=f_d=0$ .",
"Inserting Eq.",
"(REF ) into Eqs.",
"() and () yields equations for $\\tan 2\\phi $ and $\\tan 4\\phi $ respectively.",
"The compatibility of these two equations then fixes the value of $\\xi $ .",
"In this case, $\\operatorname{Im}\\bar{Z}_9=0$ is automatic, and $\\operatorname{Im}\\bar{Z}_8=0$ implies via Eq.",
"(REF ) that $\\cot 2\\chi =f_b/f_a$ .",
"At this stage, $\\chi $ , $\\xi $ and $\\phi $ are all determined prior to imposing Eq.",
"(REF ).",
"The latter is an independent condition; hence for generic values of $\\operatorname{Re}Z_8$ and $Z_9$ , it is not possible to perform a basis change such that $i\\bar{Z}_5$ , $\\bar{Z}_8$ and $\\bar{Z}_9$ are simultaneously real.",
"This completes all the subcases of case 2.",
"We conclude that if the IS scalar potential possesses at least one non-real coefficient (for generic choices of the scalar potential parameters), then no real Higgs basis exists and it is not possible to perform a U(2) transformation of the Higgs basis fields $\\lbrace H_2,H_3\\rbrace $ such that all coefficients of the scalar potential are real."
],
[
"Basis-invariant polynomial functions of the IS scalar potential parameters",
"In our analysis of the IS model, we have advocated the choice of a particular class of Higgs bases in which $\\bar{Z}_5^\\prime =0$ .",
"Nevertheless, it is instructive to show that physical observables that depend on the parameters of the IS scalar potential are independent of the choice of the scalar basis.",
"In this appendix, we introduce a number of basis invariant quantities and evaluate them in the $H23$ -basis.",
"Consider a generic basis of scalar fields, $\\lbrace \\Phi _a\\rbrace $ , where $a=1,2,3$ labels hypercharge-one, doublet fields of the 3HDM.",
"Basis transformations that leave invariant the form of the canonical kinetic energy terms correspond to global U(3) transformations, $\\Phi _a\\rightarrow U_{a{\\bar{b}}}\\Phi _b$ [and $\\Phi _{\\bar{a}}^\\dagger \\rightarrow \\Phi _{\\bar{b}}^\\dagger U^\\dagger _{b{\\bar{a}}}$ ], where the $3\\times 3$ unitary matrix $U$ satisfies $U^\\dagger _{b{\\bar{a}}}U_{a{\\bar{c}}}=\\delta _{b{\\bar{c}}}$ .",
"Here, we follow the index conventions introduced in Ref.",
"[20], in which replacing an unbarred index with a barred index is equivalent to hermitian conjugation.",
"We only allow sums over barred–unbarred index pairs, which are performed by employing the U(3)-invariant tensor $\\delta _{a{\\bar{b}}}$ .",
"In this notation, the 3HDM scalar potential in a generic $\\Phi _a$ -basis is given by, $\\mathcal {V}=Y_{a{\\bar{b}}}\\Phi _{\\bar{a}}^\\dagger \\Phi _b+\\tfrac{1}{2}Z_{a{\\bar{b}}c{\\bar{d}}}(\\Phi _{\\bar{a}}^\\dagger \\Phi _b)(\\Phi _{\\bar{c}}^\\dagger \\Phi _d)\\,,$ where $Z_{a{\\bar{b}}c{\\bar{d}}}=Z_{c{\\bar{d}}a{\\bar{b}}}$ .",
"Hermiticity of $\\mathcal {V}$ implies that $Y_{a {\\bar{b}}}= (Y_{b {\\bar{a}}})^\\ast $ and $Z_{a{\\bar{b}}c{\\bar{d}}}= (Z_{b{\\bar{a}}d{\\bar{c}}})^\\ast $ .",
"Minimizing the scalar potential, under the assumption that the vacuum preserves U(1)$_{\\rm EM}$ , yields the neutral Higgs vacuum expectation values, $\\langle \\Phi ^0_a \\rangle =v \\widehat{v}_a/\\sqrt{2}$ , where $v=246$ GeV and $\\widehat{v}_a$ is a vector of unit norm.",
"It is convenient to define the hermitian matrix[19] $V_{a{\\bar{b}}}\\equiv \\widehat{v}_a \\,\\widehat{v}_{\\bar{b}}^\\ast \\,.$ One can now construct basis-invariant quantities that depend on knowledge of the scalar potential minimum by forming products of $V_{a{\\bar{b}}}$ and $Z_{a{\\bar{b}}c{\\bar{d}}}$ such that all barred–unbarred index pairs are summed over.",
"We define six invariant quantities below, The invariants above can be evaluated in any basis.",
"In particular, in the $H23$ -basis, the only nonzero component of $V_{a{\\bar{b}}}$ is $V_{11}=1$ .",
"We thus obtain, $J_1&=&Z_1,\\\\J_2&=&Z_1+2Z_3,\\\\J_3&=&Z_1+2Z_4,\\\\J_4&=&Z_1^2+2 Z_3^2+2 Z_4^2+2 Z_5^2,\\\\J_5&=&Z_1^3+4 Z_5^2 Z_1+2 Z_3^3+6 Z_3 Z_4^2+2 Z_2 Z_5^2+4 Z_5^2 \\operatorname{Re}Z_8 ,\\\\J_6&=&Z_1^4+2 Z_3^4+2 Z_4^4+12 Z_3^2 Z_4^2+4 Z_5^4+2Z_5^2(3Z_1^2+2Z_1 Z_2+Z_2^2)\\nonumber \\\\&&+8 Z_5^2\\bigl [|Z_8|^2+(Z_1+Z_2)\\operatorname{Re}Z_8+ \\left(\\operatorname{Im}Z_9\\right)^2\\bigr ].$ Using the first four invariant quantities above, one can show that $Z_5$ can be expressed in terms of an invariant quantity.In the $H23$ -basis (where $Z_5^\\prime =0$ ), one expects that $Z^2_5$ can be expressed in terms of an invariant quantity in light of the mass relation, $M_P^2-M_Q^2=Z_5 v^2$ , which implies that $Z_5^2$ is a physical parameter.",
"In a general class of Higgs bases, the corresponding invariant quantity is $|Z_5|^2+|Z_5^\\prime |^2$ [cf. Eq.",
"(REF )].",
"In particular, $Z_5^2=-J_1^2+\\tfrac{1}{2}J_1 \\left(J_2+J_3\\right) -\\tfrac{1}{4} (J_2^2+J_3^2)+\\tfrac{1}{2}J_4\\,.$ Finally, we have discovered a remarkable invariant quantity, $\\mathcal {N}&=&32Z_5^2 J_6-16J_5^2+8J_5(3J_{21}J_{31}^2+K) -J_{31}^4(9J_{21}^2+4Z_5^2) -6KJ_{21}J_{31}^2-24Z_5^2 J_{21}^2 J_{31}^2 \\nonumber \\\\[5pt]&& -J_{21}^6 - 4Z_5^2J_{21}^4-8J_1(J_1^2+2Z_5^2)J_{21}^3 -16J_1^6-96 Z_5^2 J_1^4 -192 Z_5^4 J_1^2 -128Z_5^6\\,, $ where $J_{ij}\\equiv J_i-J_j$ , the invariant quantity $Z^2_5$ is given by Eq.",
"(REF ) and $K\\equiv 4J_1^3+8Z_5^2 J_1+J_{21}^3\\,.$ Plugging in the expressions for $J_1\\,,\\ldots \\,,J_6$ given above, we find $\\mathcal {N}=256 Z_5^4\\bigl [(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2\\bigr ]\\,.$ It follows that if $Z_5\\ne 0$ then there exists a ratio of invariant quantities, which when evaluated in the $H23$ -basis, is equal to $(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2$ .",
"In contrast, if $Z_5=0$ , then there is no invariant quantity that reduces in the $H23$ -basis to $(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2$ .",
"Nevertheless, the invariant condition, $Z_5=0$ , signals the presence of four mass-degenerate neutral scalars.",
"The significance of the invariant $\\mathcal {N}$ is as follows.",
"The CP4-conserving IS model possesses a CP2 symmetry that commutes with CP4 if and only if $\\mathcal {N}=0$ .",
"Note that the nonvanishing of $\\mathcal {N}$ does not exclude the possibility of a CP2 symmetry that does not commute with CP4.",
"Two explicit examples of this phenomenon were presented in Appendix REF : (i) $Z_9=0$ and $\\operatorname{Im}Z_8\\ne 0$ ; and (ii) $\\operatorname{Im}Z_8=\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .",
"In both these cases, a real Higgs basis exists, and the corresponding CP2 transformation does not commute with CP4.One cannot employ Eq.",
"(REF ) to compute $\\mathcal {N}$ in the real Higgs basis in cases (i) and (ii), since in both cases, the real Higgs basis lies outside the set of $H23$ bases.",
"Nevertheless, we have checked that evaluating $\\mathcal {N}$ directly in the real Higgs basis in cases (i) and (ii) reproduces the corresponding results obtained in the $H23$ basis via Eq.",
"(REF ).",
"We also noted in Section that under the assumption that $M_P\\ne M_Q$ (or equivalently for $Z_5\\ne 0$ in the $H23$ basis), the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , if kinematically allowed, is nonzero if and only if $\\mathcal {N}\\ne 0$ .",
"Note that the invariant quantity $\\mathcal {N}$ constructed above has been expressed in terms of Higgs basis parameters.",
"This means that this invariant quantity depends on the knowledge of the vacuum, i.e.",
"the minimum of the scalar potential (which is needed to formally define the Higgs basis).",
"Given an explicitly CP4-invariant scalar potential, one could ask a slightly different question: is there an invariant quantity that can differentiate between scalar potentials that explicitly preserve or violate the CP2 symmetry, independently of the vacuum.",
"This question has been recently addressed and answered in Ref. [32].",
"However, it is not clear that such an invariant quantity can be directly related in practice to a physical observable."
],
[
"An alternative Higgs basis",
"In this paper, we first defined the $H23$ -basis by employing the scalar doublet fields $\\lbrace H_1,H_2,H_3\\rbrace $ , which was one particular choice among possible Higgs bases.",
"An arbitrary Higgs basis can be obtained by performing the U(2) basis transformation given by Eqs.",
"(REF ) and (REF ).",
"The corresponding IS scalar potential is given by Eq.",
"(REF ), where the barred coefficients in terms of the unbarred coefficients are given in Eqs.",
"(REF )–().",
"In section REF , we explored another Higgs basis choice, called the $RS$ -basis, which employs the scalar doublet fields, $\\lbrace H_1,\\mathcal {R},\\mathcal {S}\\rbrace $ .",
"The relations between the $H23$ -basis and $RS$ -basis are given by, $\\mathcal {R}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2+iH_3\\bigr )=\\begin{pmatrix} R^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P+iQ^\\dagger \\bigr )\\end{pmatrix}\\,,\\quad \\qquad \\mathcal {S}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2-iH_3\\bigr )=\\begin{pmatrix} S^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P^\\dagger +iQ\\bigr )\\end{pmatrix}\\,.$ Note that the form of the CP4 transformation in this basis is $\\begin{pmatrix} H_1 \\\\ \\mathcal {R} \\\\ \\mathcal {S}\\end{pmatrix}\\longrightarrow \\begin{pmatrix} 1 & \\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & \\phantom{-}i \\\\ 0 & -i & \\phantom{-}0\\end{pmatrix}\\begin{pmatrix} H_1^\\dagger \\\\ \\mathcal {R}^\\dagger \\\\ \\mathcal {S}^\\dagger \\end{pmatrix}\\,.$ The change of basis from $\\lbrace H_1,H_2,H_3\\rbrace $ to $\\lbrace H_1,R,S\\rbrace $ corresponds to choosing $\\alpha =\\beta =\\phi =\\tfrac{1}{4}\\pi $ and $\\psi =-\\tfrac{1}{2}\\pi $ .",
"Inserting these results into Eqs.",
"(REF )–() yields, 1.25 $\\bar{Z}_2&=&Z_2+\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)\\,,\\\\\\bar{Z}_3^\\prime &=&-Z_4^\\prime +2\\operatorname{Re}Z_8\\,,\\\\\\bar{Z}_4^\\prime &=&\\tfrac{1}{2}(Z_4^\\prime -Z_3^\\prime +2\\operatorname{Re}Z_8)\\,,\\\\\\bar{Z}_5^\\prime &=&Z_5\\,,\\\\\\bar{Z}_5&=&0\\,,\\\\\\bar{Z}_8&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime +2\\operatorname{Re}Z_8)+i\\operatorname{Re}Z_9\\,,\\\\\\bar{Z}_9&=& \\operatorname{Im}Z_9+i\\operatorname{Im}Z_8\\,.$ Note that in the $RS$ -basis, the invariant $\\mathcal {N}$ defined in Eq.",
"(REF ) is $\\mathcal {N}=256\\bar{Z}_5^{\\prime \\,4}|\\bar{Z}_9|^2\\,.$ In particular, the absence [or presence] of the $Z\\rightarrow QQQQ^*$ , $QQ^*Q^*Q^*$ decay discussed in Appendix depends on the [non-]vanishing of $\\bar{Z}_9$ .",
"These decays are governed by Eq.",
"(REF ), which when expressed in the $RS$ -basis is given by, $\\delta {L}_{4h}\\ni \\tfrac{1}{2}i(PQ-P^\\dagger Q^\\dagger )\\bigl [\\bar{Z}_9(P^{\\dagger \\,2}+Q^2)+\\bar{Z}_9^*(P^2+Q^{\\dagger \\,2})\\bigr ]\\,.$"
],
[
"$Z$ decay into four inert neutral scalars",
"Consider a universe (not ours) in which the electroweak theory of elementary particles at the electroweak scale consists of the IS model, with $M_{H^\\pm ,h^\\pm }<M_Q<\\tfrac{1}{4} m_Z\\ll M_P\\,.$ In this case, the decay of the $Z$ to four neutral inert scalars would be consistent with a CP4-symmetric IS scalar potential that does not possess a real scalar basis.",
"Experimentally, the final state would be detected via the decay $Q\\rightarrow (H^\\pm , h^\\pm )+W^{*\\,\\mp }$ , with the virtual $W^{*\\,\\mp }$ decaying to quark or lepton pairs.",
"In this universe, the $H^\\pm , h^\\pm $ are the lightest particles of the inert scalar sector and hence stable.",
"Although this is not our universe, this example provides a proof in principle of the existence of an experimental distinction between the CP4-conserving/CP2-nonconserving case and the CP4/CP2-conserving case.",
"Figure: Feynman diagrams for Z→QQQQ * Z\\rightarrow QQQQ^\\ast In light of Eq.",
"(REF ), there are four contributing tree-level Feynman diagrams to the decay amplitude $Z\\rightarrow QQQQ^\\ast $ , which are shown in Fig.",
"REF .",
"Employing the Feynman rules obtained from Eq.",
"(REF ) (and including the appropriate symmetry factors in obtaining the rules for the four-scalar vertex), the invariant matrix element is given by, $i\\mathcal {M}&=&\\frac{g}{2\\cos \\theta _W}\\bigl (\\operatorname{Im}Z_8 +i\\operatorname{Im}Z_9\\bigr )\\,\\varepsilon _\\lambda (p)\\cdot \\left[\\frac{p-2k_1}{(p-k_1)^2-M_P^2}+\\frac{p-2k_2}{(p-k_2)^2-M_P^2}\\right.",
"\\nonumber \\\\[6pt]&&\\qquad \\qquad \\qquad \\left.+\\frac{p-2k_3}{(p-k_3)^2-M_P^2}-\\frac{3(p-2k_4)}{(p-k_4)^2-M_P^2}\\right]\\,,\\nonumber $ where $p$ is the four-momentum of the $Z$ and the $k_i$ are the final state momenta (with $k_4$ the momentum of $Q^*$ ).",
"We then square the matrix element and average over the initial state spins, using $|\\mathcal {M}|^2_{\\rm ave}\\equiv \\tfrac{1}{3}\\sum _\\lambda |X\\cdot \\varepsilon _\\lambda (p)|^2=-\\tfrac{1}{3} X_\\mu X^*_\\nu \\left( g^{\\mu \\nu }-\\frac{p^\\mu p^\\nu }{m_Z^2}\\right)\\,.$ where $X$ is the four vector dotted into the polarization vector in the expression for $i\\mathcal {M}$ .",
"We shall work in the approximation that $M_P\\gg m_Z$ and $M_Q=0$ .",
"In this case, $X=\\frac{g}{M_P^2\\cos \\theta _W}\\bigl (\\operatorname{Im}Z_8 +i\\operatorname{Im}Z_9\\bigr )(p-4k_4)\\,,$ where we have used conservation of momentum, $p=k_1+k_2+k_3+k_4$ .",
"It then follows after some simplification (with $p^2=m_Z^2$ ) that, $|\\mathcal {M}|^2_{\\rm ave}=\\frac{16g^2}{3m_Z^2 M_P^4\\cos ^2\\theta _W}\\bigl [(\\operatorname{Im}Z_8)^2+ (\\operatorname{Im}Z_9)^2\\bigr ] (p\\cdot k_4)^2\\,.$ The four body decay width for $Z\\rightarrow QQQQ^*$ is given by $\\Gamma =\\frac{1}{6}\\,\\frac{(2\\pi )^{-8}}{2m_Z}\\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4)|\\mathcal {M}|^2_{\\rm ave}\\,,$ where the factor of $1/6$ is due to the three identical Qs in the final state (which means we overcount by a factor of 3!",
"by integrating over the full phase space).",
"Using the above results, we obtain, $\\Gamma =\\frac{4g^2(2\\pi )^{-8}}{9m_Z^3 M_P^4\\cos ^2\\theta _W}\\bigl [(\\operatorname{Im}Z_8)^2+ (\\operatorname{Im}Z_9)^2\\bigr ]\\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4) \\,(p\\cdot k_1)^2$ after changing integration variables $k_1\\longleftrightarrow k_4$ .",
"To perform the phase space integration, we follow Ref. [82].",
"To integrate over $d^3k_1d^3k_2$ we use, $\\int \\frac{d^3k_1}{2E_1}\\frac{d^3k_2}{E_2}\\delta ^{4}(N-k_1-k_2)\\times {\\left\\lbrace \\begin{array}{ll}1& \\text{$=\\frac{1}{2}\\pi $,}\\\\k_{1\\mu }&\\text{$=\\frac{1}{4}\\pi N_\\mu $,}\\\\k_{1\\mu }k_{1\\nu }&\\text{$=-\\frac{1}{24}\\pi (N^2g_{\\mu \\nu }-4N_\\mu N_\\nu )$,}\\\\k_{1\\mu }k_{2\\nu }&\\text{$=\\frac{1}{24}\\pi (N^2g_{\\mu \\nu }+2N_\\mu N_\\nu )$,}\\end{array}\\right.",
"}$ where $N$ is an arbitrary four-vector.",
"In the present application, $N=p-k_3-k_4$ .",
"After performing this integration, we have two further integrations to do over $k_3$ and $k_4$ .",
"It is convenient to work in the $Z$ rest frame: $p=(m_Z;0,0,0);\\ \\ k_3=E_3(1;0,0,1);\\ \\ k_4=E_4(1;\\sin \\theta ,0,\\cos \\theta )\\,.$ We introduce the following scaled kinematic variables $w\\equiv \\frac{1-\\cos \\theta }{2},\\ \\ \\ y\\equiv \\frac{2E_3}{m_Z},\\ \\ \\ z\\equiv \\frac{2E_4}{m_Z}\\,.$ Then, $\\int \\frac{d^3 k_3}{2E_3}\\,\\frac{d^3k_4}{2E_4}=\\frac{\\pi ^2 m_Z^4}{4}\\int _0^{1}\\!z\\,dz\\!",
"\\left\\lbrace \\int _0^{1-z}y\\,dy\\int _0^1 dw+\\!\\int _{1-z}^1 y\\,dy\\!",
"\\int _{(y+z-1)/yz}^1 \\!dw\\right\\rbrace \\,.$ We now evaluate the integral in Eq.",
"(REF ).",
"Using the above results, $\\int \\frac{d^3k_1}{2E_1}\\frac{d^3k_2}{E_2}\\delta ^{4}(N-k_1-k_2)\\,(p\\cdot k_1)^2&=&-\\frac{\\pi }{24}\\biggl \\lbrace m_Z^2(p-k_3-k_4)^2-4\\bigl [p\\cdot (p-k_3-k_4)\\bigr ]^2\\biggr \\rbrace \\nonumber \\\\[6pt]&=&\\frac{\\pi m_Z^4}{24}\\bigl [(2-y-z)^2-1+y+z-yzw\\bigr ]\\,,$ where $N\\equiv p-k_3-k_4$ .",
"In obtaining the above result, we have used [cf. Eq.",
"(REF )], $(p-k_3-k_4)^2&=&m_Z^2-2p\\cdot (k_3+k_4)+2k_3\\cdot k_4=m_Z^2(1-y-z+yzw)\\,, \\\\[5pt]p\\cdot (p-k_3-k_4) &=&m^2_Z\\bigl [1-\\tfrac{1}{2}(y+z)\\bigr ]\\,.$ Hence, after employing Eq.",
"(REF ), we end up with $&& \\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4) \\,(p\\cdot k_1)^2\\nonumber \\\\[10pt]&& \\qquad =\\frac{\\pi ^3 m_Z^8}{96}\\int _0^{1}\\!z\\,dz\\!",
"\\left\\lbrace \\int _0^{1-z} y\\,dy\\int _0^1 dw+\\!\\int _{1-z}^1 y\\,dy\\!",
"\\int _{(y+z-1)/yz}^1 \\!dw\\right\\rbrace \\bigl [(2-y-z)^2-1+y+z-yzw\\bigr ]\\nonumber \\\\[6pt]&&\\qquad \\,\\, =\\frac{\\pi ^3 m_Z^8}{1280}\\,.$ Collecting all our results, we end up with $\\Gamma =\\frac{g^2 m_Z^5\\bigl [(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\bigr ]}{3^2\\cdot 5\\cdot 2^{14}\\, \\pi ^5 M_P^4\\cos ^2\\theta _W}\\,.$ Relative to the decay rate of $Z$ into a neutrino pair, $\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })= g^2 m_Z/(96\\pi \\cos ^2\\theta _W)$ , $\\frac{\\Gamma (Z\\rightarrow QQQQ^*)}{\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })}=\\frac{(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2}{3\\cdot 5\\cdot 2^{9}\\, \\pi ^4 }\\left(\\frac{m_Z}{M_P}\\right)^4\\,.$ Finally, we note that the decay rate for $Z\\rightarrow QQ^*Q^*Q^*$ is identical to the one given above.",
"Since $Q$ and $Q^*$ are mass degenerate, the experimentally observable width would be a factor of 2 times the one given in Eq.",
"(REF ), as quoted in Eq.",
"(REF )."
]
] | 1808.08629 |
[
[
"Quasistates and quasiprobabilities"
],
[
"Abstract The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator.",
"While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently.",
"In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates.",
"In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system.",
"Therefore, our method presents a complementary approach to prominent quasiprobability representations.",
"We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light.",
"In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement."
],
[
"Introduction",
"Quantum systems can exhibit properties which have no analog in the classical realm, opening possibilities for technological innovations beyond the limitations posed by classical physics [1].",
"For assessing the quantumness of a state, the bisection of a system into a classically accessible domain and a genuinely quantum-mechanical sector has become one of the major challenges of current research.",
"A successful approach to performing such a desired separation are quasiprobability distributions, such as the Wigner-Weyl function [2], [3].",
"Within this framework, a state is identified to be nonclassical when the corresponding quasiprobability does not have the properties of a classical probability distribution.",
"Beyond this nonclassicality aspect, another purpose of quasiprobabilities is the full characterization of a quantum state, for example, using the Glauber-Sudarshan representation to describe quantized harmonic oscillators [4], [5].",
"As such, the description of a density operator in terms of quasiprobabilities and classical reference states, e.g., coherent states, is an equally important feature of quasiprobabilities.",
"However, despite having such a vital impact on the state's representation, the density operator expansion is, by far, less frequently considered.",
"This is at least partially a result of a missing framework which focuses on the decomposition of density operators.",
"Thus, the main objective of this contributions is to devise such a missing methodology that naturally leads to the useful and complementary concept of quasistates.",
"The notion of quasiprobabilities has become one of the main hallmarks for certifying quantum features since it allows us to study resources for practical tasks.",
"For instance, negative quasiprobabilities have been identified as indicators for the state's usefulness in performing quantum computation protocols [6], [7].",
"Moreover, negativities in quasiprobabilities can be related to a broad range of concepts of nonclassicality, such as contextuality [8] and quantum entanglement [9].",
"In addition, quasiprobabilities can be helpful to determine the actual probabilities of measurement outcomes [10], [11].",
"Furthermore, quasiprobabilities can be defined for many relevant physical systems, such as continuous-variable harmonic oscillators [4], [5] or discrete-variable angular-momentum-type degrees of freedom [12], [13].",
"Typically, the underlying quasiprobabilities are obtained by generalizing a classical phase space to the quantum domain; see Ref.",
"[14] for an early study and Ref.",
"[15] for a recent generalization.",
"For example, quasiprobabilities in finite-dimensional systems can be introduced in this manner, cf., e.g., Refs.",
"[16], [17], [18], [19].",
"With regard to the quantum state representation, this typically requires us to identify pure reference states which are compatible with the properties of classical physics [20].",
"Based on such classical reference states, a general quasiprobability decomposition of a quantum state can be constructed [19].",
"Examples relevant for quantum technologies include the formulation of entanglement quasiprobabilities [21], [22], which become negative for quantum correlated states.",
"Once a quasiprobability representation is constructed, it can be further generalized.",
"Again, pioneering applications can be found in quantum optics, where the Glauber-Sudarshan and Wigner-Weyl quasiprobabilities can be unified in terms of so-called $s$ -parametrized quasiprobabilities [23], [24], which also include the well-known Husimi-Kano distributions [25], [26].",
"This unification is achieved by a convolution of the quasiprobability with a Gaussian function and can be further generalized [27], [28], [29], including non-Gaussian scenarios [30], [31].",
"A successful application of the $s$ -parameter approach in a finite-dimensional system has been reported as well [32].",
"In addition, joint non-Gaussian quasiprobabilities for multiple optical modes and points in time have been introduced to study quantum correlated light [33], [34].",
"Although such formal aspects led to profound insights into quantum physics when compared to classical statistical theories (see, e.g., [35], [36]), the method of quasiprobabilities is also of experimental importance.",
"The most frequent implementation is the identification of nonclassical light in terms of negative Wigner-Weyl functions; see Refs.",
"[37], [38] for early and recent examples.",
"While the Glauber-Sudarshan distribution can be highly singular or even ambiguous [39], [40], [41], the reconstruction of this function is feasible in certain experiments as well [42].",
"Moreover, the non-Gaussian generalization of this distribution is always experimentally accessible, such as reported for squeezed light [43].",
"Even imperfect detection schemes can be employed for the reconstruction of nonclassical phase-space distribution of light [44].",
"Conversely, a phase-space representation of the detector can experimentally verify quantum properties of the detection device used [45].",
"Another approach [46], [47] enables an experimental reconstruction of phase-space distributions and entirely circumvents a detector characterization by analyzing certain data patterns [48].",
"Beyond quantum light, equally insightful is the characterization of matter systems using quasiprobabilities, such as demonstrated for motional states of ions [49] and large atomic ensembles [50].",
"It is also worth mentioning that quantum correlations in composite hybrid systems are also accessible via generalized quasiprobabilities [51], [52].",
"Therefore, quasiprobabilities present a highly successful and versatile approach to identifying nonclassical properties of quantum states in theory and experiment.",
"Still, as outlined above, another important aspect is the quantum state decomposition.",
"Yet, in contrast to the vast number of examples for the nonclassicality certification, studies of the decomposition properties are rarely done; a gap we aim to close in this article.",
"Based on the decomposition of states in terms of quasiprobabilities, we introduce a generalized expansion of quantum states in terms of quasistates.",
"This method is complementary to the previously known approaches in which a density operator is decomposed in terms of classical reference states and a negative quasiprobability density.",
"In our general method, we can find a decomposition which results in a nonnegative (i.e., classical) probability density by overcoming the usage of physical reference states in the decomposition.",
"This naturally establishes the concept of quasistates, which then become the relevant objects for certifying the different kinds of nonclassicality.",
"The usefulness of this change of perspective from quasiprobabilities to quasistates is studied in detail.",
"One practical application relates to the experimental reconstruction of density operators.",
"Moreover, as quasiprobabilities are most frequently applied in optical systems, we also perform a detailed analysis of quasistates that correspond to prominent phase-space representations of light.",
"Finally, we investigate how quantum entanglement can be uniquely characterized via classical joint probability distributions when employing quasistates.",
"Therefore, our approach offers a toolbox for the characterization and decomposition of quantum states.",
"This article is structured as follows.",
"In Sec.",
", we discuss the general methodological framework.",
"An application to the quantum state reconstruction is developed in Sec.",
".",
"Phase-space based quasistates for quantized light are studied in Sec.",
".",
"In Sec.",
", quantum correlations are analyzed within the proposed framework.",
"Finally, concluding discussions are presented in Sec.",
"."
],
[
"Conceptual framework",
"To characterize the state of a quantum system, a representation of the density operator is required.",
"In general, a state decomposition consists of a family of pure states, defining a set $\\mathcal {S}$ , and corresponding expansion coefficients.",
"Then the density operator takes the form $\\hat{\\rho }=\\int _{\\mathcal {S}} d\\psi \\, P(\\psi ) |\\psi \\rangle \\langle \\psi |,$ where $d\\psi $ indicates an integral over the volume of $\\mathcal {S}$ and $P$ is a correspondingly defined density.",
"In the case of a discrete decomposition, the integral may be replaced by a sum.",
"Note that we deliberately choose not to specify the set $\\mathcal {S}$ to keep our treatment as broadly applicable as possible.",
"Arguably the most well-known example of a representation (REF ) is the spectral decomposition.",
"In this case, $\\mathcal {S}$ is the set of eigenstates and the probability mass function $P$ returns the corresponding eigenvalues.",
"Other examples for a decomposition (REF ) are studied in the continuation of this work and have applications, for example, in quantum optics and quantum information.",
"It is worth mentioning that a general method to construct quasiprobabilities $P$ for a given set $\\mathcal {S}$ has been recently devised [19].",
"In that approach, it is ensured that $P$ is a classical probability distribution when the state is in the convex hull $\\mathrm {conv}\\lbrace |\\psi \\rangle \\langle \\psi |:\\psi \\in \\mathcal {S}\\rbrace $ , which is a nontrivial task as the decomposition (REF ) is typically not unique [19].",
"For our purpose, we may generalize the above treatment.",
"Specifically, a convolution kernel $K:\\mathcal {S}\\rightarrow \\mathcal {S}$ is introduced together with the kernel $K^{-1}$ for the corresponding deconvolution.",
"Then we get an equivalent decomposition as $\\hat{\\rho }=\\int _{\\mathcal {S}} d\\chi \\, P_K(\\chi )\\hat{\\Delta }_{K}(\\chi ).$ Therein we have a modified density $P_K$ and a modified family of operators $\\hat{\\Delta }_{K}$ , $P_K(\\chi )=&\\int _{\\mathcal {S}} d\\psi \\, P(\\psi )K(\\psi ,\\chi ),\\\\\\hat{\\Delta }_{K}(\\chi )=&\\int _{\\mathcal {S}} d\\psi ^{\\prime }\\, K^{-1}(\\chi ,\\psi ^{\\prime }) |\\psi ^{\\prime }\\rangle \\langle \\psi ^{\\prime }|.$ Using the properties that the operations used are inverse to one another, $\\delta (\\psi ,\\psi ^{\\prime })=\\int _{\\mathcal {S}} d\\chi K(\\psi ,\\chi )K^{-1}(\\chi ,\\psi ^{\\prime })$ with $\\delta $ being the Dirac distribution, one can directly see that Eqs.",
"(REF ) and (REF ) describe the same density operator $\\hat{\\rho }$ .",
"In general, the distribution $P_K$ in Eq.",
"(REF ) is not a probability density, even if $P$ was.",
"For such generalized distributions, the name quasiprobabilities was established.",
"As discussed in the Introduction, such quasiprobability densities cover a wide range of applications, mainly for purpose of identifying quantum features.",
"In close analogy to the notion of a quasiprobability, we are going to demonstrate that the operators $\\hat{\\Delta }_K$ in Eq.",
"() are, in general, not physical density operators.",
"Consequently, we refer to such operators as quasistates.",
"A main focus of previous research was devoted to characterizing quasiprobabilities.",
"Often, the idea of a decomposition of the quantum state [cf.",
"Eqs.",
"(REF ) and (REF )] in terms of such distributions was neglected.",
"In particular, a general characterization of the distinctive features of quasistates and their applications beyond being a mathematical tool does not exist.",
"For this reason, we are going to perform the missing comprehensive investigation of quasistates as defined in Eq.",
"() and discuss useful applications of such operators."
],
[
"State reconstruction",
"One interesting application of quasistates are state reconstruction protocols as we show in this section.",
"The following considerations are based on a recent work [11] in which the dual representation of measurement operators has been introduced.",
"Here we demonstrate how this method relates to quasistates and can be used for the general reconstruction of density operators."
],
[
"Dual representation",
"Let us briefly revisit the findings in Ref.",
"[11] with regards to our method.",
"Suppose $\\lbrace \\hat{\\Pi }(j)\\rbrace _{j\\in \\mathcal {S}}$ is a positive operator-valued measure (POVM).",
"As a complementary set of operators, the notion of contravariant operator-valued measured (COVM) was introduced.",
"This defines a set $\\lbrace \\hat{\\Gamma }(j^{\\prime })\\rbrace _{j^{\\prime }\\in \\mathcal {S}}$ that satisfies the orthonormality relation $\\mathrm {tr}\\big [\\hat{\\Gamma }(j^{\\prime })\\hat{\\Pi }(j)\\big ]=\\delta (j^{\\prime },j).$ This means that the COVM represent dual basis operators to the measured POVM.",
"In contrast to the POVM, COVM operators are not necessarily positive semidefinite.",
"However, they are of particular interest, for example, when considering imperfections in the measurement process [11].",
"The construction of COVM operators is based on a kernel defined via the elements $K(l,j)=\\mathrm {tr}\\big [\\hat{\\Pi }(l)\\hat{\\Pi }(j)\\big ].$ Specifically, it was shown that $\\hat{\\Gamma }(j^{\\prime })=\\sum _{l\\in \\mathcal {S}}K^{-1}(j^{\\prime },l)\\hat{\\Pi }(l),$ for the discrete case.",
"In the following, let us explore the relation between the COVM and quasistates.",
"For simplicity, we assume that the considered complex Hilbert space is finite dimensional, $\\dim _{\\mathbb {C}}\\mathcal {H}=d<\\infty $ .",
"The set of Hermitian operators over $\\mathcal {H}$ is a real valued space with the dimension $\\dim _{\\mathbb {R}}\\mathrm {Herm}(\\mathcal {H})=d^2$ .",
"Further, rather than restricting to a POVM for a single observable as done previously, we consider an informationally complete set $\\lbrace \\hat{\\Pi }(\\psi )\\rbrace _{\\psi \\in \\mathcal {S}}$ , with $\\hat{\\Pi }(\\psi )=|\\psi \\rangle \\langle \\psi |,$ which guarantees that a reconstruction of the full quantum state is possible and implies the cardinality $|\\mathcal {S}|=d^2$ .",
"However, this also means that not all elements of $\\mathcal {S}$ can be pairwise orthogonal vectors.",
"See Refs.",
"[53], [54], [55] for introductions to informationally complete measurements.",
"Because of the properties of the now studied POVM, we can expand the quantum state as given in Eq.",
"(REF ), $\\hat{\\rho }=\\sum _{\\psi \\in \\mathcal {S}}P(\\psi ) \\hat{\\Pi }(\\psi ).$ It is important to mention that $P(\\psi )$ , in general, does not correspond to the probability of the measurement of $\\hat{\\Pi }(\\psi )$ , i.e., $\\mathrm {tr}[\\hat{\\rho }\\hat{\\Pi }(\\psi )]\\ne P(\\psi )$ .",
"However, we can employ the same formalism as discussed above, cf.",
"Eqs.",
"(REF ) and (REF ), to expand the state as [11] $\\hat{\\rho }=\\sum _{\\chi \\in \\mathcal {S}}\\rho (\\chi ) \\hat{\\Gamma }(\\chi ).$ Now we can identify the coefficient with the measurement probability, $\\rho (\\psi ^{\\prime })=\\mathrm {tr}[\\hat{\\rho }\\hat{\\Pi }(\\psi ^{\\prime })]$ for all $\\psi ^{\\prime }\\in \\mathcal {S}$ , because of relation (REF ).",
"In other words, the expansion in Eq.",
"(REF ) can be used to directly reconstruct the density operator in terms of measurement outcomes and dual operators.",
"Let us specifically relate the found reconstruction with the generalized decomposition (REF ).",
"Using the definitions (REF ) and (REF ), we find that the kernel under study is given by the $d^2\\times d^2$ Gram-Schmidt matrix $K=\\left[|\\langle \\psi |\\chi \\rangle |^2\\right]_{\\psi ,\\chi \\in \\mathcal {S}}.$ Recall that information completeness implies that this matrix is a bijection, thus invertible.",
"Further, as a result of Eqs.",
"() and (REF ), the thought-after quasistates are identical to the COVM elements, $\\hat{\\Delta }_K(\\chi )=\\hat{\\Gamma }(\\chi ).$ Consequently, it also holds true that $P_K(\\chi )=\\rho (\\chi )$ ; see Eqs.",
"(REF ) and (REF ).",
"Therefore, quasistates allow for the direct reconstruction of the density operator under study.",
"It is worth mentioning that imperfections in any measurement, meaning that $\\hat{\\Pi }(\\psi )$ is not a pure-state projector, can be treated similarly to the thorough discussion for a single observable carried out in Ref.",
"[11].",
"Furthermore, if the POVM does not allow for a full reconstruction or is over-complete (e.g., because $|\\mathcal {S}|\\ne d^2$ ), then the state in the spanned subspace can be reconstructed or a pseudo-inversion of $K$ can be performed [11]."
],
[
"Example",
"To further discuss the above findings, we study the specific example of a qubit system ($d=2$ ), defined via the orthonormal Hilbert space basis $\\lbrace |0\\rangle ,|1\\rangle \\rbrace $ .",
"The four rank-one projection operators $|\\psi _j\\rangle \\langle \\psi _j|$ in Eq.",
"(REF ), also defining the set $\\mathcal {S}$ , can be chosen according to the states $\\nonumber |\\psi _1\\rangle =|1\\rangle ,|\\psi _2\\rangle =&\\frac{\\sqrt{2} |0\\rangle +|1\\rangle }{\\sqrt{3}},|\\psi _3\\rangle =\\frac{\\sqrt{2} w|0\\rangle +|1\\rangle }{\\sqrt{3}},\\\\\\text{and }|\\psi _4\\rangle =&\\frac{\\sqrt{2} w^2|0\\rangle +|1\\rangle }{\\sqrt{3}},$ where $w=\\exp [2\\pi i/3]$ ; see the top-left panel in Fig.",
"REF for their geometric Bloch-sphere representation.",
"This allows us to construct $K$ via Eq.",
"(REF ) and compute its inverse, $K=\\frac{1}{3}\\begin{bmatrix}3 & 1 & 1 & 1\\\\1 & 3 & 1 & 1\\\\1 & 1 & 3 & 1\\\\1 & 1 & 1 & 3\\end{bmatrix}\\!,\\,K^{-1}=\\frac{1}{4}\\begin{bmatrix}5 & -1 & -1 & -1\\\\-1 & 5 & -1 & -1\\\\-1 & -1 & 5 & -1\\\\-1 & -1 & -1 & 5\\end{bmatrix}\\!.$ This then yields the quasistates as $\\hat{\\Delta }_K(\\psi _1)=&|1\\rangle \\langle 1|-\\frac{1}{2}|0\\rangle \\langle 0|\\text{ and}\\\\\\nonumber \\hat{\\Delta }_K(\\psi _j)=&\\frac{|0\\rangle \\langle 0|+\\sqrt{2}w^{j-2}|0\\rangle \\langle 1|+\\sqrt{2}w^{\\ast (j-2)}|1\\rangle \\langle 0|}{2},$ for $j=2,3,4$ .",
"As we can see from the top-right panel in Fig.",
"REF , these quasistates do not resemble physical quantum states because of the negative eigenvalue.",
"Figure: (Color online)The elements of 𝒮\\mathcal {S} given in Eq.",
"() in a tetrahedron configuration (larger green bullets) are shown in the top-left Bloch-sphere representation.The smaller red bullet indicates the state ρ ^=|0〉〈0|\\hat{\\rho }=|0\\rangle \\langle 0| to be reconstructed.The top-right panel shows the two eigenvalues of the resulting quasistates Δ ^ K (ψ j )\\hat{\\Delta }_K(\\psi _j), being identical for j=1,...,4j=1,\\ldots ,4.For the reconstruction of the state under study, the bottom panels depict the expansion coefficients P(ψ j )P(\\psi _j) and P K (ψ j )P_K(\\psi _j) to be used with |ψ j 〉〈ψ j ||\\psi _j\\rangle \\langle \\psi _j| and Δ ^ K (ψ j )\\hat{\\Delta }_K(\\psi _j) [Eqs.",
"() and ()], respectively.Let us now consider the specific state $\\hat{\\rho }=|0\\rangle \\langle 0|$ for the reconstruction.",
"Then, the measured and, therefore, nonnegative expansion coefficients are $P_K(\\psi _j)=\\mathrm {tr}(\\hat{\\rho }|\\psi _j\\rangle \\langle \\psi _j|)=\\langle \\psi _j|\\hat{\\rho }|\\psi _j\\rangle ,$ using the projectors defined via the states in Eq.",
"(REF ).",
"In addition, the corresponding COVM elements, being the quasistates in Eq.",
"(REF ), result in the expansion coefficients $P(\\psi _j)=\\mathrm {tr}[\\hat{\\rho }\\hat{\\Delta }_K(\\psi _j)]$ , which resembles a quasiprobability distribution.",
"The bottom part in Fig.",
"REF shows the values of $P_K$ and $P$ for the state $\\hat{\\rho }=|0\\rangle \\langle 0|$ .",
"Using the nonnegative $P_K$ (see Fig.",
"REF ), we can now directly confirm that the density operator $\\hat{\\rho }=|0\\rangle \\langle 0|$ is expanded as described in Eq.",
"(REF ) while using the quasistates in Eq.",
"(REF ), $\\hat{\\rho }=\\sum _{j=1,\\ldots ,4}P_K(\\psi _j)\\hat{\\Delta }_K(\\psi _j)$ .",
"The possibly negative elements of $P$ also allow for the decomposition of the state, $\\hat{\\rho }=\\sum _{j=1,\\ldots ,4}P(\\psi _j)|\\psi _j\\rangle \\langle \\psi _j|$ .",
"However, $P$ does not correspond to a directly measured quantity.",
"Thus, quasistates provide a beneficial tool to reconstruct density operators using measured quantities.",
"Let us also briefly comment on the extension of our approach to continuous-variable systems, $d=\\infty $ .",
"In fact, already in Refs.",
"[56], [57], a method was introduced to expand the state of a single-mode radiation field in terms of so-called pattern functions.",
"Using the here developed framework, we can directly identify these pattern functions with the wave functions of the corresponding quasistates.",
"Beyond this reconstruction approach, we study the expansion of nonclassical states of light in the following section."
],
[
"Quantum Light",
"We now demonstrate the application of quasistates for the description of nonclassical quantum states.",
"In particular, we consider a harmonic oscillator, which can, for example, describe a quantized radiation mode.",
"To describe the system's nonclassicality, the prominent Glauber-Sudarshan representation has been developed [4], [5], $\\hat{\\rho }=\\int _{\\mathbb {C}}d\\alpha \\,P(\\alpha )|\\alpha \\rangle \\langle \\alpha |.$ Here this decomposition implies that we identify the set $\\mathcal {S}$ with the complex coherent amplitudes $\\alpha \\in \\mathbb {C}$ , defining the coherent states $|\\alpha \\rangle $ .",
"For a thorough introduction to quantum optics in phase space, see Ref.",
"[58].",
"A state is referred to as nonclassical if $P$ does not exhibit the properties of a classical probability distribution.",
"In the following, we begin our considerations of quasistates for nonclassical light in the Fourier picture and then proceed to prominent examples of quasiprobabilities and their dual representation via quasistates."
],
[
"Fourier representation",
"The characteristic function $\\Phi (\\beta )$ is the Fourier transform $F$ of a distribution, which is accessible via the kernel $F(\\alpha ,\\beta )=\\exp \\left(\\beta \\alpha ^\\ast -\\beta ^\\ast \\alpha \\right),$ i.e., in our notation, $P_{F}=\\Phi $ .",
"The inverse Fourier transform, defined via $F^{-1}(\\beta ,\\alpha )=F(\\alpha ,\\beta )^\\ast /\\pi ^2$ , results in the corresponding quasistates [Eq.",
"()], $\\hat{\\Delta }_{F}(\\beta )=\\int _{\\mathbb {C}}d\\beta \\,\\frac{\\exp \\left(\\beta ^\\ast \\alpha -\\beta \\alpha ^\\ast \\right)}{\\pi ^2}|\\alpha \\rangle \\langle \\alpha |.$ For the evaluation of the integral, it is useful to represent the coherent-state projectors in terms of normally ordered exponential of creation ($\\hat{a}^\\dag $ ) and annihilation ($\\hat{a}$ ) operators, which reads $|\\alpha \\rangle \\langle \\alpha |={:}\\exp (-[\\hat{a}-\\alpha ]^\\ast [\\hat{a}-\\alpha ]){:}$ [58].",
"This representation yields $\\hat{\\Delta }_{F}(\\beta )=\\frac{\\exp (-|\\beta |^2)}{\\pi }{:}\\hat{D}(-\\beta ){:},$ where $\\hat{D}(\\gamma )=\\exp (\\gamma \\hat{a}^\\dag -\\gamma ^\\ast \\hat{a})=\\exp (-|\\gamma |^2/2){:}\\hat{D}(\\gamma ){:}$ is the unitary displacement operator.",
"Therefore, the quasistates $\\hat{\\Delta }_{F}(\\beta )$ correspond (up to a rescaling) to unitary operators in the case that the convolution kernel is the Fourier transformation.",
"Note that instead of the Fourier transform, the Laplace transform can be advantageous in certain scenarios as well [59].",
"One application, which also connects to the previous section, is the state reconstruction via balanced homodyne detection.",
"For this purpose, let us recall that the characteristic function of the Wigner-Weyl distribution takes the form [58] $\\Phi (\\beta )e^{-|\\beta |^2/2}=\\langle \\hat{D}(\\beta )\\rangle =\\langle \\exp \\left(i|\\beta |\\hat{x}[\\pi /2-\\arg \\beta ]\\right)\\rangle ,$ where we used the quadrature operator $\\hat{x}[\\varphi ]=e^{i\\varphi }\\hat{a}+e^{-i\\varphi }\\hat{a}^\\dag $ .",
"This means for a balanced homodyne measurement, providing a data set $\\lbrace x_j[\\varphi _j]\\rbrace _{j=1,\\ldots ,N}$ of quadratures $x_j$ for the phase values $\\varphi _j$ , we can approximate $\\begin{aligned}\\hat{\\rho }=&\\int _{\\mathbb {C}}d\\beta \\,\\Phi (\\beta )\\hat{\\Delta }_{F}(\\beta )\\\\=&\\int _{-\\infty }^\\infty dr\\int _{-\\pi /2}^{\\pi /2} d\\varphi \\,\\frac{\\pi }{\\pi }e^{r^2/2}\\langle e^{ir\\hat{x}[\\varphi ]}\\rangle \\hat{\\Delta }_{F}(ire^{-i\\varphi })\\\\\\approx &\\int _{-\\infty }^\\infty dr\\, e^{r^2/2} \\frac{\\pi }{N}\\sum _{j=1}^N e^{irx_j[\\varphi _j]}\\hat{\\Delta }_{F}(ire^{-i\\varphi _j}),\\end{aligned}$ which corresponds to a sampling approach as a consequence of a quasistate representation [see specifically the first line of Eq.",
"(REF ) and the use of $\\hat{\\Delta }_{F}$ ].",
"In fact, further analyses show that this approach is indeed equivalent to established reconstruction methods bases on balanced homodyne detection and using pattern functions [56], [57], [60].",
"In addition, it is worth emphasizing that the phases $\\varphi $ have to be uniformly distributed in the interval $[-\\pi /2,\\pi /2]$ to ensure an optimal sampling [61].",
"Figure: (Color online)The QQ function [Eq.",
"()] of the quasistate [Eq.",
"()] for different parameters ss, pp, and qq.The top-left plot corresponds to a coherent (vacuum) state.The top-center plot shows the maximally singular quasistate .The top-right plot includes, in addition, squeezing.The bottom row depicts cases in which the QQ function becomes complex;the larger and corresponding smaller plots show the amplitude |Q||Q| and phase argQ\\arg Q, respectively.Note the difference in the phases when comparing the left and right plots."
],
[
"Generalized optical quasistates",
"Translational types of convolutions in the original space, e.g., $P_{\\kappa }(\\alpha ^{\\prime })=\\int _{\\mathbb {C}}d\\alpha \\,P(\\alpha )\\kappa (\\alpha ^{\\prime }-\\alpha )$ , take a product form in the Fourier domain.",
"In this scenario, we can consider the representation $\\hat{\\rho }=\\int _{\\mathbb {C}}d\\beta \\,\\left[\\Phi (\\beta )\\Omega (\\beta )\\right]\\left[\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}{:}\\hat{D}(-\\beta ){:}\\right],$ for a kernel $\\Omega $ .",
"The inverse Fourier transform gives the distribution $P_{F^{-1}\\Omega F}$ as well as the quasistates $\\hat{\\Delta }_{F^{-1}\\Omega F}(\\alpha )=\\int _{\\mathbb {C}}d\\beta \\,e^{\\beta \\alpha ^\\ast -\\beta ^\\ast \\alpha }\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}{:}\\hat{D}(-\\beta ){:}.$ Such operators have been extensively studied in connection to operator orderings and their resulting quasiprobabilities [23], [24], [27], [28], [29].",
"Here, however, let us analyze them based on their own merit as quasistates.",
"Of particular interest are the Gaussian functions $\\Omega (\\beta )=\\exp \\left(-\\frac{1-s}{2}|\\beta |^2-\\frac{p}{4}\\beta ^2-\\frac{q}{4}\\beta ^\\ast \\right),$ where $|p|=|q|$ [27], [28], [29].",
"In Table REF , specific examples and their connection to known quasiprobabilities are given.",
"Evaluating the integrals as demonstrated in Appendix , we get the quasistates in normal ordering as $\\hat{\\Delta }_{F^{-1}\\Omega F}(0)=\\frac{2\\,{:}\\exp \\left(\\frac{-2[1+s]\\hat{a}^\\dag \\hat{a} +q\\hat{a}^{\\dag 2}+p\\hat{a}^2}{(1+s)^2-pq}\\right){:}}{\\sqrt{(1+s)^2-pq}}.$ Note that because of the convolution property, we have $\\hat{\\Delta }_{F^{-1}\\Omega F}(\\alpha )=\\hat{D}(\\alpha )\\hat{\\Delta }_{F^{-1}\\Omega F}(0)\\hat{D}(\\alpha )^\\dag $ for the displaced quasistates.",
"In Fig.",
"REF , several examples of the phase-space $Q$ -function representation, $Q(\\alpha )=\\frac{\\langle \\alpha |\\hat{\\Delta }_{F^{-1}\\Omega F}(0)|\\alpha \\rangle }{\\pi },$ are shown for different parameters that define the quasistates in Eq.",
"(REF ).",
"It is also worth mentioning that the $Q$ function is the Husimi-Kano distribution.",
"Table: Parameters [cf.",
"Eq.",
"()] for prominent quasiprobability distributions frequently used in quantum optics.Interestingly, the considered class of quasistates, given in Eq.",
"(REF ), can be related to squeezed versions of thermal-like quasistates, $\\begin{aligned}\\hat{\\Delta }_{F^{-1}\\Omega F}(0)=&\\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)\\\\&\\times \\frac{1}{\\bar{n}+1}\\sum _{n\\in \\mathbb {N}}\\left(\\frac{\\bar{n}}{\\bar{n}+1}\\right)^n|n\\rangle \\langle n|\\\\&\\times \\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)^\\dag ,\\end{aligned}$ where $\\bar{n}\\in \\mathbb {C}$ generalizes the notion of a mean thermal photon number and $\\zeta $ is the squeezing parameter.",
"A full and exact analysis is provided in Appendix , and it provides the relations between the different parameter sets as $\\zeta =e^{i\\arg \\tau }\\mathrm {artanh}\\,|\\tau |\\quad \\text{and}\\quad \\bar{n}=\\frac{\\omega }{1-\\omega },$ with $\\omega =\\frac{r-1}{r+1},\\,|\\tau |^2=\\frac{s-r}{s+r},\\,\\text{and}\\,\\frac{\\tau }{\\tau ^\\ast }=\\frac{q}{p},$ where $r=\\sqrt{s^2-pq}$ .",
"Recall that we require $|q|=|p|$ .",
"Also note that the squeezing operation is unitary which implies that we have the eigenvalues $(1-\\omega )\\omega ^n$ , following a geometric distribution.",
"In the following, let us study the resulting quasistates in more detail."
],
[
"$s$ -Parameterized quasistates",
"Arguably the most frequently used choice of parameters is obtained for $p=q=0$ , likewise $\\zeta =\\tau =0$ .",
"This choice results in the $s$ -parametrized quasiprobability distributions $P_{F^{-1}\\Omega F}$ .",
"Here, we additionally find the quasistates $\\hat{\\Delta }_{F^{-1}\\Omega F}$ .",
"In particular, those quasistates have the eigenvalues $(1-\\omega )\\omega ^n=\\frac{\\bar{n}^{n}}{(\\bar{n}+1)^{n+1}}=\\frac{2}{1+s}\\left(-\\frac{1-s}{1+s}\\right)^n$ for $n\\in \\mathbb {N}$ ; see Eq.",
"(REF ).",
"For the parameters $-1<s<1$ , the eigenvalues (REF ) of the quasistates are negative for odd $n$ , certifying that these operators are only accessible in terms of our generalization of the notion of a physical state.",
"In the limit $s\\rightarrow 1$ , we get the eigenvalue 1 for $n=0$ and 0 for $n>0$ , defining the vacuum state.",
"Its $Q$ function is shown on the left in the top row of Fig.",
"REF , and the corresponding quasiprobability distribution is the original Glauber-Sudarshan distribution.",
"Furthermore, the case $s=0$ interestingly describes an operator with the maximally possible singularities a Glauber-Sudarshan distribution can exhibit [41]; see also the center plot in the top row of Fig.",
"REF for its $Q$ function.",
"There, we observe that the spread (i.e., variance) in phase space is reduced in all directions when compared to the vacuum state, further highlighting that it is a genuine quasistate that is incompatible with the uncertainty relation which holds true for any physical state.",
"The underlying phase-space distribution $P_{F^{-1}\\Omega F}$ is the Wigner-Weyl distribution.",
"Finally, the quasistates for the Husimi-Kano distribution are obtained in the limit $s\\rightarrow -1$ , and its corresponding $Q$ function is described by a singular Dirac distribution.",
"Beyond real-valued $s$ parameters, we can additionally consider complex values as well [58].",
"Then, the eigenvalues (REF ) are complex too.",
"The example $s=i$ is shown on the left of the bottom row in Fig.",
"REF .",
"While the amplitude $|Q|$ is identical with the vacuum state, the quasistate characteristics are clearly visible through the nontrivial phases, $\\arg Q\\ne 0$ ."
],
[
"Non-$s$ -parametrized quasistates",
"Moreover, we can also consider the quasistates for cases with $p,q\\ne 0$ .",
"As we demand $|q|=|p|$ , we have two extremal scenarios, $q=p^\\ast $ and $q=-p^\\ast $ .",
"In addition to the $s$ -parametrized quasiprobabilities in Table REF , we also listed the Agarwal-Wolf distributions.",
"They correspond to the case $q=-p^\\ast =\\mp 1$ (also, $s=0$ ).",
"The complex-valued $Q$ function of the resulting quasistate $\\hat{\\Delta }_{F^{-1}\\Omega F}$ is shown in the bottom-right plot of Fig.",
"REF .",
"The amplitude $|Q|$ is again compatible with the uncertainty principle bounded by the vacuum state, similarly to the previously discussed scenario of a complex $s$ parameter.",
"Here, however, the functional behavior of the phase $\\arg Q$ does not possess a radial symmetry.",
"Rather, the quasistates to the Agarwal-Wolf distributions exhibit hyperbolic isophase contours.",
"Note that this dismisses a sometimes held believe that Agarwal-Wolf distributions are an example of a $s$ -parametrized quasiprobability.",
"Finally, we can ask ourselves what happens in the case $q=p^\\ast $ .",
"An example is shown in the top-right plot in Fig.",
"REF .",
"Interestingly, the resulting quasistates are squeezed.",
"Specifically for a squeezing parameter $\\zeta $ , we can choose $s=\\frac{1+\\tanh ^2|\\zeta |}{1-\\tanh ^2|\\zeta |}\\text{ and }q=\\frac{-2e^{i\\arg \\zeta }\\tanh |\\zeta |}{1-\\tanh ^2|\\zeta |}=p^\\ast .$ This choice implies $\\bar{n}=0$ and means that the quasistates are pure squeezed states, cf.",
"Eqs.",
"(REF ) and (REF ).",
"Consequently, the quasiprobability density $P_{F^{-1}\\Omega F}(\\alpha )$ now describes the decomposition of the density operator in terms of displaced squeezed states with a coherent amplitude $\\alpha $ and a squeezing defined by $\\zeta $ .",
"This further implies that we have a generalized Glauber-Sudarshan quasiprobability distribution which is, however, a phase-space representation that expands the state in terms of squeezed states.",
"Thus, the case $q=p^\\ast $ significantly extends the notion of $s$ -parametrized distributions to additionally include arbitrary squeezed states."
],
[
"Quantum correlations",
"In a recent work [19], we generalized the concept of quasiprobability representations to more general notions of quantum coherence [62], beyond the specific example of harmonic oscillators studied in the previous section.",
"Among the various types of quantumness, quantum correlations between multiple degrees of freedom play an outstanding role for applications in quantum technologies [1], [63].",
"For this reason, let us study the entanglement of quantum systems within the framework of quasistates.",
"A comprehensive introduction to the theory of entanglement (likewise, inseparability) can be found in Ref.",
"[63]."
],
[
"Entanglement quasiprobabilities",
"A pure separable state in a multipartite system is described by a tensor-product vector, $|(a,b,\\ldots )\\rangle =|a\\rangle \\otimes |b\\rangle \\otimes \\cdots ,$ where $a\\in \\mathcal {S}_A$ , $b\\in \\mathcal {S}_B$ , etc.",
"In addition, the inclusion of classical correlations in terms of statistical mixtures, given by a classical probability density $P$ , yields the definition of a mixed separable state [64], $\\hat{\\rho }{=}\\int _{S_A{\\times } S_B{\\times }\\cdots } da\\,db\\cdots P(a,b,\\ldots )|(a,b,\\ldots )\\rangle \\langle (a,b,\\ldots )|.$ A state is defined to be inseparable (i.e., entangled) if it cannot be written in the form of Eq.",
"(REF ).",
"However, when we allow for $P$ to be a quasiprobability, even entangled states can be expanded in terms of separable states using Eq.",
"(REF ) [21].",
"A construction approach for bipartite entanglement quasiprobabilities was introduced in Ref.",
"[22].",
"As this approach can be generalized from the bipartite scenario to the multipartite case [19], we restrict ourselves to bipartite systems in the following.",
"The optimal decomposition of entangled states in terms of separable ones warrants that the entanglement is uniquely identified through negativities in $P(a,b)$ [22], [19].",
"The separable states $|(a,b)\\rangle $ needed for the decomposition are obtained from the solution of the so-called separability eigenvalue equations (SEEs), $\\hat{\\rho }_a|b\\rangle =g|b\\rangle \\text{ and }\\hat{\\rho }_b|a\\rangle =g|a\\rangle ,$ with $\\hat{\\rho }_a=\\mathrm {tr}_A[\\hat{\\rho }(|a\\rangle \\langle a|\\otimes \\hat{1}_B)]$ and $\\hat{\\rho }_b=\\mathrm {tr}_B[\\hat{\\rho }(\\hat{1}_A\\otimes |b\\rangle \\langle b|)]$ .",
"The SEEs approach generalizes the eigenvalue problem to composite systems while respecting the tensor-product structure of the corresponding eigenvectors and was initially introduced to construct entanglement witnesses [65], [66].",
"The solutions $(a,b)\\in \\mathcal {S}$ allow us to construct the distribution $P$ by solving the linear problem $G\\vec{p}=\\vec{g},$ where $G=[|\\langle (a,b)|(a^{\\prime },b^{\\prime })\\rangle |^2]_{(a,b),(a^{\\prime },b^{\\prime })\\in \\mathcal {S}}$ is the Gram-Schmidt matrix of separability eigenvectors.",
"Furthermore, the solution vector $\\vec{p}=[P(a,b)]_{(a,b)\\in \\mathcal {S}}$ defines the quasiprobabilities, and $\\vec{g}=[g]_{(a,b)\\in \\mathcal {S}}$ is the vector of separability eigenvalues, cf.",
"Eq.",
"(REF ).",
"Let us point out that the here-used Gram-Schmidt matrix also relates to the reconstruction approach in Sec.",
"[Eq.",
"(REF )] and is a result of a underlying principle of convex decomposition, discussed in greater detail in Ref.",
"[19].",
"One can always find a set $\\mathcal {S}_0$ such that $\\mathcal {S}\\subset \\mathcal {S}_0\\times \\mathcal {S}_0$ ; we then set $P(a,b)=0$ for all $(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0\\setminus \\mathcal {S}$ .",
"Using the construction to find optimal entanglement quasiprobabilities via the SEEs, we can decompose any state, be it entangled or separable [22], [19], as $\\hat{\\rho }=\\sum _{(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(a,b)|(a,b)\\rangle \\langle (a,b)|.$ It is worth mentioning that the entanglement quasiprobability is always real valued and normalized."
],
[
"Quasistate representation",
"A universally applicable approach to diminish the negativities in quasiprobabilities for a single system is a convolution of the form $P_K(\\chi )=rP(\\chi )+\\frac{1-r}{|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}P(\\psi ),$ with $0< r\\le 1$ and $|\\mathcal {S}_0|$ being the cardinality of the set; see Appendix for details.",
"The kernel under study consequently leads to quasistates of the form $\\hat{\\Delta }_K(\\chi )=\\frac{1}{r}|\\chi \\rangle \\langle \\chi |-\\frac{1-r}{r|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}|\\psi \\rangle \\langle \\psi |.$ We emphasize that quasistates defined in this manner are both Hermitian, $\\hat{\\Delta }_K(\\chi )=\\hat{\\Delta }_K(\\chi )^\\dag $ , and normalized, $\\mathrm {tr}[\\hat{\\Delta }_K(\\chi )]=1$ .",
"Thus, the only distinction to true states is the fact that $\\hat{\\Delta }_K(\\chi )$ is, in general, not a positive semidefinite operator.",
"The convolution can be generalized to composite systems via product kernels.",
"Choosing, for example, the same $r$ for both systems, we can reformulate Eq.",
"(REF ) as $\\hat{\\rho }=\\sum _{(\\tilde{a},\\tilde{b})\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P_{K\\otimes K}(\\tilde{a},\\tilde{b})\\,\\hat{\\Delta }_K(\\tilde{a})\\otimes \\hat{\\Delta }_K(\\tilde{b}).$ In particular, we have $\\begin{aligned}P_{K\\otimes K}(\\tilde{a},\\tilde{b})=&r^2P(\\tilde{a},\\tilde{b})+\\frac{(1-r)^2}{|\\mathcal {S}_0|^2}\\\\&+\\frac{r(1-r)}{|\\mathcal {S}_0|}\\left[P(\\tilde{a})+P(\\tilde{b})\\right],\\end{aligned}$ using the marginal distributions $P(\\tilde{a})=\\sum _{b\\in \\mathcal {S}_0}P(\\tilde{a},b)$ and $P(\\tilde{b})=\\sum _{a\\in \\mathcal {S}_0}P(a,\\tilde{b})$ .",
"Eventually, we get a nonnegative distribution $P_{K\\otimes K}$ for a sufficiently large $r$ , which is also demonstrated later.",
"This means that an entangled state can be decomposed according to Eq.",
"(REF ) in terms of a classical joint probability distribution $P_{K\\otimes K}(a,b)\\ge 0$ and tensor-product quasistates $\\hat{\\Delta }_K(a)\\otimes \\hat{\\Delta }_K(b)\\ngeq 0$ [cf.",
"Eq.",
"(REF )].",
"Since we can chose $r=0$ for separable states, we find that quasistates are not required in this case.",
"Therefore, we are able to conclude that entanglement is unambiguously identified in terms of classical joint probability distributions if and only if the tensor-product operators of the decomposition have to be unphysical quasistates."
],
[
"Example",
"As a proof of concept, let us consider an entangled two-qubit quantum state parametrized as $\\hat{\\rho }=\\frac{1}{4}\\left[\\hat{1}\\otimes \\hat{1}+\\sum _{j\\in \\lbrace x,y,z\\rbrace }\\rho (j)\\,\\hat{\\sigma }_j\\otimes \\hat{\\sigma }_j\\right].$ This corresponds to a physical density operator if and only if the real coefficients satisfy $[\\rho (x),\\rho (y),\\rho (z)]\\in \\mathrm {conv}\\lbrace [-1,1,1],[1,-1,1],[1,1,-1],[-1,-1,-1]\\rbrace $ , where the extremal points represent Bell states.",
"In Ref.",
"[19], we solved the SEEs [Eq.",
"(REF )] for this family of states.",
"In particular, the separability eigenvectors are tensor products of eigenstates of Pauli operators ($\\hat{\\sigma }_x$ , $\\hat{\\sigma }_y$ , and $\\hat{\\sigma }_z$ ).",
"This implies that we have $\\mathcal {S}_0=\\lbrace x_{+},x_{-},y_{+},y_{-},z_{+},z_{-}\\rbrace ,$ where $\\hat{\\sigma }_w|w_s\\rangle =s|w_{s}\\rangle $ for $w_s\\in \\mathcal {S}_0$ .",
"The exact entanglement quasiprobability reads [19] $P(a_s,b_t)=\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right],$ with $a,b\\in \\lbrace x,y,z\\rbrace $ and $s,t\\in \\lbrace +1,-1\\rbrace $ , identifying the separability eigenvectors, $(a_s, b_t)\\in \\mathcal {S}_0\\times \\mathcal {S}_0$ .",
"Of particular importance is the parameter $q=1-|\\rho (x)|-|\\rho (y)|-|\\rho (z)|$ as it determines if the state is separable; namely, $\\hat{\\rho }$ in Eq.",
"(REF ) is separable if and only if $q\\ge 0$ [19].",
"Conversely, the negativity of the quasiprobability in the case of entanglement is given by this quantity as well; it reads $\\min _{(u, v)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(u,v)=q/12\\ge -1/6$ , and the lower bound is attained for Bell states.",
"Figure: (Color online)The top panel depicts the entanglement quasiprobabilities [Eq.",
"()] for a Bell state with ρ(x)=ρ(y)=ρ(z)=-1\\rho (x)=\\rho (y)=\\rho (z)=-1.Each patch in the AA-BB plane corresponds to a product state |a〉〈a|⊗|b〉〈b||a\\rangle \\langle a|\\otimes |b\\rangle \\langle b|, with a,b∈𝒮 0 a,b\\in \\mathcal {S}_0 according to Eq.",
"().The bottom panel shows the eigenvalues of those separable-state projectors.The negativities in the quasiprobabilities PP identify the entanglement.Now we can apply the kernel in Eq.",
"(REF ).",
"Inserting the distribution in Eq.",
"(REF ) yields $\\begin{aligned}&P_{K\\otimes K}(a_s,b_t)\\\\= & r^2\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right]+\\frac{(1-r)^2}{36}\\\\&+\\frac{r(1-r)}{6}\\left[\\frac{q}{3}+\\frac{|\\rho (a)|}{2}+\\frac{|\\rho (b)|}{2}\\right].\\end{aligned}$ After some straightforward algebra, we find that $P_{K\\otimes K}$ is always nonnegative for $0<r\\le 1/\\sqrt{7}$ .",
"In addition, we get tensor-product quasistates which are formed by the subsystem components $\\hat{\\Delta }_{K}(j)=\\frac{1}{r}|j\\rangle \\langle j|-\\frac{1-r}{2r}\\hat{1}$ for $j\\in \\mathcal {S}_0$ (see Appendix ).",
"Consequently, the eigenvalues of $\\hat{\\Delta }_{K}(j)$ are $(1+r)/(2r)$ and $-(1-r)/(2r)$ .",
"Figure: (Color online)The top panel depicts the classical joint probabilities, obtained using the kernel KK [see Eq.",
"() for r=1/7r=1/\\sqrt{7}], for the same Bell state as shown in Fig.",
".The bottom panel shows the positive and negative eigenvalues of the quasistates in Eq.",
"() which are then required for the decomposition of the entangled state.Now the entanglement is captured by the unphysical nature of the quasistates.Let us consider a Bell state $\\hat{\\rho }=|\\psi \\rangle \\langle \\psi |$ as a specific example, where $|\\psi \\rangle =\\frac{|0\\rangle \\otimes |1\\rangle -|1\\rangle \\otimes |0\\rangle }{\\sqrt{2}},$ i.e., $\\rho (x)=\\rho (y)=\\rho (z)=-1$ .",
"The quasiprobability decomposition of this state is shown in Fig.",
"REF .",
"The negativities in the joint quasiprobability, $P\\ngeq 0$ , certify the entanglement and enable us to decompose the entangled state $\\hat{\\rho }$ using nonnegative tensor-product projectors, $|a\\rangle \\langle a|\\otimes |b\\rangle \\langle b|\\ge 0$ .",
"In contrast, in Fig.",
"REF , we show the result of applying the uncorrelated and nonnegative kernel $K\\otimes K$ for $r=1/\\sqrt{7}$ ; see Eq.",
"(REF ).",
"This results in a joint classical probability distribution.",
"However, the consequence is that the tensor-product operators for the decomposition have to be unphysical quasistates (because of the negative eigenvalues) in order to expand the entangled Bell state under study."
],
[
"Conclusions and discussion",
"Since the early theoretical descriptions of quantum states in phase space, quasiprobabilities have become one of the most important tools for studying quantum phenomena.",
"Here, we complemented this treatment by introducing the concept of quasistates.",
"While nonclassical states can be expanded in terms of a quasiprobability density and physical states, we demonstrated that one can equivalently use a classical, i.e., nonnegative, distribution and quasistates to perform such an expansion.",
"Then the quantum features of the state are carried over from the necessity of negativities in the distribution to the requirement that unphysical quasistates are needed to describe the state of a system.",
"Furthermore, we elaborated a number of different aspects and applications of quasistate, rendering them a useful tool to characterize quantum systems.",
"In simple terms, quasistates almost describe a density operator—only certain properties of a physical state do not apply.",
"In particular, a physical density operator is a Hermitian and positive semidefinite operator with a unit trace.",
"Throughout this work, we discussed several examples for which one (or multiple) of the defining properties of a density operator are violated by a quasistate.",
"For instance, eigenvalues can be negative, violating the positive semidefiniteness, or a Fourier representation was shown to lead to non-Hermitian quasistates.",
"As one practical implementation, we studied the reconstruction of the density operator.",
"This was based on the findings of a recent work [11], where Born's rule was reinterpreted in terms of contravariant operator-valued measures.",
"For example, this dual concept is useful when the positive operator-valued measure describes imperfect measurement outcomes.",
"Here, we proved that the contravariant operator-valued measure directly relates to the concept of quasistates and, therefore, can be considered as a special case of our general notion.",
"Consequently, we were able to apply quasistates for a direct quantum state reconstruction in terms of measured probabilities, such as exemplified for a two-level qubit state.",
"The duality between quasiprobability densities and their corresponding quasistates was studied for prominent quantum-optical phase-space distributions.",
"For instance, we found that the quasistates of the Wigner-Weyl distribution coincide with an operator that describes the maximal singularities the Glauber-Sudarshan distribution can have [41].",
"Beyond known quasiprobabilities in quantum optics, we further showed that our approach also leads to generalized phase-space distributions which expand the state in terms of squeezed states rather than coherent ones.",
"This, for example, can be useful to characterize non-Gaussianity, which is required for universal quantum computation and, in contrast to nonclassicality in terms of the Glauber-Sudarshan distribution, not only relies on coherent states but general squeezed states [67].",
"Furthermore, we characterized quantum correlations using quasistates.",
"In particular, we showed that entanglement can be identified with a classical joint probability distribution and tensor-product quasistates.",
"This result is surprising when considering that a violation of local-hidden-variable models typically excludes such classical distributions because states are (obviously) implicitly assumed to be physical in such models.",
"Here, we proved the necessary and sufficient condition that a density operator is separable, i.e., not entangled, if and only if both the distribution and the used states are classical; conversely, entanglement requires that at least one of them is nonclassical.",
"As an example, we identified the entanglement of a Bell state using either optimal quasiprobabilities [22] and tensor-product states or classical probabilities and unphysical quasistates.",
"Based on the general construction of quasiprobabilities for other notions of quantum coherences [19], the found results can be straightforwardly generalized to other forms of quantumness.",
"In conclusion, we developed the versatile and useful framework of quasistates for decomposing density operators.",
"Previously established concepts and methods have been demonstrated to be equivalently accessible with our technique which additionally allowed us to go beyond this state of the art.",
"Thus, we believe that, analogously to quasiprobabilities, the notion of quasistates has the potential to significantly contribute to the description and reconstruction of nonclassical quantum states in theory and experiment.",
"This work has received funding from the European Union's Horizon 2020 Research and Innovation Program under Grant Agreement No.",
"665148 (QCUMbER)."
],
[
"Exponential functions of bosonic operators",
"For our treatment of quantum light in Sec.",
", some additional algebra is required.",
"In this appendix, we formulate the needed relations."
],
[
"General relations",
"We frequently apply the Gaussian integral formula $\\begin{aligned}&\\int _{\\mathbb {C}}d\\xi \\,\\exp \\left(-z|\\xi |^2-x\\xi ^2-y\\xi ^{\\ast 2}+u\\xi +v\\xi ^\\ast \\right)\\\\=&\\frac{\\pi }{\\sqrt{z^2-4xy}}\\exp \\left(\\frac{zuv-yu^2-xv^2}{z^2-4xy}\\right),\\end{aligned}$ where $\\mathrm {Re}[z\\pm (x+y)]>0$ and $\\mathrm {Re}[z^2-4xy]>0$ .",
"We use the branch with a nonnegative real part for the square root of a complex number.",
"The above relation can be extended to integrals of normally ordered expressions because under this prescription, we have an algebra of commuting operators.",
"In the case that operator ordering becomes relevant, let us formulate additional relations.",
"From the observations that $\\hat{a}^mf(\\hat{n})|n\\rangle =f(\\hat{n}+m)\\hat{a}^m|n\\rangle $ and $f(\\hat{n})\\hat{a}^{\\dag m}|n\\rangle =\\hat{a}^{\\dag m}f(\\hat{n}+m)|n\\rangle $ hold true, we conclude that $e^{x\\hat{a}^2}\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}e^{\\omega ^2x\\hat{a}^2}\\text{ and }\\omega ^{\\hat{n}}e^{y\\hat{a}^{\\dag 2}}=e^{\\omega ^2y\\hat{a}^{\\dag 2}}\\omega ^{\\hat{n}}.$ In the same manner, we get $\\exp (u\\hat{a})\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}\\exp (\\omega u\\hat{a})$ and $\\omega ^{\\hat{n}}\\exp (v\\hat{a}^\\dag )=\\exp (\\omega v\\hat{a}^\\dag )\\omega ^{\\hat{n}}$ .",
"Furthermore, using the above integral, $|\\alpha \\rangle \\langle \\alpha |={:}\\exp (-[\\hat{a}-\\alpha ]^\\dag [\\hat{a}-\\alpha ]){:}$ , and $\\pi \\hat{1}=\\int _{\\mathbb {C}}d\\alpha \\,|\\alpha \\rangle \\langle \\alpha |$ , we obtain $\\begin{aligned}&e^{x\\hat{a}^2}e^{y\\hat{a}^{\\dag 2}}=\\exp [x\\hat{a}^2]\\hat{1}\\exp [y\\hat{a}^{\\dag 2}]\\\\=&\\frac{1}{\\sqrt{1-4xy}}{:}\\exp \\left(\\frac{4xy\\hat{a}^\\dag \\hat{a}+x\\hat{a}^2+y\\hat{a}^{\\dag 2}}{1-4uv}\\right){:}\\\\=&e^{v\\hat{a}^{\\dag 2}/(1-4uv)}\\left(\\frac{1}{1-4uv}\\right)^{\\hat{n}+1/2}e^{u\\hat{a}^2/(1-4uv)}.\\end{aligned}$ The same approach yields $\\exp [u\\hat{a}]\\exp [y\\hat{a}^{\\dag 2}]=\\exp [y(\\hat{a}^\\dag +u)^2]\\exp [u\\hat{a}]$ and $\\exp [x\\hat{a}^2]\\exp [v\\hat{a}^\\dag ]=\\exp [v\\hat{a}^\\dag ]\\exp [x(\\hat{a}+v)^2]$ .",
"Finally, let us also mention the well-known formula $\\exp (u\\hat{a})\\exp (v\\hat{a}^\\dag )=\\exp (uv)\\exp (v\\hat{a}^\\dag )\\exp (u\\hat{a})$ ."
],
[
"Spectral decomposition of Gaussian quasistates",
"In the main part of this contribution, we consider filter functions of a Gaussian form [cf.",
"Eq.",
"(REF )].",
"The exact analysis for the resulting quasistates is performed here.",
"For zero displacement, the integral in Eq.",
"(REF ) yields $\\begin{aligned}\\hat{\\Delta }=&\\int _{\\mathbb {C}}d\\beta \\,\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}e^{-\\beta \\hat{a}^\\dag }e^{\\beta ^\\ast \\hat{a}}\\\\=&\\frac{2\\,{:}\\exp \\left(\\frac{-2[1+s]\\hat{a}^\\dag \\hat{a} +q\\hat{a}^{\\dag 2}+p\\hat{a}^2}{(1+s)^2-pq}\\right){:}}{\\sqrt{(1+s)^2-pq}}.\\end{aligned}$ To characterize this quasistate, let us consider its decomposition in terms of exponential operators.",
"First, the squeezing operator $\\hat{S}=\\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)$ can be equivalently given as [58] $\\hat{S}=\\exp \\left(-\\frac{\\tau }{2}\\hat{a}^{\\dag 2}\\right)\\sqrt{1-|\\tau |^2}^{\\hat{n}+1/2}\\exp \\left(\\frac{\\tau ^\\ast }{2}\\hat{a}^2\\right),$ where $\\tau =e^{i\\arg \\zeta }\\tanh |\\zeta |$ .",
"Note that $|\\tau |<1$ , and we have infinite squeezing for $|\\tau |\\rightarrow 1$ .",
"In addition, we consider thermal-state-like operator $\\begin{aligned}\\hat{W}=&(1-\\omega )\\omega ^{\\hat{n}}=\\sum _{n\\in \\mathbb {N}}(1-\\omega )\\omega ^{n}|n\\rangle \\langle n|\\\\=&(1-\\omega ){:}e^{-(1-\\omega )\\hat{n}}{:}\\\\=&\\int _{\\mathbb {C}}d\\alpha \\,\\frac{1-\\omega }{\\pi \\omega }\\exp \\left(-\\frac{(1-\\omega )|\\alpha |^2}{\\omega }\\right)|\\alpha \\rangle \\langle \\alpha |.\\end{aligned}$ See Ref.",
"[68] for a derivation and additional considerations.",
"Also note that the vacuum state $\\hat{W}=|0\\rangle \\langle 0|$ is obtained in the limit $\\omega \\rightarrow 0$ .",
"Now, we defined an operator $\\hat{T}$ to be compared with the quasistate $\\hat{\\Delta }$ , $\\hat{T}=\\hat{S}\\hat{W}\\hat{S}^\\dag .$ Using the exchange relations in Eqs.",
"(REF ) and (REF ), we find after some algebra a normally ordered expression, $\\begin{aligned}\\hat{T}=&(1-\\omega )\\sqrt{\\frac{1-|\\tau |^2}{1-|\\tau |^2\\omega ^2}}\\\\&\\times \\exp \\left(-\\frac{\\tau }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^{\\dag 2}\\right)\\\\&\\times {:}\\exp \\left(-\\frac{[1-\\omega ][1+\\omega |\\tau |^2]}{1-|\\tau |^2\\omega ^2}\\hat{n}\\right){:}\\\\&\\times \\exp \\left(-\\frac{\\tau ^\\ast }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^2\\right).\\end{aligned}$ Finally, we can compare this expression with Eq.",
"(REF ), $\\hat{\\Delta }=\\hat{T}$ .",
"Namely, equating coefficient yields $\\omega =\\frac{r-1}{r+1},\\,|\\tau |^2=\\frac{s-r}{s+r},\\,\\text{and}\\,\\frac{\\tau }{\\tau ^\\ast }=\\frac{q}{p},$ where $r=\\sqrt{s^2-pq}$ .",
"This also implies $\\tau =-q/(s+r)$ and $\\tau ^\\ast =-p/(s+r)$ .",
"Conversely, we can deduce parameters $s$ , $p$ , and $q$ from given values of $\\omega $ and $\\tau $ , $\\begin{aligned}p=\\frac{-2(1+\\omega )\\tau ^\\ast }{(1-\\omega )(1-|\\tau |^2)},\\quad q=\\frac{-2(1+\\omega )\\tau }{(1-\\omega )(1-|\\tau |^2)},\\\\\\text{and}\\quad s+1=\\frac{2(1+\\omega |\\tau |^2)}{(1-\\omega )(1-|\\tau |^2)}.\\end{aligned}$"
],
[
"Uniform attenuation",
"In the main text, we consider a specific example of a kernel.",
"For studying its properties, let us consider the corresponding convolution $P_K(\\chi )=\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )K(\\psi ,\\chi )=a\\,P(\\chi )+b\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )$ for a set $\\mathcal {S}$ of states.",
"Note that $\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )=1$ is the normalization condition.",
"To ensure that the result is normalized as well, $1=\\sum _{\\chi \\in \\mathcal {S}}P_K(\\chi )=a+b|\\mathcal {S}|$ , we find $b=\\frac{1-a}{|\\mathcal {S}|},$ using the cardinality $|\\mathcal {S}|$ of the set of states.",
"Further, when $0\\le a\\le 1$ is satisfied, the non-negativity of $P_K$ is guaranteed for any nonnegative $P$ .",
"The considered convolution can be formulated via the $|\\mathcal {S}|\\times |\\mathcal {S}|$ map $K=a\\,\\mathrm {id}+b\\,nn^\\mathrm {T},$ using the identity $\\mathrm {id}$ and the $|\\mathcal {S}|$ -dimensional vector $n=[1,\\ldots ,1]^\\mathrm {T}$ .",
"Then the inverse takes a similar form, $K^{-1}=\\frac{1}{a}\\mathrm {id}-\\frac{b/a}{a+b|\\mathcal {S}|}nn^\\mathrm {T},$ where $a+b|\\mathcal {S}|=1$ for the given $b$ .",
"Using the normalization, the convolution kernel $K$ corresponds to uniform addition of noise to the probabilities.",
"On the level of the operators, we get $\\hat{\\Delta }_K(\\chi )=\\frac{1}{a}|\\chi \\rangle \\langle \\chi |-\\frac{1-a}{a|\\mathcal {S}|}\\sum _{\\psi \\in \\mathcal {S}}|\\psi \\rangle \\langle \\psi |.$ In the continuous case, we can generalize the above relations as $P_K(\\chi )=\\int _{\\mathcal {S}} d\\psi \\,P(\\psi )\\left[a\\delta (\\psi ,\\chi )+\\frac{1-a}{|\\mathcal {S}|}\\right],$ where the volume $|\\mathcal {S}|=\\int _{\\mathcal {S}}d\\psi $ is used instead.",
"Furthermore, the projection operators transform as $\\hat{\\Delta }_{K}(\\chi )=\\int _{\\mathcal {S}} d\\psi \\left[\\frac{1}{a}\\delta (\\psi ,\\chi )-\\frac{1-a}{a|\\mathcal {S}|}\\right]|\\psi \\rangle \\langle \\psi |.$ As a discrete-variable example, we study a qubit system with orthogonal basis $\\lbrace |0\\rangle ,|1\\rangle \\rbrace $ .",
"We can identify the Pauli matrices as $\\hat{\\sigma }_z=|1\\rangle \\langle 1|-|0\\rangle \\langle 0|$ , $\\hat{\\sigma }_x=|0\\rangle \\langle 1|+|1\\rangle \\langle 0|$ , and $\\hat{\\sigma }_y=i|0\\rangle \\langle 1|-i|1\\rangle \\langle 0|$ .",
"The Hermitian operator basis can be completed with the operator $\\hat{1}=|0\\rangle \\langle 0|+|1\\rangle \\langle 1|$ .",
"The eigenvectors $|j_{\\pm 1}\\rangle $ of the Pauli matrices $\\hat{\\sigma }_j$ to the eigenvalues $\\pm 1$ for $j\\in \\lbrace z,x,y\\rbrace $ form the set $\\mathcal {S}$ .",
"Then we can decompose any density operator as $\\hat{\\rho }=\\sum _{j\\in \\lbrace x,y,z\\rbrace ,s\\in \\lbrace +1,-1\\rbrace }P(j_s)|j_{s}\\rangle \\langle j_{s}|.$ Conversely, the attenuated distribution reads $P_{K}(j_s)=aP(j_s)+(1-a)/6$ , and the transformed operators take the form $\\hat{\\Delta }_K(j_s)=a^{-1}|j_s\\rangle \\langle j_s|-(1-a)(2a)^{-1}\\hat{1}$ ."
]
] | 1808.08471 |
[
[
"Invariant Sets in Quasiperiodically Forced Dynamical Systems"
],
[
"Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.",
"We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps.",
"The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set.",
"We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction.",
"We provide a characterization of invariant sets in the quasiperiodically forced systems.",
"A theoretical result on uniform boundedness of the invariant sets is presented.",
"The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages.",
"Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability.",
"We show that our theoretical results can be used to understand stability regions in such complex systems."
],
[
"Introduction",
"Methods of ergodic theory [1], [2] have been applied for solving a number of problems in science and technology such as control of mixing of fluid flows [3], analysis of traffic jams [4] and of quantum many-body models [5], performance evaluation of ecosystem models [6].",
"A visualization method for invariant sets of discrete-time dynamical systems (or maps) based on ergodic theory was developed in [7].",
"This method is based on the ergodic partition or ergodic decomposition [8] of state space in a given dynamical system, which is associated with the eigenspace at eigenvalue 1 of the Koopman operator [9].",
"This method partitions a (compact and metric) state space $X$ using joint level sets of time-averages of a basis of functions defined on $X$ .",
"This leads to an approximation of invariant sets on which the dynamics of the system are ergodic.",
"In an ergodic invariant set, almost all points are accessible in the sense that the initial conditions in this set thoroughly sample the set.",
"The method enables the visualization of low-dimensional (in practice usually two-dimensional) slices through high-dimensional invariant structures and offers a computationally tractable method for analyzing global structures of state space in discrete-time nonlinear models.",
"The numerical methods associated with the ergodic partition theory have been developed: see [9] and references therein.",
"The purpose of this paper is to extend the ergodic partition theory and exploit it to provide global analysis of state space in quasiperiodically forced dynamical systems.",
"Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see, e.g., [10], [11].",
"Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [12], analysis of dynamics described by the Schrödinger equations [13] and of quantum chaos [14], and dynamical systems analysis with applications to fluid mixing [15], [16].",
"In the present paper, we study the global structure of state space in quasiperiodically forced systems from the viewpoint of ergodic partition.",
"First, we develop a theory of ergodic partition of state space in measure-preserving and dissipative, continuous-time dynamical systems or flows.",
"This is a natural extension of the existing theory for measure-preserving maps in [7].",
"The ergodic partition is again based on eigenspace at eigenvalue 0 of the associated Koopman group.",
"Related analyses of time-periodic flows via spectral properties of linear operators are reported in [17], [18].",
"Since an arbitrary quasiperiodically forced system with a smooth invariant measure is transformed into an autonomous system determining a measure-preserving flow (see Section ), the ergodic partition theory is used for visualization of invariant sets for the quasiperiodically forced system.",
"Next, we provide a characterization of invariant sets in quasiperiodically forced systems.",
"In general, understanding invariant structures in the quasiperiodically forced systems is not easy because the associated portraits of state space change with time in an aperiodic manner.",
"Here, we introduce a notion of uniform boundedness of invariant sets.",
"A theoretical result on uniform boundedness of invariant sets is presented in this paper: see Theorem REF , and Corollaries REF and REF .",
"This clarifies a dynamical feature of invariant sets in the quasiperiodically forced systems that is directly determined by the ergodic partition and its associated visualization technique.",
"Also, we apply the theory developed above to analysis of a nonlinear model of complex power grids that represents the short-term swing instability, which we name the Coherent Swing Instability (CSI) [19].",
"Preliminary results in this paper were published in the conference proceedings [20], [21].",
"This paper contains detailed discussion of the theory with a new proof and a set of new numerical simulations.",
"The rest of this paper is organized as follows.",
"In Section we introduce set-up and mathematical preliminaries from dynamical systems theory.",
"A theory of ergodic partition in measure-preserving flows is developed in Section .",
"Based on the result, the basin of attraction in the dissipative case is explored in ().",
"In Section we provide a new characterization of invariant sets in the quasiperiodically forced dynamical systems.",
"The developed theory is applied in Section to two simple examples of the quasiperiodically forced system and to a nonlinear model of the CSI phenomenon of a power grid.",
"Conclusions of this paper are presented in Section ."
],
[
"Quasiperiodically Forced Dynamical Systems",
"Throughout this paper, we address the following quasiperiodically forced dynamical system that evolves on a finite-dimensional metric space $M$ : for ${m}\\in {M}$ and $t\\in \\mathbb {R}$ , $\\frac{{d}{m}}{{d} t}={g}({m},t).$ The function $g$ is assumed to be quasiperiodic on $t$ in the sense of Moser [22], that is, ${g}({m},t)={G}({m},\\mathit {\\Omega }_1t,\\mathit {\\Omega }_2t,\\ldots ,\\mathit {\\Omega }_Nt),$ where ${G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)$ is assumed to be a smooth vector-valued function and of period $2\\pi $ in $\\theta _1,\\theta _2,\\ldots ,\\theta _N$ .",
"The real numbers $\\mathit {\\Omega }_1,\\mathit {\\Omega }_2,\\ldots ,\\mathit {\\Omega }_N$ are the $N$ basic angular frequencies and assumed to be rationally independent: there exists no $N$ -dimensional integer vector $(k_1,k_2,\\ldots ,k_N)^\\top $ ($\\top $ stands for the transpose operation of real-valued vectors) in which the entries do not all vanish, satisfying the resonance relation: $\\sum ^{N}_{i=1}k_i\\mathit {\\Omega }_i=0.$ The system (REF ) is non-autonomous and transformed into an autonomous system defined on the augmented state space $X:=M\\times \\mathbb {T}^N$ , in the same manner as in [16].",
"By introducing the $N$ variables $\\theta _i=\\mathit {\\Omega }_it\\in \\mathbb {T}$ ($i=1,2,\\ldots ,N$ ), we have $\\frac{d{m}}{dt}={G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N), \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N.$ In this way, by taking trajectories of the augmented system (REF ) on the augmented state variable ${x}:=({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)\\in X$ , a flow, that is, one-parameter group of diffeomorphisms is defined for the quasiperiodically forced system (REF )."
],
[
"Measure-Preserving Flows and Partition of State Space",
"In Section , as one type of flows induced by (REF ), we will investigate a measure-preserving flow defined on a class of probability spaces, where ergodic theory has been developed [8], [23].",
"The probability space considered here corresponds to the tuple $({X},\\mathfrak {B}_{X},\\mu )$ , where ${X}$ is a compact metric space, $\\mathfrak {B}_{X}$ is the Borel $\\sigma $ -algebra of ${X}$ , and $\\mu $ is a probability measure.",
"A measure-preserving flow of the probability space is a one-parameter group of measure-preserving diffeomorphisms ${S}^t: {X}\\rightarrow {X}$ , $t\\in \\mathbb {R}$ such that ${S}^t$ is measure-preserving (for all $A\\in \\mathfrak {B}_{X}$ we have $\\mu ({S}^{-t}(A))=\\mu (A)$ ), ${S}^0$ the identity, and ${S}^{t_1+t_2}={S}^{t_1}\\circ {S}^{t_2}$ for $t_1,t_2\\in \\mathbb {R}$ .",
"The important notion which we utilize throughout this paper is the partition of state space in dynamical systems.",
"One motivation behind the following definitions is that we aim to locate such a partition that plays a role in the data-driven estimation of certain statistical properties of non-ergodic measure-preserving dynamical systems; we will show this later as the ergodic partition.",
"The definitions of partition and measurable partition from [7] are the following.",
"Definition 1 A partition of ${X}$ is a family $\\zeta $ of sets satisfying $A,B\\in \\zeta \\quad \\Rightarrow \\quad \\mu (A\\cap B)=0, \\quad \\mu \\left({X}\\setminus \\bigcup _{A\\in \\zeta }A\\right)=0.$ The element of $\\zeta $ containing a point ${x}\\in {X}$ is denoted by $\\zeta ({x})$ .",
"A partition $\\zeta $ of ${X}$ is said to be measurable if there exists a countable family $\\mathfrak {D}$ of measurable sets $\\lbrace D_i\\rbrace $ such that every $D_i$ is a union of elements of $\\zeta $ , and for any pair $A_1,A_2$ of elements of $\\zeta $ there exists $D_j\\in \\mathfrak {D}$ such that $A_1\\subset D_j$ and $A_2\\subset D_j^{\\rm c}$ , where $D_j^{\\rm c}$ stands for the complement of $D_j$ in ${X}$ .",
"The family $\\mathfrak {D}$ is called a basis for $\\zeta $ .",
"In Section we use a product operation on the set of partitions of ${X}$ .",
"The product operation is defined in [23] as follows.",
"Definition 2 If $\\zeta _n$ , $n=1,\\ldots ,N$ are partitions, we then define their product $\\vee ^{N}_{n=1}\\zeta _n$ as the partition whose elements are the sets of the form $\\cap ^{N}_{i=1}{A_i}$ , for $A_i\\in \\zeta _i$ , $i=1,\\ldots ,N$ , satisfying $\\mu \\left(\\cap ^{N}_{i=1}A_i\\right)\\ne 0$ .",
"For a countable sequence of partitions, the notation $\\vee ^{\\infty }_{n=1}\\zeta _n$ will be used for the $\\sigma $ -algebra generated by $\\cup ^\\infty _{n=1}\\zeta _n$ ."
],
[
"Koopman Group",
"Consider a flow ${S}^t: {X}\\rightarrow {X}$ ($t\\in \\mathbb {R}$ ) on a finite-dimensional space ${X}$ and denote by $\\mathcal {F}$ a space of scalar-valued functions defined on ${X}$ : $\\mathcal {F}\\ni f: {X}\\rightarrow \\mathbb {C}$ , which we call the space of observables.",
"In the following, we suppose that the existence and uniqueness of solutions associated with the flow hold for all $t$ .",
"Then, we define the one-parameter group of linear operators $U^t$ ($t\\in \\mathbb {R}$ ) as a group of composition operators with ${S}^t$ : for $f\\in \\mathcal {F}$ , $(U^tf)({x}):=f({S}^t({x}))=(f\\circ {S}^t)({x}).$ where it is known as the Koopman group [24], [9].",
"For spaces as $\\mathcal {F}=\\mathcal {C}^0(X)$ and $\\mathcal {L}^p(X)$ , the Koopman group is strongly continuous.",
"Although the original flow ${S}^t$ is possibly described by a nonlinear differential equation and evolves on the finite-dimensional space ${X}$ , the operators $U^t$ are linear but evolve on the infinite-dimensional space $\\mathcal {F}$ .",
"The eigenvalue $\\lambda \\in \\mathbb {C}$ and eigenfunction $\\phi _\\lambda \\in \\mathcal {F}\\setminus \\lbrace 0\\rbrace $ of the Koopman group are defined as follows: $(U^t\\phi _\\lambda )({x})=\\exp (\\lambda t)\\phi _\\lambda ({x}).$ The notion of the Koopman group and its spectral characterization will be used in the following sections."
],
[
"Ergodic Partition in Measure-Preserving Flows",
"For completeness, in this section we extend the theory of ergodic partition in [7] to the measure-preserving flow ${S}^t$ on the probability space $(X,\\mathfrak {B}_{X},\\mu )$ , which is introduced in Section REF .",
"This section consists of a few lemmas and a theorem.",
"Their proofs are almost identical to the proofs for maps [7] and appear in Appendices and .",
"We denote by $\\mathcal {L}^1_\\mu ({X})$ the space of all $\\mu $ -integrable functions on ${X}$ and by $\\mathcal {C}({X})$ the space of all real-valued continuous functions on ${X}$ endowed with the sup norm.",
"Roughly speaking, an ergodic partition is a partition of the state space ${X}$ into invariant sets on which the dynamics are ergodic.",
"The precise definition of ergodic partition is the following.",
"Definition 3 A measurable partition $\\zeta $ of ${X}$ is said to be ergodic under the flow ${S}^t$ if for any element $A$ of $\\zeta $ , (i) $A$ is invariant under ${S}^t$ , and (ii) there exists an invariant probability measure $\\mu _A$ on $A$ such that the restriction of ${S}^t$ to $A$ , denoted by ${S}^t|_A$ , is an ergodic diffeomorphism on $A$ , and for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , $\\int _{X} f{d}\\mu =\\int _{X}\\left[\\int _{A=:\\zeta ({x})} f|_A{d}\\mu _A\\right]{d}\\mu ({x}),$ where $f|_A$ stands for the restriction of $f$ to the ergodic element $A$ , and $\\mu ({x})$ is again a probability measure on $X$ .",
"It is remarked that the ergodic partition is defined for the entire state space $X$ with $\\sigma $ -algebra, but it can be relaxed in context of the $\\sigma $ -algebra of invariant sets which can be parts of the state space.",
"Here, we denote by $f^\\ast $ the time-average of $f$ under the flow ${S}^t$ if the right-hand side of $f^\\ast ({x}):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t({x})){d} t,$ exists for almost every (a.e.)",
"point ${x}\\in {X}$ with respect to the measure $\\mu $ .",
"Birkhoff's Ergodic Theorem [25], [1] shows that for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , (i) $f^\\ast ({x})$ exists for a.e.",
"point ${x}\\in {X}$ (with respect to $\\mu $ ); (ii) $f^\\ast ({S}^t({x}))=f^\\ast ({x})$ for a.e.",
"point ${x}\\in {X}$ ; and (iii) $\\int _{X}f{d}\\mu =\\int _{X}f^\\ast {d}\\mu $ .",
"Here, we let $\\mathit {\\Sigma }$ be the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for all $f\\in \\mathcal {C}({X})$ , and $\\mathit {\\Sigma }(f)$ the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for a particular $f\\in \\mathcal {C}({X})$ .",
"The following lemma is standard (see page 129 of [23] for discrete-time dynamical systems (maps): it is obvious for flows).",
"Lemma 4 For a countable and dense set $S$ in $\\mathcal {C}({X})$ , $\\mathit {\\Sigma }=\\bigcap _{f\\in S}\\mathit {\\Sigma }(f).$ Now, we have a set $\\mathit {\\Sigma }$ such that the time-averages of all continuous functions on ${X}$ exist.",
"Then, the complimentary set $\\mathit {\\Sigma }^{\\rm c}$ is of measure zero.",
"This is because according to Birkhoff's Ergodic Theorem (i), each $\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is of measure zero, and thus $\\mathit {\\Sigma }^{\\rm c}:=\\cup _{f\\in S}\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is a countable union of sets with zero measure, which is again of measure zero.",
"The next lemma shows that the time-average of a continuous function induces a measurable partition on ${X}$ .",
"Lemma 5 Let $f$ be a continuous function on ${X}$ .",
"The family of level sets of $f^\\ast $ , $A_\\alpha :=\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace ,~~~\\alpha \\in \\mathbb {R},$ is a measurable partition of ${X}$ .",
"We denote by $\\zeta _f$ this partition and call it the partition induced by the function $f$ .",
"See Appedix .",
"Now, we can prove the first theorem stating that $\\zeta _f$ induces the ergodic partition of ${X}$ , which holds for general measure-preserving flows including the augmented system (REF ) for the quasiperiodically forced system (REF ).",
"Theorem 6 Consider the measure-preserving flow ${S}^t: X\\rightarrow X$ ($t\\in \\mathbb {R}$ ) associated with $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0} ={F}({x}) \\quad \\textrm {for each}~{x}\\in X$ where ${F}: X\\rightarrow \\mathrm {T}X$ (tangent bundle of $X$ ) is a nonlinear vector field.",
"Let $\\zeta {e}$ be the product of measurable partitions of ${X}$ induced by every $f\\in S$ : $\\zeta {e}=\\bigvee _{f\\in S}\\zeta _f.$ Then, $\\zeta {e}$ is the ergodic partition of ${X}$ .",
"See Appendix .",
"By combining the above theorem and lemma 20 in [26], we have the following corollary based on a finite number of basis functions.",
"Corollary 7 Assume there exists a complete system of functions $\\lbrace f_i\\rbrace $ , $f_i\\in \\mathcal {C}(X)$ , $i\\in \\mathbb {N}_{>0}$ (set of all natural numbers except for 0) i.e.",
"finite linear combinations of $f_i$ are dense in $\\mathcal {C}(X)$ .",
"The ergodic partition $\\zeta {e}$ is $\\zeta {e}=\\bigvee _{i\\in \\mathbb {N}_{>0}}\\zeta _{f_i}.$ For a given function $f$ , it is obvious that each level set $A_\\alpha =\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace $ ($\\alpha \\in \\mathbb {R}$ ) as an element of $\\zeta _f$ is invariant under ${S}^t$ .",
"In [7] the authors proposed to use the level sets $A_\\alpha $ , which are directly computed with the time-averaging of $f$ under ${S}^t$ , for identifying and visualizing invariant sets in discrete-time dynamical systems possessing a smooth invariant measure.",
"Theorem REF implies that the same computation can be used for visualization of invariant sets for the measure-preserving flow, namely, continuous-time dynamical systems with a smooth invariant measure.",
"Since, as shown in Section , the quasiperiodically forced system (REF ) defined on the state space $M$ is transformed to the flow defined on the augmented state space $X=M\\times \\mathbb {T}^N$ , the ergodic partition theory is applicable to visualization of invariant sets in $X$ for the original quasiperiodically forced system (REF ) possessing a smooth invariant measure.",
"However, in the quasiperiodically forced system (REF ) we are interested in dynamics on $M$ not on $X=M\\times \\mathbb {T}^N$ .",
"While for $N=1$ it is clear that every intersection of the invariant set in $M\\times \\mathbb {T}^1$ with $M$ is an invariant set of the associated Poincaré map.",
"The relationship in the quasiperiodically forced system (namely, $N\\ge 2$ ) is much less clear.",
"We make it precise in Section .",
"Here, let us introduce the connection of time-average $f^\\ast $ and the Koopman group.",
"It is obvious that for any $f\\in \\mathcal {L}^1_\\mu (X)$ its time-average $f^\\ast $ is an eigenfuction of the operators $U^t$ at eigenvalue 0: for a.e.",
"point ${x}\\in {X}$ with respect to $\\mu $ , $(U^tf^\\ast )({x})=f^\\ast ({S}^t({x}))=f^\\ast ({x})=\\exp (0t)f^\\ast ({x}).$ Since the eigenfunctions at 0 are invariant under the flow, we have a complete characterization of (possibly non-smooth) invariant sets."
],
[
"Basin of Attraction in Dissipative Flows",
"In the last of the previous section, we indicate that the time-average of an observable enables the visualization of low-dimensional slices through high-dimensional invariant sets of the quasiperiodically forced system with a smooth invariant measure.",
"Here, we consider a more general class of the quasiperiodically forced systems with dissipation and will point out that the use of time-average works for characterizing an important invariant set—basin of attraction.",
"In Section we considered the measure-preserving flow on a compact metric space.",
"Every continuous flow on a compact metric space preserves a measure.",
"This is the content of the so-called Krylov-Bogolyubov theorem [27].",
"However, this might be a useless statement if our intent is to study the behavior of trajectories from their time-averages, by the method used in the measure-preserving cases in Section .",
"For example, let us take a simple dissipative flow, $\\frac{{d} x}{{d} t}=-\\lambda x \\qquad x\\in \\mathbb {R}, \\qquad \\lambda >0.$ All of the initial conditions converge to $x=0$ as $t\\rightarrow \\infty $ along the flow $S^t(x)=\\exp (-\\lambda t)x$ .",
"The invariant measure clearly exists and is the Dirac measure supported at $x=0$ (to which any “initial\" measure defined on $\\mathbb {R}$ evolves).",
"Thus, the following statement holds: for a function $f:\\mathbb {R}\\rightarrow \\mathbb {C}$ , $f^\\ast (x)=\\int _{\\mathbb {R}}f{d}\\mu ,$ for a.e.",
"point $x\\in \\mathbb {R}$ with respect to the Dirac measure.",
"However, that excludes the whole real line except for the origin itself.",
"The situation feels better though.",
"For example, it is easy to see that the time-average of any continuous function on $\\mathbb {R}$ is just its value at 0: $f^*(x)=f(0)$ .",
"Note that for the measure-preserving case in Section , we basically considered integrable functions for which time-averages exist.",
"We can not do that in the dissipative case.",
"A function that is 1 everywhere on a finite interval $(\\underline{a},\\overline{a})$ containing 0, except at 0 where its value is 0, is clearly integrable (being a union of two simple functions).",
"Then, its time-average under the dissipative flow is identically 1, and its integral with respect to the invariant Dirac measure is 0.",
"It turns out that in dissipative systems, it is best for the time-average to work with continuous functions.",
"Adopting that idea, the whole method of ergodic partitioning can be extended to capture basins of attraction for continuous flows that are not necessarily measure-preserving.",
"Consider a continuous flow ${S}^t: X\\rightarrow X$ , where $X$ is not necessary compact, and label by $F$ the set on which time averages of continuous functions do not exist.",
"On the set $X\\setminus F$ , consider a set $C$ on which time-averages of continuous functions (or, better, a countable, separating set of continuous functions) are constant.There can be uncountably many such sets inside $X\\setminus F$ : for example, let ${S}^t$ be the identity map, mapping every point into itself.",
"Then, by the same construction described in Section and Appendix , there is an invariant measure $\\mu _C$ such that $f^*({x})=\\int _C f d\\mu _C$ for any continuous function $f$ , ${x}\\in C$ .",
"Such a measure is called a physical measure [28], provided $M$ is equipped with a measure $\\mu $ that ${S}^t$ does not preserve (but we are interested in)—say Lebesgue measure—and $C$ contains an open (and thus positive measure) set in $M$ .In light of this discussion, one might say that the notion of “ergodic measure\" should be preserved for such constructs even in dissipative systems.",
"Namely, many ergodic measures we obtain in measure-preserving system are certainly physically important, although their support does not contain an open set.",
"The tricky part is that due to Birkhoff's Ergodic Theorem, ergodic measures are associated with time averages of integrable functions.",
"In dissipative systems, this does not work (see the above example).",
"We could keep the requirement of equality of space and time averages, like in (REF ), but for continuous functions only.",
"Physical measures would then be ergodic measures with support that contains an open set.",
"Suppose that the state space of a continuous flow admits a finite number of attractors, and that the union of their basins of attraction is of full measure.",
"We state the result for flows (continuous-time systems) to emphasize that considerations here work for both discrete and continuous time.",
"Theorem 8 Let the system $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0}={F}({x}), \\quad {x}\\in X\\subset \\mathbb {R}^n$ with a flow ${S}^t$ have a finite number $N$ of attractors with basins of attraction $A_j, j=1,...,N$ , such that $\\mu (\\cup _j A_j)=\\mu (X)$ , where $\\mu $ is the Lebesgue measure.",
"Also, let the time averages of continuous functions exist everywhere on a set $X\\setminus F$ , where $\\mu (F)=0$ .",
"Then, the time-average $h^*({x})$ of a continuous function $h\\in \\mathcal {C}(X)$ is a piecewise constant eigenfunction of the Koopman operators $U^t$ at eigenvalue 0 defined a.e.",
"point with respect to $\\mu $ .",
"For any point ${x}\\in A_j,$ there exists a set $C$ such that $(U^t h^*)({x})&=U^t \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int _0^{T} h({S}^\\tau ({x}))d\\tau \\nonumber \\\\&= \\int _C (U^t h) d\\mu _C\\nonumber \\\\&=\\int _C (h\\circ {S}^t) d\\mu _C\\nonumber \\\\&=\\int _C h d\\mu _C \\nonumber \\\\&=h^*({x}), \\nonumber $ where in the second line we utilized (REF ), and transitioning from line 3 to 4 we used the fact that $\\mu _C$ is invariant under ${S}^t$ .",
"Since $(U^t h^*)({x})=h^*({x})$ is exactly the equation which $h^*({x})$ has to satisfy in order to be an eigenfunction of $U^t$ , and the basin of attraction $A_j$ is arbitrary, by taking into account that $\\mu (\\cup _j A_j)=\\mu (X)$ the proposition is proven.",
"The concept of the ergodic partition in dissipative systems is applied to a more general class of systems than those treated in Theorem REF .",
"It provides ergodic measures on invariant sets that are not necessarily attractors—because they do not have an open set of initial conditions converging to them.",
"Yet, these sets are dynamically important and can not be refined without losing the equivalence of space and time averages.",
"More precisely, the ergodic partition $\\zeta {e}$ of $M$ under ${S}^t$ (not necessarily measure-$\\mu $ preserving) is a partition into sets $C_{\\alpha }$ , where $\\alpha $ is a member of an indexing set, such that on each set $C_{\\alpha }$ (ergodic set) there exists an ergodic measure $\\mu _{C_\\alpha }$ such that $\\mu _{C_{\\alpha }}(C_{\\alpha })=1,$ For every $f\\in \\mathcal {C}(M),f^{\\ast }({x}\\in C_{\\alpha })=\\int _{C_{\\alpha }}fd\\mu _{C_{\\alpha }},$ for a.e ${x}$ point with respect to $\\mu $ .",
"The last condition emphasizes the role of the measure $\\mu $ —the time averages are equal to space averages with respect to $\\mu _{C_\\alpha }$ , but their equality is almost everywhere with respect to measure $\\mu $ that might be of our interest.",
"In this way, we have escaped the realm of the Krylov-Bogolyubov Theorem, which, in the case when dynamics are not measure-preserving, neglects dynamics on possibly large swaths of the state space."
],
[
"Sample-Based Characterization of Invariant Sets",
"In this section, we provide a characterization of invariant sets in the quasiperiodically forced system (REF ).",
"Generally speaking, understanding invariant structures of (REF ) is not easy because the associated portrait of $M$ changes with time in an aperiodic manner.",
"As shown in Section , the ergodic partition theory enables one to visualize an invariant set as its low-dimensional slice in $M$ by the time-averaging technique.",
"Also, in () it was shown that the time-averaging technique works for dissipative case to visualize the basin of attraction.",
"The obtained slice here is just a sample of the invariant set at a particular initial time (in other words, an initial condition in $\\mathbb {T}^N$ ).",
"Because of the aperiodic nature, such a sample does not seem to provide complete information on the entire structure of invariant set in the augmented state space $M\\times \\mathbb {T}^N$ .",
"However, we will prove that a boundedness property of the invariant set in $M\\times \\mathbb {T}^N$ is captured by means of one sample of it in $M$ (see Theorem REF ).",
"In the following, in order to encompass both the cases in Section and (), we consider the original state space $M$ that is metric and not necessarily compact.",
"A continuous, linear skew-product flow ${S}^t$ (diffeomorphism) on the augmented state space $X:=M\\times \\mathbb {T}^N$ derived from the quasiperiodically forced system (REF ) is described below: for ${x}=({m},{\\theta })\\in M\\times \\mathbb {T}^N$ , ${S}^t({x}):=\\left({S}^t_{M}({m},{\\theta }),{S}^t_\\mathit {\\Omega }({\\theta })\\right),$ where ${S}^t_{M}: M\\times \\mathbb {T}^N\\rightarrow M$ is the continuous map defined by trajectories of the original system (REF ), and ${S}^t_\\mathit {\\Omega }: \\mathbb {T}^N\\rightarrow \\mathbb {T}^N$ the linear (continuous) flow on the torus $\\mathbb {T}^N$ .",
"For a fixed ${\\theta }_0\\in \\mathbb {T}^N$ , we will write the orbit on $\\mathbb {T}^N$ through ${\\theta }_0$ as $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0):=\\lbrace {\\theta } : {\\theta }={S}^t_\\mathit {\\Omega }({\\theta }_0)\\in \\mathbb {T}^N, t\\in \\mathbb {R}\\rbrace $ .",
"Since the $N$ basic frequencies in the original system (REF ) are rationally independent, the following lemma is obvious.",
"Lemma 9 For any ${\\theta }_0\\in \\mathbb {T}^N$ , $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ is dense in $\\mathbb {T}^N$ .",
"Now, we investigate an invariant set in the augmented system (REF ).",
"Let $I\\subseteq X$ be a positively invariant set of (REF ) that is supposed to be closedIf $I$ is invariant and open, then its closure $\\mathrm {cl}(I)$ is still invariant..",
"It follows from Lemma REF and by closure that the following decomposition of $I$ holds: $I=\\bigcup _{\\theta \\in \\mathbb {T}^N} A_\\theta \\times \\lbrace {\\theta }\\rbrace ,$ where $A_{\\theta }$ is a closed subset of $M$ .",
"Let us denote by $A_{\\theta _0}$ an intersection or sample of the invariant set $I$ at ${\\theta }={\\theta }_0$ .",
"The next lemma provides a topological property of the intersections $A_\\theta $ .",
"Lemma 10 For each $\\epsilon >0$ there exists a positive constant $\\delta $ so that $|{\\theta }-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $d(A_\\theta ,A_{\\theta _0})<\\epsilon $ , where $d(A_{\\theta },A_{\\theta _0})$ is the Hausdorff distance that induces a topology on the family of all closed subsets of $M$ .",
"Assume not.",
"Then, for each $\\delta >0$ there exist a sequence of times $\\lbrace t_j\\rbrace $ , where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ , and a positive constant $\\epsilon _1$ such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $|{\\theta }_j-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ for every $j>J$ ($J$ is an integer), while $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ holds for some $k>J$ .",
"Here, from the continuity of ${S}^t_M$ in $t$ , for each ${m}\\in A_{\\theta _0}$ and $\\epsilon >0$ , there exists a positive constant $\\delta $ such that $|{\\theta }_k-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $|{m}_k-{m}|_{M}<\\epsilon $ , where ${m}_k:={S}^{t_k}_{M}({m},{\\theta }_0)\\in A_{\\theta _k}$ .",
"This gives us a contradiction of $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ by taking $\\epsilon =\\epsilon _1$ and from the definition of Hausdorff distance.",
"This lemma suggests that the invariant set $I$ is topologically connected with respect to ${\\theta }$ .",
"One trivial example is provided by taking as $I$ the closure of a single trajectory starting at a point $({m},{\\theta }_0)$ , for which $A_{\\theta _0}$ consists of a single point.",
"In this case, $I$ is topologically regarded as a product set of a single point in $M$ and the torus $\\mathbb {T}^N$ .",
"This is rigorously stated in the next theorem.",
"Theorem 11 Suppose that $A_{\\theta _0}$ is bounded in $M$ .",
"Then, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .",
"Now, let us construct the mapping $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ in the following manner.",
"For each $({m},{\\theta })\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , by choosing time $\\tau _\\theta $ that satisfies ${\\theta }={S}^{\\tau _\\theta }_{\\it \\Omega }({\\theta }_0)$ , we define $h_{\\theta _0}({m},{\\theta })$ as $({S}^{\\tau _\\theta }_M({m},{\\theta }_0),{\\theta })$ .",
"We here prove the continuity of $h_{\\theta _0}$ .",
"For each $\\epsilon >0$ and ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $\\left|{\\theta }_j-{\\theta }\\right|_{\\mathbb {T}^N}<\\epsilon /2$ for every $j>J$ ($J$ is an integer).",
"Furthermore, because of the continuity of ${S}^t_{M}$ in $t$ , for each ${m}_j\\in A_{\\theta _0}$ there exists a positive constant $\\delta _j$ such that $\\left|{m}-{m}_j\\right|_M<\\delta _j$ implies $\\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M<\\epsilon /2$ .",
"Here, because for every $j>J$ $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\le \\left|{m}-{m}_j\\right|_{M}+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}< \\delta _j+\\frac{\\epsilon }{2},$ we set $\\delta :=\\delta _j+\\epsilon /2$ .",
"Thus, if $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ , then $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}&\\le \\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}\\nonumber \\\\& <\\frac{\\epsilon }{2}+\\frac{\\epsilon }{2}=\\epsilon .\\nonumber $ This shows that $h_{\\theta _0}$ is continuous.",
"We next prove that $h_{\\theta _0}$ is uniformly continuous.",
"Assume not.",
"Then, for each $\\delta >0$ there exists a positive constant $\\epsilon _1$ such that $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ implies $\\left|h_{\\theta _0}({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ for some $({m}_j,{\\theta }_j) $ and $({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .",
"We here set $\\delta =1/j$ .",
"Since it is supposed that $A_{\\theta _0}$ is bounded and closed (and $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\subset \\mathbb {T}^N$ ), the two sequences of points $\\lbrace ({m}_j,{\\theta }_j)\\rbrace $ and $\\lbrace ({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\rbrace $ have convergent subsequences, denoted as $\\lbrace ({m}_{j_k},{\\theta }_{j_k})\\rbrace $ and $\\lbrace ({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})\\rbrace $ .",
"Because of $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<1/j$ , both the subsequences have the same convergent point, denoted as $({m}^\\ast ,{\\theta }^\\ast )$ .",
"That is, for each $\\delta _1>0$ , there exists a positive integer $J_k$ such that both the inequalities $\\left|({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ and $\\left|({m}_{j_k},{\\theta }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ hold for every $j_k\\ge J_k$ .",
"Here, since $h_{\\theta _0}$ is proven to be continuous at $({m}^\\ast ,{\\theta }^\\ast )$ , for each $\\epsilon >0$ , there exists a positive constant $\\delta _2$ such that $\\left|({m},{\\theta })-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _2$ implies $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon /2$ .",
"By taking $\\epsilon =\\epsilon _1$ and $\\delta _1=\\delta _2$ thus choosing $J_k$ appropriately, we see that both the inequalities $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^M}<\\epsilon _1/2$ and $\\left|h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon _1/2$ hold for every $j_k\\ge J_k$ , and $\\begin{aligned}\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^M}\\le &\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N} \\\\& + \\left|h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N} \\\\< & \\frac{\\epsilon _1}{2}+\\frac{\\epsilon _1}{2} = \\epsilon _1,\\end{aligned}$ implying the contradiction for $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ .",
"Thus, $h_{\\theta _0}$ is uniformly continuous.",
"By virtue of Theorem 3.45 in page 136 of [29], the uniformly-continuous $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ is extended to the mapping $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ that is also (uniformly) continuous.",
"We now prove that $\\hat{h}_{\\theta _0}$ is a bijection.",
"For each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , the statement is obvious from the construction of $h_{\\theta _0}$ (see its dependence on ${\\theta }$ ).",
"It is thus enough to check the case ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .",
"For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of times $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ converges to ${\\theta }$ .",
"Because $\\hat{h}_{\\theta _0}$ is continuous, the sequence $\\lbrace \\hat{h}_{\\theta _0}({m},{\\theta }_j)=h_{\\theta _0}({m},{\\theta }_j)\\rbrace $ converges to $\\hat{h}_{\\theta _0}({m},{\\theta })$ , which is represented as $\\displaystyle \\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}({m},{\\theta }_0),{\\theta }\\right)$ .",
"Namely, the limit exists for every ${m}\\in A_{\\theta _0}$ .",
"Assume that $\\hat{h}_{\\theta _0}$ is not an injection.",
"Then, there exist ${m},{m}^{\\prime }\\in A_{\\theta _0}$ satisfying ${m}\\ne {m}^{\\prime }$ such that for each $\\epsilon >0$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M<\\epsilon $ holds for every $j>J$ ($J$ is an integer and depends on $\\epsilon $ ).",
"Here, the fundamental property of uniqueness of trajectories in (REF ) (or ${S}^t$ diffeomorphsim) implies that for each $t_j$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M\\ge \\epsilon _j$ holds ($\\epsilon _j$ is a positive constant and depends on $t_j$ ), showing the contradiction by taking $\\epsilon =\\epsilon _j$ .",
"Hence, $\\hat{h}_{\\theta _0}$ is an injection.",
"Regarding $\\hat{h}_{\\theta _0}$ surjective, it is obvious that for each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , $\\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=h_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left({S}_M^{\\tau _\\theta }(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)=(A_\\theta ,{\\theta })$ .",
"For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ we see $\\displaystyle \\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)$ , and the limit converges to $A_\\theta $ from Lemma REF .",
"Hence, from (REF ), $\\hat{h}_{\\theta _0}$ is a surjection.",
"Finally, the inverse of $\\hat{h}_{\\theta _0}$ exists according to the above construction and is continuous.",
"Therefore, $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ is a homeomorphism, namely, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .",
"The following two corollaries provide a way of characterization of boundedness of $I$ by means of one sample of it.",
"Corollary 12 Suppose that $A_{\\theta _0}$ is bounded in $M$ .",
"Then, $I$ is bounded in $X$ .",
"Corollary 13 Suppose that a closed subset $B_{\\theta _0}$ of $A_{\\theta _0}$ is bounded in $M$ .",
"Then, $\\hat{h}_{\\theta _0}(B_{\\theta _0}\\times \\mathbb {T}^N)$ is a subset of $I$ and bounded in $X$ .",
"Corollaries REF and REF imply that by computing a slice of the invariant set in $M$ at one sample onset, it is possible to determine whether the invariant set or its subset is bounded in the augmented state space $X$ .",
"For a bounded invariant set $I$ in $X$ , any cross-section $I_{\\theta _0}=A_{\\theta _0}\\times \\lbrace {\\theta }_0\\rbrace $ (${\\theta }_0\\in \\mathbb {T}^N$ ) is bounded in terms of $M$ , in other words, the boundedness property of $A_{\\theta _0}$ does not depend on the choice of ${\\theta }_0$ .",
"In this way, we call the bounded invariant set $I$ in $X$ uniformly bounded invariant set."
],
[
"Example Studies",
"In this section, we provide three examples of the application of the above theoretical results.",
"The last example is related to practical problems on stability of power grids."
],
[
"Forced Linear Harmonic Oscillator",
"First, we will consider the following one-degree-of-freedom linear harmonic oscillator with quasi-periodic forcing of (multiple) $N$ frequencies: $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}),$ or $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $F_i$ ($>0$ ) ($i=1,2,\\ldots ,N$ ) are the amplitudes of periodic forces, $\\mathit {\\Omega }_i$ ($>0$ ) the angular frequencies of the forces, and $\\theta _{i0}\\in \\mathbb {T}$ the initial phases.",
"We assume there is no resonance: the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent; and $\\mathit {\\Omega }_i\\ne 1$ ($i=1,2,\\ldots ,N$ ).",
"The solution of (REF ) with initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{aligned}m_1(t) &=C_1\\cos t+C_2\\sin t+\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos (\\mathit {\\Omega }_it+\\theta _{i0}), \\\\m_2(t) &=\\displaystyle \\frac{{d}}{{d} t}m_1(t), \\\\{\\vspace{14.22636pt}}\\theta _i(t) &= \\mathit {\\Omega }_it+\\theta _{i0} \\qquad i=1,2,\\ldots ,N,\\end{aligned}\\right\\rbrace $ with $C_1:=m_{10}-\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0}, \\qquad C_2:=m_{20}+\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}.$ Now, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).",
"We define the following quadratic observable $f: \\mathbb {R}^2\\times \\mathbb {T}^N\\rightarrow \\mathbb {R}$ , related to the potential energy $m^2_1/2$ : $f(m_1,m_2,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2_1.$ Then, we calculate its time-average under the solution as $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\left\\lbrace C^2_1+C^2_2+\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}\\right\\rbrace .$ Here, as shown in (REF ), the constants $C_1$ and $C_2$ are determined by the initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .",
"The level set, parameterized by a real-valued constant $c\\in \\mathbb {R}$ , of the time-average in the two-dimensional initial plane $(m_{10},m_{20})$ at fixed $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ becomes a circle with center of $\\left(\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0},-\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}\\right),$ if the following inequality holds: $r^2:=2c-\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}>0$ Thus, the constant $r$ corresponds to the radius of the circle of the level set for $c$ .",
"The center of the level set seems to stay close to the origin and is shifted with the choice of initial phases $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .",
"Numerical results of the level sets for $N=2$ , $\\mathit {\\Omega }_1=\\pi /3$ , $\\mathit {\\Omega }_2=11/10$ , and $F_1=F_2=0.2$ are presented in Figure REF .",
"Figure: Level sets of the quasiperiodically-forced one-degree-of-freedom harmonic oscillator () for different (θ 10 ,θ 20 )(\\theta _{10},\\theta _{20}):(a) (0,0)(0,0), (b) (π/2,π/2)(\\pi /2,\\pi /2), (c) (π,π)(\\pi ,\\pi ), and (d) (3π/2,3π/2)(3\\pi /2,3\\pi /2).The horizontal (or vertical) axis for each figure is x 1 ∈[-10,10]x_1\\in [-10,10] (or x 2 ∈[-10,10]x_2\\in [-10,10])."
],
[
"One-Dimensional Dissipative System",
"Next, we consider the following linear dissipative system with quasi-periodic forcing: $\\frac{{d} m}{{d} t}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}).$ or $\\frac{dm}{dt}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $\\lambda >0$ .",
"In the same manner as above, we suppose the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent.",
"The solution of (REF ) with initial condition $(m_{0},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{array}{ccl}m(t) &=& \\displaystyle C^{-\\lambda t}+\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin (\\mathit {\\Omega }_it+\\theta _{i0})-\\mathit {\\Omega }_i\\cos (\\mathit {\\Omega }_it+\\theta _{i0})\\rbrace ,\\\\{\\vspace{11.38109pt}}\\theta _i(t) &=& \\mathit {\\Omega }_it+\\theta _{i0}~~~(i=1,2,\\ldots ,N),\\end{array}\\right\\rbrace $ with $C:=m_0-\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin \\theta _{i0}-\\mathit {\\Omega }_i\\cos \\theta _{i0}\\rbrace .$ Here, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).",
"We consider the quadratic observable as $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2.$ Then, its time-average under the solution is directly calculated as follows: $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\sum ^{N}_{i=1}\\frac{F^2_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}.$ Although this is trivial, since the transient term in the solution is filtered out, the value of the time-average is constant in the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ , in other words, does not depend on the initial condition $(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .",
"This implies that for any choice of sample at $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})^\\top \\in \\mathbb {T}^N$ , the partition of $M=\\mathbb {R}$ based on the level set of the time-averages is just one and corresponds to $\\mathbb {R}$ itself.",
"Also, the above observation on the partition holds when we pick up a polynomial observable like $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=\\sum ^{n}_{j=0}a_jm^j,$ with coefficients $a_j\\in \\mathbb {R}$ and finite integer $n$ .",
"For every continuous function defined on a closed interval in $\\mathbb {R}$ , from the Weierstrass Approximation Theorem, it can be uniformly approximated on that interval by polynomials to any degree of accuracy.",
"This implies that the above observation of the partition holds for every continuous function, which is an example of Theorem REF .",
"Thus, the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ is the basin of attraction for the system (REF ), that is to say, the torus $\\mathbb {T}^N$ is the unique attractor for the system."
],
[
"Two-Dimensional Nonlinear Model of a Power Grid",
"To show the effectiveness of the theory beyond the basic examples, we apply the developed theory for analyzing the CSI phenomenon in a loop power grid.",
"CSI is a undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance [19].",
"In the case of small dissipation, this phenomenon generally does not happen upon an infinitesimally small perturbation around a steady operating state (stable equilibrium).If we have no dissipation, nonlinear stability properties are not understood in the high-dimensional cases.",
"However, it encompasses the situation when the grid's operating state escapes a predefined, positive-measure set around the equilibrium.",
"In this way, the notion of instability that we address here is non-local.",
"In [19], we derived a reduced-order dynamical system that described averaged dynamics of machines in a simple loop power grid and explained the non-local instability.",
"The reduced-order system has an quasiperiodic forcing (so it is non-autonomous), and its solutions define a measure-preserving flow.",
"The goal of this section is to characterize the non-local instability by analyzing invariant sets of the quasiperiodically forced system, which is introduced in (REF ) below."
],
[
"Mathematical Model",
"Consider short-term (zero to ten seconds) swing dynamics of a rudimentary power grid with the loop topology shown in Figure REF , where the small gray circles denote synchronous generators.",
"The loop part of the grid consists of $N{G}$ small, identical generators, encompassed by the dotted box, which operate in the grid and are connected to the infinite busAn ideal voltage source of constant voltage and constant frequency.",
"The loss-less transmission lines joining the infinite bus and a generator are much longer than those joining two generators in the loop part.",
"Thus, the magnitude of interaction between the infinite bus and a generator is much smaller than that between two neighboring generators on the loop part.",
"We call the model of Figure REF the loop power grid in the following.",
"Here, we assume that the lengths of transmission lines between two neighboring generators are identical.",
"In [19], we showed that the CSI phenomenon in the loop power grid was accurately captured by the following dynamical system defined on the cylindrical state space $\\mathbb {T}^1\\times \\mathbb {R}$ : $\\frac{d\\delta }{dt}=\\omega , \\qquad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right)$ where $e_{ij}=\\sqrt{\\frac{2}{N{G}}}\\cos \\left(\\frac{2\\pi ij}{N{G}}+\\frac{\\pi }{4}\\right),\\qquad \\mathit {\\Omega }_j=2\\sqrt{|b{int}|}\\left|\\sin \\frac{\\pi j}{N{G}}\\right|.$ The system (REF ) represents spatially-averaged dynamics of the $N{G}$ generators in the loop power grid.",
"The variable $\\delta \\in \\mathbb {T}^1$ is the average of angular positions of rotors (with respect to the infinite bus) of the $N{G}$ generators, and $\\omega \\in \\mathbb {R}$ is the average of deviations of rotor speeds in the $N{G}$ generators relative to the system angular frequency.",
"The parameter $p{m}$ stands for the mechanical input power to a generator, $b$ for the maximum transmission power between the infinite bus and a generator, and $b{int}$ for the maximum transmission power between two neighboring generators in the loop power grid.",
"The constants $e_{ij}$ are the eigenfunctions of linear modal oscillations between coupled generators in the loop part, $\\mathit {\\Omega }_j$ their eigen-(angular) frequency, and $c_j$ the strengths of modal oscillations.",
"The finite index set $\\mathcal {J}$ determines which modes are excited in the loop part.",
"The system (REF ) is derived under the observation that the linear modal oscillations in the loop part act as perturbations on the spatially-averaged dynamics of the $N{G}$ generators: see [19] and references therein.",
"Note that (REF ) is the Hamiltonian system: $\\frac{d\\delta }{dt}=\\frac{}{\\omega }{H}(\\delta ,\\omega ,t), \\qquad \\frac{d\\omega }{dt}=-\\frac{}{\\delta }{H}(\\delta ,\\omega ,t),$ with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ , given by ${H}(\\delta ,\\omega ,t):=\\frac{1}{2}\\omega ^2-p{m}\\delta -\\frac{b}{N{G}}\\sum ^{N{G}}_{i=1}\\cos \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right).$ Because the flow defined here is divergence-free, i.e.",
"$({H}/\\delta )({d}\\delta /{d} t)+({H}/\\omega )({d}\\omega /{d} t)=0$ , the system (REF ) preserves the Liouville measure ${d}\\delta {d}\\omega $ .",
"Note that it does not conserve the Hamiltonian function ${H}$ , because of ${d}{H}/{d} t\\ne 0$ if $c_j\\ne 0$ .",
"In this way, the augmented system for (REF ), namely $\\frac{d\\delta }{dt}=\\omega , \\quad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\theta _j+\\delta \\right), \\quad \\frac{d\\theta _j}{dt}=\\mathit {\\Omega }_j \\qquad j\\in \\mathcal {J},$ defines a measure-preserving flow on $M\\times \\mathbb {T}^{|{\\cal J}|}$ with $M=\\mathbb {T}^1\\times \\mathbb {R}$ , where $|{\\cal J}|$ stands for the cardinality of ${\\cal J}$ ."
],
[
"Results and Implications",
"It was shown in [19] that the unbounded motion in the quasiperiodically forced system (REF ) corresponds to the CSI phenomenon.",
"Because of $\\delta \\in \\mathbb {T}^1$ , the unbounded motion involves the unbounded trajectory in the $\\omega $ -direction, that is, the average of deviation of rotor speeds.",
"We use Corollaries REF and REF to analyze invariant sets of the flow (defined by the system (REF )), in which all the generators show bounded deviation of rotor speeds in time.",
"The analysis is crucial to understanding the so-called stability region of the loop power grid, i.e.",
"how the grid's behavior depends on initial conditions representing failures in the grid.",
"Numerical simulations are performed for analysis of invariant sets.",
"To do so, we need to fix (i) a function $f$ , (ii) the subset of state space on which we identify invariant sets, and (iii) the exit time $T{ex}$ to obtain an approximation of each time-average $f^\\ast $ .",
"We use the function $f(\\delta )=\\sin 2\\delta $ and the grid of $401\\times 401$ of initial conditions $(\\delta ,\\omega )$ on $[1,2]\\times [-0.15,0.15]$ .",
"The averaging operation of a single function can be used for the identification of invariant sets.",
"Numerical integration of the system (REF ) is performed with the 4th-order symplectic integrator [30] with time step $h$ : see Appendix for details.",
"The parameter settings are the following: $p{m}=0.95, \\quad b=1, \\quad N{G}=20, \\quad b{int}=100, \\quad h=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\frac{1}{N},\\quad T{ex}=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\times 2000,$ where $N=8$ or 16 depending on the setting of $\\mathcal {J}$ .",
"The values of $p{m}$ , $b$ , $N{G}$ , and $ b{int}$ are the same as in [19].",
"Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–I: Initial phase θ 0 =0{\\theta }_0={0} and multiple setting of 𝒥\\mathcal {J} i.e.",
"(a) 𝒥={1}\\mathcal {J}=\\lbrace 1\\rbrace , (b) 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace , (c) 𝒥={1,2,3}\\mathcal {J}=\\lbrace 1,2,3\\rbrace , (d) 𝒥={1,2,3,4}\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace , (e) 𝒥={1,2,3,4,5}\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace , and (f) 𝒥={1,2,3,4,5,6}\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace .The horizontal axis for each figure is δ∈[1,2]\\delta \\in [1,2], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–II: 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace and multiple setting of initial phase θ 0 =(θ 10 ,θ 20 ) ⊤ =(2πkΩ 1 /Ω 2 ,0) ⊤ {\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)^\\top for (a) k=0k=0 (same as Figure (b)), (b) k=1k=1, (c) k=2k=2, (d) k=3k=3, (e) k=4k=4, and (f) k=5k=5.The horizontal axis for each figure is δ∈[0,π]\\delta \\in [0,\\pi ], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure REF shows numerical results on analysis of invariant sets of the flow defined by the quasiperiodically forced system (REF ).",
"In this figure we change the number of excitation modes, i.e.",
"$\\mathcal {J}$ ; (a) $\\mathcal {J}=\\lbrace 1\\rbrace $ , (b) $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , (c) $\\mathcal {J}=\\lbrace 1,2,3\\rbrace $ , (d) $\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace $ , (e) $\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace $ , and (f) $\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace $ .",
"The amplitude $c_j$ in the figure is common and satisfies $\\displaystyle \\sqrt{\\sum _{j\\in \\mathcal {J}}c_j^2}=1.5$ , implying that the root-means-square of the forcing term does not change for any setting of $\\mathcal {J}$ .",
"All the initial phases ${\\theta }_0$ are set to zero.",
"For the outer dark-green region in each figure, every trajectory starting from it is unbounded in time.",
"Except for the dark-green on the bottom, the color bar attached to each figure denotes the value of time-average $f^\\ast (\\delta )$ .",
"The level sets of $f^\\ast (\\delta )$ are colored by the same color.",
"That is, the set of the same color belongs to one invariant set.",
"By Corollary REF , the fact that the level set is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space $M\\times \\mathbb {T}^{|\\mathcal {J}|}$ .",
"In the figures (a,b,c), we see that the color plot of the level sets forms concentric rings.",
"However, in the figures (d,e,f), we see that the color plot does does not necessarily exhibit concentric rings and does become scattered in the bands close to the outer dark-green regions.",
"This implies that the structure of invariant sets is complicated in the bands.",
"We anticipate this results from the so-called resonance phenomenon (see, e.g., [31], [32]) as an interaction between a family of bounded oscillations in the unforced system and quasiperiodic forcing.",
"Figure REF shows other numerical results on analysis of invariant sets.",
"In this figure, we consider the forcing term with two frequencies, $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , and we change the initial phases ${\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)$ where $k=0,1,\\ldots ,5$ .",
"The color plots here are conducted in the same manner as in Figure REF , and for the outer dark-green regions every trajectory starting from them is unbounded in time.",
"By Corollary REF , the fact that the level set with same color is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space.",
"Here, under the current setting of parameters, it is conjectured that outside the outer dark-green regions (i.e., outside the computational domain of the analysis), there exists no state from which trajectory is bounded in time.",
"This is true in the unperturbed case because there exists one homoclinic orbit separating the bounded and unbounded trajectories inside the computational domain.",
"Thus, it can be inferred that the level sets discussed above are bounded in $M=\\mathbb {T}^1\\times \\mathbb {R}$ , implying by Corollary REF that the whole of the corresponding invariant sets are uniformly bounded in the augmented state space.",
"The intersection of uniformly bounded invariant sets for all ${\\theta }$ corresponds to the stability region of the loop power grid, in which all the generators show bounded deviation of rotor speeds in time."
],
[
"Conclusions",
"In this paper, we studied the ergodic partition and invariant sets of the quasiperiodically forced dynamical system (REF ).",
"The main theoretical contributions of this paper are twofold.",
"One is to provide a theory of ergodic partition of state space for smooth flows.",
"The theory is a natural extension of that in [7] and is applicable to measure-preserving and dissipative flows arising in various physical and engineering systems.",
"Examples of them include dynamical systems induced by time-dependent Hamiltonians and incompressible fluid flows with time-dependent velocity profiles.",
"The other is to provide a new characterization of invariant sets in the the quasiperiodically forced system (REF ), in which we introduced a concept of uniformly bounded invariant sets.",
"The developed theory was applied to characterize the CSI phenomenon of a rudimentary power grid.",
"We have speculated that the phenomenon can be characterized, in particular, the stability region corresponds to the intersection of uniformly bounded sets for all initial phases; or a sufficient condition for the phenomenon is that the operating state of the grid is placed outside of the bounded sets at a particular initial phase or time like $t=0$ ."
],
[
"Acknowledgments",
"Y.S.",
"thanks Dr. Marko $\\rm Budi\\check{s}i\\acute{c}$ for his introduction to theory and computation of ergodic partition and fruitful discussions.",
"The authors also appreciate the reviewers for their valuable suggestion of the manuscript.",
"During part of the work on this paper, Y.S.",
"was at the Department of Mechanical Engineering, University of California, Santa Barbara, United States, and at the Department of Electrical Engineering, Kyoto University, Japan."
],
[
"Proof of Lemma ",
"A continuous function $f$ on $X$ is measurable and, from the assumption that $X$ is compact for the theoretical analysis in Section , $f$ is bounded on $X$ .",
"Here, we note that the time-average $f^\\ast $ of the measurable function $f$ is measurable as a limit of measurable functions $f_T$ on $X$ , defined as $f_T(x):=\\frac{1}{T}\\int ^T_0 f({S}^t(x)){d}t \\qquad T>0.$ Since we consider the sets $A_\\alpha $ on $\\mathit {\\Sigma }\\subset X$ , the fact that the family of $A_\\alpha $ is a partition of $\\mathit {\\Sigma }$ is obvious.",
"Next, the fact that the partition, denoted by $\\zeta _f$ , is measurable follows by taking $\\mathfrak {D}_f$ to be the collection of pre-images under $f^\\ast $ of open intervals with rational endpoints in $\\mathbb {R}$ .",
"Because $f^\\ast $ is measurable, each pre-image $(f^\\ast )^{-1}([a,b])$ , where $a$ and $b$ are rational numbers, is measurable.",
"Every set of this type is clearly separated into sets of the form $(f^\\ast )^{-1}(\\lbrace c\\rbrace )$ , $c\\in \\mathbb {R}$ .",
"This implies that every element of $\\mathfrak {D}_f$ is a union of elements of $\\zeta _f$ .",
"Furthermore, because the set of all rational numbers is dense in $\\mathbb {R}$ , for any pair $\\alpha ,\\beta \\in \\mathbb {R}$ satisfying $\\alpha <\\beta $ , there exist two rational numbers $\\underline{a},\\overline{a}$ such that $\\alpha <\\underline{a}<\\beta <\\overline{a}$ .",
"The pre-image $(f^\\ast )^{-1}([\\underline{a},\\overline{a}])$ is an element of $\\mathfrak {D}_f$ which we denote by $D$ .",
"Obviously, we see $A_\\alpha \\subset {D}^{\\rm c}$ and $A_\\beta \\subset {D}$ .",
"Thus, it follows that $\\mathfrak {D}_f$ is a basis for $\\zeta _f$ , and we conclude that $\\zeta _f$ is measurable."
],
[
"Proof of Theorem ",
"Let $A$ be an element of $\\zeta {e}$ .",
"For a.e.",
"point $x\\in A$ , the time-average $f^\\ast (x)$ exists for all $f\\in \\mathcal {C}(X)$ .",
"Thus, the following linear functional $L_{A}$ on $\\mathcal {C}(X)$ is well-defined: $L_{A}(f):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t (x)){d} t \\qquad x\\in A.$ Then, because $L_{A}$ is a positive linear functional and $L_{A}(1)=1$ , by Riesz's Representation Theorem (I.8.4 in [23]) there exists a unique probability measure $\\mu _{A}$ on $X$ such that $\\int _{X} f{d}\\mu _A=L_{A}(f),$ for all $f\\in \\mathcal {C}(X)$ .",
"Note that $\\mu _{A}$ is invariant for ${S}^t$ .",
"To prove this, for all $t\\in \\mathbb {R}$ we have $\\int _{X} f\\circ {S}^t\\,{d}\\mu _{A}=L_{A}(f\\circ {S}^t)=L_{A}(f)=\\int _{X} f{d}\\mu _{A}.$ The second equality is a consequence of (REF ).",
"For the above operation, the continuity of ${S}^t$ is required.",
"Because $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _A}(X)$ , $\\mu _{A}$ is invariant.",
"Now, we prove that $\\mu _{A}$ is a probability measure on $A$ .",
"There is a sequence of compact sets $A^{\\rm c}_n$ , subsets of $A^{\\rm c}$ , such that ${A}^{\\rm c}_1\\subset \\cdots \\subset {A}^{\\rm c}_{n}\\subset {A}^{\\rm c}_{n+1}\\subset \\cdots , \\qquad \\mu _{A}\\left(A^{\\rm c}\\setminus \\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.",
"\\qquad $ Here, we can show $\\mu _{A}(A^{\\rm c}_n)=0$ for every $A^{\\rm c}_n$ .",
"To do this, note that by Urysohn's Lemma, for every $A^{\\rm c}_n$ , there is a continuous, positive function $f_n$ on $X$ that is equal (i) to one on $A^{\\rm c}_n$ and (ii) to zero outside of $A^{\\rm c}_{n+1}$ .",
"Clearly, we see $f_n=0$ on $A$ .",
"Therefore, because of $\\int _{X}f_n{d}\\mu _{A}=\\int _{X\\setminus A^{\\rm c}_{n+1}}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_{n+1}\\setminus A^{\\rm c}_n}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_n}f_n{d}\\mu _{A}$ and the positiveness of $f_n$ , we have $0\\le \\mu _{A}(A^{\\rm c}_n)\\le \\int _{X} f_n{d}\\mu _{A}=L_{A}(f_n)=0.$ The measure of a union of the countable number of sets with measure zero is zero: $\\mu _{A}\\left(\\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.$ Therefore, by (REF ) and (REF ), we have $\\mu _{A}(A^{\\rm c})=0$ .",
"It follows from $\\mu _{A}(X)=1$ that $\\mu _{A}$ is a probability measure on $A$ .",
"Next, let us prove that $\\mu _{A}$ is an ergodic measure on $A$ .",
"First, observe that the set of all restrictions of functions in $\\mathcal {C}(X)$ to $A$ , denoted by $\\mathcal {C}(X)|_{A}$ , is dense in the set of all $\\mu _{A}$ -integrable functions on $A$ , denoted by $\\mathcal {L}^1_{\\mu _{A}}(A)$ .",
"To show this, note that $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ .",
"Let $f$ be an element of $\\mathcal {L}^1_{\\mu _{A}}(A)$ .",
"Consider the extension of $f$ to $X$ , $\\bar{f}$ , such that $\\bar{f}=f$ on $A$ and $\\bar{f}=0$ elsewhere.",
"Then, we have $\\bar{f}\\in \\mathcal {L}^1_{\\mu _{A}}(X)$ because the following integral exists: $\\int _{X}\\bar{f}{d}\\mu _{A}=\\int _{A}f{d}\\mu _{A}.$ Here, since $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ , there is a sequence of functions in $\\mathcal {C}(X)$ , $\\lbrace f_n\\rbrace $ , converging to $\\bar{f}$ .",
"Thus, the corresponding sequence of restrictions, $\\lbrace f_n|_{A}\\rbrace $ , converges to $f$ .",
"Therefore, we observe that $\\mathcal {C}(X)|_{A}$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(A)$ .",
"Now, by the same argument as (REF ), for all $f\\in \\mathcal {C}(X)|_{A}$ we have $\\int _{A}f{d}\\mu _{A}&= L_A(f) \\nonumber \\\\&= f^\\ast (x) \\qquad x\\in A.$ Since (REF ) holds for the dense set $\\mathcal {C}(X)|_A$ in $\\mathcal {L}^1_{\\mu _{A}}(A)$ , ${S}^t|_{A}$ is ergodic: see Proposition 2.2 in Chapter II of [23] for discrete-time systems.",
"This proposition can be naturally extended to continuous-time systems.",
"Hence, we complete the proof that there indeed exists an ergodic measure $\\mu _{A}$ for any element $A$ of the partition $\\zeta {e}$ .",
"Finally, we consider (REF ) and that $A$ is invariant.",
"The equality (REF ) is obtained with the proof of Theorem 6.4 in Chapter II of [23].",
"The proof is obtained for discrete-time systems and is extended to continuous-time systems.",
"By construction, the fact that $A$ is invariant is obvious.",
"This completes the proof of Theorem REF ."
],
[
"Symplectic Integration of Time-Dependent Hamiltonian Systems",
"In Section REF , it is required to numerically simulate the Hamiltonian system (REF ) with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ .",
"Symplectic integrator [30] is normally formulated in the case of time-independent Hamiltonian functions.",
"However, one can exploit the integrator in the case of time-dependent Hamiltonian functions by augmenting the original Hamiltonian system.",
"Consider the $N$ degree-of-freedom Hamiltonian system with the Hamiltonian function ${H}(q,p,t)$ : for $i=1,2,\\ldots ,N$ , $\\frac{dq_i}{dt}=\\frac{}{p_i}{H}(q,p,t), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}{H}(q,p,t)$ where $q=(q_1,q_2,\\ldots ,q_N)^\\top $ , $p=(p_1,p_2,\\ldots ,p_N)^\\top $ , and $t\\in \\mathbb {R}$ .",
"Now, by replacing the time variable $t$ with one new variable $q_0$ and defining the other new variable $dp_0/dt:=-H/t$ , we have the augmented Hamiltonian function $\\bar{H}(q_0,p_0,q,p)$ as follows: $\\bar{H}(q_0,p_0,q,p):=p_0+{H}(q,p,q_0).$ Thus, the augmented Hamiltonian system of the time-independent Hamiltonian function $\\bar{H}$ is derived as $\\frac{dq_i}{dt}=\\frac{}{p_i}\\bar{H}(q_0,p_0,q,p), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}\\bar{H}(q_0,p_0,q,p)$ where $i=0,1,\\ldots ,N$ .",
"The flow induced by trajectories of the augmented system (REF ) is divergence-free and conserves the value of the Hamiltonian function $\\bar{H}$ .",
"Thus, by using the integrator for the augmented system, numerical simulations of the original system (REF ) are indirectly performed.",
"Note that the accuracy of numerical integration of (REF ) is checked by estimating the value of $\\bar{H}$ .",
"This idea is applicable to the case of non-periodic time-dependent Hamiltonian functions."
]
] | 1808.08340 |
[
[
"How do Convolutional Neural Networks Learn Design?"
],
[
"Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.",
"Book covers are designed in a unique way, specific to genres which convey important information to their readers.",
"By using Convolutional Neural Networks (CNN) to predict book genres from cover images, visual cues which distinguish genres can be highlighted and analyzed.",
"In order to understand these visual clues contributing towards the decision of a genre, we present the application of Layer-wise Relevance Propagation (LRP) on the book cover image classification results.",
"We use LRP to explain the pixel-wise contributions of book cover design and highlight the design elements contributing towards particular genres.",
"In addition, with the use of state-of-the-art object and text detection methods, insights about genre-specific book cover designs are discovered."
],
[
"Introduction",
"Visual design renders specific impressions to transmit information which enriches the product's value.",
"However, these visual designs despite of being important are not analyzed objectively or statistically.",
"Analyzing these visual designs enables us to understand the contained information carried by them.",
"An interesting target of visual design analysis is book cover image design where the design of a book cover can infer the genre.",
"Each book cover is carefully designed by typographers and their designs represent the book contents in an intuitive way for better sales.",
"This association of books to specific genres is based on the differences in their underlying book cover designs [1].",
"The slight change in book cover design can reflect changes in book genre which makes design learning a challenging task for book covers.",
"In order to understand the design elements used for machine aided book cover classification, we employ Convolutional Neural Networks (CNN) [2].",
"In recent years, CNNs have achieved state-of-the-art results in isolated character recognition [3], [4] and large-scale image recognition [5], [6].",
"Notably, Iwana et al.",
"[1] demonstrated that CNNs can be used for genre classification based on book cover image, although with a high level of difficulty.",
"However, that study was subjective and not enough explanation is given as to why the CNN performed as it did.",
"To interpret the reasoning behind a CNN's prediction we used a method called Layer-wise Relevance Propagation (LRP) [7].",
"LRP decomposes output function on its input variables and highlights input pixels contributing towards the network decision.",
"It produces a layer-wise relevance heatmap by recursively multiplying the relevance of higher layers by the normalized feature maps of the target layer.",
"The heatmaps can help us to discover the input image elements which have an effect on the classification result.",
"The main contributions of this paper are threefold.",
"Firstly, we classified the book cover images using one-vs-others classification with CNNs.",
"Secondly, the models built by the CNNs are analyzed using LRP.",
"With LRP, we demonstrate design elements specifically relevant to classification of the book cover images.",
"We show that certain objects have a strong relevance to particular genres.",
"Finally, we use state-of-the-art object detection and text detection methods, namely Single Shot Multibox Detector (SSD) [8] and Efficient and Accurate Scene Text Detector [9], to quantitatively enforce the results found by LRP.",
"This reveals the specific elements in which CNNs classify book cover images for genre classification.",
"The organization is as follows.",
"Section provides related works in design understanding and genre classification as well as feature visualization of CNNs.",
"Section reviews the data and tools used for understanding book cover design.",
"Section presents analysis of CNN's understanding of book cover design.",
"In Section , we demonstrate the use of LRP combined with SSD and EAST for quantitative analysis.",
"Finally, Section draws a conclusion.",
"Artistic style understanding and subjective genre classification is a budding field in machine learning.",
"For example, recent attempts have been done to identify artistic styles and quality of paintings and photographs [10], [11] with neural network models.",
"In addition, there have been trials to classify music by genre [12], [13], book covers by genre [1], movie posters by genre [14], paintings by genre [15], and text by genre [16], [17].",
"Also, in a general sense, document classification can be considered genre classification and deep CNNs are the state-of-the-art in the document classification domain [18], [19], [20]."
],
[
"Visualization inside of CNNs",
"There is a desire to visualize features and determine pixel-wise attention and relevance within the hidden layers of CNNs.",
"However, this is a not a straightforward task [21].",
"Erhan et al.",
"[21] proposed using gradient decent to maximize a node's activation to visualize the employed features.",
"Similar work has been done for large-scale image classification [22].",
"Zeiler and Fergus [23] used deconvolutional neural networks to visualize features learned by CNNs.",
"In addition, they created heatmaps by monitoring class changes systematic cover up of portions of the images.",
"Class Activation Maps (CAM) [24], GradCAM [25], and GradCAM++ [26] reveal the parts of images which are most important to a class using global average pooling (GAP).",
"Recently, LRP has been used in the fields of text [27] where classification scores were projected back to input features for extracting relevant words for a specific prediction.",
"The method has also shown successes in model understanding in fields of sentiment analysis [28], action recognition [29], and age and gender classification [30].",
"As far as the authors are aware, this is the first time LRP has been used for the understanding of genre or design classification."
],
[
"Amazon Book Cover Dataset",
"We used the Book Cover Image to Genre datasethttps://github.com/uchidalab/book-dataset Task 1.A.",
"The dataset consists of 57,000 book cover images divided into 30 classes of equal sizes.",
"In the experiments, we used the predefined training set and test set modified for one-vs-others classification.",
"In this way, genre-wise training sets were prepared with an equal distribution of positive and negative data samples."
],
[
"Convolution Neural Networks",
"CNNs are able to tackle image recognition by implementing convolutions of learned filter-like shared weights which maintain the structural qualities of images while acting as feature extractors [2].",
"For the experiment, we implement CNNs to tackle book genre classification.",
"To use the book cover images with a CNN, they were preprocessed by scaling them to 112$\\times $ 112 pixels by 3 color channel images and by normalizing the values to be between -1 and 1.",
"The CNN used for the experiments has six convolutional layers with Rectified Linear Units (ReLU) activations and a softmax output layer.",
"The convolutional layers consisted of three layers of 10 nodes with $5\\times 5$ convolutional filters, one layer of 25 nodes with a $4\\times 4$ filter, one layer of 50 nodes with a $3\\times 3$ filter, and one layer of 100 nodes with a $1\\times 1$ filter.",
"A $2\\times 2$ maxpooling layer with stride 2 was used between each convolution layer.",
"Finally, the CNNs were trained using gradient decent with a batch size of 25 at a learning rate of 0.001 for 50,000 iterations.",
"The accuracy results for each genre is summarized in Fig.",
"REF .",
"In particular, the CNNs had difficulties with the reference classes, such as \"Engineering & Transportation,\" \"Health, Fitness and Dieting,\" \"History,\" \"Medical Books,\" and \"Reference.\"",
"Conversely, \"Children Books,\" \"Romance,\" and \"Test Preparation\" had high accuracies.",
"However, more than just classification accuracy, the purpose of this paper is to understand why the CNN's performed as such and reveal the relevant parts of the images.",
"Figure: CNN accuracy by genre."
],
[
"Layer-wise Relevance Propagation",
"The LRP algorithm and the LRP toolbox [31] aims to explain the reasoning behind the decision made by a network model which allows its users to validate classifier results.",
"LRP is mainly derived from Deep Taylor Decomposition [32], a method of decomposing network's output predictions onto its input variable.",
"The results after such a decomposition is visualized in the form of a heatmap highlighting each pixel's importance for the prediction.",
"LRP explains output function, i.e.",
"classifier's decision, which helps us to derive all of the crucial pixels for a particular prediction.",
"In Fig.",
"REF , the technique is shown in which the output value given by the network is decomposed backwards layer by layer until it reaches the input.",
"This backward decomposition of network's prediction uses local redistribution rules for assigning relevance values $R_i$ to each neuron contributing towards the output, namely $\\sum _{i} R_i = \\sum _{j} R_j =\\dots = \\sum _{k}R_k = f(x),$ where $f(x)$ is the prediction function, $R_i$ is the relevance of node $i$ in the target layer, $R_j$ is the relevance of node $j$ of the previous layer, and $R_k$ is the relevance of node $k$ of the highest layer.",
"The total amount of relevance is conserved in this equation.",
"Figure: Feed forward neural network with the (left) forward pass and the (right) backward relevance calculation.",
"The function f(x)f(x) is the prediction outcome given input xx.",
"The variables a i a_i and a j a_j are the inputs for node ii and jj, respectively.",
"R i R_i is the relevance of node ii and R j R_j is the relevance of node jj.For the experiment, we used the $\\alpha -\\beta $ decomposition formula defined by $R_i = \\sum _{j} \\left(\\alpha \\frac{(a_i w_{ij})^+}{\\sum _{i} ( a_i w_{ij}) ^ +} + \\beta \\frac{(a_i w_{ij})^-}{\\sum _{i} ( a_i w_{ij}) ^ -} \\right) R_j,$ where $\\alpha $ and $\\beta $ are hyperparameters to weight the positive values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ and the negative values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ , respectively.",
"Furthermore, $w_{ij}$ is the weight between nodes $i$ and $j$ and $a_i$ is the input to node $i$ .",
"This decomposition allows for the separation of the positive connections and the negative connections.",
"Values inside positive bracket indicates propagation of activating input messages while negative weight connections indicate deactivating input values."
],
[
"Single-Shot Multibox Detector",
"To develop a better understanding of the objects within book cover images, we employed SSD [8], a state-of-the-art deep neural network based object detection method.",
"SSD is a feed forward CNN which produces a multi-scale collection of fixed size bounding boxes and scores for object detection within the boxes.",
"A final non-maximal suppression step determines the final detections.",
"The result of SSD is bounding box regions with object classification labels.",
"Using SSD, it is possible to accurately detect multiple objects of different classes within images."
],
[
"Efficient and Accurate Scene Text Detector",
"For humans, text is an important component of book covers; it is where the title, authors, and additional information is conveyed.",
"However, a CNN may place a different importance on text than humans.",
"Thus, to analyze the relevance of text in book covers, we use EAST [9] as a text detector.",
"EAST uses a multi-channel Fully Convolutional Network (FCN) and non-maximal suppression on predicted geometric shapes to detect multi-orient text-line and word boxes."
],
[
"How CNNs Understand Book Cover Design: Qualitative Analysis",
"In this section, we have presented LRP results from main genres.",
"The analysis helped us to deduce book cover design elements contributing towards a prediction by CNN.",
"We used $\\alpha -\\beta $ decomposition formula with values of $\\alpha =2$ and $\\beta =-1$ which is suggested for networks using ReLU activation functions because it emphasizes the positive elements and de-emphasizes the negative ones [7].",
"This is important due to the ReLU activation function setting negative values to zero.",
"In the heatmaps generated by LRP under this decomposition, pixels adding positive contribution are represented in red color and the ones adding negative contribution are represented by blue color."
],
[
"Sports & Outdoors",
"Under this genre, many book covers with pictures of players playing indoor and outdoor games were seen.",
"Figure REF (a) shows LRP results on these covers, which presents significance of player's picture on the cover.",
"The first image in Fig.",
"REF (a) supports this fact with LRP being centered on players who are either playing a sport or showing some player like gesture, with car in background adding no contribution.",
"The second image in Fig.",
"REF (a) emphasizes the animal's importance for this genre's prediction.",
"Figure: Correctly recognized book covers.",
"Object classes by SSD and text by EAST are highlighted."
],
[
"Engineering & Transportation",
"For this genre, almost all the covers with vehicle pictures on their covers were classified correctly by the network.",
"With LRP in Fig.",
"REF (b), part of image containing cars or motorbikes seem to add more relevance than others.",
"The last image in the Fig.",
"REF (b) presents the cases when contribution of person image was dominated by vehicle in the image."
],
[
"Romance",
"Its obvious from the genre name that pictures of couples on the cover are going to have more relevance and LRP results showed this fact to be true.",
"However, among pictures presented in Fig.",
"REF (c), LRP depicted girls to add more relevance than men or other things.",
"The reason could reside in their physical appearance, hairs, and choice of dresses.",
"The same was demonstrated in last picture of Fig.",
"REF (c) in which girl's hair are seen to add more relevance with zero relevance coming from animal part on book cover."
],
[
"Children's Books",
"Almost all the children book covers contain pictures of cartoon characters.",
"LRP on covers from this genre showed these cartoon characters to have higher relevance.",
"An interesting result is shown in first picture of Fig.",
"REF (d), where person is depicted as an adversarial identity and importance of cartoons in cover is highlighted.",
"Some covers showed more relevance for one object in the set of objects.",
"Like, in Fig.",
"REF (d) some cartoons in last picture have higher relevance.",
"It can be because of the object placement and their orientations.",
"With the help of this information, one can make smart choices for different characters, cartoons and color patterns."
],
[
"Cookbooks, Food & Wine Books",
"Book covers in this genre most commonly contained pictures of different kinds of food.",
"The results in Fig.",
"REF (e) showed these food pictures as containment of higher relevance for this genre.",
"However, carefully analyzing LRP results we discovered shapes of dishes like bowls or spoons adding significant relevance for the genre's prediction.",
"So, this marks significance of dish shape designs on covers from this genre."
],
[
"Test Preparation",
"The genre contained covers with both text and pictorial information as shown in Fig.",
"REF (f).",
"With most contribution coming from big text content on covers.",
"Images of Fig.",
"REF (f) presents big texts to add more relevance than images of people.",
"In first image of Fig.",
"REF (f), despite of big girl face, relevance is concentrated on text area of book cover.",
"Such analysis helped us to find design elements specific to the presented genres.",
"To get more familiar with design, we also presented some cases where the network was not able to correctly classify the genre.",
"Figure REF shows some of these misclassifications, mainly from the presented genres.",
"The correct genre names are written below the image.",
"From the analysis presented above, one can easily decode the reason behind their misclassification because the designs on these book covers are not aligned with their genres which makes it obvious for network to mis-classify.",
"Here, cover from \"Sports & Outdoors\" contains birds, \"Romance\" cover contains text, \"Cookbooks, Food & Wine Books\" contain no food picture and \"Test Preparation\" cover is also without any significant feature.",
"LRP justifies all these covers misclassification by highlighting these mentioned objects contributing towards the \"other\" class in one-vs-others."
],
[
"Experiment Setup",
"In order to quantitatively analyze LRP, we propose using SSD to detect objects and EAST to detect text within the book cover images.",
"We then use LRP to compare the relevance of objects bound by the detection methods.",
"The SSD was trained on the 2012 PASCAL Visual Object Classes (VOC) Challenge dataset [33].",
"The VOC dataset contains 20 classes, including \"person,\" six animal classes, eight vehicle classes, and seven indoor object classes.",
"While SSD trained with VOC is intended for natural scene images, it can be used with book cover images because book covers often contain many of the shared classes, such as \"person\" and \"car.\"",
"Similarly, EAST was trained on the 2015 ICDAR Robust Reading Competition dataset [34] meant for scene text detection.",
"Despite being trained for scene text, shown in Fig.",
"REF , EAST performs remarkably well on book covers for detecting text.",
"To extract object and text bounding box information, the book covers were prepared by scaling the images to $512\\times 512$ pixels by 3 color channels.",
"It is important to note that the images used for SSD and EAST were larger than the images used by the CNN used for genre classification.",
"This is due to the detection methods being much more effective at the higher resolution.",
"To accommodate this, the bounding boxes were scaled post detection and projected onto the LRP heatmaps.",
"The relevance of an object $R_{\\mathrm {obj}}$ is calculated using the sum of the relevance within the bounding box, or $R_{\\mathrm {obj}} = \\sum _{(n,m)\\in \\mathcal {B}}{R_{(n,m)}},$ where $R_{(n,m)}$ is the relevance at pixel coordinates $(n,m)$ within bounding box $\\mathcal {B}$ .",
"Figure: “Misclassified” book covers with correct genre names written below each book cover."
],
[
"LRP with Object Detection",
"A macro view of the genres can be seen by viewing the average relevance of object classes.",
"Figure REF illustrates the average object-wise relevance of each object class as detected by SSD and EAST for each book genre using the test set book cover images.",
"It should be noted that detected objects such as \"bottle\" and \"tvmonitor\" were overfit to certain book cover images because many books have plain covers which resemble bottle labels or televisions.",
"However, this does not mean that the information is useless.",
"For example, from Fig.",
"REF , \"bottle\" is more relevant for reference and nonfiction genres where plain covers are common.",
"Figure: Average object-wise relevance for text detected by EAST and each object class detected by SSD for each book genre.",
"Only object-genre combinations with five or more data points are shown.In addition, by examining the distribution of the $R_{\\mathrm {obj}}$ of specific object classes, such as \"person,\" it is possible to create associations between genres and detected objects.",
"For example, the relevance of \"person\" $R_{\\mathrm {person}}$ for each genre is shown in Fig.",
"REF .",
"The figure demonstrates that detected \"person\"s within certain genres are more relevant than other genres.",
"For instance, the genres of \"Romance\" and \"Mystery, Thriller & Suspense\" put a high average relevance in \"person.\"",
"This indicates that \"person\" is important for the CNNs of those categories.",
"In addition, mentioned in Section REF and shown in Fig.",
"REF (f), people are common in \"Test Preparation\" but are not necessarily relevant.",
"This is supported by Fig.",
"REF which indicates that on average, \"person\" has very little relevance.",
"Distributions for the other object classes are provided in the supplemental material.",
"Figure: Box plot of relevance of \"person\" R person R_{\\mathrm {person}} for each genre.",
"The boxes represent the first through third quartile and the mean is in red.",
"The whiskers mark the minimum and maximum datum."
],
[
"LRP with Text Detection",
"Figure REF also reveals that the average relevance of text is low.",
"The reasoning behind this phenomenon can be explained by Fig.",
"REF .",
"The figure shows that the majority of the detected text boxes have a very small relevance $R_{\\mathrm {text}}$ , but there are some text boxes have a higher relevance.",
"For most genres, the title text contains a significant amount of relevance determined by LRP, but the small descriptive text carries very little relevance.",
"Figure REF (f) in particular demonstrates this with the large title text having a high relevance and much of the smaller descriptive text having near zero relevance.",
"Figure: Box plot of relevance of \"text\" R text R_{\\mathrm {text}} for each genre.",
"The boxes represent the first through third quartile and the mean is in red.",
"The whiskers mark the minimum and maximum datum."
],
[
"Conclusion",
"In this paper, we presented importance of design in book covers belonging to a specific genre.",
"The application of LRP on the book cover dataset showed genre specific book cover features.",
"The method described most relevant parts of input book cover contributing towards a genre prediction by CNN.",
"We also presented quantitative analysis of LRP using an object detection method, SSD, and a text detection method, EAST.",
"The analysis further demonstrates that genre classification heavily relies on specific objects for each genres."
],
[
"ACKNOWLEDGEMENT",
"This research was partially supported by MEXT-Japan (Grant No.J17H06100)."
]
] | 1808.08402 |
[
[
"Towards Tight Approximation Bounds for Graph Diameter and Eccentricities"
],
[
"Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.",
"There are no known very efficient algorithms for computing the Diameter exactly.",
"Thus, much research has been devoted to how fast this parameter can be approximated.",
"Chechik et al.",
"showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\\tilde{O}(m^{3/2})$ time.",
"Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs.",
"Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$.",
"It was, however, completely plausible that a $3/2$-approximation is possible in linear time.",
"In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$.",
"Another fundamental set of graph parameters are the Eccentricities.",
"The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$.",
"Chechik et al.",
"showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\\tilde{O}(m^{3/2})$ time and Abboud et al.",
"showed that no $O(n^{2-\\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs.",
"We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH.",
"We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\\delta$ factor in $\\tilde{O}(m/\\delta)$ time for any $0<\\delta<1$.",
"This beats all Eccentricity algorithms in Cairo et al."
],
[
"Introduction",
"Among the most important graph parameters are the graph's Diameter and the Eccentricities of its vertices.",
"The Eccentricity of a vertex $v$ is the (shortest path) distance to the furthest vertex from $v$ , and the Diameter is the largest Eccentricity over all vertices in the graph.",
"The Eccentricities and Diameter measure how fast information can spread in networks.",
"Efficient algorithms for their computation are highly desired (see e.g.",
"[40], [9], [33]).",
"Unfortunately, the fastest known algorithms for these parameters are very slow on large graphs.",
"For unweighted graphs on $n$ vertices and $m$ edges, the fastest Diameter algorithm runs in $\\tilde{O}(\\min \\lbrace mn,n^\\omega \\rbrace )$ $\\tilde{O}$ notation hides polylogarithmic factors time [16] where $\\omega <2.373$ is the exponent of square matrix multiplication [52], [32], [45].",
"For weighted graphs, the fastest Eccentricity and Diameter algorithms actually compute all distances in the graph, i.e.",
"they solve the All-Pairs Shortest Paths (APSP) problem.",
"The fastest known algorithms for APSP in weighted graphs run in $\\min \\lbrace \\tilde{O}(mn),n^3/\\exp (\\sqrt{\\log n})\\rbrace $ [53], [35], [37].",
"Whether one can solve Diameter faster than APSP is a well-known open problem (e.g.",
"see Problem 6.1 in [20] and [2], [17]).",
"Whether one can solve Eccentricities faster than APSP was addressed by [50] (for dense graphs) and by [34] (for sparse graphs).",
"Vassilevska W. and Williams [50] showed that Eccentricities and APSP are equivalent under subcubic reductions, so that either both of them admit $O(n^{3-\\varepsilon })$ time algorithms for $\\varepsilon >0$ , or neither of them do.",
"Lincoln et al.",
"[34] proved that under a popular conjecture about the complexity of weighted Clique, the $O(mn)$ runtime for Eccentricities cannot be beaten by any polynomial factor for any sparsity of the form $m=\\Theta (n^{1+1/k})$ for integer $k$ .",
"Due to the hardness of exact computation, efficient approximation algorithms are sought.",
"A folklore $\\tilde{O}(m+n)$ time algorithm achieves a 2-approximation for Diameter in directed weighted graphs and a 3-approximation for Eccentricities in undirected weighted graphs.",
"Aingworth et al.",
"[2] presented an almost-$3/2$ approximation An almost-$c$ approximation of $X$ is an estimate $X^{\\prime }$ so that $X\\le X^{\\prime }\\le cX+O(1)$ .",
"algorithm for Diameter running in $\\tilde{O}(n^2+m\\sqrt{n})$ time.",
"Roditty and Vassilevska W. [41] improved the result of [2] with an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.",
"Chechik et al.",
"[21] obtained a (genuine) $3/2$ approximation algorithm for Diameter (in directed graphs) and a (genuine) $5/3$ -approximation algorithm for Eccentricities (in undirected graphs), running in $\\tilde{O}(\\min \\lbrace m^{3/2},mn^{2/3}\\rbrace )$ time.",
"These are the only known non-trivial approximation algorithms for Diameter in directed graphs.",
"So far, there are no known faster than $mn$ algorithms for approximating Eccentrities in directed graphs within any constant factor.",
"Cairo et al.",
"[15] generalized the above results for undirected graphs and obtained a time-approximation tradeoff: for every $k\\ge 1$ they obtained an $\\tilde{O}(mn^{1/(k+1)})$ time algorithm that achieves an almost-$2-1/2^k$ approximation for Diameter and an almost $3-4/(2^k+1)$ -approximation for Eccentricities."
],
[
"Our contributions.",
"We address the following natural question: Main Question: Are the known approximation algorithms for Diameter and Eccentricities optimal?",
"A partial answer is known.",
"Under the Strong Exponential Time Hypothesis (SETH), every $3/2-\\varepsilon $ approximation algorithm (for $\\varepsilon >0$ ) for Diameter in undirected unweighted graphs with $O(n)$ nodes and edges must use $n^{2-o(1)}$ time [41].",
"Similarly, every $5/3-\\varepsilon $ approximation algorithm for the Eccentricities of undirected unweighted graphs with $O(n)$ nodes and edges must use use $n^{2-o(1)}$ time [7].",
"This however does not answer the question of whether the runtimes of the known $3/2$ and $5/3$ approximation algorithms can be improved.",
"It is completely plausible that there is a $3/2$ -approximation algorithm for Diameter or a $5/3$ -approximation for Eccentricities running in linear time.",
"We address our Main Question for both sparse and dense graphs.",
"Our results are shown in Table REF .",
"Table: Our results.",
"All of the lower bounds hold even for sparse graphs.",
"SS-TT Diameter is a variant of Diameter introduced later in this section."
],
[
"Sparse graphs.",
"Our first result (restated as Theorem REF ) regards approximating Diameter in undirected unweighted sparse graphs.",
"Theorem 1 ($3/2$ -Diameter Approx.",
"is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $8/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Diameter of an undirected unweighted sparse graph.",
"In particular, any $3/2$ -approximation algorithm in sparse graphs must take $n^{3/2-o(1)}$ time.",
"Hence the $\\tilde{O}(m^{3/2})$ time $3/2$ -approximation algorithm of [41], [21] is optimal in two ways: improving the approximation ratio to $3/2-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([41]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $8/5$ .",
"Our lower bound instance says that in $O(m^{3/2-\\delta })$ time one cannot return 6 when the Diameter is 8.",
"One may be tempted to extend the above lower bound, by showing that, say, in $O(m^{4/3-\\delta })$ time one cannot even return 5 when the Diameter is 8.",
"This approach, however fails: in Theorem REF we give an $O(m^2/n)$ time algorithm that does return 5 in this case, and in general when the Diameter is $2h$ , it returns at least $h+1$ .",
"Notice that when the Diameter is $2h$ , the folklore linear time algorithm returns an estimate of only $h$ .",
"Hence for sparse graphs, our algorithm runs in linear time and outperforms the folklore algorithm.",
"Also, for constant even Diameter, it gives a better than 2 approximation.",
"We obtain stronger Diameter hardness results for weighted graphs and for directed unweighted graphs.",
"In particular, assuming SETH: For weighted sparse graphs, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $5/3-\\varepsilon $ Diameter approximation (for $\\varepsilon >0$ ) (Theorem REF ).",
"For directed unweighted sparse graphs, using a general time-accuracy tradeoff lower bound (Theorem REF ), we show that no near-linear time algorithm can achieve an approximation factor better than $5/3$ .",
"We summarize our Diameter lower bounds and compare them to the known upper bounds in Figure REF .",
"Figure: Our hardness results for Diameter.",
"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.",
"Black lines represent lower bounds.",
"Black dots represent existing algorithms.",
"Blue dots represent existing algorithms whose approximation is potentially off by an additive term (the algorithms of ).",
"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.We address our Main Question for Eccentricities as well.",
"Our main result for Eccentricities is Theorem REF .",
"Its first consequence is as follows: Theorem 2 ($5/3$ -Eccentricities Alg.",
"is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $9/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.",
"In other words, the $\\tilde{O}(m^{3/2})$ time $5/3$ -approximation algorithm of [41], [21] is tight in two ways.",
"Improving the approximation ratio to $5/3-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([7]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $9/5$ .",
"More generally, we prove (in Theorem REF ): for every $k\\ge 2$ , under SETH, distinguishing between Eccentricities $2k-1$ and $4k-3$ in unweighted undirected sparse graphs requires $n^{1+1/(k-1)-o(1)}$ time.",
"Thus, no near-linear time algorithm can achieve a $2-\\varepsilon $ -approximation for Eccentricities for $\\varepsilon >0$ .",
"The best (folklore) near-linear time approximation algorithm for Eccentricities currently only achieves a 3-approximation, and only in undirected graphs.",
"There is no known constant factor approximation algorithm for directed graphs!",
"Is our limitation result for linear time Eccentricity algorithms far from the truth?",
"We show that our lower bound result is essentially tight, for both directed and undirected graphs by producing the first non-trivial near-linear time approximation algorithm for the Eccentricities in weighted directed graphs (Theorem REF ).",
"Theorem 3 (2-Approx.",
"for Eccentricities in near-linear time.)",
"Under SETH, no $n^{1+o(1)}$ time algorithm can output a $2-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.",
"For every $\\delta >0$ , there is an $\\tilde{O}(m/\\delta )$ time algorithm that produces a $(2+\\delta )$ -approximation for the Eccentricities of any directed weighted graph.",
"The approximation hardness result is the first result within fine-grained complexity that gives tight hardness for near linear time algorithms.",
"The $2+\\delta $ approximation ratio that our algorithm produces beats all approximation ratios for Eccentricities given by Cairo et al. [15].",
"It also constitutes the first known constant factor approximation algorithm for Eccentricities in directed graphs.",
"Our approximation algorithm also implies as a corollary an approximation algorithm for the Source Radius problemThe Source Radius problem is a natural extension of the undirected Radius definition.",
"The goal is to return $\\min _x\\max _vd(x,v)$ .",
"studied in [7] with the same runtime and approximation factor ($2+\\delta $ ).",
"Abboud et al.",
"[7] showed that, under the Hitting Set Conjecture, any $(2-\\varepsilon )$ -approximation algorithm (for $\\varepsilon >0$ ) for Source Radius requires $n^{2-o(1)}$ time, and hence our Source Radius algorithm is also essentially tight.",
"Our lower bound in Theorem REF holds already for undirected unweighted graphs, and the upper bound works even for directed weighted graphs.",
"The algorithm produces a $(2+\\delta )$ -approximation, which while close, is not quite a 2-approximation.",
"We design (in Theorem REF ) a genuine 2-approximation algorithm running in $\\tilde{O}(m\\sqrt{n})$ time that also works for directed weighted graphs.",
"We then complement it (in Theorem REF ) with a tight lower bound under SETH: in sparse directed graphs, if you go below factor 2 in the accuracy, the runtime blows up to quadratic.",
"Theorem 4 (Tight 2-Approx.",
"for Eccentricities) Under SETH, no $n^{2-\\delta }$ time algorithm for $\\delta >0$ can output a $2-\\varepsilon $ approximation for the Eccentricities of a directed unweighted sparse graph.",
"There is an $\\tilde{O}(m\\sqrt{n})$ time algorithm that produces a 2-approximation for the Eccentricities of any directed weighted graph.",
"We thus give an essentially complete answer to our Main Question for Eccentricities.",
"Our results are summarized in Figures REF and REF .",
"Figure: Our algorithms and hardness results for Eccentricities.",
"The lower bounds are for unweighted graphs and the upper bounds are for weighted graphs.",
"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.",
"Black lines represent lower bounds.",
"Black dots represent existing algorithms (including our algorithm at (2,3/2)(2,3/2) in figure b).",
"Blue dots represent existing algorithms whose position may not be exactly as it appears in the figure.",
"Here, the blue dots represent our (2+δ)(2+\\delta )-approximation algorithm running in O ˜(m/δ)\\tilde{O}(m/\\delta ) time.",
"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.Our conditional lower bounds for both Diameter and Eccentricities are all based on a common construction: a conditional lower bound for a problem called $S$ -$T$ Diameter.",
"In $S$ -$T$ Diameter, the input is a graph $G=(V,E)$ and two subsets $S,T\\subseteq V$ , not necessarily disjoint, and the output is $D_{S,T}:=\\max _{s\\in S,t\\in T} d(s,t)$ .",
"$S$ -$T$ Diameter is a problem of independent interest.",
"It is related to the bichromatic furthest pair problem studied in geometry (e.g.",
"as in [29]), but for graphs (if we set $T=V\\setminus S$ ).",
"It is easy to see that if one can compute the $S$ -$T$ Diameter, then one can also compute the Diameter in the same time: just set $S=T=V$ .",
"We show that actually, when it comes to exact computation, the $S$ -$T$ Diameter and Diameter in weighted graphs are computationally equivalent (Theorem REF ).",
"We show that $S$ -$T$ Diameter also has similar approximation algorithms to Diameter.",
"We give a 3-approximation running in linear time (Claim REF based on the folklore Diameter 2-approximation algorithm), and a 2-approximation running in $\\tilde{O}(m^{3/2})$ time (Theorem REF based on the $3/2$ -approximation algorithm of [41], [21]).",
"We prove the following lower bound for $S$ -$T$ Diameter (restated as Theorem REF ), the proof of which is the starting point for all of our conditional lower bounds.",
"Theorem 5 Under SETH, for every $k\\ge 2$ , every algorithm that can distinguish between $S$ -$T$ Diameter $k$ and $3k-2$ in undirected unweighted graphs requires $n^{1+1/(k-1)-o(1)}$ time.",
"Theorem REF implies that under SETH, our aforementioned 2 and 3-approximation algorithms are optimal.",
"For all of our lower bounds, we also address the question of whether they can be extended to higher values of Diameter and Eccentricities.",
"All of our lower bounds, with the exception of directed Eccentricities, are of the form “any algorithm that can distinguish between Diameter (or Eccentricity) $a$ and $b$ requires a certain amount of time\" for small values of $a$ and $b$ .",
"This doesn't exclude the possibility of an algorithm that distinguishes between higher Diameters (or Eccentricities) of the same ratio i.e.",
"between $a\\ell $ and $b\\ell $ for some $\\ell $ .",
"For weighted Diameter, our lower bound easily extends to higher values of Diameter by simply scaling up the edge weights.",
"For $S-T$ Diameter and undirected Eccentricities, our lower bounds easily extend to higher values of Diameter and Eccentricities by simply subdividing the edges.",
"For unweighted directed Diameter, our lower bound extends to higher values of Diameter with a slight loss in approximation factor by subdividing some of the edges.",
"For unweighted undirected Diameter, our lower bound does not seem to easily extend to higher values of Diameter."
],
[
"Dense graphs.",
"Can we address our Main Question for dense graphs as well?",
"In particular, can we extend our runtime lower bounds of the form $n^{1+1/\\ell -o(1)}$ to $mn^{1/\\ell -o(1)}$ , thus matching the known algorithms for larger values of $m$ ?",
"We show that the answer is “no”.",
"For undirected unweighted graphs, we obtain $\\tilde{O}(n^2)$ time algorithms for Diameter achieving an almost $3/2$ -approximation (Theorem REF ), and for all Eccentricities achieving an almost $5/3$ -approximation algorithm (Theorem REF ).",
"These algorithms run in near-linear time in dense graphs, improving the previous best runtime of $\\tilde{O}(m\\sqrt{n})$ by Roditty and Vassilevska W. [41], and subsuming (for dense unweighted graphs) the results of Cairo et al. [15].",
"Theorem 6 There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph with Diameter $D=3h+z$ for $h\\ge 0,z\\in \\lbrace 0,1,2\\rbrace $ , returns an extimate $D^{\\prime }$ such that $2h-1\\le D^{\\prime }\\le D$ if $z=0,1$ and $2h\\le D^{\\prime }\\le D$ if $z=2$ .",
"There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph returns estimates $\\varepsilon ^{\\prime }(v)$ of the Eccentricities $\\varepsilon (v)$ of all vertices such that $3\\varepsilon (v)/5-1\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v$ .",
"We also show (in Theorem REF ) that one can improve the estimates slightly with an $O(n^{2.05})$ time algorithm."
],
[
"Related work",
"The fastest known algorithm for APSP in dense weighted graphs is by R. Williams [53] and runs in $O(n^3/2^{\\Theta (\\sqrt{\\log n})})$ time.",
"For sparse undirected graphs, the fastest known APSP algorithm is by Pettie [35] running in $O(mn+n^2\\log \\log n)$ time.",
"The fastest APSP algorithm for sparse undirected weighted graphs is by Pettie and Ramachandran [37] and runs in $O(mn\\log \\alpha (m,n))$ time.",
"For APSP on undirected unweighted graphs with $m>n\\log \\log n$ , Chan [17] presented an $O(mn \\log \\log n/\\log n)$ time algorithm.",
"In graphs with small integer edge weights bounded in absolute value by $M$ , APSP can be computed in $\\tilde{O}(Mn^\\omega )$ time (by Shoshan and Zwick [46] building upon Seidel [43] and Alon, Galil and Margalit [6]) in undirected graphs and in $\\tilde{O}(M^{0.681}n^{2.5302})$ time (by Zwick [55]) in directed graphs.",
"Zwick [55] also showed that APSP in directed weighted graphs admits an $(1+\\varepsilon )$ -approximation algorithm for any $\\varepsilon >0$ , running in time $\\tilde{O}(n^\\omega /\\varepsilon \\log (M/\\varepsilon ))$ .",
"For Diameter in graphs with integer edge weights bounded by $M$ , Cygan et al.",
"[16] obtained an algorithm running in time $\\tilde{O}(Mn^\\omega )$ .",
"The pioneering work of Aingworth et al.",
"[2] on Diameter and shortest paths approximation was the root to many subsequent works.",
"Building upon Aingworth et al.",
"[2], Dor, Halperin and Zwick [24] presented additive approximation algorithms for APSP in undirected unweighted graphs, achieving among other things, an additive 2-approximation in $\\tilde{O}(n^{7/3})$ time (notably, the best known bound on $\\omega $ is $>7/3$ ).",
"They also presented an $\\tilde{O}(n^2)$ time additive $O(\\log n)$ -approximation algorithm.",
"These algorithms were generalized by Cohen and Zwick [23] who showed that in undirected weighted graphs APSP has a (multiplicative) 3-approximation in $\\tilde{O}(n^2)$ time, a $7/3$ -approximation in $\\tilde{O}(n^{7/3})$ time, and a 2-approximation in $\\tilde{O}(n \\sqrt{mn})$ time.",
"Baswana and Kavitha [11] presented an $\\tilde{O}(m\\sqrt{n}+n^2)$ time multiplicative 2-approximation algorithm and an $\\tilde{O}(m^{2/3} n+n^2)$ time $7/3$ -approximation algorithm for APSP in weighted undirected graphs.",
"Spanners are closely related to shortest paths approximation.",
"A subgraph $H$ is an $(\\alpha , \\beta )$ -spanner of $G=(V,E)$ if for every $u,v\\in V$ , $d_H(u,v) \\le \\alpha \\cdot d_G(u,v) + \\beta $ , where $d_{G^{\\prime }}(u,v)$ is the distance between $u$ and $v$ in $G^{\\prime }$ .",
"Any weighted undirected graph has a $(2k-1,0)$ -spanner with $O(n^{1+1/k})$ edges [3].",
"Baswana and Sen [14] presented a randomized linear time algorithm for constructing a $(2k-1,0)$ -spanner with $O(kn^{1+1/k})$ edges.",
"Dor, Halperin and Zwick [24] showed that a $(1,2)$ -spanner with $O(n^{1.5})$ edges can be constructed in $\\tilde{O}(n^2)$ time.",
"Elkin and Peleg [25] showed that for every integer $k\\ge 1$ and $\\varepsilon >0$ there is a $(1+\\varepsilon , \\beta )$ -spanner with $O(\\beta n^{1+1/k})$ edges, where $\\beta $ depends on $k$ and $\\varepsilon $ but is independent of $n$ .",
"Baswana et.",
"al [12] presented a $(1,6)$ -spanner with $O(n^{4/3})$ edges.",
"Woodruff [54] presented an $\\tilde{O}(n^2)$ time algorithm that computes a $(1,6)$ -spanner with $O(n^{4/3})$ edges.",
"Chechik [18] presented a $(1,4)$ -spanner with $O(n^{7/5})$ edges.",
"Recently, Abboud and Bodwin[1] showed that there is no additive spanner with constant error and $O(n^{4/3-\\varepsilon })$ edges.",
"Thorup and Zwick [48] introduced the notion of distance oracles, a data structure that stores approximate distances for a weighted undirected graph.",
"Thorup and Zwick designed a distance oracle that for any $k$ takes $O(mn^{1/k})$ time to construct and, is of size is $O(kn^{1+1/k})$ , and given a pair of vertices $u,v \\in V$ it returns in $O(k)$ time a $(2k-1)$ -approximation for $d(u,v)$ .",
"Baswana and Sen [13] improved the construction time to $O(n^2)$ for unweighted graphs.",
"Baswana and Kavitha [11] extended the $O(n^2)$ construction time to weighted graphs.",
"Subsequently, Baswana, Gaur, Sen, and Upadhyay [10] obtained subquadratic construction time in unweighted graphs, at the price of having additive constant error in addition to the $2k-1$ multiplicative error.",
"Chechik [19] gave an oracle with space $O(n^{1+1/k})$ and $O(1)$ query time, which like previous work, returns a $(2k-1)$ -approximation.",
"Pǎtraşcu and Roditty [38] obtained a distance oracle that uses $\\tilde{O}(n^{5/3})$ space, has $O(1)$ query time, and returns a $(2k+1)$ -approximation.",
"Sommer [44] presented an $\\tilde{O}(n^2)$ time algorithm that constructs such a distance oracle.",
"The construction time was recently improved to $O(n^2)$ by Knudsen [30].",
"Pǎtraşcu et.",
"al [39] presented infinity many distance oracles with fractional approximation factors that for graphs with $m=\\tilde{O}(n)$ converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.",
"Thorup and Zwick [47] also extended their techniques from [48] to compact routing schemes.",
"The lower bounds presented in this paper were inspired by a lower bound by Pǎtraşcu and Roditty [38] who showed conditional hardness based on a conjecture on the hardness of a set intersection problem for the space usage of any distance oracle that can distinguish between distances 3 and 7."
],
[
"Organization",
"In Section we prove our lower bounds for $S$ -$T$ Diameter which serve as a basis for the rest of our lower bounds.",
"We also show equivalence between Diameter and $S$ -$T$ Diameter.",
"In Section we prove our lower bounds for Eccentricities: one for directed graphs and one for undirected graphs.",
"In Section we prove our lower bounds for Diameter.",
"This section is divided into four subsections, one for each of the results in Table REF .",
"In Section we describe our algorithms for sparse graphs: our 2-approximation and $(2+\\delta )$ -approximation for Eccentricities, our 2-approximation and 3-approximation for $S$ -$T$ Diameter, and our less than 2-approximation for Diameter.",
"In Section we describe our algorithms for dense graphs: our nearly $3/2$ -approximations for Diameter and our nearly $5/3$ -approximations for Eccentricities."
],
[
"Preliminaries",
"Let $G=(V,E)$ be a graph, where $|V|=n$ and $|E|=m$ .",
"For every $u,v\\in V$ let $d_G(u,v)$ be the length of the shortest path from $u$ to $v$ .",
"When the graph $G$ is clear from the context we omit the subscript $G$ .",
"The eccentricity $\\varepsilon (v)$ of a vertex $v$ is defined as $\\max _{u\\in V} d(v,u)$ .",
"The diameter $D$ of a graph is $\\max _{v\\in V}\\varepsilon (v)$ .",
"In a directed graph we have $\\varepsilon ^{out}(v)=\\max _{u\\in V} d(v,u)$ (resp., $\\varepsilon ^{in}(v)=\\max _{u\\in V} d(u,v)$ ).",
"Let $deg(v)$ be the degree of $v$ and let $N_s(u)$ be the set of the $s$ closest vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.",
"In a directed graph let $deg^{out}(v)$ (resp., $deg^{in}(v)$ ) be the outgoing (incoming) degree of $v$ .",
"Let $N^{\\text{out}}_s(v)$ (resp., $N^{\\text{in}}_s(v)$ ) be the set of the $s$ closest outgoing (incoming) vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.",
"For a subset $S\\subseteq V$ of vertices and a vertex $v \\in V$ we write $d(S,v):=\\min _{s \\in S}d(s,v)$ to denote the distance from the set $S$ to the vertex $v$ .",
"Let $k\\ge 2$ .",
"The $k$ -Orthogonal Vectors Problem ($k$ -OV) is as follows: given $k$ sets $S_1,\\ldots ,S_k$ , where each $S_i$ contains $n$ vectors in $\\lbrace 0,1\\rbrace ^d$ , determine whether there exist $v_1\\in S_1,\\ldots ,v_k\\in S_k$ so that their generalized inner product is 0, i.e.",
"$\\sum _{i=1}^d \\prod _{j=1}^k v_j[i]=0$ .",
"R. Williams [51] (see also [49]) showed that if for some $\\varepsilon >0$ there is an $n^{k-\\varepsilon } \\text{\\rm poly~} (d)$ time algorithm for $k$ -OV, then CNF-SAT on formulas with $N$ variables and $m$ clauses can be solved in $2^{N(1-\\varepsilon /k)} \\text{\\rm poly~} (m)$ time.",
"In particular, such an algorithm would contradict the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [28] which states that for every $\\varepsilon >0$ there is a $K$ such that $K$ -SAT on $N$ variables cannot be solved in $2^{(1-\\varepsilon )N} \\text{\\rm poly~} N$ time (say, on a word-RAM with $O(\\log N)$ bit words).",
"This also motivates the following $k$ -OV Conjectures (implied by SETH) for all constants $k\\ge 2$ : $k$ -OV requires $n^{k-o(1)}$ time on a word-RAM with $O(\\log n)$ bit words.",
"Most of our conditional lower bounds are based on the $k$ -OV Conjecture for a particular constant $k$ , and thus they also hold under SETH.",
"A main motivation behind SETH is that despite decades of research, the best upper bounds for $K$ -SAT on $N$ variables and $M$ clauses remain of the form $2^{N(1-c/K)} \\text{\\rm poly~} {(M)}$ for constant $c$ (see e.g.",
"[27], [36], [42]).",
"The best algorithms for the $k$ -OV problem for any constant $k\\ge 2$ on $n$ vectors and dimension $c\\log n$ run in time $n^{2-1/O(\\log c)}$ (Abboud, Williams and Yu [8] and Chan and Williams [22])."
],
[
"$S$ -{{formula:678000a5-c252-477a-976e-b4f5fbe676e0}} Diameter hardness",
"In the following we will prove that under SETH, our $S$ -$T$ Diameter algorithms are essentially optimal.",
"We prove the following theorem: Theorem 7 Let $k\\ge 2$ be an integer.",
"There is an $O(k n^{k-1} d^{k-1})$ time reduction that transforms any instance of $k$ -OV on sets of $n$ $d$ -dimensional vectors into a graph on $O(n^{k-1}+k n^{k-2} d^{k-1})$ nodes and $O(k n^{k-1} d^{k-1})$ edges and two disjoint sets $S$ and $T$ on $n^{k-1}$ nodes each, so that if the $k$ -OV instance has a solution, then $D_{S,T}\\ge 3k-2$ , and if it does not, $D_{S,T}\\le k$ .",
"From Theorem REF we get that if there is some $k\\ge 2$ , $\\varepsilon >0$ and $\\delta >0$ so that there is an $O(M^{1+1/(k-1)-\\varepsilon })$ time $(3-2/k-\\delta )$ -approximation algorithm for $S$ -$T$ Diameter in $M$ -edge graphs, then $k$ -OV has an $n^{k-\\gamma } \\text{\\rm poly~} (d)$ -time algorithm for some $\\gamma >0$ and SETH is false.",
"We obtain an immediate corollary.",
"Corollary 8 For $S$ -$T$ Diameter, under SETH, there is no $(2-\\varepsilon )$ -approximation algorithm running in $O(m^{2-\\delta })$ time for any $\\varepsilon >0,\\delta >0$ , no $O(m^{3/2-\\delta })$ time $7/3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0,\\delta >0$ , no $(m+n)^{1+o(1)}$ time, $3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0$ .",
"We will prove the following more detailed theorem, which will be useful for our Diameter lower bounds.",
"Theorem 9 Let $k\\ge 2$ .",
"Given a $k$ -OV instance consisting of sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.",
"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .",
"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .",
"$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .",
"Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .",
"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .",
"If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .",
"Suppose the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ .",
"Let $t=k-2$ .",
"Let $s$ be such that $0\\le s\\le t$ .",
"Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .",
"Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .",
"Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .",
"Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .",
"Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .",
"Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .",
"For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ ."
],
[
"Proof of Theorem ",
"We will prove the theorem for $k=t+2$ for any $t\\ge 0$ .",
"We will create a layered graph $G$ on $t+3$ layers, $L_0,\\ldots ,L_{t+2}$ , where the edges go only between adjacent layers $L_i,L_{i+1}$ .",
"We will set $S=L_0$ and $T=L_{t+2}$ for the $S$ -$T$ Diameter instance.",
"In particular, $D_{S,T}\\ge t+2$ because of the layering.",
"Let us describe the vertices of $G$ .",
"$L_0$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(a_0,a_1,\\ldots , a_t)$ where for each $i$ , $a_i\\in W_i$ .",
"Similarly, $L_{t+2}$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(b_1,b_2,\\ldots , b_{t+1})$ where for each $i$ , $b_i\\in W_i$ .",
"Layer $L_1$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(a_0,\\ldots ,a_{t-1}, \\bar{x})$ where for each $i$ , $a_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .",
"Similarly, $L_{t+1}$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(b_2,\\ldots ,b_{t+1}, \\bar{x})$ where for each $i$ , $b_i\\in W_i$ and $\\bar{x}$ is a $(t+1)$ -tuple of coordinates.",
"For every $j\\in \\lbrace 2,\\ldots ,t\\rbrace $ , $L_j$ consists of $n^t d^{t+1}$ vertices $(a_0, \\ldots , a_{t-j},b_{t+3-j},\\ldots , b_{t+1},\\bar{x})$ , where for each $i$ , $a_i\\in W_i$ , $b_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .",
"In other words, there is a vector from $W_i$ for every $i\\notin \\lbrace t-j+1,t-j+2\\rbrace $ .",
"Now let us define the edges.",
"Consider a node $(a_0,\\ldots ,a_t)\\in L_0$ .",
"For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(a_0,\\ldots ,a_t)$ to $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ if and only if for every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , $a_j$ is 1 in coordinates $x_0,\\ldots ,x_{t-j}$ .",
"For any $i\\in \\lbrace 1,\\ldots ,t\\rbrace $ let's define the edges between $L_i$ and $L_{i+1}$ .",
"For $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_i$ Here if $i=1$ , there are no $b$ 's in the tuple.",
"and for any $c_{t+2-i}\\in W_{t+2-i}$ , add an edge to $(a_0,\\ldots ,a_{t-i-1},c_{t+2-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_{i+1}$ .",
"Here we “forget” vector $a_{t-i}$ and replace it with $c_{t+2-i}$ , leaving everything else the same.",
"Finally, the edges between $L_{t+1}$ and $L_{t+2}$ are as follows.",
"Consider some $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .",
"For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(b_1,\\ldots ,b_{t+1})$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ if and only if for every $j\\in \\lbrace 1,\\ldots ,{t+1}\\rbrace $ , $b_j$ is 1 in coordinates $x_{t+1-j},\\ldots ,x_t$ .",
"Figure REF shows the construction of the graph for $t=2$ .",
"Figure: The reduction graph from (t+2)(t+2)-OV for t=2t=2.",
"The figure depicts when a path of length t+2t+2 exists between arbitrary a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and b 1 b 2 b 3 ∈L t+2 b_1b_2b_3\\in L_{t+2}.",
"It also shows that when there is a path of length t+2t+2 between a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and a 1 a 2 a 3 ∈L t+2 a_1a_2a_3\\in L_{t+2}, a 0 ,a 1 ,a 2 ,a 3 a_0,a_1,a_2,a_3 cannot be an orthogonal 4-tuple.An important claim is as follows: Claim 10 For every $\\bar{x}$ , each $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ is at distance $t$ to every $(b_2,\\ldots ,b_t,\\bar{x})\\in L_{t+1}$ .",
"Consider the path starting from $(a_0,\\ldots ,a_{t-1},\\bar{x})$ , and then for each $i\\ge 1$ following the edges $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1},\\bar{x})\\in L_i$ to $(a_0,\\ldots ,a_{t-1-i},b_{t+2-i},\\ldots ,b_{t+1},\\bar{x})\\in L_{i+1}$ , until we reach $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ .",
"This path exists by construction and has length $t$ .",
"Now we proceed to prove the bounds on the $S$ -$T$ Diameter.",
"Lemma 11 (Property 3 of Theorem REF ) If the $(t+2)$ -OV instance has no solution, then $D_{S,T}=t+2$ .",
"If the $(t+2)$ -OV instance has no solution, then for every $c_0\\in W_0,c_1\\in W_1,\\ldots ,c_{t+1}\\in W_{t+1}$ , there is some coordinate $x$ such that $c_0[x]=c_1[x]=\\ldots =c_{t+1}[x]=1$ .",
"Now consider the graph and any $(a_0,\\ldots ,a_t)\\in L_0$ , $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .",
"For every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , let $x_j$ be a coordinate so that $a_0,\\ldots ,a_{t-j},b_{t-j+1},\\ldots ,b_{t+1}$ are all 1 in $x_j$ .",
"Let $\\bar{x}=(x_0,\\ldots ,x_{t})$ .",
"By construction, $(a_0,\\ldots ,a_t)$ has an edge to $(a_0,\\ldots ,a_{t-1},\\bar{x})$ and $(b_2,\\ldots ,b_{t+1},\\bar{x})$ has an edge to $(b_1,\\ldots ,b_{t+1})$ .",
"Also, by Claim REF , $(a_0,\\ldots ,a_{t-1},\\bar{x})$ has a path of length $t$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})$ .",
"This shows that $D_{S,T}\\le t+2$ ; equality follows because the graph is layered.",
"Now we prove the guarantee for the case when an orthogonal tuple exists.",
"Lemma 12 (Property 4 of Theorem REF ) If there exist $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ that are orthogonal, then $D_{S,T}\\ge 3t+4$ .",
"To prove the lemma, we will actually prove a more general claim: Property 5 of Theorem REF .",
"Claim 13 (Property 5 of Theorem REF ) Suppose that $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ are orthogonal.",
"Let $s$ be such that $0\\le s\\le t$ .",
"Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .",
"Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .",
"Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .",
"Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .",
"Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .",
"Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .",
"If the claim is true, then using $s=0$ we get that the Diameter is at least $3t+4$ so Lemma REF is true.",
"The claim for $s>0$ is useful for the rest of our constructions.",
"We will show that the distance between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ is strictly more than $3t+2-2s$ .",
"Because the graph is layered and hence bipartite and $t+2\\equiv 3t+2\\mod {2}$ , the distance must be at least $3t-2s+4$ .",
"Let's assume for contradiction that the shortest path $P$ between $\\alpha $ and $\\beta $ is of length $\\le 3t+2-2s$ .",
"First let's look at any subpath $P^{\\prime }$ of $P$ strictly within $M=L_1\\cup \\ldots \\cup L_{t+1}$ .",
"All nodes on $P^{\\prime }$ must share the same $\\bar{x}$ .",
"Furthermore, if $P^{\\prime }$ starts with a node of $L_1$ and ends with a node of $L_{t+1}$ , as $P^{\\prime }$ needs to be a shortest path and by Claim REF , $P^{\\prime }$ must be of length exactly $t$ .",
"Next, notice that $P$ cannot go from $L_0$ to $L_{t+2}$ and then back to $L_0$ .",
"This is because it needs to end up in $L_{t+2}$ and any time it crosses over $M$ , it would need to pay a distance of $t+2$ , so $P$ would have to have length at least $3t+6>3t+2-2s$ .",
"Hence, $P$ must be of the following form: a path from $\\alpha $ through $L_0\\cup M$ back to $L_0$ (possibly containing only $\\alpha $ ), followed by a path crossing $M$ to reach $L_{t+2}$ , followed by a path through $L_{t+2}\\cup M$ to $L_{t+2}$ (possibly empty).",
"We will show that if $P$ has length $\\le 3t+2-2s$ then $P$ must contain a length $t+2$ subpath $Q$ between a node $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)\\in L_0$ , for some choices of the $w$ 's and some $q\\le t-s$ , and a node $(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})\\in L_{t+2}$ , for some choices of $v$ 's.",
"That is, this path traverses $M$ without weaving, by following $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_{t-1},\\bar{x})\\in L_1$ , $(a_0, \\ldots , a_{q},w_{q+1},\\ldots , w_{t-2},a_{t+1},\\bar{x})\\in L_2$ , $\\ldots $ , $(v_2, \\ldots , v_q, a_{q+1}, \\ldots , a_{t-s}, \\ldots , a_{t+1}, \\bar{x})\\in L_{t+1}$ .",
"Suppose we show that such a subpath exists.",
"Then by the construction of our graph we have that for every $i\\in \\lbrace 0,\\ldots ,q\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace 0,\\ldots ,t-i\\rbrace $ , and that for all $i\\in \\lbrace q+1,\\ldots ,t+1\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace t+1-i,\\ldots ,t\\rbrace $ .",
"That is, for all $i$ , $a_i[x_{t-q}]=1$ , and we get a contradiction since the $a_i$ were supposed to be orthogonal.",
"Now let $\\alpha ^*$ be the last node from $L_0$ on $P$ and let $\\beta ^*$ be the first node of $L_{t+1}$ of $P$ .",
"Let $a^*\\in L_1$ be the node right after $\\alpha ^*$ and let $b^*\\in L_{t+1}$ be the node right before $\\beta ^*$ .",
"Since the subpath of $P$ between $a^*$ and $b^*$ is within $M$ , it must share the same $\\bar{x}$ , and it must have length exactly $t$ by Claim REF .",
"We will show that the subpath $Q$ that we are looking for is the subpath of $P$ between $\\alpha ^*$ and $\\beta ^*$ .",
"Its length is exactly what we want: $t+2$ .",
"It remains to show that for some $q\\le t-s$ and some choices of $w$ 's and $v$ 's, $\\alpha ^*=(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)$ and $\\beta ^*=(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})$ .",
"Consider the path $P_1$ between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ and $\\alpha ^*$ and the path $P_2$ between $\\beta =(a_1,\\ldots ,a_{t+1})$ and $\\beta ^*$ .",
"Let $L_i$ be the layer in $M$ with largest $i$ that $P_1$ touches and let $L_j$ be the layer in $M$ with smallest $j$ that $P_2$ touches.",
"For convenience, let us define $j^{\\prime }=t+2-j$ .",
"The length of $P_1$ is then at least $2i$ and the length of $P_2$ is at least $2j^{\\prime }$ .",
"The length $|P|$ of $P$ equals $t+2+|P_1|+|P_2|\\ge t+2+2i+2j^{\\prime }=t+2+2(i+j^{\\prime })$ .",
"Since we have assumed that $|P|\\le 3t+2-2s$ , we must have that $t+2+2(i+j^{\\prime })\\le 3t+2-2s$ and hence $i+j^{\\prime }\\le t-s$ .",
"Now, since $P_1$ goes at most to $L_i$ , then from getting from $\\alpha $ to $\\alpha ^*$ , at most the last $i$ elements of $(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ can have been “forgotten”.",
"Hence, $\\alpha ^*=(a_0,\\ldots ,a_{t-\\max \\lbrace s,i\\rbrace },b_{t-\\max \\lbrace s,i\\rbrace +1},\\ldots ,b_{t-i},w_{t-i+1},\\ldots ,w_{t})$ for some $w$ 's.",
"(If $i\\ge s$ , the $b$ 's do not appear.)",
"Similarly, between $\\beta $ and $\\beta ^*$ , at most the first $j^{\\prime }$ elements of $\\beta $ can have been forgotten.",
"Thus, we have that $\\beta ^*=(v_1,\\ldots ,v_{j^{\\prime }},a_{j^{\\prime }+1},\\ldots ,a_{t+1})$ for some $v$ 's.",
"Now, since $i+j^{\\prime }\\le t-s$ , we must have that $j^{\\prime }\\le t-s-i\\le t-\\max \\lbrace s,i\\rbrace $ , and hence the path between $\\alpha ^*$ and $\\beta ^*$ is the path $Q$ we are searching for.",
"See Figure REF for an illustration of $L_i$ , $L_j$ etc.",
"in the case when $s=0$ .",
"Figure: Here PP contains at least 2 nodes in L 0 L_0 and at least 2 in L t+2 L_{t+2}, and s=0s=0."
],
[
"Equivalence between Diameter and $S$ -{{formula:bdb52e68-e562-4944-b84c-7c1359b488ce}} Diameter",
"Here we will prove that when it comes to exact computation, $S$ -$T$ -Diameter and Diameter in weighted graphs are equivalent.",
"The proof for directed graphs is much simpler, so we focus on the equivalence for undirected graphs.",
"Also, it is clear that if one can solve $S$ -$T$ Diameter, one can also solve Diameter in the same running time since one can simply set $S=T=V$ .",
"We prove: Theorem 14 Suppose that there is a $T(n,m)$ time algorithm that can compute the Diameter of an $n$ node, $m$ edge graph with nonnegative integer edge weights.",
"Then, the $S$ -$T$ Diameter of any $n$ node $m$ edge graph with nonnegative integer edge weights can be computed in $T(O(n),O(m))$ time.",
"Let $G=(V,E),S,T$ be the $S$ -$T$ Diameter instance; let $w:E\\rightarrow \\lbrace 0,\\ldots ,M\\rbrace $ be the edge weights.",
"First, we can always assume that $M$ is even: if it is not, multiply all edge weights by 2; all distances (and hence also the $S$ -$T$ Diameter) double.",
"Let $S=\\lbrace s_1,\\ldots ,s_k\\rbrace $ and $T=\\lbrace t_1,\\ldots ,t_{\\ell }\\rbrace $ .",
"Now, let $W=Mn$ .",
"First add $|S|=k$ new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ .",
"For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .",
"Let $G_S$ be this new graph.",
"Let's consider the Diameter of $G_S$ .",
"For every pair of nodes $u,v\\notin S^{\\prime }$ , the distance is the same as in $G$ .",
"For $v_i\\in S^{\\prime }$ and $x\\notin S^{\\prime }$ , the distance is $W+d_G(s_i,x)\\le W+M(n-1)<2W$ .",
"For $v_i,v_j\\in S^{\\prime }$ , the distance is $2W+d(s_i,s_j)$ .",
"Hence the Diameter of $G_S$ is $2W+\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .",
"Hence by computing the Diameter of $G_S$ , we can compute $D_S=\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .",
"We can create a similar graph $G_T$ whose Diameter will allow us to compute $D_T=\\max _{t_i,t_j\\in S} d(t_i,t_j)$ .",
"After this, let's create a graph $G_{S,T}$ as follows.",
"Add new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ and $T^{\\prime }=\\lbrace u_1,\\ldots ,u_\\ell \\rbrace $ .",
"For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .",
"For each $j\\in \\lbrace 1,\\ldots ,\\ell \\rbrace $ add a new edge $(u_j,t_j)$ of weight $W$ .",
"With a similar argument as above, the Diameter $D^{\\prime }$ of $G_{S,T}$ is $D^{\\prime }=2W+\\max _{u,v\\in S\\cup T} d_G(u,v)$ .",
"Let's assume without loss of generality that $D_S\\ge D_T$ .",
"If $D^{\\prime }> D_S$ , then $D^{\\prime }=2W+\\max _{u\\in S,v\\in T} d_G(u,v)$ , and we can compute the $S$ -$T$ Diameter of $G$ by subtracting $2W$ .",
"Now suppose that we get $D^{\\prime }\\le D_S$ ; we must have then actually gotten $D^{\\prime }=D_S$ .",
"The $S$ -$T$ Diameter of $G$ might be strictly smaller than $D_S$ .",
"We add two new nodes $x$ and $y$ to $G_{S,T}$ .",
"We add an edge $(x,y)$ of weight $2W$ , edges $(x,v_i)$ for every $v_i\\in S^{\\prime }$ of weight $D_S/2$ and (symmetrically) edges $(y,u_j)$ for every $u_j\\in T^{\\prime }$ of weight $D_S/2$ .",
"The construction of $G^{\\prime }$ is depicted in Figure REF .",
"Figure: A depiction of the construction of G ' G^{\\prime }.Let us consider the distances in this new graph $G^{\\prime }$ .",
"For every $a,b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(a,b)=d_G(a,b)<W$ as any path not in $G$ would have to use an edge of weight $W>d(a,b)$ .",
"For every $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(x,b)=W+D_S/2 +\\min _{a\\in S} d_G(a,b)\\le 2W+D_S/2$ .",
"Similarly, $d(y,b)=W+D_S/2 +\\min _{a\\in T} d_G(a,b)\\le 2W+D_S/2$ .",
"For every $v_i\\in S^{\\prime }$ , $d(x,v_i)=D_S/2$ , and $d(y,v_i)=2W+D_S/2$ .",
"For every $u_i\\in T^{\\prime }$ , $d(y,u_i)=D_S/2$ , and $d(x,u_i)=2W+D_S/2$ .",
"For every $v_i,v_j\\in S^{\\prime }$ , $d(v_i,v_j)=D_S$ .",
"For every $u_i,u_j\\in T^{\\prime }$ , $d(u_i,u_j)=D_S$ .",
"For every $v_i\\in S^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(v_i,b)\\le W+d(s_i,b)\\le 2W.$ For every $u_i\\in T^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(u_i,b)\\le W+d(t_i,b)\\le 2W.$ For every $v_i\\in S^{\\prime }$ and $u_j\\in T^{\\prime }$ , $d(v_i,u_j)$ is the minimum of $D_S+2W, D_S+2W+\\min _{s\\in S} d_G(s,t_j), D_S+2W+\\min _{t\\in t} d_G(t,s_i)$ and $2W+d_G(s_i,t_j)$ .",
"The middle two terms are $\\ge D_S+2W$ , and hence $d(v_i,u_j)=2W+\\min \\lbrace D_S,d_G(s_i,t_j)\\rbrace $ .",
"Consider $s_i\\in S, t_j\\in T$ that are the end points of the $S$ -$T$ Diameter $D$ in $G$ .",
"Then $D=d_G(s_i,t_j)$ .",
"Now, we have from before that $D\\le D_S$ , as otherwise we have computed $D$ already.",
"Hence in $G^{\\prime }$ , the distance $d(u_i,v_j)$ equals $2W+\\min \\lbrace D_S,D\\rbrace =2W+D$ We note that for any $s_i,s_j\\in S$ , and any $t\\in T$ , $d_G(s_i,s_j)\\le d_G(s_i,t)+d_G(t,s_j)\\le 2\\max _{s\\in S,t\\in T} d_G(s,t) = 2D$ .",
"Thus, $D\\ge D_S/2$ .",
"The distances in cases (1) to (5) are all $\\le 2W+D_S/2\\le 2W+D$ .",
"Hence the Diameter of $G^{\\prime }$ is actually exactly $2W+D$ ."
],
[
"Undirected graphs",
"Theorem 15 Let $k\\ge 2$ .",
"Under the $k$ -OV conjecture, every algorithm that can distinguish between eccentricity at most $2k-1$ and eccentricity at least $4k-3$ for every vertex in an $O(n)$ edge and node undirected graph, requires at least $n^{1+1/(k-1)-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.",
"Figure: The undirected Eccentricities lower bound for k=5k=5.Let's start with the $S$ -$T$ -diameter construction for $k$ obtained from a given $k$ -OV instance.",
"We remove any internal nodes if they don't have edges to one of their adjacent layers - they don't hurt the instance.",
"We have a graph on $O(n^{k-1}d^{k-2})$ vertices and edges with the following properties: (1) Suppose that the $k$ -OV instance has no $k$ -OV solution.",
"Then for every $s\\in S,t\\in T$ , $d(s,t)=k$ .",
"Also, for every $s\\in S$ and $u\\notin S\\cup T$ , $d(s,u)\\le (k-1)+k=2k-1$ since we can take a $\\le (k-1)$ length path from $u$ to some node $t\\in T$ and since $d(s,t)=k$ .",
"(2) If there is a $k$ -OV solution, there are two nodes $s\\in S,t\\in T$ with $d(s,t)\\ge 3k-2$ .",
"We modify the construction as follows.",
"For every $s\\in S$ , we create an undirected path on $k-2$ new vertices $s_1\\rightarrow s_2\\rightarrow \\ldots \\rightarrow s_{k-2}$ and add an edge $(s,s_1)$ ; let's call $s$ by $s_0$ .",
"Now, the distance between $s_0$ and $s_i$ is $i$ .",
"Add a new node $y$ and create edges $(s_{k-2},y)$ for every $s\\in S$ .",
"Now, $d(y,s_0)=k-1$ for every $s\\in S$ , and also for every $s,s^{\\prime }\\in S$ and all $i,j\\in \\lbrace 0,\\ldots ,k-2\\rbrace $ , we have that $d(s_i,s^{\\prime }_j)\\le 2k-2$ .",
"For every $s\\in S$ , $t\\in T$ , there is now potentially a new path between them, from $s$ to $y$ in $k-1$ steps, then to some other $s^{\\prime }$ in $k-1$ steps and then to $t$ using $\\ge k$ steps.",
"The length is $\\ge 2(k-1)+k=3k-2$ , so when there is a $k$ -OV solution, there is still a pair $s,t$ at distance at least $3k-2$ .",
"Now, we also attach paths to the nodes in $T$ .",
"In particular, for each $t$ , add an undirected path $t\\rightarrow t_1\\rightarrow \\ldots \\rightarrow t_{k-1}$ .",
"The distance between any $s\\in S$ and any $t_i$ is $i+d(s,t)$ .",
"Hence when there is no $k$ -OV solution, the Eccentricities of all $s_0$ for $s\\in S$ are $\\le k+(k-1)=2k-1$ , and when there is a $k$ -OV solution, there is $s\\in S, t\\in T$ so that $d(s_0,t_{k-1})\\ge (3k-2)+(k-1)=4k-3$ ."
],
[
"Directed graphs",
"Theorem 16 Under the 2-OV conjecture, for any $\\delta >0$ , any $(2-\\delta )$ -approximation algorithm for all Eccentricities in an $n$ node, $O(n)$ -edge directed unweighted graph, requires $n^{2-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.",
"Figure: The directed Eccentricities lower bound.Suppose we are given an instance of 2-OV: two sets of vectors $U,V$ over $\\lbrace 0,1\\rbrace ^d$ and we want to know whether there are $u\\in U,v\\in V$ with $u\\cdot v=0$ .",
"Let $L\\ge 1$ be any integer.",
"Let us create a directed unweighted graph $G$ ; an illustration can be found in Figure REF .",
"$G$ will have a vertex $u$ for every $u\\in U$ and a vertex $c$ for every $c\\in [d]$ .",
"Every $v\\in V$ will be represented by a directed path $v_0\\rightarrow v_1\\rightarrow \\ldots \\rightarrow v_L$ .",
"In addition, there is a directed path $P_x$ on $L$ extra nodes, $x_1\\rightarrow \\ldots \\rightarrow x_L$ so that every $u\\in U$ has directed edges $(u,x_1)$ and $(x_L,u)$ .",
"For every $u\\in U$ and every $c$ for which $u[c]=1$ , we add a directed edge $(u,c)$ .",
"For every $v$ and every $c$ for which $v[c]=1$ , we add a directed edge $(c,v_0)$ .",
"Remove any $c$ that does not have at least one edge coming from $U$ .",
"Let us consider the eccentricity of any node $u\\in U$ .",
"First, for all $u^{\\prime }\\in U$ , $d(u,u^{\\prime })\\le L+1$ since one can go through the path $P_x$ .",
"For every $c\\in [d]$ , there is at least one edge coming from some $u^{\\prime }\\in U$ , and so one can reach $c$ from $u$ by first taking $P_x$ to $u^{\\prime }$ and then using the edge $(u^{\\prime },c)$ .",
"Hence, $d(u,c)\\le L+2$ for all $c\\in [d]$ .",
"For any $v\\in V$ and $i\\in \\lbrace 1,\\ldots , L\\rbrace $ , the distance $d(u,v_i)=i+d(u,v_0)$ , and so we consider $d(u,v_0)$ .",
"If there is a $c$ for which $u[c]=v[c]=1$ , then $d(u,v_0)=2$ , and hence for all $i$ , $d(u,v_i)\\le L+2$ .",
"If no such $c$ exists and so if $u$ and $v$ are orthogonal, the only way to reach $v_0$ is potentially via $P_x$ to some other $u^{\\prime }\\in U$ which is at distance 2 to $v_0$ .",
"Hence if $u$ and $v$ are orthogonal, $d(u,v_0)=L+3$ , and hence $d(u,v_L)=2L+3$ .",
"In other words, the eccentricity of $u$ is $L+2$ if it is not orthogonal to any vectors in $V$ and it is $\\ge 2L+3$ if there is some $v$ that is orthogonal to $u$ .",
"The number of vertices in the graph is $O(nL+d)$ and the number of edges is $O(nL+nd)$ .",
"Suppose that there is a $(2-\\varepsilon )$ -approximation algorithm for all Eccentricities in graphs with $O(m)$ nodes and edges running in $O(m^{2-\\delta })$ time for some $\\varepsilon ,\\delta >0$ .",
"Then, we construct the above instance for $L=\\lceil 1/\\varepsilon \\rceil $ and run the algorithm on it.",
"The approximation returned is at least as good as a $(2-1/L)$ -approximation.",
"Hence if the diameter is at least $2L+3$ , the algorithm will return an estimate that is at least $L(2L+3)/(2L-1) > L+2$ .",
"Thus the algorithm can solve 2-OV in time $O((nL+nd)^{2-\\delta }) = O(n^{2-\\delta }d^{2-\\delta })$ , contradicting the 2-OV conjecture."
],
[
"Diameter lower bounds",
"For all of our constructions we begin with the $S$ -$T$ diameter lower bound construction from Theorem REF .",
"Here, if the $k$ -OV instance has no solution, $D_{S,T}\\le k$ and if the instance has a solution $D_{S,T}\\ge 3k-2$ .",
"To adapt this construction to Diameter, we need to ensure that if the OV instance has no solution then all pairs of vertices have small enough distance.",
"We begin by augmenting the $S$ -$T$ Diameter construction by adding a matching between $S$ and a new set $S^{\\prime }$ as well as a matching between $T$ and a new set $T^{\\prime }$ .",
"Without any further modifications, pairs of vertices $u,v\\in S\\cup S^{\\prime }$ (or $u,v\\in T\\cup T^{\\prime }$ ) could be far from one another.",
"The challenge is to add extra gadgetry to make these pairs close for “no\" instances while maintaining that in “yes\" instances the distance between the diameter endpoints $s^{\\prime }\\in S^{\\prime },t^{\\prime }\\in T^{\\prime }$ is large.",
"That is, for “yes\" instances, we want a shortest path between the diameter endpoints $s^{\\prime }$ and $t^{\\prime }$ to contain the vertex $s\\in S$ matched to $s^{\\prime }$ and the vertex $t\\in T$ matched to $t^{\\prime }$ so that we can use use the fact that $d(s,t)\\ge 3k-2$ .",
"In other words, we do not want there to be a shortcut from $s^{\\prime }$ to some vertex in $S$ that allows us to use a path of length $k$ from $S$ to $T$ .",
"For example, we cannot simply create a vertex $x$ and connect it to all vertices in $S\\cup S^{\\prime }$ because this would introduce shortcuts from $S^{\\prime }$ to $S$ .",
"We will describe some intuition for the augmentations to the graph regarding 3-OV for simplicity.",
"Recall that $s^{\\prime }\\in S^{\\prime }, t^{\\prime }\\in T^{\\prime }$ are the endpoints of the diameter and let $t$ be the vertex matched to $t^{\\prime }$ .",
"To solve the problem outlined in the above paragraph, we observe that in the “yes\" case there are three types of vertices $s\\in S$ .",
"(1) close: $d(s,t)=3$ , (2) far: $d(s,t)\\ge 7$ (property 4 of Theorem REF ), and (3) intermediate: $d(s,t)\\ge 5$ (property 5 of Theorem REF ).",
"For close $s$ , we need $d(s^{\\prime },s)$ to be large so that there is no shortcut from $s^{\\prime }$ to $t^{\\prime }$ through $s$ .",
"For far $s$ , it is ok if $d(s^{\\prime },s)$ is small because $d(s,t)$ is large enough to ensure that paths from $s^{\\prime }$ to $t^{\\prime }$ through $s$ are still long enough.",
"For intermediate $s$ , $d(s^{\\prime },s)$ cannot be small, but it also need not be large.",
"To fulfill these specifications, we add a small clique (the graph is still sparse) and connect each of its vertices to only some of the vertices in $S$ and/or $S^{\\prime }$ according to the implications of property 5 of Theorem REF .",
"When $s$ is close, we ensure that $d(s^{\\prime },s)$ is large by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique, uses an edge inside of the clique, and then goes from the clique to $s$ .",
"When $s$ is intermediate, we ensure that $d(s^{\\prime },s)$ is not too small by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique and then from the clique to $s$ (without using an edge inside of the clique).",
"These intermediate $s$ are important as they allow every vertex in the clique to have an edge to some vertex in $S$ and thus be close enough to the $T$ side of the graph in the “no\" case."
],
[
"5 vs 8 unweighted undirected construction",
"In this section we show that under the 3-OV Hypothesis, any algorithm that can distinguish between Diameter 5 and 8 in sparse undirected unweighted graphs, requires $\\Omega (n^{3/2-o(1)})$ time.",
"Theorem REF gives us the following theorem.",
"Theorem 17 Given a 3-OV instance consisting of three sets $A,B,C \\subseteq \\lbrace 0,1\\rbrace ^d$ , $|A|=|B|=|C|=n$ , we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following properties.",
"The graph consists of 4 layers of vertices $S,L_1,L_2,T$ .",
"The number of nodes in the sets is $|S|=|T|=n^2$ and $|L_1|,|L_2|\\le nd^2$ .",
"$S$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .",
"Similarly, $T$ consists of all tuples $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .",
"If the 3-OV instance has no solution, then $d(u,v)=3$ for all $u \\in S$ and $v \\in T$ .",
"If the 3-OV instance has a solution $a \\in A, b \\in B, c \\in C$ with $a,b,c$ orthogonal, then $d((a,b)\\in S,(b,c) \\in T)\\ge 7$ .",
"Setting $k=3,s=1$ in Property 5 of Theorem REF : for any $b^{\\prime } \\in B$ we have $d((a,b) \\in S,(b^{\\prime },c) \\in T)\\ge 5$ and $d((a,b^{\\prime }) \\in S,(b,c) \\in T)\\ge 5$ .",
"For any vertex $u \\in L_1$ there exists a vertex $s \\in S$ that is adjacent to $u$ .",
"Similarly, for any vertex $v \\in L_2$ there exists a vertex $t \\in T$ that is adjacent to $v$ .",
"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the other properties.",
"In the rest of the section we use Theorem REF to prove the following result.",
"Theorem 18 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.",
"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .",
"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 8$ ."
],
[
"Construction of the graph",
"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.",
"Figure REF illustrates the construction of the graph.",
"Figure: The illustration for the 5 vs 8 construction.",
"The edges between sets S,L 1 ,L 2 S,L_1,L_2 and TT are not depicted.",
"The edges between vertices in S ' S^{\\prime } and SS (TT and T ' T^{\\prime }) form a matching.",
"Vertices in S '' S^{\\prime \\prime } (T '' T^{\\prime \\prime }) form a clique.We start by adding a set $S^{\\prime }$ of $n^2$ vertices.",
"$S^{\\prime }$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .",
"We connect every $(a,b) \\in S^{\\prime }$ to its counterpart $(a,b) \\in S$ .",
"Thus, there is a matching between the sets of vertices $S$ and $S^{\\prime }$ .",
"We also add another set $S^{\\prime \\prime }$ of $n$ vertices.",
"$S^{\\prime \\prime }$ contains one vertex $a$ for every $a \\in A$ .",
"For every pair of vertices from $S^{\\prime \\prime }$ we add an edge between the vertices.",
"Thus, the $n$ vertices form a clique.",
"Furthermore, for every vertex $a \\in S^{\\prime \\prime }$ we add an edge to $(a,b) \\in S$ for all $b \\in B$ .",
"In total we added $n^2+n=O(n^2)$ vertices and $\\binom{n}{2}+2n^2=O(n^2)$ edges.",
"We do a similar construction for the set $T$ of vertices.",
"We add a set $T^{\\prime }$ of $n^2$ vertices - one vertex for every tuple $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .",
"We connect every $(b,c)\\in T^{\\prime }$ to $(b,c) \\in T$ .",
"Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.",
"$T^{\\prime \\prime }$ contains one vertex for every vector $c \\in C$ .",
"For every pair of vertices from $T^{\\prime \\prime }$ we add an edge between the vertices.",
"We connect every $c \\in T^{\\prime \\prime }$ to $(b,c) \\in T$ for all $b \\in B$ .",
"This finishes the construction of the graph.",
"In the rest of the section we show that the construction satisfies the promised two properties."
],
[
"Correctness of the construction",
"We need to consider two cases."
],
[
"Case 1: the 3-OV instance has no solution",
"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .",
"We consider three subcases."
],
[
"Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$ and {{formula:e49ba0d1-2dce-4e1e-bc34-0f08bcbafce8}}",
"We observe that there exists $s \\in S$ that has $d(u,s)\\le 1$ .",
"Indeed, if $u \\in S$ , then $s=u$ works.",
"If $u \\in S^{\\prime } \\cup S^{\\prime \\prime }$ , then we are done by the construction.",
"On the other hand, if $u \\in L_1$ , then there exists such an $s \\in S$ by property 6 from Theorem REF .",
"Similarly we can show that there exists $t \\in T$ such that $d(v,t)\\le 1$ .",
"Finally, by property 3 we have that $d(s,t)=3$ .",
"Thus, we can upper bound the distance between $u$ and $v$ by $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le 1+3+1=5$ as required."
],
[
"Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$",
"From the previous case we know that there are two vertices $s_1,s_2 \\in S$ such that $d(u,s_1)\\le 1$ and $d(s_2,v)\\le 1$ .",
"To show that $d(u,v)\\le 5$ it is sufficient to show that $d(s_1,s_2)\\le 3$ .",
"This is indeed true since both vertices $s_1$ and $s_2$ are connected to some two vertices in $S^{\\prime \\prime }$ and every two vertices in $S^{\\prime \\prime }$ are at distance at most 1 from each other."
],
[
"Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup L_2$",
"The case is analogous to the previous case."
],
[
"Case 2: the 3-OV instance has a solution",
"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 8$ .",
"Let $a \\in A, b \\in B, c \\in C$ be a solution to the 3-OV instance.",
"We claim that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 8$ .",
"Let $P$ be an optimal path between $u=((a,b) \\in S^{\\prime })$ and $v=((b,c) \\in T^{\\prime })$ that achieves the smallest distance.",
"We want to show that $P$ uses at least 8 edges.",
"Let $t \\in T$ be the first vertex from the set $T$ that is on path $P$ .",
"Let $s \\in S$ be the last vertex on path $P$ that belongs to $S$ and precedes $t$ in $P$ .",
"We can easily check that, if $s \\ne ((a,b) \\in S)$ , then $d(u,s)\\ge 3$ and, similarly, if $t \\ne ((b,c) \\in T)$ , then $d(t,v)\\ge 3$ .",
"We consider three subcases."
],
[
"Case 2.1: $s \\ne ((a,b) \\in S)$ and {{formula:bf5aaac7-5b1b-4304-83ad-0a1fd19bda3a}}",
"Since $s$ and $t$ are separated by two layers of vertices, we must have $d(s,t)\\ge 3$ .",
"Thus we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+3+3=9>8$ as required."
],
[
"Case 2.2: $s = ((a,b) \\in S)$ and {{formula:a28d868f-f38c-4534-b9c1-df1ca7ddd683}}",
"In this case we use property 4 and conclude $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)=1+d((a,b) \\in S,(b,c) \\in T)+1\\ge 1+7+1=9>8$ as required."
],
[
"Case 2.3: either $s = ((a,b) \\in S)$ or {{formula:1d8b5c54-e4d4-4cf8-8777-ab4e928d855f}} holds but not both",
"W.l.o.g.",
"$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .",
"If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 4$ and we get that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 4+3+1=8$ .",
"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .",
"By property 5 we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 5$ .",
"We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+5+1=9>8$ as required."
],
[
"6 vs 10 weighted undirected construction",
"In this section we change the construction from Theorem REF to show that under the 3-OV Hypothesis, any algorithm that can distinguish between diameter 6 and 10 in sparse undirected weighted graphs requires $\\Omega (n^{3/2-o(1)})$ time.",
"We get the following theorem.",
"Theorem 19 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an weighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.",
"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .",
"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 10$ .",
"Each edge of the graph has weight either 1 or 2."
],
[
"Construction of the graph",
"The construction of the graph is the same as in Theorem REF except all edges connecting vertices between sets $L_1$ and $L_2$ have weight 2 and all edges inside the cliques on nodes $S^{\\prime \\prime }$ and $T^{\\prime \\prime }$ have weight 2.",
"All the remaining edges have weight 1."
],
[
"Correctness of the construction",
"The correctness proof is essentially the same as for Theorem REF .",
"As before we consider two cases."
],
[
"Case 1: the 3-OV instance has no solution",
"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .",
"In the analysis of Case 1 in Theorem REF we show a path between $u$ and $v$ such that the path involves at most one edge from the cliques or between sets $L_1$ and $L_2$ .",
"Since we added weight 2 to the latter edges, the length of the path increased by at most 1 as a result.",
"So we have upper bound $d(u,v)\\le 6$ for all pairs $u$ and $v$ of vertices."
],
[
"Case 2: the 3-OV instance has a solution",
"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 10$ .",
"Similarly to Theorem REF we will show that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 10$ , where $a \\in A, b \\in B, c \\in C$ is a solution to the 3-OV instance.",
"The analysis of the subcases is essentially the same as in Theorem REF .",
"For cases 2.1 and 2.2 in the proof of Theorem REF we had $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 9$ .",
"Since we increased edge weights between $L_1$ and $L_2$ to 2 and every path from $(a,b) \\in S^{\\prime }$ to $(b,c) \\in T^{\\prime }$ must cross the layer between $L_1$ and $L_2$ , we also increased the lower bound of the length of the path from 9 to 10 for cases 2.1 and 2.2.",
"It remains to consider Case 2.3.",
"As in the proof of Theorem REF , w.l.o.g.",
"$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .",
"If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 5$ and we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 5+4+1=10$ .",
"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .",
"By property 5 and because we increased edge weights between $L_1$ and $L_2$ to 2, we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 6$ .",
"We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+6+1=10$ as required."
],
[
"$3k-4$ vs {{formula:cef9204d-0a6e-4579-9476-41520f256b04}} unweighted directed construction",
"In this section, we show that under SETH, for every $k\\ge 3$ , every algorithm that can distinguish between Diameter $3k-4$ and $5k-7$ in directed unweighted graphs requires $\\Omega (n^{1+1/(k-1)-o(1)})$ time.",
"Theorem REF gives us the following theorem.",
"Theorem 20 Given a $k$ -OV instance consisting of $k\\ge 2$ sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.",
"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .",
"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .",
"$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .",
"Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .",
"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .",
"If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .",
"Setting $s=k-2$ in Property 5 of Theorem REF : If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then for any tuple $(b_1,\\dots ,b_{k-2})$ , if $\\alpha =(a_0,b_1,\\dots ,b_{k-2})\\in S$ and $\\beta =(a_1,\\dots ,a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .",
"Symmetrically, if $\\alpha =(a_0,a_1,\\dots ,a_{k-2})\\in S$ and $\\beta =(b_1,\\dots ,b_{k-2},a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .",
"For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .",
"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the previous three properties.",
"In the rest of the section we use Theorem REF to prove the following result.",
"Theorem 21 Given a $k$ -OV instance, we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, directed graph with $O(k n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(kn^{k-1}d^{k-1})$ edges that satisfies the following two properties.",
"If the $k$ -OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .",
"If the $k$ -OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 5k-7$ ."
],
[
"Construction of the graph",
"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.",
"First we will construct a weighted graph and then we will make it unweighted.",
"Figure REF illustrates the construction of the graph for the special case $k=4$ .",
"Figure: The 3k-43k-4 vs 5k-75k-7 construction for the special case k=4k=4.",
"The edges between sets S,L 1 ,L 2 ,L 3 S,L_1,L_2,L_3 and TT are not depicted.",
"The matching between sets SS and S ' S^{\\prime } consists of unweighted paths of length k-2=2k-2=2.",
"The edges between sets SS and S '' S^{\\prime \\prime } consists of unweighted paths of length k-2=2k-2=2.",
"Similarly for the right side.We start by adding a set $S^{\\prime }$ of $n^{k-1}$ vertices.",
"$S^{\\prime }$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .",
"We connect every $(a_0,a_1,\\ldots , a_{k-2}) \\in S^{\\prime }$ to its counterpart $(a_0,a_1,\\ldots , a_{k-2}) \\in S$ with an undirected edge of weight $k-2$ to form a matching.",
"We also add another set $S^{\\prime \\prime }$ of $n$ vertices.",
"$S^{\\prime \\prime }$ contains one vertex $a_0$ for every $a_0 \\in W_0$ .",
"For every pair of vertices in $S^{\\prime \\prime }$ we add an undirected edge of weight 1 between the vertices.",
"Thus, the $n$ vertices form a clique.",
"Furthermore, for every vertex $a_0 \\in S^{\\prime \\prime }$ we add an undirected edge of weight $k-2$ to $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .",
"Finally for every vertex $a_0 \\in S^{\\prime \\prime }$ we add a directed edge of weight 1 towards $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .",
"Some of the edges that we added have weight $k-2$ .",
"We make those unweighted by subdividing them into edges of weight 1.",
"Let $S^{\\prime \\prime \\prime }$ be the set of newly added vertices.",
"In total we added $O(kn^{k-1})$ vertices and $O(kn^{k-1})$ edges.",
"We do a similar construction for the set $T$ of vertices.",
"We add a set $T^{\\prime }$ of $n^{k-1}$ vertices — one vertex for every tuple $(a_1,\\dots ,a_{k-1})$ where for each $i$ , $a_i\\in W_i$ .",
"We connect every $(a_1,\\dots ,a_{k-1})\\in T^{\\prime }$ to $(a_1,\\dots ,a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ .",
"Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.",
"$T^{\\prime \\prime }$ contains one vertex for every vector $a_{k-1} \\in W_{k-1}$ .",
"We connect every pair of vertices in $T^{\\prime \\prime }$ by an undirected edge of weight 1.",
"We connect every vertex $a_{k-1} \\in T^{\\prime \\prime }$ to $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ for all $b_1,\\dots ,b_{k-2}$ .",
"Also, for every vertex $a_{k-1} \\in T^{\\prime \\prime }$ we add a directed edge of weight 1 from $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T^{\\prime }$ to $a_{k-1}$ for all $b_1,\\dots ,b_{k-2}$ .",
"Some of the edges that we just added have weight $k-2$ .",
"We make those unweighted by subdividing them into edges of weight 1.",
"Let $T^{\\prime \\prime \\prime }$ be the set of newly added vertices.",
"This finishes the construction of the graph.",
"In the rest of the section we show that the construction satisfies the promised two properties stated in Theorem REF ."
],
[
"Correctness of the construction",
"We need to consider two cases."
],
[
"Case 1: the $k$ -OV instance has no solution",
"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .",
"We consider subcases."
],
[
"Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_i$ for {{formula:341c2d42-7167-4db5-bd6a-4b4996ef5fb1}} and {{formula:b26c11ee-44d0-4a3f-8d06-2943a8aaf115}} for {{formula:b2ebe27d-5f57-42a2-b20b-50932506480e}}",
"We observe that there exists $s \\in S$ that has $d(u,s)\\le k-2$ .",
"Similarly, there exists $t \\in T$ with $d(t,v)\\le k-2$ .",
"By property 3 from Theorem REF we have that $d(s,t)\\le k$ .",
"This gives us upper bound $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le (k-2)+k+(k-2)=3k-4$ as required.",
"The proof when the sets for $u$ and $v$ are swapped is identical since we only use paths on unweighted edges."
],
[
"Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_1$",
"We note that there is some vertex $s\\in S^{\\prime \\prime }$ with $d(u,s)\\le 2(k-2)$ (via undirected edges).",
"Also, there is some vertex $s^{\\prime }\\in S^{\\prime \\prime }$ with $d(s^{\\prime },v)\\le k-1$ (possibly using directed edges).",
"$S^{\\prime \\prime }$ is a clique so $d(s,s^{\\prime })\\le 1$ .",
"Thus, $d(u,v)\\le d(u,s)+d(s,s^{\\prime })+d(s^{\\prime },v)\\le 2(k-2)+1+(k-1)=3k-4$ ."
],
[
"Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup T^{\\prime \\prime \\prime } \\cup L_{k-1}$",
"This case is similar to the previous case.",
"We note that there is some vertex $t\\in T^{\\prime \\prime }$ with $d(t,v)\\le 2(k-2)$ (via undirected edges).",
"Also, there is some vertex $t^{\\prime }\\in T^{\\prime \\prime }$ with $d(u,t^{\\prime })\\le k-1$ (possibly using directed edges).",
"$S^{\\prime \\prime }$ is a clique so $d(t^{\\prime },t)\\le 1$ .",
"Thus, $d(u,v)\\le d(u,t^{\\prime })+d(t^{\\prime },t)+d(t,v)\\le (k-1)+1+2(k-2)=3k-4$ ."
],
[
"Case 2: the $k$ -OV instance has a solution",
"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 5k-7$ .",
"Let $(a_0,a_1,\\ldots , a_{k-1})$ be a solution to the $k$ -OV instance where for each $i$ , $a_i\\in W_i$ .",
"We claim that $d((a_0,\\dots ,a_{k-2}) \\in S^{\\prime },(a_1,\\dots ,a_{k-1}) \\in T^{\\prime })\\ge 5k-7$ .",
"Let $P$ be an shortest path between $u=((a_0,\\dots ,a_{k-2}) \\in S^{\\prime })$ and $v=((a_1,\\dots ,a_{k-1}) \\in T^{\\prime })$ .",
"We want to show that $P$ uses at least $5k-7$ edges.",
"Let $s \\in S$ be the first vertex on path $P$ that belongs to $S$ and let $t \\in T$ be the last vertex from the set $T$ that is on path $P$ .",
"We observe that due to the directionality of the edges, $s$ and $t$ must be the counterparts of $u$ and $v$ respectively; that is, $s=((a_0,\\dots ,a_{k-2}) \\in S)$ and $t=((a_1,\\dots ,a_{k-1}) \\in T)$ .",
"Note that these definitions of $s$ and $t$ differ from the definitions of $s$ and $t$ in previous proofs.",
"We consider three subcases."
],
[
"Case 2.1: A vertex in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ appears after {{formula:6406b760-7ede-4dc4-b0f6-a59f46a09387}} on the path {{formula:a6c998cf-9396-4ea3-832a-1c6dca88cdb2}}",
"We observe that if $s_1,s_2\\in S$ is a pair of vertices on the path $P$ such that no vertex in $S$ appears between them on $P$ , then the portion of $P$ between $s_1$ and $s_2$ either contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ or contains no vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .",
"Let $s_1,s_2\\in S$ be such that the portion of $P$ between them contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .",
"Such $s_1,s_2$ exist by the specification of this case.",
"If $s_1=s_2$ then $P$ is not a shortest path.",
"Otherwise, the portion of $P$ between $s_1$ and $s_2$ must include a vertex in $S^{\\prime \\prime }$ .",
"Thus, $d(s_1,s_2)\\ge 2(k-2)$ .",
"We consider three subcases.",
"$s_1\\ne s$ .",
"The distance between any pair of vertices in $S$ is at least 2 so $d(s,s_1)\\ge 2$ .",
"Then, $d(u,v)\\ge d(u,s)+d(s,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2+2(k-2)+k+(k-2)=5k-6$ .",
"$s_1=s$ and $s_2=((a_0,b_1,\\dots ,b_{k-2}) \\in S)$ for some $b_1,\\dots ,b_{k-2}$ .",
"In this case, by property 5 we have $d(s_2,t)\\ge k+2$ .",
"Thus, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2(k-2)+(k+2)+(k-2)=5k-6$ .",
"$s_1=s$ and $s_2=((b_0,\\dots ,b_{k-2} \\in S)$ for some with $b_0\\ne a_0$ .",
"In this case, the path from $s_1$ to $s_2$ must include an edge in the clique $S^{\\prime \\prime }$ since these are the only edges among vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ for which adjacent tuples can differ with respect to their first element.",
"Thus, $d(s_1,s_2)\\ge 2(k-2)+1\\ge 2k-3$ .",
"Therefore, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+(2k-3)+k+(k-2)=5k-7$ ."
],
[
"Case 2.2: A vertex in $T^{\\prime }\\cup T^{\\prime \\prime }\\cup T^{\\prime \\prime \\prime }$ appears before {{formula:2e52092d-2023-4c53-a764-a41805329e43}} on the path {{formula:d86d9c20-8168-4be0-8d10-99eb7834ec04}}",
"This case is analogous to the previous case."
],
[
"Case 2.3: The portion of the path $P$ between {{formula:558fd34a-9236-4d24-8a2c-f5022f0a499a}} and {{formula:c0c71b02-54cb-42b0-8842-f210ddc962a7}} contains no vertices in {{formula:7c23b5b2-7499-4d20-8946-fe3d3dacb80d}}",
"By property 4, $d(s,t)\\ge 3k-2$ .",
"Thus, $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge (k-2)+(3k-2)+(k-2)=5k-6$ .",
"We note that a slight modification of this construction gives a lower bound for higher values of Diameter.",
"For any L, we can get an $L(3k-4)$ vs $L(5k-8)+1$ construction by subdividing all of the edges in the construction (into paths of length $L$ ) except for the directed edges and the edges within the cliques."
],
[
"2-Approximation for Eccentricities in $\\tilde{O}(m\\sqrt{n})$ time",
"Theorem 22 Given a weighted, directed $m$ edge $n$ node graph, in $\\tilde{O}(m\\sqrt{n})$ time we can output quantities $\\epsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have $\\epsilon (v)/2\\le \\epsilon ^{\\prime }(v)\\le \\epsilon (v)$ .",
"The algorithm is inspired by the 2-approximation algorithm for directed radius of Abboud, Vassilevska W., and Wang [7].",
"We claim that the following algorithm achieves the above guarantees.",
"Sample a random subset $S \\subset V$ of size $|S|=\\Theta (\\sqrt{n} \\log n)$ .",
"With high probability for every $u \\in V$ we have $N^{\\text{in}}_{\\sqrt{n}}(u)\\cap S \\ne \\emptyset $ .",
"Let $w$ be a vertex that maximizes $d(S,w)$ , which we find using Dijkstra's algorithm.",
"Let $S^{\\prime }:=N^{\\text{in}}_{\\sqrt{n}}(w)$ .",
"For every vertex $v \\in S^{\\prime }$ we output $\\epsilon ^{\\prime }(v)=\\epsilon (v)$ by running Dijkstra's algorithm and following the outgoing edges.",
"For every vertex $v \\notin S^{\\prime }$ we output the estimate $\\epsilon ^{\\prime }(v)=\\max _{s \\in S\\cup \\lbrace w\\rbrace }d(v,s)$ .",
"We can determine all these quantities by running Dijkstra's algorithm out of all vertices in $S \\cup \\lbrace w\\rbrace $ and following the incoming edges."
],
[
"Correctness",
"Consider an arbitrary vertex $v\\notin S^{\\prime }$ (if $v \\in S^{\\prime }$ , then we are done by the third step).",
"If there exists $s \\in S$ such that $d(v,s)\\ge \\epsilon (v)/2$ , then we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,s)\\ge \\epsilon (v)/2$ .",
"Otherwise, we have $d(v,s)<\\epsilon (v)/2$ for all $s \\in S$ .",
"Let $v^{\\prime }$ be a vertex that achieves $d(v,v^{\\prime })=\\epsilon (v)$ .",
"By the triangle inequality we have $d(s,v^{\\prime })>\\epsilon (v)/2$ for all $s \\in S$ .",
"Equivalently, $d(S,v^{\\prime })>\\epsilon (v)/2$ .",
"This implies that $d(S,w)>\\epsilon (v)/2$ by our choice of $w$ .",
"Since $d(S,w)>\\epsilon (v)/2$ and $S^{\\prime }=N^{\\text{in}}_{\\sqrt{n}}(w)$ intersects $S$ , we must have that $S^{\\prime }$ contains all vertices $u$ with $d(u,w)\\le \\epsilon (v)/2$ .",
"Since $v \\notin S^{\\prime }$ , we must have $d(v,w)>\\epsilon (v)/2$ and we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,w)>\\epsilon (v)/2$ ."
],
[
"Almost 2-Approximation for Eccentricities in almost linear time",
"In contrast to our $\\tilde{O}(m\\sqrt{n})$ time algorithm from the previous section, our near-linear time $(2+\\delta )$ -approximation algorithm is very different from all previously known algorithms.",
"Our algorithm proceeds in iterations and maintains a set $S$ of nodes for which we still do not have a good eccentricity estimate.",
"In each iteration either we get a good estimate for many new vertices and hence remove them from $S$ , or we remove all vertices from $S$ that have large eccentricities, and for the remaining nodes in $S$ we have a better upper bound on their eccentricities.",
"After a small number of iterations we have a good estimate for all vertices of the graph.",
"Theorem 23 Suppose that we are given a weighted, directed $m$ edge $n$ node graph.",
"The weights of all edge are non-negative integers bounded by $n^{O(1)}$ .",
"For any $1>\\tau >0$ we can in $\\tilde{O}(m/\\tau )$ time output quantities $\\varepsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have ${1-\\tau \\over 2}\\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ .",
"We maintain a subset $S\\subseteq V$ of vertices $v$ for which we still do not have an estimate $\\varepsilon ^{\\prime }(v)$ .",
"Initially $S=V$ and we will end with $|S|\\le O(1)$ .",
"When $|S|\\le O(1)$ we can evaluate $\\varepsilon (v)$ for all $v \\in S$ in the total time of $O(m)$ .",
"Also we maintain a value $D$ that upper bounds the largest eccentricity of a vertex in $S$ .",
"That is, $\\varepsilon (v)\\le D$ for all $v \\in S$ .",
"Initially we set $D=n^C$ for some large enough constant $C>0$ (we assume that the input graph is strongly connected).",
"The algorithm proceeds in phases.",
"Each phase takes $\\tilde{O}(m)$ time and either $|S|$ decreases by a factor of at least 2 or $D$ decreases by a factor of at least $1/(1-\\tau )$ .",
"After $O(\\log (n)/\\tau )$ phases either $|S|\\le O(1)$ or $D<1$ .",
"For a subset $S \\subseteq V$ of vertices and a vertex $x \\in V$ we define a set $S_x\\subseteq S$ to contain those $|S_x|=|S|/2$ vertices from $S$ that are closest to $x$ (according to distance $d(\\cdot ,x)$ ).",
"The ties are broken by taking the vertex with the smaller id.",
"Given a subset $S \\subseteq V$ of vertices and a threshold $D$ , a phase proceeds as follows.",
"[itemsep=0mm] We sample a set $A \\subseteq S$ of $O(\\log n)$ random vertices from the set $S$ .",
"With high probability for all $x \\in V$ we have $A \\cap S_x \\ne \\emptyset $ .",
"Let $w \\in V$ be a vertex that maximizes $d(A,w)$ .",
"We can find it using Dijkstra's algorithm.",
"We consider two cases.",
"Case $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ .",
"For all $x \\in S\\setminus S_w$ we have ${1-\\tau \\over 2} D \\le \\varepsilon (x)\\le D$ and we assign the estimate $\\varepsilon ^{\\prime }(x)={1-\\tau \\over 2}D$ .",
"This gives us that ${1-\\tau \\over 2} \\varepsilon (x)\\le {1-\\tau \\over 2}D=\\varepsilon ^{\\prime }(x)\\le \\varepsilon (x)$ for all $x\\in S\\setminus S_w$ .",
"We update $S$ to be $S_w$ .",
"This decreases the size of $S$ by a factor of 2 as required.",
"Case $d(S\\setminus S_w,w)<{1-\\tau \\over 2}D$ .",
"Set $S^{\\prime }=S$ .",
"For every vertex $v \\in S$ evaluate $r_v:=\\max _{x \\in A}d(v,x)$ .",
"We can evaluate these quantities by running Dijkstra's algorithm from every vertex in $A$ and following the incoming edges.",
"If $r_v\\ge {1-\\tau \\over 2}D$ , then assign the estimate $\\varepsilon ^{\\prime }(v)={1-\\tau \\over 2}D$ and remove $v$ from $S^{\\prime }$ .",
"Similarly as in the previous case we have ${1-\\tau \\over 2} \\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v \\in S\\setminus S^{\\prime }$ .",
"Below we will show that for every $v \\in S^{\\prime }$ we have $\\varepsilon (v)\\le (1-\\tau )D$ .",
"Thus we can update $S=S^{\\prime }$ and decrease the threshold $D$ to $(1-\\tau )D$ as required.",
"Correctness We have to show that, if there exists $v \\in S^{\\prime }$ such that $\\varepsilon (v)>(1-\\tau )D$ , then we will end up in the first case (this is the contrapositive of the claim in the second case).",
"Since $v \\in S^{\\prime }$ we must have that $d(v,x)\\le {1-\\tau \\over 2}D$ for all $x \\in A$ .",
"Since $\\varepsilon (v)>(1-\\tau )D$ , we must have that there exists $v^{\\prime }$ such that $d(v,v^{\\prime })>(1-\\tau )D$ .",
"By the triangle inequality we get that $d(x,v^{\\prime })>{1 - \\tau \\over 2}D$ for every $x \\in A$ .",
"By choice of $w$ we have $d(A,w)>{1-\\tau \\over 2}D$ .",
"Since $A \\cap S_w \\ne \\emptyset $ , we have $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ and we will end up in the first case.",
"The guarantee on the approximation factor follows from the description.",
"As a corollary, we get an algorithm for Source Radius with the same runtime and approximation ratio as Theorem REF .",
"First, run the Eccentricities algorithm and let $v$ be the vertex with the smallest estimated eccentricity $\\epsilon ^{\\prime }(v)$ .",
"Then run Dijkstra's algorithm from $v$ and report $\\epsilon (v)$ as the Radius estimate $R^{\\prime }$ .",
"Let $R$ be the true radius of the graph and let $x$ be the vertex with minimum Eccentricity i.e.",
"$\\epsilon (x)=R$ .",
"If $\\alpha $ is the approximation ratio for the Eccentricities algorithm then $\\epsilon (v)\\le \\alpha \\epsilon ^{\\prime }(v)\\le \\alpha \\epsilon (v)$ and $\\epsilon (x)\\le \\alpha \\epsilon ^{\\prime }(x)\\le \\alpha \\epsilon (x)$ .",
"By choice of $v$ , $\\epsilon ^{\\prime }(v)\\le \\epsilon ^{\\prime }(x)$ .",
"Thus, $\\alpha R=\\alpha \\epsilon (x)\\ge \\alpha \\epsilon ^{\\prime }(x)\\ge \\alpha \\epsilon ^{\\prime }(v)\\ge \\epsilon (v)=R^{\\prime }$ .",
"Clearly $R^{\\prime }\\ge R$ , so $R\\le R^{\\prime }\\le \\alpha R$ .",
"$S$ -$T$ Diameter algorithms Recall that the $S$ -$T$ diameter problem is as follows: Given an undirected graph $G=(V,E)$ and two sets $S\\subseteq V, T\\subseteq V$ , determine $D_{S,T}=\\max _{s\\in S, t\\in T} d(s,t)$ .",
"Here we will outline two algorithms for the problem.",
"Let us first consider a fast 3-approximation algorithm.",
"Claim 24 There is an $O(m+n)$ time algorithm that for any $n$ node $m$ edge graph $G=(V,E)$ and $S\\subseteq V, T\\subseteq V$ , computes an estimate $D^{\\prime }$ such that $D_{S,T}/3\\le D^{\\prime }\\le D_{S,T}$ and two nodes $s\\in S$ , $t\\in T$ such that $d(s,t)=D^{\\prime }$ .",
"In graphs with nonnegative weights, the same estimate can be achieved in $O(m+n\\log n)$ time.",
"The algorithm is extremely simple: pick arbitrary nodes $s\\in S$ and $t\\in T$ , compute BFS($s$ ) and BFS($t$ ) and return $\\max \\lbrace \\max _{t^{\\prime }\\in T} d(s,t^{\\prime }),\\max _{s^{\\prime }\\in S} d(s^{\\prime },t)\\rbrace $ (also returning the two nodes achieving the maximum).",
"For weighted graphs, run Dijkstra's algorithm instead of BFS.",
"Let's see why this algorithm provides the promised guarantee.",
"Suppose that for every $t^{\\prime }\\in T$ , $d(s,t^{\\prime })<D_{S,T}/3$ (otherwise we are done).",
"Then for every $t^{\\prime },t^{\\prime \\prime }\\in T$ , $d(t^{\\prime },t^{\\prime \\prime })\\le d(t^{\\prime },s)+d(s,t^{\\prime \\prime })<2D_{S,T}/3$ .",
"In particular, for all $t^{\\prime }\\in T$ , $d(t,t^{\\prime })<2D_{S,T}/3$ .",
"If we also had that for every $s^{\\prime }\\in S$ , $d(t,s^{\\prime })<D_{S,T}/3$ , then we'd get that for all $s^{\\prime }\\in S, t^{\\prime }\\in T$ , $d(s^{\\prime },t^{\\prime })\\le d(s^{\\prime },t)+d(t,t^{\\prime })<D_{S,T}$ , contradicting the definition of $D_{S,T}$ .",
"Thus, $\\max \\lbrace \\max _{t^{\\prime }\\in T} d(s,t^{\\prime }),\\max _{s^{\\prime }\\in S} d(s^{\\prime },t)\\rbrace \\ge D_{S,T}/3$ .",
"We will now show an analogue to the $\\tilde{O}(m\\sqrt{n})$ time almost-$3/2$ -approximation diameter algorithm of Roditty and Vassilevska W. [41] for $S$ -$T$ Diameter giving a 2-approximation.",
"Using a trick from Chechik et al.",
"[21] we also obtain a true 2 approximation algorithm running in $\\tilde{O}(m^{3/2})$ .",
"2-Approximation for $S$ -$T$ Diameter [1] 2-Approx $X$ - random sample of nodes, $|X|=\\Theta (\\sqrt{n} \\log n)$ $D_1:=0$ every $x\\in X$ Run BFS($x$ ) Let $t_x$ be the closest node to $x$ in $T$ Run BFS($t_x$ ) $D_1 = \\max \\lbrace D_1,\\max _{s\\in S} d(s,t_x)\\rbrace $ [t]@X@ Let $\\bar{t}$ be the furthest node of $T$ from $X$ (computed above) Run BFS($\\bar{t}$ ) $D_2=\\max _{s\\in S} d(s,\\bar{t})$ .",
"Let $Y$ be the closest $\\sqrt{n}$ nodes to $\\bar{t}$ .",
"every $y\\in Y$ Run BFS($y$ ) Let $s_y$ be the closest node to $y$ in $S$ Run BFS($s_y$ ) $D_2 = \\max \\lbrace D_2,\\max _{t\\in T} d(s_y,t)\\rbrace $ $\\max \\lbrace D_1,D_2\\rbrace $ We use Algorithm 2-Approx to prove: Theorem 25 In $\\tilde{O}(m\\sqrt{n})$ time one can obtain an estimate $D^{\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node unweighted undirected graph such that $2\\lfloor D/4\\rfloor \\le D^{\\prime }\\le D$ .",
"In $\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\prime \\prime }$ such that $D/2\\le D^{\\prime \\prime }\\le D$ .",
"First we analyze Algorithm 2-Approx.",
"Let $s^*\\in S$ and $t^*\\in T$ be the end points of the $S$ -$T$ Diameter path so that $d(s^*,t^*)=D$ .",
"Let $d=\\lfloor D/4\\rfloor $ .",
"Suppose first that for some $x\\in X$ , $d(x,t^*)\\le d$ .",
"Then, $d(x,t_x)\\le d(x,t^*)\\le d$ and hence $d(t_x,t^*)\\le d(t_x,x)+d(x,t^*)\\le 2d$ .",
"However, then $d(t_x,s^*)\\ge d(t^*,s^*)-d(t^*,t_x)\\ge D-2d\\ge D/2$ .",
"Thus, if $D_1<D/2$ , it must be that for every $x\\in X$ , $d(x,t^*)\\ge d+1$ .",
"Hence, for every $x \\in X$ , $d(x,\\bar{t})\\ge d(x,t^*)\\ge d+1$ by the definition of $\\bar{t}$ .",
"If $d(\\bar{t},s^*)\\ge D/2$ , then $D_2\\ge D/2$ and we are done, so let us assume that $d(\\bar{t},s^*)\\le D/2$ .",
"Now, as $X$ is random of size $c\\sqrt{n}\\log n$ for large enough $c$ , with high probability, $X$ hits the $\\sqrt{n}$ -neighborhoods of all nodes.",
"In particular, $X\\cap Y\\ne \\emptyset $ .",
"However, since $d(x,\\bar{t})\\ge d+1$ for every $x\\in X$ , it must be that $Y$ contains all nodes at distance $d$ from $\\bar{t}$ as it contains all nodes closer to $\\bar{t}$ than $x\\in Y\\cap X$ .",
"If $s^*\\in Y$ , then we would have run BFS from $s^*$ and returned $D$ .",
"Hence $d(\\bar{t},s^*)>d$ .",
"Let $a$ be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d$ .",
"We thus have that $a\\in Y$ .",
"Also, since $d(\\bar{t},s^*)\\le D/2$ , $d(a,s^*)\\le D/2-d$ and hence $d(a,s_a)\\le D/2-d$ , so that $d(s_a,t^*)\\ge D-2(D/2-d)\\ge 2d$ .",
"This finishes the argument that 2-Approx returns an estimate $D^{\\prime }$ with $2\\lfloor D/4\\rfloor \\le D^{\\prime }\\le D$ .",
"It is not too hard to see that the only time that we might get an estimate that is less than $D/2$ is in the last part of the argument and only if the diameter is of the form $4d+3$ .",
"(We will prove the algorithm guarantees formally soon.)",
"The analysis fails to work in that case because $Y$ is guaranteed to contain only the nodes at distance $d$ from $\\bar{t}$ .",
"In particular, if $Y$ contains all nodes at distance $d+1$ from $\\bar{t}$ instead of just those at distance at most $d$ , we could consider $a$ to be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d+1$ , and $a\\in Y$ .",
"Now since $d(\\bar{t},s^*)\\le 2d+1$ (as otherwise we'd be done), $d(a,s_a)\\le d(a,s^*)\\le 2d+1-d-1 =d$ , so that $d(s_a,t^*)\\ge 2d+3$ .",
"Hence everything would work out.",
"We handle this issue with a trick from Chechik et al. [21].",
"First, we make graph have constant degree by blowing up the number of nodes and adding 0 weight edges as follows.",
"Let $v$ be an original node and suppose it has degree $d(v)$ .",
"Replace $v$ with a $d(v)$ -cycle of 0 weight edges so that each of the cycle nodes is connected to a one of the neighbors of $v$ , where each neighbor has a cycle node corresponding to it.",
"This makes every node have degree 3 and increases the number of vertices to $O(m)$ .",
"Now, we run algorithm 2-Approx with two changes.",
"The first is that instead of BFS we use Dijkstra's algorithm because the edges now have weights.",
"The second change is that we redefine $Y$ as follows.",
"Let $Z$ be the closest $\\sqrt{m}$ nodes $Z$ to $\\bar{t}$ .",
"Define $Y$ to be $Z$ , together with all nodes that have an edge of weight 1 to some node of $Z$ .",
"Now, consider the shortest path $P$ between $\\bar{t}$ and $s^*$ .",
"Consider the last node $a$ of $P$ (in the direction from $\\bar{t}$ to $s^*$ ) that has distance $\\le d$ from $\\bar{t}$ .",
"The node $a^{\\prime }$ after $a$ on $P$ must be connected to $a$ by an edge $(a,a^{\\prime })$ of weight 1, as otherwise $a^{\\prime }$ would be at distance at most $d$ from $\\bar{t}$ .",
"Since $a\\in Z$ , we must have $a^{\\prime }\\in Y$ and we know also that $d(\\bar{t},a^{\\prime })=d+1$ .",
"In the last case when $d(\\bar{t},s^*)\\le 2d+1$ , we get that $Y$ contains a node $a^{\\prime }$ of distance $\\le (2d+1)-(d+1) = d$ from $s^*$ and hence $d(a^{\\prime },s_{a^{\\prime }})\\le d$ and hence $d(s_{a^{\\prime }},t^*)\\ge 2d+3$ .",
"Since every node has degree 3, the number of nodes in $Y$ is $\\le 4|Z|\\le O(\\sqrt{m})$ and hence we can afford to run Dijkstra from each of them and complete the algorithm in $\\tilde{O}(m^{3/2})$ time.",
"Let us now formally analyze the guarantees of the algorithm.",
"Suppose that $D=4d+z$ where $z\\in \\lbrace 0,1,2,3\\rbrace $ ; we will show that the estimate that the algorithm gives is always at least $\\lceil D/2\\rceil =2d+\\lceil z/2\\rceil $ .",
"If some node $x\\in X$ has $d(x,t^*)\\le d$ , we get that $d(t_x,s^*)\\ge D-2d=2d+z\\ge \\lceil D/2\\rceil $ .",
"If we are not done, all nodes of $X$ have $d(x,\\bar{t})\\ge d(x,t^*)\\ge d+1$ and $Z$ contains all nodes at distance $\\le d$ from $\\bar{t}$ .",
"If $s^*\\in Z$ , we are done so we must have $d(s^*,\\bar{t})\\ge d+1$ .",
"Consider the last node $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path (in the direction towards $s^*$ ) for which $d(\\bar{t},a^{\\prime })\\le d$ .",
"We have that $a^{\\prime }\\in Z$ .",
"Also, the node $a$ after $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path must be in $Y$ since by the choice of $a^{\\prime }$ , $d(\\bar{t},a)=t+1$ and $(a,a^{\\prime })$ is an edge of weight 1.",
"If $d(\\bar{t},s^*)\\ge 2d+\\lceil z/2\\rceil $ , we are done, so we get that $d(a,s^*)\\le 2d+\\lceil z/2\\rceil -1 - (d+1) =d+\\lceil z/2\\rceil -2$ .",
"Hence, $d(s_a,t^*)\\ge D-2(d+\\lceil z/2\\rceil -2)=2d + (z+4-2\\lceil z/2\\rceil )\\ge 2d+z$ .",
"It is not hard to extend the $S$ -$T$ Diameter algorithms to work for weighted undirected graphs as well.",
"The basic idea is to use Dijkstra's algorithm instead of BFS in Algorithm 1.",
"This gives an almost 2-approximation.",
"In particular, let $a^{\\prime }$ be the last node on the $\\bar{t}$ to $s^*$ shortest path that is at distance $\\le d$ from $\\bar{t}$ and let $a$ be the node after $a^{\\prime }$ .",
"The weighted version of Algorithm 1 achieves a guarantee $D^{\\prime }$ of the $S$ -$T$ Diameter, such that $D/2-2w(a,a^{\\prime })\\le D^{\\prime }\\le D$ .",
"We can obtain an $\\tilde{O}(m^{3/2})$ true 2-approximation algorithm similar to the proof above.",
"Let $Z$ be the closest $\\sqrt{m}$ nodes to $\\bar{t}$ and extend $Z$ to $Y$ by adding all nodes that have a non-zero edge to a node of $Z$ .",
"With this modification, the node $a$ is guaranteed to be in $Y$ and hence the estimate for the diameter is at least $D/2$ .",
"Corollary 26 In $\\tilde{O}(m\\sqrt{n})$ time one can obtain an estimate $D^{\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node undirected graph with nonnegative edge weights such that $D/2 - 2w(a,a^{\\prime }) \\le D^{\\prime }\\le D$ for some edge $(a,a^{\\prime })$ .",
"In $\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\prime \\prime }$ such that $D/2\\le D^{\\prime \\prime }\\le D$ .",
"It is an easy exercise to see that when $D=2h+1$ then the value $\\max \\lbrace \\epsilon ^{in}(v), \\epsilon ^{out}(v) \\rbrace $ of an arbitrary vertex $v \\in V$ is an estimation to the diameter which is at least $h+1$ and at most $D$ .",
"In this section we present a deterministic algorithm that gets a directed unweighted graph $G$ with $D=2h$ and computes in $O(m^2/n)$ time an estimation $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .",
"The algorithm works as follows.",
"A variable $\\hat{D}$ is set to zero.",
"The algorithm searches for a vertex $v$ that its sum of in and out degree is minimal.",
"Then the algorithm computes the in and out eccentricity of $v$ and every vertex that has an edge with $v$ (incoming or outgoing).",
"The algorithm outputs the maximum of all the in and out eccentricities that were computed.",
"Theorem 27 Let $G=(V,E)$ an unweighted directed graph and let $D=2h$ .",
"Algorithm REF returns in $O(m^2/n)$ time an estimate $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .",
"We start with the running time analysis.",
"Consider the graph $G$ and ignore the edge directions.",
"For every $v\\in V$ let $deg(v) = deg^{in}(v)+deg^{out}(v)$ .",
"Since $m = \\frac{1}{2}\\sum _{v\\in V} deg(v)$ a vertex $v$ of minimal degree satisfies $deg(v)\\le 2m/n$ .",
"Therefore, the cost of computing in and out eccentricities for all vertices in the set $N(v) \\cup \\lbrace v \\rbrace $ is $O(\\frac{m}{n} \\times m)$ .",
"We now turn to bound $\\hat{D}$ .",
"Let $a, b\\in V$ and let $d(a,b)=2h$ .",
"Let $P(a,b)$ be a shortest path from $a$ to $b$ and let $v\\in P(a,b)$ .",
"If $d(a,v)\\le h-1$ then $\\epsilon ^{out}(v)\\ge h+1$ .",
"Similarly, if $d(v,b)\\le h-1$ then $\\epsilon ^{in}(v)\\ge h+1$ .",
"Consider now the case that $d(a,v)=h$ and $d(v,b)=h$ .",
"Let $u\\in P(a,b)$ be the vertex that precedes $v$ on the shortest path from $a$ to $b$ .",
"Since $u$ has an incoming edge to $v$ it follows that $u\\in N(v)$ and $\\epsilon ^{out}(u)$ and $\\epsilon ^{in}(u)$ are computed.",
"Since $d(a,v)=h$ it follows that $d(a,u)=h-1$ , $\\epsilon ^{out}(u)\\ge h+1$ and $\\hat{D}$ is at least $h+1$ .",
"Fast approximation of the diameter [1] Diam-Approx$G$ $\\hat{D} = 0$ $v = \\arg \\min _{x\\in V} deg^{in}(x)+deg^{out}(x)$ every $w \\in N^{\\text{in}}(v) \\cup N^{\\text{out}}(v) \\cup \\lbrace v \\rbrace $ compute $\\epsilon ^{in}(w)$ and $\\epsilon ^{out}(w)$ $\\hat{D}= \\max \\lbrace \\hat{D},\\epsilon ^{in}(w),\\epsilon ^{out}(w)\\rbrace $ $\\hat{D}$ Algorithms for dense graphs Algorithm overview The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.",
"[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.",
"Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.",
"For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.",
"Therefore, it is interesting to consider the opposite question to the one considered by [41].",
"Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?",
"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.",
"For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .",
"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.",
"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.",
"To enable this approach we can no longer sample $A$ naively.",
"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.",
"The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .",
"We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.",
"The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w)<d(u,A) \\rbrace $ is of size $O(n/ |A|)$ .",
"Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .",
"For any such pair we can efficiently compute their exact distance.",
"Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .",
"This, combined with some other ideas, gives our approximation guarantees.",
"We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.",
"The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4].",
"A simple approach with additive error Let's first consider a simple approach obtaining an $\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.",
"Suppose that we have an algorithm ALG that can compute in $\\tilde{O}(m\\sqrt{n})$ time, for any graph $G^{\\prime }$ , an estimate $D^{\\prime }$ of its Diameter $D$ such that $p\\cdot D - q\\le D^{\\prime }\\le D$ , estimates $e(v)$ of $\\epsilon (v)$ for all $v$ so that $r\\epsilon (v)-s\\le e(v)\\le \\epsilon (v)$ , and an estimate $D^{\\prime \\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\cdot D_{S,T}-u\\le D^{\\prime \\prime }\\le D_{S,T}$ .",
"Now, Dor, Halperin and Zwick [24] showed that in $\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\tilde{O}(n^{1.5})$ edges.",
"In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.",
"he removed all logs!).",
"Let's compute $H$ for our given graph and run ALG on $H$ .",
"The runtime is $\\tilde{O}(n^{1.5}\\cdot \\sqrt{n})\\le \\tilde{O}(n^2)$ since $H$ has $\\le O(n^{1.5})$ edges.",
"Let $D^{\\prime }_H,e_H(\\cdot ),D^{\\prime \\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\epsilon _H(\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .",
"Let's return $D^{\\prime }_H-2,e_H(\\cdot )-2,D^{\\prime \\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .",
"Notice that $pD-q\\le p\\cdot D_H - q\\le D^{\\prime }_H\\le D_H\\le D+2$ and so $pD-2-q \\le D^{\\prime }_H-2\\le D$ .",
"Similarly since $r\\epsilon (v)-s\\le r\\epsilon _H(v)-s\\le e_H(v)\\le \\epsilon _H(v)\\le \\epsilon (v)+2$ , we get $r\\epsilon (v)-s-2\\le e_H(v)-2\\le \\epsilon (v)$ .",
"Finally since $t\\cdot D_{S,T}-u\\le t\\cdot D^H_{S,T}-u\\le D^{\\prime \\prime }_H\\le D^H_{S,T}\\le D_{S,T}+2$ , we get $t\\cdot D_{S,T}-u-2\\le D^{\\prime \\prime }_H-2\\le D_{S,T}$ .",
"Thus, in $\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.",
"Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.",
"This is especially important when these parameters are constant, which is the hard case of the problems anyway.",
"Near linear almost 3/2-approximation for Diameter Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.",
"Among the tools they use are clusters and bunches.",
"Let $A\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .",
"For every $v\\in V$ , let $B_A(u)=\\lbrace v\\in V\\mid d(u,v) <d(u,A \\rbrace $ be the bunch of $u$ .",
"For every $w\\in V\\setminus A$ , let $C_A(w)= \\lbrace v \\mid w\\in B_A(v)\\rbrace $ be the cluster of $w$ .",
"Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\in V$ .",
"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.",
"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.",
"Their algorithm works as follows.",
"It sets $A$ to the empty set and $W$ to $V$ .",
"Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .",
"The algorithm computes $C_A(w)$ for every $w\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .",
"The pseudo-code is given in Algorithm REF .",
"Thorup and Zwick center algorithm [1] center$G,p$ $A= \\emptyset $ $W =V$ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\cup X$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .",
"For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .",
"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .",
"They did not provide the details and refer the reader to [48].",
"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.",
"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .",
"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .",
"This can be done as follows.",
"Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .",
"The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).",
"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.",
"To this end, more ideas are needed.",
"The following simple observation helps us to achieve our goal.",
"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .",
"Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .",
"For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .",
"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .",
"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].",
"This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.",
"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.",
"see [48]).",
"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.",
"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .",
"In more detail, our algorithm works as follows.",
"For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.",
"Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .",
"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .",
"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .",
"Finally, it computes the clusters and $W$ using the bunches as we described above.",
"The rest of the algorithm is almost identical to Algorithm REF .",
"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .",
"The pseudo-code is given in Algorithm REF .",
"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .",
"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .",
"compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .",
"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .",
"every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .",
"The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].",
"The cost of computing $A$ is $O(n/p)$ time [48].",
"Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .",
"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .",
"As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .",
"Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .",
"This completes the analysis of the part that precedes the while loop.",
"Next, we analyze the cost of the while loop.",
"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.",
"From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .",
"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.",
"Thorup and Zwick [47] proved that the expected number of iterations is $O(\\log n)$ .",
"The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.",
"For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\log n)$ iterations in expectation.",
"This implies that a set $A$ of expected size $(np \\log n)$ is returned in $O(n/p^2 \\log n)$ expected time.",
"The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.",
"As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.",
"almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\sqrt{n})$ every $w\\in V$ Step 1 every $\\langle u, v\\rangle \\in C_A(w) \\times C_A(w)$ , s.t.",
"$u\\ne v\\;$ $M(u,v) = \\min (M(u,v), d(u,w)+d(v,w))$ every $\\langle u, v\\rangle \\in V \\times V$ , s.t.",
"$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\in A$ compute shortest paths tree for $u$ in $H$ and set $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ $D_2 = \\max _{u\\in A} \\epsilon _H(u)$ $\\hat{D} = \\max (D_1,D_2-2)$ $\\hat{D}$ We can now turn to describe the new Diameter algorithm.",
"The algorithm works as follows.",
"All entries of an $n \\times n$ matrix $M$ are set to $n$ .",
"A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].",
"For every vertex $w\\in V$ and every pair $\\langle u, v\\rangle \\in C_A(w)\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).",
"Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .",
"Given a pair $\\langle u, v\\rangle \\in V \\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).",
"Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\in A$ it computes $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).",
"The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ and $D_2$ is $\\max _{u\\in A} \\epsilon _H(u)$ .",
"Next, we bound the value returned by Algorithm REF .",
"Theorem 31 Let $D=3h+z$ , where $z\\in [0,1,2]$ .",
"The value $\\hat{D}$ returned by Algorithm REF satisfies: $\\begin{array}{ll}2h-1 & \\mbox{if } z \\in [0,1] \\\\2h & \\mbox{if } z=2\\end{array}\\le \\hat{D} \\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .",
"If $B_A(u) \\cap B_A(v)\\ne \\emptyset $ then $(B_A(u) \\cap B_A(v)) \\cap P(u,v) \\ne \\emptyset $ .",
"If $v\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\notin B_A(u)$ .",
"Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\cap P(u,v)$ .",
"From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .",
"Assume, towards a contradiction, that $(B_A(u) \\cap B_A(v)) \\cap P(u,v) = \\emptyset $ .",
"This implies that $w\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .",
"However, since $B_A(u) \\cap B_A(v)\\ne \\emptyset $ there is a vertex $w^{\\prime }$ such that $d(u,w^{\\prime })\\le d(u,w)$ and $d(v,w^{\\prime })\\le d(v,A)-1 < d(u,v)-d(u,w)$ .",
"This implies that $d(u,w^{\\prime })+d(v,w^{\\prime })< d(u,v)$ , a contradiction to the triangle inequality.",
"Lemma 33 Let $u,v\\in V$ .",
"If $B_A(u) \\cap B_A(v)= \\emptyset $ then $d(u,A)+d(v,A)-1 \\le d(u,v)$ .",
"Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).",
"Let $P(u,v)$ be a shortest path between $u$ and $v$ .",
"Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .",
"Similarly, let $w^{\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .",
"Since $B_A(u) \\cap B_A(v)= \\emptyset $ it holds that $w\\ne w^{\\prime }$ .",
"Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\prime })\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\in [0,1,2]$ .",
"Let $P(a,b)$ be a shortest path between $a$ and $b$ .",
"Assume first that $B_A(a) \\cap B_A(b)\\ne \\emptyset $ .",
"It follows from Lemma REF that there is a vertex $w\\in P(a,b)$ such that $\\langle a, b\\rangle \\in C(w) \\times C(w)$ .",
"Therefore, $M(a,b)=D$ after Step 1.",
"After the update in Step 2 it follows from Lemma REF that $M(u,v)\\le d(u,v)$ for every $u ,v\\in V$ .",
"Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .",
"Let $x = \\operatornamewithlimits{arg\\,max}_{y\\in A} \\epsilon _H(y)$ .",
"Since $H$ is an additive 2 spanner it holds that $\\epsilon _H(x)\\le D+2$ , hence, we have $D_2 \\le D$ and the algorithm returns the exact value of the Diameter.",
"Assume now that $B_A(a) \\cap B_A(b) = \\emptyset $ .",
"From the discussion of the previous case it follows that in this case $\\hat{D}\\le D$ as well.",
"Thus, it is only left to prove the lower bound.",
"Assume that $z\\in [0,1]$ .",
"Consider first the case that $d(a,A)\\ge h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .",
"If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).",
"Assume, wlog, that $d(a,A)< h$ .",
"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .",
"Assume now that $z=2$ .",
"If either $d(a,A)\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .",
"If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).",
"Assume, wlog, that $d(a,A)\\le h$ .",
"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .",
"We now turn to we analyze the running time of Algorithm REF .",
"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .",
"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .",
"From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.",
"For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .",
"Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .",
"Step 2 takes $O(n^2)$ time as well.",
"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.",
"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.",
"We also compute $|A|$ shortest paths trees in $H$ .",
"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time.",
"Near linear almost 5/3-approximation for Eccentricities almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.",
"We run lines 2-11 of Algorithm REF .",
"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .",
"Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .",
"The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .",
"The pseudo-code is given in Algorithm REF .",
"We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.",
"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .",
"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.",
"The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.",
"Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .",
"We now turn to bound $\\epsilon ^{\\prime }(u)$ .",
"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .",
"It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .",
"Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .",
"We distinguish between two cases.",
"Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .",
"It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .",
"From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.",
"Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .",
"Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .",
"Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .",
"Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .",
"From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .",
"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .",
"Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .",
"As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .",
"Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .",
"This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .",
"Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .",
"Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .",
"If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .",
"Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .",
"As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .",
"Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .",
"From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows.",
"Algorithms for dense graphs using matrix multiplication Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.",
"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.",
"In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].",
"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].",
"The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .",
"After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.",
"The main idea of our algorithms is as follows.",
"If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .",
"Small distances are easy to compute with matrix multiplication.",
"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .",
"Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].",
"If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.",
"[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .",
"The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.",
"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.",
"Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].",
"Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .",
"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .",
"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .",
"Let us compare to our $O(n^2\\log n)$ time algorithms.",
"For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .",
"The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .",
"For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .",
"We will rely on two known algorithms.",
"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).",
"Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .",
"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .",
"The algorithm uses fast matrix multiplication and is quite straightforward.",
"Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.",
"$A$ is the adjacency matrix added to the identity matrix.",
"Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .",
"For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.",
"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .",
"Define $A^0$ as the identity matrix.",
"Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).",
"Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .",
"Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .",
"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .",
"Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.",
"Armed with these two algorithms, let us recap Roditty et al.",
"'s (and Cairo et al.",
"'s) approximation algorithm and see how to modify it.",
"The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.",
"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .",
"Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .",
"For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .",
"Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .",
"Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .",
"The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .",
"Let us describe how to modify the algorithm.",
"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.",
"Our Modified Approximation.",
"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .",
"Let $X=3a\\log n$ .",
"Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .",
"For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .",
"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .",
"Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .",
"[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .",
"For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .",
"Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.",
"Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .",
"Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .",
"[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .",
"Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .",
"[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .",
"Consider our modified algorithm, FasterApproximation.",
"Now we will prove several claims.",
"Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .",
"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.",
"Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.",
"Recall that $X+a\\log n = O(\\log n)$ .",
"Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.",
"Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .",
"Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.",
"The rest of the steps run in linear time.",
"Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .",
"Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .",
"Suppose that $\\tilde{D}_1\\ge X$ .",
"The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .",
"By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .",
"Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .",
"Suppose now that $\\tilde{D}_1< X$ .",
"This means that $D<X+a\\log n$ and every distance in the graph is $\\le X+a\\log n$ .",
"In the second part of the algorithm we set $Q=2(X+a\\log n)$ , and hence every distance is computed exactly: for every $s\\in S$ , $v\\in V$ , $d_\\le (s,v)=d(s,v)$ .",
"Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\le \\tilde{D}\\le D$ .",
"Claim 40 For every node $v$ , $\\frac{3\\epsilon (v)-1}{5} \\epsilon (v)\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .",
"Fix $v$ .",
"Suppose first that there exists some $x$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ .",
"Then $\\epsilon (v)\\ge \\tilde{\\epsilon }_1(v)=\\max _u d^{\\prime }(u,v) - a\\log n \\ge \\epsilon (v)-a\\log n = \\epsilon (v)(1-a\\log n/\\epsilon (v))$ .",
"Since $\\epsilon (v)\\ge d(x,v)\\ge d^{\\prime }(x,v)-a\\log n \\ge X$ , we get that $\\tilde{\\epsilon }_1(v)\\ge \\epsilon (v)(1-a\\log n / X) = 2\\epsilon (v)/3$ .",
"Now suppose that for all $x\\in V$ , $d^{\\prime }(x,v)<X+a\\log n$ .",
"Then, also for all $x\\in V$ , $d(x,v)<X+a\\log n$ and $\\epsilon (v)<X+a\\log n$ .",
"Consider all the quantities needed in the second part of the algorithm to compute $\\tilde{\\epsilon }_2(v)$ : $d_\\le (w,v)$ : since $\\forall x\\in V$ , $d(x,v)<X+a\\log n$ , $d_\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\le (x,v)$ for every $x\\in W$ : as above, $d_\\le (x,v)=d(x,v)$ ; $\\epsilon _\\le (x)-d_\\le (x,v)$ for all $x\\in T$ : here, $\\epsilon (x)\\le \\epsilon (v)+d(x,v)\\le 2\\epsilon (v)<2(X+a\\log n)$ .",
"Since we compute all distances from nodes in $S$ up to $2(X+a\\log n)$ and $x\\in S$ , $\\epsilon _\\le (x)=\\epsilon (x)$ .",
"Also as in the above bullets, $d_\\le (x,v)=d(x,v)$ .",
"Thus all the quantities needed are the correct ones and $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.",
"From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\sqrt{n},n,n)\\le O(n^{2.044183})$ .",
"We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\epsilon (v)$ : $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .",
"For every node $v$ , $\\frac{3\\epsilon (v)-1}{5}\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .",
"Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.",
"The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .",
"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."
],
[
"Algorithm overview",
"The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.",
"[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.",
"Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.",
"For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.",
"Therefore, it is interesting to consider the opposite question to the one considered by [41].",
"Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?",
"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.",
"For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .",
"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.",
"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.",
"To enable this approach we can no longer sample $A$ naively.",
"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.",
"The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .",
"We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.",
"The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w)<d(u,A) \\rbrace $ is of size $O(n/ |A|)$ .",
"Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .",
"For any such pair we can efficiently compute their exact distance.",
"Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .",
"This, combined with some other ideas, gives our approximation guarantees.",
"We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.",
"The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4]."
],
[
"A simple approach with additive error",
"Let's first consider a simple approach obtaining an $\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.",
"Suppose that we have an algorithm ALG that can compute in $\\tilde{O}(m\\sqrt{n})$ time, for any graph $G^{\\prime }$ , an estimate $D^{\\prime }$ of its Diameter $D$ such that $p\\cdot D - q\\le D^{\\prime }\\le D$ , estimates $e(v)$ of $\\epsilon (v)$ for all $v$ so that $r\\epsilon (v)-s\\le e(v)\\le \\epsilon (v)$ , and an estimate $D^{\\prime \\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\cdot D_{S,T}-u\\le D^{\\prime \\prime }\\le D_{S,T}$ .",
"Now, Dor, Halperin and Zwick [24] showed that in $\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\tilde{O}(n^{1.5})$ edges.",
"In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.",
"he removed all logs!).",
"Let's compute $H$ for our given graph and run ALG on $H$ .",
"The runtime is $\\tilde{O}(n^{1.5}\\cdot \\sqrt{n})\\le \\tilde{O}(n^2)$ since $H$ has $\\le O(n^{1.5})$ edges.",
"Let $D^{\\prime }_H,e_H(\\cdot ),D^{\\prime \\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\epsilon _H(\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .",
"Let's return $D^{\\prime }_H-2,e_H(\\cdot )-2,D^{\\prime \\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .",
"Notice that $pD-q\\le p\\cdot D_H - q\\le D^{\\prime }_H\\le D_H\\le D+2$ and so $pD-2-q \\le D^{\\prime }_H-2\\le D$ .",
"Similarly since $r\\epsilon (v)-s\\le r\\epsilon _H(v)-s\\le e_H(v)\\le \\epsilon _H(v)\\le \\epsilon (v)+2$ , we get $r\\epsilon (v)-s-2\\le e_H(v)-2\\le \\epsilon (v)$ .",
"Finally since $t\\cdot D_{S,T}-u\\le t\\cdot D^H_{S,T}-u\\le D^{\\prime \\prime }_H\\le D^H_{S,T}\\le D_{S,T}+2$ , we get $t\\cdot D_{S,T}-u-2\\le D^{\\prime \\prime }_H-2\\le D_{S,T}$ .",
"Thus, in $\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.",
"Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.",
"This is especially important when these parameters are constant, which is the hard case of the problems anyway."
],
[
"Near linear almost 3/2-approximation for Diameter",
"Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.",
"Among the tools they use are clusters and bunches.",
"Let $A\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .",
"For every $v\\in V$ , let $B_A(u)=\\lbrace v\\in V\\mid d(u,v) <d(u,A \\rbrace $ be the bunch of $u$ .",
"For every $w\\in V\\setminus A$ , let $C_A(w)= \\lbrace v \\mid w\\in B_A(v)\\rbrace $ be the cluster of $w$ .",
"Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\in V$ .",
"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.",
"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.",
"Their algorithm works as follows.",
"It sets $A$ to the empty set and $W$ to $V$ .",
"Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .",
"The algorithm computes $C_A(w)$ for every $w\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .",
"The pseudo-code is given in Algorithm REF .",
"Thorup and Zwick center algorithm [1] center$G,p$ $A= \\emptyset $ $W =V$ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\cup X$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .",
"For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .",
"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .",
"They did not provide the details and refer the reader to [48].",
"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.",
"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .",
"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .",
"This can be done as follows.",
"Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .",
"The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).",
"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.",
"To this end, more ideas are needed.",
"The following simple observation helps us to achieve our goal.",
"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .",
"Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .",
"For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .",
"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .",
"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].",
"This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.",
"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.",
"see [48]).",
"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.",
"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .",
"In more detail, our algorithm works as follows.",
"For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.",
"Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .",
"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .",
"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .",
"Finally, it computes the clusters and $W$ using the bunches as we described above.",
"The rest of the algorithm is almost identical to Algorithm REF .",
"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .",
"The pseudo-code is given in Algorithm REF .",
"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .",
"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .",
"compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .",
"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .",
"every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .",
"The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].",
"The cost of computing $A$ is $O(n/p)$ time [48].",
"Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .",
"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .",
"As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .",
"Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .",
"This completes the analysis of the part that precedes the while loop.",
"Next, we analyze the cost of the while loop.",
"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.",
"From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .",
"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.",
"Thorup and Zwick [47] proved that the expected number of iterations is $O(\\log n)$ .",
"The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.",
"For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\log n)$ iterations in expectation.",
"This implies that a set $A$ of expected size $(np \\log n)$ is returned in $O(n/p^2 \\log n)$ expected time.",
"The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.",
"As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.",
"almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\sqrt{n})$ every $w\\in V$ Step 1 every $\\langle u, v\\rangle \\in C_A(w) \\times C_A(w)$ , s.t.",
"$u\\ne v\\;$ $M(u,v) = \\min (M(u,v), d(u,w)+d(v,w))$ every $\\langle u, v\\rangle \\in V \\times V$ , s.t.",
"$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\in A$ compute shortest paths tree for $u$ in $H$ and set $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ $D_2 = \\max _{u\\in A} \\epsilon _H(u)$ $\\hat{D} = \\max (D_1,D_2-2)$ $\\hat{D}$ We can now turn to describe the new Diameter algorithm.",
"The algorithm works as follows.",
"All entries of an $n \\times n$ matrix $M$ are set to $n$ .",
"A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].",
"For every vertex $w\\in V$ and every pair $\\langle u, v\\rangle \\in C_A(w)\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).",
"Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .",
"Given a pair $\\langle u, v\\rangle \\in V \\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).",
"Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\in A$ it computes $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).",
"The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ and $D_2$ is $\\max _{u\\in A} \\epsilon _H(u)$ .",
"Next, we bound the value returned by Algorithm REF .",
"Theorem 31 Let $D=3h+z$ , where $z\\in [0,1,2]$ .",
"The value $\\hat{D}$ returned by Algorithm REF satisfies: $\\begin{array}{ll}2h-1 & \\mbox{if } z \\in [0,1] \\\\2h & \\mbox{if } z=2\\end{array}\\le \\hat{D} \\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .",
"If $B_A(u) \\cap B_A(v)\\ne \\emptyset $ then $(B_A(u) \\cap B_A(v)) \\cap P(u,v) \\ne \\emptyset $ .",
"If $v\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\notin B_A(u)$ .",
"Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\cap P(u,v)$ .",
"From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .",
"Assume, towards a contradiction, that $(B_A(u) \\cap B_A(v)) \\cap P(u,v) = \\emptyset $ .",
"This implies that $w\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .",
"However, since $B_A(u) \\cap B_A(v)\\ne \\emptyset $ there is a vertex $w^{\\prime }$ such that $d(u,w^{\\prime })\\le d(u,w)$ and $d(v,w^{\\prime })\\le d(v,A)-1 < d(u,v)-d(u,w)$ .",
"This implies that $d(u,w^{\\prime })+d(v,w^{\\prime })< d(u,v)$ , a contradiction to the triangle inequality.",
"Lemma 33 Let $u,v\\in V$ .",
"If $B_A(u) \\cap B_A(v)= \\emptyset $ then $d(u,A)+d(v,A)-1 \\le d(u,v)$ .",
"Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).",
"Let $P(u,v)$ be a shortest path between $u$ and $v$ .",
"Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .",
"Similarly, let $w^{\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .",
"Since $B_A(u) \\cap B_A(v)= \\emptyset $ it holds that $w\\ne w^{\\prime }$ .",
"Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\prime })\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\in [0,1,2]$ .",
"Let $P(a,b)$ be a shortest path between $a$ and $b$ .",
"Assume first that $B_A(a) \\cap B_A(b)\\ne \\emptyset $ .",
"It follows from Lemma REF that there is a vertex $w\\in P(a,b)$ such that $\\langle a, b\\rangle \\in C(w) \\times C(w)$ .",
"Therefore, $M(a,b)=D$ after Step 1.",
"After the update in Step 2 it follows from Lemma REF that $M(u,v)\\le d(u,v)$ for every $u ,v\\in V$ .",
"Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .",
"Let $x = \\operatornamewithlimits{arg\\,max}_{y\\in A} \\epsilon _H(y)$ .",
"Since $H$ is an additive 2 spanner it holds that $\\epsilon _H(x)\\le D+2$ , hence, we have $D_2 \\le D$ and the algorithm returns the exact value of the Diameter.",
"Assume now that $B_A(a) \\cap B_A(b) = \\emptyset $ .",
"From the discussion of the previous case it follows that in this case $\\hat{D}\\le D$ as well.",
"Thus, it is only left to prove the lower bound.",
"Assume that $z\\in [0,1]$ .",
"Consider first the case that $d(a,A)\\ge h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .",
"If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).",
"Assume, wlog, that $d(a,A)< h$ .",
"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .",
"Assume now that $z=2$ .",
"If either $d(a,A)\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .",
"If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).",
"Assume, wlog, that $d(a,A)\\le h$ .",
"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .",
"We now turn to we analyze the running time of Algorithm REF .",
"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .",
"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .",
"From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.",
"For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .",
"Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .",
"Step 2 takes $O(n^2)$ time as well.",
"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.",
"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.",
"We also compute $|A|$ shortest paths trees in $H$ .",
"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time."
],
[
"Near linear almost 5/3-approximation for Eccentricities",
"almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.",
"We run lines 2-11 of Algorithm REF .",
"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .",
"Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .",
"The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .",
"The pseudo-code is given in Algorithm REF .",
"We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.",
"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .",
"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.",
"The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.",
"Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .",
"We now turn to bound $\\epsilon ^{\\prime }(u)$ .",
"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .",
"It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .",
"Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .",
"We distinguish between two cases.",
"Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .",
"It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .",
"From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.",
"Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .",
"Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .",
"Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .",
"Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .",
"From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .",
"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .",
"Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .",
"As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .",
"Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .",
"This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .",
"Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .",
"Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .",
"If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .",
"Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .",
"As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .",
"Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .",
"From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows."
],
[
"Algorithms for dense graphs using matrix multiplication",
"Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.",
"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.",
"In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].",
"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].",
"The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .",
"After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.",
"The main idea of our algorithms is as follows.",
"If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .",
"Small distances are easy to compute with matrix multiplication.",
"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .",
"Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].",
"If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.",
"[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .",
"The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.",
"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.",
"Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].",
"Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .",
"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .",
"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .",
"Let us compare to our $O(n^2\\log n)$ time algorithms.",
"For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .",
"The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .",
"For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .",
"We will rely on two known algorithms.",
"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).",
"Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .",
"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .",
"The algorithm uses fast matrix multiplication and is quite straightforward.",
"Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.",
"$A$ is the adjacency matrix added to the identity matrix.",
"Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .",
"For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.",
"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .",
"Define $A^0$ as the identity matrix.",
"Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).",
"Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .",
"Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .",
"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .",
"Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.",
"Armed with these two algorithms, let us recap Roditty et al.",
"'s (and Cairo et al.",
"'s) approximation algorithm and see how to modify it.",
"The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.",
"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .",
"Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .",
"For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .",
"Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .",
"Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .",
"The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .",
"Let us describe how to modify the algorithm.",
"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.",
"Our Modified Approximation.",
"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .",
"Let $X=3a\\log n$ .",
"Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .",
"For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .",
"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .",
"Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .",
"[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .",
"For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .",
"Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.",
"Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .",
"Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .",
"[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .",
"Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .",
"[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .",
"Consider our modified algorithm, FasterApproximation.",
"Now we will prove several claims.",
"Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .",
"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.",
"Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.",
"Recall that $X+a\\log n = O(\\log n)$ .",
"Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.",
"Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .",
"Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.",
"The rest of the steps run in linear time.",
"Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .",
"Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .",
"Suppose that $\\tilde{D}_1\\ge X$ .",
"The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .",
"By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .",
"Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .",
"Suppose now that $\\tilde{D}_1< X$ .",
"This means that $D<X+a\\log n$ and every distance in the graph is $\\le X+a\\log n$ .",
"In the second part of the algorithm we set $Q=2(X+a\\log n)$ , and hence every distance is computed exactly: for every $s\\in S$ , $v\\in V$ , $d_\\le (s,v)=d(s,v)$ .",
"Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\le \\tilde{D}\\le D$ .",
"Claim 40 For every node $v$ , $\\frac{3\\epsilon (v)-1}{5} \\epsilon (v)\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .",
"Fix $v$ .",
"Suppose first that there exists some $x$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ .",
"Then $\\epsilon (v)\\ge \\tilde{\\epsilon }_1(v)=\\max _u d^{\\prime }(u,v) - a\\log n \\ge \\epsilon (v)-a\\log n = \\epsilon (v)(1-a\\log n/\\epsilon (v))$ .",
"Since $\\epsilon (v)\\ge d(x,v)\\ge d^{\\prime }(x,v)-a\\log n \\ge X$ , we get that $\\tilde{\\epsilon }_1(v)\\ge \\epsilon (v)(1-a\\log n / X) = 2\\epsilon (v)/3$ .",
"Now suppose that for all $x\\in V$ , $d^{\\prime }(x,v)<X+a\\log n$ .",
"Then, also for all $x\\in V$ , $d(x,v)<X+a\\log n$ and $\\epsilon (v)<X+a\\log n$ .",
"Consider all the quantities needed in the second part of the algorithm to compute $\\tilde{\\epsilon }_2(v)$ : $d_\\le (w,v)$ : since $\\forall x\\in V$ , $d(x,v)<X+a\\log n$ , $d_\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\le (x,v)$ for every $x\\in W$ : as above, $d_\\le (x,v)=d(x,v)$ ; $\\epsilon _\\le (x)-d_\\le (x,v)$ for all $x\\in T$ : here, $\\epsilon (x)\\le \\epsilon (v)+d(x,v)\\le 2\\epsilon (v)<2(X+a\\log n)$ .",
"Since we compute all distances from nodes in $S$ up to $2(X+a\\log n)$ and $x\\in S$ , $\\epsilon _\\le (x)=\\epsilon (x)$ .",
"Also as in the above bullets, $d_\\le (x,v)=d(x,v)$ .",
"Thus all the quantities needed are the correct ones and $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.",
"From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\sqrt{n},n,n)\\le O(n^{2.044183})$ .",
"We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\epsilon (v)$ : $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .",
"For every node $v$ , $\\frac{3\\epsilon (v)-1}{5}\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .",
"Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.",
"The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .",
"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."
]
] | 1808.08494 |
[
[
"NavigationNet: A Large-scale Interactive Indoor Navigation Dataset"
],
[
"Abstract Indoor navigation aims at performing navigation within buildings.",
"In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor scenes to complete various tasks in order to serve human.",
"In most scenarios, we expected an intelligent device capable of navigating itself in unseen environment.",
"Although several solutions have been proposed to deal with this issue, they usually require pre-installed beacons or a map pre-built with SLAM, which means that they are not capable of working in novel environments.",
"To address this, we proposed NavigationNet, a computer vision dataset and benchmark to allow the utilization of deep reinforcement learning on scene-understanding-based indoor navigation.",
"We also proposed and formalized several typical indoor routing problems that are suitable for deep reinforcement learning."
],
[
"Introduction",
"Indoor navigation aims at preforming navigation within buildings, such as train stations, airports, shopping centers, offices and museums.",
"Most intelligent devices require an indoor routing functionality to guide itself precisely through rooms to complete various tasks.",
"In most of the scenarios, we expect an intelligent device capable of navigating itself in new environments to serve human.",
"In the past years, several solutions have been proposed to deal with this problem [1].",
"Although they can be applied in some special cases, none of them can work smartly in novel environments or are based on scene understanding.",
"Those solutions require either dense pre-installed beacons (e.g.",
"WiFi/Bluetooth chips) over the scene or a map pre-built with SLAM in which human workers are required to carry cameras to scan the scene in advance.",
"Such requirements significantly limit the applications of these solutions.",
"New scenes have to be well prepared before the robots can work properly in it, which makes it impossible to be put into large-scale application in various indoor scenarios.",
"Additionally, neither beacons nor maps are helpful in semantically understanding indoor scenes and thus will lead to the failure of the robot to perform many practical tasks.",
"For example, to complete a regular command like 'to take a cup', a housekeeping android should understand what the scene is like, what a cup is and where a cup usually be, to navigate properly.",
"We expected a more general solution of indoor navigation problems that requires no preparation of the scene in advance.",
"Such a high demand requires a more human-like robot that is much more intelligent to implement self-exploration in novel environments.",
"Recent years, we have witnessed the rapid development of deep reinforcement learning (DRL) [2], [3] and computer vision [4], [5].",
"Therefore, smart indoor navigation enabled by DRL with visual inputs has been introduced [6].",
"In this setting, pre-installed beacons or pre-built map are no longer required.",
"A smart robot with vision will be enough to do the navigation job in different scenes.",
"To apply reinforcement learning methods on navigation problems, massive trial-and-error is necessary in training the model.",
"A straight-forward way is to construct a real robot and train Reinforcement Learning (RL) models on real-world scenes.",
"However, such process is extremely slow and costly.",
"Additionally, it is often accompanied by robot collision and damage, which means that more time and resources would be wasted.",
"Even worse, due to the bottleneck of physical movements, to execute any action in the real world requires at least several seconds, which is unacceptable for any machine learning methods since that they usually require millions of times of trial-and-error.",
"Aside from those drawbacks, such a process raises the problem of reproducibility.",
"Even if all the background settings of a research are published in detail, it is still very hard for third parties to reproduce the experiment result since it is costly and time-consuming to find or construct a new environment that is identical to the one the original author used.",
"That makes it impossible for third parties to examine and evaluate a new research result.",
"We believe, an open-source, low-cost, large-scale dataset and corresponding benchmarks would be a key to largely advance this research field.",
"Some previous work has tried to train the robot in computer graphics (CG) scenes like AI2-THOR[6], SUNCG[7].",
"However, we do have noticed that training models on CG environments raised the problem of model transferring as the gap between hand-crafted scenes and the real world is significant.",
"The visual appearance of images rendered by CG system always look non-realistic.",
"Existing 3D models are limited in representing real-world scenes and objects.",
"To deal with this problem, we proposed NavigationNet, a large-scale, real-world, interactive dataset.",
"In each scene, we use human-size robot with 8 cameras to capture photos towards 8 different directions at all walk-able positions.",
"Given all the collected images, we can easily tour the room like a CG system by moving among those walk-able positions.",
"However, different from CG systems, what we observed from this dataset are all real images captured from the real world.",
"Therefore, we can virtually command a robot to walk in the scene and get the first-perspective view of the robot.",
"Our dataset is interactive.",
"That is to say, as we send a command of movement (e.g.",
"moving forward, turning right), the robot will perform as required and return a new view it sees from the new position.",
"To our best knowledge, our NavigationNet is the first real-world indoor scene dataset that is interactive in computer vision field.",
"Our dataset covers about 1500 $m^2$ indoor area with various bedrooms, studies, meeting room and etc.",
"This makes the bias less in experiments on our dataset.",
"Based on NavigationNet, we proposed four applications that require indoor navigation.",
"We will explained them in detail in chapter .",
"To summarize, we presented a large-scale dataset consisting of visual data collected from real scenarios to allow a low-cost, reproducible training of indoor navigation robots based on reinforcement learning methods.",
"We also proposed and formalized several practical indoor navigation tasks in the framework of reinforcement learning."
],
[
"Related Work",
"In this chapter, we will discuss some of the previous works that are related to our work or act as a tool in this project.",
"First we will talk about the technology of Visual Semantic Planning.",
"A brief introduction to the popular datasets in the field of computer vision, after which we will explain three techniques used in our work in detail: Reinforcement Learning, Convolutional Neural Network(CNN) and Simultaneous Localization and Mapping."
],
[
"Visual Semantic Planning",
"Fundamental tasks in mobile robot controlling include navigation, mapping, grasping, path planning and so on.",
"Such task of interacting with a visual world and planning a sequence of actions to achieve a certain goal is addressed as Visual Semantic Planning [8].",
"A series of work [9], [10], [11], [12] has addressed the task-level planning and learning from dynamics.",
"Classical approaches [13], [14] of motion planning require building map purely geometrically using Light Detection and Ranging (LiDAR) or Structure from Motion (SfM) [15], [16], [17].",
"Srivastava et al.",
"[18] provides an interface between task and motion planning, such that the task planner can effectively operate in an abstracted state space that ignores geometry.",
"Recently proposed end-to-end learning-based approaches [19], [10], [20] can go directly from pixels to actions.",
"For instance Zhu et al.",
"[19] adopt feed-forward neural network to target-driven visual navigation in indoor scenes.",
"Gupta et al.",
"[10] use online Cognitive Mapping and Planning (CMP) approach for goal direction visual navigation without requiring a pre-constructed map.",
"Jang et al.",
"[9] train a deep neural network to perform the semantic grasping task inspired by the \"two-stream hypothesis\" of human vision.",
"In terms of Complete Path Planning, Kollar et al.",
"[21] use reinforcement learning to find trajectories and the learnt policy transfers successfully to a real environment."
],
[
"Datasets for Computer Vision",
"Datasets and corresponding benchmarks have played a significant role in many areas such as computer vision and speech recognition.",
"The evolution from WordNet[22] to ImageNet[23] was one of the many successful examples that proved the power of an effective dataset to accelerate the development of one area.",
"On the contrary, the lack of appropriate data has become one of the most crucial challenges for deep reinforcement learning, especially when it is dealing with real world problems like visual semantic learning.",
"Since training and quantitatively evaluating DRL algorithms in real environments is either impractical or costly, this problem has become more urgent.",
"For RL algorithms dealing with virtual-world problems, the Arcade Learning Environment (ALE) [24] exposed Atari 2600 games as reinforcement learning problems.",
"Works such as Duan et al.",
"[25] and Brockman et al.",
"[26] propose toolkits to qualify progress in reinforcement learning.",
"Other benchmarks [27], [28], [29], [30], [31], simulators [32] or physics engines [33] are also designed for the development of deep reinforcement learning.",
"Also, scientists are trying to model the real world for DRL or other visual tasks.",
"Chang et al.",
"[34] provides 90 building-scale reconstructed scenes for supervised and self-supervised computer vision tasks such as keypoint matching, view overlap prediction semantic segmentation, scene classification and so on.",
"AI2-THOR[19], SUNCG[7] and House3D are all CG scenes designed especially for deep reinforcement learning.",
"The advanced version of AI2-THOR[35] even provides the functionality to let the robot directly interact with the objects in the scenarios.",
"Different from our NavigationNet, these datasets are either synthetic or reconstructed using computer graphics techniques, which introduce inconsistency distribution between the real-world scenarios and the produced ones."
],
[
"Reinforcement Learning",
"Reinforcement learning (RL) [36] method was proposed in the late 90s.",
"It provides a technique to allow the robot or any other intelligent systems to build a value function to evaluate policies in given situations from an interactive way of trail-and-error.",
"Pioneering works from Mnih et al.",
"[37], Lillicrap et al.",
"[38], Schulman et al.",
"[39] and Silver et al [40] introduced and ignited the idea of join Reinforcement Learning and Deep Learning into Deep Reinforcement Learning.",
"In learning complex behavior skills and solving challenging control tasks in high-dimensional raw sensory state-space, DRL methods have shown tremendous success [40], [41], [3], [42].",
"Trust Region Policy Optimization (TRPO) [41] makes a series of approximations to the theoretically-justified procedure and has robust performance on a wide range of tasks.",
"Asynchronous Advantage Actor-Critic (A3C) Algorithm [3], which uses asynchronous gradient descent for optimization, succeeds on task of navigating random 3D mazes as well as a wide variety of continuous motor control problems.",
"UNsupervised REinforcement and Auxiliary Learning (UNREAL) [43] brings together the A3C framework with auxiliary control tasks and auxiliary reward tasks.",
"Actor Critic using Kronecker-factored Trust Region (ACKTR) [42] Algorithm uses a Kronecker-factored approximation to natural policy gradient that allows the covariance of the gradient to be inverted efficiently.",
"Proximal Policy Optimization (PPO) [44] Algorithms use multiple epochs of stochastic gradient ascent to perform each policy update.",
"Reinforcement Learning (RL) provides a powerful and flexible framework to several applications.",
"Andrew Y. Ng et al.",
"[45] describe a successful application of autonomous helicopter.",
"Finn et al.",
"[12] present an approach that automates state-space construction by learning a state representation directly from camera images with deep spatial autoencoder.",
"Heess et al.",
"[46] explore a rich environment can help to promote the learning of complex behavior.",
"They normalize observations, scale the reward and use per-batch normalization of the advantages.",
"Gu et al.",
"[47] demonstrate a DRL algorithm based on off-policy training of deep Q-function can scale to complex 3D manipulation tasks and can learn deep neural network policies efficiently enough to train on real physical robots.",
"Denil et al.",
"[48] find DRL methods can learn to perform the experiments necessary to discover properties such as mass and cohesion of objects."
],
[
"Convolutional Neural Network",
"Deep convolutional neural networks [49], [50] have led to a series of breakthrough for visual tasks.",
"He et al.",
"[5] present a Residual Network (ResNet) to ease the training of deep neural network.",
"We extract feature map with ResNet 101.",
"Huang et al.",
"[51] presents Dense Convolutional Network (DenseNet) which connects each layer to every other layer in a feed-forward fashion.",
"Chen et al.",
"[52] propose Dual Path Net (DPN) by revealing ResNet and DenseNet within the higher order recurrent neural network framework.",
"Deep Learning models have been successful in learning powerful representations and understanding stereo context [53], [54].",
"Shaked et al.",
"[55] present a three-step pipeline for the stereo matching problem and a network architecture (ResMatch) for computing the matching cost at each possible disparity.",
"Kendall et al.",
"[56] propose an end-to-end model (GC-Net) for regressing disparity from a rectified pair of stereo images."
],
[
"Simultaneous Localization and Mapping",
"Visual SLAM plays an important role in autonomous navigation of mobile robots [57].",
"The emergence of MonoSLAM [58] makes the idea of utilizing one camera become popular.",
"Parallel Tracking and Mapping (PTaM) [59] uses an approach based on keyframes with two parallel processing threads.",
"ORB [60] becomes a computationally-efficient replacement to SIFT keypoint detector and descriptor that has similar matching performance.",
"Keyframe bundle adjustment outperforms filtering, since it gives the most accuracy per unit of computing time [61].",
"S-PTAM [62] allows accuracy improvement of the mapping process with respect to monocular SLAM and avoiding the well-known bootstrapping problem.",
"ORB-SLAM2 [63] is a complete SLAM system for monocular, stereo and RGB-D cameras.",
"RGB-D SLAM [64] can producing high quality globally consistent surface reconstruction with only a low-cost commodity RGB-D sensor.",
"Mur-Artal et al.",
"[65] propose a visual-inertial monocular SLAM which is able to close loops and reuse its map to achieve zero-drift localization in already mapped areas."
],
[
"NavigationNet",
"NavigationNet is specifically designed for applying reinforcement learning methods to indoor navigation tasks.",
"We collected data from the real world and organized it properly to allow the robot to roam in the dataset as if they are in the corresponding real-world scenario.",
"In this chapter, we will explain its organization and applications in detail to show you the world of NavigationNet."
],
[
"Data Organization",
"Images in NavigationNet are organized hierarchically.",
"The hightest node is the root of NavigationNet.",
"The second level nodes, also the primary elements, are scenes.",
"A scene is a collection of images collected from the same indoor space.",
"In the next level, a scene is divided into rooms, space connected with doors.",
"In the current version of NavigationNet, we have 15 scenes, each with 1-3 rooms.",
"The origin room for each scene is at least 50 $m^2$ in area.",
"The fourth level is called postion.",
"A position is from where images are collected.",
"A room contains hundreds of postions.",
"When being trained, the robot could move among the adjacent positions to 'see' from that perspective.",
"When constructing the dataset, our mobile robot iterated over all the walkable points in the room with the granularity of 20cm.",
"Hence, in the dataset, two adjacent positions are 20cm away from each other.",
"With such a granularity, one room usually contains thounsands of positions.",
"From the perspective of information, a position can be seen as a bundle of information that the robot will receive when it is placed at that position.",
"Usually, we offer 8 images in this bundle.",
"These images are divided into four directions: front, back, left and right.",
"Towards each direction, two parallel images are taken to complete a binocular vision system.",
"Such a setting enables the possibilities like stereo matching and panorama reconstruction.",
"In addition to the images, we also provide a ground-truth map attached to the scene.",
"The map can be considered as a binary map indicating where is walkable or non-walkable.",
"This is essential especially to the Auto-SLAM problem."
],
[
"Robot Moving Control",
"To satisfy the different requirements of potential tasks, we tried our best to provide a more generalizable moving control SDK for NavigationNet.",
"As is stated in section REF , the fundamental element in NavigationNet is scene, virtual robots are allowed to roam in scenes.",
"To serve well as an action in a reinforcement learning task, movements should be standard, discrete, limited and complete.",
"An action should be standard, so that the robot should move the same distance towards the same direction or turning the same angle when a specific instruction of a movement is given.",
"An action should be discrete, so that the action space can be finite and discrete.",
"Actions should be limited, so that the algorithm do not need to choose actions from an infinite action space.",
"Actions should be complete, so that all the actions joined together could describe the whole real-world action space.",
"In practice, it is impossible to use a finite action space to cover an infinite one.",
"We need some sort of compromise.",
"The one we take in this case is to discrete the movement length and the turning angle.",
"So that we defined six types of movements.",
"MOVE FORWARD Ask the robot to move forward one position.",
"For example, go from (9, 12) to (9, 13) when facing north but go from (11, 8) to (10, 8) when facing west.",
"Since two adjacent nodes are 20cm away, this is to move forward 20cm in real-world scenes.",
"MOVE BACKWARD Ask the robot to move backward one position.",
"This is the opposite movement of MOVE FORWARD and can revert the effect of MOVE FORWARD.",
"MOVE LEFT Ask the robot to move left one position.",
"For example, go from (5, 4) to (4, 4) when facing north but go from (7, 4) to (7, 3) when facing west.",
"The real world distance would be the same as MOVE FORWARD and MOVE BACKWARD.",
"It should be paid attention to that MOVE LEFT is very different from TURNING LEFT as it will move the position of the robot but will not change the facing direction.",
"MOVE RIGHT Ask the robot to move right one position.",
"This is the opposite movement of MOVE LEFT and can revert the effect of MOVE LEFT.",
"TURN LEFT Ask the robot to turn left at 90-degree.",
"For example, when the robot receives this request when at (5, 7) facing north, it should then facing west still at (5, 7).",
"It should be paid extra attention that this movement is different from MOVE LEFT that the former changes the facing while the latter change the position.",
"TURN RIGHT Ask the robot to turn right at 90-degree.",
"This is the opposite movement of TURN LEFT.",
"Figure: MovementsAt the same time, we should make it clear that, in any given tasks, according to the specific setting, not all the actions (movement) need to be taken into consideration.",
"For example, when we would like to simulate a two-wheel one-eye robot, TURN LEFT and TURN RIGHT are a must, otherwise seventy-five percent of the images can never be perceived while MOVE LEFT and MOVE RIGHT should be eliminated since a two-wheel robot can not make the exact movement of moving left 20cm.",
"Meanwhile, when simulating a robot with panorama vision, we could only take MOVE LEFT and MOVE RIGHT but not TURN LEFT and TURN RIGHT as they are not necessary but can only enlarge the action space unnecessarily and make the task more difficult."
],
[
"NavigationNet Construction",
"NavigationNet is a large-scale dataset with hundreds of thousands of images.",
"The large amount makes the data collection a task of impossible.",
"What is worst is that, as to each image, it is required that the position, angle and height where it is taken should be exactly where it is expected so long that the simulation of reality will not offset.",
"These two factors together make the task of collection much more difficult than expected.",
"In this chapter, we will talk about how we constructed this dataset efficiently and accurately."
],
[
"Collector Mobile Robot",
"Robots are designed to liberate human beings from the repetitive boring work.",
"NavigationNet is built for intelligent robots, also built with the help of intelligent robots.",
"Collecting data for NavigationNet requires heavy labor and high accuracy but little flexibility (most of the possible conditions are under control), which is surprisingly suitable for robots.",
"In order to reduce the labor requirement, we exploited the possibilities of smart hardware.",
"Our team developed a dedicated data-collecting mobile robot with Arduino Mega2560 and Raspberry Pi 3 Model B codenamed GoodCar.",
"Arduino[66] is an open-source physical computing platform based on a simple I/O board and a development environment that implements the Processing/Wiring language.",
"Arduino boards can be programmed in the same name program language Arduino, which is a C-like area-specific language to receive and send signals from tens of low-voltage I/O ports.",
"It is specifically suitable for robot controlling.",
"In this project, we use a Arduino Mega2560 to control the movement of the robot.",
"Raspberry Pi[67] is a cheap single-board computer system running a specificialy made Linux distribution.",
"It is originally built for computer programming education but we find it suitable for robot controlling.",
"It consumes little electricity which allows us to strip the 220V power wires.",
"It contains all kinds of I/O ports such that we could plug it with Arduinos, cameras, other full-sized computers and many other modules to make it the center of the system.",
"It runs standard linux distribution such that the possibility of software extension is beyond imagine.",
"Last but not least, it is cheap so that we could us as many as we want on the robot.",
"In practice, we use two Raspberry Pi 3B for controlling all other modules and communicating with the upper computer.",
"To build this robot, we used a two-level structure.",
"The upper-level is as high as 1.4m to simulate the the height of the eyes of an average adult.",
"We plugged-in eight cameras on this level, two for each direction to make a stereo vision system and to avoid unnecessary turning-around.",
"The lower-level is for the motion systems.",
"The Raspberry Pi and Arduino mentioned above are all on this level.",
"The Arduino is to control the mobile devices and sensors.",
"It is linked with four motors to control the movement.",
"In the meantime, it is linked with a sonar and code plate counters to avoid collision and measure the distance of movement.",
"Also it is linked with a Raspberry Pi and under its control.",
"We used two Raspberry Pi 3B in each robot, one is master and the other is slave.",
"The master computer controls many things.",
"First it communicates with the Arduino to control the movement indirectly.",
"Second it is responsible for evaluating the signal Arduino sent back to avoid collision.",
"Third it is master computer's duty to control four of the cameras to take photo at given position and store the images on the SD card.",
"Fourth the slave computer is also under its control to take photos when required.",
"Lastly it communicates with the upper computer, which is controlled by human, via Wi-Fi connection.",
"That is the control core of our robot.",
"Under the lower-level is the mobile devices.",
"We have tried on many types of mobile devices and at last we used four track structures to reduce error.",
"We had constructed six mobile robots before moving on to collect data.",
"Figure: Robot"
],
[
"Data Collection",
"To make the data collecting process efficient and accurate, also to make the whole process manageable and under control, we designed and followed the following instructions: Step 1 Measure the size of the target room(s).",
"Find the most southwestern corner as the starting point.",
"Step 2 Move the robot to the starting point.",
"Evaluate and record its distance towards the borders.",
"Step 3 Let the robot face north and start to collect required photos at this point.",
"Step 4 After photos are taken, let the robot run 20cm towards north and collect data.",
"Step 5 Keep iterating over Step 3-4 until the robot reaches the end of the line.",
"Step 6 Move the robot to the starting point of the line and then move 20cm towards east.",
"Step 7 Iterating over Step3-6 until the robot reaches the east corner.",
"Step 8 Complete this scenario by taking photos from those points that the robot did not reach.",
"Data Processing After the collection, the original data should be pre-processed before taken into production.",
"The data processing steps we taken includes: Data Cleaning When collecting data with robot, there are always inevitable errors occurring especially when dealing with room corners.",
"For each room we collect, we will scan over all the collected photos and find errors.",
"If necessary, we would go and retry collecting data to complete the set.",
"Map Building When collecting data, we also recorded whether a point is accessible.",
"With this information, we are able to rebuild a walkable-or-not map for reference.",
"Stereo Vision For some of the task, to simulate a two-eye system or a RGB-D system, it would be of much help if we could produce depth channel in the very beginning.",
"To serve this purpose, we use MC-CNN[54] to produce some pre-processed depth channel data.",
"Applications NavigationNet is designed for indoor navigation problems.",
"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.",
"Apart from these four applications, we believe our dataset will spawn more applications.",
"Target-driven Navigation Motivation In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.",
"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.",
"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.",
"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.",
"This usually falls in the field of visual semantic learning.",
"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.",
"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based) Formalization The formalization of this issue is inspired by Yuke Zhu et al.[6].",
"To complete the task of target-driven navigation, a robot control system is given two images.",
"One is the target image, the other is the current first-person perspective image perceived from the environment.",
"The robot can choose to walk forward, backward, turn left or right according to the input images.",
"The environment will update the states accordingly and give out a reward.",
"This process loops until the target image is the same with the perceived one.",
"To formalize that: Goal The goal is straight-forward, to find the given object.",
"In practice, it can be converted into make the perceived image and the given image identical.",
"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.",
"One thing that is sure is that a large positive reward will be given when the goal is achieved.",
"To improve time efficiency, each step it taken should be given a small negative reward.",
"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.",
"Action Space To find an object, the understanding of the scene is the most important factor of the task.",
"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.",
"Hence the State space would be the set of all possible image pairs.",
"Evaluation To evaluate such a system, many aspects should be taken into consideration.",
"The most important two are, trajectory should be short and the robot should not run into the obstacles.",
"Collisions should be limited strictly as only one collision could greatly damage the robot.",
"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.",
"To calculate a overall score on all the test results, we utilize the idea of histogram.",
"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .",
"A overall score is calculated by finding the average over this histogram.",
"Multi-target Navigation Motivation The target-driven navigation task is focused on finding one object efficiently and safely.",
"However, in practical scenarios, this is often not enough.",
"In many cases, we need to find more than one items and we do not care about the order.",
"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.",
"This has become a rather different problem since it usually requires the knowledge of adjacency.",
"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.",
"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based) Formalization We also formalized this problem into a reinforcement learning problem.",
"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.",
"The rest will be similar with the target-driven task.",
"A first-person perspective image is given, an action is executed and then state is updated, reward is given.",
"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.",
"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.",
"We believe that such a design would be better for deep reinforcement training[43].",
"Also a minor negative time time punishment is necessary.",
"Action Space Since the task type is similar with the target-driven problem.",
"The same action space remains.",
"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.",
"However, a set images with unknown-length is hard to be processed.",
"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Evaluation Although the task is different from the previous task, the two major meters are the same.",
"Hence, we use the same metric system in this task as is in the target-driven navigation task.",
"Sweeper Route Planning Motivation Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.",
"One widely-discussed but never well-addressed problem is the sweeper route planning problem.",
"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.",
"The current solutions often lead to stuck at some small space.",
"Besides sweeper robot, it is general problem we will meet in many scenarios.",
"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.",
"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.",
"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.",
"It must see and understand to avoid collisions.",
"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based) Formalization Again, we formalized this problem into a reinforcement learning task.",
"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.",
"The environment should give out the reward then according to how much area has been covered by the sweeper.",
"Typically, the robot should maintain a map of the room inside its system.",
"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.",
"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.",
"Reward Reward design is simple in this problem.",
"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.",
"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.",
"Action Space In this task, a sweeper does not really care about which direction it is facing.",
"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).",
"So we will not give the robot the ability to turn left/right.",
"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.",
"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.",
"Evaluation The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.",
"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.",
"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.",
"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.",
"An average over scores of all steps in a trajectory is the final score of the trajectory.",
"Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Auto-SLAM with Deep Reinforcement Learning Motivation Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.",
"Many solutions have been proposed to increase the mapping accuracy.",
"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.",
"Additionally, applying SLAM for some large-scale scenes (e.g.",
"Museum) is effort consuming, which requires us to carry camera walked about all corners.",
"So, we hope that robots can preform SLAM autonomously.",
"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.",
"Formalization This task is also formalized as a reinforcement learning problem.",
"On each step, the robot is fed with images perceived at its current location.",
"The system could use the visual inputs and history information to improve the map.",
"Reward should be calculated according to the difference between the constructed map and the ground truth.",
"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.",
"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.",
"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.",
"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.",
"By talking about progress, we are assuming total trust in legacy SLAM algorithms.",
"We would like to give a positive reward for any new reconstructed area after each steps.",
"In the meantime, a fruitless step is also punished with a minor negative reward.",
"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.",
"In the meantime, we need to add the function of moving left/right to the robot.",
"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.",
"Most of it is about 3D reconstruction.",
"Thus only a first-person single-eye perception is far from enough.",
"In this setting, the environment should provide stereo vision from the four direction with depth information.",
"Evaluation SLAM tasks are divided into two part, mapping and positioning.",
"Building maps is a way for positioning.",
"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.",
"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.",
"After that, 10 images are given as query to the positioning system.",
"The score is calculated by average over the positioning success rate.",
"Conclusion We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.",
"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.",
"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.",
"We extract robot controlling system out from robot itself.",
"We hope to eliminate the long and high-cost preparation process before really training the system.",
"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.",
"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.",
"We also proposed four possible tasks that can be conducted on our dataset.",
"We are hoping that more tasks can be proposed to exploit its maximum potential.",
"Our future work includes increasing the number of scenes in the dataset.",
"Also we will construct more attributes in the dataset (object segmentation for example)."
],
[
"Applications",
"NavigationNet is designed for indoor navigation problems.",
"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.",
"Apart from these four applications, we believe our dataset will spawn more applications."
],
[
"Motivation",
"In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.",
"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.",
"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.",
"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.",
"This usually falls in the field of visual semantic learning.",
"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.",
"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based)The formalization of this issue is inspired by Yuke Zhu et al.[6].",
"To complete the task of target-driven navigation, a robot control system is given two images.",
"One is the target image, the other is the current first-person perspective image perceived from the environment.",
"The robot can choose to walk forward, backward, turn left or right according to the input images.",
"The environment will update the states accordingly and give out a reward.",
"This process loops until the target image is the same with the perceived one.",
"To formalize that: Goal The goal is straight-forward, to find the given object.",
"In practice, it can be converted into make the perceived image and the given image identical.",
"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.",
"One thing that is sure is that a large positive reward will be given when the goal is achieved.",
"To improve time efficiency, each step it taken should be given a small negative reward.",
"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.",
"Action Space To find an object, the understanding of the scene is the most important factor of the task.",
"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.",
"Hence the State space would be the set of all possible image pairs.",
"To evaluate such a system, many aspects should be taken into consideration.",
"The most important two are, trajectory should be short and the robot should not run into the obstacles.",
"Collisions should be limited strictly as only one collision could greatly damage the robot.",
"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.",
"To calculate a overall score on all the test results, we utilize the idea of histogram.",
"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .",
"A overall score is calculated by finding the average over this histogram.",
"The target-driven navigation task is focused on finding one object efficiently and safely.",
"However, in practical scenarios, this is often not enough.",
"In many cases, we need to find more than one items and we do not care about the order.",
"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.",
"This has become a rather different problem since it usually requires the knowledge of adjacency.",
"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.",
"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based)We also formalized this problem into a reinforcement learning problem.",
"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.",
"The rest will be similar with the target-driven task.",
"A first-person perspective image is given, an action is executed and then state is updated, reward is given.",
"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.",
"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.",
"We believe that such a design would be better for deep reinforcement training[43].",
"Also a minor negative time time punishment is necessary.",
"Action Space Since the task type is similar with the target-driven problem.",
"The same action space remains.",
"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.",
"However, a set images with unknown-length is hard to be processed.",
"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Although the task is different from the previous task, the two major meters are the same.",
"Hence, we use the same metric system in this task as is in the target-driven navigation task.",
"Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.",
"One widely-discussed but never well-addressed problem is the sweeper route planning problem.",
"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.",
"The current solutions often lead to stuck at some small space.",
"Besides sweeper robot, it is general problem we will meet in many scenarios.",
"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.",
"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.",
"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.",
"It must see and understand to avoid collisions.",
"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based)Again, we formalized this problem into a reinforcement learning task.",
"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.",
"The environment should give out the reward then according to how much area has been covered by the sweeper.",
"Typically, the robot should maintain a map of the room inside its system.",
"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.",
"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.",
"Reward Reward design is simple in this problem.",
"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.",
"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.",
"Action Space In this task, a sweeper does not really care about which direction it is facing.",
"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).",
"So we will not give the robot the ability to turn left/right.",
"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.",
"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.",
"The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.",
"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.",
"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.",
"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.",
"An average over scores of all steps in a trajectory is the final score of the trajectory.",
"Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.",
"Many solutions have been proposed to increase the mapping accuracy.",
"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.",
"Additionally, applying SLAM for some large-scale scenes (e.g.",
"Museum) is effort consuming, which requires us to carry camera walked about all corners.",
"So, we hope that robots can preform SLAM autonomously.",
"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.",
"This task is also formalized as a reinforcement learning problem.",
"On each step, the robot is fed with images perceived at its current location.",
"The system could use the visual inputs and history information to improve the map.",
"Reward should be calculated according to the difference between the constructed map and the ground truth.",
"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.",
"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.",
"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.",
"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.",
"By talking about progress, we are assuming total trust in legacy SLAM algorithms.",
"We would like to give a positive reward for any new reconstructed area after each steps.",
"In the meantime, a fruitless step is also punished with a minor negative reward.",
"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.",
"In the meantime, we need to add the function of moving left/right to the robot.",
"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.",
"Most of it is about 3D reconstruction.",
"Thus only a first-person single-eye perception is far from enough.",
"In this setting, the environment should provide stereo vision from the four direction with depth information.",
"SLAM tasks are divided into two part, mapping and positioning.",
"Building maps is a way for positioning.",
"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.",
"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.",
"After that, 10 images are given as query to the positioning system.",
"The score is calculated by average over the positioning success rate."
],
[
"Conclusion",
"We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.",
"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.",
"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.",
"We extract robot controlling system out from robot itself.",
"We hope to eliminate the long and high-cost preparation process before really training the system.",
"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.",
"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.",
"We also proposed four possible tasks that can be conducted on our dataset.",
"We are hoping that more tasks can be proposed to exploit its maximum potential.",
"Our future work includes increasing the number of scenes in the dataset.",
"Also we will construct more attributes in the dataset (object segmentation for example)."
]
] | 1808.08374 |